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1.19M
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=
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n2
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= −23.5 kPa, where the third-order constants l and n are kept identical for all four layers and the second-order constants λ, μ and third-order constant m for Layer 2 (and Layer 4) are to be varied in the parametric study.(a) and (b) plot λ2-μ2-m2 surfaces for constant values of ur and Tr, respectively. The single point marked in each figure, where λ2
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=
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λ1, μ2
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=
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μ1 and m2
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=
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m1, indicates a homogeneous sphere. Specifically, (a) plots the surfaces of constant ur at the outermost boundary, i.e., ur(r
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=
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r4)/r1
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= 1.5, 2, 4. This reveals: (1) the λ2,μ2,m2 values on the curved surfaces offer multiple choices of elastic constants for a targeted amount of swelling, (2) the bounds on λ2,μ2,m2 can be determined if the displacement is to lie within or outside a certain range, (3) the surface shrinks with increasing swelling, which requires smaller values for λ2,μ2 (linearly compliant) but increasingly large negative values for m2 (nonlinearly compliant), (4) larger swelling can, in general, be achieved by stronger layer dissimilarity, in contrast to the smaller swelling (ur(r
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=
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r4)/r1
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= 1.372) of the homogeneous sphere, (5) the third-order constant m2, just as the second-order constants, can be effectively used to modify the swelling response, and (6) it becomes increasingly more difficult to achieve large swelling as the surfaces shrink and steepen with increasing outer displacement.(b) shows the evolution of the λ2-μ2-m2 surfaces at the constant interfacial stress Tr at r
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=
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r2, i.e., Tr(r
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=
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r2) = −5, 0, 5 kPa. It is evident that: (1) there exists a zero stress curved surface which compartmentalizes the domain into tensile and compressive regions, (2) this compartmentalization may guide the selection of elastic constants yielding compressive stress to avoid interfacial delamination, or a small tensile stress to avoid compressive instability, and (3) the closer proximity of the surfaces representing zero and compressive stress implies a greater stress sensitivity to elastic parameter control in compression than in tension. The stress at the corresponding location in the homogeneous sphere is 3.87 kPa. (b) shows only the radial stress at r
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=
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r2; the stresses throughout the capsule should be determined for the proper selection of elastic constants during biomaterial design. suggests the possibility of rationally designing for the elastic fields using second- and third-order elastic constants. The stress field has a crucial influence on the material integrity and the functionality of the composite, e.g., cell proliferation and orientation will likely be affected by the stress magnitude, sign and multiaxiality.To further illustrate the influence of elastic dissimilarity, consider the four-layer sphere with Layers 1 and 3, and Layers 2 and 4, respectively identical in their elastic constants. The dilatation loading is ϑp1. Further, the constants for Layers 1 and 3 are selected as either λ1
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= 35.7, μ1
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= 10.3, l1
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= −35.6, m1
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= −24.2 and n1
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= −23.5 kPa, or λ1
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= 35.7, μ1
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= 0.103, l1
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= −35.6, m1
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= −500 and n1
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= −23.5 kPa. The elastic constants for Layers 2 and 4 are those that have been determined for the polymer gels (chosen from the data sets 18–34) through Eq. , it can be seen that the radial stress at r
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=
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r3 varies from 1 to 9 kPa if the elastic constants for Layers 1 and 3 assume the first set of values and those for Layers 2 and 4 assume the values corresponding to the data sets listed in . If instead the constants for Layers 1 and 3 assume the second set of values (while the remaining layers have the constants listed in ), the radial stresses are significantly different and can be compressive. Thus, emphasizes the influence of elastic dissimilarity on the stress state.The effect of modifying the layer thicknesses of a four-layer composite, with Layers 1, 3 and Layers 2, 4 respectively identical, is next investigated. The chosen dilatation profile is ϑp2, where ϑ1
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=
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ϑ3
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= 0.2 and ϑ2
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=
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ϑ4
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= 0.1. plots the variation of ur(r
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=
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r4)/r1 vs. r2 and r3 (>r2), holding r1 and r4
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= 4r1 fixed. The elastic constants employed are those for an agar-gelatain phantom, measured via the method of transient elastography (): λ1
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= 22.5 × 106
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= 100λ2, μ1
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= 80 = 100μ2, l1
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=
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l2
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= −2 × 106, m1
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= −2 × 106
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= 0.01m2, n1
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=
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n2
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= −100 kPa. Layer 2 (or 4) is both linearly and nonlinearly more compliant than Layer 1 (or 3). Note the five to six orders difference in magnitude between λ and μ, and the four to six orders difference between n and l, m. That λ
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≫
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μ shows that this material is nearly incompressible. shows that a maximum in ur occurs at r2/r1
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∼ 1 and r3/r1
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∼ 3.2. The composite will appear like that indicated in , which shows the “near-elimination” of Layer 2. It suggests a competition between the effects of dilatation and material compliance. The dilatation in Layers 2 and 4 is half that in Layers 1 and 3, while the former are more compliant than the latter. Hence, for maximum ur the size of Layers 2 and 4 should be reduced due to the dilatation effect, while they should be increased due to the compliance effect. It turns out that the dilatation effect is more dominant for Layer 2 while the compliance effect is more dominant for Layer 4, resulting in a configuration with a much reduced size for Layer 2 and a fairly large size for Layer 4. In contrast, if a very small swelling of the composite is desired, suggests that either r3 should be just a little larger than r2 (regardless of their values), or r3 should be just a little less than r4 (regardless of r2). This implies the “near-elimination” of Layers 3 and 4, respectively, as also indicated in It is also useful to investigate the influence of the chemical concentrations on the mechanical behavior. Two spherically symmetric dilatation profiles are selected for this investigation: ϑp2 and ϑp3, where r1
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=
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r4/4 =
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r3/3 =
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r2/2. The profile ϑp3 is similar to ϑp2, with the dilatation values in the layers interchanged, i.e., ϑ1
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=
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ϑ3
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= 0.1 and ϑ2
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=
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ϑ4
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= 0.2. To eliminate the effect of material dissimilarity, a homogeneous sphere of radius r4 is considered here with the following elastic constants: λ
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= 35.7, μ
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= 10.3, l
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= −35.6, m
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= −24.2, and n
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= −23.5 kPa. plots the variation of (a) the radial displacement ur(r)/r1, (b) radial stress and (c) meridional stress against the radial coordinate r for the two dilatation profiles. It can be observed that even though the average dilatation value is the same for both profiles, the outer displacement (swelling) of the homogenous sphere subjected to ϑp3 is ∼30% larger than that subjected to ϑp2. The kinks in the displacement curves reflect the dilatation jumps at these locations. Moreover, ϑp2 and ϑp3 induce stresses which differ in sign. If the dilatation value at a certain radial location experiences a drop in value, then the surrounding regions experience a compressive radial stress, as at r1 and r3 when the profile is ϑp2, and at r2 when it is ϑp3. The radial stress vanishes at the outermost boundary in both cases, as required by the boundary condition. Similar opposing meridional stress distributions are predicted for the two profiles. In addition, the meridional stress experiences discontinuities at the locations of the dilatation jumps. show that the concentration profile in a multilayer capsule exerts a rather complex effect on the stress distributions. Coupled with dissimilarity in the layer elastic constants and thicknesses, the mechanical response is expected to be of even greater complexity. Hence, simulations with relatively simple nonlinear elastic models may provide useful qualitative and quantitative assessments which will aid the design of multilayer gels.The current work is motivated by prior results, e.g. , which show that the mechanical properties of the bioenvironment have a profound influence on the biological processes occurring in it. It highlights the importance of nonlinear elasticity in the mechanical modeling of both synthetic and bio-polymer gels. There exist many nonlinear models for soft matter, e.g., microscopic models based on internal structural/chemical variables or molecular dynamics simulations, and macroscopic models derived from the phenomenology of the nonlinear behavior (). Microscopic models are generally used for investigating the internal structure and its dynamics, but may be computationally prohibitive if the desired outputs are to be macroscopic variables. Although multiscale models that aim to bridge the microscopic and the macroscopic have been developed, simple phenomenological models can be used to study the response of architecturally complex gels (multiple layers with different elastic properties) without significant computational cost. This is demonstrated in this work, which shows that solutions can be obtained for an N-layer composite gel.The second-order elastic model requiring five elastic constants is generic and describes a typical weakly/moderately nonlinear elastic material. The five constants can be obtained indirectly through fitting the relevant equations to experimental data, or directly through measurements. Acousto-elastic methods have been used to determine the elastic constants of soft gels (). The constants used in the simulations of were directly determined via such methods in the framework of second-order elasticity. Atomic force microscopic methods have also been used to study the mechanical properties of cross-linked polyelectrolyte multilayer films in which the Young’s modulus was usually the only elastic parameter determined (). However, much research remains to be done on the relationship between the elastic constants, especially the higher-order ones, and the biological parameters, e.g., the dependence of cell differentiation and growth characteristics on the third-order elastic constants. The current work predicts the mechanical state for a given bioenvironment, but the link of the mechanical state to the biological activities is not yet established. The interface between mechanics and biology remains an interesting challenge.The solutions for the stresses and displacements in an arbitrary N-layer cylindrical and spherical gel have been obtained for constant stepped dilatations. The problem can be extended to other loading and boundary conditions. For instance, Eqs. can be solved analytically for non-constant dilatations within the layers and for pressure loading on the outer/inner surfaces. For a more general shape other than spherical or cylindrical, the second-order elastic model can be incorporated as a constitutive model in a numerical strategy to solve complex boundary-value problems.Naturally, the current model has its limits. It does not consider time and rate dependence phenomena such as viscoelasticity and how they may modulate the mechanical state over time. Detailed theoretical models accounting for finite viscosity, plasticity and rate effects have been developed () and the visco-elasto-plastic effects have been probed experimentally via indentation methods (). Factors such as the elastic anisotropy of the soft materials, e.g., biological tissues with aligned fibers, have often been ignored. It is however, possible to account for elastic anisotropy through the use of additional strain invariants in the strain energy functions (). The modeling of the anisotropic viscoelasticity of soft tissues for medical image computing has also been attempted (In this paper, power-law solutions for the stresses and displacements in multilayer spherical and cylindrical gels subjected to stepped dilatations are developed in the framework of second-order elasticity. This requires the use of five elastic constants for each layer. It is shown that universal relations can be derived for uniaxial tension and the swelling of the gels due to the effect of pH. Reasonably good fit of these universal relations to experimental data has been obtained for a wide range of synthetic and biological gels. The importance of elastic nonlinearity in the mechanical behavior of such gels is emphasized.Further simulations for a four-layer composite show that dissimilarity in the layer elasticities, thicknesses and dilatations can strongly influence the mechanical response. Consequently, the biomaterial design may benefit from such simulations, which can be used to search for the loading, geometrical and material parameters most suitable for a desired mechanical state in a specific application. This may enable multiple functionalities, enhance material integrity and improve the bio-mimicry.Correlations between microstructures and properties of Cu-Ni-Si-Cr alloyA commercial Cu-Ni-Si-Cr alloy used as a model material was annealed and then peak-aged. The detailed microstructure and property analyses were performed to determine the correlations between the microstructure characteristics and properties. Theoretical calculations, which gave results in good agreement with experimental results, demonstrated that solid-solution scattering and Orowan bypass of precipitate strengthening were the key mechanisms for electrical and mechanical properties, respectively. Furthermore, the effects of composition on the aging behaviors and then electrical and mechanical properties for Cu-Ni-Si series alloys were systematically investigated, showing that both the precipitation extent and mechanical properties improved simultaneously with nominal or effective concentration of the precipitates. Quick estimation on the basis of the present analysis procedure indicates that our work is applicable with high accuracy in real cases.Cu-Ni-Si alloys are some of the most widely used materials in electrical and electronic industries for applications such as lead frames, contactors and electrical connectors where a combination of good electrical and mechanical properties are required Except the composition manipulation, other variables like processing methods that affect the macro- and microscopic properties have also been researched intensively Understanding the respective contributions of the different mechanism can be very useful for achieving desirable properties through optimal balance among the different mechanisms. Therefore, the present work has three basic objectives: first, to characterize all microstructural characteristics of the model Cu-Ni-Si-Cr alloy in detail; second, to quantify the individual mechanisms that affect the electrical and mechanical properties; third, to expand the present discussion into the full composition range to systematically investigate the effects of Cu-Ni-Si alloys' composition on the properties and build up the specific correlations between the composition and properties for direct use in practice.Commercial Cu-2.69Ni-1.14Si-0.45Cr at% alloys were annealed at 930 ℃ for 1 h and then quenched in water. The annealed alloys were aged at 510 ℃ for 3 h to obtain a good combination of strength and electrical conductivity First, the metallographic examination and the typical tensile true stress-true strain curve of the commercial alloys before annealing are given as a reference in a, the grains are severely textured which is caused by forging. The average spacing in the direction perpendicular to the elongated direction (the forging direction) is mostly around 30–70 µm. There are a number of regions where particles in light blue aggregate (as indicated by the red dashed circles) with many other isolated but still large ones distributing in the matrix too, indicating the alloy is overaged to some extent. The yield strength is about 600 MPa and the strain to failure lies typically around 0.04, not stable in the tests. Since the alloys will be of limited use with the poor plasticity and are not ideal as a model material, they are annealed again in the laboratory and the later investigation will mainly focus on the annealed and peak-aged samples. displays the results of the metallographic examination through OM and SEM with the grain diameter distribution obtained by the line intercept method a as indicated by the red arrows. Further observation on these particles by SEM shows that their diameters vary from sub-micrometers to about ten micrometers. The EDS analysis conducted on the relatively large ones verifies that these particles are Cr/Si compounds. Since the present annealing temperature falls into the proposed temperature range of Ref. shows the TEM micrographs and statistical diameter distribution of the precipitates. As seen from a, a high number density of nano-scaled precipitates with strain field contrast are distributed homogenously in the matrix, as was also reported in previous studies Even though two major Ni/Si precipitates are present in the Cu-Ni-Si-Cr alloy, namely, β-Ni3Si and δ-Ni2Si (as indicated in c through [0 0 1]cu), their diameters may lie in the same range in the present aging conditions d. From the distribution, the average radius of the precipitates can be determined to be approximately 4.5 ± 1 nm with the volume fraction of 2.5 ± 0.5%. The ratio of the number of the two kinds of precipitates is roughly assumed to be 1:1 for simplicity. In addition to the nickel silicides, some precipitates different are shown in e and f. As indicated by the red dashed circles in e, the precipitates are mostly spherical and are a little larger than the Ni/Si precipitates, that is, over ten nanometers. From the size and shape, they can be recognized as the newly precipitated Cr particles f, the diameter is slightly smaller than one micrometer and its appearance is just as same as that of Cr3Si in Ref. So far, Ni2Si particles, Ni3Si particles and Cr particles have been found except the very fine Cr3Si particles (formed during aging), which may be due to the low content and complex existing forms of Cr.The electrical conductivity of the specimens before and after peak aging is measured to be 21 ± 1% IACS and 44 ± 1% IACS, i.e., approximately 82.1 ± 0.4 nΩ·m and 39.2 ± 0.1 nΩ·m respectively, showing dramatic increase in the conductivity after aging due to the depletion of the solute atoms in the matrix.The typical tensile true stress-true strain curve of the aged samples is shown in . The yield and ultimate tensile strength are respectively 559 ± 8 MPa and 757 ± 13 MPa, and the strain-to-failure is 18.3 ± 2.0%, exhibiting apparent recovery in the plasticity but a little reduction in the yield strength as in comparison with those of the original state.Comparing with the Cu-3Ni-0.63Si-0.12Ti wt% alloy without