On the Stress Field of a Nonlinear Elastic Solid Torus with a Toroidal Inclusion

In this paper we analyze the stress field of a solid torus made of an incompressible isotropic solid with a toroidal inclusion that is concentric with the solid torus and has a uniform distribution of pure dilatational finite eigenstrains. We use a perturbation analysis and calculate the residual stresses to the first order in the thinness ratio (the ratio of the radius of the generating circle and the overall radius of the solid torus). In particular, we show that the stress field inside the inclusion is not uniform. This is in contrast with the corresponding results for infinitely-long and finite circular cylindrical bars and spherical balls with cylindrical and spherical inclusions, respectively. We also show that for a solid torus of any size made of an incompressible linear elastic solid with an inclusion with uniform (infinitesimal) pure dilatational eigenstrains the stress inside the inclusion is not uniform.


Introduction
Eigenstrains are the anelastic part of the total strain tensor and represent referential rearrangements, changes, distortions, etc. When deformations (more precisely displacement gradients) are large different measures of strain may be considered and an eigenstrain would explicitly depend on the choice of a strain measure. Eigenstrains model many different phenomena, e.g., plasticity [2,14], thermal strains [25,37], swelling [26,27], and bulk growth B A. Yavari 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  [31,38,39,42]. For a detailed discussion of finite eigenstrains see Yavari and Goriely [45] and Golgoon et al. [6].
In a seminal paper, Eshelby [4] showed that for an ellipsoidal inclusion in an infinite linear elastic solid, for uniform eigenstrains the stress inside the inclusion is uniform as well. There have been many investigations in recent years on the validity of this uniformity property for nonlinear elastic solids and inclusions with finite eigenstrains. There are several results in 2D in the case of harmonic solids [9-11, 32, 33]. In 3D, recently Yavari and Goriely [45] showed that in the case of cylindrical bars (finite or infinitely-long) and spherical balls with cylindrical and spherical inclusions, respectively, with pure dilatational finite eigenstrains, the stress uniformity property holds for both incompressible isotropic solids and some special classes of compressible isotropic solids. Note that these geometries are simply-connected. Perhaps the simplest example of a non-simply connected body is a hollow cylinder. However, in that case one can only have an annular inclusion. Another simple example of a non-simply connected body is a solid torus. To our best knowledge, finite (or infinitesimal) eigenstrains in a solid torus and their induced residual stresses have not been studied in the literature. In this paper, we investigate this problem in the case of incompressible solids (see Fig. 1).
Kydoniefs and Spencer [17] and Kydoniefs [15] studied the finite deformation of a torus made of a homogeneous, isotropic, incompressible elastic solid under inflation by uniform internal pressure and under inflation and rotation with a constant angular velocity, respectively. They assumed that the torus in its deformed state is generated by rotating two concentric circles about a line in their plane. Assuming that the radii of the generating circles are small compared to the overall radius of the torus, they obtained approximate solutions for the stress and deformation fields in the torus. Their work was further extended to a solid torus inflated from a torus in its undeformed state by Hill [7]. He too assumed that the ratio of the radius of the generating circles and the overall radius of the torus (thinness ratio) is small and obtained the solutions to the first order of this small ratio. Under the same assumption, Kydoniefs and Spencer [18] explored the finite inflation of an elastic toroidal membrane due to uniform internal pressure such that it has a circular cross section in its reference configuration. They obtained the solutions to the second order in the thinness ratio and presented some numerical results for a toroidal membrane made of a Mooney-Rivlin material, describing the dependence of the deformation and the generated stresses on the internal pressure. Krokhmal [13] studied the displacement boundaryvalue problem of a linear elastic torus. He reduced the boundary-value problem to an infinite system of linear algebraic equations and developed an analytical technique for solving it.
Toroidal inclusions and inhomogeneities have been observed in the microstructures of both natural and engineered materials [29]. Onaka et al. [24] investigated the problem of elastic toroidal inclusions in an infinite linear elastic medium using an averaged Eshelby tensor. They found that the averaged Eshelby tensor of toroidal inclusions on an arbitrary plane is nearly the same as the average of the Eshelby tensors of randomly oriented rodlike inclusions on that plane. Onaka [22] considered an infinitely extended body having a doughnut-like inclusion with purely dilatational eigenstrains. He observed that near the inclusion there are two points at which all the components of the strain tensor vanish. Note that this is not the case for spherical inclusions with purely dilatational eigenstrains placed in an infinite linear elastic medium, for which strains become null only at infinitely far distances from the inclusion. In another paper by Onaka [23], the strain field generated by elongated toroidal inclusions were studied and compared with that of doughnut-like and spherical inclusions. It was observed that for an infinitely elongated tubular inclusion, all the strain tensor components in the matrix region surrounded by the inclusion vanish. The reinforcing effects of rigid toroidal inhomogeneities in a linear elastic medium was studied by Argatov and Sevostianov [1]. They observed that there is no noticeable difference in the reinforcing properties of toroidal and spheroidal inhomogeneities with the same volume and diameter. Kirilyuk [12] investigated the effects of a toroidal inhomogeneity on the stress concentration in an infinite isotropic medium. They considered two cases: perfect bonding and slipping at the inhomogeneity-matrix interface. As an example, they showed that the difference in the maximum stress could differ up to 40 % for the two cases.
In the setting of linearized elasticity, it is known that for a single inclusion with uniform eigenstrain in an infinite domain to have a uniform stress field the inclusion must be an ellipsoid [8,19]. Earlier, Rodin [30] had shown that the remarkable property of ellipsoidal inclusions is not shared by polygonal inclusions in 2D or polyhedral inclusions in 3D. In particular, a toroidal inclusion with uniform eigenstrain in an infinite solid would have a nonuniform stress field. One may now consider a solid torus with an inclusion whose generating circle is concentric with the boundary circle of the solid torus (see Fig. 1). Is the stress field inside such an inclusion with uniform and pure dilatational eigenstrain uniform? We solve this problem for finite dilatational eigenstrains in the case of a "thin" solid torus made of an incompressible isotropic nonlinear elastic solid. We will show that to the first order in the thinness ratio, stress inside the inclusion is not uniform. We then study the same problem for a solid torus made of an incompressible linear elastic solid with a toroidal inclusion with a uniform infinitesimally small pure dilatational eigenstrain. We show that for any size of the solid torus (not necessarily thin) the stress inside the inclusion is not uniform.
This paper is organized as follows. In Sect. 2 we briefly review some basic concepts of the geometric theory of nonlinear elasticity. In Sect. 3 we formulate the governing equilibrium equations of a solid torus with an axially-symmetric distribution of finite eigenstrains. In Sect. 3.1 we consider a toroidal inclusion that is concentric with the solid torus and calculate the residual stress field using a perturbation analysis. We then present some numerical examples for neo-Hookean solids. Finally, we solve the corresponding problem in linear elasticity in Sect. 3.2. Conclusions are given in Sect. 4.

