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= 8 in the above equations. These iso-parametric elements with quadratic shape functions use only boundary nodes (). The appropriate derivatives of the displacement field and the effective plastic strain field can be expressed ase˙ij=∑n=12kNi,jnD˙nϵ˙ij=∑n=12kEijnD˙nϵ˙,iP=∑n=1lM,inϵ˙nPUsing these relations in the principle of virtual work (Eq. ), which must hold for arbitrary variations of δD˙n and δϵ˙nP, the discretized equations can now be written in the following formKenm=∫V(EijnRijklEklm+σij(Nk,jmNk,in-2EikmEkjn))dVKpnm=∫VmijRijklmklMmMn+hMmMn+AijM,jmM,in+12Bi(M,imMn+MmM,in)+CMmMndVWhen the nodal displacement increments and effective plastic strain increments have been determined, the elastic strains are calculated, and then the Jaumann rate of the stress and the convected rate of the higher order stress can be found according to Eqs. . The increments of the stress tensor and the higher order stress vector are then calculated byPlastic yielding is initiated when σ(e) becomes larger than the initial yield stress, σy. Elastic unloading at an integration point starts, when the solution has found a negative increment of the effective plastic strain at that integration point in the previous increment. Plastic yielding starts again when Q reaches Qy, which is updated using Eq. . The present strain gradient plasticity theory reduces to J2 finite strain theory when all material length parameters are set to zero.Boundary conditions on internal elastic–plastic boundaries are either applied as vanishing higher order traction rate, t˙=0, or conversely vanishing plastic strain rate, ϵ˙P=0. For these two boundary conditions the work on the internal boundaries vanish, which would not be the case if intermediate boundary conditions were considered. When imposing a vanishing higher order traction rate on internal boundaries Kp is multiplied by 10−4 for Gauss points in the elastic region. When imposing a vanishing plastic strain rate on internal boundaries Kp is multiplied by 108 for Gauss points in the elastic region.In all cases studied it has been checked that a sufficiently refined mesh has been used. It is noted that the use of gradients here is motivated by physical properties of the material studied, and is not considered a numerical tool to avoid discretization sensitivity. The typical meshes employ eight elements through the half width of the specimen. The number of elements along the length of the specimens depend on the aspect ratio. For some of the shorter specimens in the necking studies 16 and even 32 elements through half the width have been used, so that a sufficient number of elements along the length could be used keeping the aspect ratio of the elements reasonable. Along the length of the specimen, stretching of the mesh has been used for the necking studies to ensure that the aspect ratio of the elements in the neck is reasonable throughout the analyses.The plane strain tension test and the plane strain compression test are here analyzed for specimens with dimensions in the micron range. a shows the geometry for the plane strain tension test. The specimen has the width 2a0 and the length 2b0, and an initial symmetric cosine imperfection of initial amplitude δ0. Along the horizontal edges displacements are prescribed to enforce tension in the vertical direction, while shear stresses vanish. The vertical edges are stress free. After some elongation the deformation begins to localize, and the onset of localization is defined as the point when elastic unloading initiates somewhere in the specimen.b shows the geometry for the plane strain compression test. An initial anti-symmetric cosine imperfection is used so that anti-symmetric buckling modes are triggered. Along the horizontal edges displacements are prescribed to enforce compression in the vertical direction, while shear stresses vanish. The vertical edges are stress free. At some level of the overall deformation plastic buckling will occur, when elastic unloading initiates at a material point in the specimen.For both the tension and the compression problem the maximum width and the length of the deformed specimens are denoted 2a and 2b, respectively. The amplitude of the neck and the buckling mode amplitude are denoted by δ.The higher order boundary condition on the outer surface of the specimen is t |
= 0. On internal elastic–plastic boundaries the higher order traction rate is here taken to vanish (t˙=0). Whether this internal boundary condition or the alternative boundary condition of a full constraint on the increment of the plastic flow (ϵ˙P=0) is used, has been found to have a negligible effect on the problems studied within the present paper.Results for a rather short and stubby plane strain tensile test specimen, with b0/a0 |
= 3.0, are shown in . Here, the average nominal tensile stress is denoted σn and the average logarithmic strain is denoted ϵav. The small initial imperfection in the width is specified by δ0/a0 |
= 0.005 and the material parameters are σy/E |
= 0.004, ν |
= 0.3, and n |
= 10. The star on the curves indicate the maximum load point, while the circles show the onset of localization (elastic unloading). The curve for conventional theory shows the behavior known from many previous analyses of the plane strain tensile test, i.e., the maximum load occurs at ϵav |
= 1/n and the onset of localization occurs with a little delay after the maximum. Subsequently, as the localized neck develops, the load drops down below the level corresponding to continued uniform deformation. also shows results for gradient dependent materials. One curve corresponds to the single parameter theory with l∗/a0 |
= 0.5, and results for the multi-parameter theory are shown with each of the three length parameters set equal to 0.5a0, while the other two are zero. For this case the gradient effect gives a significant extra delay of the onset of localization, after the load maximum, and also the subsequent load decay during neck evolution occurs more slowly than found for the conventional theory. It is clear in that there is rather little difference between the four curves representing gradient effects, when the values of the four different material lengths are the same., with the main difference that they considered a linear hardening material with the tangent modulus ET |
= |
E/40. This gave a load maximum at a much larger strain, ϵav |
≈ 0.7, and subsequently a decaying load as localized necking developed. But apart from the much larger strains the behavior was similar to that shown in in that the gradient effect gave a significant delay of the onset of localization and there was little difference between the four curves representing gradient effects analogous to those in , for n |
= 10, two of the curves were also shown in , but in addition to the curve for l∗/a0 |
= 0.5 according to the single parameter gradient theory results are also shown for the values 0.25, 1.0, and 2.0. This comparison shows a very strong sensitivity to the value of the characteristic material length relative to specimen dimensions, particularly in this range where so small specimens are considered that the material length exceeds the specimen size. For increasing values of the material length parameter, the onset of localization is further delayed relative to the maximum load point. For l∗/a0 |
= 2.0 the onset of localization occurs much after the load maximum, and also the subsequent necking develops slowly. also shows the effect of the strain hardening exponent by including curves for n |
= 5 and 20. For each of these strain hardening levels the curves behave essentially as described for n |
= 10. As expected, both the load levels and the strains at the maximum and at localization increase much with the level of strain hardening.With the initial aspect ratio b0/a0 |
= 3, analyzed in the current aspect ratio at the onset of localization for n |
= 10 is b/a |
≈ 3.8 for the conventional material and even higher for the gradient dependent materials, so that the wave-length of the corresponding bifurcation mode is several times the current specimen width at bifurcation. In , for larger values of the initial aspect ratio b0/a0, the wave-length of the bifurcation mode governing the onset of localization is larger than that in , relative to the current width. All these cases are in the range of conventional necking behavior, where the limit of b0/a0 |
→ ∞ gives the Considére condition that necking initiates at the load maximum, but where the onset of localization is delayed for a finite aspect ratio, with a larger delay beyond the load maximum the smaller value of b0/a0. In all the cases the onset of localization is further delayed by the effect of a material length scale.Smaller values of the initial aspect ratio b0/a0 are considered in , where the smallest value, b0/a0 |
= 0.125, starts to look more like a surface wave instability than a necking mode instability. We see that for the two smaller values of b0/a0 there is a very significant delay of the onset of localization when gradient effects are present. In the case of b0/a0 |
= 0.125 the onset of localization for the conventional material occurs at the current aspect ratio b/a |
≈ 0.40, but for l∗/a0 |
= 0.5 this current aspect ratio has grown much more, to 3.0, when localization occurs, and at this point l∗/a |
≈ 2.5. The bifurcation analyses of have shown that for the smaller wave-lengths gradient effects significantly increase the value of the true stress at bifurcation. This is not illustrated in , as the stress σn shown is an average nominal stress, but the large increases of ϵav at the onset of localization do reflect the increases of the true stress.The effect of the magnitude of the initial geometric imperfection δ0/a0 is illustrated in . A 10 times smaller imperfection does delay localization a little, both for the conventional solid and the gradient dependent solid, but rather little additional difference is found for 102 or 103 times smaller imperfections, indicating that here the curves have converged towards the post-bifurcation behavior of the perfect solid.The development of the neck as a function of overall strain is shown in for the same initial imperfection amplitudes as in . It is seen that the small imperfections give almost identical results, which implies that they essentially show the post-bifurcation behavior of a perfect specimen. The solid curves in the figure show results for conventional materials, while the dashed curves show results for gradient dependent materials with l∗/a0 |
= 0.5. It is seen that strain gradient hardening has the effect of limiting strain localization and neck development for a given overall strain, when compared to conventional predictions. a short, stubby plane strain specimen, b0/a0 |
= 4, is analyzed for compressive loading, so that it acts as a stubby column. The initial imperfections considered are in the anti-symmetric mode, as illustrated in b, and the diagram shows the average nominal compressive stress −σn vs. the buckling mode amplitude δ. In the two smallest imperfections analyzed are small enough to closely follow the bifurcation and post-bifurcation behavior (). It is characteristic that plastic buckling occurs under increasing load and that a load maximum higher than the bifurcation load is reached on the post-bifurcation path. When the imperfection amplitude is larger, the curves show the characteristic buckling behavior that the load maximum is reduced and that the maximum occurs at a larger value of the mode amplitude (). The curves for the single parameter gradient theory with l∗/a0 |
= 1.0 show that the buckling load is significantly increased by the gradient effects, as compared with the predictions for the conventional solid.The choice of the internal higher order boundary condition would have a small but noticeable influence on the results presented , whereas the influence is negligible in the other figures shown until now. For the gradient dependent solid the buckling load as well as the maximum load point are virtually unaffected by the specific choice of higher order boundary condition, but the strain level at the end of the analyses would be increased by a few percent if the full constraint on the plastic strain increment was used instead of the constraint on the higher order traction rate (see the fourth paragraph of Section , for a small initial imperfection (δ0/a0 |
= 5 × 10−5), results of the multi-parameter theory are also shown, with each of the three length parameters set equal to a0, while the other two are zero. There is rather little difference between the four curves representing strain gradient plasticity, while they all show higher buckling strength than the conventional solid, as in Three different values of the initial specimen aspect ratio are considered in . For b0/a0 |
= 4 an additional curve is included, l∗/a0 |
= 0.5, besides the curves already shown in and 9. For the larger aspect ratio, b0/a0 |
= 8, the behavior is affected by the abrupt jump in the tangent modulus at the onset of plastic yielding. Thus, for a perfect column the instability occurs immediately when yielding initiates, and for the small initial imperfection considered in buckling develops rapidly when a small part of the column material has entered the plastic range.For the smaller initial aspect ratio, b0/a0 |
= 2, no bifurcation is found at all when l∗/a0 |
= 1.0, even at values of −σn/σy far above those shown in . Also, the post-bifurcation behavior for the conventional solid and for l∗/a0 |
= 0.5 are untypical of buckling, in that no load maximum is found after bifurcation. Here, the current aspect ratio b/a of the specimen at bifurcation is 1.87 for the conventional solid and 1.73 for l∗/a0 |
= 0.5. It is seen that the current aspect ratio at bifurcation decays rapidly for increasing values of l∗/a0, so that the first critical bifurcation moves further and further into the short wave range. In this range it is known from that the value of the critical bifurcation stress is highly sensitive to a reduction of the wave-length or an increase of the critical material length, and this explains why no bifurcation is found for l∗/a0 |
= 1.0 in the range considered.Relative to the study of plane strain sheet-necking by the only improvement of the numerical procedure here lies in the use of higher order elements. The procedure based on triangular elements worked very well for a linear hardening material, but did not show satisfactory converge for a power hardening material, even in the absence of a material length. This surprising problem, which is introduced due to the two-field finite element interpolation, is removed by using the higher order elements, and the results presented in the present paper for a power hardening material have shown excellent convergence.Both tensile and compressive loading on a plane strain specimen have been analyzed here. For the tensile loading, where bifurcation results in necking, the results show much analogy to those found by , with the difference that the present results are based on a power law representation of the uniaxial tensile test, which can often be used to obtain a good representation of experimental observations. For the plane strain tension–compression test the effect of a gradient plasticity model on the occurrence of bifurcation has been analyzed by , and the numerical results obtained in the present paper are related to these bifurcation results, even though the strain gradient plasticity theories used here differ from that in the bifurcation study. Small initial geometrical imperfections are assumed in all the present numerical studies, so that bifurcations do not occur, and the onset of localization or buckling develops gradually from the initial imperfections. However, in several of the studies the imperfections are so small that the numerical solutions initially follow closely along the pre-bifurcation path, before the bifurcation point is approached, and subsequently the post-bifurcation path is closely followed, so that in fact these computations give a very good representation of the bifurcation and post-bifurcation behavior of the specimens.For larger values of the initial aspect ratio of the specimen, both tensile and compressive tests follow standard behavior. In tension, the onset of necking is delayed beyond the load maximum due to the finite aspect ratio, and is further delayed by the effect of gradient hardening. In compression, a load maximum is reached after bifurcation, on the post-bifurcation path, and as usual the buckling load is higher the shorter the column, but the buckling load is further increased by the effect of gradient hardening.For small values of b0/a0 the bifurcation mode wave-length is forced to be small relative to the specimen width, and it is known from the earlier bifurcation results that this significantly delays the onset of bifurcation in the presence of gradient hardening. Here tension or compression give a different trend, as illustrated in . In tension, for b0/a0 |
= 0.125 and gradient hardening with l∗/a0 |
= 0.5, the onset of localization is delayed to such a large strain that the current aspect ratio of the deformed specimen is increased by a factor 20, which changes the behavior of the specimen to one with a much higher aspect ratio but with a relatively stronger gradient effect. By contrast, in compression with b0/a0 |
