Line and Point Defects in Nonlinear Anisotropic Solids
Ashkan Golgoon Arash Yavari
cond-mat.soft cond-mat.mtrl-sci

abbrvnat




1]Ashkan Golgoon
1,2]Arash YavariCorresponding author, e-mail: arash.yavari@ce.gatech.edu
[1]School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA
[2]The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA




Line and Point Defects in Nonlinear Anisotropic Solids
    [
     December 30, 2023
======================================================







In this paper, we present some analytical solutions for the stress fields of nonlinear anisotropic solids with distributed line and point defects. In particular, we determine the stress fields of i) a parallel cylindrically-symmetric distribution of screw dislocations in infinite orthotropic and monoclinic media, ii) a cylindrically-symmetric distribution of parallel wedge disclinations in an infinite orthotropic medium, iii) a distribution of edge dislocations in an orthotropic medium, and iv) a spherically-symmetric distribution of point defects in a transversely isotropic spherical ball.



	
  Keywords: Transversely isotropic solids; orthotropic solids; monoclinic solids; defects; disclinations; dislocations; nonlinear elasticity.
	
  Mathematics Subject Classification 74B20 · 70G45 · 74E10 · 15A72 · 74Fxx









§ INTRODUCTION

 
In anelasticity, any measure of strain has both an elastic and a non-elastic part. Given a pair of thermodynamically conjugate stress and strain, locally a non-vanishing strain does not necessarily correspond to a non-vanishing stress. Elastic strain refers to the part of strain that is locally related to the corresponding stress. The remaining part is referred to as eigenstrain, a term that was first used by <cit.>. Defects are one source of anelasticity. Vito Volterra, in his seminal work <cit.>, pioneered the mathematical study of defects many years before the first experimental observations of defects in solids. He classified line defects into six types, three of which are now called dislocations or translational defects, and the other three are called disclinations or rotational defects. 
<cit.> and <cit.> independently explored the profound connections between the mechanics of defects and non-Riemannian geometries in the 1950s. <cit.> discovered that the reference configuration of a solid is not necessarily Euclidean in the presence of defects. He realized that the curvature and the torsion of the reference manifold are measures of incompatibility and the density of dislocations, respectively. 
Defects due to plastic deformations naturally occur in most of the known problems in mechanics and tribology, e.g., contact mechanics <cit.>, mechanical impact <cit.>, and dislocation-boundary interactions <cit.>.
Other examples of anelastic sources include swelling and cavitation <cit.>, bulk and surface growth <cit.>, thermal strains <cit.>, and the presence of inclusions and inhomogeneities <cit.>. There have been some theoretical investigations on the effects of eigenstrains in linear anisotropic media, e.g., <cit.>, and references therein.

Very little is known about the effects of material anisotropies on the stress field and energetics of defects in solids.
The dynamical response of uniformly moving dislocations in linear anisotropic media was studied by <cit.>. It was observed that both edge and screw dislocations are prone to exhibiting anomalous dynamical behavior such that the interaction force between two parallel dislocations (on the same slip plane) changes sign when dislocation velocity increases. <cit.> predicted instabilities of dislocations in some anisotropic metallic crystals. It was found that a straight dislocation may decrease its energy if it changes to a zig-zag shape, i.e., a straight dislocation may be unstable.
In the setting of the linear theory of elasticity, <cit.> analyzed dislocations in anisotropic media (see also <cit.>). Particularly, the displacement fields of infinite straight dislocations and plane curvilinear dislocation loops were obtained. <cit.> investigated edge dislocations with an infinite straight axis in linear anisotropic solids. He extended Nabarro's calculation of the width of a dislocation to the anisotropic case. His results are limited to edge dislocations with an axis that is an infinite straight line, but there is no restriction on the type of anisotropy of the medium. 
<cit.> investigated the elastic interaction of point defects in linear isotropic and anisotropic cubic media using Green's function approach. They specifically discussed the differences between the interactions in isotropic and anisotropic materials and the effects of anisotropy on the interaction potential. Some basic developments in the linear theory of dislocations in anisotropic media was given in <cit.>. Methods for obtaining the induced linear elastic fields of defects in transversely isotropic bimaterials and orthotropic bicrystals (in 2D) were proposed in <cit.> and <cit.>, respectively. In particular, some closed-form solutions for inclusions and dislocation lines were presented.

A successive-approximation method was proposed in <cit.> to study the nonlinear screw dislocation problem using the linear elasticity solution. Nonetheless, the method fails to find the correct solution near the dislocation axis. Only a handful of exact solutions for defects in nonlinear elastic solids exist in the literature, and they are all restricted to isotropic materials. We should mention <cit.> for dislocations, <cit.> for disclinations, and <cit.> for point defects and discombinations. 



To the best of our knowledge, despite the known importance of the anisotropic behavior of solids, especially at finite strains, the study of defects in the setting of nonlinear elasticity has been limited to isotropic solids. In this paper we study several examples of line and point defects in nonlinear anisotropic solids and present some analytical solutions for their stress fields. We consider an arbitrary cylindrically-symmetric distribution of parallel screw dislocations in orthotropic and monoclinic media, along with a parallel cylindrically-symmetric distribution of wedge disclinations in an infinite orthotropic medium. As the geometry of the material manifold explicitly depends on the distribution of defects, the material preferred directions (that identify the type of anisotropy) in the reference configuration explicitly depend on the defect distribution as well, and, in general, are different from those of the material in its current configuration. For instance, for the distributed  screw dislocations that we consider, the assumption that the dislocated body is orthotropic in the reference (current) configuration implies that the body is monoclinic in the current (reference) configuration. 

The boundedness of the stress components on the dislocation and disclination axes will be discussed. In particular, for an arbitrary cylindrically-symmetric distribution of parallel screw dislocations the stress exhibits a logarithmic singularity on the dislocation axis unless the axial deformation is suppressed. Note that these singularities arise due to the anisotropic effects (e.g., radial fiber-reinforcement), and, in particular, do not occur when the material is isotropic. Exploiting the so-called standard reinforcing model (see, e.g., <cit.>), we obtain conditions under which the energy per unit length and the resultant longitudinal force of a single screw dislocation for a fiber-reinforced material are finite provided that the isotropic base material has a finite axial force and a finite energy per unit length. Employing Cartan's moving frames approach, for a given distribution of edge dislocations we will construct the material manifold and obtain explicit solutions for the stress field when the medium is orthotropic. We will also consider a spherically-symmetric distribution of point defects in a finite transversely isotropic spherical ball. We will show that for an arbitrary incompressible transversely isotropic material with the radial material preferred direction a uniform point defect distribution induces a uniform hydrostatic stress inside the region the distribution is supported.

The rest of the paper is structured as follows. In <ref> we tersely review some fundamentals of geometric nonlinear anisotropic elasticity and some related topics on nonlinear defect mechanics. We consider a cylindrically-symmetric distribution of parallel screw dislocations in orthotropic and monoclinic media in <ref> and <ref>, respectively. A cylindrically-symmetric distribution of parallel wedge disclinations in an orthotropic medium is studied in <ref>. In <ref> edge dislocations in an orthotropic medium are considered. In <ref> we calculate the residual stresses due to a spherically-symmetric distribution of point defects in a transversely isotropic ball. We end the paper with some remarks in <ref>.






§ GEOMETRIC ANELASTICITY FOR ANISOTROPIC SOLIDS



In this section we briefly review some fundamental elements of the geometric theory of nonlinear elasticity for anisotropic solids. For more detailed discussions, see <cit.>.



  
Kinematics.A body ℬ is identified with a Riemannian manifold (ℬ,𝐆), and a configuration of ℬ is a smooth embedding φ:ℬ→𝒮, where (𝒮,𝐠) is a Riemannian manifold —the ambient space. An affine connection ∇ on a smooth manifold ℬ is a linear mapping ∇:𝒳(ℬ)×𝒳(ℬ)→𝒳(ℬ), where 𝒳(ℬ) represents the set of all smooth vector fields on ℬ, such that the following properties are satisfied  ∀ 𝐗,𝐘,𝐗_1,𝐗_2,𝐘_1,𝐘_2∈𝒳(ℬ),∀ f,f_1,f_2∈ C^∞(ℬ),∀ a_1,a_2∈ℝ (see <cit.> for more details): a) ∇_f_1𝐗_1+f_2𝐗_2𝐘=f_1∇_𝐗_1𝐘+f_2∇_𝐗_2𝐘, b) ∇_𝐗(a_1𝐘_1+a_2𝐘_2)=a_1∇_𝐗(𝐘_1)+a_2∇_𝐗(𝐘_2), c) ∇_𝐗(f𝐘)=f∇_𝐗𝐘+(𝐗f)𝐘.
It can be shown that there is a unique torsion-free and compatible affine connection associated with any Riemannian manifold that is called a Riemannian connection. Let us denote the Levi-Civita connection associated with the Riemannian manifolds (ℬ,𝐆) and (𝒮,𝐠) by ∇^𝐆 and ∇^𝐠, respectively. We denote the set of all configurations of ℬ by 𝒞. A motion is a curve c:ℝ^+→φ_t∈𝒞 such that φ_t assigns a spatial point x=φ_t(X)=φ(X,t)∈𝒮 to every material point X∈ℬ at any time t. The body is assumed to be   stress-free in its reference configuration, which may have a nontrivial geometry, in general, e.g., in the presence of eigenstrains. The deformation gradient 𝐅 is the tangent map of φ defined as 𝐅(X,t)=dφ_t(X): T_Xℬ→ T_φ_t(X)𝒮.
The adjoint of 𝐅 is defined as 𝐅^𝖳(X,t):T_φ_t(X)𝒮→ T_Xℬ, 𝐠(𝐅𝐕,v)=𝐆(𝐕,𝐅^𝖳v), ∀ 𝐕∈ T_Xℬ, v∈ T_φ_t(X)𝒮.
The right Cauchy-Green deformation tensor is defined as
𝐂(X,t)=𝐅^𝖳(X,t)𝐅(X,t):T_Xℬ→ T_Xℬ. The Finger deformation tensor is defined as 𝐛(x,t)=𝐅(X,t)𝐅^𝖳(X,t):T_xφ(ℬ)→ T_xφ(ℬ), in components, b^ab=F^a_AF^b_BG^AB. Another measure of strain is the Lagrangian strain tensor given as 𝐄=1/2(φ^*_t𝐠-𝐆). The Jacobian of deformation J relates the Riemannian volume element of the material manifold dV(X,𝐆) to that of the spatial manifold dv(φ_t(X),𝐠), written as


    J=√(det 𝐠/det 𝐆) det 𝐅 ,    dv(x,𝐠)=J dV(X,𝐆) .