Elements of Geometric Anelasticity
In this section, we tersely review some fundamental elements of the geometric theory of nonlinear elasticity and anelasticity. For more detailed discussions, see [20,43].
Kinematics A body B is assumed to be identified with a Riemannian manifold (B, G), and a configuration of B is a smooth embedding ϕ : B → S, where (S, g) is also assumed to be a Riemannian manifold. An affine connection ∇ on a smooth manifold M is a linear map ∇ : indicates the set of all smooth vector fields on M, that has to satisfy some specific properties (see do Carmo [3] for details). It turns out that there is a unique torsion-free and compatible affine connection associated with any Riemannian manifold, referred to as Riemannian connection (see, for example, [3,28]). We denote the Levi-Civita connection associated with the Riemannian manifold (S, g) by ∇ g . The set of all configurations of B is denoted 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 . It is assumed that the body is stress-free in its reference configuration, which may have a nontrivial geometry, e.g., in the presence of eigenstrains. The deformation gradient F is the derivative map of ϕ defined as The adjoint of F is defined as follows The Finger deformation tensor is defined as b( Another measure of strain is the Lagrangian strain tensor that is defined as E = 1 2 (ϕ * t g−G). The Jacobian of deformation J relates the Riemannian volume elements of the material manifold dV (X, G) and the spatial manifold dv(ϕ t (X), g) and is written as