Cr addition in Ref. The dislocation density in the alloy was roughly estimated based on the Williamson-Hall method where β is the true FWHM, λ is the wavelength of Cu Kα radiation, K is a constant equal to ~ 0.9, ε is the microstrain and θ is the Bragg angle.The value of micro-strain ε deduced from the slope of the fitted line is 0.083, with a small negative intercept (~ −0.07) which may be due to the experimental errors where d is the grain diameter, 65 µm and b is the Burgers vector, 0.255 nm. Therefore, the dislocation density is 1.7 × 1013 /m2, which is reasonable based on the comparison to other annealed copper samples First of all, the effects of Cr in Cu-Ni-Si-Cr alloy have to be specified. As stated before, there are three existing forms for Cr in present alloy, that is, solute atom, undissolved Cr particles and chromium silicides (Cr3Si) There are two major scattering mechanisms, namely, lattice defects and impurities. The former mainly includes dislocations, grain boundaries and precipitates while the latter basically consists of solute atoms. Without considering the temperature effect, the electrical resistivity ρalloy of Cu-Ni-Si-Cr alloy in this work can be expressed by Mattiessen's law as where ρalloy is the overall electrical resistivity of the alloy after aging, ρ0 is the resistivity of pure copper, 17.2 nΩ·m Precipitates reduce the effective conducting volume and thus lead to an increase in electrical resistivity. Under the assumption that the precipitates are nearly non-conducting compared to the matrix where f is the volume fraction of the precipitates, equal to a total of 2.5% for Ni3Si and Ni2Si. Thus, the increase in electrical resistivity from the precipitate is 0.6 nΩ·m. This result indicates that the electrical resistivity increases due to the precipitate scattering after peak aging is rather small and can also be neglected in some sense.Solid-solution scattering has a significant impact on the electrical conductivity of the alloy due to lattice disruption. The contribution to electrical resistivity from the solid-solution scattering can be calculated as where x is the solubility of a specific element in the Cu matrix. The coefficient (Δρ/Δx) varies depending on the identity of the element, as described in In addition to the matrix, it is reasonable to assume that the electrical resistivity of the alloy in this work is mainly affected by the solute atoms η=1−ρalloy−ρ0−ρSS′(NiorSi)−ρSS′(Cr)ρS−ρ0−ρSS′(NiorSi)−ρSS(Cr)where ρs is the resistivity of the alloy after annealing, and ρSS′(NiorSi) is the resistivity induced by the residual unpaired atomic concentration of Ni or Si in the matrix after matching Ni/Si as Ni2Si or Ni3Si. ρSS(Cr) and ρSS′(Cr) is respectively the resistivity increment induced by the solute Cr in the matrix at the annealed and aged states. Therefore, η ≈ 0.72 ± 0.02. Then, based on the knowledge of the density and precipitation extent, the volume fraction of the precipitates can be approximately calculated as:where Mp and MCu are the molar masses, 146 g/mol for Ni2Si or 205 g/mol for Ni3Si and 64 g/mol for the matrix. ceffec=cT*η is the effective atomic concentration of the precipitates and cT is the average nominal concentration ratio between Ni2Si and Ni3Si from the alloy composition, and η is the corresponding precipitate extent calculated from Eq. . ρp and ρCu are the densities of the precipitates and the matrix, 8.0 g/cm3Finally, the atomic concentration of the residual element in the Cu matrix is given by Taking unmatched atoms along with unaged Ni/Si atom pairs into account, the electrical resistivity increase due to the solid-solution scattering can be calculated according to Eqs. . With the resistivity compensation from the residual Cr, the result is 21.0 nΩ m.It is important to note that the decrease in the electrical conductivity caused by the residual Co in the matrix is almost equal to that of Si and is about six times larger than that of Ni, even though their other properties are quite similar. Much attention should therefore be devoted to the determination of the content of these elements in the present alloy.Based on the obtained data and the above equations, the electrical resistivity of the alloys is calculated to be 38.8 nΩ·m and the respective contributions from different mechanisms are summarized in It can be seen that the experimental and calculated results are in excellent agreement and it can conclude that the residual elements (mainly Ni and Si) in the matrix after aging are responsible for the moderate electrical conductivity of the present alloy. The minor differences may be due to the error of the testing method or averaging treatment in determining the electrical resistivity caused by the solid solution, that is the assumption of the 1:1 ratio between Ni2Si and Ni3Si, whereas this ratio may actually deviate slightly from 1:1.Strengthening mechanisms in polycrystalline materials are usually divided into four categories: solid-solution hardening, grain-boundary hardening, dislocation hardening, and precipitation hardening. Since the four mechanisms operate independently, the yield strength can be calculated by linearly adding the four individual contributions as follows where σ0 is the lattice friction stress of the Cu matrix, equal to 20 MPa Precipitate strengthening arises either from the dislocation shearing or the Orowan dislocation bypass mechanism. The shearing mechanism occurs for coherent precipitates with small radii, while the Orowan bypass mechanism operates when the coherent precipitate radius exceeds a critical value or when the precipitate is incoherent When precipitates are bypassed through the Orowan dislocation looping mechanism, the corresponding increase of yield strength is given by where M= 3.06 is the mean orientation factor for the fcc polycrystalline matrix where f is the volume fraction of the precipitates and is equal to 2.5%.When the precipitates are sheared by the dislocations, three effects should be taken into account, namely, coherency strengthening (Δσcs), modulus mismatch strengthening (Δσms) and order strengthening (Δσos) The strength increment due to the coherency strengthening Δσcs is due to strain-field interactions between a coherency precipitate and a dislocation and is given by Modulus strengthening occurs due to the mismatch between the shear modulus of the precipitate and matrix phases. The strength increment is given by Order strengthening is caused by the formation of an antiphase boundary (APB), which occurs when a matrix dislocation shears an ordered precipitate. The strength increment is given by where γAPB is the APB energy of the precipitate phase and is taken to be that of Ni3Si, which is equal to 250 mJm−2Using the above data and equations, the values of total Δσor, Δσcs, Δσms and Δσos are calculated to be 530 MPa, 1380 MPa, 331 MPa and 208 MPa, respectively; Therefore, the dominant mechanism for precipitate strengthening is the Orowan dislocation bypass, that is, the strength increment from the precipitates Δσp is equal to 530 MPa.It is well-known that grain boundaries or interfaces can impede the movement of dislocations and thus lead to increased strength. The grain boundary strengthening can be described well by the classical Hall-Petch equation where KHP is the Hall-Petch coefficient equal to 112 MPa·um1/2The strength increment due to dislocation strengthening is given by where α is a constant which is equal to 0.2 for fcc metals The increase in the strength caused by solid-solution strengthening is calculated by where c is the atomic concentration of residual elements (Ni or Si) in the matrix, which can be calculated according to Eq. given by the precipitating extent of approximately 72%. β is a constant, 3 The relevant parameters and calculated results of the above equations are summarized in . The obtained strength increment due to the solid-solution ΔσSS is 21 MPa.Using the obtained data and the above equations, the overall strength of the alloy is calculated to be 613 MPa and the individual contributions from the corresponding mechanisms are summarized in The calculated result is in good agreement with the experimental result and it reveals that the precipitate-hardening arising from the fine nickel silicides contributes absolutely the largest part of the overall strength. The slight difference may arise from the error in estimating the dislocation density and the size and volume fraction of the precipitates. In this case, if the volume fraction (f) in Eqs. is replaced with the theoretical results from Eq. , then the overall calculated yield strength should be 587 MPa, even closer to the testing results; this further demonstrates the equations are reasonable.Since most alloying elements are introduced into the Cu-Ni-Si-X (X represents the micro-alloying elements) alloys for grain refinement, precipitation improvement, property optimization and microstructure stability at high temperature . A coefficient of 2.4 is adopted to transform the hardness value into the yield strength value.First, the effect of composition variation on aging behavior and electrical property is considered. The precipitate extent and atomic concentration of the residual elements are obtained according to Eqs. . All residual elements are changed into Ni equivalents (cr) according to for consistency. The results are shown in Second, the effect of composition variation on mechanical strength is taken into account. The strength increment due to the precipitate strengthening is plotted versus effective precipitates concentration (ceffec) in The analysis procedure based on the results above is summarized as:Even though the precipitate extent grows continuously with increasing nominal atomic concentration of the precipitates as shown in a, it is generally below 100%, which may be an underlying mechanism for the moderate electrical conductivity of Cu-Ni-Si-X series alloys (approximately 20–50% IACS). The curve in a shows a sigmoidal shape and can roughly be divided into three regions based on the atomic concentration range: first, from 0 to 0.5 at%, the degree of precipitation stays at a relatively low level (less than 10%), and grows slowly with increasing concentration; Second, for 0.5–0.75%, the precipitate extent rises steeply with growing concentration; finally, for 0.75–4 at%, the degree of precipitation still improves with the concentration but in a more gentle way compared to the second region and progressively approaches 100%. When the atomic concentration of Ni/Si becomes much higher, the mechanical properties will degrade rapidly b presents the data for the variation of the electrical resistivity with the atomic concentration of residual Ni equivalents and a line fitted to the experimental results. The graph indicates that a strong correlation exists between these two variables; the slope of the fitted line is 7.4 ± 0.6 nΩ·m/at%, close to the solid-solution scattering coefficient of Ni. The result validates the hypothesis that the residual atoms in the matrix play a critical role in the overall electrical resistivity of the alloys.c that the strength increment changes mildly with the effective precipitates concentration above approximately 1 at% and approaches some upper limit. The fitting curve also agrees well with the experimental data in the entire range. Given the good agreement between experimental results and fitted curves, we may be able to rewrite Eqs. to avoid the exhaustive microstructure characterization work in an attempt to gain a comprehensive knowledge of properties of Cu-Ni-Si-X alloys with a reasonable accuracy:σy=σ0+Δσp+Δσd+ΔσGB+ΔσSS=20+712(1−e−120ceffec)+7*10−6*ρ1/2+112d1/2+192εSS3/2c1/2where ceffec, ρ, d, εSS and c have the same meaning as above.If more background information is provided, the overall electrical resistivity and mechanical strength will readily be evaluated for different Cu-Ni-Si-X alloys after varying treatment using the results in based on the analysis procedure shown in . To test the efficiency of this approach, a complete application of the procedure to a complex combined aging treatment condition in Ref. From the alloy composition, the nominal precipitates concentration is approximately 2.6% and then the precipitation extent can be directly read from a and is approximately 0.82. Thus, the effective precipitates concentration and residual atomic concentration of Ni equivalents are calculated to be 2.13% and 4.32% respectively. According to Eq. , the strength increment due to the precipitate strengthening is deduced to be 657 MPa. Taking heavy cold deformation and short duration of secondary aging into consideration, the grain size and dislocation density are estimated to be approximately 100 nm and 1015 /m2, and therefore, the strength increments due to grain boundary and dislocation are 354 MPa and 221 MPa, respectively. Finally, the solid solution strengthening is calculated to be 27 MPa. With compensation of fine grain ; the yield strength is 1280 MPa according to Eq. . Comparison between the experimental and calculated results shows that our analysis applies to the referred case fairly well. On the basis of this analysis, we can further explain the minor strength difference shown in A commercial Cu-Ni-Si-Cr alloy is annealed and peak aged to gain the electrical resistivity of 39.2 nΩ·m and the yield strength of 559 MPa. Detailed microstructure examinations are conducted in order to determine the relationships between the internal characteristics and the macroscopic properties. The results show that the matrix has the average diameter of 65 µm and dislocation density of 1.7*1013 /m2 while the precipitates have the average radius of 4.5 nm and volume fraction of 2.5% with the precipitating extent of nearly 72%. Quantitative analyses indicate that solid-solution scattering and Orowan bypass of precipitate strengthening dominate among the different scattering and strengthening mechanisms, respectively. Further investigation shows that both the precipitation extent and mechanical properties improve simultaneously with increased nominal or effective concentration of precipitates. A quick estimate indicates that the present analysis procedure is quite applicable for Cu-Ni-Si-X alloys in real cases. Taken together, the present work provides useful guidance for the optimization of the combination between the composition adjustments and processing methods to achieve ideal properties.Electrochemical behavior of a Ti-based bulk metallic glassIn this study, potentiodynamic experiments were conducted with a Ti-based BMG alloy with a nominal composition of Ti43.3Zr21.7Ni7.5Be27.5 [atomic percent (at.%)], commonly known as LM-010. Electrochemical characterization was performed in a phosphate-buffered saline (PBS) electrolyte at 37 °C with a physiologically relevant dissolved oxygen content. This BMG exhibited passive behavior at the open-circuit potential with a low mean corrosion-penetration rate. A susceptibility to localized corrosion was observed but is not a concern at the open-circuit potentials. The resistance of the LM-010 alloy to localized corrosion was statistically equivalent to, or better than, all of the BMG materials and the 316L stainless steel for which direct statistical comparisons were possible. Microscopic examination revealed that the samples predominantly exhibited many scattered, small pits (diameter ⩽100 μm) in addition to several larger pits. Based upon the pit morphology and comparisons with the literature, it appears that localized corrosion initiated at clusters of inhomogeneities within the amorphous matrix.The term metallic glass or amorphous alloy describes a metastable class of materials that have no long-range, periodic atomic order Although Ti-based metallic glasses were reported as early as 1977 Crystalline titanium and its alloys have excellent biocompatibility with outstanding corrosion resistance, a relatively low modulus, low density, and high strength that results in good specific strength. This combination of properties has resulted in the widespread use of titanium and Ti-based alloys in the aerospace and biomedical industries, among others. The ease of fabrication and unparalleled collection of properties found in Ti-based BMGs could be advantageous in biomedical applications such as pacemaker cases, housings for ventricular-assist devices, implantable infusion drug pumps, screws, bone plates, etc. However, before biomedical or structural applications can be considered, the electrochemical behavior of these alloys should be investigated.All biomedical implant devices interact to some extent with the tissue environment in which they are placed This study is focused on the evaluation of the material response of a Ti43.3Zr21.7Ni7.5Be27.5 (atomic percent, at.%) BMG alloy to a physiologically relevant environment. Although the host response is not addressed directly in this study, the electrochemical interaction between the material and the environment inevitably results in the release of metal ions into the surrounding tissue While Ti and Zr are recognized as some of the most biocompatible elements, Ni and Be are not. Ni is one of the most commonly known allergens that affects an estimated 10–15% of the population Although a Be-free alloy would likely produce improved biocompatibility, the few cell-culture studies of Be found in the literature yielded surprising results. Wataha and co-authors This particular alloy was selected because, like Zr-based alloys, Ti-based alloys naturally form a tenacious, passive oxide layer that usually demonstrates good corrosion resistance and could significantly limit metal-ion release of the less biocompatible elements in the bulk alloy. While several electrochemical studies have been reported on Ti-based, melt-spun metallic glasses A Ti-based alloy plate with a nominal composition of Ti43.3Zr21.7Ni7.5Be27.5 (at.%), known as LM-010, was fabricated by Liquidmetal Technologies (Lake Forest, CA). The alloy was prepared by combining commercial-purity elemental metals (with total oxygen content less than 1000 ppm) and arc-melting on a cooled copper hearth. The homogeneous alloys were subsequently cast into 2.5 mm thick plates with dimensions of 55 mm × 85 mm by a proprietary permanent-mold casting technique. Corrosion samples (10 × 10 mm) were extracted from these plates, and X-ray diffraction (XRD) was performed prior to the electrochemical study to verify the amorphous structure of the samples (Philips X’pert X-Ray Diffractometer) within the resolution limits of XRD.Electrochemical cyclic-anodic-polarization tests were conducted using an EG&G Princeton Applied Research 263A potentiostat with EG&G Powercorr software (Princeton Applied Research, Oak Ridge, TN). The electrochemical cell consisted of the corrosion sample (working electrode), a saturated calomel reference electrode, and a platinum-foil counter electrode. Thus, all potentials cited in this study will henceforth be in reference to the saturated calomel electrode (SCE).One test was conducted per sample for a total of 20 corrosion tests. Electrochemical characterization was performed in a phosphate-buffered saline (PBS) electrolyte at 37 °C. The composition of the electrolyte is presented in . This electrolyte is intended to simulate surgical implant conditions similar to those found in vivo. The mean pH (±95% confidence intervals) of the PBS electrolyte was 7.44 ± 0.02, which is near that commonly found in the human body (7.0–7.4) Immediately prior to electrochemical testing, each sample was ground to a 600 grit SiC surface finish. A polymer sample holder was utilized to repeatedly expose 50 mm2 of each sample to the electrolyte during testing. Before each polarization scan was initiated, the corrosion sample was allowed to stabilize in the electrolyte until the open-circuit potential (Eoc) changed by no more than 2 mV over a 5 min time period. The Eoc at which this occurred was considered to be the open-circuit corrosion potential (Ecorr). The scan was started at 20 mV below Ecorr and continued in the positive direction at a scan rate of 0.17 mV/s until a current density (i) of 104
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mA/m2 was attained. At this point, the scan direction was reversed, and the potential was decreased at the same rate until a potential of 50 mV below Ecorr was reached. Following electrochemical testing, each sample was examined by stereomicroscopy, and select samples were examined by a scanning electron microscope (SEM) with a secondary-electron detector.Multiple corrosion-related parameters were derived from the cyclic-anodic-polarization curves as previously delineated in the literature At potentials below Epp, the material is immune to localized corrosion, and any damage to the passive film will repassivate. Above Epp but below Epit, pitting is possible after an incubation time or due to damage to the passive film. In terms of the overall resistance to pitting corrosion, two parameters are important: (a) the pitting overpotential (ηpit), or (Epit
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−
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Ecorr), and (b) the repassivation overpotential (ηpp), or (Epp
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−
