Anisotropic Elastic Materials that Uncouple All Three Displacement Components, and Existence of One-Displacement Green's Function

It was shown in an earlier paper that, under a two-dimensional deformation, there are anisotropic elastic materials for which the antiplane displacement u ₃ and the inplane displacements u ₁, u ₂ are uncoupled but the antiplane stresses σ₃₁, σ₃₂ and the inplane stresses σ₁₁, σ₁₂, σ₂₂ remain coupled. The conditions for this to be possible were derived, but they have a complicated expression. In this paper new and simpler conditions are obtained, and a general anisotropic elastic material that satisfies the conditions is presented. For this material, and for certain monoclinic materials with the symmetry plane at x ₃ = 0, we show that the unnormalized Stroh eigenvectors a _k for k = 1, 2, 3 are all real. The matrix A =[a ₁, a ₂, a ₃] is a unit matrix when the material has a symmetry plane at x ₂ = 0. Thus any one of the u ₁, u ₂, u ₃ can be the only nonzero displacement, and the solution is a one-displacement field. Application to the Green's function due to a line of concentrated force f and a line dislocation with Burgers vector v in the infinite space, the half-space with a rigid boundary, and the infinite space with an elliptic rigid inclusion shows that one can indeed have a one-displacement field u ₁, u ₂ or u ₃. One can also have a two-displacement field polarized on a plane other than the (x ₁, x ₂)-plane. The material that uncouples u ₁, u ₂, u ₃ is not as restrictive as one might have thought. It can be triclinic, monoclinic, orthotropic, tetragonal, transversely isotropic, or cubic. However, it cannot be isotropic. This revised version was published online in August 2006 with corrections to the Cover Date.     

Bifurcation From Stability to Instability for a Free Boundary Problem Arising in a Tumor Model

We consider a time-dependent free boundary problem with radially symmetric initial data: σ_t − Δσ + σ = 0 if and σ(r,0)=σ₀(r) in {r < R(0)} where R(0) is given. This is a model for tumor growth, with nutrient concentration (or tumor cells density) σ(r,t) and proliferation rate then there exists a unique stationary solution (σ_S(r), R_S), where R_S depends only on the number . We prove that there exists a number μ_*, such that if μ < μ_* . . . then the stationary solution is stable with respect to non-radially symmetric perturbations, whereas if μ > μ_* then the stationary solution is unstable.     

Effects of material anisotropy and inhomogeneity on cavitation for composite incompressible anisotropic nonlinearly elastic spheres

The effects of material anisotropy and inhomogeneity on void nucleation and growth in incompressible anisotropic nonlinearly elastic solids are examined. A bifurcation problem is considered for a composite sphere composed of two arbitrary homogeneous incompressible nonlinearly elastic materials which are transversely isotropic about the radial direction, and perfectly bonded across a spherical interface. Under a uniform radial tensile dead-load, a branch of radially symmetric configurations involving a traction-free internal cavity bifurcates from the undeformed configuration at sufficiently large loads. Several types of bifurcation are found to occur. Explicit conditions determining the type of bifurcation are established for the general transversely isotropic composite sphere. In particular, if each phase is described by an explicit material model which may be viewed as a generalization of the classic neo-Hookean model to anisotropic materials, phenomena which were not observed for the homogeneous anisotropic sphere nor for the composite neo-Hookean sphere may occur. The stress distribution as well as the possible role of cavitation in preventing interface debonding are also examined for the general composite sphere.     

Measurement of length-scale and solution of cantilever beam in couple stress elasto-plasticity

Owing to the absence of proper analytical solution of cantilever beams for couple stress/strain gradient elasto-plastic theory, experimental studies of the cantilever beam in the micro-scale are not suitable for the determination of material length-scale. Based on the couple stress elasto-plasticity, an analytical solution of thin cantilever beams is firstly presented, and the solution can be regarded as an extension of the elastic and rigid-plastic solutions of pure bending beam. A comparison with numerical results shows that the current analytical solution is reliable for the case of σ0 〈〈 H 〈〈 E, where σ0 is the initial yield strength, H is the hardening modulus and E is the elastic modulus. Fortunately, the above mentioned condition can be satisfied for many metal materials, and thus the solution can be used to determine the material length-scale of micro-structures in conjunction with the experiment of cantilever beams in the micro-scale.     