Constitutive Equations
For isotropic solids the energy function W depends only on the principal invariants of b, denoted by I 1 , I 2 , and I 3 . In the case of incompressible solids, I 3 = 1, and hence, W = W (X, I 1 , I 2 ). We restrict our attention to isotropic incompressible hyperelastic solids, for which the Cauchy stress has the following representation [21,35] where p is the Lagrange multiplier associated with the internal incompressibility condition, and W I 1 := ∂W ∂I 1 , W I 2 := ∂W ∂I 2 . Note that b and C have the same principal invariants, and b = ϕ * (G ). We assume that the body in the absence of eigenstrains is isotropic. Eigenstrains are modeled by a material metric G that explicitly depends on the distribution of eigenstrains [5,45,47]. In other words, stress-free configuration of a body with a distribution of eigenstrains may not be globally realizable in the Euclidean ambient space.

Equilibrium Equations
The localized spatial balance of linear momentum of a body in static equilibrium in the absence of body forces in terms of the Cauchy stress reads divσ = 0. In components (divσ ) a = σ ab |b = ∂σ ab ∂x b + σ ac γ b cb + σ cb γ a cb , (2.5) where γ a bc denotes the Christoffel symbols of the connection ∇ g in the local charts {x a }, defined as ∇ g ∂ b ∂ c = γ a bc ∂ a . Moreover, the Christoffel symbols of the Levi-Civita connection In this paper we model finite eigenstrains in a nonlinear elastic solid by defining a Riemannian material manifold, which has a metric that explicitly depends on the distribution of eigenstrains. This idea has been discussed in detail in our previous works [6,45,47,48].

An Incompressible Isotropic Solid Torus with Axially-Symmetric Finite Eigenstrains
In this section we consider a solid torus generated by rotating a circle with radius R o about a line in its plane such that the distance from the origin to the center of the circle is B. Let (R, Θ, Φ) and (r, θ, φ) be the material and spatial toroidal coordinates as illustrated in Fig. 2. In the toroidal coordinates (R, Θ, Φ), the metric of the eigenstrain-free torus is written as We assume an axially-symmetric (Θ-independent) eigenstrain (pre-strain) distribution in the torus. Following the construction suggested by Yavari and Goriely [45] to model eigenstrains, we consider the following material metric 1 where Ω(R, Φ) is an arbitrary function describing the inhomogeneous dilatational eigenstrain distribution in the torus. The ambient space is endowed with the Euclidean metric, which in the toroidal coordinates (r, θ, φ) has the following representation Let us consider an axially-symmetric class of deformations of the following form The deformation gradient for this class of deformations reads We assume an incompressible solid, i.e., J = detg detG detF = 1, which gives us 2 The Finger deformation tensor reads The first two principal invariants of b are (I 3 = 1) where .
The inverse of Finger tensor b −1 = c is written as Following (2.4), the non-zero components of the Cauchy stress read The physical components of the Cauchy stress are calculated using the relationσ ab = σ ab √ g aa g bb [40]. Thuŝ The non-zero components of the first Piola-Kirchhoff stress tensor, i.e., P aA = J (F −1 ) A b σ ab are written as The Christoffel symbol matrices of g read (cf. (2.6)) (3.14) In the absence of body forces, the non-trivial equilibrium equations are σ rb |b = 0 and σ φb |b = 0, which after simplification read (the equilibrium equation in the θ -direction gives p = p(R, Φ)) ∂σ rr ∂r + ∂σ rφ ∂φ (3.17)

Boundary Conditions
As we are interested in finding the residual stress field, we assume that the boundary of the torus is traction-free, i.e., Finding an exact solution of the PDEs (3.15) and (3.16) does not seem feasible. Therefore, we seek approximate solutions assuming that the radius of the cross section generating the torus is small compared to the radius of revolution [7,[15][16][17][18]. Hence, we find the solution assuming that R/B and r/b, which are of the same order, are sufficiently small so that the second and higher powers of R/B (and r/b) can be neglected. In doing so, the problem is essentially a perturbation of the problem of finite eigenstrains in an infinitely-long circular cylindrical bar, which was discussed in [45] for the special case of cylindrically-symmetric distribution of eigenstrains.