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Ecorr). Higher, positive values of these quantities are desirable The corrosion-penetration rate (CPR) is an estimate of the corrosion rate of a material at Ecorr and can also be derived from cyclic-anodic-polarization tests. The CPR is dependent upon inherent material and electrolyte properties and can be determined from icorr by the application of Faraday’s law The parameters derived from the LM-010 corrosion tests were statistically compared to the data available for other corrosion studies in the literature. All statistical analyses were conducted with JMP 5.1.1 software (SAS Institute Inc., Cary, NC). Within each corrosion parameter, the parametric assumptions of residual normality and equivalence of variance were tested by the Shapiro–Wilk In several cases, the initial data did not meet the parametric assumptions. Thus, the Box–Cox transformation was used to improve the compliance with these assumptions Ninety-five percent confidence intervals (95% CIs) were calculated for all of the means presented in this study based on the Student’s t distribution. Finally, box-and-whisker plots were constructed with Origin 7.5 software (OriginLab Corporation, Northampton, MA), which uses Tukey’s hinges A representative XRD spectrum from the LM-010 corrosion samples is shown in . Each spectrum demonstrated a broad, diffuse peak that is characteristic of amorphous alloys. The average cyclic-anodic-polarization curve is plotted in . All of the electrochemical parameters derived in the cyclic-anodic-polarization testing are summarized in . Furthermore, the Ecorr data is plotted in . The original corrosion data was available for the LM-001 BMG The mean (±95% CIs) and median CPRs for the LM-010 alloy were calculated to be 2.9 ± 2.6 and 0.8 μm/yr, respectively (). However, the CPR distribution was positively skewed due to four data points that were determined to be extreme outliers. In this case, the median, which is more resistant to outlier effects, is a better measure of the CPR.The LM-010 alloy exhibited a susceptibility to localized corrosion at elevated potentials with mean (±95% CIs) and median Epit values of 217 ± 17 and 206 mV (SCE), respectively (). These values resulted in mean (±95% CIs) and median ηpit values of 589 ± 57 and 550 mV, respectively (). Furthermore, repassivation was observed to occur prior to reaching Ecorr on the reverse scan. The mean (±95% CIs) and median Epp values were 38 ± 32 and 70 mV (SCE), respectively (), resulting in mean (±95% CIs) and median ηpp values of 411 ± 70 and 366 mV, respectively (For a comparison to other corrosion-resistant materials in the same, or similar, electrolytes, these corrosion parameters and those reported in the literature are summarized in . Comparing these values to those reported in the literature for which direct statistical comparisons were possible, no statistically significant differences were found between the CPRs, at the tested levels. The mean LM-010 ηpit value was approximately 44% and 24% greater than those reported for the LM-001 (p
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= 0.0360) All of the corrosion samples were examined by stereomicroscopy after corrosion testing. Every sample exhibited corrosion in the form of localized pitting. With the exception of one sample, crevice corrosion at the sample/sample holder interface was not prevalent. In general, the samples predominantly exhibited many scattered, small pits (diameter ⩽100 μm) in addition to several larger pits. The number of pits on each sample was quantified, and the pits were subjectively classified into one of the following qualitative categories: extra-large (XL), large (L), medium (M), and small (S). A pitting score was calculated for each sample by assigning values to each pit size (i.e., XL = 4, L = 3, M = 2, and S = 1) and summing the product of the pitting score and the quantity of pits within each category. The pitting scores were not correlated to any of the corrosion parameters examined in this study (coefficient of determination =
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r2
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⩽ 0.35).Furthermore, several corrosion samples with both high and low CPRs were selected for SEM analyses. Representative images of the pit morphology are presented in (a) is approximately 90 μm in diameter and is typical of the pits observed on all of the samples. A magnified image inside this pit at location A is shown in (b) and illustrates the small, hemiellipsoidal features approximately 1–6 μm in diameter that were observed on the walls of the larger pits.The average polarization curve demonstrates that this alloy spontaneously forms an barrier-type, passive oxide layer as demonstrated by the low icorr. These low corrosion current densities resulted in low CPRs that are well within the expected range for passive behavior. In general, CPRs of less than about 76 μm/yr are considered acceptable for chemical and industrial applications Since the corrosion resistance of Ti-based BMGs has not been previously evaluated, this material was compared to Zr-based BMGs and other corrosion-resistant, crystalline materials (). In a majority of the cases, the parameters from other reports in the literature were estimated based upon analyses of the published polarization curves. Again, it should be noted that comparisons between these results and those conducted in other laboratories should be made with the knowledge that they are dependent upon differing test parameters. Nevertheless, it appears that the CPR for the LM-010 BMG alloy is comparable to these other materials. In terms of the localized corrosion resistance, the LM-010 BMG was equivalent, or superior, to all of the BMGs but inferior to both CoCrMo and Ti–6Al–4V materials, which did not pit in this environment.Although several of the studies by Hiromoto et al. It is difficult to make comparisons with the reports by Hiromoto et al. on the corrosion resistance of the Zr65Al7.5Ni10Cu17.5 (at.%) in Hanks’ solution, Eagle’s minimum essential medium (MEM), and MEM with fetal bovine serum (MEM + FBS) BMG alloys are ideally single-phase, homogeneous materials without grain boundaries, dislocations, inclusions, dispersed second phases, or other physical imperfections. On the exposed surface, these imperfections can lead to local defects in the passive film and commonly serve as initiation sites for localized corrosion in crystalline materials We propose that this process is also responsible for the hemiellipsoidal features on the surfaces of the pits observed in this study (). The development of the large pit was the result of a large cluster of micrometer-sized inhomogeneities that successively dissolved as the pit propagated down from the sample surface. Pit propagation continued into the surrounding single-phase amorphous matrix until another inhomogeneity was encountered and the process was begun again. Upon reaching Epp near the end of the polarization test, the actively propagating surfaces of the pits repassivated and the process was halted, leaving these small hemiellipsoidal features on the walls of the larger pit. The hemiellipsoids could be the result of elongated inhomogeneities or simply the result of many hemispherical pits growing in close proximity and impinging.Based upon the results of this investigation, the following conclusions were made about the corrosion resistance of the LM-010 BMG alloy in a PBS electrolyte:The LM-010 BMG alloy exhibited passive behavior at the open-circuit potential with a low corrosion rate.The mean CPR (2.9 ± 2.6 μm/yr) was low, well within the expected range for corrosion-resistant materials, and statistically equivalent to those of other corrosion-resistant crystalline and amorphous materials reported in the literature, at the tested levels, for which direct statistical comparisons were possible.The LM-010 BMG exhibited a susceptibility to localized corrosion in the form of pitting. However, both the ηpit and ηpp were relatively high, positive values; therefore, localized corrosion is not a concern at the open-circuit potential with this alloy in this environment.The resistance of the LM-010 alloy to localized corrosion was statistically equivalent to, or better than, all of the BMG materials for which direct statistical comparisons were possible. Furthermore, the localized corrosion resistance was statistically equivalent to, or better than, the 316L stainless steel.The LM-010 BMG alloy appears to be comparable, or superior, to the other BMG alloys from the literature in terms of CPRs and localized corrosion resistance.Microscopic examinations revealed that the samples predominantly exhibited many scattered, small pits (diameter ⩽ 100 μm) in addition to several larger pits. Based upon the pit morphology and comparisons with the literature, it would appear that localized corrosion initiated at clusters of inhomogeneities within the amorphous matrix.Surface displacements resulting from magma-chamber roof subsidence, with application to the 2014–2015 Bardarbunga–Holuhraun volcanotectonic episode in IcelandThe conditions which lead to caldera collapse are still poorly constrained. As there have only been four, possibly five, well-documented caldera forming events in the past century, the geodetic signals produced during chamber roof subsidence, or chamber volume reduction (shrinkage) in general, are not well documented or understood. In particular, when two or more geodetic sources are operating and providing signals at the same time, it is important to be able to estimate the likely contribution of each. Simultaneous activities of different geodetic sources are common and include pressure changes in magma chambers/reservoirs occurring at the same time as dyke emplacement. Here we present results from numerical models designed to simulate the subsidence of a magma-chamber roof, either directly (chamber shrinkage) or through ring-fault displacement, and the induced surface deformation and crustal stresses. We consider chamber depths at 3 km, 5 km, and 7 km below the crustal surface, using both non-layered (isotropic) and layered (anisotropic) crustal models. We also model the effects of a caldera lake and of a thick ice cover (ice sheet) on top of the caldera. The results suggest that magma-chamber roof subsidences between 20 m and 100 m generate large (tens of centimetres) vertical and, in particular, horizontal displacements at the surfaces of the ice and the crust out to distances of up to tens of kilometres from the caldera/chamber centre. Crustal layering tends to reduce, but increasing chamber depth to enlarge, the horizontal and vertical surface displacements. Applying the results to the ice subsidence in the Bardarbunga Caldera during the 2014–2015 Bardarbunga–Holuhraun volcanotectonic episode indicates that the modelled ice displacements are less than those geodetically measured. Also, the geodetically measured crustal displacements are less than expected for a 60 m chamber-roof subsidence. The modelling results thus suggest that only part of the ice subsidence is due to chamber-roof subsidence, the other part being related to flow in and down-bending of the ice. We show that such a flow is likely within the caldera as a result of the stress induced by the 45-km-long regional dyke emplaced (primarily in vertical magma flow) during the episode. This conclusion is further supported by the model results suggesting that the ring-fault (piston-like) displacements must have been much less than the total 60 m ice subsidence, or else faults with tens-of-metres displacements would have cut through the ice (these are not observed). We suggest that the ring-fault subsidence was triggered by small doming of the volcanic field and system hosting the Bardarbunga Caldera and that this doming occurred as a result of magma inflow and pressure increase in a deep-seated reservoir. The doming is confirmed by GPS measurements and supported by the seismicity results. The magmatic pressure increase in the reservoir was, in terms of the present model, responsible for the regional dyke emplacement, the Holuhraun eruption, and part of the stress concentration around, and displacement of, the Bardarbunga Caldera.Caldera collapses are a common occurrence in the evolution of major volcanic systems (). While many of these events are catastrophic and associated with the expulsion of large volumes of magma and ignimbrite formation (), perhaps the more prevalent situation involves relatively small or no magma expulsion (). Well-documented caldera collapses occurred in 2000 and 2007 at the summits of Miyakejima (). These events have been referred to as periodic () or slow collapses. These terms relate to the total caldera growth occurring over periods of perhaps as much as one month (). Much of the longer-period caldera growth was due to mass wasting, a process which likely also shaped lake Öskjuvatn (Iceland) following the 1875 caldera forming eruption (). A mechanism of ‘slow caldera collapse’ has also been suggested as an explanation for the measured ice subsidence during the 2014–2015 Bardarbunga episode (The timescale of deformation at calderas ranges from events of hours to days () to longer events taking months or years (). Collapse may occur along pre-existing structures, such as regional faults or earlier-formed ring-faults (), but the shape and size of collapse are significantly influenced by the depth, size, and shape of an underlying magma chamber (The movement of large crustal segments, as occurs during the formation or reactivation of collapse calderas, must produce significant crustal deformation. However, the magnitude and type of the deformation are poorly constrained. This is partly due to the lack of geophysical measurements syn-collapse, the exceptions being Piton de la Fournaise (), although measurements at these locations were predominantly limited to the central edifice and vent area. Therefore, understanding the far-field effects of crustal subsidence due to caldera formation or chamber shrinkage is useful for constraining geophysical observations at volcanoes where the summit region cannot be observed, either due to cloud cover, inaccessibility, or ice cover. The last point is salient because many, if not most, of the central volcanoes in Iceland are ice covered (). In addition, understanding the timing and development of collapse is important for hazard and risk estimation, partly because many calderas are associated with the formation of ring-dikes (When magma leaves or flows out of a chamber/reservoir during an eruption and/or dyke injection, the volume of the chamber/reservoir decreases. The same may happen during caldera collapse (). The volume decrease or shrinkage affects the crustal segment hosting the chamber, primarily through changes in stress and associated displacement and strain. The effects of chamber shrinkage are most easily detected through surface deformation. The aim of this work is to understand better (1) how the surface deformation associated with chamber shrinkage, in particular during roof subsidence, is reflected in horizontal and vertical displacements (and stresses) at the surface of the hosting crustal segment (as well as at the surface of the ice cover), (2) how the surface deformation changes with distance from the chamber, and (3) how much surface deformation can feasibly be accommodated in an elastic crust before ring-faults will form or reactivate, resulting in a normal caldera collapse. The results, while completely general, are here applied to the 2014–15 Bardarbunga–Holuhraun volcano-tectonic episode.Many analogue models of caldera collapse indicate initial ground surface slumping () followed by the formation of peripheral faults that ultimately control the majority of vertical subsidence (). As many of these models use dry sand or other similar granular materials to simulate the crust, it is often impossible to determine surface displacements far from the deformation centre. This follows partly because a dry sand pack lacks cohesion (which corresponds to rock tensile strength) and normally does not transmit tensile stresses as solid linear elastic material. By contrast, the crust behaves approximately as linear elastic solid material with a non-zero tensile strength. More specifically, the range of in-situ tensile strength of solid rocks is 0.5–9 MPa, the most common values being 2–4 MPa (). Numerical models which simulate an elastic crustal segment hosting a magma chamber therefore provide a reasonable approximation of surface ground deformation (In order for a caldera to form, or for slip to occur on a pre-existing ring-fault, there must be suitable state of stress within the crust. The initiation of sub-vertical, normal ring-faults depends on three stress field conditions which must be satisfied simultaneously (The minimum value of σ3, the maximum tensile (minimum compressive) principal stress, must be at the surface.The maximum value of (σ1–σ3)/2, the shear stress, must occur above the outer margins or lateral edges of the magma chamber, that is, in a zone extending from the lateral edge of the chamber to the surface and within which the ring-fault forms (or slips).The maximum tensile stress at the surface must peak at a radial distance approximately equal to the lateral dimension, the diameter, of the magma chamber.These stress conditions are most likely to be induced by a double magma chamber, where the shallow chamber is sill-like and (1) the crustal segment hosting the double chamber is subject to horizontal extension, or (2) the deeper chamber, a large reservoir, is subject to slight increase in magma pressure so as to dome the crustal segment hosting the shallower chamber (). Other predominant collapse trigger mechanisms () include (a) internal magma chamber overpressures initiating roof and surface fractures (e.g., ) and (b) internal magma chamber underpressure following chamber rupture (e.g., ). Here we consider in detail a situation more compatible with the second of these two mechanisms, namely an inferred underpressure in the shallow chamber, particularly in view of the suggestions that the ice subsidence during the Bardarbunga–Holuhraun episode being related to pressure decrease in the chamber (e.g., As indicated above, shallow chambers within crustal segments undergoing slight doming, regional extension, or both are the ones most likely to generate stress concentrations favourable for ring-fault formation (). Prime examples of this type of regional settings are the volcanoes of the Eastern Volcanic Zone (EVZ) in Iceland () and the Taupo Volcanic Zone (TVZ) in New Zealand ( shows the stresses around a sill-like magma chamber with negative internal pressure, an underpressure, of 5 MPa (e.g., ), located at 5 km depth in a 40-km wide and 20-km thick crustal segment but simultaneously subject to excess (doming) pressure of 10 MPa from a deep-seated reservoir. In this example, the doming pressure largely controls the stress/displacement fields and the maximum tensile stress concentrates at the free surface in a zone above the lateral margins or edge of the shallow sill-like chamber. In addition, the maximum shear stress concentrates at the lateral margins of the magma chamber at depth. These conditions are ideal for the formation of, initially, tension fractures at the surface that propagate down towards the chamber and change at a critical depth – normally less than 0.5 km (). If the tensile stresses are higher at the magma chamber margin than at the free surface above the margin, then a ring-dyke would be more likely to form (When a caldera forms, it is common for the depression to be filled with water, generating in a caldera lake. Well-known examples include Crater Lake, USA (). The occurrence of sub-glacial lakes within calderas has also been noted, such as in the Grimsvötn volcanic system in Iceland (). A caldera lake is important because the solid contact with water gives rise to a free surface, that is, a surface of zero shear stress. Therefore, a caldera lake will reduce the mechanical coupling between bedrock and glacier which in-turn will influence stresses and displacements within the ice.The finite element program Comsol was used to investigate the crustal and ice-sheet response to the vertical displacement of the magma-chamber roof (). In these models the magma chamber is modelled as a cavity (). In the first model the chamber is residing within a homogeneous, isotropic elastic half-space with a Young's modulus (E) of 40 GPa and Poisson's ratio (ν) of 0.25 (). In this model, the focus is on the typical conditions for ring-fault formation or reactivation in rift-zone environment. Thus, the loading condition is a combination of doming excess pressure of 10 MPa in the deep-seated reservoir and a horizontal tension of 5 MPa. The results are in agreement with earlier results suggesting that doming, horizontal tension, or both are loading conditions that favour the concentration of shear stress in a zone above the lateral ends or edges of the shallow chamber (C). Furthermore, the induced tensile stress peaks where these zones meet the free surface (D). The zones of high shear stress are thus likely to develop ring-faults, for the given loading conditions.Horizontal tension, however, will not be much discussed here. In the later part of the paper, we discuss the effects of doming by the deep-seated reservoir. The main focus here is on the effects of shallow-chamber roof subsidence on the associated surface deformation and stresses. The roof (upper boundary) of the chamber is supposed to subside, so that, the loading is prescribed negative vertical (z-axis) displacement. The vertical roof displacements tested in the models range from 20 m to 100 m. To make the models realistic, the crustal segment hosting the chamber is also modelled as anisotropic, that is, layered (). In the anisotropic models, directly above the chamber there are six layers, each with thickness t and of varying stiffness (Young's modulus, E) but constant density (ρ
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= 2500 kg/m3) and constant Poisson's ratio (ν
