Surface effects in the deformation of an anisotropic elastic material with nano-sized elliptical hole

We examine the effect of surface energy on an anisotropic elastic material weakened by an elliptical hole. A closed-form, full-field solution is derived using the standard Stroh formalism. In particular, explicit expressions for the hoop stress, normal, in-plane tangential and out-of-plane displacement components along the edge of the hole are obtained. These expressions clearly demonstrate the effect of elastic anisotropy of the bulk material on the corresponding field variables. When the material becomes isotropic, the hoop stress agrees with the well-known result in the literature while both the in-plane tangential and out-of-plane displacements vanish and the normal displacement is constant along the entire boundary of the elliptical hole.     

Effet de l'anisotropie élastique cristalline sur la distribution des facteurs de Schmid à la surface des polycristauxInfluence of crystalline elasticity anisotropy on Schmid factor distribution at the free surface of polycrystals

Experimental studies of the plasticity mechanisms of polycrystals are usually based on the Schmid factor distribution supposing crystalline elasticity isotropy. A numerical evaluation of the effect of crystalline elasticity anisotropy on the apparent Schmid factor distribution at the free surface of polycrystals is presented. Cubic elasticity is considered. Order II stresses (averaged on all grains with the same crystallographic orientation) as well as variations between averages computed on grains with the same crystallographic orientation but with different neighbour grains are computed. The Finite Element Method is used. Commonly studied metals presenting an increasing anisotropy degree are considered (aluminium, nickel, austenite, copper). Concerning order II stresses in strongly anisotropic metals, the apparent Schmid factor distribution is drifted towards small Schmid factor values (the maximum Schmid factor is equal to 0.43 instead of 0.5) and the slip activation order between characteristic orientations of the crystallographic standard triangle is modified. The computed square deviations of the stresses averaged on grains with the same crystallographic orientation but with different neighbour grains are a bit higher than the second order ones (inter-orientation scatter). Our numerical evaluations agree quantitatively with several observations and measures of the literature concerning stress and strain distribution in copper and austenite polycrystals submitted to low amplitude loadings. Hopefully, the given apparent Schmid factor distributions could help to better understand the observations of the plasticity mechanisms taking place at the free surface of polycrystals. To cite this article: M. Sauzay, C. R. Mecanique 334 (2006).     

On displacement methods in planar anisotropic elasticity

Two displacement formulation methods are presented for problems of planar anisotropic elasticity. The first displacement method is based on solving the two governing partial differential equations simultaneously/ This method is a recapitulation of the orignal work of Eshelby, Read and Shockley [7] on generalized plane deformations of anisotropic elastic materials in the context of planar anisotropic elasticity.The second displacement method is based on solving the two governing equations separately. This formulation introduces a displacement function, which satisfies a fourth-order partial differential equation that is identical in the form to the one given by Lekhnitskii [6] for monoclinic materials using a stress function. Moreover, this method parallels the traditional Airy stress function method and thus the Lekhnitskii method for pure plane problems. Both the new approach and the Airy stress function method start with the equilibrium equations and use the same extended version of Green's theorem (Chou and Pagano [13], p. 114; Gao [11]) to derive the expressions for stress or displacement components in terms of a potential (stress or displacement) function (see also Gao [10, 11]). It is therefore anticipated that the displacement function involved in this new method could also be evaluated from measured data, as was done by Lin and Rowlands [17] to determine the Airy stress function experimentally.The two different displacement methods lead to two general solutions for problems of planar anisotropic elasticity. Although the two solutions differ in expressions, both of the depend on the complex roots of the same characteristic equation. Furthermore, this characteristic equation is identical to that obtained by Lekhnitskii [6] using a stress formulation. It is therefore concluded that the two displacement methods and Lekhnitskii's stress method are all equivalent for problems of planar anisotropic elasticity (see Gao and Rowlands [8] for detailed discussions).     

Stress concentration and surface instability of anisotropic solids with slightly wavy boundary

This paper presents a first order perturbation analysis of stress concentration and surface morphology instability of elastically anisotropic solids. The boundary of the solids under consideration is periodic along two orthogonal directions. The magnitude of the undulation is sufficiently small so that a half-space model can be used for simplification. We derive expressions for the stress concentration factors and the critical wavelength of the perturbation in terms of the remote stresses, surface energy anisotropy and the elastic anisotropy of the solid. Numerical applications to cubic materials using Barnett–Lothe integrals are also given.     

