Journal of Elasticity最新文献

Following Hill and Leblond, the aim of our work is to show, for isotropic nonlinear elasticity, a relation between the corotational Zaremba–Jaumann objective derivative of the Cauchy stress (sigma ), i.e.

and a constitutive requirement involving the logarithmic strain tensor. Given the deformation tensor (F = mathrm {D}varphi ), the left Cauchy-Green tensor (B = F , F^{T}), and the strain-rate tensor (D = operatorname{sym}(dot{F} , F^{-1})), we show that

where (operatorname{Sym}^{++}_{4}(6)) denotes the set of positive definite, (minor and major) symmetric fourth order tensors. We call the first inequality of (1) “corotational stability postulate” (CSP), a novel concept, which implies the True-Stress True-Strain strict Hilbert-Monotonicity (TSTS-M⁺) for (B mapsto sigma (B) = widehat{sigma}(log B)), i.e.

A similar result, but for the Kirchhoff stress (tau = J , sigma ) has been shown by Hill as early as 1968. Leblond translated this idea to the Cauchy stress (sigma ) but only for the hyperelastic case. In this paper we expand on the ideas of Hill and Leblond, extending Leblond calculus to the Cauchy elastic case.

X-ray diffraction methods are an established technique to analyze residual stresses in polycrystalline materials. Using diffraction, lattice plane distances are measured, from which residual stresses can be calculated by using diffraction elastic constants which can be inferred from experimental measurements or calculated based on micromechanical model assumptions. We consider two different generalizations of existing micromechanical models for the case of texture-free, i.e. statistically isotropic, single-phase polycrystals. The first is based on the singular approximation method of classical micromechanics, from which existing Voigt, Reuss, Hashin-Shtrikman and self-consistent methods are recovered. The second approach, which is newly proposed in this work, is based on the micromechanical Maximum Entropy Method. Both approaches are applied to the problem of calculating diffraction elastic constants of texture-free cubic polycrystals and are found to be consistent with each other in that case. Full-field FFT simulations are used to validate the analytical models by simulating X-ray diffraction measurements of copper. In the simulative setting, many sources of experimental measurement error are not present, which results in a particularly accurate validation of theoretical bounds and approximations. The first core result of the paper is a formulation of diffraction elastic constants for texture-free polycrystals in terms of the macroscopically measurable effective shear modulus. These diffraction elastic constants can be adapted to the properties of a given material sample. The second core result is the validation of the Maximum Entropy Method for X-ray diffraction stress analysis of texture-free single-phase materials as a preliminary step before extending the method to textured and multi-phase materials.

Shallow shells are widely encountered in biological structures, especially during embryogenesis, when they undergo significant shape variations. As a consequence of geometric frustration caused by underlying biological processes of growth and remodeling, such thin and moderately curved biological structures experience initial stress even in the absence of an imposed deformation. In this work, we perform a rigorous asymptotic expansion from three-dimensional elasticitiy to obtain a nonlinear morphoelastic theory for shallow shells accounting for both initial stress and large displacements. By application of the principle of stationary energy for admissible variation of the tangent and normal displacement fields with respect to the reference middle surface, we derive two generalised nonlinear equilibrium equations of the Marguerre-von Kármán type. We illustrate how initial stress distributions drive the emergence of spontaneous mean and Gaussian curvatures which are generally not compatible with the existence of a stress free configuration. We also show how such spontaneous curvatures influence the structural behavior in the solutions of two systems: a saddle-like and a cylindrical shallow shell.

Necessary and sufficient elastic stability conditions for single crystals for the seven crystal systems are specified for both stiffness and compliance tensors. For Laue classes of four of these crystal systems, conditions for the positive-definite forms of suitably chosen 4 × 4 real symmetric matrices are shown to be both useful and relevant. Other supposedly equivalent and often simpler conditions proposed in the literature for tetragonal, orthorhombic and monoclinic crystal systems are analysed; all are shown to be incorrect.

Harmonic inclusions are defined as those that do not disturb the mean stress component of an initial stress field existing in a homogeneous elastic matrix when they are introduced into the matrix. The design of harmonic inclusions in the literature mainly focuses on the common cases in which the initial stress field has a constant mean stress component (while the corresponding deviatoric stress component may be either constant or non-constant). To identify the configuration of harmonic elastic inclusions in the common cases, researchers consistently assumed that the internal stresses inside the inclusions are hydrostatic and uniform although no rigorous justification was given. In this paper, we present a rigorous proof for the necessity of this assumption in the design of harmonic elastic inclusions in plane deformations. Specifically, we show that the internal stresses inside any elastic inclusion meeting the harmonicity condition must be uniform and (in-plane) hydrostatic (except for trivial cases in which the inclusion and matrix have the same shear modulus). We develop also a general analytic procedure to determine the desired shape for an isolated harmonic elastic inclusion for an arbitrary deviatoric component of the initial stress field, which is illustrated via a few numerical examples.