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= 0.25), simulating an anisotropic crust. The number of layers used in the models is arbitrary as most volcanic systems are presumably made of hundreds of layers, while many of these may group into larger units of internally similar mechanical properties. Here we choose to include six layers or units simply to investigate the effects of crustal anisotropy on the local stresses and displacements. The uppermost layer of thickness (2 t) represents an elastic body of ice; namely a glacier, with a Young's modulus of 4 GPa. We use ice as the top-most layer primarily because many volcanoes are located beneath ice sheets, particularly in Iceland, including recently erupting volcanoes in Iceland such as Bardarbunga (For the purpose of this study we model the ice as a brittle layer which behaves elastically through its entire thickness (). Other studies assume that only certain parts of an ice layer behave elastically, with the remaining parts behaving as ductile – using, for example Glen's flow law (e.g. ). Ice behaves elastically at high strain rates and comparatively low stresses or pressures (). The brittle deformation of ice is exemplified in the formation or fractures, crevasses, as are common during subsidence associated with volcanism (). The assumption of linear elastic behaviour of the ice is thus reasonable and does not significantly affect the calculated displacements and stresses in the crust (the surface rock) itself outside the ice sheet. The crustal layering or anisotropy is of much greater significance than the assumed elastic behaviour of the ice as regards surface deformation (e.g. ). The mechanical properties of ice are variable (). For example, typical laboratory values of stiffness or Young's modulus (E) can range from as high as 15 GPa () to more commonly 8–9 GPa, depending on temperature, and grain size and orientation (). The stiffness values are only moderately anisotropic (). These are dynamic values, however. Static values are more difficult to measure because of time-dependent deformation in ice. Estimated typical static or field values for Young's modulus of ice are around 1 GPa (). Poisson's ratios for ice are commonly between 0.2 and 0.4 (). In the modelling we use a Young's modulus somewhere between typical field and (dynamic) laboratory values, or 4 GPa. Also, we use a Poisson's ratio of 0.3 and a density of 920 kg m− 3. The general crustal and ice parameters used in the numerical models are given in In all models we assume a strong coupling between glacier and bedrock or crustal surface (except at the location of the caldera lake) using the same assumptions as . More specifically, if the coupling between the ice and the bedrock is of sufficient strength, stresses within the crust are transmitted to the ice. Then the ice can be considered to act mechanically as part of the layered crust. The other mechanical situation is where the ice and crust are weakly bonded, in which case slip may occur along the weak boundary and stresses would not be transferred from the bedrock surface to the ice. We consider one such scenario where the ice and crust are not directly coupled, designed to represent a caldera lake. The lake depth is 0.5 t (half the thickness of a typical crustal layer) and its width is a (the radius of the magma chamber/caldera). The lake is modelled as a free surface at all edges. This follows because the contact between water and the bedrock below as well as the contact with the ice above are surfaces of zero shear stress. As previously stated, many if not most calderas develop a caldera lake at some point, particularly those calderas formed under ice (If the stresses within a volcano are suitable for the formation of a caldera then displacement would be likely to occur along a bounding ring-fault (circumferential fault). In order to incorporate the mechanical response to ring-faulting we include in one of the models a soft (low-Young's modulus) vertical zone directly above the magma-chamber edge. This zone is supposed to represent a typical caldera fault, a ring-fault (without a ring-dyke) consisting of a highly fractured and mechanically soft damage zone with respect to the host rock (). The magnitudes of the vertical and horizontal displacements depend on the magma chamber size – in this case the chamber radius a is 4 km (its horizontal diameter thus 8 km). All models assume uniform vertical displacement of the roof, that is, a piston-like subsidence irrespective of the absence (as in most models) or the presence (as in one model) of the ring-fault itself.), although the nucleus-of-strain solution with application to volcanoes was initially derived by . Mogi's analytical solution can be replicated using the finite element method (e.g., ). If a Poisson's ratio of 0.25 is assumed for the elastic half-space – generally a reasonable assumption – then the basic equations of the where uz and ur are the vertical and horizontal (radial) displacements at the surface above the magma chamber, respectively. Also, pe is the magmatic excess pressure in the chamber, a is the radius of the chamber, μ is shear modulus, d is the depth to the centre of the chamber below the surface of the earth (), and r is the radial coordinate at the surface. At the point right about the centre of the magma chamber, we have r = 0, and the maximum vertical displacement uz becomes (Magmatic underpressure, that is, pressure less than lithostatic, is often regarded as the condition for ring-fault and ring-dyke formation. In fact, an underpressure or contracting nucleus-of-strain was original model for the formation of ring-dykes and the connection with the Mogi model is straightforward (cf. we show the numerical results of a two-dimensional (circular) chamber subject to an underpressure of 10 MPa, a common underpressure value when considering ring-fault formation (). We model the horizontal and vertical displacement at the surface of the ice and at the surface of the bedrock (the crust under and outside the ice sheet) for two magma-chamber depths: 3 km and 5 km. The modelled chamber radius is 1 km and is thus small in relation to the chamber depth below the surface, as it should be for a “Mogi model”. There are two basic model configurations. The first one (A) has no caldera lake, but the second one (B) has a caldera lake between the bottom of the ice and the bedrock surface. The lake is included in several of the models in this paper because, as indicated above. Such lakes are common in the many calderas located beneath ice in Iceland (The surface displacements, vertical and horizontal, are very small (less than 3 cm) for this type of loading (), suggesting that a ‘Mogi model’ is, as a rule, not very suitable for generating large (tens-of-metre scale) subsidences. The geometries of the displacement curves (), however, are in excellent agreement with those obtained from the Mogi model () are shown both for the surface of the rock (the crust under and outside the ice) as solid lines as well as for the surface of the ice itself, as broken lines. As is also seen in subsequent models, the caldera lake has great effects on the displacement curves for the surface of the ice. The other main results as regards the surface-displacement curves will be discussed in context of the later and more realistic models, to which we turn now.Here we present the results of the stresses and surface displacement induced by a given subsidence of the roof of a sill-like magma chamber. The chamber, modelled as a cavity within a crustal segment, is given a zero excess pressure condition at its lower boundary and prescribed a vertical displacement at the upper boundary in all models apart from those simulating slip on the ring-fault. Thus, in the models the chamber is in lithostatic equilibrium with its surroundings prior to the prescribed vertical displacement, that is, the roof subsidence. While these models are partly “inspired” by the events in Bardarbunga 2014–15, they are completely general and apply to all central volcanoes – collapse calderas in particular – under ice. We explore two main types of models, namely where the magma chamber is located (1) in a homogeneous, isotropic crustal segment, and (2) in a layered, anisotropic, crustal segment. Based on information from Bardarbunga, where the maximum subsidence of the ice surface is estimated at around 60 m (), we explore vertical roof displacements or subsidences from 20 m to 100 m. To cover the likely shallow chamber depths, we consider chambers with roofs at depths of 3 km, 5 km, and as an extreme shallow-chamber depth, 7 km below the crustal or rock surface.These models are somewhat similar in set-up as the elastic half-space or the Mogi model (). There are, however, three main differences between the present models (. First, the shallow magma chamber (cavity) has here () a sill-like geometry in contrast to the spherical or point-like Mogi source (circular in ). Also, here the radius of the chamber a is 4 km (), and thus 4-times the radius of the previous circular chamber (), and with a maximum thickness 2b of 2 km. Second, the displacements at the surface of the bedrock (the crust) and the ice result here from prescribed chamber-roof vertical displacement or subsidence rather than the underpressure in the models in . Third, the subsequent sill-like chamber models analyse chambers in a layered (anisotropic) crustal segment rather than in an elastic half-space as is done in the Mogi model (), and some of the sill-like models also include a lake beneath the ice, thereby forming a free surface. shows the vertical (uz) and horizontal or radial (ur) surface displacements of the ice and the bedrock (or crust) resulting from a vertical chamber-roof displacement or subsidence of 100 m. Here there is no ring-fault. The chamber roof is prior to the displacement at different depths below the bedrock or crustal surface d, namely at depths of 3, 5 and 7 km. Salient model results are shown in , but here we summarise some of the basic results (The maximum vertical displacement (shown as negative displacement or surface subsidence), both of the ice and the crust, is above the centre of the magma chamber. The subsidence reaches about 97 m in the bedrock/crust and about 78 m in the ice (). The subsidence changes to uplift or doming at distances of 15–18 km (depending on the chamber depth) from the surface point right above the chamber or caldera centre (). Unless otherwise stated, chamber/caldera centre in the discussion that follows refers to this surface point.The horizontal displacement towards the centre (above the centre of the chamber), shown as negative, reaches its maximum at 4–5 km from chamber/caldera centre. The horizontal displacement reaches a maximum of about 25 m in the crust and about 32 m in the ice (). For the chamber at 3 km depth, however, the horizontal displacement becomes positive (movement away from the centre) at about 25 km distance from the centre.The vertical surface displacement, both in the ice and in the bedrock/crust, is less than that of the chamber roof. There is thus not a one-to-one correspondence between the displacement at the surface either of the ice or the crust and the chamber roof subsidence.The vertical and horizontal displacements extend to distances far from the chamber/caldera centre. Thus, in both the bedrock/crust and the ice the vertical displacement is in excess of 0.5 m out to distances of about 14 km, whereas the horizontal displacements are in excess of 0.5 m out to distances of about 19 km (Generally, significant surface displacements associated with the chamber-roof subsidence of 100 m occur at lateral distances of up to 40–50 km (in the ice as well as in the bedrock/crust) from the chamber/caldera centre. For example, a chamber located at 3 km depth produces horizontal surface displacements of 20 cm at approximately 21 km from the chamber/caldera centre, while a chamber at 7 km depth produces the same displacement at approximately 33 km from the centre. Thus, for the imposed vertical displacement of the chamber roof, large horizontal displacements are expected out to tens of kilometres from the chamber, and these should be easily detected in the ice or at the bedrock/crustal surface by geodetic measurements.In the second set of homogeneous crustal models, we consider the effects of a pressurised deep-seated reservoir, such as are common as magma sources for shallow chambers in Iceland (). Doming is modelled as being the effect of 10 MPa excess magmatic pressure acting on the roof (a boundary load) of the reservoir. The general effect of doming is to reduce the magnitude of vertical and horizontal surface displacements but increase the surface area where those displacements are significant. In other words, the subsidence becomes much less concentrated at the surface immediately above the shallow chamber.In the third set of homogeneous crustal models, we added vertical faults (). These are supposed to represent a two-dimensional version of a caldera ring-fault. The fault is modelled as a soft elastic inclusion, that is, as a zone with a low Young's modulus. This is because active or recently active faults have generally lower Young's moduli than most of the host rock because the fault is composed of a fractured damage zone and breccia fault core (). The precise relationship between damage and Young's modulus evolution in caldera settings is, as yet, poorly constrained. The results () are similar to those of the previous models without a fault (Two layered crustal-segment models were run (). One model (A) has two soft layers in-between stiffer crustal units, while the other model (B) has three soft layers, including the top layer. All the soft layers have a stiffness of 1 GPa, which corresponds to the stiffnesses of soft hyaloclastites (basaltic breccias) and of glacial sediments, such as are common in most active volcanoes in Iceland. The layers are modelled as soft to explore the maximum effects that sediments and soft breccias could have on the displacement fields. Introducing mechanical heterogeneities and anisotropies through soft layers with low Young's moduli into the model setup has the following effects:There is a general reduction in magnitudes of the far-field displacements. That is, the horizontal and vertical displacements far from the chamber/caldera centre are smaller in the layered models than in the non-layered models (The maximum vertical displacements are also smaller in the layered models than in the non-layered models. More specifically, the maximum surface vertical displacements in the bedrock/crust are 79–84 m in the layered models but 97–98 m in the non-layered models (). Similarly, the maximum surface displacement in the ice in the layered models is 64–65 m, but 70–72 m in the non-layered models.The maximum horizontal surface displacements are much smaller in the layered models than in the non-layered models. In the layered models the maximum surface horizontal displacement is 2–6 m but ~25 m in the non-layered models.The general effect of layering is to reduce the displacements measured at the surface of the bedrock/crust and the ice. The reasons for the reductions are partly that the stresses become “dissipated” at the contacts with the soft layers. Similar results have been obtained in general studies of surface deformation associated with various pressure sources, such as dykes (). Crustal segments with alternating stiff and very soft layers generally transport less stress and deformation to the surface than non-layered segments, or segments where all the layers have similar mechanical properties. Well-known examples of the reducing effects of mechanically contrasting layers on surface stresses and deformation/displacement relate to emplacement of dykes and other vertically fluid-driven fractures (We also modelled the effects of a piston-like subsidence along a ring-fault on the surface displacement fields. In view of the results from Bardarbunga, where inferred vertical maximum displacement in the ice inside the collapse caldera is about 60 m (), we impose 50 m vertical displacement on the ring-fault (). The ring-fault, the fault zone, is modelled as a soft inclusion, with a Young's modulus of 0.1 GPa. We tried other stiffnesses for the fault zone, such as 0.01 GPa, but the overall results remained similar. The crust itself is non-layered in this model with the properties used in the earlier non-layered models () show that the displacement, both the vertical and the horizontal, becomes more concentrated at and within the caldera (the ring-fault) than in the previous roof-subsidence models without ring-fault. The maximum subsidence of the bedrock/crust is the same as that of the fault, namely 50 m, but the ice subsidence is greater, or 60 m. This is because the ice can bend or subside somewhat into the caldera lake (and/or soft sediments) at the contact between the ice and the crust, whereas the crust clearly cannot do so. For the same reason, the horizontal displacement (towards the centre) at the surface of the ice also exceeds that of the crust. Both reach a maximum at the location of the ring-fault, the vertical displacement of the ice (the fault throw) being up to about 17 m and that of the bedrock/crust up to 10 m.These results illustrate various aspects of the effect of ring-fault subsidence in a caldera located beneath ice, including the following:The crustal displacements, the horizontal and, in particular, the vertical, reflect strongly the ring-fault geometry. This means that both displacements are maximum at the caldera fault. In fact, the vertical displacement reaches its maximum of 50 m at the fault and stays the same throughout the roof of the chamber.The caldera lake magnifies the surface displacement of the ice. The horizontal displacement in the ice is considerably larger than that in the crust. And, most importantly, the total vertical displacements in the ice exceed that imposed on the ring-fault by about 10 m. This is because of the caldera lake beneath the ice into which the ice can subside.Displacement of 50 m is so large that it would certainly cut through the ice as a fault. The inferred vertical displacement at the location of the ring-fault are about 17 m. No tensile or shear strength is given to ice in the model, so it does not fracture. But 50 m vertical displacement in an ice sheet of thickness, say, 800–1000 m would become very clearly through-faulted.The underpressure or withdrawal-of-magmatic-support model is often favoured when explaining the formation of calderas, both in analogue-model setups () and for explaining geophysical observations (). Most recently this model has been invoked to explain ice surface subsidence above the Bardarbunga Caldera (). The assumption is then that a volume of magma was removed from a chamber by lateral magma propagation, eventually forcing an eruption some 45 km from the central volcano. Similar ideas have been offered to explain the occurrence of lavas outside the main central volcanoes and within the active rift zone of Iceland (), although more recent studies have shown that alternative explanations with predominating vertical magma propagation are equally plausible (). The competing hypothesis is that the Holuhraun lavas, and many other large and rather primitive basaltic fissure eruptions in Iceland, are fed by regional dykes which are injected from magma reservoirs at a much greater depths (15–25 km) than the shallow chambers (The models presented in this paper have certain implications for volcano-tectonic processes in central volcanoes in general. Further implications apply primarily to calderas located in ice sheets such as many calderas in Iceland – Bardarbunga in particular. We consider first the implication for the magnitude of the surface displacements and the size of the area affected (the surface area showing significant displacement). Both aspects of the deformation are very important, particularly when trying to separate the deformation associated with a caldera and/or a shallow magma chamber from that associated with simultaneous dyke emplacement.Vertical surface displacements of 10–20 cm extend out to distances of 15–16 km from the centre of the caldera, and horizontal displacements of similar magnitude to 20–30 km (). For the horizontal displacement, 10 cm displacements occur out to 40–50 km from the centre, depending on the depth of the chamber. These refer to the non-layered models and the exact distances for the displacement mentioned depend on the depth of the chamber: the surface displacements induced by the deepest chamber, at 7 km, extend for the greatest distances from the centre. If the surface displacement would relate partly to a deep-seated reservoir, as we propose here, say a reservoir at the depth of 15–20 km, then significant surface displacements would extend still further from the centre. Using the same model configuration and properties but roof-subsidence varying from 20 m to 100 m, the results are similar – significant displacements extend to 15–20 km from the centre () and are only slightly less when a vertical ring-fault is introduced (The layered models produce less displacements, both in magnitude and lateral extension from the centre (). However, these models are with somewhat extreme layering since the soft layers have stiffness or Young's modulus of only 1 GPa, which is low for hyaloclastites and sedimentary rocks. Nevertheless, there are still large surface displacements of 50 cm at distances of about 12 km (for the vertical displacement) and 12–17 km (for the horizontal displacement) in the bedrock/crust, and somewhat larger distances in the ice (). Larger displacements are obtained from the 50-m-fault displacement model (Overall the displacement results indicate that, for the models considered, large displacements, of the order of tens of centimetres or hundreds of millimetres, should be detected out to distances of 10–20 km, for the vertical displacement, and 20–30 km or more for the horizontal displacements. Even for a small roof-subsidence of 20 m, the horizontal displacement at 10–12 km distance from the centre is still of the order of tens of centimetres (). Results of this kind show clearly the effect of nearby subsidence of a magma-chamber roof, or a collapse caldera displacement, and should make it possible to distinguish between displacements induced by such a subsidence and those induced by a dyke formed in the same volcano-tectonic episode.The displacement field associated with the subsidence of the ice in the Bardarbunga episode in 2014–15 is educational in this respect. For the period up to 6 September 2014 the GPS-estimated maximum displacement or subsidence in the ice in the Bardarbunga Caldera was about 16 m (), and the entire cumulative displacement during the episode 2014–15 is estimated at over 60 m (). Dyke emplacement was essentially completed by 31 August when the main eruption began (), and no significant horizontal dyke-induced displacements were detected after 4 September 2014 (). The horizontal displacements induced by the dyke can thus largely be separated from those