  
Equilibrium Equations.The localized balance of linear momentum in spatial and material forms are written as


    divσ+ρ𝐛=ρ𝐚,   Div𝐏+ρ_0𝐁=ρ_0𝐀,


where σ and 𝐏 are the Cauchy stress and the first Piola-Kirchhoff stress, respectively. Note that the material and spatial divergence operators in components are given as  


    (divσ)^a=   σ^ab_|b=∂σ^ab/∂ x^b+σ^acγ^b_cb +σ^cbγ^a_cb ,
    (Div𝐏)^a=   P^aA_|A=∂ P^aA/∂ X^A+P^aBΓ^A_AB +P^cAF^b_Aγ^a_bc ,


where γ^a_bc and Γ^A_BC denote the Christoffel symbols of the connections ∇^𝐠 and ∇^𝐆, respectively. Note that in the local coordinate charts {x^a} and {X^A}, one has ∇^𝐠_ ∂_b∂_c=γ^a_bc∂_a and ∇^𝐆_ ∂_B∂_C=Γ^A_BC∂_A, respectively.




  
Constitutive Equations. In this paper our calculations are restricted to incompressible transversely isotropic, orthotropic, and monoclinic solids. To establish a materially covariant strain energy density function, structural tensors corresponding to the symmetry group of the material are used. For detailed discussions on structural tensors and the determination of the integrity basis and the corresponding invariants of a set of tensors, see <cit.>. 




  
Transverse Isotropy. Let us assume a compressible transversely isotropic material such that the unit vector 𝐍(X) identifies the material preferred direction at a point X in the reference configuration. The strain energy density per unit volume of the reference configuration is given as (see, e.g., <cit.>) W=W(X,𝐆,𝐂^♭, 𝐀), where 𝐀=𝐍⊗𝐍 is a structural tensor representing the transverse isotropy of the material symmetry group. The second Piola-Kirchhoff stress tensor is given by 


    𝐒=2∂ W/∂𝐂^♭ .


The energy function W depends on the following five independent invariants defined as 


    I_1=tr 𝐂 ,   I_2=det 𝐂 tr 𝐂^-1 ,   I_3=det 𝐂 ,   I_4=𝐍·𝐂·𝐍 ,   I_5=𝐍·𝐂^2·𝐍 .


In components they read


    I_1=C^A_A ,  I_2=det(C^A_B)(C^-1)^D_D ,  I_3=det(C^A_B) ,  I_4=N^AN^BC_AB ,   I_5=N^AN^BC_BQC^Q_A .


Using (<ref>), one obtains[For the sake of brevity, we do not assume an explicit dependence of W on X, which in the case of inhomogeneous bodies is needed. We suppose instead that the material is piece-wise homogeneous and model an inhomogeneity using different energy functions in different regions of the body.]


    𝐒=∑_n=1^52W_I_n∂ I_n/∂𝐂^♭ ,     W_I_n
    	:=∂ W/∂ I_n ,    n=1,…,5 .


Note that


    ∂ I_1/∂𝐂^♭=𝐆^♯ ,  ∂ I_2/∂𝐂^♭=I_2𝐂^-1-I_3𝐂^-2 ,  ∂ I_3/∂𝐂^♭=I_3𝐂^-1 ,  ∂ I_4/∂𝐂^♭=𝐍⊗𝐍 ,   ∂ I_5/∂𝐂^♭=𝐍⊗𝐂·𝐍+𝐍·𝐂⊗𝐍 .


Thus, from (<ref>) and (<ref>), one obtains the following representation for the second Piola-Kirchhoff stress tensor


    𝐒=2{W_I_1𝐆^♯+W_I_2(I_2𝐂^-1-I_3𝐂^-2)+W_I_3I_3𝐂^-1+W_I_4(𝐍⊗𝐍)+W_I_5(𝐍⊗𝐂·𝐍+𝐍·𝐂⊗𝐍)} .
 

If the material is incompressible, then I_3=1, and thus, W=W(X,I_1,I_2,I_4,I_5). Therefore, from (<ref>), 𝐒 is expressed as


    𝐒=2{W_I_1𝐆^♯+W_I_2(I_2𝐂^-1-𝐂^-2)+W_I_4(𝐍⊗𝐍)+W_I_5(𝐍⊗𝐂·𝐍+𝐍·𝐂⊗𝐍)}-p𝐂^-1 ,
 

in which p is the Lagrange multiplier associated with the incompressibility condition J=1. The Cauchy stress tensor σ^ab=1/JF^a_A F^b_BS^AB is represented in component form as[Note that one can use the Cayley-Hamilton theorem and obtain


    ∂ I_2/∂𝐂^♭=I_2(𝐂^-1)^♯-I_3(𝐂^-2)^♯=I_1𝐆^♯-𝐂^♯.
]


    σ^ab=2 F^a_A F^b_B [(W_I_1+I_1W_I_2)G^AB-W_I_2C^AB+W_I_4N^A N^B+W_I_5(N^QN^AC^B_Q+N^PN^BC_P^A)]-pg^ab.
  





  
Orthotropy. Next, we consider a compressible orthotropic material with three 𝐆-orthonormal vectors 𝐍_1(X), 𝐍_2(X), and 𝐍_3(X) specifying the orthotropic axes in the reference configuration at a point X. A choice of structural tensors is given by 𝐀_1=𝐍_1⊗𝐍_1, 𝐀_2=𝐍_2⊗𝐍_2, and 𝐀_3=𝐍_3⊗𝐍_3, where only two of which are independent as 𝐀_1+𝐀_2+𝐀_3=𝐈. Hence, the energy function is given as <cit.>


    W=W(X,𝐆,𝐂^♭, 𝐀_1,𝐀_2) .


The energy function W is represented in terms of the following seven independent invariants


    I_1=tr 𝐂 ,   I_2=det 𝐂 tr 𝐂^-1 ,   I_3=det 𝐂 ,   I_4=𝐍_1·𝐂·𝐍_1 ,
     I_5=𝐍_1·𝐂^2·𝐍_1 ,   I_6=𝐍_2·𝐂·𝐍_2 ,   I_7=𝐍_2·𝐂^2·𝐍_2 .


Using (<ref>), one obtains


    𝐒=∑_n=1^72W_I_n∂ I_n/∂𝐂^♭ ,     W_I_n
    	:=∂ W/∂ I_n ,    n=1,…,7 .


Substituting (<ref>) into (<ref>), the second Piola-Kirchhoff stress tensor is given by


    𝐒=2{W_I_1𝐆^♯+W_I_2(I_2𝐂^-1-I_3𝐂^-2)+W_I_3I_3𝐂^-1+W_I_4(𝐍_1⊗𝐍_1)+W_I_5(𝐍_1⊗𝐂·𝐍_1+𝐍_1·𝐂⊗𝐍_1)
    +W_I_6(𝐍_2⊗𝐍_2)+W_I_7(𝐍_2⊗𝐂·𝐍_2+𝐍_2·𝐂⊗𝐍_2)} .


In the case of incompressible solids I_3=1 and W=W(X,I_1,I_2,I_4,I_5,I_6,I_7). Therefore, using (<ref>), one obtains the following representation for the second Piola-Kirchhoff stress tensor


    𝐒=2{W_I_1𝐆^♯+W_I_2(I_2𝐂^-1-𝐂^-2)+W_I_4(𝐍_1⊗𝐍_1)+W_I_5(𝐍_1⊗𝐂·𝐍_1+𝐍_1·𝐂⊗𝐍_1)
    +W_I_6(𝐍_2⊗𝐍_2)+W_I_7(𝐍_2⊗𝐂·𝐍_2+𝐍_2·𝐂⊗𝐍_2)}-p𝐂^-1 .


In components, the Cauchy stress tensor is given as


    σ^ab=2 F^a_A F^b_B [(W_I_1+I_1W_I_2)G^AB-W_I_2C^AB+W_I_4N_1^A N_1^B+W_I_5(N_1^QN_1^AC^B_Q+N_1^PN_1^BC_P^A)
    +W_I_6N_2^A N_2^B+W_I_7(N_2^SN_2^AC^B_S+N_2^KN_2^BC_K^A)]-pg^ab .





  
Monoclinic Symmetry. One of the preferred directions of a material with a monoclinic symmetry (say 𝐍_3(X)) is perpendicular to the plane of the other two (denoted by 𝐍_1(X) and 𝐍_2(X)), which are not orthogonal. As an example one can consider an isotropic base material reinforced with two families of fibers such that the fibers are not at right angles, nor are they mechanically equivalent. In this case, the energy function is similar to that of orthotropic materials given by (<ref>), where 𝐀_1=𝐍_1⊗𝐍_1 and 𝐀_2=𝐍_2⊗𝐍_2. Nonetheless, an extra invariant I_8=(𝐍_1·𝐍_2)𝐍_1·𝐂·𝐍_2 that models the coupling between the fibers (in 𝐍_1 and 𝐍_2 directions) 


is needed to express the energy function for monoclinic materials as 𝐍_1 and 𝐍_2 are not perpendicular (see <cit.>). Therefore   


    𝐒=∑_n=1^82W_I_n∂ I_n/∂𝐂^♭ ,     W_I_n
    	:=∂ W/∂ I_n ,    n=1,…,8 .