Perturbation Formula for Phase Velocity of Rayleigh Waves in Prestressed Anisotropic Media

Herein we consider Rayleigh waves propagating along the free surface of a macroscopically homogeneous, anisotropic, prestressed half-space. We adopt the formulation of linear elasticity with initial stress and assume that the deviation of the prestressed anisotropic medium from a comparative ‘unperturbed’, unstressed and isotropic state, as formally caused by the initial stress and by the anisotropic part of the incremental elasticity tensor, be small. No assumption, however, is made on the material anisotropy of the incremental elasticity tensor. With the help of the Stroh formalism, we derive a first-order perturbation formula for the shift of phase velocity of Rayleigh waves from its comparative isotropic value. Our perturbation formula does not agree totally with that which was derived some years ago by Delsanto and Clark, and we provide another argument as further support for our version of the formula. According to our first-order formula, the anisotropy-induced velocity shifts of Rayleigh waves, taken in totality of all propagation directions on the free surface, carry information only on 13 elastic constants of the anisotropic part of the incremental elasticity tensor. The remaining eight elastic constants are those which would become zero if were monoclinic with the two-fold symmetry axis normal to the free surface of the material half-space in question.     

End effects for anti-plane shear deformations of sandwich structures

The purpose of this research is to further investigate the effects of material inhomogeneity and the combined effects of material inhomogeneity and anisotropy on the decay of Saint-Venant end effects. Saint-Venant decay rates for self-equilibrated edge loads in symmetric sandwich structures are examined in the context of anti-plane shear for linear anisotropic elasticity. The problem is governed by a second-order, linear, elliptic, partial differential equation with discontinuous coefficients. The most general anisotropy consistent with a state of anti-plane shear is considered, as well as a variety of boundary conditions. Anti-plane or longitudinal shear deformations are one of the simplest classes of deformations in solid mechanics. The resulting deformations are completely characterized by a single out-of-plane displacement which depends only on the in-plane coordinates. They can be thought of as complementary deformations to those of plane elasticity. While these deformations have received little attention compared with the plane problems of linear elasticity, they have recently been investigated for anisotropic and inhomogeneous linear elasticity. In the context of linear elasticity, Saint-Venant's principle is used to show that self-equilibrated loads generate local stress effects that quickly decay away from the loaded end of a structure. For homogeneous isotropic linear elastic materials this is well-documented. Self-equilibrated loads are a class of load distributions that are statically equivalent to zero, i.e., have zero resultant force and moment. When Saint-Venant's principle is valid, pointwise boundary conditions can be replaced by more tractable resultant conditions. It is shown in the present study that material inhomogeneity significantly affects the practical application of Saint-Venant's principle to sandwich structures.     

Effect of elastic anisotropy on surface pattern evolution of epitaxial thin films

Stress-induced surface instability and evolution of epitaxial thin films leads to formation of a variety of self-assembled surface patterns with feature sizes at micro- and nanoscales. The anisotropy in both the surface and bulk properties of the film and substrate has profound effects on the nonlinear dynamics of surface evolution and pattern formation. Experimentally it has been demonstrated that the effect of anisotropy strongly depends on the crystal orientation of the substrate surface on which the film grows epitaxially. In this paper we develop a nonlinear model for surface evolution of epitaxial thin films on generally anisotropic crystal substrates. Specifically, the effect of bulk elastic anisotropy of the substrate on epitaxial surface pattern evolution is investigated for cubic germanium (Ge) and SiGe films on silicon (Si) substrates with four different surface orientations: Si(0 0 1), Si(1 1 1), Si(1 1 0), and Si(1 1 3). Both linear analysis and nonlinear numerical simulations suggest that, with surface anisotropy neglected, ordered surface patterns form under the influence of the elastic anisotropy, and these surface patterns clearly reflect the symmetry of the underlying crystal structures of the substrate. It is concluded that consideration of anisotropic elasticity reveals a much richer dynamics of surface pattern evolution as compared to isotropic models.     

