Finite strain solutions in compressible isotropic elasticity

Three classes of compressible isotropic elastic solids are introduced, for each of which the strain energy, expressed as a function of the three principal invariants of the stretch tensors, is linear in two of its arguments and nonlinear in the third argument. One of these is the class of harmonic materials. Several deformation fields are examined, for which the governing equations, for general compressible isotropic elastic response, reduce to a nonlinear ordinary differential equation. For the three special classes of materials, this differential equation may be solved in closed form, giving a deformation field which is independent of the material response function, or its solution may be reduced to a single quadrature, involving the nonlinear material response function.     

Physically Based Invariants for Nonlinear Elastic Orthotropic Solids

Hydrostatic loading causes an isotropic elastic solid to be in a state of pure dilatation with no distortion relative to its unstressed reference configuration. Similarly, hydrostatic loading causes a general orthotropic solid to be distorted relative to its unstressed reference configuration. This paper introduces physically based invariants for orthotropic nonlinear elastic solids which are measures of distortions that cause deviatoric Cauchy stress. Specifically, these invariants allow for the modeling of the distortion in a hydrostatic state of stress independently of the form of the strain energy function. Consequently, use of these invariants may lead to simpler forms of the strain energy function which adequately model specific orthotropic solids.      

An energy-based anisotropic yield criterion for cellular solids and validation by biaxial FE simulations

The initial yield surface of 2D lattice materials is investigated under biaxial loading using finite element analyses as well as by analytical means. The sensitivity of initial yield surface to the dominant deformation mode is explored by using both low- and high-connectivity topologies whose dominant deformation mode is either local bending or strut stretching, respectively. The effect of microstructural irregularity on the initial yield surface is also examined for both topologies. A pressure-dependent anisotropic yield criterion, which is based on total elastic strain energy density, is proposed for 2D lattice structures, which can be easily extended for application to 3D cellular solids. Proposed criterion uses elastic constants and yield strengths under uniaxial loading, and does not rely on any arbitrary parameter. The analytical framework developed allows the introduction of new scalar measures of characteristic stresses and strains that are capable of representing the elastic response of anisotropic materials with a single elastic master line under multiaxial loading.     

On the Lagrangian and instability of medium with defects

The article presents the Lagrangian of defects in the solids, equipped with bending and warp. The deformation of such elastic medium with defects is based on Riemann-Cartan geometry in three dimensional space. In the static theory for the media with dislocations and disclinations the possible choice of the geometric Lagrangian yield the equations of equilibrium. In this article, the assumed expression for the free energy leading is equal to a volume integral of the scalar function (the Lagrangian) that depends on metric and Ricci tensors only. In the linear elastic isotropic case the elastic potential is a quadratic function of the first and second invariants of strain and warp tensors with two Lame, two mixed and two bending constants. For the linear theory of homogeneous anisotropic elastic medium the elastic potential must be quadratic in warp and strain. The conditions of stability of media with defects are derived, such that the medium in its free state is stable. With the increasing strain the stability conditions could be violated. If the strain in material attains the critical value, the instability in form of emergence of new topological defects occurs. The medium undergoes the spontaneous symmetry breaking in form of emerging topological defects.     

Physical invariants for nonlinear orthotropic solids

Seven invariants, with immediate physical interpretation, are proposed for the strain energy function of nonlinear orthotropic elastic solids. Three of the seven invariants are the principal stretch ratios and the other four are squares of the dot product between the two preferred directions and two principal directions of the right stretch tensor. A strain energy function, expressed in terms of these invariants, has a symmetrical property almost similar to that of an isotropic elastic solid written in terms of principal stretches. Ground state and stress–strain relations are given. Using principal axes techniques, the formulation is applied, with mathematical simplicity, to several types of deformations. In simple shear, a necessary and sufficient condition is given for Poynting relation and two novel deformation-dependent universal relations are formulated. Using series expansions and the symmetrical property, the proposed general strain energy function is refined to a particular general form. A type of strain energy function, where the ground state constants are written explicitly, is proposed. Some advantages of this type of function are indicated. An experimental advantage is demonstrated by showing a simple triaxial test can vary a single invariant while keeping the remaining invariants fixed.     