induced by the subsidence measured in the Bardarbunga Caldera.Our model results suggest that vertical displacement of about 16–20 m, corresponding to period up to about 6 September, should generate horizontal displacements of the order of tens of centimetres towards the Bardarbunga Caldera within 10–12 km from the centre of the caldera. Similarly, horizontal displacements of many tens of centimetres are expected out to distances of up to tens of kilometres, depending on the model used – in particular, the assumed depth of the shallow chamber and mechanical properties of the host rock and the ring-fault. The measured displacements at the GPS stations in the crust outside the ice, some of which are at 13–17 km from the centre of the caldera, are significantly less than expected from the models (). The difference may be partly related to the modelling procedure but most likely indicates that only part of the subsidence in the ice within the Bardarbunga Caldera is actually directly related the magma-chamber roof subsidence, or ring-fault displacement.The last point is also of importance when interpreting the subsidence measured in the ice within the caldera. There are several remarkable features of the ice subsidence, as shown in maps by The maximum subsidence is about 3 km from the northern caldera rim and about 5 km from the southern and south-eastern rims.The subsidence at the rims, at the ring fault itself, is much smaller than the maximum subsidence – in fact about zero in the early stage of the subsidence.Fracture development at the surface of the ice is comparatively small, with no major caldera-related fault cutting through the ice.These observations and measurements, when compared with the ring-fault subsidence model (), suggest the following interpretations. First, the displacement along the ring-fault is small in comparison with the overall subsidence in the ice. In particular, displacements of the order of 20–60 m along the ring fault would without doubt have propagated faults through the ice (say, vertical displacements of 10–20 m; ) – and these faults are not observed. From standard fracture mechanics () and the mechanical properties of a typical ice () with displacement of up to tens of metres in ice of thickness of several hundred metres – in fact, the ice thickness is only 200–300 m above part of the caldera rims – would become a through crack, that is, reach the bottom and surface of the ice sheet. This conclusion is the same even if there is a caldera lake beneath the ice (). Since normal faults with these throws are not observed, the cumulative vertical ring-fault displacements cannot be of the of the order of tens of metres, and is most likely of the order of metres or less.Second, for a porous-media chamber, as most chambers presumably are (), the maximum subsidence, if caused by magma flow out of an underlying magma chamber, would normally be, initially at least, close to the ‘outlet’, that is, the intersection of the dyke or sheet transporting the magma with the boundary of the chamber and the active ring-fault. propose that the subsidence of the ice is directly related to chamber roof-subsidence associated with magma flowing laterally along a dyke that dissects a chamber along its southeast margin. It is not clear from the subsidence data, however, why the maximum subsidence is then not at the outlet and the active ring-fault but rather close to the northern margin of the chamber/caldera.In fact, the inferred segmentation of the dyke, with distances between nearby tips of segments up to kilometres (), is a strong argument against lateral flow of magma from a chamber beneath Bardarbunga and to the volcanic fissures in Holuhraun and an argument for vertical flow of magma from a deep-seated reservoir (). The arguments against the lateral flow between dyke segments are many, including the following. (1) There is no seismicity between the nearby ends of some of the 8 segments, particularly between segments 1 and 2 and 5 and 6 (), suggesting that no magma migrated laterally between them. The zones connecting many of the segments, being highly oblique to the overall strike of the dyke, are zones of high shear stress making it highly unlikely that a magma-driven fractures could propagate along the zones without triggering earthquakes. Earthquakes are, in fact, used as criteria for identifying magma paths. It follows that absence of earthquakes, that is, seismically quiet zones, would normally mean absence of magma paths. (2) In the unlikely event of aseismic magma-path formation at shallow depths from segment 1 to 2, and from segment 5 to 6, then the same magma would have to flow from the shallow depths vertically down to at least 10 km depth in segments 2 and 6. Downward flow of magma on this scale is not supported by any observations and does not agree with well-established physical principles of fluid dynamics and dyke propagation (). Thus, the segments of the regional Bardarbunga–Holuhraun dyke were presumably formed primarily through vertical flow of magma from the proposed deep-seated magma reservoir (). Such a doming, as small as of the order of centimetres, is known to be one of the principal mechanisms for generating caldera collapses (), particularly along normal ring-faults (). Focal mechanisms suggest that the slip on the ring-fault of Bardarbunga in the 2014–15 episode was primarily through normal faulting (). Most of the ring-fault seismicity occurred at shallow depths (< 3 km) (), in agreement with the ring-fault seismicity being related to stress concentration above the margins of a proposed shallow magma chamber, which is the model suggested here (cf. ) reduces the effective thickness of the crustal segment or plate above the shallow magma chamber (). The reduction in the effective plate thickness de encourages further doming of the volcanic field hosting the Bardarbunga caldera even if the magmatic excess pressure at the deep-seated reservoir remains constant or decreases slightly for a while (This last point brings us to the 45-km-long regional dyke and associated eruption, and how they relate to the subsidence in the Bardarbunga Caldera. The first thing to notice is that the strike of the dyke close to the caldera/chamber is in perfect agreement with the local trajectories of the maximum horizontal principal stress around a circular or slightly elliptical cavity under tension (). Further from the chamber/caldera the regional stress field took over, and the dyke followed the field that has existed in this part of Iceland for at least 8–10 Ma (). The main dyke was injected when the excess pressure in the deep-seated reservoir reached the conditions (where pl is the lithostatic stress or overburden pressure at the reservoir rupture site (in the reservoir roof), pe
|
=
|
pt
|
−
|
pl is the difference between the total fluid pressure pt in the reservoir and the lithostatic stress at the time of reservoir rupture, σ3 is the minimum compressive or maximum tensile principal stress, and T0 the local in situ tensile strength at the rupture site. When the dyke became injected into the roof of the reservoir and began to propagate up into the crustal layers above, its overpressure po changed as:where pe is the magmatic excess pressure in the reservoir at the time of rupture (and equal to the in-situ tensile strength of the roof at the rupture site, T0), ρr is the host-rock density, ρm is the magma density, g is acceleration due to gravity, h is the dip dimension or height of the dyke above the rupture site, and σd is the differential stress at a particular depth in the crust (the depth of interest). At the magma-chamber rupture site itself, the stress difference is included in the excess pressure term, so that there σd
|
= 0. Also, at the rupture site, before the dyke has propagated and reached any significant height, we have h = 0, so that the third term in Eq. , the buoyancy term, becomes zero. It follows that the only pressure available to rupture the reservoir roof and drive the magma out at and close to the roof contact with the magma is the excess magmatic pressure pe. We also know that pe
|
= T0, that is, the excess pressure at the time of roof rupture is equal to the in-situ tensile strength, with in-situ (field) values ranging from 0.5 to 9 MPa (), the most common values being 2–4 MPa (It follows that during the rupture and initial propagation of the resulting dyke, the only driving pressure is pe, of the order of several mega-pascal. As the dip dimension (height) of the dyke increases, however, positive buoyancy adds to the driving pressure, so long as the average magma density is less than the average density of the rock layers through which the dyke propagates. The magma is olivine tholeiite () so that its density may be taken as about 2700 kg m− 3 (). The erupted magma originated at depths somewhere between 10 and 20 km (), that is, from a deep-seated magma reservoir as have been proposed under most volcanic systems in Iceland, and the Bardarbunga System in particular (). Given the crustal density in Iceland, then from Eq. the magmatic overpressure p0 or driving pressure of the dyke, at different crustal depths (and thus with different σd values) could easily have reached 10–15 MPa (cf. In the model presented here, the injection of the main dyke from the deep-seated reservoir, as well as the subsidence in the Bardarbunga Caldera, were both primarily the consequence of the same process: namely inflow of magma into the deep-seated reservoir. This inflow may have started many years before the 2014 episode, particularly from 2006 and onwards as indicated by seismicity (), and was certainly noticeable as widespread doming or uplift on GPS instruments for months before the regional dyke injection began in August 2014 (). There may have been magma flow into the shallow chamber associated with the caldera, and several smaller dykes may have been emplaced during the early stages of the episode – some from the deep-seated reservoirs, others (small radial dykes) from the shallow chamber. The only dyke to develop into a major dyke, however, was the 45-km-long regional dyke emplaced over a period of 2 weeks in August 2014 (The regional dyke presumably came from depths of at least 15–20 km, perhaps deeper. This is suggested partly by the chemistry of the erupted lavas (), partly by the widespread doming detected in the months before the episode, discussed above, and partly by the earthquake distribution in the area. From 2012 there were many earthquakes north and northeast of the Bardarbunga Caldera (), where one of the earthquake swarms (and possible dyke injection) occurred during the first days of the August 2014 episode. Even more importantly, deep earthquakes occurred in a vertical zone southeast of the Bardarbunga Caldera from about this time and extended until August 2014 () at roughly the location of the first segment of the regional dyke, as formed in the first days of the August 2014 episode.The regional dyke had enormous stress effects on the Bardarbunga Caldera and, by implication, the associated shallow magma chamber (). The stress field induced by the dyke around the caldera contributed to three important aspects of the 2014 episode; (1) normal faulting along the caldera ring-fault, (2) elongation of the caldera in a roughly north-south direction, and (3) ductile deformation and flow of the ice, primarily inside the caldera.Normal faulting is the dominating mechanism on the Bardarbunga ring-fault during the present episode (). This is in agreement with the two main mechanisms of caldera slip proposed here, namely: (1) a combination of stresses concentrating at the ring-fault as a consequence of slight doming due to excess pressure increase in the deep-seated reservoir () and (2) dyke-induced stress concentration, particularly at the northern and southern sectors of the ring-fault (). Both encourage normal faulting on the ring-fault itself, while the dyke-induced stresses also encourage strike-slip and reverse-faulting on differently oriented faults away from the ring-fault (The elongation of the ring-fault in the roughly north-south direction is due to the compressive and shear stresses that concentrate in the “breakout areas” around the caldera (). Elongation of collapse calderas due to “breakouts” is well known from other areas (). The elongation would encourage flow of magma to the ring-fault in these sectors, possible ring-dyke formation – which may partly explain the common non-double couple earthquakes () – and contribute to the subsidence of the caldera roof. The main reason why the earthquake activity along the Bardarbunga Caldera has been so concentrated in the north and south parts of the caldera is presumably related to the dyke-induced stresses in these sectors (While no attempts were made to measure or monitor ice flow in the ice-sheet cover of the Bardarbunga Caldera and its vicinity during the 2014–15 episode, such flow is likely to have occurred. During the emplacement of the regional dyke east and northeast of the caldera, the magmatic overpressure in the dyke (Eq. ) may easily have reached 10–15 MPa. The dyke induced major displacements and thus stresses, within the caldera, and the high mountains of the caldera rim must have transmitted those compressive stresses (σH) from the dyke into the ice (). Depending somewhat on the strain rate, ice flows at pressures or stresses of less than 1 MPa, so that stresses of up to 10 MPa – somewhat diminishing with distance from the dyke – would certainly have caused flow in the ice within the caldera. The main flow would have been within the caldera because that is where the mountains are high – the caldera rim – and can thus most easily transmit the dyke-induced stresses to shallow levels in the ice sheet (). Since ice flows from higher to lower pressure, the dyke-induced stresses would have encouraged ice flow out of the caldera.How much the ice flow may have contributed to the measured 60 m subsidence in the ice is unknown. The comparatively minor fracturing at the surface of the ice during its subsidence would suggest that the ice was flowing right up to the surface. Flow in and down-bending of the ice may have contributed significantly to the measured subsidence. In this model, the flow or creep or strain rate was highest just after the emplacement of the regional dyke, and then became gradually lower, as is typical for a creeping response to sudden load or displacement (here the dyke emplacement). Down-bending would be encouraged by a caldera lake and/or soft sediments (subject to earthquake shaking) existing beneath the glacier (As the excess pressure pe in the deep-seated reservoir declined, the doming-related ring-fault displacement also gradually decreased and, as pe approached zero, the subsidence stopped altogether. It is clear that long before the eruption came to an end on 27 February 2015 the earthquake activity associated with the Bardarbunga Caldera had greatly diminished. Also, the subsidence measured in the Bardarbunga Caldera ceased several weeks before the end of the eruption. These and other observations suggest that the pressure decrease in the deep-seated reservoir was partly responsible for the ring-fault slips and associated earthquakes.We present general numerical models on the effects of “shallow” magma chamber contraction at various depths, namely the result of chamber roof subsidence at depths of 3 km, 5 km, and 7 km. In all the models, the magma chambers are associated with a collapse caldera which is located beneath a thick glacier or ice sheet. The models are general, and apply to many volcanoes in Iceland and elsewhere, but the results are here applied to the 2014–15 volcano-tectonic episode in Bardarbunga–Holuhraun in Iceland.Several models were tested for the shrinkage of the magma chamber through vertical downward displacement of its roof. Some of the models use a simple elastic crust (elastic half space) hosting the chamber, with an ice sheet on the top. Others use layered (anisotropic) crust above the shallow magma chamber, that is, layers with different stiffnesses (Young's moduli). And still other models have a caldera lake between the ice sheet and the rock or crustal surface. The simplest loading used is 100 m vertical downward displacement of the chamber roof. Other models include different vertical displacement of the roof (in steps from 20 m to 100 m), as well as displacement of 50 m along a vertical caldera fault (the ring-fault). Some of the main results are as follows:For chamber-roof displacements in the range of 20–100 m, the models suggest large vertical and particularly horizontal displacements in the ice and in the bedrock/crust surface under the ice out to distances of 10–40 km from the caldera centre, depending on the depth of the chamber and the exact type of modelling used. The vertical displacements in all models reach maximum at the surface of the bedrock/crust and the surface of the ice right above the centre of the subsiding magma-chamber roof. The horizontal displacements at the surface, however, reach their maximum values (maximum displacement towards the chamber or caldera centre) at a distance of 4–5 km from the centre.For a non-layered (isotropic) crustal model with a 100 m roof subsidence, the vertical displacement exceeds 50 cm to a distance of 14 km from the centre and the horizontal displacement exceeds 50 cm to a distance of 19 km from the centre. A chamber located at 3 km depth produces horizontal displacement of 20 cm to a distance of 21 km from the centre, and for a chamber at 7 km depth horizontal displacement of 20 cm is produced to a distance of 33 km from the centre. Similar results are obtained if a vertical non-slipping ring-fault is added to the model, but the displacements show an abrupt change (a break) at the location of the fault.The general effect of crustal layering (using mechanically layered or anisotropic models) is to reduce the displacements measured at the surface in comparison with those generated in the non-layered (isotropic) models. The reasons are partly that the stresses become "dissipated" at contacts between still and soft layers.In a model where the subsidence is related to vertical downward piston-like displacement by 50 m of the ring-fault, the results show that the vertical displacement in the crust/chamber roof exactly reflects that of the ring-fault and reaches a maximum of 50 m. By contrast, the vertical displacement in the ice follows a curve that reaches its maximum of 60 m in the centre of the caldera. This "extra" vertical displacement in the ice is partly because it can bend or subside somewhat into the caldera lake below. Displacement of 50 m along the ring-fault is so large that the fault would most definitely cut through the ice, forming a through fault with displacements of up to tens of metres (which is not observed in Bardarbunga, however).The modelling results have several implications for the interpretation of the 2014–15 Bardarbunga–Holuhraun episode.First, the measured horizontal displacements in the surface rocks outside the ice appear to be significantly less than expected from modelling 60 m vertical displacement. At stations west of the Bardarbunga Caldera, horizontal displacements towards the caldera of the order of tens of centimetres would be expected but are not observed. This indicates that the vertical displacement in the bedrock/crust, and thus the chamber roof-subsidence, is significantly less than the maximum of about 60 m measured in the ice.Second, a 50 or 60 m piston-like displacement along the ring-fault is ruled out. The ring-fault would, for such a large displacement, definitely cut through the ice to form a large and easily visible fault, but this has not happened. By contrast, there has been comparatively little fracturing in the ice within the Bardarbunga Caldera during the subsidence, which suggests that the ice behaved as ductile, was flowing, right up to its surface. The results seem to limit the actual ring-fault (piston-like) subsidence to, at most, a few metres.Which brings us to the third implication, namely that the 45-km-long regional dyke generated compressive stresses in the ice within the caldera which resulted in ice flow out of the caldera, thereby contributing to the measured subsidence in the ice. How large factor the ice flow (and possible down-bending into lake/sediments) may have been is unknown since no measurements of the ice flow were made. What is known, however, is that ice flows easily at low pressures, of the order of 1 MPa, and our calculations suggest magmatic overpressure in the regional dyke of the order of 10–15 MPa.We interpret the geochemical, seismic, and geodetic data so that the regional dyke was injected from a large reservoir at 15–20 km depth, perhaps deeper. Earthquake data suggest that the reservoir received new magma over many years before the beginning (in August 2014) of the Bardarbunga–Holuhraun episode, particularly from the year 2006 and onwards. The magma injection resulted in widespread doming (uplift), as detected by GPS instruments in the months prior to August 2014 when the dyke emplacement began (In our interpretation, the August 2014 reservoir rupture and regional dyke injection as well as the ring-fault displacement (caldera subsidence) are all primarily the consequence of the associated reservoir magmatic pressure increase and doming. The conditions for reservoir rupture, dyke injection, as well as the overpressure change with vertical propagation of the dyke, are presented in Eqs. . The effects of ring-fault formation or reactivation as a result of reservoir-pressure increase, slight doming, and stress concentration around the chamber/caldera are discussed in the paper with reference to earlier numerical models (), all of which suggest doming as a main mechanism of ring-fault displacement. This mechanism is also in agreement with the dominating normal-fault focal mechanisms of the ring-fault earthquakes (We interpret the seismic and geodetic data so that, in addition to the 45-km-long regional dyke, there may have been several other dyke injections, including a northwest-striking dyke emplaced some 15 km north of the