Hence[Note that ∂ I_8/∂𝐂^♭=𝐍_1⊗𝐍_2+𝐍_2⊗𝐍_1.]


    𝐒=2{W_I_1𝐆^♯+W_I_2(I_2𝐂^-1-I_3𝐂^-2)+W_I_3I_3𝐂^-1+W_I_4(𝐍_1⊗𝐍_1)+W_I_5(𝐍_1⊗𝐂·𝐍_1+𝐍_1·𝐂⊗𝐍_1)
    +W_I_6(𝐍_2⊗𝐍_2)+W_I_7(𝐍_2⊗𝐂·𝐍_2+𝐍_2·𝐂⊗𝐍_2)+W_I_8/2(𝐍_1⊗𝐍_2+𝐍_2⊗𝐍_1)} ,


and for incompressible solids


    𝐒=   2{W_I_1𝐆^♯+W_I_2(I_2𝐂^-1-𝐂^-2)+W_I_4(𝐍_1⊗𝐍_1)+W_I_5(𝐍_1⊗𝐂·𝐍_1+𝐍_1·𝐂⊗𝐍_1)
       +W_I_6(𝐍_2⊗𝐍_2)+W_I_7(𝐍_2⊗𝐂·𝐍_2+𝐍_2·𝐂⊗𝐍_2)+W_I_8/2(𝐍_1⊗𝐍_2+𝐍_2⊗𝐍_1)}-p𝐂^-1 .


The Cauchy stress is given in components as


    σ^ab   =2 F^a_A F^b_B [(W_I_1+I_1W_I_2)G^AB-W_I_2C^AB+W_I_4N_1^A N_1^B+W_I_5(N_1^QN_1^AC^B_Q+N_1^PN_1^BC_P^A)
       +W_I_6N_2^A N_2^B+W_I_7(N_2^SN_2^AC^B_S+N_2^KN_2^BC_K^A)+W_I_8/2(N_1^A N_2^B+N_2^A N_1^B)]-pg^ab .





  
Cartan's Moving Frame.At a point X of a manifold ℬ consider an orthonormal frame field {𝐞_α}_α=1^N forming a basis for T_Xℬ. This frame field is not necessarily a coordinate basis for the tangent space. However, given a coordinate basis {∂/∂ X^A}, one can obtain an arbitrary frame field {𝐞_α} using an SO(N,ℝ)-rotation of the coordinate basis such that 𝐞_α=𝖥^A_α∂/∂ X^A. For a coordinate frame [∂/∂ X^A,∂/∂ X^B]=0,[Note that for any pair of vector fields 𝐔 and 𝐕 on ℬ, one can define a new vector field —the commutator —given by [𝐔,𝐕]_Xf:=𝐔_X(𝐕f)-𝐕_X(𝐕f), for any smooth function at X on ℬ.] 
whereas for the non-coordinate frame, [𝐞_α,𝐞_β]=-c^γ_αβ𝐞_γ, where c^γ_αβ are the componenets of the object of anhonolomy. 
One can show that c^γ_αβ=𝖥^A_α𝖥^B_β(∂_A𝖥^γ_B-∂_B𝖥^γ_A), where 𝖥^γ_A is the inverse of 𝖥^A_γ. 
Connection 1-forms are defined by ∇𝐞_α=𝐞_γ⊗ω^γ_α, and in components, ∇_𝐞_β𝐞_α=<ω^γ_α,𝐞_β>𝐞_γ=ω^γ_βα𝐞_γ. In terms of the co-frame field {ϑ^α}_α=1^N corresponding to {𝐞_α}, one has ω^γ_α=ω^γ_βαϑ^β. Similarly, one obtains ∇ϑ^α=-ω^α_γϑ^γ and ∇_𝐞_βϑ^α=-ω^α_βγϑ^γ. The metric tensor is represented as 𝐆=δ_αβϑ^α⊗ϑ^β. Metric compatibility of ∇ gives the following constraints on the connection 1-forms δ_αγω^γ_β+δ_βγω^γ_α=0.
In a non-coordinate basis, the torsion and curvature have the following components T^α_βγ =ω^α_βγ-ω^α_γβ+c^α_βγ and ℛ^α_βλμ =∂_βω^α_λμ-∂_λω^α_βμ+ω^α_βξω^ξ_λμ-ω^α_λξω^ξ_βμ+ω^α_ξμc^ξ_βλ, respectively.
Torsion and curvature 2-forms are, respectively, given by 𝒯^α=dϑ^α+ω^α_β∧ϑ^β and ℛ^α_β =dω^α_β+ω^α_γ∧ω^γ_β.
These are called Cartan's first and second structural equations. The density of Burgers' vector 𝐛 at a point X of ℬ is related to torsion 2-from as follows 


    b^α(X;C_s)=∫_Ω_s𝖯^α_β𝒯^β ,


where Ω_s∈ℬ is a smooth surface with a boundary given by the curve C_s, and 𝖯(C_s)_τ^t:T_C_s(τ)ℬ→ T_C_s(t)ℬ parallel transports vectors tangent to the manifold at C_s(τ) to C_s(t) (see <cit.> for more details).


 



§ EXAMPLES OF ANISOTROPIC BODIES WITH DISTRIBUTED DEFECTS


In this section, we consider several examples of distributed defects in cylindrical bars made of orthotropic and monoclinic solids as well as distributed defects in spherical balls made of transversely isotropic solids. Particularly, we consider cylindrically-symmetric distributions of parallel screw dislocations and disclinations in an orthotropic medium, a spherically-symmetric distribution of point defects in a transversely isotropic spherical ball, and a cylindrically-symmetric distribution of screw dislocations in a monoclinic medium. We also discuss the effects of the constitutive parameters on the induced stress fields for different types of defects.  










 §.§ A Cylindrically-Symmetric Distribution of Parallel Screw Disclocations in an Orthotropic Medium



Let us consider a cylindrically-symmetric distribution of screw dislocations parallel to the Z-axis with a radially-symmetric Burgers' vector density b(R) (in a cylindrical coordinate system (R,Θ,Z)) in an infinite orthotropic medium. We assume that in the reference configuration the dislocated body is orthotropic. The material preferred directions at a material point X are denoted by 𝐍_1(X), 𝐍_2(X), and 𝐍_3(X) in the reference configuration. In the current configuration, the preferred directions are given by 𝐧_1(x), 𝐧_2(x), and 𝐧_3(x) at the ambient point x corresponding to the material point X. We assume that 𝐍_1 and 𝐍_2 are in the radial and axial directions, respectively. Note that 𝐍_3, which is perpendicular to 𝐍_1 and 𝐍_2, explicitly depends on the distribution of screw dislocations as will be seen in the following. This is because the geometry of the material manifold has an explicit nontrivial dependence on the dislocations distribution (see (<ref>)). In the current configuration, the body will have monoclinic anisotropy as 𝐧_1 will be perpendicular to the plane of 𝐧_2 and 𝐧_3, which will not be orthogonal in the ambient space. 

It turns out that the material manifold for a nonlinear solid with distributed dislocations is a Weitzenböck manifold, i.e., a manifold with torsion having a flat connection and vanishing non-metricity (see <cit.> for more details). Therefore, the material metric for the dislocated body is written as


    𝐆=
    (
    [          1          0          0;          0 R^2+f(R)^2       f(R);          0       f(R)          1 ]) ,


where f(R) is related to the Burgers' vector density b(R) such that f'(R)=R/2πb(R). 
Let us endow the ambient space with the Euclidean metric 𝐠=diag{1,r^2,1}. We then assume an embedding of the material manifold into the ambient space of the form (r,θ,z)=(r(R),Θ,α Z), where α is a positive constant denoting the longitudinal stretch. Hence, 𝐅=diag{r'(R),1,α}. Assuming incompressibility, i.e., J=√(det𝐠/det𝐆)det𝐅=1, one obtains r(R)/Rr'(R)α=1. Eliminating the rigid body translation by setting r(0)=0, one obtains r(R)=1/√(α)R. Therefore, the right Cauchy-Green deformation tensor is written as[The symbolic computations in this paper were performed using Mathematica <cit.>.]


    𝐂=
    (
    [                 1/α                   0                   0; ;                   0                 1/α        -α^2f(R)/R^2; ;                   0             -f(R)/α α^2/R^2(R^2+f(R)^2) ]) .




Note that 𝐍_1=𝐄_R, 𝐍_2=𝐄_Z, and 𝐍_3=1/R𝐄_Θ-f(R)/R𝐄_Z. Note also that 𝐍_3 is obtained using the orthonormality of the material preferred directions, and   𝐄_R=∂/∂ R, 𝐄_Z=∂/∂ Z, and 𝐄_Θ=∂/∂Θ form a basis for T_Xℬ. Using (<ref>), the invariants of the strain energy function are simplified and are written as


    I_1=   tr𝐂=2/α+α^2/R^2(R^2+f(R)^2) ,   
    I_2=
    1/2[tr(𝐂^2)-(tr𝐂)^2]=1/α^2+2α+αf(R)^2/R^2 ,
    	
    I_4=   1/α ,    
    I_5=1/α^2 ,    
    I_6=α^2 ,   
    I_7= α^4/R^2(R^2+f(R)^2)  .


The non-zero components of the Cauchy stress tensor following (<ref>) read 


    σ^rr=   2/α ^2[W_I_2(α
    ^3+α ^3 f(R)^2/R^2+1)+α 
    W_I_4+2 W_I_5]+2 W_I_1/α-p(R) ,
    σ^θθ=   2 α  W_I_1+2 (α^3+1) W_I_2-α^2p(R)/α  R^2 ,
    σ^zz=   2 α/R^2[ (f(R)^2+R^2) (α W_I_1+W_I_2+2 α^3W_I_7)+R^2 (W_I_2+α   W_I_6)]-p(R) ,
    σ^θ z=   -2 f(R)/R^2(α  W_I_1+W_I_2+α ^3 W_I_7) .


