Degenerate weakly non-linear elastic plane waves

Weakly non-linear plane waves are considered in hyperelastic crystals. Evolution equations are derived at a quadratically non-linear level for the amplitudes of quasi-longitudinal and quasi-transverse waves propagating in arbitrary anisotropic media. The form of the equations obtained depends upon the direction of propagation relative to the crystal axes. A single equation is found for all propagation directions for quasi-longitudinal waves, but a pair of coupled equations occurs for quasi-transverse waves propagating along directions of degeneracy, or acoustic axes. The coupled equations involve four material parameters but they simplify if the wave propagates along an axis of material symmetry. Thus, only two parameters arise for propagation along an axis of twofold symmetry, and one for a threefold axis. The transverse wave equations decouple if the axis is fourfold or higher. In the absence of a symmetry axis it is possible that the evolution equations of the quasi-transverse waves decouple if the third-order elastic moduli satisfy a certain identity. The theoretical results are illustrated with explicit examples.     

Inversion for weak triclinic anisotropy from acoustic axes

Acoustic axes are directions in anisotropic elastic media, in which phase velocities of two or three plane waves (P$P$, S1$S1$ or S2$S2$ waves) coincide. Acoustic axes are important, because they can cause singularities in the field of polarization vectors and anomalies in the shape of the slowness surface. The maximum number of acoustic axes in triclinic anisotropy is 16, and their directions depend on anisotropy parameters in a complicate way. Under weak anisotropy approximation this dependence simplifies and the directions of acoustic axes can be used for the inversion for anisotropy parameters. The maximum acoustic axes under weak anisotropy is 16, the minimum number of acoustic axes is zero. In the inversion, we can retrieve 13 combinations of anisotropy parameters provided we use directions of 7 acoustic axes at least. Under weak anisotropy approximation, the directions of acoustic axes are insensitive to strength of anisotropy; hence we cannot invert for absolute values of weak anisotropy parameters, but only for their relative values. Numerical tests have shown that the inversion is applicable only to very weak anisotropy with strength of less than 5%, provided that the acoustic axes used in the inversion are determined with an accuracy of 0.1^°$0.1^{°}$ or better. In this case the inversion yields an average error for elastic parameters of less than 10%. In order to invert for the total set of 21 anisotropy parameters it is necessary to combine the measurements of the directions of the acoustic axes with measurements of other attributes of elastic waves in anisotropic media.     

Optimization of the strain energy density in linear anisotropic elasticity

The problem considered here is that of extremizing the strain energy density of a linear anisotropic material by varying the relative orientation between a fixed stress state and a fixed material symmetry. It is shown that the principal axes of stress must coincide with the principal axes of strain in order to minimize or maximize the strain energy density in this situation. Specific conditions for maxima and minima are obtained. These conditions involve the stress state and the elastic constants. It is shown that the symmetry coordinate system of cubic symmetry is the only situation in linear anisotropic elasticity for which a strain energy density extremum can exist for all stress states. The conditions for the extrema of the strain energy density for transversely isotropic and orthotropic materials with respect to uniaxial normal stress states are obtained and illustrated with data on the elastic constants of some composite materials. Not surprisingly, the results show that a uniaxial normal stress in the grain direction in wood minimizes the strain energy in the set of all uniaxial stress states. These extrema are of interest in structural and material optimization.     

平行微裂纹削弱的弹性体的各向异性有效性能与双周期裂纹模型

平行微裂纹损伤模型被用于构建各向异性损伤理论.当施加在代表性体积单元上的边界条件满足Hill条件时,基于平均场理论论证了由平行穿透裂纹损伤的弹性体仅有6个独立有效弹性常数.除了原各向同性基体的2个弹性常数外,与损伤相关的另外4个常数中,3个描述有效弹性常数的折减,1个描述损伤导致的拉剪耦合效应.结合单胞模型和有限元方法分析了双周期阵列平行裂纹问题,数值结果显示:裂纹呈一般双周期阵列时,拉剪耦合参数相比其它模量小很多;当裂纹密度一定时,改变裂纹的排列形式,面内剪切模量和面外剪切模量的折减呈现出不同的规律.     