= 2 the current aspect ratio grows smaller as the load is increased, so that the specimen is deformed into a range where the delay due to the gradient effect keeps increasing, and this is the reason why no bifurcation was found for the value of l∗/a0 as large as 1. A common feature of tension and compression is that with gradient hardening present we have not found instabilities corresponding to very short wave modes, approaching the so-called surface wave instabilities. Trans. Nonferrous Met. Soc. China 23(2013) 756−764 Hot deformation behavior and processing maps of Mg−Zn−Cu−Zr magnesium alloy Hui YU 1,2 , Hua-shun YU 1 , Young-min KIM 2 , Bong-sun YOU 2 , Guang-hui MIN 1 1. Key Laboratory for Liquid−Solid Evolution and Processing of Materials, Ministry of Education, School of Materials Science and Engineering Shandong University, Ji’nan 250061, China; 2. Light Metals Group, Korea Institute of Materials Science, Changwon 642831, Korea Received 21 May 2012; accepted 5 November 2012 Abstract: The deformation behaviors of a new quaternary Mg−6Zn−1.5Cu−0.5Zr alloy at temperatures of 523−673 K and strain rates of 0.001−1 s −1 were studied by compressive tests using a Gleeble 3800 thermal-simulator. The results show that the flow stress increases as the deformation temperature decreases or as the strain rate increases. A strain-dependent constitutive equation and a feed-forward back-propagation artificial neural network were used to predict flow stress, which showed good agreement with experimental data. The processing map suggests that the domains of 643−673 K and 0.001−0.01 s −1 are corresponded to optimum conditions for hot working of the T4-treated Mg−6Zn−1.5Cu−0.5Zr alloy. Key words: Mg alloy; Cu addition; flow stress; deformation behavior; constitutive equation; artificial neural network; processing map 1 Introduction As the lightest metallic structural materials, magnesium and its alloys recently show the most promising development due to their low density, high specific strength and specific stiffness, good damping capacity, excellent machinability and high recycling rate. Accordingly, Mg alloys have great potential in the aerospace, aircraft and automobile industries [1]. Copper (Cu) is cost-effective, compared with some rare earth elements such as Ce, Nd, Y and Gd. According to reports by ZHU et al [2] and BUHA and OHKUBO [3], Mg−Zn−Cu alloys (ZC series) that have good castability show a significant increase in age-hardening response compared with binary Mg−Zn alloy. As Cu addition can increase eutectic temperatures, Cu-added alloys can be solution-treated at higher temperatures. The addition of minor alloying elements such as Mn, Al, Zr to Mg−Zn−Cu alloys can make significant influences on mechanical properties; for instance, a favorable combination of high ductility, yield strength and hardness can be achieved in naturally-aged cast Mg−6Zn−2Cu−0.1Mn alloy [4]. Zirconium (Zr) can significantly refine grain size of cast products, resulting in improving tensile properties and corrosion resistance [5]. Meanwhile, designers of metal forming process are eager to understand how metals and alloys behave under certain hot-working conditions. The constitutive equation is a powerful tool which can describe the plastic flow behaviors of metals and alloys. Based on the principle of the dynamic material model (DMM), processing maps are constructed to figure out the optimum conditions for hot working processes [6]. Although many researchers have attempted to utilize these tools with a wide range of materials including Mg−Zn−Zr and Mg−Al−Zn alloys [1,7−13], there have been rare studies on the hot deformation behavior and workability of Mg−Zn−Cu−Zr magnesium alloy. In this study, therefore, we investigated the plastic flow behaviors of Mg−6Zn− 1.5Cu−0.5Zr magnesium alloy. Two approaches (regression and artificial neural network) were used to develop the strain-dependent models to predict the flow stress. In addition, the processing maps of Mg−6Zn− 1.5Cu−0.5Zr magnesium alloy at different strain levels were constructed to determine the optimum conditions for hot working process, using the calculated strain-rate sensitivity. Corresponding author: Young-min KIM; Tel: +82-55-2803537; Fax: +82-55-2803599; E-mail: [email protected] DOI: 10.1016/S1003-6326(13)62526-1 Hui YU, et al/Trans. Nonferrous Met. Soc. China 23(2013) 756−764 757 2 Experimental The nominal composition of the studied alloy is Mg−6%Zn−1.5%Cu−0.5%Zr (referred as ZCK620). The alloy was prepared by a permanent mold casting method at 993 K under a protective mixture of CO 2 and SF 6 atmosphere. The composition of the billet (100 mm in diameter and 200 mm in length) was measured using an inductively coupled plasma spectrometer, which was very close to the nominal value. The billet was homogenized at 673 K for 8 h. Cylindrical specimens (8 mm in diameter and 12 mm in length) were machined from the homogenized billet. Hot compressive tests were conducted with a Gleeble 3800 thermo-simulator at a temperature range of 523−673 K and interval of 50 K. Four typical strain rates were selected from 0.001 to 1 s −1 . The temperature was controlled and measured with a thermocouple welded to the mid-height of the sample. All specimens were heated at 10 K/s up to deformation temperature, held for 5 min to homogenize the temperature of the sample, and then deformed up to a true strain of 0.8. A three-layer feed-forward back- propagation ANN with Levenberg-Marquardt learning algorithm was used to predict the flow stress, and then compared with those calculated by the regression method. 3 Results and discussion 3.1 True stress—true train curves The true stress—true strain curves of the ZCK620 magnesium alloy obtained at different strain rates are shown in Fig. 1. Generally, the flow stress increases to a peak and then decreases to a steady state. For instance, the curve at strain rate of 1 s −1 shows a hardening stage, a softening stage, and a steady stage. The different stages come from the combined effects of work hardening (WH), dynamic recovery (DRV) as well as dynamic recrystallization (DRX) [14]. In addition, the flow stress increases as the deformation temperature decreases or as the strain rate increases. 3.2 Strain-dependent constitutive equation The relationship between the flow stress, the strain rate, and the temperature is commonly expressed as follows [14−17]: 1 1 n Aσε = & (1) )exp( 2 βσε A= & (2) Fig. 1 True stress—true strain curves of ZCK620 magnesium alloy at various temperatures with different strain rates: (a) ε& =0.001 s −1 ; (b) ε & =0.01 s −1 ; (c) ε & =0.1 s −1 ; (d) ε & =1 s −1 Hui YU, et al/Trans. Nonferrous Met. Soc. China 23(2013) 756−764 758 ⎟ ⎠⎞ ⎜ âŽ� ⎛ −= RT Q A n exp)][sinh(ασε & (3) where n 1 , A 1 , β, A 2 , A and α (α=β/n 1 ) are material constants. Equations (1) and (2) are usually applied to low stress and high stress, respectively. Equation (3) in the hyperbolic sine law is generally used to describe the flow stress and deformation activation behavior over a wide range of temperature and strain rate. Q is the activation energy for deformation; R is the mole gas constant (8.314 J/mol K); σ is the flow stress; T is the temperature. ZENER and HOLLOMON [18] proposed a flow stress model, where the relationship between temperature and strain rate is denoted by a parameter Z: n A RT Q Z )][sinh(exp ασε = ⎟ ⎠⎞ ⎜ âŽ� ⎛ = & (4) Considering the definition of the hyperbolic law, the flow stress can be expressed as the function of the Zener- Hollomon parameter: ⎪ ⎠⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎠⎞ ⎜ âŽ� ⎛ + ⎟ ⎠⎞ ⎜ âŽ� ⎛ = 2/1 /2/1 1ln 1 nn A Z A Z α σ (5) Due to similar behavior of peak stress and steady- state stress, the constitution equation was calculated using the peak stress (σ p ). The value of n 1 and β can be obtained from the slopes of the lnε &—lnσ p (Fig. 2(a)) and ln ε &—σ p (Fig. 2(b)) plots, respectively. The calculated mean values of n 1 , β, and α (=β/n 1 ) are 6.636, 0.0904 MPa −1 , and 0.0136 MPa −1 , respectively. For a given strain rate, the activation energy Q can be calculated as follows: () ε ασ ασ ε & & T RQ T /1 )]ln[sinh( )]ln[sinh( ln ∂ ∂ â‹… ∂ ∂ = (6) Therefore, by substituting the values of temperature and peak flow stress, the value of Q can be obtained from the slope of the plots of ln ε & vs ln[sinh(ασ)] (Fig. 2(c)) and ln[sinh(ασ)] vs 1/T (Fig. 2(d)), respectively. The calculated Q value is 176.8293 kJ/mol. From the natural logarithm used in Eq. (4), the linear-relationship between ln[sinh(ασ)] and ln Z was obtained, as shown in Fig. 3. From the above results, the peak stress can be expressed by substituting Eq. (5) with the following parameters: α=0.0136 MPa −1 , Z= ε & exp[176829.3/(RT)], A=2.0926×10 12 , and n=5.0692. Then, the material constants (α, n, Q, ln A) of the constitutive equations were computed under different deformation strains within the range of 0.05−0.8 at the interval of 0.05. The relationships between α, n, Q, ln A and true strain for ZCK620 magnesium alloy (Fig. 4) can Fig. 2 Relationships among peak stress, strain rate and temperature: (a) lnσ p —ln ε& ; (b) σ p —ln ε& ; (c) ln[sinh(ασ p )]—ln ε& ; (d) ln[sinh(ασ p )]—1/T Hui YU, et al/Trans. Nonferrous Met. Soc. China 23(2013) 756−764 759 Fig. 3 Relationship between flow stress and Zener−Hollomon parameter be fitted by the fifth-order polynomials. 5 5 4 4 3 3 2 210 εεεεεα AAAAAA +++++= (7) 5 5 4 4 3 3 2 210 εεεεε BBBBBBn +++++= (8) 5 5 4 4 3 3 2 210 εεεεε CCCCCCQ +++++= (9) 5 5 4 4 3 3 2 210 ln εεεεε DDDDDDA +++++= (10) The α, n, Q, and ln A can be measured through the least square method, and they are summarized in Table 1. Table 1 Results of α, n, Q, ln A of ZCK620 Mg alloy in polynomial fit Equation Parameter Eq. (7) A 0 =0.0491, A 1 =−0.373, A 2 =1.6693, A 3 =−3.6911, A 4 =3.9299 A 5 =−1.6022 Eq. (8) B 0 =5.3119, B 1 =−3.0863, B 2 =−4.5364, B 3 =56.2357, B 4 =−95.1206, B 5 =48.3322 Eq. (9) C 0 =250.017, C 1 =−860.9199, C 2 =3583.5383, C 3 =−7355.0188, C 4 =7325.7735, C 5 =−2838.0401 Eq. (10) D 0 =32.1887, D 1 =−53.8861, D 2 =199.0253, D 3 =−322.3007, D 4 =238.9525, D 5 =68.3331, In addition, four kinds of standard statistical performance evaluation methods were used to evaluate the accuracy of the developed constitutive equation, namely, the correlation coefficient (R), the average absolute relative error (E), the relative error (δ) and the standard deviation (S.D.). They are expressed as follows: ∑∑ ∑ == = −− −− = n i n i ii n i ii PPEE PPEE R 11 22 1 )()( ))(( (11) Fig. 4 Relationships between α (a), n (b), Q (c), ln A (d) and true strain by polynomial fit of ZCK620 magnesium alloy Hui YU, et al/Trans. Nonferrous Met. Soc. China 23(2013) 756−764 760 %100 1 1 × − = ∑ = n i i ii E PE n E                         (12) %100× ⎟ ⎟ ⎠⎞ ⎜ ⎜ âŽ� ⎛ − = i ii E PE δ (13) 1 )( S.D. 1 2 − − = ∑ = n XX n i i (14) where E i and P i are experimental and predicted values, respectively; E and P are the mean values of E i and P i , respectively; X and X are the original data and mean value of X, respectively; N is the total number employed in this study. As shown in Figs. 5(a) and (a′), the experimental and predicted results show a good agreement. 3.3 Artificial neural network model In the regression method, however, the response of the deformation behaviors of the materials under elevated temperatures and increased strain rates is highly nonlinear, and many factors affecting the flow stress are also nonlinear, which make the accuracy of the flow stress predicted by the regression methods lower and the applicable range limited [19]. In most cases, one hidden layer is found to be adequate, and this reaffirms the universal approximation theorem that a single layer of non-linear hidden units is sufficient to approximate any continuous function [20]. In this study, we determined the optimal number of neurons in the hidden layer by comparing the performance of the network, with 7−20 hidden neurons, and found that 8 neurons produced the greatest network performance. The schematic diagram of Fig. 5 Comparison between experimental and predicted flow stress of ZCK620 magnesium alloy using ANN: (a) Regression data; (b) Training data; (c) Testing data; (a′), (b′), (c′) Corresponding statistical analyses of relative error of (a), (b), (c), respectively Hui YU, et al/Trans. Nonferrous Met. Soc. China 23(2013) 756−764 761 the ANN structure for flow stress prediction of the ZCK620 alloy is shown in Fig. 6. The inputs of the network are temperature (T), strain rate ( ε & ), and strain (ε), respectively, and stress (σ) is the output. A total of 656 input-output data were selected from the true stress—true strain curves. The 256 points at true strains between 0.05 and 0.8 with interval of 0.05 were chosen to test the predictability of ANN model, while the remaining 400 points were used to train. Before the training of the network, both input and output variables were normalized with the range of 0−1 as follows: ⎟ ⎟ ⎠⎞ ⎜ ⎜ âŽ� ⎛ − − ×+= minmax min 8.01.0' XX XX X (15) where X′ is the normalized value of a certain parameter (T, ε, σ); X is the measured value for this parameter; X min and X max are the minimum and the maximum values for this parameter, respectively. Since ε & often changes by an order of magnitude, the following logarithm equation is adopted to normalize: ⎟ ⎟ ⎠⎞ ⎜ ⎜ âŽ� ⎛ − − ×+= minmax min lnln lnln 8.01.0' εε εε ε && && & (16) The results of the statistical analyses of the variables used to develop the ANN model for ZCK620 magnesium alloy are listed in Table 2. Fig. 6 Schematic illustration of artificial neural network Table 2 Statistical analysis of variables used to develop ANN model Va r i a b l e Minimum value Maximum value Average Standard deviation ε & /s −1 0.001 1 0.2848 0.4249 T/K 523 673 598.9909 56.0718 ε 0.0036 0.8009 0.3855 0.2334 σ/MPa 25.6378 179.214 91.7104 41.4517 After training and testing, the neural network performance was checked as shown in Figs. 5(b) and (c). The diagrams show an analysis of the network response in a form of linear regression analysis between the network outputs (predicted data) and the corresponding targets (experimental data). It is obvious that the predicted values from the trained neural network outputs track the targets very well. Additionally, we calculated the δ of neural network (Figs. 5(b′) and (c′)), and the δ showed a typical Gaussian distribution with the mean value at about zero. As shown in Fig. 7, the effects of deformation temperature and strain rate on the flow behaviors of ZCK620 magnesium alloy are simulated using the developed ANN model. Obviously, the predicted results agreed well with the experimental data. Both the deformation temperature and strain rate have pronounced influences on the flow stress. The flow stress decreases with an increase in deformation temperature and a decrease in strain rate, which can be explained by the terms of dynamic recrystallization and dislocation mechanism [19,21]. 3.4 Processing map In DMM, the work piece is considered to be a dissipater of power and the characteristics of power dissipation through microstructural changes are expressed in terms of an efficiency of power dissipation given by η=2m/(m+1), where m ( )ln(/)(ln εσ & ∂∂ ) is the strain rate sensitivity of flow stress. According to the criterion developed in Refs. [1,6, 13], flow instability will occur if 0 )ln( )]1/(ln[ )( <+ ⎠⎬ ⎫ ⎩ ⎨ ⎧ ∂ +∂ = m mm ε εξ & & (17) Figures 8 (a)−(d) present the processing maps of T4-treated ZCK620 magnesium alloy at different strains. The processing map is superimposed by the instability map on the power dissipation map, the contour represents constant efficiency (in percentage), and the gray areas indicate the regimes of flow instability. The strain levels of 0.1, 0.3, 0.5, and 0.8 correspond to the pre-peak strain, the peak strain, the post-peak strain, and the steady-state strain, respectively. It is obvious that the strain level has a significant effect on the processing maps. As the strain increases, the region with a low temperature and a low strain rate changes to a region with a low temperature and a medium strain rate, and a region with a medium-high temperature and a high strain rate. In the instable region, the efficiency of the power dissipation decreases rapidly, which should be avoided for hot working processes. In addition, the optimum parameters for subsequent processing have been suggested at the peak efficiency, where the dynamic Hui YU, et al/Trans. Nonferrous Met. Soc. China 23(2013) 756−764 762 Fig. 7 Comparisons between experimental and predicted flow stress of ZCK620 magnesium alloy using ANN at different strains: (a, b) effects of deformation temperature; (c, d) Effects of strain rate Fig. 8 Processing maps for ZCK620 magnesium alloy at different strains: (a) ε=0.1; (b) ε=0.3; (c) ε=0.5; (d) ε=0.8 Hui YU, et al/Trans. Nonferrous Met. Soc. China 23(2013) 756−764 763 recrystallization is operating to reduce the tendency for flow localization [13]. Thus, the optimized condition to conduct the hot working is Domain I (temperature range of 643−673 K and strain rate range of 10 −3 −10 −2 s −1 ) for ZCK620 magnesium alloy. 4 Conclusions 1) A strain-dependent constitutive equation and a feed-forward back-propagation artificial neural network are developed to predict the flow stress for the entire strain range. Good agreements between the experimentally measured values and the calculated ones of the flow stress are observed in both of the models. 2) A comparison of the results of predicting the flow stress using the regression and the ANN model shows that the latter provides better agreement with the experimentally measured data. 3) According strain-dependent constitutive analysis using the ANN model, processing maps of the ZCK620 alloy at different strain levels were constructed. The optimum processing conditions for hot working of the ZCK620 alloy are in the range of 643−673 K and 0.001− 0.01 s −1 . Acknowledgements This study was financially supported by the R&D Program of Korea Institute of Materials Science and the World Premier Materials Program funded by The Ministry of Knowledge Economy, Korea. One of the authors (Hui YU) is grateful for the financial support from China Scholarship Council (CSC) and Dr. Chao ZHENG for ANN discussion. References [1] WANG C Y, WANG X J, CHANG H, WU K, ZHENG M Y. Processing maps for hot working of ZK60 magnesium alloy [J]. Materials Science and Engineering A, 2007, 464(1−2): 52−58. [2] ZHU H M, SHA G, LIU J W, WU C L, LUO C P, LIU Z W, ZHENG P K, RINGER S P. Microstructure and mechanical properties of Mg−6Zn−xCu−0.6Zr (wt.%) alloys [J]. Journal of Alloys and Compounds, 2011, 509(8): 3526−3531. [3] BUHA J, OHKUBO T. Natural aging in Mg−Zn(−Cu) alloys [J]. Metallurgical and Materials Transactions A, 2008, 39(9): 2259−2273. [4] BUHA J. Mechanical properties of naturally aged Mg−Zn−Cu−Mn alloy [J]. Materials Science and Engineering A, 2008, 489(1−2): 127−137. [5] SUN M, WU G, WANG W, DING W. Effect of Zr on the microstructure, mechanical properties and corrosion resistance of Mg−10Gd−3Y magnesium alloy [J]. Materials Science and Engineering A, 2009, 523(1−2): 145−151. [6] PRASAD Y V R K. Processing maps: A status report [J]. 