We assume that the stress vanishes when the body is dislocation-free and the longitudinal stretch α=1 (see also <cit.>). Thus


    (W_I_4+2W_I_5)|_I_1=I_2=3, I_4=I_5=I_6=I_7=1=0 ,  and  (W_I_6+2W_I_7)|_I_1=I_2=3, I_4=I_5=I_6=I_7=1=0 .


In the absence of body and inertial forces, the only non-trivial equilibrium equation is σ^rb_|b=0, implying[Note that p=p(R) is implied from the other equilibrium equations.] that (cf. (<ref>)) σ^rr_,r+σ^rr/r-rσ^θθ=0. Therefore, p'(R)=h(R), where


    h(R)=   2 /α  R^5[2 R^3 f(R) f'(R) (α
    	^2 W_I_2+α ^2 W_I_1I_1+α
    	 (2+α^3)W_I_1I_2 +α ^2 W_I_1I_4+2 α
    	W_I_1I_5+α ^4 W_I_1I_7
       +(α
    	^3+1) W_I_2I_2+α 
    	W_I_2I_4+2 W_I_2I_5+α ^3(α ^3+1)
    	W_I_2I_7+α
    	^4 W_I_4I_7+2 α ^3
    	W_I_5I_7)
       +2 α ^3 R f(R)^3 f'(R)
    	(α 
    	W_I_1I_2+W_I_2I_2+α ^3
    	W_I_2I_7)-R^2 f(R)^2 {2 α  W_I_1I_2(2+α^3)
       +2 α ^2
    	W_I_1I_4+4 α  W_I_1I_5+2 α
    	^4 W_I_1I_7+α ^2 W_I_2+2(α ^3+1)
    	 W_I_2I_2+2
    	α  W_I_2I_4+4 W_I_2I_5
       +2 α
    	^3 (α ^3+1)W_I_2I_7+2
    	α ^4 W_I_4I_7+4 α ^3
    	W_I_5I_7+2 α ^2
    	W_I_1I_1}
       -2 α ^3 f(R)^4
    	(α 
    	W_I_1I_2+W_I_2I_2+α ^3
    	W_I_2I_7)+R^4 W_I_4]+4
    	 W_I_5/α ^2 R .


If one assumes that the medium is a cylinderical bar with a finite radius R_o and the surface R=R_o is traction-free, one obtains


    p(R)=∫_R_o^Rh(ζ)dζ+2/α ^2[ (α
    ^3+α ^3 f(R_o)^2/R_o^2+1)W_I_2|_R=R_o+α 
    W_I_4|_R=R_o+2 W_I_5|_R=R_o]+2/α W_I_1|_R=R_o .


Let us employ the so called standard reinforcing model for compressible materials, which is defined as <cit.>


    W=W(I_1,I_2,I_4,I_5,I_6,I_7)=W_iso(I_1,I_2)+W_fib^R(I_4,I_5)+W_fib^Z(I_6,I_7) ,


where W_iso denotes the strain energy function for the isotropic base material, whereas W_fib^R and W_fib^Z represent the anisotropic effects due to the fiber reinforcement in the radial and longitudinal directions, respectively. Consider as an example a cylindrical body made of a Mooney-Rivlin solid reinforced with fibers in the radial and longitudinal directions such that 


    W(I_1,I_2,I_4,I_5,I_6,I_7) =   μ_1/2(I_1-3)+μ_2/2(I_2-3)+γ_1/2(I_4-1)^2
       +γ_2/2(I_5-1)^2+ξ_1/2(I_6-1)^2+ξ_2/2(I_7-1)^2 .
 

Using (<ref>), we have


    h(R)=αμ_2 f(R)/R^3[ 2  R  f'(R)- f(R)]+2/R1/α^2(1/α-1)  [αγ_1 +2 γ_2
    	(1+1/α)] .


Thus, from (<ref>)


    p(R)=   αμ_2∫_R_o^Rf(ζ)/ζ^3[ 2  ζ  f'(ζ)- f(ζ)]dζ+2/α^2(1/α-1) [αγ_1 
    	+2 γ_2(1+1/α)]
    	[1+lnR/R_o]
       + μ_1/α
    	+μ_2/α ^2(α^3+α ^3 f(R_o)^2/R_o^2+1)  .


The physical components of the Cauchy stress read[The physical components of the Cauchy stress tensor, i.e., σ̂^ab=σ^ab√(g_aag_bb) (no summation) <cit.> are given as σ̂^rr=σ^rr, σ̂^θθ=r^2(R)σ^θθ, σ̂^zz=σ^zz, and σ̂^θ z=r(R)σ^θ z.
]


    σ̂^rr=   αμ_2 ( f(R)^2/R^2- f(R_o)^2/R_o^2)+αμ_2∫_R^R_of(ζ)/ζ^3[ 2  ζ  f'(ζ)- f(ζ)]dζ
       -2/α^2
    (1/α-1) {αγ_1 +2 γ_2
    (1+1/α)}lnR/R_o ,
    σ̂^θθ=   αμ_2∫_R^R_of(ζ)/ζ^3[ 2  ζ  f'(ζ)- f(ζ)]dζ -αμ_2f(R_o)^2/R_o^2
       -2/α^2
    (1/α-1) {αγ_1 +2 γ_2
    (1+1/α)}[1+lnR/R_o] ,
    σ̂^zz=   2α^2(α^2-1)ξ_1+4α^4ξ_2/R^2(f(R)^2+R^2) [  α^4/R^2(f(R)^2+R^2)-1]
       +αμ_2 ( f(R)^2/R^2- f(R_o)^2/R_o^2)-2/α^2(1/α-1) {αγ_1 +2 γ_2
    (1+1/α)}[1+lnR/R_o]
       +αμ_2∫_R^R_of(ζ)/ζ^3[ 2  ζ  f'(ζ)- f(ζ)]dζ+(αμ_1+μ_2)(α-1/α^2)+α^2μ_1f(R)^2/R^2 ,
    
    σ̂^θ z=   - f(R)/α^1/2R(αμ_1 +μ_2+2α ^3 ξ_2[α^4/R^2(R^2+f(R)^2)-1]) .





From (<ref>), for an arbitrary cylindrically-symmetric distribution of parallel screw dislocations, the pressure p(R), and hence, σ̂^rr, σ̂^θθ, and σ̂^zz exhibit a logarithmic singularity on the dislocation axis (R=0) unless α=1. Note that this singularity is inherent to the anisotropic effects due to the reinforcement in the radial direction. In particular, the singularity does not occur when γ_1=γ_2=0, e.g., when the material is isotropic. 





Note that in the case of fiber-reinforced neo-Hookean materials (μ_2=0) and a given arbitrary cylindrically-symmetric distribution of screw dislocations supported on a cylinder of radius R_i, the stress field for R>R_i is independent of b(R) and is identical to that of a single screw dislocation with Burgers vector b_0=∫_0^R_iη b(η)dη. <cit.> and <cit.> observed that this result holds for isotropic neo-Hookean solids.



As an example, let us assume the following Burgers' vector density distribution:


    b(R)=
    
    b_0   0<R≤ R_i ,
    
    0   R_i<R≤ R_o ,


where R_i≤ R_o. Thus


    f(R)=1/2π∫_0^Rη b(η)dη=b_0/4π
    R^2   0<R≤ R_i ,
    
    R_i^2   R_i<R≤ R_o .


Fig. <ref> depicts the variation of the different components of the Cauchy stress for the Burgers' vector density distribution (<ref>) such that R_i/R_o=0.5 and b_0R_o=20. Notice that the σ̂^rr and σ̂^θθ vanish for a neo-Hookean solid.







As noted by <cit.>, the energy per unit length (along the dislocation line) of a single screw dislocation in a Mooney-Rivlin solid is unbounded.[Note, however, that the energy of distributed screw dislocations is not necessarily unbounded (see also <cit.>). In particular, a Mooney-Rivlin reinforced material with the energy function (<ref>) and the Burgers' vector distribution (<ref>) has a finite energy per unit length.] This is also the case for a fiber-reinforced Mooney-Rivlin material due to the standard reinforcing model considered here (cf. (<ref>)). Let us consider incompressible isotropic base materials, for which the energy per unit length of a single screw dislocation remains bounded, i.e., 2π∫_0^R_oW_iso(I_1(ξ),I_2(ξ))ξ dξ<∞, for finite R_o (examples include Varga <cit.>, incompressible power-law <cit.>, generalized incompressible neo-Hookean <cit.> materials, and Hencky material <cit.>). Exploiting the standard reinforcing model, the energy function for the fiber-reinforced material with the isotropic base with the energy function W_iso(I_1,I_2) is assumed to be given as


    W=W_iso(I_1,I_2)+γ_1/2(I_4-1)^2
    	+γ_2/2(I_5-1)^2+ξ_1/2(I_6-1)^2+ξ_2/2(I_7-1)^β .


Then the energy per unit length along a single screw dislocation line is finite if β<1. To see this, we need to show that 2π∫_0^R_oξ_2/2(I_7(ζ)-1)^βζ dζ<∞ as the finiteness of the contribution of the other terms in the energy per unit length is trivial (cf. (<ref>)). Noting that for a single screw dislocation with Burgers vector b_i, b(R)=2π b_i δ^2(R), and hence, f(R)=b_i/2πH(R), we have


    πξ_2∫_0^R_o(I_7(ζ)-1)^βζ dζ=πξ_2∫_0^R_o[α^4/ζ^2(ζ^2+f(ζ)^2)-1]^βζ dζ=πξ_2∫_0^R_o[α^4-1+α^4b_i^2/4π^2ζ^2]^βζ dζ<∞ ,


provided that β<1. Similarly, one can show that if the resultant longitudinal force, i.e., F_Z=2π∫_0^R_oσ̂^zz(ζ)ζ dζ, induced by a single screw dislocation is finite for the isotropic base material with the energy function W_fib(I_1,I_2), so is the axial force for the fiber-reinforced material with the energy function (<ref>) when β<1.  

