Theory of images in plane-layered axisymmetric elastostatics

A fundamental theorem which automatically transforms the solution due to the presence of an arbitrary axisymmetric singularity in an unbounded homogeneous isotropic elastic solid into the corresponding solution for two perfectly bonded isotropic semi-infinite elastic solids is systematically applied in a stepwise fashion to obtain the complete image system when an arbitrary axisymmetric singularity is operative in or near a thick elastic layer which separates two other dissimilar isotropic semi-infinite elastic solids. The solution is closed and well structured. As an illustration, the distant effect in the interface layer produced by an influencing normal point force is luminously revealed to be two-dimensional, consisting of a combination at the origin of a bending hot spot and an infinite line of centres of dilatation. We conclude the paper with a complete theory of images for a free elastic layer under the influence of both axisymmetric and asymmetric singularities. We find that if the influencing displacement field is the gradient of a harmonic function, then the calculation of the induced elastic field reduces to the operation of differentiation only.     

A finite element approach to study cavitation instabilities in non-linear elastic solids under general loading conditions

This paper proposes an effective numerical method to study cavitation instabilities in non-linear elastic solids. The basic idea is to examine—by means of a 3D finite element model—the mechanical response under affine boundary conditions of a block of non-linear elastic material that contains a single infinitesimal defect at its center. The occurrence of cavitation is identified as the event when the initially small defect suddenly grows to a much larger size in response to sufficiently large applied loads. While the method is valid more generally, the emphasis here is on solids that are isotropic and defects that are vacuous and initially spherical in shape. As a first application, the proposed approach is utilized to compute the entire onset-of-cavitation surfaces (namely, the set of all critical Cauchy stress states at which cavitation ensues) for a variety of incompressible materials with different convexity properties and growth conditions. For strictly polyconvex materials, it is found that cavitation occurs only for stress states where the three principal Cauchy stresses are tensile and that the required hydrostatic stress component at cavitation increases with increasing shear components. For a class of materials that are not polyconvex, on the other hand and rather counterintuitively, the hydrostatic stress component at cavitation is found to decrease for a range of increasing shear components. The theoretical and practical implications of these results are discussed.     

EIGEN THEORY OF VISCOELASTIC MECHANICS FOR ANISOTROPIC SOLIDS

Anisotropic viscoelastic mechanics is studied under anisotropic subspace. It is proved that there also exist the eigen properties for viscoelastic medium. The modal Maxwell's equation, modal dynamical equation (or modal equilibrium equation) and modal compatibility equation are obtained. Based on them, a new theory of anisotropic viscoelastic mechanics is presented. The advantages of the theory are as follows: 1) the equations are all scalar, and independent of each other. The number of equations is equal to that of anisotropic subspaces, 2) no matter how complicated the anisotropy of solids may be, the form of the definite equation and the boundary condition are in common and explicit, 3) there is no distinction between the force method and the displacement method for statics, that is, the equilibrium equation and the compatibility equation are indistinguishable under the mechanical space, 4) each model equation has a definite physical meaning, for example, the modal equations of order one and order two express the volume change and shear deformation respectively for isotropic solids, 5) there also exist the potential functions which are similar to the stress functions of elastic mechanics for viscoelastic mechanics, but they are not man-made, 6) the final solution of stress or strain is given in the form of modal superimposition, which is suitable to the proximate calculation in engineering.     

Asymptotic fields for dynamic crack growth in non-associative pressure sensitive materials

Asymptotic near-tip fields are analyzed for a plane strain Mode I crack propagating dynamically in non-associative elastic–plastic solids of the Drucker–Prager type with an isotropic linear strain hardening response. Eigen solutions are obtained over a range of material parameters and crack speeds, based on the assumption that asymptotic solutions are variable-separable and fully continuous. A limiting speed, beyond which a tendency to slope discontinuity in angular distributions of stresses and velocities is detected, is found to deviate from the associative models. At low strain-hardening rates, the onset of the plastic potential corner zone ahead of the crack-tip imposes another limit to the crack speed. Correspondingly, those limits imply the limits to the degree of non-associativity at a given crack speed. In addition, a tendency to slope discontinuity in the angular radial stress distribution sets another limit on the non-associativity at vanishing hardening rates.     