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Plastic deformation and dynamic recrystallization behaviors of Mg−5Gd− 4Y−0.5Zn−0.5Zr alloy [J]. Materials Science and Engineering A, 2008, 485(1−2): 487−491. [18] ZENER C, HOLLOMON J H. Effect of strain rate upon plastic flow of steel [J]. Journal of Applied Physics, 1944, 15: 22−32. [19] QIN Y J, PAN Q L, HE Y B, LI W B, LIU X Y, FAN X. Artificial neural network modeling to evaluate and predict the deformation behavior of ZK60 magnesium alloy during hot compression [J]. Materials and Manufacturing Processes, 2010, 25(7): 539−545. [20] MANDAL S, SIVAPRASAD P V, VENUGOPL S, MURTHY K P N, RAJ B. Artificial neural network modeling of composition-process- property correlations in austenitic stainless steels [J]. Materials Science and Engineering A, 2008, 485(1−2): 571−580. [21] LIN Y C, ZHANG J, ZHONG J. Application of neural networks to predict the elevated temperature flow behavior of a low alloy steel [J]. Computational Materials Science, 2008, 43(4): 752−758. Hui YU, et al/Trans. Nonferrous Met. Soc. China 23(2013) 756−764 764 Mg−Zn−Cu−Zr é•�å�ˆé‡‘çš„çƒå�˜å½¢è¡Œä¸ºå’ŒåР工图 ä½™ æ™– 1,2 , 于化顺 1 , Young-min KIM 2 , Bong-sun YOU 2 , 闵光辉 1 1. å±±ä¸œå¤§å¦ æ��料科å¦ä¸Žå·¥ç¨‹å¦é™¢ï¼Œæ��料液固结构演å�˜ä¸ŽåŠ å·¥æ•™è‚²éƒ¨é‡�点实验室,济å�— 250061ï¼› 2. Light Metals Group, Korea Institute of Materials Science, Changwon 642831, Korea 摘 è¦�:在温度 523~673 K,应å�˜çއ 0.001~1 s −1 æ�¡ä»¶ä¸‹ï¼Œä½¿ç”¨ Gleeble 3800 çƒæ¨¡æ‹Ÿæœºç ”究一ç§�新的四元 Mg−6Zn−1.5Cu−0.5Zrå�ˆé‡‘çš„å�˜å½¢è¡Œä¸ºã€‚结果表明,æµ�å�˜åº”力éš�ç�€å�˜å½¢æ¸©åº¦çš„å�‡é«˜æˆ–éš�ç�€åº”å�˜çŽ‡çš„ä¸‹é™�而å‡�å°�。 采 用ä¾�赖于应å�˜çš„æœ¬æž„方程和å‰�馈å��å�‘ä¼ æ’人工神ç»�网络æ�¥é¢„测æµ�å�˜åº”力,其结果与实验数æ�®å�»å�ˆå¾ˆå¥½ã€‚çƒåР工图 表明,对于ç»� T4 处ç�†çš„ Mg−6Zn−1.5Cu−0.5Zr å�ˆé‡‘çš„çƒåŠ å·¥ï¼Œå…¶æœ€ä½³å·¥ä½œæ�¡ä»¶ä¸ºæ¸©åº¦ 643~673 K,应å�˜é€Ÿçއ 0.001~0.01 s −1 。 关键è¯�:é•�å�ˆé‡‘ï¼›é“œæ·»åŠ ï¼›æµ�å�˜åº”力;å�˜å½¢è¡Œä¸ºï¼›æœ¬æž„方程;人工神ç»�ç½‘ç»œï¼›åŠ å·¥å›¾ (Edited by Xiang-qun LI) Engineering Computational Materials Science and Numerical ModellingHot deformation behavior and processing maps of Mg–Zn–Cu–Zr magnesium alloyThe deformation behaviors of a new quaternary Mg–6Zn–1.5Cu–0.5Zr alloy at temperatures of 523–673 K and strain rates of 0.001–1 s−1 were studied by compressive tests using a Gleeble 3800 thermal-simulator. The results show that the flow stress increases as the deformation temperature decreases or as the strain rate increases. A strain-dependent constitutive equation and a feed-forward back-propagation artificial neural network were used to predict flow stress, which showed good agreement with experimental data. The processing map suggests that the domains of 643–673 K and 0.001–0.01 s−1 are corresponded to optimum conditions for hot working of the T4-treated Mg–6Zn–1.5Cu–0.5Zr alloy.Fracture strength of silicon solar wafers with different surface texturesSurface texturing is widely used in modern photovoltaic converters to improve light capture and increase efficiency. This paper focuses on the influence of the surface texture on the mechanical strength of silicon wafers. Various existing test methods and approaches for measuring the mechanical strength of silicon wafers are discussed and reviewed. The surface morphology of silicon wafers with different textures is characterized by scanning electron microscopy (SEM). The mechanical strength of thin (<200 μm) silicon wafers with different surface textures is measured with a ring-on-ring bending test. A finite element method (FEM) is used to calculate the maximum stress. Weibull statistical theory is applied to the measured fracture stresses. The mean fracture strength, its standard deviation, the Weibull characteristic strength, and the Weibull modulus are obtained for silicon wafers with various surface textures. The fracture strength of silicon wafers is strongly affected by the surface texture. It is found that the fracture strength of silicon wafers with pyramidal textures increases with decreasing size of the pyramids. A new type of surface texture produced by a V2O5 selective oxidation exhibited fracture comparable or higher than that of the wafers with conventional pyramidal textures produced by alkaline etching.Photovoltaic (PV) technology is attracting increasing attention as a major source of renewable green energy. Nowadays, monocrystalline silicon is the most widely used PV material. The efficiency of heterojunction with intrinsic thin-layer (HIT) solar cells (SC) reaches 26% [] which is very close to the theoretical maximum of about 30%. As the ultimate efficiency of silicon SC has been almost reached, other factors such as cost, durability, reliability, and lifespan enter the foreground of photovoltaic research and development. The mainstream in solar cell design is to use thinner silicon substrates to increase the material utilization efficiency and reduce the production cost. For example, most of the modern HIT solar cells use thin (<200 μm) silicon wafers. Indeed, moving to thinner wafers may reduce the light absorption efficiency in the near-infrared spectral range. However, the efficiency drop is insignificant compared to the potential economic benefits of thin silicon. A decrease in the wafer thickness from 150 μm to 90 μm results in an efficiency loss of only 5% []. It should also be noted that the efficiency penalty due to incomplete absorption in thin wafers can be reduced or even eliminated with further advancement of light management schemes.Surface texturing is an efficient way to increase light capture in a broad spectral range. Most of the commercial solar cells employ silicon wafers with textured surfaces to minimize the light reflection from the front surface and maximize the light trapping. On the other hand, the use of thin textured silicon raises the issues of mechanical stability and strength. According to Köntges et al., cracking is one of the main reasons (∼20%) for failure of solar cells []. It is well known that the fracture strength of brittle materials depends on the surface finish. For example, chemically polished silicon wafers are substantially stronger compared to silicon wafers with rough surface after mechanical slicing or grinding []. Thus, the optimization of the texturing process must take into account both the optical properties of the produced surface finish and its impact on the fracture strength of the wafers.At present, there are two main techniques for slicing silicon ingots into thin wafers. The first one is loose abrasive sawing (LAS) where silicon blocks are cut by steel wires using abrasive slurry. The second one is a fixed abrasive sawing with a diamond wire (DWS). The use of diamond abrasive sawing leads to a change in the material removal mechanism and an increase in the strength of as-cut samples. The DWS method has become mainstream due to the high cutting speed, smaller kerf loss, environmental friendliness, and reduced wafer cost.The LAS process yields a homogeneous rough surface consisting of pits and bumps produced by free rolling abrasive particles with no indication of the sawing direction. In contrast, the DWS process produces long parallel grooves oriented in the direction of the wire sawing and randomly distributed pits. The depth of grooves and pits and their density for the DWS wafers is less than for the wafers sliced by the LAS method. This means better surface cleanliness, which makes it possible to reduce the loss of expensive silicon in the subsequent stages of the technological process. A significant advantage of DWS is the possibility of mass production of thin (<100 μm) silicon wafers. Another important benefit of the DWS process is that silicon can be cut in a ductile mode. This cutting mode results in a smoother surface and a sharp decrease in the depth and number of surface cracks and defects [Silicon wafers can be textured by a variety of methods such as anisotropic wet etching, plasma etching, photolithography etc. Etching in an alkaline solution is the most widely used technique to produce random upright pyramid textures []. Inverted pyramid textures also have attracted lots of research interest due to their superior light-absorbing ability and lower surface area. However, despite simplicity, cost-efficiency, and technical elegance, pyramidal texturing has a number of serious drawbacks. In particular, pyramidal structures do not produce ideal Lambertian surface scattering light randomly in all directions in a wide range of wavelengths []. Also, pyramidal patterns act as stress concentrators and reduce the mechanical strength of the wafers [Numerous alternative strategies for improving the light trapping in solar cells have been proposed. One of the promising approaches is the creation of spherical cavities embedded in the silicon layer. An important advantage of this approach is that voids of spherical shape are very small stress concentrators and do not reduce the fracture strength of silicon wafers. An array of such cavities can be used as light absorbers, scatterers, and back reflectors. According to theoretical studies, the scattering efficiency of spherical voids strongly increases with the size of the voids and saturates when the void diameter approximates the wavelength in the host medium []. On the other hand, smaller voids produce more isotropic scattering []. Thus, the optimum size of voids lies somewhere in between of 50 nm and 200 nm. Voids can be produced by a variety of methods such as anodization, stain-etching, bottom-up synthesis, and others. One of the novel approaches to produce voids in silicon is selective oxidation under a thin layer of V2O5 []. The V2O5 oxidation process yields a surface layer with numerous voids of spherical or elliptical shape which effectively scatter light in a wide spectral range. At the same time, the produced texture is free of sharp edges and notches acting as stress concentrators. This technique is easy to implement and does not require expensive equipment.The conventional approach to evaluate the mechanical strength of thin plates is to conduct uniaxial (three- or four-point bending of bar-shaped samples) [] or biaxial (ring-on-ring or ball-on-ring tests) flexural tests [However, stress analysis and calculation of the strength value for biaxially deformed thin plates are not that straightforward. A closed-form analytical solution for stress can be obtained in the framework of the linear theory of elasticity []. The analytical formula to obtain peak stress during the ring-on-ring test was originally introduced by Vitman and Pukh [where ν is Poisson's ratio, b and a are the radii of the support and the loading rings, respectively; c and h are the radius and thickness of the plate; F is the load, and σr and σθ are the radial and tangential stresses in the region inside the small ring, respectively. This equation is listed in the ASTM C 1499-19 and ISO 17167 standards and has been used by some authors to analyze the fracture strength of silicon wafers. was derived under the assumption of small loads and deflections. For example, the ASTM standard restricts the use of only for deflections less than 25% of the test piece thickness. In the case of thin wafers the deflections are comparable or larger than the test-piece thickness and the force-displacement and force-stress relationships are no longer linear. Therefore, in most cases, the linear theory of elasticity is unsuitable for the calculation of the real strength of silicon wafers and alternative approaches should be used. Up to now, there is no standard test method for the strength evaluation of thin silicon wafers. The number of studies on mechanical strength of thin textured silicon wafers is also very limited. Various methods to calculate stresses were considered in detail in [], stress distributions were simulated using a finite element method (FEM). Numerical simulation techniques such can be used to calculate the stress and strain fields in the wafer for any arbitrary loading force and test geometry.There are experimental studies that employed X-ray diffraction [] methods to measure the wafer curvature and stress distribution during a biaxial flexure test. X-ray diffraction is a useful technique to gain an insight into the real stress distribution during the biaxial flexure test. The strain distribution can be acquired without any sensors or reference marks simply by measuring the diffraction angle of an X-ray beam directed to various points of the surface. By measuring the change in the diffraction angle under different loads for crystal planes of different orientations, it is possible to reconstruct the full strain tensor and calculate the stress tensor components []. However, the experimental approaches are technically demanding and require a special design of test fixtures providing access of light or X-rays to the wafer region with the highest stresses. Also, it is difficult to change from one wafer size to another or change the diameters of the fixture rings.In this paper, we use ring-on-ring bending tests to study the fracture strength of silicon wafers with various surface finishes. The results are analyzed using a numerical simulation and Weibull statistics.The samples were prepared using commercial 156 × 156 mm pseudo-square Czochralski (CZ) single crystalline DWS silicon wafers produced by LONGi. The thickness of the damaged layer was about 3–4 μm. The wafers were (100) oriented with an as-cut surface. The thickness of the wafers ranged from 0.14 mm to 0.19 mm. The wafers were n-type doped with phosphorus to a resistivity of about 1.5 Ω cm. The oxygen concentration was approximately 8·1017 cm–3, the carbon concentration was ∼5·1016 cm–3, and the dislocation density was about 500 cm–2. The wafers were randomly divided into five groups and subjected to various treatments to produce different surface finishes.The specimens of Group 2 underwent the RCA cleaning followed by a saw damage etching (SDE) process using a 15% KOH solution at 80°C for 4 min. The SDE-treated group represents an example of silicon wafers with smooth surface finish and low number of critical defects.The specimens in Groups 3 and 4 were subjected to the RCA cleaning procedure followed by the surface damage etching. After that the specimens were etched in 2% KOH solution with commercial texturing additives at 80 °C for 12 min. The specimens of Group 3 were prepared using the CellTex Ultra M additive provided by ICB GmbH & Co. KG. The specimens of Group 4 were etched using the monoTex-F additive produced by RENA Technologies GmbH. With other conditions being the same, the etch recipe formulated with the CellTex Ultra additive yields pyramidal textures with the nominal size of 1.5–2.5 μm and the monoTex-F additive produces larger pyramids with the nominal size of 3–6 μm. Group 4 (large pyramids) represents the optimal pyramidal texture from the point of view of light trapping. Solar cells produced using this texturing process showed an efficiency of over 21%. Group 3 (small pyramids) represents a suboptimal pyramidal texture.Similar to the previous groups, the specimens in Group 5 were treated with RCA and SDE solutions. After that a fine structure was produced on one side of the wafers using the V2O5 assisted selective oxidation process. The texturing process involved the formation of a thin V2O5 layer on the wafer followed by oxidation of silicon wafer via a heterogeneous reaction between Si and V2O5 layer and dissolution of SiOх oxide in a 40% HF solution. A detailed description of the V2O5 assisted texturing process and optical properties of the produced textures can be found in []. The specimens with a fine surface texture produced by the V2O5 assisted process were included in the study as an example of a new type of surface texture that combines good optical performance with high mechanical strength.After preparation, the silicon wafers were laser cut into 30 × 30 mm squares and rinsed in deionized water. The surface morphology of the wafers was analyzed by scanning electron microscopy (SEM) using a JSM 7001F microscope (JEOL).Loading tests were performed using an Instron 1342 machine equipped with a ring-on-ring fixture (see ). Silicon wafers were placed on a support ring with a diameter of 18 mm. A biaxial flexure was induced by applying load via a loading ring with a diameter of 6 mm. The loading and support rings were coaxial to each other and to the test specimen. Samples of Groups 1 to 4 had the same surface finish on both sides and therefore were agnostic to orientation during the loading test. In contrast, Group 5 samples had different textures on the front and back surfaces. These samples were mounted in the set-up in such a way that the tensile stresses were located at the side with a fine texture produced by V2O5 assisted selective oxidation. The load was applied with a constant rate of 0.2 mm/min. During the experiment, the loading force was recorded as a function of the displacement of the rod of the testing machine and the maximum load at wafer fracture was recorded for analysis.A finite element method was employed to simulate the behavior of silicon wafers during the ring-on-ring biaxial fracture strength test. A three-dimensional numerical model was created and analyzed using the Comsol Multiphysics package. The loading and supporting rings were modeled as rigid bodies. Non-slip contact between the rings and the specimen was assumed. The specimen was considered to be isotropic and linear elastic. A mesh with rectangular parallelepiped elements was used. The optimal size of the elements was found to be about 0.2 of the plate thickness. Using smaller element size did not prove to be beneficial because it resulted in increased convergence time, while the calculated stress values were no more than 0.2% different. For comparison, the accuracy of stress measurements by XRD was only about 2%.The elastic modulus E and the Poisson ratio ν were calculated as mean values averaged over all directions in the (100) plane []. The model accounted only for elastic deformations; other effects such as fracture mechanics and change of the contact area during deformation were neglected. The developed FEM model can predict stress and deflection at any arbitrary point of the test plate and for any given applied force.The FEM model was validated against the experimental results which had been obtained in the previous works []. In those publications, round silicon wafers were chemically polished by a mixed solution of HNO3 and HF acids. After that, the wafers were subjected to biaxial deformation in the ring-on-ring test. Strain and stress distributions in