caldera and several smaller radial dykes/inclined sheets injected from the shallow chamber beneath the caldera. The shallow chamber may have received magma from the reservoir during the episode before the radial dyke injection; alternatively, stress concentration around the shallow chamber, through the external loading (doming), can have triggered the radial dyke injection (The regional dyke induced stress concentration at the caldera/shallow chamber, in addition to that generated by the doming. The dyke-induced stress concentration contributed to three processes during the 2014–15 episode. First, normal-fault slip along the ring-fault. Focal mechanisms indicate dominating normal-fault slip along the ring-fault (). Much of the faulting occurred at the northern and southern sectors of the ring-fault, exactly in the areas where numerical and analytical models suggest that dyke-induced stress encourages normal faulting (). Second, caldera elongation and “breakout” mechanisms at the northern and southern sectors of the caldera/chamber were induced by the dyke. These may have encouraged ring-dyke emplacement. Third, ductile deformation and flow in the ice inside the caldera. The caldera rim is composed of tall mountains that transmitted the compressive stress induced by the dyke to the ice, resulting in ice flow out of the caldera. The rate of flow of ice was greatest immediately following the dyke emplacement, and then gradually declined, as is typical of creeping material response to a sudden load (here the dyke emplacement). How much the ice flow contributed to the measured 60-m-subsidence in the ice is as yet unknown.As the excess pressure in the reservoir pe decreased below a certain level, the stress concentration around the ring-fault became too small for further significant to large slips and associated earthquakes to occur. This follows because the main slips were through normal faulting, so that the slips were controlled by the available driving stress at any time. Thus both the ice flow and the ring-fault subsidence gradually decreased with time. Significant subsidence in the caldera had apparently stopped in early February, several weeks before the eruption in Holuhraun came to an end on 27 February 2015. That the eruption continued for several more weeks indicates that it ceased only when the excess pressure in the deep-seated reservoir had vanished completely, that is, its excess pressure pe had become zero.3D simulation of dependence of mechanical properties of porous ceramics on porosity3D computer simulation of mechanical behavior of a brittle porous material under uniaxial compression is considered. The movable cellular automaton method, which is a representative of particle methods in solid mechanics, is used for computation. In an initial structure the automata are positioned in fcc packing. The pores are set up explicitly by removing single automata from the initial structure. The computational results show that dependence of strength and elastic properties of the modeled material on porosity below percolation threshold (material with closed pores) differs from the dependence for porosity above the threshold (permeable material). The results obtained are in close agreement with available experimental data.pair part of the force acting between the automata i and jshear modulus of the material of the ith automatontorque caused by relative rotation of automata in the pair i–jbulk modulus of the material of the ith automatonrelative tangential displacement in the pair i–jtotal torque acting between the automata i and junit vector directed from the ith automaton to the jth onepressure in the volume of the jth automatondistance from the center of the ith automaton to the point of its interaction with the jth onearea of interaction of the automata i and junit vector of the direction of tangential part of the force F→ijpairtranslation velocity of the ith automatonvalue of specific force of central interaction in the pair i–jnormal strain of the ith automaton in the direction of the jth onestrength of material of the ith automatoncomponents of the average stress tensor in the ith automatonspecific force of tangential interaction in the pair i–jrotational velocity of the pair i–j as a whole (rigid body)subscripts denoting components of tensors and vectorsLet us consider a porous material in which all pores have the same size. If porosity of the material is small, each pore is closed and isolated form the others. With porosity increasing the number of pores becomes greater and the distance among pores decreases. When porosity reaches percolation threshold the pores are not closed anymore, connect with the others in a spanning cluster and consequently form a new morphology. It is well known that dependence of strength and elastic properties of porous materials on porosity is determined by the pore morphology (this problem has a long history Notice that experimental investigation of the problem is very difficult, because of stochastic nature of the pore size and morphology in real materials. For example, it is extremely hard to fabricate samples with various porosity values and the same porous structure, for example the same pore size. Of course, the modern technology like selective laser sintering allows producing very complex 3D structures but only for special materials, mainly polymers. At the same time potentialities of the modern computational mechanics allow to study in detail not only deformation of complex heterogeneous materials but also their fracture.The most of research in computational mechanics are performed using finite element method. But for modeling severe distortion and failure of a material, converting finite elements into particles is more convenient. For example the authors of work Thus, the most advantage in modeling fracture of heterogeneous materials seems to belong to particle method. That is why the purpose of this study was to develop a computational model of porous ceramics based on particle approach and to study elastic and strength properties of the modeled material in wide range of porosity.Method of movable cellular automata (MCA) The most well-known representative of this group of methods is discrete element method (DEM) Our research shows that many basic problems of DEM, including those with description of consolidated solids at various scales, can be solved by using many-particle interaction. In This force is represented as a superposition of the pair components F→ijpair dependent on the spatial position or displacement of the element i relative to the neighbor j and the volume-dependent component F→iΩ related to collective effects of the surroundings. This many-particle approach for notation of interaction forces is an essential part of the movable cellular automaton method.Within the frame of MCA, it is assumed that any material is composed by a certain amount of elementary objects of finite size (automata) which interact among each other and can move from one location to another, thereby simulating a real deformation process. The automaton motion is governed by the Newton–Euler equations:mid2R→idt2=∑j=1NiF→ijpair+F→iΩJ^id2θ→idt2=∑j=1NiM→ij,where R→i,θ→i,mi and J^i are the location vector, rotation vector, mass and moment of inertia of ith automaton respectively, F→ijpair is the interaction force of the pair of ith and jth automata, F→iΩ is the volume-dependent force acting on ith automaton and depending on the interaction of its neighbors with the remaining automata. In the latter equation, M→ij=qij(n→ij×F→ijpair)+K→ijrot, here qij is the distance from the center of ith automaton to the point of its interaction (“contact”) with jth automaton, n→ij=(R→j-R→i)/rij is the unit vector directed from the center of ith automaton to the jth one and rij is the distance between automata centers (), K→ijrot is the torque caused by relative rotation of automata in the pair (see below).Note, that automata of the pair may represent the parts of different bodies or of one consolidated body. Therefore its interaction is not always really contact one. That is why we put the word “contact” in quotation marks. More of that, as it shown in size of the automaton is characterized by one parameter di, but it does not mean that the shape of the automaton is sphere. When we compute the volume of an automaton, its shape is determined by area of its “contacts” with neighbors. For example, if we use initial fcc packing then the automata are shaped like a rhombic dodecahedron, but if we use cubic packing then the automata are cube-shaped. But when we compute interautomation forces or torques it is enough to use only one size parameter di and value of the “contact” area Sij. In this case we can imagine the automaton like a sphere.For locally isotropic media the volume-dependent component can be expressed in terms of the pressure Pj in the volume of the neighboring automaton j as follows where Sij is the area of interaction surface of automata i and j and A is a material parameter.The total force acting on automaton i can be represented as a sum of explicitly defined normal Fijn and tangential (shear) Fijt components:F→i=∑j=1NiF→ijpair,n(hij)-APjSijn→ij+F→ijpair,t(l→ijshear)t→ij=∑j=1NiF→ijn+F→ijt.where F→ijn and F→ijt are the normal and tangential pair interaction forces depending respectively on the automata overlap hij (a) and their relative tangential displacement l→ijshear (b) calculated with taking into account rotation of the both automata formally corresponds to the form of element interaction in conventional discrete element models Using homogenization procedure for stress tensor in a particle described in where α and β denote the axes X,Y,Z of the laboratory coordinate system, Vi is the current volume of automaton i,nij,α is the α-component of unit vector n→ij and Fij,β is β-component of the total force acting at the point of “contact” between automata i and j.The pressure Pi, or what is the same, the mean stress in the automaton volume can be determined from thus calculated stress tensor components:Tensor components allow us to calculate the other tensor invariants in the automaton volume, in particular stress intensity:σiint=12(σ‾xx-σ‾yy)2+(σ‾yy-σ‾zz)2+(σ‾zz-σ‾xx)2+6(σ‾xy2+σ‾yz2+σ‾xz2). it follows that the specific form of the expressions for F→ijn and F→ijt determines rheological behavior of a model medium.For further convenience, the interaction parameters of movable cellular automata are considered in relative (specific) units. Thus, the central and tangential interaction of the automata i and j are characterized by the corresponding stresses ηij and τij :Note, that in the most of papers devoted to the description and use of MCA the equations and formulae of the method are written for two-dimensional case. This study uses three-dimensional version of the MCA method. That is why herein shear stress τ→ij is a vector in the plane which is normal to n→ij.To characterize deformation of automaton i under its normal interaction with automaton j we can use the following dimensionless parameter (normal strain)In general case each automaton of a pair represents different material and overlap of the pair is distributed among ith and jth automata:where symbol Δ denotes increment of a parameter per time step Δt of numerical integration of the motion equations Eq. . The rule of strain distribution in the pair is intimately associated with the expression for computing interaction forces of the automata. This expression for central interaction is similar to Hooke’s relations for diagonal stress tensor components:where Ki is the bulk modulus, and Gi is the shear modulus of the material of ith automaton, Pi is the pressure of automaton i that may be computed using formulae Eqs. at previous time step or by predictor–corrector scheme.To determine a parameter characterizing shear deformation in pair of automata i–j we start with kinematics formula for free motion of the pair as a rigid bodywhere r→ij=R→j-R→i,v→i is the translation velocity of the ith automaton centroid, ω→ij is the rotational velocity of the pair as a whole (rigid body). If we multiply both sides of equation Eq. on the left by r→ij and neglect rotation about the axis connecting centers of the automata of the pair (i.e. let ω→ij·r→ij=0 because rotation about the axis of the pair does not produce shear deformation) then we get the following formulaω→ij=r→ij×(v→j-v→i)rij2=n→ij×(v→j-v→i)rij.Besides such rotation of the pair as a whole (defined by the difference in translational velocities of the automata), each automaton rotates with its own rotational velocity (a). The difference between these rotational velocities produces shear deformation. Thus, increment of shear deformations of automata i and j per time step Δt is defined by the relative tangential displacement at the contact point Δl→ijshear divided by the distance between the automataγ→ij+γ→ji=Δl→ijshearrij=qij(ω→ij-ω→i)×n→ijΔtrij.The expression for tangential interaction of movable cellular automata is similar to Hooke’s relations for non-diagonal stress tensor components:The difference in automaton rotation leads also to deformation of relative “bending” and “torsion” (only in 3D) of the pair b. It is obvious that resistance to relative rotation in the pair cause the torque, which value is proportional to the difference between the automaton rotations: describe mechanical behavior of a linearly elastic body in the framework of MCA method. Note, that relations Eqs. are written in increments, i.e., in the hypoelastic form. In paper We propose to apply MCA for studying brittle materials herein, but in for the system of movable cellular automata are numerically integrated with the use of velocity Verlet algorithm R→in+1=R→in+v→inΔt+F→in2mi(Δt)2θ→in+1=θ→in+ω→inΔt+M→in2J^i(Δt)2v→in+1=v→in+F→in+F→in+12miΔtω→in+1=ω→in+M→in+M→in+12J^iΔt,modified by introducing a predictor for estimation of σ‾i,αβ at the current time step n
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+ 1 in order to calculate Pin+1 and then F→in+1.A pair of elements might be considered as a virtual bistable automaton (bound and unbound states), which permits simulation of fracture by the MCA. Within the framework of MCA method crack initiation is imitated by bound → unbound transition of the pair state. Fracture criterion depends on physical mechanisms of material deformation. An important advantage of the described above formalism is that it makes possible direct application of conventional fracture criteria (Huber–Mises–Hencky, Drucker–Prager, Mohr–Coulomb, Podgorski, etc.), which are written in tensor form. Note, that switching of a pair of automata to an unbound state would result in a changeover in the forces acting on the elements; in particular, they would not resist moving away from one another.Thus, a fracture criterion has to be calculated with use of mean values of stress/strain tensor components in the considered pair i–j:The condition of fracture based on Huber–Mises–Hencky criterion will take the form:where σic is assigned threshold value (strength) for the material of automaton i, stress intensity σ‾ijint is calculated with use of σ‾ij,αβ.Herein we used MCA method for 3D computer simulation of mechanical behavior of a porous material under uniaxial compression. Response function of automata used in this study corresponded to ZrO2 ceramics with average size of pores commensurable with the grain size of the material and porosity equal to 2% (i.e. Young’s modulus of intact material was equal to 80 GPa and Poisson’s ratio 0.3). According to the pore distribution diagram of this material All the modeled samples were bricks with a square base. In a one can see a solid sample as fcc packing of automata. To simulate loading, one and the same velocity in the vertical direction was assigned for all the automata of the upper layer (dark gray particles at the top of the sample in ), while the automata of the lower layer (dark gray particles at the bottom of the sample in ) were fixed in the vertical direction. For the stress-state to be uniformly distributed, all the automata were allowed to move along the horizontal plane. In order to preclude dynamic effects Pores were generated by removing single automata from the initial fcc structure (b). The porosity values were varied from 0% to 65%. Note, that the maximal porosity on the assumption of pore disconnection in closed packing of spheres for 2D is 1/3 = 33% (b only one closed packing plane is shown for fcc). In this case the pores are situated regularly.The maximal value of closed porosity for random removing of single automata from fcc packing is 17.6%. The site percolation threshold for 3D fcc packing is 19.9% At the first stage it is required to determine the representative volume of the modeled material. For material without pores the representative volume was determined based on four solid samples. For this purpose the convergence of effective elastic modulus of the sample Eeff to the material elastic modulus E0 with increasing the sample base size a was analyzed.The convergence was assumed to be satisfied if the difference between Eeff and E0 was not greater than 3%. The computations suggested that the representative volume corresponded to the brick of 10 μm base size. The fracture pattern (cracks) in this sample is shown in . The cracks are generated from the stress concentrators (edges) and propagate along the maximum tangential stress direction.If a material contains pores the cracks may initiate not in the edges but in the bulk regions of the highest local porosity. The direction of cracks propagation in such material is defined by the porous topology. shows the first cracks in porous samples with various values of porosity. One can see that the failure behavior of the porous material with large porosity and the failure of a solid material differ dramatically. In particular, the path of crack propagation in porous material is rather crinkly.The fracture pattern of the modeled brittle porous 3D samples qualitative corresponds to 2D simulation results To analyze numerically the dependence of mechanical properties of the modeled material on its porosity it is necessary to determine the size of representative sample (volume) for each porosity value. For this purpose the base size of the samples, a, varied in the range from 10 to 45 μm. This made it possible to determine the representative volume of simulated materials, which would be an indication that any further increase in the sample size would cause no significant change in the elasticity modulus and strength limit of the simulated samples under uniaxial compression.On the base of calculation results the convergence of elastic and strength properties was analyzed for the simulated samples as a function of increasing sample size. To do this, the elastic moduli were calculated for the samples under compression (the slope of the first linear part of the loading diagram); the strengths (the maxima on the diagrams) were also calculated for each sample. The relative deviation from the mean Young’s modulus, E, of the calculated values, Ei, observed for all the samples is demonstrated in . The presented data suggests that for porosity value less than 15% the base size of representative sample is 20 μm (). For porosity value varying from 15% up to 35% the size of representative volume is found to be 30 μm (b). For porosity varying from 35% up to 50% the size of representative volume is 40 μm (c). Note that for porous material the convergence criterion was not as strong as for intact material. It was assumed to be enough if the difference between Ei and E was not greater than 5%.The loading diagrams of the modeled samples are shown in . They are in close qualitative agreement with 2D simulation results First, it has to be noted that porous 3D samples can demonstrate quasi-viscous regime of fracture which previously was observed for 2D models Let us consider the dependence of the effective elastic modulus of 3D samples on its porosity. In a each point with error bar represents the value averaged on five representative samples with various pore distributions in space. This plot obviously can be divided into two characteristic parts connected with porous structure: the first corresponds to closed pores (5–20%), the second corresponds to interconnected pores (20–65%). To substantiate that let us fit the computational results with phenomenological equationwhere Cmax and m are adjustable parameters. The fitting curve with the parameter values Cmax=0.7213 and m=1.9026 is plotted with a solid line in a. One can see that this curve fits very well the calculated points for porosity not greater than 50%. But if we zoom this plot and look at porosity range of 0–20% (b) then we can see that the calculated points actually are placed right between the curve and the fitted exponent E=E0exp(-mC) with value of the parameter m=3.0872 (straight dashed line in The calculated points in porosity range of 20–65% fit much better the following equation: with the parameter values EP=0.9472,Cmax=0.6761 and m=1.6292 is plotted in a as a dashed line. Physical meaning of the adjustable parameters of Eqs. is clear. Cmax means the maximal value of porosity of a sample that have any strength for the given pore morphology. For example, the procedure used for generating porosity in this work has a limit beyond which the remaining automata would not interact with each other. They would “hang” in the air and not resist to any load. Indeed, it is well known that real materials having large porosity have pore wall thickness much less than pore size. Thus, to generate larger porosity for samples based on fcc packing of automata it is necessary to remove not a single automata, but also all its neighbors. Factor EP in Eq. means that at zero porosity the modulus is not equal to the value of intact material, but it allows to get exact value at other porosity value, for example at percolation limit.The influence of percolation threshold is observed even better from dependence of strength of the simulated ceramics on its porosity. In a again each point with error bar represents the value averaged on five representative samples with various pore distributions