A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media

The conditions for the strong ellipticity of the equilibrium equations of compressible, isotropic, nonlinearly elastic solids (established by Simpson and Spector [1]) are expressed in terms of the stored-energy function regarded as a function of the principal stretches. The applicability of this reformulation is illustrated with the help of two specific examples.     

Material symmetry group of the non-linear polar-elastic continuum

We extend the material symmetry group of the non-linear polar-elastic continuum by taking into account microstructure curvature tensors as well as different transformation properties of polar and axial tensors. The group consists of an ordered triple of tensors which makes the strain energy density of polar-elastic continuum invariant under change of reference placement. An analog of the Noll rule is established. Four simple specific cases of the group with corresponding reduced forms of the strain energy density are discussed. Definitions of polar-elastic fluids, solids, liquid crystals and subfluids are given in terms of members of the symmetry group. Within polar-elastic solids we discuss in more detail isotropic, hemitropic, cubic-symmetric, transversely isotropic, and orthotropic materials and give explicitly corresponding reduced representations of the strain energy density. For physically linear polar-elastic solids, when the density becomes a quadratic function of strain measures, reduced representations of the density are established for monoclinic, orthotropic, cubic-symmetric, hemitropic and isotropic materials in terms of appropriate joint scalar invariants of stretch, wryness and undeformed structure curvature tensors.     

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.     

各向同性弹性损伤的双标量描述

损伤状态的描述是损伤力学中仍未完善解决的基本问题．我们旨在对此问题就最简单的一种情形——各向同性弹性损伤，进行较为全面的研究．首先，指出了古典各向同性损伤理论中，基于应变等效假设，用单个标量损伤变量描述损伤状态的局限性．然后，建立了一个用两个标量损伤变量描述的各向同性弹性损伤模型．此模型解除了古典理论的局限，能完全描述各向同性弹性损伤，并且得到本文数值实验的验证．最后，将本文模型与现有细观力学结果连接，给出了宏细观损伤变量之间的关系，使得细观量可以通过宏观量来反映，建立了一个用细观损伤材料常数描述细观缺陷特征的损伤本构模型     

Stress concentration around a nano-scale spherical cavity in elastic media: effect of surface stress

Influence of surface effect on stress concentration around a spherical cavity in a linearly isotropic elastic medium is studied on the basis of continuum surface elasticity. Following Goodier's work, a close form solution of the elastic field created by biaxial uniform load is presented. The stress concentration factors under different load combinations are obtained. It is concluded that consideration of surface effect leads to dependence of stress concentration factors on cavity size. Besides, numerical result indicates that stress concentration factors around the cavity are mainly affected by residual surface tension. The result is significant in the understanding of relevant mechanical phenomena in solids with nano-sized cavities.     

Finite amplitude,periodic motion of a body supported by arbitrary isotropic,elastic shear mountings

The undamped, finite amplitude, periodic motion of a load supported symmetrically by arbitrary isotropic, elastic shear mountings is investigated. Conditions on the shear response function sufficient to guarantee periodic motions for finite shearing with arbitrary initial data are provided. Some general results applicable for all simple shearing oscillators in the class are derived and illustrated graphically. The mechanical response of the general nonlinear shearing oscillator is compared with the response of a certain linear oscillator of comparable design. As consequence, certain static and dynamic aspects of the motion of an arbitrary nonlinear oscillator supported by shear springs are compared with those of a simple, linear oscillator for which the response is well-known and readily determined for the same initial data. The effect of a finite static shear deformation on the frequency equation for superimposed, small amplitude vibrations of the load is examined. The general analysis is applied to a class of hyperelastic biological tissues; and the frequency relation for finite amplitude oscillations of a load supported by soft tissue is derived. The finite amplitude oscillatory shearing of a general isotropic elastic continuum is described; and three universal relations connecting the stress and the oscillatory shearing deformation for every isotropic elastic material are presented.     

