Journal of Elasticity最新文献

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.

In this paper, we present the first study of plane-strain problems within the framework of complete thermo-flexoelectric theory, incorporating strain-gradient elasticity, direct and converse flexoelectricity, as well as thermoelasticity. We derive the exact solutions for three typical thermoelastic plane strain problems, which are the mechanical-electrical-thermal coupling problem for an infinite-length strip, the mechanical-electrical-thermal coupling problem for a hollow cylinder, and the thermal eigenstrain problem for a cylindrical inclusion. We develop the mixed finite element framework for the plane-strain thermo-flexoelectric problems, benchmarked against the three analytical solutions. This study reveals that the electric field induced by inhomogeneous heating in thermo-flexoelectric solids exhibits a pronounced size effect. Notably, an increase in the strain-gradient length scale parameter diminishes the thermo-flexoelectric effects. This study not only deepens the understanding of the mechanisms of multiphysical fields coupling in thermo-flexoelectric solids, but also provides insights for designing nano thermo-electric converters based on the principle of thermo-flexoelectricity.

A thermodynamically consistent theory for finite deformation size-dependent elastic-inelastic response of a Cosserat material with a deformable director triad ({mathbf{d}}_{i}) and a single absolute temperature (theta ) has been developed by the direct approach. A unique feature of the proposed theory is the Eulerian formulation of constitutive equations, which do not depend on arbitrariness of reference or intermediate configurations or definitions of total and plastic deformation measures. Inelasticity is modeled by an inelastic rate tensor in evolution equations for microstructural vectors. These microstructural vectors model elastic deformations and orientation changes of material anisotropy. General hyperelastic anisotropic constitutive equations are proposed with simple forms in terms of derivatives of the Helmholtz free energy, which depends on elastic deformation variables that include elastic deformations of the directors relative to the microstructural vectors. An important feature of the model is that the gradients of the elastic director deformations in the balances of director momentum control size dependence and are active for all loadings. Analytical solutions of the small deformation equations for simple shear are obtained for elastic response and strain-controlled cyclic loading of an elastic-viscoplastic material.

The primary objective of this study is to investigate the stochastic plasma-mechanical-elastic wave propagation at the boundary of an elastic half-space in a semiconductor material using photo-thermoelasticity theory. The novelty of this work lies in the combination of stochastic simulation with temperature-dependent electrical conductivity and variable thermal conductivity, applied to a two-dimensional (2D) electromagnetic problem based on the electron-hole interaction model. Unlike previous studies, this work incorporates white noise as the randomness factor, providing a more realistic representation of random processes in semiconductor materials. The normal mode analysis technique is used to derive both deterministic and stochastic wave behaviors, focusing on short-time dynamics. The results, which are numerically analyzed and graphically represented, provide new insights into the differential behavior of stochastic versus deterministic distributions in magneto-photo-thermoelastic wave propagation, contributing to a more comprehensive understanding of semiconductor behavior under random influences.

In 1978, Murdoch presented a direct second-gradient hyperelastic theory for thin shells in which the strain-energy density associated with a deformation (boldsymbol{eta }) of a surface (mathcal{S}) is allowed to depend constitutively on the three kinematical descriptors (boldsymbol{C}), (boldsymbol{H}), and (boldsymbol{F}^{scriptscriptstyle top }boldsymbol{G}), where (boldsymbol{F}=text{Grad} _{scriptscriptstyle mathcal{S}} boldsymbol{eta }), (boldsymbol{C}=boldsymbol{F}^{scriptscriptstyle top }boldsymbol{F}), (boldsymbol{H}=boldsymbol{F}^{scriptscriptstyle top }boldsymbol{L}_{ scriptscriptstyle mathcal{S}'}boldsymbol{F}) is the covariant pullback of the curvature tensor (boldsymbol{L}_{scriptscriptstyle mathcal{S}'}) of the deformed surface (mathcal{S}'), and (boldsymbol{G}=text{Grad} _{scriptscriptstyle mathcal{S}} boldsymbol{F}). On the other hand, in Koiter’s direct thin-shell theory, the strain-energy density depends constitutively on only (boldsymbol{C}) and (boldsymbol{H}). Due to the popularity of Koiter’s theory, the second-order tensors (boldsymbol{C}) and (boldsymbol{H}) are well understood and have been extensively characterized. However, the third-order tensor (boldsymbol{F}^{scriptscriptstyle top }boldsymbol{G}) in Murdoch’s theory is largely overlooked in the literature. We address this gap, providing a detailed characterization of (boldsymbol{F}^{scriptscriptstyle top }boldsymbol{G}). We show that for (boldsymbol{eta }) twice continuously differentiable, (boldsymbol{F}^{scriptscriptstyle top }boldsymbol{G}) depends solely on (boldsymbol{C}) and its surface gradient (text{Grad} _{scriptscriptstyle mathcal{S}}boldsymbol{C}) and does not depend on (boldsymbol{L}_{scriptscriptstyle mathcal{S}'}). For the special case of a conformal deformation, we find that a suitably defined strain measure corresponding to (boldsymbol{F}^{scriptscriptstyle top }boldsymbol{G}) depends only the conformal stretch and its surface gradient. For the further specialized case of an isometric deformation, this strain measure vanishes. An orthogonal decomposition of (boldsymbol{F}^{scriptscriptstyle top }boldsymbol{G}) reveals that it belongs to a ten-dimensional subspace of the space of third-order tensors and embodies two independent types of non-local phenomena: one related to the spatial variations in the stretching of (mathcal{S}') and the other to the curvature of (mathcal{S}).

Many biological materials exhibit the ability to actively deform, essentially due to a complex chemical interaction involving two proteins, actin and myosin, in the myocytes (the muscle cells). While the mathematical description of passive materials is well-established, even for large deformations, this is not the case for active materials, since capturing its complexities poses significant challenges. This paper focuses on the mathematical modeling of active deformation of biological materials, guided by the important example of skeletal muscle tissue. We will consider an incompressible and transversely isotropic material within a hyperelastic framework. Our goal is to design constitutive relations that agree with uniaxial experimental data whenever possible. Finally, we propose a novel model based on a coercive and polyconvex elastic energy density for a fiber-reinforced material; in this model, active deformation occurs solely through a change in the reference configuration of the fibers, following the mixture active strain approach. This model assumes a constant active parameter, preserving the good mathematical features of the original model while still capturing the essential deformations observed in experiments.