A Nonlinear Solid Torus with Finite Eigenstrains and r/b 1 and R/B 1
In this section, we restrict our attention to the radially-symmetric dilatational eigenstrain distributions, for which Ω = Ω(R). Moreover, we assume that r/b and R/B are sufficiently small and find the solutions to the first order in the thinness ratio ε = R o /B 1. For the zero-order problem (ε → 0), the torus becomes a cylinder with the cylindrically-symmetric distribution of purely dilatational eigenstrains, for which, in the cylindrical coordinates, r = r(R), φ = Φ, z = b B Z, and p = p(R). Therefore, we consider the following asymptotic expansions 3 Substituting (3.19) into (3.6) and equating the zero and the first-order terms on both sides one gets Similarly Denoting α = dr (0) dR , one has Assuming that the material is piecewise homogeneous 4 one can write where W (0) 11d) one obtains the following expressions for the non-zero Cauchy stress components Using (3.17) Assuming that r(0) = 0, (3.20) has the following solution It follows from (3.28) that dp (0) (R) and k = B/b. Note that (3.33)

Example: A Toroidal Inclusion with Uniform Pure Dilatational Eigenstrains in a Neo-Hookean Solid Torus
Let us consider the following distribution of eigenstrains We assume that the torus is made of an incompressible homogeneous neo-Hookean solid, i.e., W = μ 2 (I 1 − 3), where μ is the shear modulus at the ground state. Therefore, it follows from (3.31) that where γ o = e 3Ωo 2 − 1. Using (3.32), we find the zero-order pressure field as 5 where c o and c i are constants to be determined after enforcing the boundary conditions (3.18) and the continuity of the traction vector on the inclusion-matrix interface. The continuity of the traction vector on the boundary of the inclusion implies that σ rr (0) must be continuous at R = R i . Therefore, c o and c i are computed as (3.37) The zero-order stress components are simplified to read σ rr (3.38c) Substituting (3.35) into (3.21), one obtains the following relations in the inclusion and the matrix (3.39) The expressions (3.27a)-(3.27g) for the first-order stress components are simplified to read σ rr (1) kμ 1+γo (3.40d) We now seek a solution of the following form (see Appendix B for a proof of this representation) 6 Substituting (3.41) into (3.29) and (3.30) one obtains 44a) 6 Solutions with a similar form were discussed in [15,17].
It then follows that (3.45a) has the following solution After some simplifications, the boundary conditions (3.18) give one the following relations The continuity of the displacement field at the inclusion-matrix interface implies that The traction vector is defined as t = σ , n g , which in components reads t a (x, n) = σ ac g bc n b . The unit normal vector to the inclusion-matrix interface to the first order in ε reads n =r + n We now use the method of variation of parameters to find the particular solution of (3.45b), which is denoted by f 57) 7 Note that one can easily verify that (3.55) and (3.56) are indeed the homogeneous solutions of (3.45b) for R ∈ [R i , R o ], and therefore, using the power series method is justified. 8 The dilogarithm function is defined as: (3.63d) Remark 3.1 Note that when R i = R o , i.e., when the entire solid torus has a uniform pure dilatational eigenstrain, no residual stresses are generated. In this case, we recover the exact solution, for which r/R = b/B = e Ωo 2 . Note that this is indeed the exact solution, as it is stress-free, and thus, the equilibrium equations are trivially satisfied. Also, it satisfies the incompressibility condition (3.6).