the silicon wafers were measured by X-ray diffraction. We employed the developed FEM model to simulate the deformation of the silicon wafer under the same test conditions as described in the previous experimental works. The input parameters of the FEM model were the following: wafer thickness h = 0.22 mm, wafer diameter 2c = 30 mm, inner ring diameter 2a = 6 mm, outer ring diameter 2b = 18 mm, load F = 250 N. shows experimental and calculated distributions of radial σr and tangential σθ stress components along the radius in a silicon wafer under ring-on-ring test.Although there is a minor discrepancy between calculated and experimental results in the central part of the plate and near the support ring, the model yields a fairly accurate prediction for the maximum stress. Thus, the developed FEM model can be used to calculate maximum stress versus applied load. shows experimental (1) and calculated (3) dependencies of the radial stress versus load. The straight line (2) represents the dependence of the radial stress calculated by the analytical Equation One can see that experimental and calculated values of the maximum radial stress for the aforementioned test geometry are in fairly good agreement, while the analytical formula results in a grossly overestimated value of stress. Good agreement between the numerical simulation and experiment convincingly demonstrates the validity of the FEM model. However, experimental measurements of stress distribution by the XRD method are complicated and in some cases, for example for small plates, even impossible. A simpler approach to verification and validation of the FEM model used for stress calculations can be proposed. As it follows from theoretical and experimental data, the maximum radial stress in the plate at the moment of fracture always occurs under the small loading ring. The deflection of the plate at these points is equal to the displacement of the test machine head. Thus, the validity of the model can be assessed through the comparison of the experimentally measured displacement of the machine’s head and the calculated deflection of the plate points under the loading ring at the same value of the applied force. This approach is rather straightforward and can be applied to the test plates and rings of any arbitrary size. shows numerically calculated and experimentally measured deflection of the loading ring for silicon wafers with different surface textures. As one can see, experimentally measured displacements for the two wafers with different surface finish and the numerically simulated load-displacement curve are essentially the same. Good agreement between experimental and calculated values of displacement was observed for all tested wafers with different surface structures. Therefore we can conclude that the model yields trustworthy results for stress values. represents the dependence of displacement versus load calculated using analytical formulas. It can be seen that the analytical expression gives reasonably good agreement with experiment only for very small loads and displacements. Wafer fracture occurs at much higher loads and displacements where the analytical expression yields totally wrong results.Fracture of brittle materials is by nature a probabilistic process. The probability of failure and the scatter of the strength measurements can be quantified by the Weibull analysis which is customarily used to describe the fracture statistics of brittle materials []. The Weibull distribution has the following form:where σ is the strength of the specimen represented by the maximum stress occurring during testing of that specimen, P(σ) is the failure probability under the stress σ; σ0 and m are the Weibull parameters. The characteristic strength, σ0 is the tensile stress at which ∼63.2% of the samples fail. The Weibull modulus m describes the width of the strength distribution. The parameters of the Weibull representation of strength data are found by making a plot of ln{ln[1/(1–P)]} as a function of ln(σ). In this scaling equation appears as a straight line with a slope of m and an intercept of –mln(σ0).Representative SEM images of silicon wafers with various surface treatments are shown in a, the as-cut silicon wafer (Group 1) has a large portion of parallel smooth scratches generated by abrasive particles. The typical surface morphology of the SDE-treated wafer (Group 2) is shown in b. As one can see, SDE treatment removed most of the scratch marks and resulted in a macroscopic texture on both sides of the wafer with a characteristic size of 30–45 μm. The etch pattern is mainly composed of depressions with a distorted rectangular shape. The formation of such texture can be attributed to the anisotropic nature of alkaline etching and is described in detail in [ a and 6b show SEM images of samples after anisotropic wet etching in alkaline solution (Groups 3 and 4). The produced surface finish can be described as a superposition a random pyramidal texture on a larger scale (30–45 μm) pattern after SDE. The formation of such surface morphology is explained by the strong dependence of the etching rate of the crystal on the orientation of the crystallographic planes []. The characteristic size of pyramids is 1.5–2.5 μm and 3–6 μm for the wafers from Groups 3 and 4, respectively, reflecting the difference in the etching recipe. shows SEM images of a silicon wafer after the SDE treatment followed by the V2O5 fine texturing process (Group 5). The surface morphology is a nanostructure consisting of numerous concave voids with sizes from 100 to 400 nm superimposed on a typical microstructure with characteristic of size of 30–45 μm produced by SDE. Concave voids of spheroidal shape are separated by relatively smooth regions of 300–800 nm in size. Since the V2O5 treatment was performed only on one side of the wafer, the reverse side of these samples had a typical macroscopic texture formed after SDE. compares the dependence of the maximum stress on applied load calculated using equation and predicted by the numerical model. It can be seen that the analytical solution yields grossly overestimated values of stress and is practically unusable. The numerical model yield different values of tangential and radial stresses. According to FEM calculation, the tangential component of stress is always lower that the radial component. Therefore the strength of the wafers was taken as the maximum radial stress σR at the moment of fracture. shows the Weibull plots for silicon wafers with different surface textures. The mean value of strength, its standard deviation, the Weibull characteristic strength, and the Weibull modulus for each group of wafers are listed in As expected, due to the presence of a large number of scratches and micro-cracks on the surface, the as-cut wafers exhibited the lowest mean strength of 160 MPa. Other authors reported similar values of fracture strength for as-cut silicon wafers ranging from 100 MPa to 200 MPa []. It should be noted here that uniaxial bending tests used by other researchers are very sensitive to the surface anisotropy of the test pieces. Fracture strength in the direction parallel to the scratch marks can be up to twice as high as that in the perpendicular direction [SDE treated samples exhibited the mean strength value of 874 MPa which is more than fivefold higher than that of as-cut silicon wafers. The increase of strength is related to the removal of the surface layer containing the most critical defects. Also the obtained value of fracture strength is higher than most of the reported values for chemically polished silicon wafers which rarely exceed a few hundred of MPa. The Weibull modulus for this series of samples is larger indicating a narrower distribution of defects in the SDE-treated wafers.Wafers with large pyramidal texture showed the mean strength of 318 MPa. In contrast, wafers with small pyramids exhibited considerably higher strength of 479 MPa. The Weibull moduli are 4.9 and 15.1 for the textures with small and large pyramids, respectively. One can see a clear trend of Weibull modulus increasing with increasing mean size of pyramidal structure. This indicates that pyramids of a larger size form deeper and sharper notches with minimal variation in size and shape. On the one hand, these notches reduce the fracture strength of the wafer. On the other hand, they make the fracture more reproducible resulting in narrower distribution of the strength values. In contrast, small pyramids do not form deep notches and strong stress concentrators which results in increased fracture strength and lower Weibull modulus.A closer examination of the Weibull plot (b) for Group 4 (large pyramids) reveals that the distribution has a pronounced kink at which the slope of the fitting line changes abruptly indicating that the data cannot be adequately described with a single Weibull modulus shown in Wafers with the texture obtained by V2O5 selective oxidation revealed high fracture strength of 432 MPa which is only slightly less than that for the wafers with small pyramids. High fracture strength of this group of samples is to be expected because the V2O5 texturing process produces surface finish without notches and sharp edges.Although direct comparison is difficult, it seems that fracture strength values obtained in our experiments for all groups of samples are about the same or higher compared to those reported by other researches for silicon wafers with similar surface textures []. Higher fracture strength of our samples can be attributed to better optimization of the texturing process and the use of the ring-on-ring test method which is less sensitive to edge defects, and thus yields higher values of fracture strength.S.E. Nikitin: Methodology, Investigation, Writing - Original Draft. V.V. Shpeizman: Conceptualization, Formal analysis, Visualization, Writing - Original Draft. A.O. Pozdnyakov: Methodology, Investigation. S.I. Stepanov: Writing - Review & Editing. R.B. Timashov: Methodology, Investigation. V.I. Nikolaev: Conceptualization, Supervision. E.I. Terukov: Conceptualization, Resources. A.V. Bobyl: Conceptualization, Resources.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Analytical and numerical study of plastic overstrength of shear linkseccentricity of N about the centroidal link axissecond moment of area of the cross-sectionincrease of the first-order bending moment after shear yieldingincrease of the axial force after shear yieldingSteel eccentric bracing (EB) was generated from an idea which is dated back to the 1970s–1980s In EB systems, the use of short links is often preferred, because of the larger stiffness and ductility. Short links yield in shear and dissipate the earthquake input energy through cyclic plastic deformation, while developing some hardening. A good estimate of the level of hardening developing prior than buckling or fracture phenomena producing strength degradation is essential at the design stage for a reliable application of capacity design principles. Former tests carried out in the 1980s consistently showed that failure of shear links started as local shear buckling of panel zones at link ends, ultimately leading to fracture because of excessive local plastic deformation. The same tests consistently showed that the peak shear strength of short links is, in average, 1.4–1.5 times the yielding shear strength of the link web. Recent test results This paper presents a deeper and wider theoretical investigation of the plastic shear overstrength of short links made of typical European shapes (HE and IPE). In order to bound the possible response, different end restraint conditions are examined. The theoretical investigation is based on the results of finite element (FE) model simulations, but a simple analytical model is also presented for computing the plastic overstrength in such cases. The FE model has been verified by comparison with some experimental test results available from the literature.a small hollow rhombus is used for indicating shapes tested by Hjelmstad and Popov hollow triangles refer to the tests carried out by Kasai and Popov bold triangles refers to the cross-sections tested by Okazaki and Engelhardt a large hollow rhombus indicates shapes tested by McDaniel et al. a bold circle indicates shapes tested by Dusicka et al. a bold small rhombus refers to the test on the built-up shape presented in Mazzolani et al. a bold square indicates the European IPE shape tested by Stratan and Dubina It is noted that shapes common to two or more experimental campaigns are attributed just to one of them.The figure close to the acronym of each shape is the number of tests found available in the literature. For example, the string of characters “W8 × 10, 7” indicates that at least 7 tests have been carried out on W8 × 10 shapes. lines are also displayed, illustrating the ratio Af/Aw characterising European shapes IPE, HE A, HE B, HE M with depth d of the cross section in the range (100 mm, 600 mm). The range of link depths shown in is intended to cover both the case of shear links used as primary lateral load resisting elements in multi-storey frames (larger shapes) and the case of shear links used as hysteretic dampers (smaller shapes).One interesting observation, emphasised by A careful look at the experimental setup used in various experimental investigations allows a fundamental difference being discovered between some of the tests: the presence of a restraint to the link axial deformations. In the tests by McDaniel et al. Finite element models have been developed using ABAQUS ver. 6.10 The shell element type “S4R” has been used. It has four nodes with six degrees of freedom per node, 3 translational and 3 rotational. The geometry of each shell model corresponded to the centreline dimensions of a prototype link. Mesh refinement studies were conducted to determine the level of refinement necessary to accomplish the objectives.Steel yielding has been modelled by means of the von Mises yield criteria. Plastic hardening was represented using a nonlinear kinematic hardening law calibrated on the basis of the cyclic material properties derived from cyclic coupon tests performed by Kaufmann et al. Modelling of strength deterioration due to buckling has been taken into account by using the large displacements option. Previous studies . Nodes belonging to cross-sections at the ends of the link were slaved to reference points: RP-A is the master node at one end and RP-B is the master node at the other end. The link shear deformation was imposed by applying a displacement at RP-A in the 3-direction (transverse to the link axis).In order to reproduce accurately some test results, linear rotational springs were needed at link ends, as shown in a. However, for the parametric analysis subsequently described in this paper, perfect end restraints were assumed (b). The cases u1 |
= 0 (“with axial restraint”) and u1 |
≠ 0 (“without axial restraint”) were alternatively considered.The effects of residual stresses, geometric imperfections and fracture phenomena (low-cycle fatigue effects) have not been taken into account. This is deemed to be not a significant limitation for the main scope of this study, because of the following reasons:Residual stresses change the first yield point, but they do not affect the fully plastic response.Geometrical imperfections can be important if the buckling and post-buckling link response are of concern. But they generally have negligible influence on the response when buckling occurs after significant inelastic deformation has taken place. Besides, the scope of this study is to capture the link response prior to buckling (see also the discussion in Low-cycle fatigue should be considered if the aim of the numerical simulation is to capture failure due to fracture.The accuracy of modelling assumptions was verified by comparison of theoretical outcomes with experimental results obtained by Okazaki and Engelhardt Material properties, stiffeners and loading protocols in the simulations are those of the experimental tests. Boundary conditions have been simulated to be as close as possible to the experimental setup. Hence, flexural springs at both link ends have been included (a) to simulate additional flexibility from the boundary elements to which links were connected in the experimental setup.The results of numerical analyses show a good agreement between the experimental and simulated response, both in terms of collapse mechanism (). From comparison between numerical and experimental curves in a,b,c it can be easily recognised that the monotonic response curves underestimate only slightly the shear strength at a link rotation of 0.08 rad (see also the next section).The experimental evidence consistently indicates that shear links exhibit a stable response if web stiffeners are properly designed and the peak value of the link rotation (γ) does not exceed a limit value. Namely, recent test results The influence of web stiffeners on links made of European hot rolled shapes was numerically investigated for different boundary conditions. Indeed, owing to geometric non-linearity axial end restraints produce the development of a tension axial force, as far as the transverse relative displacement between the end sections is increased. This axial force might appreciably modify the shear buckling and post-buckling behaviour.Both monotonic and cyclic loading protocols were used in the numerical analyses. Two cyclic loading protocols were considered: the former AISC 2002 protocol and the Richards and Uang proposal Results obtained for the link more susceptible to shear buckling are presented hereinafter. Such a link is characterised by the maximum web slenderness λ¯w (Eurocode 3 Numerical analyses showed that unstiffened shear links made of European hot-rolled shapes can exhibit strength and stiffness degradation under cyclic loading even in the case of perfectly restrained axial deformations. clearly shows the strong effects of buckling on the link response. In particular, in case of IPE 600, web buckling occurred. It is noted that in case of a link with axial restraint the cyclic response curves are characterised by a positive slope even in the post-buckling range (b). This is due to the formation of an increasing tensile axial force generated by the second order effects for increasing transverse displacements.In case of stiffened links without or with axial restraint, numerical analyses have shown that hysteresis loops are stable in the range of ± 0.08 rad of plastic shear deformation (). It is noted that in both cases the difference between monotonic and cyclic envelope curves was minor to negligible.Therefore, similar to the case shown in , it is noted that in case of stiffened links, either without or with axial restraint, the strength developing at a link plastic rotation angle of 0.08 rad can reasonably be estimated by monotonic analysis. The possibility of using monotonic analysis is clearly advantageous because of the reduced computational cost. In light of these results, monotonic analyses are used in the following to evaluate the plastic shear overstrength.A parametric analysis has been carried out on shear links made of European hot-rolled shapes, namely HE A, HE B, HE M and IPE (with cross section depth ranging from 100 mm to 600 mm). The average stress–strain curve for the S 275 steel grade has been considered illustrates the link shear force (V) vs. rotation (γ) relationship obtained from the FE model analysis results for an ensemble of links with axial restraint. Results are relevant to HE A shapes with depth d of the cross section in the range (100 mm, 600 mm) and link lengths from 575 mm to 2112 mm.The axial force (N) developed by the selected ensemble of HE A shapes is illustrated in a and b, it can be noted that the axial force is very small for small link rotations (let say smaller than 0.01 rad). But, the axial force rapidly increases for larger rotations reaching an intensity which is comparable to that of the shear force. Hence, it is argued that axial forces and their effects are not negligible in the plastic range of response.Results similar to those illustrated here for HE A shapes can be obtained for other cross section types and link lengths. Further results from the parametric analysis are shown in the following sections and utilised to derive a simple analytical model to capture the link plastic overstrength.The link elastic stiffness might be analytically computed using basic concepts of beam elastic mechanics. The second-order axial force developing as a consequence of the shear deformation is considered negligible in the elastic range of response (Consequently, the elastic stiffness (ke=dVdγ) might be computed by means of Eq. shows the ratio between the theoretical stiffness (kth) corresponding to Eq. and the numerical value (knum) obtained from the results of FE model analyses. Results for two alternative definitions of shear area are illustrated: Av,1 |