Perturbation Formulas for Polarization Ratio and Phase Shift of Rayleigh Waves in Prestressed Anisotropic Media

This paper completes an earlier study (Tanuma and Man, Journal of Elasticity, 85, 21–37, 2006) where we derive a first-order perturbation formula for the phase velocity of Rayleigh waves that propagate along the free surface of a macroscopically homogeneous, anisotropic, prestressed half-space. We adopt the formulation of linear elasticity with initial stress and assume that the deviation of the prestressed anisotropic medium from a suitably-chosen, comparative, unstressed and isotropic state be small. No assumption, however, is made on the material anisotropy of the incremental elasticity tensor. With the help of the Stroh formalism, here we derive first-order perturbation formulas for the changes in polarization ratio and phase shift of Rayleigh waves from their respective comparative isotropic value. Examples are given, which show that the perturbation formulas for phase velocity and polarization ratio can serve as a starting point for investigations on the possible advantages of using Rayleigh-wave polarization, as compared with using wave speed, for acoustoelastic measurement of stress.      

The scattering of a scalar beam from isotropic and anisotropic two-dimensional randomly rough Dirichlet or Neumann surfaces: The full angular intensity distributions

By the use of Green’s second integral identity we determine the field scattered from a two-dimensional randomly rough isotropic or anisotropic Dirichlet or Neumann surface when it is illuminated by a scalar Gaussian beam. The integral equations for the scattering amplitudes are solved nonperturbatively by a rigorous computer simulation approach. The results of these calculations are used to calculate the full angular distribution of the mean differential reflection coefficient. For isotropic surfaces, the results of the present calculations for in-plane scattering are compared with those of earlier studies of this problem. The reflectivities of Dirichlet and Neumann surfaces are calculated as functions of the polar angle of incidence, and the reflectivities for the two kinds of surfaces of similar roughness parameters are found to be different. For an increasing level of surface anisotropy, we study how the angular intensity distributions of the scattered waves are affected by this level. We find that even small to moderate levels of surface anisotropy can significantly alter the symmetry, shape, and amplitude of the scattered intensity distributions when Gaussian beams are incident on the anisotropic surfaces from different azimuthal angles of incidence.     

Out-of-plane loading of anisotropic bimaterials and semi-infinite materials

Summary Analytical closed-form solutions are proposed in a rather compact form for the stress and displacement fields induced by out-of-plane loading of a semi-infinite anisotropic material with inclined strata. The solutions are then extended to include the case of a bimaterial with a planar interface. Several boundary conditions are considered for the interface which may be between two anisotropic half-planes with different elastic properties, or two different orientations of the strata in the same material.     

各向异性非线性固体力学的规范空间理论

本文在弹性规范空间概念基础上，利用非平衡态热力学理论，证明了各向异性固体力学非线性问题规范空间场以及不可逆过程本征解的存在。损伤对结构刚度的弱化效应和损伤诱发各向异性效应分别反映在本征弹性和相应的模态向量中。在简正坐标中考察各向异性体变形时，材料的行为以六个普通的粘弹性Ｍａｘｗｅｌｌ方程描述，总的响应由模态叠加得到。以此为基础给出的非线性本构方程具有坐标转换不变性，最后给出了二个具体的算例。     

Theory of interfacial dynamics of nematic polymers

 The interfacial momentum and torque balance equations for deforming interfaces between nematic polymers and isotropic viscous fluids are derived and analyzed with respect to shape selection and interfacial nematic ordering. It is found that the interfacial momentum balance equation for nematic interfaces involves bending forces that act normal to the interface, and that interfacial pressure jumps may exist even for planar surfaces. In addition tangential forces on nematic interfaces arise in the presence of surface gradients of the tensor order parameter. The torque balance equation shows that couple stress jumps are balanced by the surface molecular field. The interfacial balance equations are shown to be coupled such that nematic ordering depends on shape and vice versa. The governing dimensionless numbers for deforming nematic polymer interfaces are identified and the limiting regimes are discussed in reference to related experimental data. It is found that the ratio of Frank elasticity to surface anchoring controls whether the surface tensor order parameter deviates from its preferred equilibrium value. Whether the shape is affected, depends on the relative magnitudes of the isotropic surface tension, Frank bulk elasticity, and anchoring energy, and capillary number. Received: 16 April 1999/Accepted: 19 August 1999     