Remark 3.2
One can simply check that in the first-order approximation with respect to the thinness ratio the deformed shapes of the outer boundaries of the inclusion and the matrix remain circular with radii equal to those of their corresponding zero-order approximations, but they become eccentric with eccentricity E = f o (R o ) − f i (R i ). Note that the inclusion and the matrix outer boundary points rotate with respect to one another after deformation such that their relative rotation is (Φ) = (g o (R o ) − g i (R i )) sin Φ for any pair of points located at an angle Φ in the initial configuration on the outer boundaries of the inclusion and the matrix. Notice that all the first-order stress components vary linearly with the material radial coordinate in the inclusion. As expected all the stress components undergo a jump at the inclusion-matrix boundary except the radial stress component, which is continuous at the interface. For the positive eigenstrain case (Fig. 3), the maximum shear stress in the inclusion and matrix is first increasing, then decreasing as R i /R o increases from zero (a torus without eigenstrain) such that the maximum shear stress for a torus with R i /R o = 0.4 is greater than that of a torus with R i /R o = 0.2 and R i /R o = 0.6 in both the inclusion and the matrix. For the case of negative eigenstrain, the maximum shear stress in the torus increases with the increase in R i /R o ratio from 0 to 0.8. After that, however, as R i /R o increases the maximum shear stress decreases until it becomes zero when R i /R o = 1 (Fig. 4). Note that R i /R o = 1 corresponds to the entire torus having a uniform pure dilatational eigenstrain distribution, which is stress-free as was discussed earlier in Remark 3.1.
The contour plots of the first-order residual stress components for a torus with R o /B = 0.1 are depicted in Figs. 5, 6, 7. A torus with an inclusion with a negative pure dilatational eigenstrain and R i /R o = 0.4 is shown in Fig. 5. One observes that the shear stress concentrates across the inclusion-matrix interface with its maximum attained at the top and the bottom. For the selected parameters, the first-order circumferential stress componentσ φφ (1) /μ is negligible in the inclusion, and hence, the circumferential stress component remains uniform to the first order in the inclusion. This is also true for the positive eigenstrain cases Ω o = 0.5 and Ω o = 0.7 with R i /R o = 0.4 and 0.2, respectively (Figs. 6, 7).  inclusion from the origin in the initial and the deformed configurations, respectively (see Fig. 2).) For positive eigenstrain values, b/B monotonically increases as Ω o increases, and as expected, the higher the R i /R o ratio, the more rapid the increase. For negative eigenstrains, nevertheless, b/B reaches a minimum, which decreases as R i /R o increases, and is attained at lower eigenstrain values. As was mentioned earlier for both negative and positive eigenstrains, as R i /R o approaches 1, the b/B curve gets closer to e Ωo 2 (see Remark 3.1). The variation of the eccentricity ratio Fig. 8b for several values of R i /R o . As Ω o increases from 0, the eccentricity decreases until it reaches its minimum, which increases as R i /R o increases, and is attained at lower values of Ω o . 11 For Ω o < 0, the eccentricity ratio is first increasing, then decreasing as Ω o decreases starting from zero. The maximum eccentricity corresponds to the lower values of eigenstrains as R i /R o increases. Moreover, the maximum eccentricity first increases as R i /R o increases, then it decreases. For instance, the maximum eccentricity for a torus with R i /R o = 0.7 is greater than that of a torus with R i /R o = 0.5 and R i /R o = 0.9 when 11 Note that positive and negative eccentricity values correspond to the inclusion moving to the left and right relative to the matrix, respectively. Ω o < 0. As R i /R o approaches 1, the eccentricity ratio tends to zero for any value of the eigenstrain Ω o .