 §.§ A Cylindrically-Symmetric Distribution of Parallel Screw Disclocations in a Monoclinic Medium



In the previous section, we assumed that the dislocated body is orthotropic in the reference configuration.
Instead, let us assume that the medium with the cylindrically-symmetric distribution of parallel screw dislocations is orthotropic in its current configuration such that the orthotropic axes are in the radial, circumferential, and axial directions in the ambient space. 



In the reference configuration, the material will be monoclinic such that 𝐍_3=𝐑̂ is perpendicular to the plane of 𝐍_1=Θ̂ and 𝐍_2=𝐙̂.[Note that 𝐍_1 and 𝐍_2 are not orthogonal in the nontrivial geometry of the reference configuration.]
We assume the same class of deformations as was assumed in the previous section, and thus, r(R)=1/√(α)R. Hence, the right Cauchy-Green deformation tensor is given by (<ref>). 






The invariants of the strain energy function for the monoclinic material are given as


    I_1=   tr(𝐂)=2/α+α^2/R^2(R^2+f(R)^2) ,   
    I_2=
    1/2[tr(𝐂^2)-(tr𝐂)^2]
    =1/α^2+2α+αf(R)^2/R^2 ,
    	
    I_4=   R^2/α(R^2+f(R)^2) , 
    I_5=R^2/α^2(R^2+f(R)^2) , 
    I_6=α^2 , 
    I_7= α^4/R^2(R^2+f(R)^2)  ,  I_8=0 .


From (<ref>), the non-zero components of the Cauchy stress tensor read



    σ^rr=   2W_I_2/α ^2(α
    ^3+α ^3 f(R)^2/R^2+1)+2 W_I_1/α-p(R) ,
    σ^θθ=   2 α  W_I_1+2 (α^3+1) W_I_2-α^2p(R)/α  R^2+2(α W_I_4+2W_I_5)/α(R^2+f(R)^2) ,
    σ^zz=   2 α/R^2[ (f(R)^2+R^2) (α W_I_1+W_I_2+2 α^3W_I_7)+R^2 (W_I_2+α   W_I_6)]-p(R) ,
    σ^θ z=   -2 f(R)/R^2(α  W_I_1+W_I_2+α ^3 W_I_7)-2f(R)/R^2+f(R)^2W_I_5+α/(R^2+f(R)^2)^1/2W_I_8 .


Note that for the stress to vanish when α=1 and the body is dislocation-free, i.e., f(R)=0 (identically), one needs to have (W_I_4+2W_I_5)=(W_I_6+2W_I_7)=W_I_8=0, evaluated at I_1=I_2=3, I_4=I_5=I_6=I_7=1, I_8=0. The equilibrium equation implies that p'(R)=S(R), where


    S(R)=   -1/α ^4 R^3[-4 α f(R)(Rf'(R)-f(R)) (α ^4 W_I_1I_1
    	+α ^3W_I_1I_2+α ^6 W_I_1I_7-R^4(α W_I_1I_4
    	+W_I_1I_5)/(f(R)^2+R^2)^2) 
       -4 f(R) (α ^3+α ^3f(R)^2/R^2+1) (R f'(R)-f(R)) 
    	{α^4 W_I_1I_2+α ^3W_I_2I_2+α ^6 W_I_2I_7
        -  R^4(α W_I_2I_4+W_I_2I_5)/(f(R)^2+R^2)^2}
    	+4 α ^5 f(R) W_I_2(f(R)-Rf'(R)) 
    	+2α ^2 R^4 (α  W_I_4+2W_I_5)/f(R)^2+R^2
       -2α^5 f(R)^2 W_I_2] .


Assuming that the surface R=R_o is traction-free the pressure is obtained as


    p(R)=∫_R_o^RS(ζ)dζ+2/α ^2(α
    ^3+α ^3 f(R_o)^2/R_o^2+1)W_I_2|_R=R_o+2/α W_I_1|_R=R_o .


Let us consider the following model for the strain energy function


    W=W(I_1,I_2,I_4,I_5,I_6,I_7,I_8)=W_iso(I_1,I_2)+W_fib^Θ(I_4,I_5)+W_fib^Z(I_6,I_7)+W_fib^ZΘ(I_8) ,


where W_iso describes that part of the energy function pertaining to the isotropic base material, while W_fib^Θ and W_fib^Z represent the reinforcement effects in the circumferential and axial directions. W_fib^ZΘ(I_8) models the coupling between the axial and circumferential fibers. Note, however, that I_8=0, and for the stress to vanish for the dislocation-free body, one needs W_I_8=0 at I_8=0, which implies that W_fib^ZΘ(I_8)=0, i.e., the coupling term must vanish. A way out would be to require that the coupling term depend on some other invariants as well, e.g., one can define W_fib^ZΘ(I_1,I_8)=η I_8(I_1-3) for some positive constant η.
For the sake of simplicity, as an example, we consider a fiber-reinforced Mooney-Rivlin material with the following energy function


    W(I_1,I_2,I_4,I_5,I_6,I_7) =   μ_1/2(I_1-3)+μ_2/2(I_2-3)+λ_1/2(I_4-1)^2
       +λ_2/2(I_5-1)^2+ξ_1/2(I_6-1)^2+ξ_2/2(I_7-1)^2 .
 

Therefore, one obtains


    S(R)=   1/α ^2 R^3 (f(R)^2+R^2)[ α ^3 μ_2  f(R) (R^2+f(R)^2)(2Rf'(R)-f(R))
       +2 R^4 {αλ_1-
    R^2/f(R)^2+R^2(2 λ_2/α^2+λ_1)+2 λ_2}] .


Thus, the physical components of the stress are given as


    σ̂^rr   =αμ_2(   f(R)^2/R^2-f(R_o)^2/R_o^2)+∫_R^R_oS(ζ)dζ ,
    σ̂^θθ   =∫_R^R_oS(ζ)dζ-αμ_2f(R_o)^2/R_o^2+2R^2/α^2(R^2+f(R)^2)[αλ_1(R^2/α(R^2+f(R)^2)-1)
       +2λ_2(R^2/α^2(R^2+f(R)^2)-1)] ,
    
    σ̂^zz   =2α^2ξ_1(α^2-1)+4α^4ξ_2(1+f(R)^2/R^2)[α^4(1+f(R)^2/R^2)-1]+∫_R^R_oS(ζ)dζ
       +αμ_2(   f(R)^2/R^2-f(R_o)^2/R_o^2)+(αμ_1+μ_2)(α-1/α^2)+α^2μ_1f(R)^2/R^2 ,
    
    σ̂^θ z   =-1/√(α)f(R)/R(αμ_1+μ_2)-2ξ_2α^5/2f(R)/R[α^4(1+f(R)^2/R^2)-1]
       -2λ_2/√(α)Rf(R)/R^2+f(R)^2(R^2/α^2(R^2+f(R)^2)-1) .





For an arbitrary cylindrically-symmetric distribution of parallel screw dislocations with a smooth Burgers' vector density b(R) in a monoclinic material, the pressure, and hence, σ̂^rr, σ̂^θθ, and σ̂^zz have a logarithmic singularity on the dislocation axis unless α=1. Nevertheless, the shear component σ̂^θ z is finite and vanishes at R=0. This is because as R→ 0, we have f(R)=b(0)/4πR^2+𝒪(R^3), and thus, from (<ref>)


    S(R)=2(α-1)/α^2[λ_1+2λ_2/α(1+1/α)]1/R+𝒪(R) .


Therefore, p(R)=C-2(α-1)/α^2[λ_1+2λ_2/α(1+1/α)]lnR/R_o+𝒪(R^2) as R→ 0, where C is a constant. It is easy to see that when α=1, the stress is finite and σ̂^rr=σ̂^θθ=σ̂^zz at R=0.



In Fig. <ref> the stress field is shown for the dislocation distribution (<ref>), where R_i/R_o=0.5 and b_0R_o=20 for different values of the constitutive parameters given by (<ref>).


































 





 §.§ A Parallel Cylindrically Symmetric Distribution of Wedge Disclinations in an Orthotropic Medium




Let us consider a parallel cylindrically-symmetric distribution of wedge disclinations in an infinite orthotropic medium in the reference configuration. 

In the cylindrical coordinates (R,Θ,Z), assume that the material orthotropic axes are in the R, Θ, and Z directions. The radial density of the wedge disclinations is denoted by w(R). The material manifold for a body having a distribution of wedge disclinations is a Riemannian manifold with a non-vanishing curvature. The material metric for the disclinated body is given by <cit.> 


    𝐆=
    (
    [      1      0      0;      0 f(R)^2      0;      0      0      1 ]) ,


where f”(R)=-R/2πw(R). 

The ambient space is endowed with the Euclidean metric 𝐠=diag{1,r^2,1}. We embed the material manifold into the ambient space by looking for mappings[Note that for the class of deformations that is considered, the material will be orthotropic in its current configuration as well. The orthotropic axes in the current configuration will be in the radial, circumferential, and axial directions (similar to those in the reference configuration).] of the form (r,θ,z)=(r(R),Θ,α Z), where α is a constant representing the axial stretch of the bar that depends on the axial boundary conditions. Therefore, the deformation gradient reads 𝐅=diag(r'(R),1,α). Incompressibility constraint dictates that J=√(det𝐠/det𝐆)det𝐅=αr(R)/f(R)r'(R)=1.
Thus, imposing r(0)=0, we have r(R)=(2/α∫_0^Rf(ξ)dξ)^1/2. The right Cauchy-Green deformation tensor reads 𝐂=diag{1/α^2f(R)^2/r(R)^2,r(R)^2/f(R)^2,α^2}. From (<ref>), the invariants of the strain energy function are simplified to read


    I_1=   tr(𝐂)=α^2+1/α^2f(R)^2/r(R)^2+r(R)^2/f(R)^2 ,   
    I_2=
    1/2(tr(𝐂^2)-tr(𝐂)^2)=1/α^2+α^2r(R)^2/f(R)^2+f(R)^2/r(R)^2 ,
    	
    I_4=   1/α^2f(R)^2/ r(R)^2 ,    
    I_5=1/α^4f(R)^4/ r(R)^4 ,    
    I_6=α^2 ,   
    I_7= α^4  .