Thermoelastic stress intensity factors for a cracked layer between two different anisotropic solids

Steady-state anisotropic thermoelasticity equations are used to obtain the stress intensity factors for a cracked layer sandwiched between two different anisotropic elastic solids. The anisotropy is assumed to arise from discrete fibers whose orientation could alter with reference to the crack edges. A generalized plane deformation prevails in the dissimilar media domain with a line of discontinuity disturbing a uniform heat flow. The flexibility/stiffness matrix approach is used such that the crack problem reduces to solving two sets of singular integral equations. Numerical values of the crack tip stress-intensity factors are obtained for various crack size, crack location, crack surface insulation, fiber volume fraction and orientation angles. The results are displayed graphically.     

Restrictions on dynamically propagating surfaces of strong discontinuity in elastic-plastic solids

For dynamic three-dimensional deformations of elastic-plastic materials, we elicit conditions necessary for the existence of propagating surfaces of strong discontinuity (across which components of stress, strain or material velocity jump). This is accomplished within a small-displacement-gradient formulation of standard weak continuum-mechanical assumptions of momentum conservation and geometrical compatibility, and skeletal constitutive assumptions which permit very general elastic and plastic anisotropy including yield surface vertices and anisotropic hardening. In addition to deriving very explicit restrictions on propagating strong discontinuities in general deformations, we prove that for anti-plane strain and incompressible plane strain deformations, such strong discontinuities can exist only at elastic wave speeds in generally anisotropic elastic-ideally plastic materials unless a material's yield locus in stress space contains a linear segment. The results derived seem essential for correct and complete construction of solutions to dynamic elastic-plastic boundary-value problems.     

Elliptical inclusion in an anisotropic plane: non-uniform interface effects

We study the plane deformation of an elastic composite system made up of an anisotropic elliptical inclusion and an anisotropic foreign matrix surrounding the inclusion. In order to capture the influence of interface energy on the local elastic field as the size of the inclusion approaches the nanoscale, we refer to the Gurtin-Murdoch model of interface elasticity to describe the inclusion-matrix interface as an imaginary and extremely stiff but zero-thickness layer of a finite stretching modulus. As opposed to isotropic cases in which the effects of interface elasticity are usually assumed to be uniform (described by a constant interface stretching modulus for the entire interface), the anisotropic case considered here necessitates non-uniform effects of interface elasticity (described by a non-constant interface stretching modulus), because the bulk surrounding the interface is anisotropic. To this end, we treat the interface stretching modulus of the anisotropic composite system as a variable on the interface curve depending on the specific tangential direction of the interface. We then devise a unified analytic procedure to determine the full stress field in the inclusion and matrix, which is applicable to the arbitrary orientation and aspect ratio of the inclusion, an arbitrarily variable interface modulus, and an arbitrary uniform external loading applied remotely. The non-uniform interface effects on the external loading-induced stress distribution near the interface are explored via a group of numerical examples. It is demonstrated that whether the nonuniformity of the interface effects has a significant effect on the stress field around the inclusion mainly depends on the direction of the external loading and the aspect ratio of the inclusion.     

A plasticity model for pressure-dependent anisotropic cellular solids

The initial and subsequent yield surfaces for an anisotropic and pressure-dependent 2D stochastic cellular material, which represents solid foams, are investigated under biaxial loading using finite element analysis. Scalar measures of stress and strain, namely characteristic stress and characteristic strain, are used to describe the constitutive response of cellular material along various stress paths. The coupling between loading path and strain hardening is then investigated in characteristic stress–strain domain. The nature of the flow rule that best describes the plastic flow of cellular solid is also investigated. An incremental plasticity framework is proposed to describe the pressure-dependent plastic flow of 2D stochastic cellular solids. The proposed plasticity framework adopts the anisotropic and pressure-dependent yield function recently introduced by Alkhader and Vural [Alkhader M., Vural M., 2009a. An energy-based anisotropic yield criterion for cellular solids and validation by biaxial FE simulations. J. Mech. Phys. Solids 57(5), 871–890]. It has been shown that the proposed yield function can be simply calibrated using elastic constants and flow stresses under uniaixal loading. Comparison of stress fields predicted by continuum plasticity model to the ones obtained from FE analysis shows good agreement for the range of loading paths and strains investigated.     