Stationarity of the strain energy density for some classes of anisotropic solids

Homogeneous, anisotropic and linearly elastic solids, subjected to a given state of strain (or stress), are considered. The problem dealt with consists in finding the mutual orientations of the principal directions of strain to the material symmetry axes in order to make the strain energy density stationary. Such relative orientations are described through three Euler’s angles. When the stationarity problem is formulated for the generally anisotropic solid, it is shown that the necessary condition for stationarity demands for coaxiality of the stress and the strain tensors. From this feature, a procedure which leads to closed form solutions is proposed. To this end, tetragonal and cubic symmetry classes, together with transverse isotropy, are carefully dealt with, and for each case all the sets of Euler’s angles corresponding to critical points of the energy density are found and discussed. For these symmetries, three material parameters are then defined, which play a crucial role in ordering the energy values corresponding to each solution.     

On the conditions for the existence of a plane of symmetry for anisotropic elastic material

We consider the problem of determining necessary and sufficient conditions for the existence of symmetry planes of an anisotropic elastic material. These conditions are given in several equivalent forms, and are used to determine special coordinate systems where the number of non-zero components in the elasticity tensor is minimized. By the method presented here it is also shown that an elastic solid has at least six coordinate systems with respect to which there are only 18 non-zero elastic constants and cannot possess more then ten traditional and distinct symmetrics by planes of symmetry.     

Anisotropic elastic materials capable of a three-dimensional deformation (static or dynamic) with only one displacement component and uncoupling of all three displacement components

It is shown that there are anisotropic elastic materials that are capable of a non-uniform three-dimensional deformation with only one displacement component. For wave propagation, the equation of motion can be cast in the form of the differential equation for acoustic waves. For elastostatics, the equation of equilibrium reduces to Laplace’s equation. The material can be monoclinic, orthotropic, tetragonal, hexagonal or cubic. There are also anisotropic elastic materials that uncouple all three displacement components. The governing equation for each of the uncoupled displacement can be cast in the form of the differential equation for acoustic waves in the case of dynamic or Laplace’s equation in the case of static. The material can be orthotropic, tetragonal, hexagonal or cubic.     

Convex Fung-type potentials for biological tissues

The anisotropic, non-linear elastic behavior of biological soft tissue is typically accounted for by the hypothesis of hyperelasticity, i.e., the existence of an elastic potential. Fung-type potentials, based on the exponential of a quadratic form in the components of the Green-Lagrange strain, have been widely used in soft tissue modeling, and have inspired potentials in which the exponential was replaced by other monotonically increasing functions. It has been shown that simple fitting of the parameters of a Fung-type potential to experimental stress-strain curves may lead to non-convexity, with undesirable effects on the reliability of the algorithms used in Finite Element simulations. In this paper, we prove that the necessary and sufficient condition for the strict convexity of a Fung-type potential is that the quadratic form in the exponential is positive definite. This result provides a clear physical meaning for the parameters featuring in the quadratic form, and their relationship with the small-strain elastic moduli. This consistency relationship must be respected in order to guarantee that the Fung-type potential correctly reduces to the quadratic potential of classic linear elasticity in the small-strain approximation. Furthermore, we show that, when the conditions of convexity and consistency with the linear theory are respected, Fung-type potentials become a one-parameter family, and we discuss the consequences of this result for when fitting experimental data. An erratum to this article can be found at     

Crack growth resistance for anisotropic plasticity with non-normality effects

For a plastically anisotropic solid a plasticity model using a plastic flow rule with non-normality is applied to predict crack growth. The fracture process is modelled in terms of a traction–separation law specified on the crack plane. A phenomenological elastic–viscoplastic material model is applied, using one of two different anisotropic yield criteria to account for the plastic anisotropy, and in each case the effect of the normality flow rule is compared with the effect of non-normality. Conditions of small scale yielding are assumed, with mode I loading conditions far from the crack-tip, and various directions of the crack plane relative to the principal axes of the anisotropy are considered. It is found that the steady-state fracture toughness is significantly reduced when the non-normality flow rule is used. Furthermore, it is shown that the predictions are quite sensitive to the value of the maximum angle of deviation from normality in the non-normality flow rule.     