= |
dtw and Av,2 |
= (d |
− |
tf)tw. It can be seen that Av,1 |
= |
dtw gives the best approximation. Similar results have been obtained for other cross section shapes and lengths. However, it has been found that the approximations worsen when the link length reduces. This is illustrated by b, where an HE A 100 shape and three different lengths are considered. This plot shows that the shear stiffness given by Eq. tends to underestimate the actual stiffness when very short links are considered. The correction of Eq. with the second order geometric stiffness due to the axial force has been checked to be negligible. Hence, the progressive deviation of the theoretical value from the numerical one must be attributed to the inapplicability of the beam theory for very small link lengths. Based on the numerical results, the approximation of a shear area equal to Av |
= |
dtw is considered acceptable (ratio kth/knum |
> 0.95) for link lengths larger than two times the cross section depth, i.e. e |
> 2d.Several definitions are available in the technical literature for the shear yielding strength: is used for the same purposes by Eurocode 8 gives the plastic shear strength of I-shaped beams according to Eurocode 3 might be significant. This is illustrated for HE A shapes in , assuming that the same definition of τy is used in all the equations. The ratios between the shear area linked to Eqs. are plotted versus the cross section depth (d) of shapes. It is apparent that Eq. would provide an estimate of the elastic limit shear strength, while Eq. should be taken as an estimate of the fully plastic shear strength. Eq. provided by Eurocode 3 gives appreciably larger values of the shear strength, allowing some plastic deformation and consequent strain hardening to be developed. As a general remark, it can be also observed that for all European shapes the difference of shear strength given by the various equations reduces when increasing the link cross section depth (a). In particular, the difference between Eq. A sample comparison between the yielding strengths given by Eqs. and the numerical shear force vs. rotation relationships obtained using the FE models is illustrated in b are relevant to a cross section shape HE A 100 and two different link lengths e1=1.6MpVy1 and e4=1.6MpVy4<e1, while the upper curve is relevant to a HE M 100 shape with length e=1.6MpVy1. For a given cross section shape (HE A 100 in b) the shear yielding strength is independent of the link length, as implied by any of the equations previously introduced. But the link stiffness, both elastic and post-yielding, increases with decreasing values of the link length. Consequently, the plastic shear overstrength increases when the link length is decreased. Besides, for a given link length, both the strength and the stiffness strongly depend on the cross section shape and size for a given cross section depth (compare the response of the HE A 100 and HE M 100 shapes with length e=1.6MpVy1). As shown by can give a yielding strength which intercepts the real (numerical) curve for a relatively large rotation (as in the case of HE M 100 shape), thus implying that some plastic deformation must develop before that strength is attained. Instead, Eq. always intercepts the curve on the elastic response branch, but with appreciable underestimation of the shear force corresponding to significant reduction of stiffness. Eq. obviously yields results intermediate between Eqs. , always staying quite close to the knee part of the curve. b shows that the ability of a given equation to predict the yield strength is not uniform, but depends upon the cross section geometry. Indeed, Eq. gives a very good prediction of the yield strength in case of a HE A 100 shape, while it significantly overestimates the yielding strength of an HE M 100 shape having the same normalised length. Therefore, it is impossible to select a unique definition of the yield strength working optimally in every case. If Eq. is adopted, then there will be some underestimation of strength in some cases, whether using Eq. overestimation is obtained in other cases.Previous comments are more exhaustively presented in , showing the normalised shear force (V/Vy) vs. link rotation (γ) relationship, considering a constant normalised link length eMp/Vy1=1.6. As can be observed, two alternative definitions of the yield value (Vy) are considered: Vy2 (Eq. c and d); and two types of cross section shape (IPE, b and d). For both IPE and HE M shapes the curves normalised by Vy2 are all coincident, but Vy2 appears to slightly underestimate the location of the knee part of the curve. A similar behaviour has been found for the other shapes (HE A and HE B) and smaller lengths. Using Vy3 instead of Vy2 the stiffness strongly reduces at a normalised shear force of about 1 and all curves are again almost coincident for IPE shapes, but this is not true for HE M shapes where appreciable scattering appears and overestimation of the knee point is obtained for shapes of smaller depth (e.g. HE M 100, which is the lowest curve in d). For HE A and HE B shapes, graphs can be obtained similar to those illustrated in for IPE shapes. The difference between the HE M shapes and the other types (IPE, HE A and HE B shapes) is the value of the ratio (d |
− |
tf)/d.In the following, the definition of the yield strength according to EN 1998-1 (2005) (Eq. , Vy2) will be used for the calculation of strength ratios.In consideration of the available experimental results (e.g Based on the results of finite element model analyses, the effects of the axial tension force developing as a consequence of the axial restraint are considered non-negligible in the plastic range of deformation. The post-yield stiffness exhibited by the shear links is affected by the geometrically non-linear effect of this axial force shows a free body diagram of link forces ΔN is the increase of the axial force after yielding (i.e. for V |
> |
Vy)x is the eccentricity of ΔN about the centroidal axis due to the corresponding increase of the first-order bending moment (ΔM=ΔVe2) acting on the cross section at the link end (namely x=ΔMΔN).It is noted that in case of no axial restraint, the axial force N is zero while the eccentricity x is infinite, and the product (ΔM) is finite. the link overstrength factor can be computed by means of Eq. Using results of FE model analyses, it can be shown that both the normalised increase of axial force Δn=ΔN2Nfy and its normalised location ξ=xd are function of the normalised link length ρ |
= |
e/(Mp/Vy). The product Δm |
= |
Δn |
⋅ |
ξ represents the normalised increase of bending moment. As previously noted, if the stiffness of the axial restraint approaches zero, then Δn approaches zero but ξ approaches infinite and Δm is still a finite quantity. requires knowledge of the product Δm rather than the individual factors (Δn and ξ), this product has been plotted in for links with axial restraint. Very interestingly, a shows that the variation of Δm is relatively small when links of length in the range 1.6MpVy4≤e≤1.6MpVy1 are considered. For example, at a link plastic rotation of 0.08 rad, the minimum value recorded for all the HE A shapes considered is 0.22 and the maximum value is 0.25. The average value is equal to 0.24 for link lengths equal to 1.6MpVy4 and 0.25 for link lengths equal to 1.6MpVy1, thus demonstrating that the product Δm is quite insensitive to the normalised link length.An attempt has been made to explore whether this property of an almost constant value of the product Δm remains valid outside of the investigated range of link lengths. To this end, a HE A 100 shape was considered, with length equal to 0.75×1.6MpVy4. a shows that, unfortunately, this property seems to be not valid for shorter links. In other words, the ratio Δm appears to be reducing with the normalised link length, at a given plastic rotation, but with an initially negligible variation when the length is close to the limit value 1.6MpVy1. Indeed, the stiffness given by Eq. tends to underestimate the stiffness coming from FE models as far as the link length is reduced, the approximation being acceptable when the length is e |
> 2d. Analogously, the yielding shear strength given by one of Eq. tends to be an overestimation of the yielding point coming from FE model results, as far as the link length reduces. However, the approximations related to the beam theory are acceptable when links are considered with length in the range 1.6MpVy4≤e≤1.6MpVy1, where the ratio Δm appears to be a rather stable quantity, whose variation may be neglected.In case of links with permitted axial deformations (without axial restraint), variations of Δm are relatively larger. Indeed, for a link plastic rotation of 0.08 rad, the minimum value recorded for all the considered HE A shapes is 0.16 and the maximum value is 0.22 (b). The average value is equal to 0.17 for link lengths equal to 1.6MpVy4 and 0.19 for link lengths equal to 1.6MpVy1.Therefore, for each link shape a constant value of Δm, for γp |
= 0.08 rad, is proposed to be assumed, with difference made between links with or without axial restraint. Thus Eq. with the coefficient β generally depending on (i) the stiffness of the axial restraints, (ii) the cross section shape, (iii) the material strain-hardening properties. for all European HE shapes and comparing the analytical predictions with the FE model analysis results previously summarised, for γp |
= 0.08 rad, values of β |
= 1.70 for links with axial restraint and β |
= 1.35 for links without axial restraint have been found effective. For IPE shapes smaller variations of Δm have been observed for links with or without axial restraint, leading to suggest a value of β |
= 1.70 for links with axial restraint and β |
= 1.60 for links without axial restraint. Proposed values of β are summarised in and coefficients β as given previously () are compared with FE analysis results in . Link lengths e1 |
= 1.6Mp/Vy1 and e4 |
= 1.6Mp/Vy4 are considered for each shape. It can be clearly noted that the influence of axial restraints is significant for shorter links, while becoming less important when link lengths approach the upper bound to short links (e1). could easily be exploited in order to derive a value of the coefficient β as a function of the plastic deformation demand γp. Hence, for any given value of the rotation capacity the maximum overstrength could also be calculated.Values of plastic shear overstrength experimentally measured during tests on short links having axial restraint have been collected from the available literature. Obviously, axial restraints in the experimental tests were characterised by finite yet large stiffness. Therefore, experimental values are compared with the range of the analytical predictions (with and without axial restraint), as shown in . It is worth noting that the link plastic shear strengths reported in references for the purposes of this comparison. Moreover, the comparison has been made using the shear force experienced by the tested links at 0.08 radians of plastic deformation. As shown in , experimental results are within the range of analytical model predictions.A theoretical study about the response of shear links has been carried out. The study is essentially based on FE model analysis, but comparison with available experimental test results has also been considered. Based on both the experimental evidence and the numerical results of FE model analysis, the following conclusions are drawn:It has been noted that tensile axial forces develop due to restraints to axial deformations and nonlinear geometric effects. These tensile forces acting in combination with the other two parameters may significantly modify the link shear overstrength.In usual cases, a shear overstrength equal to 1.5 has been confirmed by the theoretical investigation for a link plastic rotation of 0.08 rad. However, for very short links, with compact cross sections and perfect axial restraints, values of shear overstrength up to 2 have been obtained in the range of shapes and lengths investigated. For built up links with very compact shape and short length, even larger values could be obtained, especially if larger rotation capacity is achievable.Comparison of theoretical predictions and experimental test results indicates the ability of the proposed model to correctly capture the range of shear overstrength values corresponding to different degrees of axial restraint.Surface wrinkling of nanostructured thin films on a compliant substrateThe wrinkling of films on a soft substrate is a critical issue of many technologically important applications and thus has attracted considerable attentions. However, the effect of the surface roughness on the buckling mode of the film remains unclear. In the present paper, the buckling of a rough film, resting on a compliant substrate is theoretically investigated. Dimensional analysis and large-scale finite element computations are performed to explore the wrinkling behavior of a film with periodically triangular or sinusoidal nanostructured (rough) surface patterns. The dependence relationship of the wrinkling wavelength on the geometric parameters of surface nanostructures and the elastic properties of the film and substrate is established. Our study shows that the effects of various surface nanostructures on the film buckling can be well described by using the concept of the equivalent thickness. A relation between the equivalent thickness and the geometric parameters of the system are derived by fitting our computational results. The results reported here may be instructive for surface patterning and biomimetic design of novel materials and devices with specific surface properties. To demonstrate the potential application of the buckling method in the fabrication of hierarchical surface structures, we provide two examples inspired by the micro/nano-patterns on lotus leaves and mosquito legs.Surface wrinkling is a critical issue in a diversity of technologically significant applications. On one hand, various techniques have been established in the past decades to prevent from surface roughening or wrinkling in some fields. On the other hand, the wrinkling of a stiff film on a compliant substrate holds a great promise for applications in, for instance, thin-film metrology Theoretical efforts directed toward understanding the wrinkling behavior of thin layers resting on a soft substrate may date back to the early investigations in the context of structural sandwich panels All the above investigations are limited to the buckling of films with ideally smooth surfaces. However, many films in reality are coated with rough surfaces which occur naturally during synthesis. In addition, rough (nanostructured) surfaces are designed on the films to render some specific properties for the needs of applications. To our best knowledge, the buckling of rough films on a compliant substrate has not been systematically addressed as yet. Therefore, the present paper is aimed to investigate the effects of the surface roughness on the film wrinkling. Periodically distributed surface nanostructures are introduced to model the surface roughness of the film. The influences of the wavelength and amplitude of surface roughness on the buckling modes of the film are examined by using dimensional analysis and finite element computations in Sections , respectively. The computational results are analyzed and discussed in Section . As the applications of the obtained results, a hybrid fabrication approach is suggested to engender hierarchical surface structures. Finally, we provide two examples inspired by the micro/nano-patterns of lotus leaves and mosquito legs to illustrate the potential applications of the buckling of rough films.First, we apply the dimensional analysis method to the buckling problem of a thin-film bonded on a compliant substrate, as shown in . The film has periodically distributed triangular nanostructure, where λ0 and A0 are the wavelength and amplitude of the triangular surface patterns, respectively, h0 the thickness of the film without triangular patterns. Let H denote the equivalent film thickness with the effect of surface nanostructure being considered. Only plane strain problems are studied here. The dimensional analysis can provide not only insights into the dependence of the equivalent thickness on the geometric parameters of surface nanostructures and material properties, but also a guideline for the finite element computations described below.For a smooth film lying on a soft elastic substrate, the existing theoretical models give the expression of the buckling wavelength as where Ef∗=Ef/(1-νf2) and Es∗=Es/(1-νs2) are the plane strain moduli of the film and substrate, respectively, and h is the film thickness. Ef, νf, Es, and νs are the Young’s moduli and Poisson’s ratios of the film and substrate, respectively. The film with surface roughness in can be deemed as a smooth film with an equivalent thickness H. So by substituting h for H, Eq. which may be used to characterize the buckling wavelengths of surface nanostructured films. Analogously, the wrinkling amplitude A of the rough film can be written as A=Hεa/εc-1 according to the mechanics model The equivalent thickness H is defined in the sense of the equivalence of the wrinkling wavelength in the analysis. For the considered configuration in , H is a function of the following independent parameterswhich involves two dimensionally independent quantities. Without loss of generality, we take them as Es∗ and h0, and then the remaining variables are dimensionally dependent. Consequently, applying the π theorem of dimensional analysis to Eq. represents the dependence relation of the equivalent thickness H on the parameters λ0, A0, h0, Ef∗, and Es∗. However, this relation is implicit since the unknown dimensionless function Π1 is difficult to be determined analytically. The effects of the dimensionless parameters λ0/h0,A0/h0, and Ef∗/Es∗ on the equivalent thickness of the nanostructured film will be investigated by finite element analysis in the next section.A large number of finite element computations are performed to study the wrinkling behavior of surface nanostructured films lying on an elastic substrate by using the computational model in . In particular, we will determine the dimensionless function Π1 in Eq. Refer to a Cartesian coordinate system (o–xy). Both the ends of x |