The non-zero physical components of the Cauchy stress are as follows[When the body is disclination-free f(R)=R, and the stress vanishes if the energy function satisfies (<ref>).] 


    σ̂^rr=   2/α^2f(R)^2/r(R)^2(W_I_1+α^2W_I_2+W_I_4)+4/α^4f(R)^4/r(R)^4W_I_5+2/α^2W_I_2-p(R) ,
    σ̂^θθ=   2r(R)^2/f(R)^2(W_I_1+α^2W_I_2)+2/α^2W_I_2-p(R) ,
    σ̂^zz=   2α^2(W_I_1+W_I_6+2α^2W_I_7)+2W_I_2(f(R)^2/r(R)^2+α^2r(R)^2/f(R)^2)-p(R) .


The equilibrium equation implies that p'(R)=k(R), where


    k(R)    =2/α ^8 f^3 r^9[2 α ^4 f^4 r^7 f' (α ^2
    	W_I_1+W_I_1I_2-2 W_I_1I_5+α ^4 W_I_2
    	+α^2 W_I_2I_2+W_I_2I_4-2 α ^2 W_I_2I_5+α ^2	W_I_4)
       +2 α ^2 f^6 r^5 f' (α ^2W_I_1I_1+2 α ^4 W_I_1I_2+2 α ^2 W_I_1I_4
    	+α^6 W_I_2I_2+2 α ^4 W_I_2I_4+2 W_I_2I_5+α ^2W_I_4I_4
    	+4 α ^2 W_I_5)
       -2 α ^6 f^2 r^9 f'(W_I_1I_1+2 α ^2 W_I_1I_2+W_I_1I_4
    	+α ^4W_I_2I_2+α ^2 W_I_2I_4)
       +8 α ^2 f^8 r^3f' (W_I_1I_5+α ^2 W_I_2I_5+W_I_4I_5)
    	+8f^10 W_I_5I_5 r f'-2 α ^6 r^11 f'(W_I_1I_2+α ^2 W_I_2I_2)
       -α ^3 f^6 r^5 (α ^2W_I_1+2 W_I_1I_2-4 W_I_1I_5+α ^4 W_I_2
    	+2α ^2 W_I_2I_2+2 W_I_2I_4-4 α ^2 W_I_2I_5
    	+α^2 W_I_4)
       -α ^5 f^2 r^9 (α ^2 W_I_1
    	-2W_I_1I_2+α ^4 W_I_2-2 α ^2 W_I_2I_2)-8 α  f^10 r (W_I_1I_5
    	+α ^2W_I_2I_5+W_I_4I_5)
       -2α  f^8 r^3 (α ^2 W_I_1I_1+2 α ^4W_I_1I_2+2 α ^2 W_I_1I_4
    	+α ^6 W_I_2I_2+2 α^4 W_I_2I_4+2 W_I_2I_5+α ^2 W_I_4I_4
    	+3 α ^2W_I_5)
       +2 α ^5 f^4 r^7 (W_I_1I_1
    	+2 α ^2W_I_1I_2+W_I_1I_4+α ^4 W_I_2I_2+α ^2W_I_2I_4)
    	-8 f^12 W_I_5I_5/α r] .


Assuming (<ref>) for the energy function, one obtains


    p'(R)=   -1/α ^9 f r^10[-2 α ^7 f^2 r^8 f' (α ^2 μ_2-2
    	γ_1+μ_1)+α ^5 f^4 r^6 {α(α ^2 μ_2-2 γ_1+μ_1)-8
    	(γ_1-2 γ_2) f'}
       -32 αγ
    	_2 f^8 r^2 f'+6 α ^4 (γ_1-2 γ_2)
    	f^6 r^4+28 γ_2 f^10+α ^8 r^10(α ^2
    	μ_2+μ_1)] .


Knowing that the traction vanishes on the outer boundary R=R_o, one finds


    p(R_o)=1/α^2f(R_o)^2/r(R_o)^2[μ_1+α^2μ_2+2γ_1(1/α^2f(R_o)^2/r(R_o)^2-1)]
    	+4γ_2/α^4f(R_o)^4/r(R_o)^4(1/α^4f(R_o)^4/ r(R_o)^4-1)
    	+μ_2/α^2 .


Therefore, p(R)=∫_R_o^Rp'(ξ)dξ+p(R_o). The stress components are simplified and read


    σ̂^rr=   1/α^2f(R)^2/r(R)^2[μ_1+α^2μ_2+2γ_1(1/α^2f(R)^2/r(R)^2-1)]
    	+4γ_2/α^4f(R)^4/r(R)^4(1/α^4f(R)^4/ r(R)^4-1)
    	+μ_2/α^2-p(R) ,
    σ̂^θθ=   r(R)^2/f(R)^2(μ_1+α^2μ_2)+μ_2/α^2-p(R) ,
    σ̂^zz=   α^2[μ_1+2ξ_1(α^2-1)+4α^2ξ_2(α^4-1)]
    	+μ_2(f(R)^2/r(R)^2+α^2r(R)^2/f(R)^2)-p(R) .





For a uniform disclination  distribution w(R)=w_o, one has f”(R)=-R/2πw_o, and thus, f(R)=R-w_o/12πR^3. Therefore

	
    r(R)=R/α^1/2(1-w_o/24πR^2)^1/2 ,
 

provided that w_o<24π/R_o^2. 



For the uniform disclination distribution, the stress field exhibits a logarithmic singularity on the disclinations axis unless the axial stretch α=1. Moreover, when α=1, the stress is finite and hydrostatic at R=0. To see this, as R→ 0, we have r(R)=R/α^1/2+𝒪(R^3), and f(R)=R+𝒪(R^3). From (<ref>), therefore


    p(R)=C+2(1-α)/α^4 [α ^2 γ_1+2 γ_2(1+α)]lnR+𝒪(R^2) ,


where C is a constant. Hence, the stress is logarithmically unbounded at R=0 unless α=1. Similar to the case of parallel screw dislocations in an orthotropic medium (cf. Remark. <ref>), the singularity arises as a result of radial reinforcement effects, and does not, in particular, occur in isotropic materials. Note that for α=1, at R=0, one has σ̂^rr=σ̂^θθ=σ̂^zz=μ_1+2μ_2+∫_0^R_op'(ξ)dξ-p(R_o).



In Fig. <ref>, we show the variation of the stress components for the uniform disclination distribution with w_o=8π/R_o^2 and for some different values of the constitutive parameters.[Note that the numerical values shown in  <cit.>'s Fig. 4 are not correct. This was caused by a typo in the sign of the integral term in the numerical evaluation of the pressure function from Eq. (4.23). In other words, the numerical values in that figure correspond to the following (incorrect) relation for the pressure with a positive sign for the integral term


    p(R)=μf^2(R_o)/r^2(R_o)+μ∫_R^R_o[f(η)f'(η)/∫_0^ηf(ξ)dξ-f^3(η)/4(∫_0^ηf(ξ)dξ)^2-1/f(η)]dη .


] 









For a single wedge disclination ω(R)=2πΘ_oδ^2(R), where Θ_o is the angle of the wedge shape region that is removed in Volterra's cut-and-weld operation (see <cit.> for more details). Therefore, f”(R)=-Θ_o/2πδ (R), which implies that f(R)=R(1-Θ_o/2π), and thus, r(R)=R/α^1/2(1-Θ_o/2π)^1/2. Fig. <ref> illustrates the stress distribution for different values of the reinforcement and the base material parameters in the case of a single wedge disclination of positive sign with Θ_o=π/2.












 §.§ Distributed Edge Dislocations in an Orthotropic Medium



Next, we consider a distribution of edge dislocations in an orthotropic medium such that the material preferred directions are parallel to the Cartesian axes in the Cartesian coordinates (X,Y,Z). Let us consider the orthonormal frame field {𝐞_α(X,Y,Z)}_α=1^3, where 𝐞_1, 𝐞_2, and 𝐞_3 are in the X, Y, and Z-directions, respectively. We assume that the edge dislocation distribution consists of dislocations with i) the dislocation line parallel to the Z-axis such that the Burgers' vector density is given by b_1(Z)𝐞_1+c_1(Z)𝐞_2, ii) X-oriented Burgers' vector density b_2(X,Y,Z)𝐞_1 such that the dislocation line is parallel to the Y-axis, iii) Y-oriented Burgers' vector c_2(X,Y,Z)𝐞_2 with the dislocation line parallel to the X-axis. Let us consider the following co-frame field  


    ϑ^1=e^ξ(Z)+γ(Y)dX ,  ϑ^2=e^η(Z)+λ(X)dY ,   ϑ^3=e^ψ(Z)dZ ,


where ξ(Z), γ(Y), η(Z), λ(X), and ψ(Z) are scalar functions to be determined. The corresponding frame field reads


    𝐞_1=e^-ξ(Z)-γ(Y)∂_X ,  𝐞_2=e^-η(Z)-λ(X)∂_Y ,  𝐞_3=e^-ψ(Z)∂_Z .


Note that 𝐆=δ_αβϑ^α⊗ϑ^β, and thus


    𝐆=diag{e^2(ξ(Z)+γ(Y)),e^2(η(Z)
    	 +λ(X)),e^2ψ(Z)}.