Electroacoustic waves in piezoelectric crystals: Certain limiting cases

The Christoffel equations for electroacoustic waves in unbounded piezoelectric crystals are solved in the limits of weak and strong electromechanical coupling and for the case where the unstiffened elastic constants satisfy the conditions of elastic isotropy. Lyubimov's proof that piezoelectric stiffening of the elastic constants increases acoustic velocities is extended to cover degenerate modes. It is shown that when the elastic contribution to the stiffened elastic constants is zero, only one acoustic branch survives with a finite velocity. When the unstiffened elastic constants satisfy the conditions of elastic isotropy one of the acoustic branches is unaffected by the piezoelectric stiffening and, moreover, Lyubimov's theorem holds nonperturbatively.     

Formation fronts of a nonlinear elastic medium from a medium without shear stresses

The fronts of phase transition of a medium without shear stresses to a nonlinear incompressible anisotropic elastic medium are considered. The mass flux through unit area of a front is assumed to be known. The variation of the tangential components of the medium’s velocity and the variation of the arising shear stresses are studied. An explicit form of boundary conditions is found using the existence condition of a discontinuity front structure. The Kelvin–Voight viscoelastic model is adopted for this structure.     

Elliptic cavity in a three-dimensional anisotropic body with inclined strata deformed by line loads and dislocations

Summary Analytical solutions are proposed for the stress and displacement fields in a quasi three-dimensional elastic anisotropic body containing an elliptic cavity or rigid inclusion. The directions of the principal elastic axes are allowed to be inclined arbitrarily with respect to the axes of the elliptic cavity. As an application, expressions for the stress intensity factors are formulated when the cavity reduces to a colinear crack.     

On the non-conservativeness of a class of anisotropic damage models with unilateral effectsSur le caractère non-conservatif de quelques modèles d'endommagement anisotrope avec conditions unilatérales

The aim of this Note is to show that a class of anisotropic elastic-damage models including unilateral effects can be considered, for constant damage values, as non-linear and non-conservative elastic. The conservative character of corresponding constitutive models is related to the symmetry of the Hessian tensor. For the models under consideration, it is shown that the condition of conservativeness (existence of the elastic potential energy function) is obtained only when there is coaxiality of the strain and damage tensors. To cite this article: N. Challamel et al., C. R. Mecanique 334 (2006).     

On Compressible Materials Capable of Sustaining Axisymmetric Shear Deformations. Part 4: Helical Shear of Anisotropic Hyperelastic Materials

Conditions on the form of the strain energy function in order that homogeneous, compressible and isotropic hyperelastic materials may sustain controllable static, axisymmetric anti-plane shear, azimuthal shear, and helical shear deformations of a hollow, circular cylinder have been explored in several recent papers. Here we study conditions on the strain energy function for homogeneous and compressible, anisotropic hyperelastic materials necessary and sufficient to sustain controllable, axisymmetric helical shear deformations of the tube. Similar results for separate axisymmetric anti-plane shear deformations and rotational shear deformations are then obtained from the principal theorem for helical shear deformations. The three theorems are illustrated for general compressible transversely isotropic materials for which the isotropy axis coincides with the cylinder axis. Previously known necessary and sufficient conditions on the strain energy for compressible and isotropic hyperelastic materials in order that the three classes of axisymmetric shear deformations may be possible follow by specialization of the anisotropic case. It is shown that the required monotonicity condition for the isotropic case is much simpler and less restrictive. Restrictions necessary and sufficient for anti-plane and rotational shear deformations to be possible in compressible hyperelastic materials having a helical axis of transverse isotropy that winds at a constant angle around the tube axis are derived. Results for the previous case and for a circular axis of transverse isotropy are included as degenerate helices. All of the conditions derived here have essentially algebraic structure and are easy to apply. The general rules are applied in several examples for specific strain energy functions of compressible and homogeneous transversely isotropic materials having straight, circular, and helical axes of material symmetry.     