General Expressions of Constitutive Equations for Isotropic Elastic Damaged Materials

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A domain-independent integral for computation of stress intensity factors along three-dimensional crack fronts and edges by BEM

The present work deals with an evaluation of stress intensity factors (SIFs) along straight crack fronts and edges in three-dimensional isotropic elastic solids. A new numerical approach is developed for extraction, from a solution obtained by the boundary element method (BEM), of those SIFs, which are relevant for a failure assessment of mechanical components. In particular, the generalized SIFs associated to eigensolutions characterized by unbounded stresses at a neighbourhood of the crack front or a reentrant edge and also that associated to T-stress at the crack front can be extracted. The method introduced is based on a conservation integral, called H-integral, which leads to a new domain-independent integral represented by a scalar product of the SIF times some element shape function defined along the crack front or edge. For sufficiently small element lengths these weighted averages of SIFs give reasonable pointwise estimation of the SIFs. A proof of the domain integral independency, based on the bi-orthogonality of the classical two-dimensional eigensolutions associated to a corner problem, is presented. Numerical solutions of two three-dimensional problems, a crack problem and a reentrant edge problem, are presented, the accuracy and convergence of the new approach for SIF extraction being analysed.     

Transient dynamic fracture analysis by an extended meshfree method with different crack-tip enrichments

Investigation of transient dynamic stress intensity factors (DSIFs) of two-dimensional fracture problems of isotropic solids and orthotropic composites by an extended meshfree method is described. We adopt the recently developed extended meshfree radial point interpolation method (X-RPIM), which combines either the standard branch functions or the new linear ramp function associated with Heaviside functions to capture crack-tip behaviors. It is the first time the linear ramp function integrating into meshfree X-RPIM has been presented in a dynamical fracture context. We are particularly interested in exploring insight into the behaviors of DSIFs under dynamic impact loadings (e.g., step, blast and sine loading types) using our meshfree method. For some of these problems numerical examples have been performed using the new ramp functions, and the obtained results of DSIFs have also been compared with those using the standard enrichment functions under which the two schemes have the same setting. In each case it is found that the numerical solutions delivered using the X-RPIM associated with the ramp enrichments are in good agreement with those with the standard functions. The paper first describes formulations and then provides verification of our developed approach through a series of numerical examples in transient dynamic fracture for both solids and orthotropic composites. Illustration of scattered elastic stress waves propagating in the cracked body is depicted to take an insight look at the behavior of dynamic response.     

Phase equilibria in isotropic solids

The paper determines the forms of equations of equilibrium for stable coherent phase interfaces in isotropic solids. If the first phase satisfies the Baker Ericksen inequalities strictly and the principal stretches of the second phase differ from those of the first phase, one obtains the equality of three specific generalized scalar forces and of a generalized Gibbs function. The forms of these quantities depend on the signs of the increments of the principal stretches across the interface. The proof uses the rank 1 convexity condition for isotropic materials (?ilhavý in Proc. R. Soc. Edinb 129A:1081–1105, 1999) and is available only if the two phases are not too far from each other or if one of the two phases is a fluid (liquid). The result does not follow from the representation theorems for isotropic solids.     

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

In the conventional theory of finite deformations of fibre-reinforced elastic solids it is assumed that the strain-energy is an isotropic invariant function of the deformation and a unit vector A that defines the fibre direction and is convected with the material. This leads to a constitutive equation that involves no natural length. To incorporate fibre bending stiffness into a continuum theory, we make the more general assumption that the strain-energy depends on deformation, fibre direction, and the gradients of the fibre direction in the deformed configuration. The resulting extended theory requires, in general, a non-symmetric stress and the couple-stress. The constitutive equations for stress and couple-stress are formulated in a general way, and specialized to the case in which dependence on the fibre direction gradients is restricted to dependence on their directional derivatives in the fibre direction. This is further specialized to the case of plane strain, and finite pure bending of a thick plate is solved as an example. We also formulate and develop the linearized theory in which the stress and couple-stress are linear functions of the first and second spacial derivatives of the displacement. In this case for the symmetric part of the stress we recover the standard equations of transversely isotropic linear elasticity, with five elastic moduli, and find that, in the most general case, a further seven moduli are required to characterize the couple-stress.