Localized Thickening of a Compressed Elastic Band

An infinite elastic band is compressed along its unbounded direction, giving rise to a continuous family of homogeneous configurations that is parameterized by the compression rate β < 1 (β = 1 when there is no compression). It is assumed that, for some critical value β ₀, the compression force as a function of β has a strict local extremum and that the linearized equation around the corresponding homogeneous configuration is strongly elliptic. Under these conditions, there are nearby localized deformations that are asymptotically homogeneous. When the compression force reaches a strict local maximum at β ₀, they describe localized thickening and they occur for values of β slightly smaller than β ₀. Since the material is supposed to be hyperelastic, homogeneous and isotropic, the localized deformations are not due to localized imperfections. The method follows the one developed by A. Mielke for an elastic band under traction: interpretation of the nonlinear elliptic system as an infinite dimensional dynamical system in which the unbounded direction plays the role of time, its reduction to a center manifold and the existence of a homoclinic solution to the reduced finite dimensional problem in [A. Mielke, Hamiltonian and Lagrangian fiows on center manifolds, Lecture Notes in Mathematics 1489. Springer, Berlin Heidelberg New York, 1991]. The main difference lies in the fact that Agmon's condition does not hold anymore and therefore the linearized problem cannot be analyzed as in Mielke's work.     

Cavitation for incompressible anisotropic nonlinearly elastic spheres

In this paper, the effect ofmaterial anisotropy on void nucleation and growth inincompressible nonlinearly elastic solids is examined. A bifurcation problem is considered for a solid sphere composed of an incompressible homogeneous nonlinearly elastic material which is transversely isotropic about the radial direction. Under a uniform radial tensile dead-load, a branch of radially symmetric configurations involving a traction-free internal cavity bifurcates from the undeformed configuration at sufficiently large loads. Closed form analytic solutions are obtained for a specific material model, which may be viewed as a generalization of the classic neo-Hookean model to anisotropic materials. In contrast to the situation for a neo-Hookean sphere, bifurcation here may occur locally either to the right (supercritical) or to the left (subcritical), depending on the degree of anisotropy. In the latter case, the cavity has finite radius on first appearance. Such a discontinuous change in stable equilibrium configurations is reminiscent of the snap-through buckling phenomenon of structural mechanics. Such dramatic cavitational instabilities were previously encountered by Antman and Negrón-Marrero [3] for anisotropiccompressible solids and by Horgan and Pence [17] forcomposite incompressible spheres.     

Symmetric Trajectories for the 2N-Body Problem with Equal Masses

We consider the problem of 2N bodies of equal masses in for the Newtonian-like weak-force potential r ^−σ, and we prove the existence of a family of collision-free nonplanar and nonhomographic symmetric solutions that are periodic modulo rotations. In addition, the rotation number with respect to the vertical axis ranges in a suitable interval. These solutions have the hip-hop symmetry, a generalization of that introduced in [19], for the case of many bodies and taking account of a topological constraint. The argument exploits the variational structure of the problem, and is based on the minimization of Lagrangian action on a given class of paths.     

An internal crack problem for an infinite elastic layer

The elastostatic plane problem of an infinite elastic layer with an internal crack is considered. The elastic layer is subjected to two different loadings, (a) the elastic layer is loaded by a symmetric transverse pair of compressive concentrated forces P/2, (b) it is loaded by a symmetric transverse pair of tensile concentrated forces P/2. The crack is opened by an uniform internal pressure p ₀ along its surface and located halfway between and parallel to the surfaces of the elastic layer. It is assumed that the effect of the gravity force is neglected. Using an appropriate integral transform technique, the mixed boundary value problem is reduced to a singular integral equation. The singular integral equation is solved numerically by making use of an appropriate Gauss–Chebyshev integration formula and the stress-intensity factors and the crack opening displacements are determined according to two different loading cases for various dimensionless quantities.     

Reversible Bifurcation of Homoclinic Solutions in Presence of an Essential Spectrum

We consider bifurcations of a class of infinite dimensional reversible dynamical systems which possess a family of symmetric equilibria near the origin. We also assume that the linearized operator at the origin L_ɛ has an essential spectrum filling the entire real line, in addition to the simple eigenvalue at 0. Moreover, for parameter values ɛ < 0 there is a pair of imaginary eigenvalues which meet in 0 for ɛ = 0, and which disappear for ɛ > 0. The above situation occurs for example when one looks for travelling waves in a system of superposed perfect fluid layers, one being infinitely deep. We give quite general assumptions which apply in such physical examples, under which one obtains a family of bifurcating solutions homoclinic to every equilibrium near the origin. These homoclinics are symmetric and decay algebraically at infinity, being approximated at main order by the Benjamin–Ono homoclinic. For the water wave example, this corresponds to a family of solitary waves, such that at infinity the upper layer slides with a uniform velocity, over the bottom layer (at rest).     