Interface crack growth for anisotropic plasticity with non-normality effects

A plasticity model with a non-normality plastic flow rule is used to analyze crack growth along an interface between a solid with plastic anisotropy and an elastic substrate. The fracture process is represented in terms of a traction-separation law specified on the crack plane. A phenomenological elastic–viscoplastic material model is applied, using an anisotropic yield criterion, and in each case analyzed the effect of non-normality is compared with results for the standard normality flow rule. Due to the mismatch of elastic properties across the interface the corresponding elastic solution has an oscillating stress singularity, and with conditions of small scale yielding this solution is applied as boundary conditions on the outer edge of the region analyzed. Crack growth resistance curves are calculated numerically, and the effect of the near-tip mode mixity on the steady-state fracture toughness is determined. It is found that the steady-state fracture toughness is quite sensitive to differences in the initial orientation of the principal axes of the anisotropy relative to the interface.     

Finite strain analysis of simple shear using recent anisotropic yield criteria

The finite strain response of a rectangular block subjected to constrained simple shearing deformations is considered in order to evaluate the predictive capability of some recently proposed anisotropic yield functions. It is shown that, in the presence of plane anisotropy, the prediction of realistic second order normal stresses cannot be expected since every different initial orientation of the material axes relative to the loading axes results in a different response due to the rotation of the material axes with shear. Parametric studies are performed in order to determine possible limits on the material constants so that the predicted normal stresses remain second order with respect to the shear stress itself. Our numerical results indicate that, in particular, the commonly employed range of one parameter associated with grain related anisotropy renders results which no longer imply predicting a proper second order effect but rather introduces errors of the first order. The results suggest that the modelling of anisotropy with such phenomenological anisotropic yield functions should be limited to near-quadratic yield surfaces for applications involving stress states outside the biaxial tensile stress range.     

Analysis of 2D infinite anisotropic elastic strips and semi-strips

Summary  Using Stroh's formalism and the theory of analytic functions, simple and explicit solutions for a line dislocation in an infinite anisotropic elastic strip are obtained. The two boundaries of the strip are free of traction. The problem of a dislocation in an anisotropic elastic semi-infinite strip with traction-free boundaries is also studied. A set of singular integral equations governing the unknown functions is derived. When the medium is orthogonal anisotropic and the coordinate axes x ₁ x ₂ x ₃ are coincident with the material principal axes, all the eigenvalues of the material coefficient matrix are pure imaginary. Explicit expressions of the unknown functions are given for this case. The results obtained are valid not only for plane and anti-plane problems but also for coupled problems between in-plane and out-of-plane deformations. Received 30 October 2000; accepted for publication 28 March 2001     

Homogenized elastic–viscoplastic behavior of anisotropic open-porous bodies with pore pressure

Constitutive modeling is studied for the homogenized elastic–viscoplastic behavior of pore-pressurized anisotropic open-porous bodies made of metallic base solids at small strains and rotations. For this purpose, by describing micro–macro relations relevant to periodic unit cells of anisotropic open-porous bodies subjected to pore pressure, constitutive features are discussed for the viscoplastic macrostrain rate in steady states. On the basis of the constitutive features found, the viscoplastic macrostrain rate is represented as an anisotropic function of Terzaghi’s effective stress, which is shown using Hill’s macrohomogeneity condition. The resulting viscoplastic equation is used to simulate the homogenized elastic–viscoplastic behavior of an ultrafine plate-fin structure subjected to uniaxial/biaxial loading in addition to pore pressure. The corresponding finite element homogenization analysis is also performed for comparison. It is demonstrated that the developed viscoplastic equation simulates well the anisotropic effect of pore pressure in the viscoplastic range in spite of there being no anisotropic factor and no fitting parameter in Terzaghi’s effective stress itself.     

Scattering of an incident wave from an interface separating two fluids

We consider the scattering of an incident pulse from an interface separating two fluids. The interface can be either an elastic membrane or a two-fluid interface with surface tension. By considering the limit where the ratio of acoustic wavelength to the surface wavelength is small, we systematically derived a boundary condition relating the scattered wave and the surface deformation. This condition is local and can be used to derive a partial differential equation for the deformation of the interface. This equation includes the contribution of the acoustic waves induced by the motion of the interface and once it is solved it can be used to determine the scattered field. At leading order in our analysis we find the plane wave approximation. The addition of the next order terms results in an on surface condition equivalent to that of Kriegsmann and Scandrett. We present numerical calculations to show that our results are in good agreement with the exact numerical solution as well as that of Kriegsmann and Scandrett. Physical situations where the conditions of our analysis are valid are presented.