Journal of Elasticity最新文献

In this article, the proposed model analyzed shear wave propagation through an orthotropic strip with an edge crack. Dual integral equations have been developed for solution of the governing mixed boundary value problem with the aid of Hankel transform technique. Then, the dual integral equations have been transformed into a second kind Fredholm integral equation employing Abel’s transformation. The numerical calculations of stress intensity factor and crack opening displacement are performed utilizing the Fox & Goodwin method and displayed graphically. Elastic constants of two orthotropic materials have been used to illustrate the influence of material orthotropy and normalized strip width on SIF and COD.

We present a class of models of elastic phase transitions with incompatible energy wells in an arbitrary space dimension, where in a hard device an abundance of Lipschitz global minimizers coexists with a complete lack of strong local minimizers. The analysis is based on the proof that every strong local minimizer in a hard device is also a global minimizer which is applicable much beyond the chosen class of models. Along the way we show that a new demonstration of sufficiency for a subclass of affine boundary conditions can be built around a novel nonlinear generalization of the classical Clapeyron theorem.

Within the framework of linear elasticity, we show that any displacement of a straight rod is the sum of a Bernoulli–Navier displacement and two terms, one for shearing and the other for warping. Then, we load a straight rod so that bending and shear contribute the same to the rotations of the cross-section.

In a large class of porous elastic solids such as cement concrete, rocks, ceramics, porous metals, biological materials such as bone, etc., the material moduli depend on density. When such materials undergo sufficiently small deformations, the usual approach of appealing to a linearized elastic constitutive relation to describe their response will not allow us to capture the dependence of the material moduli on the density, as this would imply a nonlinear relationship between the stress and the linearized strain in virtue of the balance of mass as dependence on density implies dependence on the trace of the linearized strain. It is possible to capture the dependence of the material moduli on the density, when the body undergoes small deformations, within the context of implicit constitutive relations. We study the stress concentration due to a rigid elliptic inclusion within a new class of implicit constitutive relations in which the stress and the linearized strain appear linearly, that allows us to capture the dependence of the material moduli on the density. We find that the stress concentration that one obtains employing the constitutive relation wherein the material moduli depend on the density can be significantly different from that obtained by adopting the classical linearized elastic constitutive relation to which it reduces to when the density dependence of the material moduli are ignored.

Lattice materials formed by hinged springs or linear elastic bonds may exhibit diverse anisotropy and asymmetry features of the overall elastic behavior depending on their unit cell configuration. The recently developed singum model transfers the force-displacement relationship of the springs in the lattice to the stress-strain relationship in the continuum particle, and provides the analytical form of tangential elasticity. When a pre-stress exists in the lattice, the stiffness tensor significantly changes due to the effect of the configurational stress; existing methods like the lattice spring method, relying on a scalar energy equivalence, are insufficient in such situations. Instead, a tensorial homogenization method with the new definition of singum stress and strain, should be preferred. Different lattice structures lead to different symmetries of the stiffness tensors, which are demonstrated by five lattices. When all bonds exhibit the same length, regular hexagonal, honeycomb, and auxetic lattices demonstrate that the stiffness changes from an isotropic to anisotropic, from symmetric to asymmetric tensor. When the central symmetry of the unit cell is not satisfied, the primitive cell will contain more than one singums and the Cauchy–Born rule fails by the loss of equilibrium of the single singum. A secondary stress is induced to balance the singums. Displacement gradient (d_{ij}=u_{j,i}) is proposed to replace strain in the constitutive law for the general case because (d_{12}) and (d_{21}) can produce different stress states. Although the hexagonal and honeycomb lattices may exhibit isotropic behavior, for general auxetic lattices, an anisotropic and asymmetric elastic tensor is obtained with the loss of both minor and major symmetry, which is also demonstrated in a square lattice with unbalanced central symmetry and a chiral lattice. The modeling procedure and results can be generalized to three dimensions and other lattices with the anisotropic and asymmetric stiffness.

A single elliptical or ellipsoidal inclusion with an arbitrary uniform eigenstrain is known to achieve a constant stress field when embedded in an elastic medium provided the edge of the medium is sufficiently far from the inclusion (i.e. the interaction between the inclusion and the edge of the medium is negligible). In this paper, we aim to answer the question as to whether there exists an inclusion of certain configuration (with a uniform eigenstrain) that remains to possess a constant stress when embedded in a bounded medium whose edge interacts significantly with it. Specifically, we consider the anti-plane shear case of an inclusion with a uniform eigenstrain in a circular medium with a traction-free edge. We derive a sufficient and necessary condition ensuring the uniformity of the stress within the inclusion, which further leads to a nonlinear system of equations with respect to an infinite group of parameters characterizing the shape of the inclusion. We obtain convergent solutions for the truncated version of the nonlinear system using numerical techniques, and illustrate the corresponding shape of the inclusion in a few numerical examples. Our results for the case corresponding to small inclusion size and small edge-inclusion distance (relative to the radius of the medium) are well-consistent with the existing results for an inclusion with uniform stress in a semi-infinite medium with a traction-free surface, while those for centrally placed inclusions achieving uniform stress capture the classical case of centric circular inclusion accurately. The results presented in this paper provide a strong evidence for the existence of inclusions possessing uniform stress in an elastic bounded domain subjected to common external boundary conditions under anti-plane shear deformation.

Mechanical fields over thin elastic surfaces can develop singularities at isolated points and curves in response to constrained deformations (e.g., crumpling and folding of paper), singular body forces and couples, distributions of isolated defects (e.g., dislocations and disclinations), and singular metric anomaly fields (e.g., growth and thermal strains). With such concerns as our motivation, we model thin elastic surfaces as von Kármán plates and generalize the classical von Kármán equations, which are restricted to smooth fields, to fields which are piecewise smooth, and can possibly concentrate at singular curves, in addition to being singular at isolated points. The inhomogeneous sources to the von Kármán equations, given in terms of plastic strains, defect induced incompatibility, and body forces, are likewise allowed to be singular at isolated points and curves in the domain. The generalized framework is used to discuss the singular nature of deformation and stress arising due to conical deformations, folds, and folds terminating at a singular point.