= 0 and the opposite are applied an increasing displacement. The bottom surface is constrained in the y direction, and all the other boundaries are traction-free. The simulation is carried out by using the commercial finite-element software, ABAQUS (Version 6.8.1). We use the “tie” constraint in this software such that the nodes on the lower surface of the film have the same displacements as those on the top surface of the substrate. Eight-node biquadratic plane strain quadrilateral, reduced integration elements (CPE8R) are used for both the film and substrate. represents the mesh of computational model.Convergence analysis is carried out for each given values of λ0/h0,A0/h0, and Ef∗/Es∗, respectively. The influence of the substrate dimensions is carefully examined, which are chosen to be sufficiently large such that the solutions are insensitive either to the substrate sizes or the adopted boundary conditions.According to the computational results, the dependences of the dimensionless function Π1 on the corresponding dimensionless parameters are summarized as follows., the equivalent thickness H is a function of dimensionless parameters λ0/h0,A0/h0, and Ef∗/Es∗. The contribution of material properties on H is investigated in this subsection. In the computations, the Young’s modulus and Poisson’s ratio of the substrate are taken as Es |
= 10 MPa and νs |
= 0.45, respectively. The Poisson’s ratio of the film is set as νf |
= 0.3, and we take several representative Young’s moduli of the film, Ef= 20, 40, 60, 80, 100, 140, and 180 GPa. The value of λ0/h0 is taken as 10/3, and A0/h0 varies from 1/6 to 8/3. The equivalent thickness H for each value of Ef, λ0/h0, and A0/h0 is determined according to the equivalence of the buckling wavelength obtained by the finite element method and that for a smooth film, which means determining from Eq. presents the variation of the equivalent thickness H, normalized by the parameter h0, with respect to the modulus ratio Ef∗/Es∗. It is seen that H almost holds constant with the increase of modulus ratio Ef∗/Es∗. Therefore, the effects of the material properties on the equivalent thickness are negligible. It should be emphasized that this conclusion holds true under the condition that the buckling wavelength λ is much larger than the wavelength λ0 of the nanostructures (e.g., λ>10λ0 according to our computational results). In this case, Eq. which indicates that the equivalent thickness depends mainly on the geometric parameters and is basically independent of the material parameters when λ>10λ0. In the following discussions, the precondition λ>10λ0 always holds true throughout this paper.The effects of the geometric parameters λ0, A0, and h0 on dimensionless functions Π in Eq. is further analyzed via systematic finite element computations. The Young’s modulus and Poisson’s ratio of the substrate are set as Es |
= 10 MPa and νs |
= 0.45, respectively, and the Poisson’s ratio of the film is taken as νf |
= 0.3. To further illustrate the effectiveness of neglecting the effects of material properties on the equivalent thickness when λ>10λ0, the Young’s moduli of the films are taken as Ef= 20 GPa and Ef= 400 GPa, respectively. In the simulations, A0/h0 and λ0/h0 vary in a wide range and the detailed values are listed in . More than 70 examples of large-scale simulations are carried out.a and b present the variation of the normalized equivalent thickness H/h0 with respect to the normalized amplitude A0/h0 and wavelength λ0/h0 of the surface nanostructure of the film, respectively. The results indicate that the equivalent thickness obtained by the buckling wavelengths corresponding to Es= 20 GPa and Ef= 400 GPa, respectively, is almost the same (data not shown), which further proves that the effects of material properties on the equivalent thickness can be neglected when λ>10λ0. The averaged value H for Ef being 20 GPa and 400 GPa is taken as the equivalent thickness corresponding to each specified group values of A0/h0 and λ0/h0. It is seen that the effects of λ0/h0 and A0/h0 on the equivalent thickness can be neglected without introducing an evident error when the values of A0/h0 or λ0/h0 is small, (e.g., A0/h0 |
< 1/6 or λ0/h0<0.5). However, for large values of A0/h0 or λ0/h0, their influences should be taken into account, and the equivalent thickness increases with the increment of A0/h0 or λ0/h0 .To further facilitate the application of the obtained results, we give an explicit expression to correlate the equivalent thickness with the parameters A0/h0 and λ0/h0 by fitting the datum in Hh0=p0+p1A0h0+p2λ0h0+p3A0h02+p4A0λ0h02+p5λ0h02,where the values of the coefficients pi (i |
= 1 − 5) are listed in can fit well with the computational results, and the relative errors are smaller than 2.0%. Thus, one can easily use Eqs. to predict the buckling wavelength of a thin-film with rough surface structures in the given ranges of the parameters A0/h0 and λ0/h0 .Besides the triangular shape of surface nanostructure, we will discuss the effects of sinusoidal shaped roughness on the buckling of nanostructured films in order to address the effect of geometric morphology of surface structure on dimensionless function Π. We assume that the results in are applicable for the film with sinusoidal shaped structure. This hypothesis will be validated by using finite element analysis in the sequel.Several groups values of A0/h0 and λ0/h0 are simulated for sinusoidal shaped roughness and the corresponding value of H/h0 are given by the “stars” in . Obviously, the equivalent thickness of the film with sinusoidal surface structure is almost identical with that of triangular nanostructure under given values of A0/h0 and λ0/h0. The errors are no more than 5%, which is acceptable for most practical engineering applications and fabrications of hierarchical surface patterns. The results demonstrate that for sinusoidal and triangular shaped surface structures with identical wavelengths and amplitudes, the equivalent thickness can be well described by Eq. However, cautions should be taken for other shapes (e.g., rectangular) of surface structures though the relative error of this fitting expression is also expected to be acceptable.The above analysis about the buckling of rough films bonded to a compliant substrate may provide helpful information for thin-film metrology as suggested by Stafford et al. The approaches for generating complex micro- and nanopatterns on polymeric surfaces have been reviewed by del Campo and Arzt Besides the potential application in surface patterning, the buckling of rough film could also engender some special structures by imitating the organisms in nature. In fact, many biomaterials in plants and animals have hierarchical surface structures, which are of critical importance for their biological functions. The special microstructures endow the leaves, e.g. lotus, ladymantle, leymus arenarius, and colocasia esculenta Biomimetic design and fabrication of materials and devices with similar hierarchical surface structure as lotus leaves and mosquito legs have been reported in recent years a, which is quite similar to the surface structures of lotus leaves. Although the simulation is two dimensional, it can indeed give some instructive enlightenments for a three dimensional design. In the second example, we consider the buckling of a film with triangular nanostructures bonded to a cylindrical substrate. The obtained wrinkling pattern, as shown in b, resembles the morphology of mosquito legs. The two examples show that the buckling of surface nanostructured films may be effective and convenient for surface patterning and biomimetic fabrication.We have investigated the buckling of rough or nanostructured films resting on a compliant substrate by using dimensional analysis and finite element simulations. It is shown that the effects of surface nanostructures on the wrinkling behavior can be accounted for in terms of the equivalent thickness H, which depends mainly on the geometric features of surface structures. A fitting expression is provided to correlate the equivalent thickness H with the geometric parameters A0/h0 and λ0/h0. The results may be adopted as a guideline not only for the practical experiments to produce desired hierarchical surface patterns, but also for determining the elastic properties of films with rough surfaces. We further use two examples inspired by the micro/nano-patterns of the lotus leaves and mosquito legs to demonstrate the potential usefulness of the buckling of nanostructured films in biomimetic fabrication.Finally, it is worth mentioning that the investigation focuses on the computational studies, and we call for experiments to validate the effectiveness of fabricating hierarchical surface structures by using the buckling technique of nanostructured (rough) films lying on a compliant substrate.Effects of vacancies on the mechanical properties of zirconium: An ab initio investigationThe irradiation-induced defects strongly influence the mechanical behaviors of zirconium (Zr) and its alloys in nuclear reactors. In this work, we focus on how the vacancies change the mechanical properties of α-Zr through density functional theory (DFT) calculations. Both uniformly distributed vacancies and vacancy clusters were considered. And a wide range of vacancy concentrations from 0.005 to 0.063 (molar fraction) was modeled. The most stable configurations of di- and trivacancy clusters were predicted, which correspond to the most compact distribution of vacancies. Mechanical properties were explored in terms of single-crystal elastic constants, based on which the polycrystalline elastic moduli, Pugh's ratio for ductility and Vickers hardness were derived. Our results show that the existence of uniformly distributed vacancies can reduce the ductility, while enhance the hardness in general. However, when the vacancy concentration is larger than a critical value, a rise in the ductility and a reduction in the hardness occur, which may contribute to the degeneration of the material. Compared with the uniform distribution of vacancies, clustering of vacancies strengthens the above changes of ductility and hardness. Moreover, it was found that the anisotropy of Young's modulus decreases with increasing vacancy concentration.Zirconium (Zr) and its alloys are widely used as the fuel claddings in nuclear reactors owing to their low capture cross section to thermal neutron, good mechanical properties, and high corrosion resistance in water environment. While in service, they are exposed to a fast neutron flux, leading to collisions between the neutrons and the atoms. Then the displaced atoms collide with other atoms, thus causing a displacement cascade. This process will produce high concentration of point defects in the region of cascade, including vacancies and self-interstitials (SIAs). Driven by the bulk diffusion, vacancies and SIAs recombine or aggregate. After a long time evolution, the concentration of point defects will saturate at a very low level, while larger defects, such as the dislocation loops and cavities, will appear Recently, several calculations based on density functional theory (DFT) have been conducted to investigate the effect of vacancies on the mechanical properties of Zr. Zheng et al. The work of this manuscript was dedicated to a systematic investigation of the impact of vacancies on the mechanical properties of α-Zr through DFT calculations. To exclude the influence of alloying elements, pure Zr was used for the modeling. Both uniformly distributed vacancies and vacancy clusters were simulated for investigating the vacancy effect and clustering effect, respectively. Considering the fact that the concentration of vacancies can vary in a wide range during irradiation process and can be high in local regions containing vacancy clusters, a wide range of vacancy concentrations from 0.005 to 0.063 (molar fraction) was modeled. For vacancy clusters, only di- and trivacancies were considered in the present work. The configurations of these vacancy clusters were determined at first. Then the elastic constants, the mechanical stability, polycrystalline elastic moduli, Pugh's ratio for ductility, Vickers hardness, and elastic anisotropy were calculated. The variations of these mechanical properties with respect to the vacancy concentration were obtained and discussed.This paper is organized as follows. Details of the computational model and methods are presented in , the stability of di- and trivacancy clusters is investigated. Then the effects of vacancies on the mechanical properties are shown and discussed in . Finally, the paper ends with a short summary of the conclusions in Our DFT calculations were performed using the Vienna Ab-initio Simulation Package (VASP) The calculations of this paper were performed according to the following procedures. Firstly, the lattice constants and mechanical properties of perfect Zr were calculated and compared with the experimental values to check the reliability of the settings. Then, the same procedure was employed for Zr with vacancies of different concentrations and distributions. Both uniformly distributed vacancies and vacancy clusters were considered. For the vacancy clusters, the vacancy formation energy and binding energy were calculated in advance to determine the most stable configurations.The lattice constants of α-Zr were obtained by relaxing the primitive unit cell. As shown in , the calculated lattice constants are in good agreement with experiments and previous DFT calculations. The simulation boxes used for the following calculations were constructed based on the optimized primitive unit cell.To model the uniformly distributed vacancies, we used the simulation boxes with 2 × 2 × 2, 2 × 3 × 2, 3 × 3 × 2, 3 × 3 × 3, 4 × 4 × 2, 5 × 5 × 2, 5 × 5 × 4 unit cells containing one vacancy, respectively. The box is periodic in all three dimensions. Thus, the vacancy concentration ranges from 0.005 to 0.063. Here, the vacancy concentration was defined as the ratio of the number of vacancies to the total number of lattice sites, which is a non-dimensional quantity. We modeled di- and trivacancy clusters in the simulation boxes with 5 × 5 × 4 unit cells. Both the geometry of the simulation box and the positions of the atoms were optimized.Vacancy formation energy describes the energy required for the formation of vacancies and was calculated bywhere N is the total number of atoms in the perfect lattice, n is the number of vacancies, E(N |
− |
n) is the total energy of lattice with n vacancies, and E(N) is the total energy of the perfect lattice. To characterize the interaction between vacancies, we further calculated the binding energy of the vacancies, which is defined as the difference between the formation energy of the isolated vacancies and that of the clusterwhere E1f is monovacancy formation energy. Positive Enb indicates that the interaction between vacancies is attractive and the configuration of the cluster is stable.The elastic constants were calculated by the second derivative of the energy with respect to the strainwhere V0 is the equilibrium volume, E is the total energy and εi is the strain in Voigt notation. Due to the hexagonal symmetry, α-Zr has five independent elastic constants (C11, C12, C13, C33, C44). Therefore, we applied five independent deformations to the equilibrium lattice: (ε, ε, ε, 0, 0, 0), (0.5ε, 0.5ε, − |