Symmetry of Ground States of Quasilinear Elliptic Equations

. We consider the problem of radial symmetry for non‐negative continuously differentiable weak solutions of elliptic equations of the form under the ground state condition Using the well‐known moving plane method of Alexandrov and Serrin, we show, under suitable conditions on A and f, that all ground states of (1) are radially symmetric about some origin O in . In particular, we obtain radial symmetry for compactly supported ground states and give sufficient conditions for the positivity of ground states in terms of the given operator A and the nonlinearity f. (Accepted September 21, 1998)     

Uniqueness of Positive Solutions of Δu+f(u)=0 in ℝN, N≦3

We study the uniqueness of radial ground states for the semilinear elliptic partial differential equation in ℝ ^N . We assume that the function f has two zeros, the origin and u ₀>0. Above u ₀ the function f is positive, is locally Lipschitz continuous and satisfies convexity and growth conditions of a superlinear nature. Below u ₀, f is assumed to be non-positive, non-identically zero and merely continuous. Our results are obtained through a careful analysis of the solutions of an associated initial‐value problem, and the use of a monotone separation theorem. It is known that, for a large class of functions f, the ground states of (*) are radially symmetric. In these cases our result implies that (*) possesses at most one ground state. (Accepted July 3, 1996)     

On the operator of symmetric differentiation on a compact Riemannian manifold with boundary

We state a particular case of one of the theorems which we shall prove. Let Ω be a bounded open set in ℝ ⁿ with smooth boundary and let σ=(σ _ij )be a symmetric second-order tensor with components σ _ij εH ^k(Ω) for some (positive or negative) integer k; H ^k are Sobolev spaces on Ω. Then we have for some u _i εH ^k +1(Ω),i=1,...,n, if and only if (if k<0, the integral is in fact a duality) for any symmetric tensor (ω with components and such that ). Some applications in the theory of elasticity are also given.     

Eshelby Tensor of a Polygonal Inclusion and its Special Properties

Nozaki and Taya (ASME J. Appl. Mech. 64 (1997) 495–502) analyzed the elastic field in a convex polygonal inclusion in an infinite body. By numerical analysis, they found that, when the shape of the inclusion is a regular polygon, “the strain at the center of inclusion” and “the strain energy per unit volume of inclusion” have strange and remarkable properties: these values are the same as those of a circular inclusion and are invariant for inclusion's orientation if the shape of the inclusion is not a square. In this paper, we first derive a simple, exact expression of the Eshelby tensor for an arbitrary polygonal inclusion. Using the expression, we then show a mathematical explanation why these special properties appear. This revised version was published online in July 2006 with corrections to the Cover Date.     

Void nucleation in tensile dead-loading of a composite incompressible nonlinearly elastic sphere

In this paper, the effect of material inhomogeneity on void formation and growth in incompressible nonlinearly elastic solids is examined. A bifurcation problem is considered for a solid composite sphere composed of two neo-Hookean materials perfectly bonded across a spherical interface. Under a uniform radial tensile dead-load, a branch of radially symmetric configurations involving a traction-free internal cavity bifurcates from the underformed configuration. Such a configuration is the only stable solution for sufficiently large loads. In contrast to the situation for a homogeneous neo-Hookean sphere, bifurcation here may occur either locally to the right orto the left. In the latter case, the cavity has finite radius on first appearance. This discontinuous change in stable equilibrium configurations is reminiscent of the snap-through buckling phenomenon observed in certain structural mechanics problems.Since this paper was written, the authors have carried out further analysis of the class of problems of concern here [11]. In particular the stress distribution in the composite neo-Hookean sphere has been described in [11].Paper presented at the 17th International Congress of Theoretical and Applied Mechanics, Grenoble, France, August 1988.     

Threshold Solutions and Sharp Transitions for Nonautonomous Parabolic Equations on $${\mathbb{R}^N}$$

This paper is devoted to a class of nonautonomous parabolic equations of the form u _t = Δu + f(t, u) on \mathbbR^N{\mathbb{R}^N} . We consider a monotone one-parameter family of initial data with compact support, such that for small values of the parameter the corresponding solutions decay to zero, whereas for large values they exhibit a different behavior (either blowup in finite time or locally uniform convergence to a positive constant steady state). We are interested in the set of intermediate values of the parameter for which neither of these behaviors occurs. We refer to such values as threshold values and to the corresponding solutions as threshold solutions. We prove that the transition from decay to the other behavior is sharp: there is just one threshold value. We also describe the behavior of the threshold solution: it is global, bounded, and asymptotically symmetric in the sense that all its limit profiles, as t → ∞, are radially symmetric about the same center. Our proofs rely on parabolic Liouville theorems, asymptotic symmetry results for nonlinear parabolic equations, and theorems on exponential separation and principal Floquet bundles for linear parabolic equations.     