An analytical solution is presented for adhesive contact of a rigid disc inclusion embedded in a penny-shaped crack in a transversely isotropic piezoelectric material. By virtue of Hankel transforms and a method of potentials, the mixed boundary-value problem is formulated as dual and triple integral equations, which, in turn, are reduced to Fredholm integral equations. The results of primary interest to engineering applications, namely, the total indentation load, the total electric charge, and stress intensity factor at the tip of the crack are evaluated as integral equations in terms of dimensionless parameters. Finally, to reveal the efficacy of the proposed method and also to verify it, comparison is made with indentation solutions in transversely isotropic and isotropic media.

This article focuses on the derivation of explicit descriptions of networks in large deformation through the homogenization method of discrete media. Analytical models are established for the in-plane behavior of a planar periodic truss, whose cell contains a single node, as frequently encountered in practice. The cell is composed of bars that support only axial forces and are connected by perfect hinges. For the considered type of trusses, (given that the equilibrium conditions of the node and of the cell coincide) closed-form expressions for the local behaviour in the case of large deformations can be derived. This case makes it possible to combine the non-linearities arising from large deformations on the one hand and rheological characteristics on the other, and to compare their respective effects as a function of cell morphology. The results are illustrated by the shear and extension responses of specific trusses. The analysis is carried out for bars with stiffening, linear or softening behavior. The combination of the effects of geometrical non-linearities, rheological non-linearities and anisotropy results in particularly rich behaviors of the network.

In this note, we address several issues, including some raised in recent works and commentary, related to bending measures and energies for plates and shells, and certain of their invariance properties. We discuss overlaps and distinctions in results arising from two different definitions of stretching, correct an error and citation oversights in our prior work, reiterate some of the early history of dilation-invariant bending measures, and provide additional brief observations regarding the relative size of energetic terms and the symmetrization of bending measures. A particular point of emphasis is the distinction between dilation-invariant measures and a recently introduced non-dilation-invariant measure for shells and curved rods. In the course of this discussion, we provide a simpler presentation of the elementary, but much neglected, fact that the through-thickness derivative of tangential stretch of material near the mid-surface of a thin body is the product of the mid-surface stretch and change in curvature.

A continuum mechanical theory incorporating an extension of Finsler geometry is formulated for fibrous soft solids. Especially if of biologic origin, such solids are nonlinear elastic with evolving microstructures. For example, elongated cells or collagen fibers can stretch and rotate independently of motions of their embedding matrix. Here, a director vector or internal state vector, not always of unit length, in generalized Finsler space relates to a physical mechanism, with possible preferred direction and intensity, in the microstructure. Classical Finsler geometry is extended to accommodate multiple director vectors (i.e., multiple fibers in both a differential-geometric and physical sense) at each point on the base manifold. A metric tensor can depend on the ensemble of director vector fields. Residual or remnant strains from biologic growth, remodeling, and degradation manifest as non-affine fiber and matrix stretches. These remnant stretch fields are quantified by internal state vectors and a corresponding, generally non-Euclidean, metric tensor. Euler-Lagrange equations derived from a variational principle yield equilibrium configurations satisfying balances of forces from elastic energy, remodeling and cohesive energies, and external chemical-biological interactions. Given certain assumptions, the model can reduce to a representation in Riemannian geometry. Residual stresses that emerge from a non-Euclidean material metric in the Riemannian setting are implicitly included in the Finslerian setting. The theory is used to study stress and damage in the ventricle (heart muscle) expanding or contracting under internal and external pressure. Remnant strains from remodeling can reduce stress concentrations and mitigate tissue damage under severe loading.

Contractile biopolymer networks, such as the actomyosin meshwork of animal cells, are ubiquitous in living organisms. The active gel theory, which provides a thermodynamic framework for these materials, has been mostly used in conjunction with the assumption that the microstructure of the biopolymer network is based on rigid rods. However, experimentally, crosslinked actin networks exhibit entropic elasticity. Here we combine an entropic elasticity kinetic theory, in the spirit of the Green and Tobolsky model of transiently crosslinked networks, with an active flux modelling biological activity. We determine this active flux by applying Onsager reciprocal relations to the corresponding microscopic dynamics. We derive the macroscopic active stress that arises from the resulting dynamics and obtain a closed-form model of the macroscopic mechanical behaviour. We show how this model can be rewritten using the framework of multiplicative deformation gradient decomposition, which is convenient for the resolution of such problems.