ε, 0, 0, 0), (0, 0, ε, 0, 0, 0), (0, ε, 0, 0, 0, 0), and (0, 0, ε, 2ε, 0, 0). The values of ε were taken as − 0.01, − 0.005, 0, 0.005, 0.01, respectively, to obtain the energy-strain relationship. The atom positions of all distorted cells have been optimized. Based on these optimized cells, static calculations were performed to obtain the accurate energy. The above five deformations result in five independent energy-strain relationships, which were fitted to polynomial functions. From the second-order term of the polynomial functions, the elastic constants were extracted. The calculated elastic constants of α-Zr were summarized in , which are in good agreement with experiments and previous DFT calculations.Based on the elastic constants of single crystal, the isotropic elastic moduli of polycrystal can be calculated using the Voigt BV=2C11+C12+4C13+C339,GV=130C11+C12+2C33−4C13+12C44+12C66BR=C33C11+C12−2C132C11+C12+2C33−4C13,GR=52C44C66C33C11+C12−2C1323BVC44C66+C44+C66C33C11+C12−2C132,Through the bulk and shear moduli, the Young's modulus E and Possion's ratio ν can be obtained byAt first, the vacancy formation energy E1f was calculated through Eq. . By simulating one vacancy in supercells with different sizes (2 × 2 × 2, 2 × 3 × 2, 3 × 3 × 2, 3 × 3 × 3, 4 × 4 × 2, 5 × 5 × 2, 4 × 4 × 4, 5 × 5 × 4, 5 × 5 × 5), E1f as a function of the vacancy concentration was obtained as shown in . The obtained E1f lies in between 1.920 eV and 2.064 eV. This is consistent with experimental values of E1f |
≥ 1.5 eV by positron annihilation spectroscopy (PAS) Under neutron or ion irradiation, vacancy clusters can be formed from the migration and aggregation of vacancies. Therefore, understanding how the vacancies interact and determining the configurations of the vacancy clusters should be a prerequisite before exploring the effect of vacancies on the mechanical properties or the evolutions of the defect microstructures under radiation. In this manuscript, we only investigated the properties of di- and trivacancies, which are the basis for establishing larger vacancy clusters.To characterize the packing density of vacancies in the cluster, we defined a dimensionless number χ called compactness, which is the ratio of the lattice constant a to the average distance between the cluster center and the vacancies r:Larger χ indicates higher packing density.For divacancies, the configurations were chosen such that the distance d between two vacancies is less than the eighth nearest neighbors of Zr as shown in . Variations of the binding energy E2b and the compactness χ with respect to the configuration were displayed in , respectively. As the index of the configuration goes from a to g, χ decreases, which corresponds to the increase of d. With increasing d, E2b first decreases from being positive (configurations a and b) to negative (configurations from c to g), then increases towards zero. This indicates that the interaction between two vacancies is attractive only when the two vacancies are within the second nearest neighbors. The most stable configuration of divacancy lies along 1/622¯03 direction (configuration a), which corresponds to the largest χ. When d is larger than or equal to the third nearest neighbors, the two vacancies start to repel each other and the divacancy becomes unstable. The most unstable configuration of divacancy corresponds to the fourth nearest neighbors and lies along [0001] direction (configuration d). For divacancy with d larger than the fourth nearest neighbors, the repulsive interaction becomes weaker. Therefore, we can conclude that two neighboring vacancies tend to aggregate.From the above discussions, we can explain the concentration dependence of the vacancy formation energy shown in . The side lengths of our simulated supercells are all larger than 6.47 Å, which indicates that the distances between the vacancy and its periodic images are all larger than the fourth nearest neighbors of vacancies (5.19 Å). Therefore, the interactions between vacancies are all repulsive. With the increase of the vacancy concentration, the distance between vacancies becomes smaller, which results in the stronger repulsive interaction. This indicates that more energy would be required for the formation of vacancies, thus leading to larger vacancy formation energy.The stability of trivacancy was further investigated. Based on the above results, only the configurations which contain divacancy within the second nearest neighbors were chosen for calculations as shown in . All the calculated binding energies are positive, which indicates that the configurations of these trivacancies are all stable. The variation of the binding energy with respect to the configuration was plotted in (c), which is in positive correlation with the variation of the compactness χ ((d)). And the most stable trivacancy (configuration a) corresponds to the largest χ. Therefore, we can conclude that three neighboring vacancies tend to aggregate to form the most compact distributions.However, χ is not absolutely decreasing with the configuration, but accompanied with some oscillations. This indicates that compactness is not the only factor that influences the binding energy. Other factors, such as the local atomic arrangements and the stacking topology of vacancies, can also affect the binding energy. For example, the compactness of configurations b and c is equal, but their binding energies are different. This is attributed to their different local atomic arrangements, although they have the same stacking topology.Actually, the aggregation of di- and trivacancies has been verified by PAS experiments Comparison has been made between the binding energy of di- and trivacancy clusters obtained by us and that by Varvenne et al. as shown in . The general variation trend of the binding energy with respect to the cluster configuration is similar. However, the most stable configuration of trivacancy predicted by Varvenne et al. corresponds to configuration b in . And the values of the binding energy by Varvenne et al. are larger than those by us in general. We consider these differences mainly originate from different schemes used for the lattice relaxations and energy calculations. They kept the volume constant during relaxations and applied an elastic correction to calculate the energy, while we relaxed both the atomic positions and the simulation box, and calculated the energy without any corrections. In addition, the DFT package, pseudopotential, smearing width, and grid of k-points used by Varvenne et al. are all different from our uses, which may also contribute to the differences.Two factors that could influence the mechanical properties were considered. One is the distribution of vacancies, and the other is the vacancy concentration. For the distribution of vacancies, we modeled two limiting cases: the uniformly distributed vacancies and the most stable vacancy clusters predicted in . The concentration of the uniformly distributed vacancies was controlled by the size of the simulation box, while the concentration of the vacancy cluster was controlled by the number of constituent vacancies. The modeling details have been described in The calculated elastic constants of Zr with vacancies of different distributions and concentrations are summarized in . Through these elastic constants, the mechanical stability, polycrystalline elastic moduli, Pugh's ratio for ductility, Vickers hardness, and elastic anisotropy were further derived. In the following discussion, we first explore the variations of these mechanical properties with respect to the vacancy concentration for Zr with uniformly distributed vacancies, and then consider the effect of vacancy clustering on these properties.For hexagonal crystals, the Born elastic stability conditions All of the calculated elastic constants in satisfy the above conditions. This indicates that the existence of vacancies does not destroy the mechanical stability of Zr.In industrial applications, Zr is commonly prepared in polycrystalline form. To examine the vacancy effect on the elastic properties of polycrystalline Zr, the isotropic elastic moduli have been calculated by Eqs. . Calculations based on Hill approximations are used for the following discussions.The variations of the bulk, shear, Young's moduli and Possion's ratio with respect to the vacancy concentration cv are displayed in . With the increase of cv, the bulk modulus B decreases monotonically, which implies the reduction of polycrystalline Zr's resistance to uniform compression. However, the shear modulus G, Young's modulus E, and Possion's ratio ν do not change monotonically with cv. For cv |
≤ 0.028, G and E increase with cv, while ν decreases. These indicate that the existence of vacancies with low concentration could enhance polycrystalline Zr's deformation resistance to shear stress, uniaxial stress, and the transverse contraction under tensile deformation. When cv |
> 0.028, a reduction of G and E, and a rise of ν occur. These indicate that high concentration of vacancies can contribute to the degeneration of Zr.The above discussions were focused on the influence of vacancies on the elastic properties. However, irradiation damage not only leads to the elastic deformation, but also to the plastic deformation. Therefore, we further explored the vacancy effects on the ductility and hardness.Plastic deformation of metals is usually accompanied by the formation and evolution of dislocations. In the irradiated Zr, mainly dislocation loops with 〈a〉 Burgers vector were found, the diameter of which ranges from 5 to 20 nm In 1954, Pugh proposed a non-dimensional number called Pugh's ratio for characterizing the ductility of pure metals Positive correlation between k and elongation has been validated for fcc, hcp and bcc metals with high melting points. And Zr has been included in the range of validity.The material is ductile for k |
> 1.75, while brittle for k |
< 1.75. With a smaller k, the material will behave in a less ductile way. As shown in , all the k values are larger than 1.75. And k values for cv |
> 0 are all smaller than that for cv |
= 0. These indicate that although Zr with vacancies is still ductile, but should deform in a less ductile way compared with perfect Zr. The ductility does not decrease monotonically with the vacancy concentration. It decreases to a minimum when cv |
= 0.028, followed by which the ductility increases.The hardness can also be calculated based on B and G. Through the introduction of shear modulus G and Pugh's ratio k, Chen et al. where the units of HV and G are GPa. Chen's model has been validated for a wide range of crystalline materials and bulk metallic glasses. Recently, several investigations based on ab initio calculations have used this model for predicting the Vickers hardness (b), the Vickers hardness for cv |
> 0 is larger than that for cv |
= 0. This indicates the hardening of Zr due to the existence of vacancies. With increasing vacancy concentration, the hardness first increases, and then decreases., we have concluded that two and three neighboring vacancies tend to aggregate to form clusters. The red lines in show the effect of vacancy clusters on the mechanical properties. The general variation trend is similar to that for the uniformly distributed vacancies. However, the clustering of vacancies obviously enhances the changes of the mechanical properties. And this enhancement becomes more obvious with increasing vacancy concentration. This indicates that vacancy clustering may be a factor that exacerbates the changes of the mechanical properties under irradiation.Furthermore, the effect of vacancies on the elastic anisotropy of single crystal Zr was explored, which plays a key role in the microstructure evolutions of polycrystalline Zr . For the perfect lattice, Young's modulus along c direction is larger than that along a direction. With the increase of vacancy concentration, Young's modulus along a direction gradually increases, while that along c direction does not change much until cv |
= 0.019. This indicates a more isotropic distribution of Young's modulus. When cv is increased to 0.063, Young's moduli along a and c directions both decrease, and the energy distribution range becomes narrow. These phenomena indicate a further decrease in the anisotropy. Therefore, we can conclude that the introduction of vacancies will decrease the anisotropy of Young's modulus.Employing DFT calculations, the properties of vacancy clusters and the effect of vacancies on the mechanical properties of α-Zr have been systematically investigated. It was found that neighboring vacancies tend to aggregate to form vacancy clusters. And the most stable di- and trivacancy clusters correspond to the most compact configurations.In general, the introduction of vacancies will decrease the bulk modulus, but increase the shear and Young's moduli. For the vacancy concentration ranging from 0.005 to 0.063, the maximum changes of the bulk, shear, and Young's moduli are 9.52%, 25.53%, and 21.68%, respectively. With these moduli, the ductility via Pugh's ratio and Vickers hardness were further derived. The presence of vacancies contributes to a reduction in ductility, while a rise in hardness. With the continuous increase of the vacancy concentration, the bulk modulus monotonically decreases. However, the shear modulus, Young's modulus, and hardness first increase to a maximum value, and then decrease. Inversely, the ductility decreases to a minimum value followed by a slight increase. Compared with the uniform distribution of vacancies, the clustering of vacancies will enhance the above changes of elastic moduli, ductility and hardness. Moreover, the elastic anisotropy was explored. It was found that the distribution of Young's modulus becomes more isotropic due to the introduction of vacancies.In summary, our results have shown the effect of vacancies and their clustering behavior on the mechanical properties of Zr. We hope these results could give some useful knowledge for understanding the irradiation-induced changes of the mechanical properties.A double mixing process to greatly enhance thermal conductivity of graphene filled polyamide 6 compositesTremendous investigations have been recently conducted to increase the thermal conductivity (TC) of nanofiller filled thermoplastic composites. However, the agglomeration of nanofillers limits their loading and is a bottleneck to further increase the TC of composites. In this work, a preparation technique combining the benefits of both solution mixing and melt blending is proposed, with which high loading and good dispersion of graphene sheets (GSs) filled polyamide 6 (PA6) matrix composites are achieved. Noteworthily, the through-plane TC of the composites reaches 3.55 W m−1 K−1 with 20 wt% GSs, corresponding to a TC enhancement of 1167% compared to pure PA6. The excellent performance can be attributed to high loading and uniform dispersion of GSs, which facilitate the formation of thermally conductive network for efficient heat transfer. An effective medium theory model by taking the filler-to-matrix and filler-to-filler interfacial effects into consideration is also proposed to perceive the behind mechanism. The theoretical calculation agrees well with our experimental results. Moreover, we find the Young’s modulus of the composite increases significantly and the tensile strength nearly remains unchanged even at a GSs filler loading of 20 wt%. The results provide an effective and facile route for developing low-cost and highly conductive thermoplastic composites.Undesired heat accumulation problem originated from the miniaturization and integration of electronic devices has become more and more serious for long-term operation of various power and storage systems Recently, graphene-based composites have received increasing attention acting as heat dissipation materials due to the ultrahigh TC of ~5300 Wm−1 K−1 and excellent mechanical properties of graphene In this work, a facile fabrication process combining solution mixing with melt blending is proposed, called double mixing process. As for solution mixing, it is a method of dispersing polymer matrix and thermally conductive filler into a certain solvent, through ultrasonic, rapid mechanical stirring, etc., to get a uniformly dispersed solution. Compared with melt mixing method, the solution mixing method provides a low viscosity environment for the mixing of matrix and filler, thus facilitating the dispersion of filler. Through the double mixing process, the as-synthesized composites exhibit high TC and maintain good mechanical properties. It is believed that the solution mixing process gives rise to good particle dispersion as reported previously PA6 powders with an average size of ~400 μm were bought from Macklin Biochemical Co., Ltd, China. The density and melt flow rate of PA6 were 1.084 g/cm3 and 2.6 g/10 min, respectively. GSs were purchased by Deyang Carbonene Technology Co., Ltd., China.The frabrication process of GSs/PA6 composites is shown in . Firstly, PA6 powders were dissolved in formic acid by magnetic stirring, and a calculated amount of GSs were sonicated in formic acid with a power of 200 W for 30 min. Secondly, PA6 and GSs were mixed together under stirring for 2 h to get GSs/PA6 solution. The formic acid was removed in an oven at 100 °C for 8 h to obtain GSs/PA6 compound. It is worth mentioning that mechanical stirring was employed at the same time to avoid the formation of GSs/PA6 film during the volatilization of formic acid at the beginning time. The sample shape were in the chunk form with small size. Prior to the extrusion, the compound was thoroughly dried in a vacuum oven at 80 °C to remove water absorbed in PA6. The compound were melt-blending by twin screw extruder with a temperature around 235 °C for 5 min. Finally, the composites samples were moulded using a mini injection moulding machine (XSIM WZS10G) to prepare dumbbell-shaped specimens for mechnical tests and disks with 6.35 mm in radius for thermal diffusivity tests. It should be emphasized that in solution process, formic acid instead of ethanol or deionized water is used to dissolve the PA6 aiming to improve the dispersity of GSs and the interfacial adhesion between matrix and filler. In melt blending process, twin-screw extrusion plus injection molding further improve the filler dispersion and compactness. For comparison, PA6 composites filled with the distributed GSs were also prepared via a direct melt mixing method, hereinafter labeled as ‘single mixing’. Correspondingly, a process that incorporates both solution mixing and melt blending is referred to as ‘double mixing’.Wrinkles of GSs are clearly