in space. In this plot the solid line presents the fitted curve using Eq. with values of the adjustable parameters Cmax=0.7277 and m=1.9156; the dashed line corresponds to Eq. with the parameter values EP=0.9522,Cmax=0.6528 and m=1.5161. The main distinction of strength variation with porosity from elastic modulus variation can be observed for small porosity. Thus, it is shown in b that in porosity range of 0–25% the calculated points for strength are fitted much better using equationThe corresponding curve with the parameter values Cmax=0.3481,m=4.8947,C1=0.4018 and n=0.6903 is plotted in the calculation results are plotted as fitted curves, the experimental data taken from as points with error bars for comparison. One can see that the computational elastic modulus is in very close agreement with the experimental data. Calculated strength differs from experimental evidence significantly in the range of porosity of 10–40%. It can be explained by ideal structure of the modeled material, because it contains only equiaxed pores and no stress concentrators specific for real ceramics. Besides, according to Thus, simulation of uniaxial compression of brittle porous 3D samples by the movable cellular automaton method allow to conclude that percolation transition from the closed pores to interconnected pores in the porous material leads to change in the dependence of its elastic and strength properties on porosity.It is necessary to note that this result could be obtained only in three-dimensional simulation because two-dimensional samples with interconnected porosity are not topologically connected and do not resist to mechanical loading. Besides, the porosity value at which formation of interconnected clusters occurs in a two-dimensional case (22.4%) is considerably less than the corresponding percolation threshold (69.6%).Deformation and fracture of porous ceramics have been successfully simulated using the movable cellular automaton method in wide range of porosity. The method based on discrete element approach allows taking into account pores explicitly and modeling crack initiation and development. To reveal the influence of porosity percolation on the porosity dependence of mechanical properties of the material we consider only equiaxed pores of one and the same size stochastically distributed in space. We use this simplification to avoid the influence of other structure parameters on the studied problem. The main results are summarized as follows:Use of simple fracture criterion based on threshold for stress intensity (von Mises stress) shows that cracks in porous ceramics initiate in the regions of highest local porosity, which could be far away from the usual stress concentrators for solid samples.The simulations for wide range of porosity show that the plots of strength and elastic modulus of the brittle porous samples versus porosity have a change at the porosity value corresponding to percolation threshold. The obtained results are in close agreement with available experimental data.The proposed 3D MCA model of porous ceramics can be obviously extended for the case of hierarchical porous structure using the approach proposed for 2D computations Novel measurement technique of crack length for indentation fracture (IF) method using high contrast image of crack tips through thin film coatingIn order to improve the reproducibility of the indentation fracture (IF) method, a novel visualizing technique of the crack tips was developed. Coating a visualizing solution on the indented surface of silicon carbide could enhance the contrast of the crack tips. In order to verify our new approach, a round-robin test on the indentation fracture resistance, KIFR, of silicon carbide was conducted with nine laboratories. The clear identification of the crack tips could reduce the errors in reading crack lengths markedly even when measured with the ordinal metallurgical microscope at a low total magnification of 100×. Thus, good matching of KIFR was obtained between laboratories. Two main mechanisms of the enhanced contrast of the crack tips were proposed as follows; (a) the reduction of diffuse scattering from the smooth surface of the coating, (b) the interference color due to the slight change in thickness of the coating around the cracks.The indentation fracture (IF) method, which was proposed by Lawn and his co-workers However, the IF method has been regarded as an improper technique because of the poor between-laboratory consistency in the round-robin tests (e.g. VAMAS, In this study, a new measurement technique which improves the visibility of crack tip by coating a transparent solution was proposed. The visualizing solution was applied to indentations on the silicon carbide ceramics to enhance the contrast of the crack tips. In order to verify the effect of the coating on the precision of the crack length, a domestic round-robin test with nine laboratories was conducted (). The coated indentations were observed with the ordinal metallurgical microscope at both high and low magnifications. The measured crack length was compared with that obtained in the previous round robin test with and without high powered microscopy Silicon carbide ceramics sintered with B and C as sintering additives were purchased from IBIDEN Co., Ltd (IBICERAM SC-850). The bulk density of the sample was 3.024 g/cm3 and the relative density was calculated to be 94% by using the theoretical density of 3.217 g/cm3. The Young's modulus obtained by the ultrasonic pulse echo method was 365 GPa. Rectangular specimens with dimensions of 4 mm × 3 mm × 38 mm were machined from the sintered samples. The larger 4 mm × 38 mm surface was polished to a mirror finish for indentations. The batch of the samples and the polishing procedure were the same as those used in our previous round-robin test A visualization solution for crack tips was prepared by dissolving commercially available phthalic acid resin (Varnish for exterior use, ROCK PAINT Co., Ltd.) into petroleum hydrocarbon (Paint thinner, ROCK PAINT Co., Ltd.). The concentration of phthalic acid resin was ca. 12 wt%. The solution was transparent and the color was too light. The solution was filled into a refillable felt pen. All the SiC samples and the felt pens were prepared by the authors and then delivered to nine laboratories listed in for indentation and crack length measurements.Each laboratory made more than eight Vickers indentations with a hardness tester. The indentation force was 196 N except laboratory no. 9 whose indentation force was 98 N. The indentation contact time was 15 s. After unloading, the visualization solution was applied to the whole indentation area with the felt pen, just the same manner as painting with a usual felt pen, and dried for ca. 30 s at the ambient temperature. The thickness of the film measured with the ellipsometer was ca. 100 nm. Then, both diagonal sizes and surface crack lengths were measured twice at two different magnifications, one at a low magnification with the objective lens of 10× or 13× and the other at a high magnification with the object lens of 40× or 50×. For the first measurement at the low magnification, all laboratories employed an ordinal microscope furnished with the hardness tester. An eyepiece with a magnification of 10× was used in the most laboratories, so that the total magnification was 100× or 130×, whereas laboratory nos. 4, 8 and 9 observed the impressions with both CCD camera and black-and-white monitor so that the total magnification ranged from 250× to 325×. For the second measurement at the high magnification, the laboratory nos. 1, 2 and 7 measured both diagonal size of impression, 2a, and crack length, 2c, directly using the traveling microscopes with the object lens of 50× and the eyepiece of 10×. Thus, the total magnification was 500×. The rest laboratories, nos. 3–6, 8 and 9 attached the objective lens of 40× to the same optics as before and the distance between the two opposite crack tips, 2c, was calculated from the movement of the stage of the optics. The eyepiece with the magnification of 10× was used in laboratory nos. 3 and 6 so that the total magnification was 400×, whereas the crack tip image captured with the CCD camera was further enlarged on the black-and-white monitor in the laboratory nos. 4, 5, 8 and 9 and the total magnification was in the range of 1100× to 1300×. Although two measurements at the low and high magnifications took more than 10 min for a single indentation, the measured two values were compared directly with each other in the following sections since SCG is hardly expected for the solid-phase sintered SiC with B and C additives and the visualization solution hardly contained water.Only indentations whose four primary cracks emanated symmetrically and straight forward from each corner were accepted (). Indentations with badly split cracks or with gross chipping were rejected as well as those whose horizontal crack length differed by more than 10% from the vertical one.KIFR was determined from the as-indented crack lengths using Niihara's equation for the median crack system as follows where E and H are Young's modulus and the Vickers hardness, respectively, P is the indentation force, and c is the half-length of the as-indented surface crack length. In this study, Young's modulus mentioned above was used. KIFR was calculated for each indentation using the hardness value obtained for each impression. The calculated KIFR together with the raw data were collected by the test organizer. shows a typical optical microscope photo of an indentation at 196 N both (a) before coating and (b) after coating. The faint image of crack tips observed with the conventional method became clear and visible enough to measure the crack length by the coating, whereas the image of the impression hardly changed after the coating. It is reasonable to suppose that such a clear image of the crack tip makes it easy to read the crack length precisely. shows the diagonal sizes and crack lengths measured by each laboratory. The diagonal sizes, 2a, measured at the low magnification by each laboratory were almost the same as those observed at the high magnification and both variations were in the range of ca. 134–151 μm. Both grand averages and standard deviations of 2a obtained at low and high magnifications were consistent with those reported in our previous round-robin test using the same material shows indentation fracture resistance, KIFR, reported by each laboratory at both low magnification (open circles) and high magnification (closed triangles). KIFR from the participants were almost constant except laboratory no. 1 and the difference due to the magnification seemed negligible, suggesting that the coating technique could give the similar KIFR as those reported using the precise observation method with the high resolution image.In order to assess the precision of the measurement techniques, the results were analyzed numerically in accordance with the Japanese Industrial Standard Z8402-2 ( shows an enlarged image of the crack tip before and after coating. Before coating, the signal from the crack was weak as compared with the noisy background such as small black dots, making it too vague to detect the real crack tips. By contrast, some noisy small black dots disappeared after coating and the background became clear enough to find the crack tip. Thus, the increment of S/N ratio of crack image is one of the origins of the improved visibility of the crack tip. Such phenomenon can be explained from the reduction of the defuse scattering. In the conventional systems, the diffuse scattering of the incident ray occurs from the slight roughness which remains even on the mirror finished surfaces, which produces a noise to inhibit the clear identification of the crack, that is, the S/N ratio is low (). In the developed technique, a crack visualizing solution was coated on the indented surface to make a thin transparent film on it (). It is likely that the diffuse scattering is reduced by the smooth surface of the thin film so that S/N ratio increases to make the image of cracks clear (The second mechanism for the improved visibility is the coloring of the crack region, which is clearly displayed in . The coloring of the crack region is attributable to the different interference color appeared in the vicinity of the crack since the solution itself was transparent and had little color and the thickness of the film was ca. 100 nm. It is probable that the thickness of the film near the crack differs from that of other part, resulting in the difference between interference color of the crack vicinity and that of other parts. The origin of such a variation of the thickness of the film near the crack can be explained as follows. It is likely that some amounts of the solution around the crack infiltrates into the crack just after the coating (). Then, the solution is solidified during drying, leading to the slight change in the thickness of the coating especially in the vicinity of the crack ( reveals that the real crack tip located ca. 15 μm away from the end of the colored crack region, indicating that the length of the red line corresponds to the misreading of the half of the crack length, c. It is reasonable to suppose that the infiltration of the solution near the crack tip can be negligible since the depth of the crack is very shallow at the crack tip. In that case, the thickness of the film doesn’t change at the crack tip, resulting in the insufficient coloring of the real crack tip. Such a phenomenon occurred occasionally and most participants might identify the end of colored region as the crack tips when the low magnification was employed, which is a most probable reason of the shorter 2c obtained at the low magnification as mentioned in Section . Thus, it seems necessary to optimize the visualization solution and/or coating method to realize full coloring of the crack tips even when it is applied at different painting condition e.g. speed of painting, etc.In order to improve the visibility of the crack tip, a transparent organic solution was applied to the indentation on the SiC sample and was dried to make a thin film on it. Both diagonal size and crack length of the indentation on the SiC were measured using this coating technique by the nine laboratories in Japan. It was found that the misreading of the crack length was significantly reduced even when observed at the low magnification of 100×. Thus, most of the participants gave almost the same KIFR value regardless of the magnification, indicating that this technique can become a fascinating accurate IF method since it only needs the visualization solution. However, there still remained some error in reading crack length of ca. 10 μm, so that the technique was not perfect at this stage. The origin of the enhanced contrast of the crack tips was explained by the two main mechanisms. The first one is the increment of the S/N ratio due to the reduction of the diffuse scattering from the smooth surface of the film. The second one is the interference color around the crack tip due to the slight change of the film thickness. The occasionally-appeared insufficient coloring of the crack tips, which was a plausible reason of the slightly short 2c, was accounted for by the second mechanism, which implied the strategy to make this method more powerful.Tensile and stress corrosion cracking behavior of ferritic–martensitic steels in supercritical waterTensile and stress corrosion cracking behavior of ferritic–martensitic steels in supercritical water were studied in order to evaluate the suitability of these steels for supercritical water nuclear reactor concept. The ferritic–martensitic steels tested in this study consisted of T91, HCM12A, HT-9, weld T91, and weld HCM12A. A series of constant extension rate tensile (CERT) tests at a strain rate of 3 × 10−7
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s−1 were conducted in supercritical water over a temperature range of 400–600 °C and pressure 24.8 ± 0.07 MPa. CERT tests in argon and in supercritical water with 100 and 300 appb dissolved oxygen also were performed at 500 °C to compare the effect of environment. The results show that HT-9 exhibited the highest yield and maximum stresses, followed by HCM12A, and T91. The reduction in area of T91 is the highest, followed by HCM12A, and HT-9. Temperature has a great effect on tensile behavior of these steels. An increase in test temperature from 400 to 600 °C reduces the yield stress by ∼50%. Both T91 and HCM12A weld steels exhibited a slightly lower yield and maximum stresses than the base steels. Increased dissolved oxygen in the water resulted in a significant reduction of ductility. Fractography showed that all of the specimens exhibited ductile rupture except for HT-9 that showed evidence of intergranular cracking. Intergranular cracking in HT-9 is affected by temperature and oxygen concentration in supercritical water.The supercritical water-cooled reactor (SCWR) is one of the Generation IV advanced nuclear reactor concepts that utilizes water in the supercritical state (T
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> 374 °C, P
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> 22.1 MPa) in a direct, once-through mode. Operation above the critical pressure eliminates boiling, so the coolant remains in single-phase. Viability of the concept requires operation at temperatures up to 620 °C, at pressure greater than 22.1 MPa and core materials that can withstand neutron doses of 15–30 dpa (E
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> 1 MeV) F–M steels are candidates for structural components in the SCWR by virtue of their performance in supercritical fossil plants and the numerous advantages they have over austenitic stainless steels, such as higher thermal conductivity, lower susceptibility to SCC and reduced swelling under irradiation. However, F–M steels have limitations due to high corrosion rates, low creep strength at high temperatures, and radiation embrittlement at low temperatures F–M steels were developed to have high creep rupture strength at elevated temperature and high pressure This paper focuses on tensile and SCC properties of F–M steels in SCW. The CERT tests were conducted at a slow strain rate (3 × 10−7
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s−1) in deaerated SCW at 400, 500 and 600 °C. CERT tests were also conducted in argon, and in SCW containing 100 and 300 appb dissolved oxygen at 500 °C to determine the effect of dissolved oxygen on SCC.Three F–M steels: T91, HCM12A, and HT-9 were used in the as-received (normalized and tempered) condition. The heat treatment condition was selected for each steel in order to obtain the optimum microstructure of phase, precipitation and grain size. The chemical compositions in wt.% and heat-treatments are given in . The microstructure of all three steels consisted of tempered martensite laths forming subgrains in a ferrite matrix, with (V, Nb)-carbonitrides precipitated mainly on dislocations within the subgrains, and M23C6 precipitated on the prior austenite grain (PAG) boundaries and on subgrain boundaries Samples in the form of round tensile bars 39 mm long with threaded ends, a gage length of 21 mm and square cross-section of 2 mm × 2 mm were fabricated via electric discharge machining (EDM). The SCC bars were mechanically polished using SiC paper FEPA P#4000 (US grit 1200) and then electropolished in a solution of perchloric acid (10%) and methanol at −30 °C with an applied voltage of 35 V for 5–20 s to obtain a mirror finish. Electropolishing ensured the removal of the remelt layer from the EDM process, as the thickness of the remelt layer is ∼5 μm on each side, while ∼70 μm were removed by polishing on each side.To study the effect of welds in T91 and HCM12A, plates were joined by Gas-Tungsten-Arc-Weld (GTAW) at Idaho National Laboratory (Courtesy of Dr. James Cole). To avoid hydrogen cracking, the steel plates were first pre-heated to 250 °C followed by GTAW using P91 (Blue Max LNT 9Cr®) filler. The composition of the filler is listed in shows how the SCC bars were machined through the thickness of the weld and across the width of the plate.During welding, austenite forms in the heat affected zone (HAZ) and grain coarsening occurs close to the fusion boundary. After cooling, the austenite transforms to martensite, and sometimes the δ-ferrite phase remains at the grain boundary Although this paper will report on the tensile and SCC behavior of samples in the welded and heat treated condition, all welded samples were proton irradiated on one side for another study (see Ref. CERT tests were conducted in a multi-sample supercritical water system at the University of Michigan . Additional CERT tests were conducted in argon on HT-9 to isolate the effect of environment on the SCC susceptibility. Because the experiments were conducted at a pressure of 24.1–24.8 MPa, the total stress applied to the sample is the sum of the stress applied by the motor and that due to the system pressure Following CERT tests, scanning electron microscopy (Philips XL-30) was used to characterize the fracture surfaces and to determine the amount of cracking on the sample gage sections. Crack depth was determined from cross-sections of SCC samples prepared by mounting in an epoxy resin followed by mechanical polishing. The mounted specimens were etched in a solution of 1 part HCl to three parts HNO3 in order to observe the grain boundaries. The cracks were investigated in SEM in both secondary and back scattered electron modes. The crack depth was measured from the deepest crack to the interface of inner and outer oxide (since it is the original surface of steel). Cracks in HT-9 occurred in three areas of the SCC bars; marked as A, B and C in . These areas are defined by the ratio of the width in the necked region to that in the un-necked region. Area A is from the fracture surface to 80% of the original width, area B is from 80% to 90%, and area C is from 90% to the full un-necked width. Since area A is characterized by extensive localized plastic deformation, no crack analysis was performed in this