Passing stiffness anisotropy in multilayers and its effects on nanoscale surface self-organization

A binary monolayer adsorbed on a solid surface can separate into distinct phases that further self-assemble into various two-dimensional patterns. The surface stresses in the two phases are different, causing an elastic field in the substrate. The self-organization minimizes the combined free energy of mixing, phase boundary, and elasticity. One can obtain diverse patterns by using substrates with various crystalline symmetries. Consider the pattern of a set of periodic stripes. The stripe orientation depends on the anisotropy in surface stress, substrate stiffness, and phase boundary energy. A more powerful and flexible way is to use a layered substrate. Surface properties designed for the applications of those patterns can be obtained by choosing appropriate materials and structures for the monolayer and the top layer of the substrate. The subsequent layers of the substrate provide the required stiffness anisotropy, the effect of which is passed to the monolayer patterns through the elastic field. We solve the elastic field in the anisotropic, heterogeneous, three-dimensional half-space by using the Eshelby–Stroh–Lekhnitskii formalism and the Fourier transformation. Depending on the thicknesses and the degrees of the stiffness anisotropy of the substrate layers, the lowest energy stripes can have tunable equilibrium size and orientation. We also discuss other possibilities of manipulating the phase patterns by engineering the elastic field.     

Partial Uniqueness and Obstruction to Uniqueness in Inverse Problems for Anisotropic Elastic Media

We consider the inverse problem of identifying the density and elastic moduli for three-dimensional anisotropic elastic bodies, given displacement and traction measurements made at their surface. These surface measurements are modelled by the dynamic Dirichlet-to-Neumann map on a finite time interval. For linear or nonlinear anisotropic hyperelastic bodies we show that the displacement-to-traction surface measurements do not change when the density and elasticity tensor in the interior are transformed tensorially by a change of coordinates fixing the surface of the body to first order. Our main tool, a new approach in inverse problems for elastic media, is the representation of the equations of motion in a covariant form (following Marsden and Hughes, 1983) that preserves the underlying physics.In the case of classical linear elastodynamics we then investigate how the type of anisotropy changes under coordinate transformations. That is, we analyze the orbits of general linear, anisotropic elasticity tensors under the action by pull-back of diffeomorphisms that fix the surface of the elastic body to first order, and derive a pointwise characterization of parts of the orbits under this action. For example, we show that the orbit of isotropic elastic media, at any point in the body, consists of some transversely isotropic and some orthotropic elastic media. We then derive the first uniqueness result in the inverse problem for anisotropic media using surface displacement-traction data: uniqueness of three elastic moduli for tensors in the orbit of isotropic elasticity tensors. ^†Partially supported by an MSRI Postdoctoral Fellowship. Research at MSRI is supported in part by NSF grant DMS-9850361. This work was conducted while the first author was a Gibbs Instructor at Yale University. ^‡Partially supported by an MSRI Postdoctoral Fellowship, and by NSF grant DMS-9801664 (9996350).     

The evolution of Hooke's law due to texture development in FCC polycrystals

Initially isotropic aggregates of crystalline grains show a texture-induced anisotropy of both their inelastic and elastic behavior when submitted to large inelastic deformations. The latter, however, is normally neglected, although experiments as well as numerical simulations clearly show a strong alteration of the elastic properties for certain materials. The main purpose of the work is to formulate a phenomenological model for the evolution of the elastic properties of cubic crystal aggregates. The effective elastic properties are determined by orientation averages of the local elasticity tensors. Arithmetic, geometric, and harmonic averages are compared. It can be shown that for cubic crystal aggregates all of these averages depend on the same irreducible fourth-order tensor, which represents the purely anisotropic portion of the effective elasticity tensor. Coupled equations for the flow rule and the evolution of the anisotropic part of the elasticity tensor are formulated. The flow rule is based on an anisotropic norm of the stress deviator defined by means of the elastic anisotropy. In the evolution equation for the anisotropic part of the elasticity tensor the direction of the rate of change depends only on the inelastic rate of deformation. The evolution equation is derived according to the theory of isotropic tensor functions. The transition from an elastically isotropic initial state to a (path-dependent) final anisotropic state is discussed for polycrystalline copper. The predictions of the model are compared with micro–macro simulations based on the Taylor–Lin model and experimental data.