visible, as presented in (b)) further indicates that about nine layers of GSs are used. On the basis of AFM image of GSs ((c)), the average thickness and lateral size are 3–5 nm and 0.5–3 μm, respectively. Raman spectra of GSs is given in (d). The peaks appeared at around 1350, 1580 and 2650 cm−1 correspond to D peak (defect mode owing to disruption of planar configuration), G peak (E2g stretching mode of the graphitic crystalline structure), and 2D peak (signature of sp2 materials) of the carbonic materials, respectively It is well known that the dispersion of nanofillers in composites plays a key role in their performance. However, effective methods to determine the distribution of nanofillers inside the composites are scarce. Here, three different methods are adopted to give a clear description of GSs distribution. The first method is the traditional observation of the cryo-fractured surfaces of composite samples under SEM as given in . The surface of neat PA6 is smooth. Flakes of the GSs are well dispersed without preferred orientation and obviously protruding from the surfaces in PA6 matrix as presented in (b–e). As a comparison, the SEM images of GSs/PA6 composites prepared by single mixing are shown in ). The GSs tend to aggregate on the surface of fractured surfaces, which visually gives rise to a bigger size of GSs to a minor degree. However, due to the very close gray color of PA6 and GSs, it is not easy to clearly identify all the GSs by SEM images.Then Raman imaging technique is used to depicted the dispersion of GSs in the surface of matrix. The visualizations in (a) (4 wt% GSs composite) exhibits a minor part of intense bright color signals which is strongest and presents the existence of the GSs. When the loading of GSs increases to 20 wt%, the bright area increased greatly as shown in (b). The image results show a quite good dispersion of the GSs in GSs/PA6 composites. The image results can only show the surface information of the filler which might be influenced by the posture of the filler in the matrix, and further characterization is needed to analyze the dispersion of GSs in a more accurate way.Then micro-CT is employed in this work to obtain a 3D vision of the GSs dispersion in the matrix. High-resolution X-rays was used in micro-CT technology to create cross-sections of a physical object that is further employed to recreate a virtual model (3D model) and represent fine internal structure of object without destroying it. More intuitive images of filler dispersion were obtained as shown in . The gray value of the micro-CT is determined by the density of each component. Small white pieces represent GSs in this work, which is a component with relatively higher density. It can be seen from (a) that GSs are distributed in the gray PA6 matrix without preferential alignment and visible aggregation, proving the effectiveness of double-mixing process for good dispersion of GSs. Compared with the imaging result shown in (b), an obvious aggregation of GSs is observed in the matrix fabricated by single mixing, and thereby gives rise to a much bigger size of GSs.The TC and thermal diffusivity of studied composites as a function of GSs weight fraction are given in . In the case of double mixing, the TC increases obviously from 0.28 W m−1 K−1 of pure PA6 to 3.55 W m−1 K−1 of 20 wt% GSs/PA6 composite, presenting an enhancement of 1167%. It should be noted that the increase of TC with GSs loading shows an accelerated tendency. This may be ascribed to the good dispersion of GSs even at as high as 20 wt% loading, which forms denser networks of heat transportation. The density of GSs/PA6 composites with various loading in the case of double mixing and single mixing are measured, as shown in . It is found the double mixing could effectively increase the density of our prepared composites, which enabled the densification of a thermally conductive network in the composites. This trend seems to be different from other publications in which TC usually saturates when filler loading is high because of the worse agglomeration of fillers. However, a degraded behavior concerning the enhancement of TC and thermal diffusivity of the composites prepared by single mixing process was experimentally observed compared with double mixing process, as indicated in Here we also used TC enhancement efficiency η (TC enhancement per 1 wt% loading) to compare the efficiency of thermally conductive fillers. η is describled in the following form:where Kc and Km are the TC of composites and matrix, respectively, W is the mass fraction of filler. The η values as a function of filler content for PA matrix materials according to published reports are summarized in (c) illustrates the infrared thermal images of the samples captured during the heating process. All samples are put on the same heating stage and the temperature distribution is taken as a function of heating time. From the temperature change of the samples, the addition of GSs enhances the heat transport speed. 20 wt% GSs/PA6 composite presents the highest temperature during the same heating time. The time-temperature profiles during heating are described in (d). Specifically, the temperature of 20 wt% GSs/PA6 composite reaches 118 °C at 200 s, which is almost identical to the stage temperature. In contrast, the surface temperature of pure PA6 is only 95 °C, 23 °C lower than 20 wt% GSs/PA6 composites. This result shows that thermal transport ability of PA6 is significantly improved by introducing GSs via double mixing process.A comparison of out-of-plane TC among different fillers filled in thermoplastic composites is summarized in . It should be particularly pointed out that only thermoplastic composites are compared because the frabrication technique and properties of thermosetting composites are quite different. Although filler selection and processing methods have some effects on TC under the same filler loading, few previous works employed a double mixing process to further enhance the TC. In , composites with thermoplastic polymers as the matrix and with high thermally conductive carbon materials or BN as fillers are selected to be the comparable system. Due to the relatively low TC, BN filled polymer composites have lower TC. The hybrid filling of CNTs and graphite seems to be a more favorable way to enhance the TC due to the synergic effect between one dimensional CNTs and two dimensional graphite. Besides, melt-blending and melt-compounding methods are effective processing to get high TC. The high loading and good dispersion of fillers in matrix are also important to guarantee the high efficiency of TC enhancement. GSs can contribute more importantly to the increase of TC than graphite because graphene has larger TC than graphite. For example, Liang et alThe weight loss curves of pure PA6 and GSs/PA6 composites are obtained as shown in (a). It is observed that the addition of GSs improves the thermal stability of the composites. Taking T30% (temperature at 30% weight loss) for example, T30% of 20 wt% GSs/PA6 composite increases about 5 °C compared with pure PA6, and the change can be seen in the insert of (a) clearly. The decomposition of polymer at high temperature gives rise to the weight loss. Differential thermogravimetric (DTG) curves ((b)) show that the GSs/PA6 composites decomposed with an apparent one-step process, independent of GSs loading values. Moreover, the incorporation of GSs brings about an obvious increase in the temperature of the maximum weight loss rate compared with matrix. The GSs existing in the composites hinders the movement of the PA6 molecular chains, impedes the escape of small molecular decomposition products and then leads to the better thermal stability.The high TC of the composites can be rationalized from two major aspects as follows: (1) High filler loading. When the GSs loading is relatively low, the GSs in the matrix exists in an isolated condition just like a “sea-island” structure. In this case, the inter-particle distance and the phonon scattering in the composites increase considerably, which adds to the interfacial thermal resistanceThe modified effective-medium theory (EMT) Different methods are used to compare the experiment data, and the results are shown in . The green curve gives the calculation results based on the EMT without interfacial effect. The value reachs 90 Wm−1 K−1 when the loading of the fillers is 20 wt%. Obviously, the result without interfacial effect overestimats the effective TC compared to experimental results, which is shown in the black curve. When taking the interfacial effect (Kapitza resistance) into account, a much lower TC is obtained. When the mass fraction increases from 0 to 2, the enhancement in the effective TC is limited. This is mainly due to the strong thermal resistance at the GSs-PA6 interface, which greatly inhibits the heat transfer between the filler and matrix. On the basis of considering the interfacial effect of filler-to-matrix and filler-to-filler interface, the calculated results are depicted by the blue curve, which is consistent with the experimental results. As stated above, interfaces between matrix-to-filler and filler-to-filler have significant influence on the TC of the composites, which inevitably produces in experimental process. So, interfacial modification may be an effective method to improve the thermal conductance efficiency.(a) depicts the stress-strain curves of GSs/PA6 composites under room temperature, from which tensile strength, Young's modulus and elongation at break can be acquired as showed in (b–d). It is known that the modulus and the ratio components significantly influence the modulus of composite. According to tensile tests, the adding of GSs brings an outstanding increase in Young's modulus compared with PA6 matrix, at the meantime, large decrease of elongation at break. Particularly, it shows 271% enhancement in Young’s modulus for 20 wt% GSs composite compared with pure PA6. This indicates that restraining the movement of polymer molecular chains and bearing the force themselves greatly contribute to the reinforcement effect of rigid fillers on matrix. Unlike the modulus, the strength reinforcement of GSs behaves differently. At a lower loading of the filler, the addition of GSs can increase the tensile strength. However, when the loading exceeds 4 wt%, the tensile strength of the composites decreases gradually. This can be explained by the following reasons. Two mian factors that determine the strength of the composites is the interfacial adhesion between the PA6 matrix and the reinforcement phase (GSs) The tensile strength of GSs/PA6 composites is greater than that of pure PA6 when GSs loading is below 12 wt%. However, compared to pure PA6, it is remarked that the tensile strength decreases only 3.57% when GSs loading is as high as 20 wt%. This is almost negligible and the strength is higher than most of reported results in literature with the loading of 20 wt% A shortage of mechanical properties is the large decrease of elongation at break. This is a common problem encountered for rigid filler filled polymer composites. The solution to this problem may be using elastic coating on filler surface to extend the deformation, and thus to increase the elongation at break of polymer composites.In summary, we have proposed a processing technique that combines the benefits of both solution mixing and melt blending to obtain highly thermally conductive and mechanically robust graphene sheets (GSs)/polyamide 6 (PA6) composites. The dispersion of GSs is characterized by conventional SEM observation, Raman spectra image and X-ray micro CT as well. Experimental results show that the TC of the composite reaches 3.55 W m−1 K−1 with 20 wt% GSs, corresponding to a TC enhancement of 1167% compared to pure PA6, representing the highest TC enhancement efficiency for PA6 composites reported previously. In addition, the Young’s modulus of composite increases from 0.83 GPa to 2.25 GPa and the tensile strength remains almost unchanged even at a GSs filler loading of 20 wt%. Possible mechanisms are analyzed from the perspective of high content and uniform dispersion of GSs. These factors facilitate the formation of efficient heat transfer pathways and conductive networks. In addition, an effective medium theory model by taking both the filler-matrix and filler-filler interfacial effects into consideration is also provided, which gives an theoretical insight into the mechanism behind the TC enhancement. The GSs/PA6 composites are endowed with good comprehensive performance and show a promising prospect in heat management area. And we believe these results allow a broader investigation and offer an effective and facile route to further develop low-cost and high TC thermoplastic composites.The authors declared that there is no Conflict of Interest.Supplementary data to this article can be found online at The following are the Supplementary data to this article:B3–B1 phase transition and pressure dependence of elastic properties of ZnSWe have performed the ab initio calculations based on density functional theory to investigate the B3–B1 phase transition and mechanical properties of ZnS. The elastic stiffness coefficients, C11, C12, C44, bulk modulus, Kleinman parameter, Shear modulus, Reuss modulus, Voigt modulus and anisotropy factor are calculated for two polymorphs of ZnS: zincblende (B3) and rocksalt (B1). Our results for the structural parameters and elastic constants at equilibrium phase are in good agreement with the available theoretical and experimental values. Using the enthalpy–pressure data, we have observed the B3 to B1 structural phase transition at 18.5 GPa pressure. In addition to the elastic coefficients under normal conditions, we investigate the pressure dependence of mechanical properties of both phases: up to 65 GPa for B1-phase and 20 GPa for B3-phase.ZnS is a wide band gap semiconductor and an attractive material due to its use in various advanced technological applications in infrared optics, ultraviolet laser devices and electronic image displays, etc. Nanobelts, nanoclusters and various specimens in thin film form have been prepared for this material but the fabrication of large crystal of ZnS is a big problem in many studies due to critical growth condition. This semiconductor compound crystallizes in the cubic zincblende (B3) and wurtzite (B4) structures at ambient pressure. It is well known that there is a phase transition from B3 structure to B1 structure, when the pressure is applied. The pressure induced polymorphic structural transformation as well as unique electro-optical properties of ZnS and its nanostructures present researchers with several challenging problems to study. Over the past few years, several experimental and theoretical studies have been carried out to understand the mechanical and electronic properties of ZnS When the phase transition occurs, the mechanical properties of material change. The most common assessment of mechanical properties can be made by the determination of its elastic constants. Especially, the elastic constants of materials at high pressures are essential in order to predict and understand material response, strength, mechanical stability, and phase transition. However, only a few studies have conducted on the mechanical properties of ZnS at elevated pressures, as well as the pressure dependence of its elastic constants, so far. The accurate measurement of these quantities is a difficult task due to difficult experimental conditions at high pressure. However, ab initio quantum mechanical methods are quite suitable for a systematic study of the elastic properties at ambient or elevated pressure conditions.With the advances in ab initio methods, it has become possible to compute with a great accuracy the structural, mechanical, electronic and optical properties of materials. Therefore, a large number of ab initio studies on the structural and mechanical properties of ZnS have been carried out. Recently, Wang The rest of the paper is organized as follows. The method of calculation is introduced and the computational details are given in Section . We present and discuss the simulation results for structural and mechanical properties of ZnS in Section . Finally, the summary of our main results and conclusion are given in Section Total energy calculations have been carried out by using the Vienna ab initio simulation package (VASP) shows the corresponding structures of B3 and B1, respectively. A series of different values of lattice constant are set to calculate the total energy and corresponding primitive cell volumes for both B3 and B1 structures of ZnS.The calculated total energies and pressures as a function of the primitive cell volume for ZnS are used to determine the structural properties, phase behavior and isotropic mechanical response functions (bulk modulus and its pressure derivative) of ZnS by fitting the data to a third order Birch–Murnaghan equation of states (EOS). In the 3rd-order Birch–Murnaghan EOS, the total energy and pressure as a function of volume are given as E(V)=E0+9V0B016V0V2/3−13B′+V0V2/3−126−4V0V2/3P(V)=3B02V0V−V0V5/37/31+34(B′−4)V0V2/3−1In these expressions, E0 is the total energy, V0 is the equilibrium volume, B0 is the bulk modulus at P |
= 0 GPa, and B′ is the first derivative of the bulk modulus with respect to pressure.The second order elastic stiffness coefficients determine the first order contribution to the strain energy-content as the material is strained. Furthermore, these properties provide important criteria on the thermo-mechanical stability of the material in a given crystal structure. Here, the elastic stiffness constants at normal and under hydrostatic pressure conditions have been calculated to investigate the mechanical stability of ZnS in the B3 and B1 crystal structures. The cubic crystals have only three independent elastic constants, namely C11, C12 and C44. Therefore, we need only a set of three equations to calculate these three constants. This implies that only three types of strain must be applied to the crystal structure. The Equation of state directly yields the behavior under isotropic-volumetric strain and hence it can be used to determine the bulk modulus B of the crystals. For a cubic crystal, the bulk modulus is defined as B |
= (C11 |
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