region. Cracks were recorded and reported only in areas B and C.Tensile behavior was observed by the stress–strain response examined in CERT tests. SCC was characterized by documenting cracking on the fracture and gage surfaces of the failed specimens. The effects of steel type, temperature and environment on stress–strain behavior will be discussed first, followed by a discussion of fracture behavior.A summary of the results obtained from CERT tests at 400, 500, and 600 °C in DSCW, at 500 °C with 100 and 300 appb DO, and at 500 °C in Ar is presented in . Plots of stress–strain curves of T91, HCM12A and HT-9 from CERT experiments in DSCW at each temperature are shown in . All of the stress–strain curves showed work-softening behavior. The uniform elongation of these steels ranged between 5% and 10%, and total elongation was 12–15%. There was substantial necking in all samples. The yield strengths were 355–490 MPa for the tests at 400 °C, 375–471 MPa at 500 °C, but only 155–198 MPa at 600 °C. At all temperatures the strengths were in the order HT-9 > HCM12A > T91. shows a comparison of the stress–strain curves of T91 in different environments at 500 °C, which consists of the tests in Ar, in deaerated SCW, and in SCW containing 100 or 300 appb DO. Note that there is a significant drop in yield strength between Ar and DSCW. With increasing oxygen content in SCW, the yield strength drops slightly and the total elongation decreases more significantly. Comparison of uniform and total elongations is shown in . Comparison of reductions of area (RA) as a function of temperature in DSCW, or as a function of environment at 500 °C is presented in Under all conditions, the steels exhibited similar work-softening behavior. For each temperature, HT-9 had the highest yield and maximum stresses, followed by HCM12A, and T91, . The results agree with those reported in the literature The three F–M steels exhibited a similar range of total elongation (10–15%) under all test conditions. Low uniform elongation is a typical feature associated with the tempered martensite structure . Fractography showed that all steels failed by ductile rupture. There was no evidence of IG in T91 and HCM12A in any of the test conditions.Temperature has a significant effect on the stress–strain behavior of F–M steels in SCW. Both yield and maximum stresses decreased with increasing temperature, . The trend agrees closely with literature data on typical F–M steels tested at these temperatures Uniform elongation (strain to maximum stress) increases as a result of softening, which agrees with results reported by Klueh . The plastic deformation in the necked region may have decreased as a result of oxide constraint on the surface, not a behavior of steel. A result on oxidation of F–M steels in SCW shows a comparison of the high plastic deformation in the necking area of 500 °C sample and the low necking in 600 °C sample. Localized necking was also observed at each crack, implying that the oxide has a significant effect on the reduction of area but not on the yield and maximum stresses of steels.Comparisons of yield stress, maximum stress, elongation, and RA in different environments () show that dissolved oxygen concentration appears to have only a small effect on tensile behavior. Both yield and maximum stresses of T91 and HT-9 were higher in argon as compared to that in the SCW environment, but they still fall in the same range with standard deviation. Total elongation slightly decreased as the DO concentration increased, . These results showed that DO concentration does not have a significant effect on yield and maximum stresses, which is expected. Two significant results observed from the test at 500 °C with 300 appb were the reduction in total elongation for three steels, and the cracking behavior of HT-9. The decrease in total elongation probably was affected from cracking initiated in oxide that formed on the surface, not the behavior of steel. The SCC susceptibility increased at higher oxygen concentration. Further detail about SCC will be discussed in Section The welded samples of T91 and HCM12A exhibited values of yield and maximum stresses, total elongation, and RA that were close to those of the base steels tested in the same conditions, . Note that while irradiated samples were used in this comparison, irradiation is not expected to affect the tensile behavior because radiation damage affects only the top 15 μm of the irradiated surface Among the three F–M steels and the weld samples tested in this study, only HT-9 exhibited IG cracking. The gage surface of HT-9 showed IG cracking in both the necked region and in regions away from the neck. Cross-sections of the gage region were investigated in order to evaluate crack penetration. The maximum crack depth was recorded as this is important in component failure and is a more meaningful characterization of cracking in a small dataset. Cross-section images of crack on HT-9 occurred at 400–600 °C deaerated SCW, and at 500 °C in Ar and 300 appb SCW are showed in . A plot of crack density and maximum crack depth is shown in . The results show that both environment (SCW and DO) and temperature have influence on the crack density and the maximum crack depth.The effects of environment and DO concentration were determined at 500 °C by performing CERT tests in Ar, in deaerated SCW, and in SCW with 300 appb DO. Results indicate that cracking behavior is influenced by both environment and DO, which the cracking in SCW with 300 appb DO is highest, followed by those in SCW and in Ar. The cracking in Ar is quite low (crack density is ∼3 cracks/mm2 and maximum crack depth is 8.3 μm). However this evidence showed that HT-9 also has susceptibility to IG cracking in non-oxidizing environment. This implied that microstructure of HT-9 plays a role in the inherent susceptibility to IG cracking. The previous work reported by Gupta et al. The environment has a significant effect to increase the cracking susceptibility from the Ar environment to the deaerated SCW to the SCW containing 300 appb DO. The trend also follows the oxidation potential of the environment (see ) also showed that the oxides exerted the pressure to crack tip and induced the crack growth. The result from 300 appb DO SCW test, which the crack density and maximum crack depth increased dramatically, demonstrated that the oxidation generally affected the cracking behavior. Furthermore, note that the crack density of sample tested in 300 appb DO SCW is higher than that in 600 °C deaerated SCW indicating that the oxidation promotes the crack initiation.Temperature has a major effect on cracking susceptibility in which the crack density is least at 400 °C and highest at 600 °C under deaerated condition. The result of crack density also followed the oxidation potential where oxidation at 600 °C deaerated SCW is the highest ). Large crack on 600 °C specimen associated with the cracking in oxide since the oxide is very brittle, and it formed thick layers at this temperature. This result also supports that the oxidizing environment of SCW increases the IG cracking susceptibility of HT-9, both for the crack density and the maximum crack depth. However, the microstructure of HT-9 has a major effect on the inherent susceptibility to cracking since two other F–M steels T91 and HCM12A did not crack under the same environment.In CERT tests under all conditions, the yield and maximum stresses for HT-9 are the highest, followed by HCM12A, and T91. RA and total elongation are highest for T91, followed by HCM12A, and HT-9. These trends correspond to the amount of Cr in the steels.Temperature has a significant effect on mechanical properties of F–M steels. The yield and maximum stresses decreased rapidly with an increase in test temperature. Conversely, the total elongation increased at higher temperatures. These results are consistent with changes in microstructure at higher temperature; coarsening of carbides and higher dislocation mobility.The reductions of area of 600 °C samples are lower than expected. This could be a result of high uniform elongation and low plastic deformation. The formation of a thick oxide layer on the tensile bar limits the deformation in transverse direction.The dissolved oxygen concentration in 500 °C test did not show a significant effect on yield and maximum stresses. However, the elongation reduced in 300 appb DO tests.Both welded T91 and HCM12A samples tested in 500 °C deaerated SCW exhibited a slight decrease of yield and maximum stresses. This reduction in strength may be a result from the grains coarsening in HAZ. No IG cracking was observed on both weld steels.HT-9 was the only steel that displayed evidence of IG cracking. Cracking was observed on both the plastic and uniform deformation regions. Cracks penetrated through both outer and inner oxide layers, and into the steel matrix. Both crack density and maximum crack depth increased with temperature and oxidizing environment. Crack density and depth were highest in 300 appb DO test and lowest in Ar test, implying that increased dissolved oxygen promotes IG cracking susceptibility in HT-9. However the microstructure of the steel plays an important role on the inherent susceptibility to cracking.Investigation of the role of Ti oxide layer in the size-dependent superelasticity of NiTi pillars: Modeling and simulationRecent compression tests of NiTi pillars of a wide range of diameters have shown significant size dependency in the strain recovered upon unloading. In this paper, we propose a numerical model supporting the previously proposed explanation that the external Ti oxide layer may be responsible for the loss of superelasticity in the small pillars. The shape memory alloy at the center of the pillar is described using a nonlocal superelastic model, whereas the Ti oxide layer is modeled as elastoplastic. Voigt average analysis and finite element calculations are compared to experiments for the available range of pillar sizes. The simulation results also suggest a size-dependent strain hardening due to the constraint on the phase transformation effected by the confining Ti oxide layer.The unique feature of shape recovery upon thermomechanical loading cycle makes shape memory alloys (SMAs) very popular for applications in aerospace industry, medical devices, consumer products and other engineering fields. The underlying mechanism for the shape memory and superelastic effects, martensitic phase transformation, has been studied both theoretically and experimentally ) the fixed-thickness Ti oxide layer and Ti-depleted zone take most of the pillar volume, and the suppression of superelasticity can be expected. In this work, we attempt to provide a model-based quantitative study on how this Ti oxide layer affects the mechanical behavior of NiTi pillars under compression, giving special emphasis to the size-dependent incomplete strain recovery observed experimentally.In the past, a large number of thermomechanical material models have been developed for SMAs. A comprehensive review of these efforts can be found in Ref. The modeling approach adopted in this paper treats the NiTi pillars as a composite material comprising a uncontaminated NiTi core and an external Ti oxide layer. We propose a nonlocal superelastic model for the NiTi core, and an elastoplastic model for the Ti oxide layer. Through Voigt average analysis and finite element simulations, these models are used to investigate the quantitative influence of the Ti oxide layer on the mechanical responses of NiTi pillars under cyclic compression loading. The simulation results show that the plastic deformation in the Ti oxide layer constrains the recovery of deformation in the whole pillar, and the effect becomes severe with diminishing pillar size. The agreement with experimental results suggests that the size-dependent strain recovery and the loss of superelasticity in small pillars are likely to be associated with the plastic deformation in the Ti oxide layer.The NiTi pillar consists of a Ti oxide layer (mainly TiO2). The Ti-depleted zone is expected to behave as a smooth transition from Ti oxide to NiTi SMA. Due to the lack of material properties for this region, we investigate the two bounding cases in which the Ti-depleted zone is either full NiTi or full TiO2. The TiO2 layer has, respectively, a thickness of 15 and 65 nm. Material models for the NiTi SMA and TiO2 will be discussed in the following subsections.We assume for simplicity isotropic response for both elastic and superelastic effects. Specifically, we ignore the dependency of the elastic moduli, the critical stresses for phase transformation, the maximum phase transformation strain and the phase transformation strain hardening on crystal orientation. For definiteness, we calibrate our model parameters to one specific composition and orientation. In our model, the displacement u and the martensitic volume fraction ξ are the two primary unknown fields. The total strain E=12(∇u+(∇u)τ), where ∇ denotes the spatial gradient and ( )τ denotes the transpose, is decomposed into an elastic part Ee and a phase transformation part Et:The evolution of the phase transformation strain Et is assumed to follow the relationwhere Λt is the phase transformation flow direction and ()̇ denotes temporal derivatives. Following Ref. Λt=32ε¯tSdev‖Sdev‖,forξ̇>032ε¯tEt,r‖Et,r‖,forξ̇<0where Sdev is the deviatoric part of the Cauchy stress tensor S, the scalar ε¯t is the maximum transformation strain along the loading direction and Et,r is the phase transformation strain tensor upon unloading.The free energy per unit volume consists of the elastic, chemical, hardening and nonlocal terms:ψNiTi=12(C:Ee):Ee-Δseq(T-Teq)ξ+12Ht(ξ)2+12S0ℓe2‖∇ξ‖2The elastic tensor C is the arithmetic average of the corresponding elastic moduli for austenite and martensite, i.e. C=(1-ξ)CA+ξCM. Teq is the equilibrium temperature between the two phases in the stress-free state, Δseq is the entropy for phase transformation from austenite to martensite at Teq, and T is the temperature at which the experiments are performed. The hardening parameter Ht has dimensions of stress and characterizes the classic strain hardening during phase transformation. The nonlocal term accounts for the interfacial energy between the two phases. S0 is a model parameter with dimensions of stress, and ℓe is the energetic length scale. Introducing the gradient of the martensitic volume fraction ξ in the free energy leads to an extra governing partial differential equation (microforce equilibrium), in addition to the classic force balance equation, both of which follow directly from the principle of virtual power where k and knl are the work-conjugate variables to ξ̇ and ∇ξ̇, respectively. The external power expended on V can be expressed bywhere ∂V is the surface of V,tˆ and kˆ are the traction and microtraction, respectively. The principle of virtual power states thatfor any general velocity (ũ̇,ξ̃̇,E∼̇e) satisfying the kinematic requirement E∼̇=12(∇ũ̇+(∇ũ̇)τ)=E∼̇e+E∼̇t=E∼̇e+ξ̃̇Λt Integrating by parts and using the symmetry of the stress tensor S, the left-hand side of Eq. Pint(E∼̇e,ξ̃̇)=∫V(-∇·S)·ũ̇+(-S:Λ+k-∇·knl)ξ̃̇dx+∫∂V(S·n)·ũ̇+(knl·n)ξ̃̇dxwhere n is the unit outer normal to the surface ∂V. Since Eq. must hold for any admissible field ũ̇ and ξ̃̇, the following two partial differential equations are obtained: the macroforce balance equationas well as the two boundary conditions S·n=tˆ and knl·n=kˆ on ∂V. The second law of thermodynamics requires that the temporal increase in the free energy cannot exceed the externally expended power:The temporal increment in free energy density can be expressed as ψ̇NiTi=∂ψNiTi∂Ee:Ėe+∂ψNiTi∂ξξ̇+∂ψNiTi∂(∇ξ)·∇ξ̇. It then follows that0⩽S-∂ψNiTi∂Ee:Ėe+k-∂ψNiTi∂ξξ̇+knl-∂ψNiTi∂(∇ξ)·∇ξ̇Inspired by the strain gradient plasticity theory in Ref. With the newly derived constitutive relations, Eq. Ysign(ξ̇)=S:Λt-12∂C∂ξ:Ee:Ee+Δseq(T-Teq)-Htξ+S0ℓe2(∇·∇)ξwhere the left-hand side can be viewed as the resistance to the phase transformation, i.e. ±Y for the forward and reverse transformations, respectively, and the right-hand side can be viewed as the driving force for the phase transformation. In the absence of the gradient term, Eq. represents the conventional local phase transformation conditions The TiO2 layer is modeled as an isotropic elastic–perfectly plastic material. The decomposition of the total strain tensor now readswhere Ep is the plastic strain tensor. The evolution of Ep follows the flow rulewhere ε̇p denotes the equivalent plastic strain rate and Λp is the plastic flow direction, which takes the normality ruleThe constitutive relations include Hooke’s law,where CO is the elastic moduli of TiO2 and the conventional J2 plastic yield condition,where σ¯y is the compressive yield strength.The plastic hardening of TiO2 is ignored because it is expected to be much smaller than the strain-hardening rate of NiTi SMA. The fixed-thickness TiO2 layer is supposed to dominate in the small pillars, while it has been observed that pillars with diameter smaller than 200 nm exhibit less strain hardening than pillars with larger diameters, and the 162 nm [2 1 0]-oriented pillar even shows a perfect plateau The values of the SMA model parameters are determined for [1 1 1]-oriented Ti–50.9 at.% Ni, for which size dependence of the strain recovery is observed where σ is the stress along the loading direction. At T
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= 298 K, a stress value of 800 MPa has been reported as the point at which the forward martensitic phase transformation initiates , one obtains ∂σ∂εε¯t-Ht∂ξ∂ε=0 by taking the derivative with respect to the total strain ε. From Eq. , one obtains ∂σ∂ε=E1-∂ξ∂εε¯t with assumption E
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=
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EA
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=
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EM. Combining these two equations leads to Ht=∂σ∂ε(ε¯t)2/1-1E∂σ∂ε. By replacing ∂σ∂ε with the experimentally reported value 20 GPa The group of parameters S0ℓe2 has the effect of enhancing the strain-hardening rate for nonuniform phase transformations Material parameters for TiO2, including the Young’s modulus EO= 287 GPa, the Poisson’s ratio νO
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= 0.268 and the compressive yield strength σ¯y=3GPa, are obtained from Ref. In the analysis of composite materials, the Voigt average, which assumes uniform strains, is commonly used to estimate the stiffness and the stresses. In this work, we also employ it to analyze the response of the composite NiTi/TiO2 pillars. Consider an NiTi pillar with diameter D that contains a TiO2 layer with thickness tO. The strain along the loading direction ε is assumed to be identical in the two materials. Given the strain history, the stress along the loading direction within each material, σNiTi and σO, can be calculated independently using its constitutive relations, Eqs. , where the gradient term in the NiTi SMA model is ignored. plots the stress–strain curves of NiTi SMA and TiO2 during a compressive loading cycle with a maximum strain of 3%. Complete strain recovery and stress hysteresis in the strain-loading cycle can be observed in the response of NiTi SMA. For TiO2, one can observe the typical strain-cycle response for an elastic–perfectly plastic material leading to a residual stress when the strain goes back to zero. The reaction force from the pillar cross-section, f, is the sum of the reaction forces from the two materials, i.e.The average stress response of the composite can be obtained as follows:The results using this model are shown in Section The composite model presented in the previous section does not consider the interaction between the TiO2 layer and the NiTi core, and in particular ignores the constraint from the TiO2 layer on the martensitic phase transformation in NiTi SMA. In addition, due to the locality of the constitutive models for the TiO2 plasticity and the SMA superelasticity, the homogenized approach can only capture size effects through the volume ratio of the two components, but will be insensitive to a change of the spatial scale.In order to explore the role of the interaction between the two components, including gradient effects at the TiO2–NiTi interface produced by the internal constraint to the phase transformation, three-dimensional finite element calculations are performed using the full nonlocal SMA model. The pillar is modeled as a cylinder of diameter D and height h. Due to symmetry, only a quarter of the pillar is considered in the computation (). In reality, the top surface is also covered by the TiO2 layer, which could significantly affect the mechanical response if the aspect ratio h/D is small. It has been reported that the aspect ratio of all samples ranges between 1.6 and 3.9 , is a partial differential equation of the martensitic volume fraction, and is coupled with the macroforce balance equation, Eq. . With proper boundary conditions, these two equations for NiTi SMA, and the governing equation for TiO2 (which is the same as Eq. ), complete the formulation of the pillar compression test boundary value problem. A finite element discretization with a staggered coupled scheme is used to approximate the resulting coupled macro- and microforce balance equations in weak form.For both the composite Voigt average and finite element models, the experiments are simulated as follows. Since both the superelasticity and the plasticity are history dependent, the strain history is applied in increments of 0.1%, and at each strain increment the constitutive models are integrated numerically. Following the experimental conditions, the strain is first increased to −3%, then decreased until the reaction force becomes zero. The pillar is then reloaded to −5% strain and unloaded until the reaction force becomes zero again. The evolutions of the stress, the martensitic volume fraction (NiTi SMA) and the plastic strain (TiO2) are recorded during the entire procedure. The strain history is applied at a constant temperature T
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