A Linear Elastic Solid Torus with Small Eigenstrains
In this section we derive the governing equations of a solid torus made of an incompressible linear elastic material that has a distribution of small eigenstrains. In geometric elasticity, in order to linearize one starts with a reference motion • ϕ and a one-parameter family of motions ϕ such that ϕ =0 = • ϕ [20,49]. Let us consider a one-parameter family of motions ϕ such that ϕ (R, Θ, Φ) = (r (R, Φ), Θ, φ (R, Φ)). We will linearize about the stressfree configuration The variation field is defined as where u and w are the non-zero displacement components. Linearization of the Incompressibility Constraint For any motion in the given oneparameter family we have Taking derivative with respect to of both sides and evaluating at = 0, one obtains Similarly, from (3.9), one obtains the variation of I and β as Therefore, it follows from (3.67) that δI 1 = δI 2 = 0. To simplify the calculations, we assume that the material is piecewise homogeneous and use (3.11a)-(3.11d) and (3.67) to find the

Linearization of the Equilibrium Equations
Substituting the linearized stress components given by (3.68a)-(3.68d) into the above equations one finds

Linearization of the Boundary Conditions
Similarly, boundary conditions (3.18) are linearized and are written as (3.73b)

Example: A Toroidal Inclusion with Uniform Pure Dilatational Eigenstrains in an Incompressible Linear Elastic Solid Torus
Let us consider the following distribution of eigenstrains in the torus Pressure and displacement fields in the torus are written as (3.75) Therefore, from (3.66) it follows that From the continuity of the displacement field at the inclusion-matrix interface, we know that Also, we eliminate the rigid body motion by setting u i (0, Φ) = 0 and w(R, 0) = 0. We next show that for a torus made of an isotropic incompressible linear elastic solid with a toroidal inclusion having a non-zero uniform pure dilatational eigenstrain distribution, the stress field inside the inclusion cannot be uniform. Let us assume that the stress field inside the inclusion is uniform, i.e., each physical Cauchy stress component is constant. Thus where c 1 to c 4 are some constants. After some simplifications, it follows from (3.69) that which implies that c 1 = c 3 = c 4 = C and c 2 = 0. Therefore (note that the equilibrium equation (3.70) has already been satisfied) From (3.68a)-(3.68d) and δσ rr − R 2 δσ φφ = 0, one obtains (3.82) Using the above relation in δσ rr − (B + R cos Φ) 2 δσ θθ = 0, one finds This is a contradiction because δb has to depend on the radius of the inclusion R i . In other words, the above relation is telling us that the change in the overall radius of the solid torus after deformation is independent of the size of the inclusion. In particular, when R i → 0 we expect δb → 0, which is not what (3.86) predicts. This contradiction shows that the stress field inside the inclusion cannot be uniform.

Conclusions
In this paper, we studied the residual stress and deformation fields of a solid torus containing a toroidal inclusion with finite eigenstrains that is concentric with the solid torus. We used a perturbation analysis and obtained the stress and displacement fields to the first order in the thinness ratio. We showed that the stress field in the toroidal inclusion is nonuniform, unlike cylindrical and spherical inclusions in infinitely-long and finite circular cylindrical bars and spherical balls, respectively, in which the stress field inside the inclusion is uniform. We presented some numerical results for a neo-Hookean solid torus having an inclusion with a uniform pure dilatational eigenstrain distribution. In particular, we observed that all the first-order stress components in the inclusion have a linear dependence on the referential radial coordinate. Moreover, the maximum shear stress in the torus is first increasing, then decreasing as the relative size of the inclusion increases from zero. We observed shear stress concentration regions across the inclusion-matrix interface for a torus with a negative pure dilatational eigenstrain distribution. Interestingly, the torus exhibits different responses for positive and negative eigenstrain values. It was observed that for the positive eigenstrains, b/B monotonically increases as the eigenstrain Ω o increases, and the increase is more rapid for inclusions with larger relative sizes. For negative eigenstrains, nonetheless, b/B reaches a minimum, the value of which decreases as the relative size of the inclusion becomes larger. We noticed that in the first-order approximation with respect to the thinness ratio the deformed shapes of the outer boundaries of the matrix and the inclusion are eccentric circles with radii equal to those of their corresponding zero-order approximations. Finally, we proved that the stress field inside a toroidal inclusion with nonzero uniform pure dilatational (infinitesimal) eigenstrains in an isotropic incompressible linear elastic solid torus is always nonuniform for any size of the solid torus.