The above dislocation distribution corresponds to the following torsion 2-forms (cf. (<ref>))


    𝒯^1=b_1(Z)ϑ^3∧ϑ^1+b_2(X,Y,Z)ϑ^1∧ϑ^2 ,  𝒯^2=c_1(Z)ϑ^2∧ϑ^3+c_2(X,Y,Z)ϑ^1∧ϑ^2 ,  𝒯^3=0 .


This represents a distribution of edge dislocations with the following total Burgers' vector density   


    𝐛(X,Y,Z)   =(b_1(Z)+b_2(X,Y,Z))𝐞_1+(c_1(Z)+c_2(X,Y,Z))𝐞_2
       =e^-ξ(Z)-γ(Y)[b_1(Z)+b_2(X,Y,Z)]∂_X
    +e^-η(Z)-λ(X)[c_1(Z)+c_2(X,Y,Z)]∂_Y .


From (<ref>), one obtains


    dϑ^1=   e^-ψ(Z)ξ'(Z)ϑ^3∧ϑ^1+e^-η(Z)-λ(X)γ'(Y)ϑ^2∧ϑ^1 ,
     dϑ^2=   e^-ψ(Z)η'(Z)ϑ^3∧ϑ^2
    +e^-ξ(Z)-γ(Y)λ'(X)ϑ^1∧ϑ^2 ,     dϑ^3=0 .




Metric compatibility implies the following connection 1-forms matrix 


    ω=[ω^α_β]=
    (
    [      0  ω^1_2 -ω^3_1; -ω^1_2      0  ω^2_3;  ω^3_1 -ω^2_3      0 ]) .


Cartan's first structural equation gives the following connection 1-forms 


    ω^1_2   =(b_2(X,Y,Z)+γ'(Y)e^-η(Z)-λ(X))ϑ^1
    +(c_2(X,Y,Z)-λ'(X)e^-ξ(Z)-γ(Y))ϑ^2 ,
    ω^2_3   =(c_1(Z)+η'(Z)e^-ψ(Z))ϑ^2 ,    ω^3_1=(b_1(Z)-ξ'(Z)e^-ψ(Z))ϑ^1 .


The second structural equation, i.e., ℛ^α_β=0 is trivially satisfied if one assumes that


    ξ'(Z)=b_1(Z)e^ψ(Z) ,  γ'(Y)   =-b_2(X,Y,Z)e^η(Z)+λ(X) ,  η'(Z)=-c_1(Z)e^ψ(Z) ,
    λ'(X)   =c_2(X,Y,Z)e^ξ(Z)+γ(Y) .


Thus


    η'(Z) =-1/b_2∂ b_2/∂ Z,  λ'(X)
    	=-1/b_2∂ b_2/∂ X,  γ'(Y)
    	=-1/c_2∂ c_2/∂ Y,  ξ'(Z)
    	=-1/c_2∂ c_2/∂ Z,  ψ(Z)
    	=ln(-1/b_1c_2∂ c_2/∂ Z) ,


where one needs to have ∂ b_2/∂ Z=-c_1 b_2/b_1 c_2∂ c_2/∂ Z and -1/b_1c_2∂ c_2/∂ Z>0. If we assume that b_2 and c_2 are separable in X, Y, and Z, i.e., b_2=b_2X(X)b_2Y(Y)b_2Z(Z) and c_2=c_2X(X)c_2Y(Y)c_2Z(Z), then


    e^η(Z)=C_1/b_2Z(Z) ,   e^λ(X)=C_2/b_2X(X) ,   e^γ(Y)=C_3/c_2Y(Y) ,   e^ξ(Z)=C_4/c_2Z(Z) ,   e^ψ(Z)=-c_2Z(Z)'/b_1(Z)c_2Z(Z) ,


where C_i, i=1,…,4 are constants of integration. The compatibility conditions are written as


    C_1C_2 b_2Y(Y)=c'_2Y(Y)/c_2Y(Y) ,   
    	C_3C_4 c_2X(X)=-b'_2X(X)/b_2X(X) ,  b'_2Z(Z)/b_2Z(Z)=-c_1(Z)/b_1(Z)c'_2Z(Z)/c_2Z(Z) .


Therefore, we have the material manifold (<ref>) for the edge dislocation distributions with the Burgers' vector density (<ref>). For the sake of simplicity of calculations, in the remaining of this section we consider two simplified cases of the distribution (<ref>): (i) b_2(X,Y,Z)=c_2(X,Y,Z)=0, γ(Y)=0, λ(X)=0, ψ(Z)=0, and (ii) b_2(X,Y,Z)=c_2(X,Y,Z)=0, γ(Y)=0, λ(X)=0, c_1(Z)=0, η(Z)=0.




  
Case (i).From (<ref>), the Burgers' vector density reads 𝐛=𝐛(Z)=b_1(Z)𝐞_1+c_1(Z)𝐞_2=b_1(Z)e^-ξ(Z)∂_X+e^-η(Z)c_1(Z)∂_Y, where, using (<ref>), ξ'(Z)=b_1(Z) and η'(Z)=-c_1(Z). The material metric (<ref>) is simplified as 𝐆=diag{e^2ξ(Z),e^2η(Z),1}. Looking for solutions of the form (x,y,z)=(X,Y,α Z), the incompressibility constraint implies that J=α/e^ξ(Z)+η(Z)=1, and thus, ξ(Z)+η(Z)=lnα. This means that ξ'(Z)+η'(Z)=0, and hence, c_1(Z)=b_1(Z). Choosing orthonormal vectors 𝐍_1=e^-ξ(Z)∂_X,  𝐍_2=e^-η(Z)∂_Y, and 𝐍_3=∂_Z as the orthotropic axes, the invariants of the energy function are obtained from (<ref>) as follows


    I_1=α^2+e^-2ξ(Z)+1/α^2e^2ξ(Z) ,   I_2=1/α^2+e^2ξ(Z)+α^2e^-2ξ(Z) ,   I_5=I_4^2=e^-4ξ(Z) ,   I_7=I_6^2=e^4ξ(Z)/α^4 .


Therefore, the non-zero components of the Cauchy stress tensor read


    σ̂^xx   =2e^-2ξ(Z)[W_I_1+(e^2ξ(Z)/α^2+α^2)W_I_2+W_I_4]+4e^-4ξ(Z)W_I_5-p(Z) ,
    σ̂^yy   =2/α^2e^2ξ(Z)[W_I_1+(e^-2ξ(Z)+α^2)W_I_2+W_I_6]+4/α^4e^4ξ(Z)W_I_7-p(Z) ,
    σ̂^zz   =2α^2[W_I_1+(e^-2ξ(Z)+e^2ξ(Z)/α^2)W_I_2]-p(Z) .


Equilibrium equations imply that σ̂^zz=C, where C is a constant. Vanishing of the traction vector on surfaces parallel to the X-Y plane gives the pressure as


    P(Z)=2α^2[W_I_1+(e^-2ξ(Z)+e^2ξ(Z)/α^2)W_I_2] .













  
Case (ii). The Burgers' vector density is given by 𝐛=𝐛(Z)=b_1(Z)𝐞_1=b_1(Z)e^-ξ(Z)∂_X, where ξ'(Z)=b_1(Z). From (<ref>), the material metric is 𝐆=diag{e^2ξ(Z),1,e^2ψ(Z)}. We then look for solutions of the form (x,y,z)=(X,Y,α Z). Incompressibility implies that J=α/e^ξ(Z)+ψ(Z)=1, and hence, ξ(Z)+ψ(Z)=lnα. The orthotropic axes are 𝐍_1=e^-ξ(Z)∂_X, 𝐍_2=∂_Y, and 𝐍_3=e^-ψ(Z)∂_Z. The invariants of the strain energy function read


    I_1=1+e^-2ξ(Z)+e^2ξ(Z) ,   I_2=1+e^2ξ(Z)+e^-2ξ(Z) ,   I_5=I_4^2=e^-4ξ(Z) ,   I_6=I_7=1 .


The non-zero components of the Cauchy stress are given as


    σ̂^xx   =2W_I_2+2e^-2ξ(Z)(W_I_1+W_I_2+W_I_4)+4e^-4ξ(Z)W_I_5-p(Z) ,
    σ̂^yy   =2W_I_1+2W_I_2(e^-2ξ(Z)+e^2ξ(Z))+2W_I_6+4W_I_7-p(Z) ,
    σ̂^zz   =2(W_I_2+e^2ξ(Z)(W_I_1+W_I_2))-p(Z) .


The equilibrium equation and the vanishing of traction vector on surfaces parallel to X-Y plane yield


    p(Z)=2(W_I_2+e^2ξ(Z)(W_I_1+W_I_2)) .
 









 §.§ A Spherically-Symmetric Distribution of Point Defects in a Transversely Isotropic Ball



In this section, we calculate the stress field of a spherically-symmetric distribution of point defects in a transversely isotropic ball of radius R_o. The material manifold of a medium with distributed point defects is a flat Weyl manifold <cit.>. Let us assume that the material preferred direction is radial, i.e., 𝐍=𝐑̂,[Note that 𝐑̂=1/f(R)𝐄_R is the unit vector identifying the material preferred direction, where 𝐄_R=∂/∂ R such that ⟨⟨𝐄_R,𝐄_R⟩⟩_𝐆=G_RR.] where 𝐑̂ is a unit vector in the radial direction. The material metric for the body with a radial distribution of point defects in the spherical coordinates (R,Θ,Φ) reads 𝐆=diag{f^2(R),R^2,R^2sin^2Θ}, where


    f(R)=1-𝗇(R)/1-1/R^3∫_0^R3y^2𝗇(y)dy .


We endow the ambient space with the flat Euclidean metric 𝐠=diag{1,r^2,r^2sin^2θ} in the spherical coordinates (r,θ,ϕ). Given an embedding of the form (r,θ,ϕ)=(r(R),Θ,Φ), the deformation gradient is written as 𝐅=diag{r'(R),1,1}. The right Cauchy-Green deformation tensor reads 𝐂=diag{r'^2(R)/f^2(R),r^2(R)/R^2,r^2(R)/R^2}. Assuming incompressibility, the Jacobean is expressed as


    J=√(det𝐠/det𝐆)det𝐅=r^2(R)r'(R)/R^2f(R)=1 .