The Cosserat Subspace ũ(−1) for Bodies of Spherical Geometry

We construct the orthogonal bases of the Cosserat eigenvectors ũ⁽⁻¹⁾ for the first boundary value problem of an elastic solid sphere and an infinite elastic space containing a spherical rigid inclusion. These orthogonal bases are expressed in terms of the Jacobi and Legendre polynomials. An example of a nonharmonic heat source shows the convergence of the sequence of the eigenvectors ũ⁽⁻¹⁾. This revised version was published online in August 2006 with corrections to the Cover Date.     

The remarkable nature of radially symmetric equilibrium states of aeolotropic nonlinearly elastic bodies

This paper treats the radially symmetric equilibrium states of aeolotropic nonlinearly elastic solid cylinders and balls under constant normal forces on their boundaries. It is shown that the aeolotropy gives rise to solutions describing both intact and cavitating states, which exhibit an array of remarkable new phenomena, not suggested by the solutions for isotropic bodies. E.g., it is shown that there are materials having a critical pressure such that for applied pressures on the boundary below the critical value, the normal pressures at the center of the body are zero and for applied pressures above the critical value, the normal pressures at the center are infinite. There are also materials for which there is no equilibrium state with center intact when the boundary is subjected to uniform tension. It is also shown that the equilibrium states treated here are the only radially symmetric equilibrium states. Thus the strange phenomena discovered here must be present in such stable equilibrium states.     

Analytical Solution for Radial Deformations of Functionally Graded Isotropic and Incompressible Second-Order Elastic Hollow Spheres

We analytically analyze radial expansion/contraction of a hollow sphere composed of a second-order elastic, isotropic, incompressible and inhomogeneous material to delineate differences and similarities between solutions of the first- and the second-order problems. The two elastic moduli are assumed to be either affine or power-law functions of the radial coordinate R in the undeformed reference configuration. For the affine variation of the shear modulus μ, the hoop stress for the linear elastic (or the first-order) problem at the point R=(R _ou R _in (R _ou +R _in )/2)^1/3 is independent of the slope of the μ vs. R line. Here R _in and R _ou equal, respectively, the inner and the outer radius of the sphere in the reference configuration. For μ(R)∝R ⁿ , for the linear problem, the hoop stress is constant in the sphere for n=1. However, no such results are found for the second-order (i.e., materially nonlinear) problem. Whereas for the first-order problem the shear modulus influences only the radial displacement and not the stresses, for the second-order problem the two elastic constants affect both the radial displacement and the stresses. In a very thick homogeneous hollow sphere subjected only to pressure on the outer surface, the hoop stress at a point on the inner surface depends upon values of the two elastic moduli. Thus conclusions drawn from the analysis of the first-order problem do not hold for the second-order problem. Closed form solutions for the displacement and stresses for the first-order and the second-order problems provided herein can be used to verify solutions of the problem obtained by using numerical methods.     

The Evolution of Stress Intensity Factors and the Propagation of Cracks in Elastic Media

When a crack Γ _s propagates in an elastic medium the stress intensity factors evolve with the tip x(s) of Γ _s . In this paper we derive formulae which describe the evolution of these stress intensity factors for a homogeneous isotropic elastic medium under plane strain conditions. Denoting by ψ=ψ(x,s) the stress potential (ψ is biharmonic and has zero traction along the crack Γ _s ) and by κ(s) the curvature of the crack at the tip x(s), we prove that the stress intensity factors A ₁(s), A ₂(s), as functions of s, satisfy:

Finite Deformations and Motions of Radially Inextensible Hollow Spheres

Radial inflation–compaction and radial oscillation solutions are presented for hollow spheres of isotropic elastic material that are radially inextensible. The solutions for radial inflation–compaction and radial oscillation are obtained also for everted radially inextensible hollow spheres of isotropic elastic material. The static and dynamic results for everted and uneverted radially inextensible hollow spheres are then compared. Harmonic and compressible Varga materials are used to demonstrate the solutions.