This gives r(R)=(∫_0^R3ξ^2f(ξ)dξ)^1/3. Using (<ref>), the invariants are written as


    I_1=tr(𝐂)=R^4/r^4(R)+2r^2(R)/R^2 ,   
    	I_2=1/2(tr(𝐂^2)-tr(𝐂)^2)
    	=r^4(R)/R^4+2R^2/r(R)^2 ,   I_5=I_4^2=R^8/ r^8(R) .


Using (<ref>), the non-zero stress components read[When the body is defect-free, f(R)=1, and thus, I_1=I_2=3 and I_4=I_5=1. If one assumes that the stress vanishes in this case, one has (see  <cit.> for similar conditions)
	
	
    (2 
    	W_I_5+ W_I_4)|_I_1=I_2=3, I_4=I_5=1=0 .

	
]


    σ̂^rr   =2R^4/r^4(R)(W_I_1+W_I_4)
    	+4R^2/r^2(R)W_I_2+4R^8/r^8(R)W_I_5-p(R) ,
    σ̂^θθ   =σ̂^ϕϕ=2r^2(R)/R^2W_I_1
    	+2R^2/r^2(R)W_I_2+2r^4(R)/R^4W_I_2-p(R) .


The non-trivial equilibrium equation is simplified to read 1/r'(R)σ^rr_,R+2/rσ^rr-2rσ^θθ=0. This gives p'(R)=q(R), where


    q(R)=   -4/R^3 r^19[f {R^9 r^12(W_I_1+4 W_I_2I_2-4W_I_2I_5+W_I_4)
    	+R^3 r^18(W_I_1-4W_I_2I_2)
       +R^7 r^14[W_I_2
    	-2(W_I_1I_1+W_I_1I_4)]+2 R^13 r^8(W_I_1I_1+2W_I_1I_4+W_I_4I_4
    	+3W_I_5)
       +2 R^11 r^10(3 W_I_1I_2-2 W_I_1I_5+3W_I_2I_4)
    	-2 R^5 r^16(3W_I_1I_2+W_I_2I_4)+8 R^17 r^4(W_I_1I_5+W_I_4I_5)
       +R W_I_2 r^20+12 R^15W_I_2I_5 r^6+8 R^21W_I_5I_5}-2 r^3 
    	{R^6 r^12(W_I_1+2W_I_2I_2-2W_I_2I_5+W_I_4)
       +R^4r^14(W_I_2-W_I_1I_1-W_I_1I_4)+R^10 r^8(W_I_1I_1+2W_I_1I_4+W_I_4I_4
    	+4W_I_5)
       +R^8 r^10(3W_I_1I_2-2 W_I_1I_5+3W_I_2I_4)-R^2 r^16(3W_I_1I_2+W_I_2I_4)+4R^14 r^4(W_I_1I_5+W_I_4I_5)
       -2 W_I_2I_2 r^18+6 R^12W_I_2I_5 r^6+4 R^18W_I_5I_5}] .


Next, we assume an energy function corresponding to a radially reinforced Mooney-Rivlin spherical ball of the following form


    W(I_1,I_2,I_4,I_5) =μ_1/2(I_1-3)+μ_2/2(I_2-3)+γ_1/2(I_4-1)^2
    +γ_2/2(I_5-1)^2 ,
 

where μ_1 and μ_2 are constants of the Mooney-Rivlin base material, while γ_1 and γ_2 are non-negative material constants pertaining to the reinforcement strength in the radial direction. Thus, (<ref>) is simplified to read


    p'(R)=   -2/R^2
    	r^19[f {6 R^12 (γ_1-2 γ_2) r^8+R^8 (μ_1-2 γ_1) r^12+μ_2
    R^6 r^14+μ_1 R^2
    r^18+μ_2 r^20+28
    γ_2 R^20}
       -2 R^3 r^3
    (4 R^6 (γ_1-2 γ_2) r^8+R^2 (μ_1-2 γ_1) r^12+μ_2 r^14+16
    γ_2 R^14)] .


The stress components are also simplified and read


    σ̂^rr   =R^4/r^4[μ_1+2γ_1(R^4/r^4-1)]
    	+2μ_2R^2/r^2+4γ_2R^8/r^8(R^8/r^8-1)-p ,
    σ̂^θθ   =σ̂^ϕϕ=μ_1r^2/R^2+μ_2R^2/r^2+μ_2r^4/R^4-p .


Assuming that the boundary of the ball is traction-free, one obtains


    p(R_o)=R^4_o/r^4(R_o))[μ_1+2γ_1(R^4_o/r^4(R_o)-1)]
    	+2μ_2R^2_o/r^2(R_o)+4γ_2R^8_o/r^8(R_o)(R^8_o/r^8(R_o)-1) .


Thus


    p(R)=   2∫_R^R_o1/ξ^2
    	r^19(ξ)[f(ξ) {6 ξ^12 (γ_1-2 γ_2) r^8(ξ)
    	+ξ^8 (μ_1-2 γ_1) r^12(ξ)+μ_2ξ^6 r^14(ξ)+μ_1 ξ^2r^18(ξ)
       +μ_2 r^20
    	+28γ_2 ξ^20}-2 ξ^3 r^3(ξ)(4 ξ^6 (γ_1-2 γ_2) r^8(ξ)
    	+ξ^2 (μ_1-2 γ_1) r^12+μ_2 r^14(ξ)
       +16γ_2 ξ^14)]dξ+p(R_o) .


Let us consider the following distribution of point defects in the ball


    𝗇(R)=
    𝗇_o     0≤ R≤ R_i ,
    
    0     R_i<R≤ R_o .


Therefore, from (<ref>)


    f(R)=
    
    1 ,                   0≤ R≤ R_i ,
    
    (1-𝗇_o(R_i/R)^3)^-1 ,  R_i<R≤ R_o ,


and hence


    r(R)=
    
    R ,                                     0≤ R≤ R_i ,
    [R^3+𝗇_oR_i^3ln(R/R_i)^3-𝗇_o/1-𝗇_o]^1/3 ,  R_i<R≤ R_o .


Fig. <ref> shows the stress field variation for the point defect distribution (<ref>), where R_i/R_o=0.3 and 𝗇_o=-0.1 for different values of the reinforcement and base material constants in (<ref>).








Consider an arbitrary nonlinear incompressible transversely isotropic spherical ball of radius R_o such that the material preferred direction is radial. Suppose that the ball is subject to a uniform pressure on its boundary and has the point defect distribution (<ref>). Then, in the ball R≤ R_i, the stress is uniform and hydrostatic. Interestingly, the value of the hydrostatic stress inside the ball R≤ R_i has an explicit dependence on the reinforcement parameters (see Fig. <ref>). To show this, for R≤ R_i, f(R)=1 and r(R)=R, following (<ref>) and (<ref>), respectively. Therefore, after some simplification, (<ref>) implies that p'(R)=q(R)=4/R(W_I_4+2W_I_5)|_I_1=I_2=3,I_4=I_5=1=0, where we used the relation (<ref>). Hence, for R≤ R_i, p(R)=C, where C is a constant depending on the reinforcement and base material parameters. From (<ref>), σ̂^rr=σ̂^θθ=σ̂^ϕϕ=2(W_I_1+2W_I_2)|_I_1=I_2=3,I_4=I_5=1-C for R≤ R_i.









































































































































§ CONCLUDING REMARKS



Despite the crucial role that anisotropy plays in the overall response of materials in the presence of large strains, the study of defects in nonlinear solids has been overwhelmingly restricted to isotropic materials to this date. In this paper, we presented a few analytical solutions for the stress fields induced by distributed line and point defects in nonlinear anisotropic solids. We considered a parallel cylindrically-symmetric distribution of screw dislocations in infinite orthotropic and monoclinic media, and also, a cylindrically-symmetric distribution of parallel wedge disclinations in an orthotropic medium. 
Because the material manifold is endowed with a nontrivial Riemannian metric that explicitly depends on the defect distribution, the material preferred directions, and hence, the class of anisotropy of the defective body are, in general, different in the reference and current configurations. We observed, in particular, that for a cylindrically-symmetric distribution of screw dislocations, assuming that the body is orthotropic in the reference (current) configuration, it is monoclinic in its current (reference) configuration. 
We found that for an arbitrary cylindrically-symmetric distribution of parallel screw dislocations and a uniform wedge disclination distribution, the stress field is logarithmically singular on the dislocation and disclination axes, respectively, unless the axial deformation is suppressed. These stress singularities are inherent to the anisotropic effects due to the radial fiber-reinforcement, and do not, in particular, arise in isotropic materials. This observation demonstrates the significance of taking material anisotropy into consideration in the analysis of solids with distributed defects.
For a single screw dislocation, we employed the standard reinforcing model and discussed the conditions that guarantee that the energy per unit length and the resultant axial force are finite for a fiber-reinforced material as long as the isotropic base material has a finite energy per unit length and a finite axial force. For a distribution of edge dislocations the resulting stress are calculated when the medium is orthotropic. Finally, we studied a spherically-symmetric distribution of point defects in a transversely isotropic spherical ball. We showed that for an arbitrary incompressible transversely isotropic ball with the radial material preferred direction, a uniform point defect distribution results in a uniform hydrostatic stress field inside the spherical region the distribution is supported in.     
The role that anisotropy plays in the dynamics, stability, and interactions of defects at finite strains are exciting problems that will be the subjects of future communications.  





§ ACKNOWLEDGEMENT


This work was partially supported by NSF – Grant No. CMMI 1561578,  ARO Grant No. W911NF-18-1-0003, and AFOSR – Grant No. FA9550-12-1-0290.  


abbrvnat






