Linear elasticity

About: Linear elasticity is a research topic. Over the lifetime, 9080 publications have been published within this topic receiving 258684 citations.

Fracture initiation at three-dimensional bimaterial interface corners

Abstract: We propose an approach to correlate fracture initiation at three-dimensional bimaterial interface corners based on the existence of a universal singular stress field in the context of linear elasticity. The idea is to correlate fracture initiation with critical values of the stress intensities of the singular fields calculated from a linear elastic stress analysis. The approach is in the spirit of interface fracture mechanics but applicable to a different class of problems, specifically, situations without a preexisting crack and situations where subsequent crack propagation does not necessarily occur along the interface. In order to validate the proposed approach, we designed and fabricated a series of aluminum/epoxy bimaterial specimens that possess well-defined three-dimensional interface corners. They consist of two perfectly bonded square prisms with varying interface dimensions and surface finishes. The specimens were loaded in four-point flexure until fracture initiated at the three-dimensional interface corner. The nominal stress at failure varied significantly with interface dimensions, thus invalidating its use as a fracture criterion. From a rigorous asymptotic analysis of the three-dimensional interface corner stress state, we determined the order of the singularity and the angular variation of the stress and displacement fields. We determined the corresponding stress intensities via full-field finite element analyses of the aluminum/epoxy specimens. Although the measured failure stress varied significantly with interface dimensions, the corresponding critical stress intensity did not, although as expected it varied with interface surface finish. These findings support the use of critical stress intensities to correlate fracture initiation at three-dimensional bimaterial interface corners.

Finite Element Methods in Continuum Mechanics

Abstract: Publisher Summary The chapter presents a brief introduction to the different finite element formulations for linear elastic solids and discusses similar formulations for several other field problems. The chapter presents detailed illustrations for several typical finite element formulations. In the finite element formulation, displacement and stress fields are assumed to be continuous within each discrete element. This formulation calls for modified variational principles for which the continuity or equilibrium conditions along the interelement boundaries are introduced as conditions of constraint and appropriate boundary variables are used as the corresponding Lagrangian multipliers. The chapter presents the several variational principles and the corresponding models used in the finite element formulation. The large majority of the existing finite element formulations are based on the assumed displacement approach. The chapter discusses equilibrium problems of linear elastic solids. There are several other problems in solid mechanics, which can be formulated by means of variational principles and hence can be solved by finite element methods. The finite element methods have also been extended to nonlinear problems resulting from elastic-plastic material properties or from large deflections or finite strains.

Analysis of an HDG method for linear elasticity

Abstract: Summary We present the first a priori error analysis for the first hybridizable discontinuous Galerkin method for linear elasticity proposed in Internat. J. Numer. Methods Engrg. 80 (2009), no. 8, 1058–1092. We consider meshes made of polyhedral, shape-regular elements of arbitrary shape and show that, whenever piecewise-polynomial approximations of degree k≥0 are used and the exact solution is smooth enough, the antisymmetric part of the gradient of the displacement converges with order k, the stress and the symmetric part of the gradient of the displacement converge with order k + 1/2, and the displacement converges with order k + 1. We also provide numerical results showing that the orders of convergence are actually sharp. Copyright © 2014 John Wiley & Sons, Ltd.

On the functional form of non-local elasticity kernels

Abstract: The functional form of non-local elasticity kernels is studied within the context of the integral formalism. The study is limited to linear isotropic elasticity. The kernels are derived analytically based on the discrete structure of the material at the atomic scale. Atomistic simulations are used to validate the results. Materials in which the interatomic interactions are represented by pair, as well as embedded atom-type potentials are considered. The derived kernels have a range which extends up to the cut-off radius of the interatomic potential, are positive at the origin, and become negative approximately one atomic distance away, thus departing from the commonly assumed Gaussian functional form. The functional form of the potential and the radial distribution function of interacting neighbors about a representative atom fully define their shape. This new continuum model involves two material length scales that are both derived from atomistics for a Morse solid and for Al. Two applications are considered in closure. It is shown that in strained superlattices, the non-local model predicts maximum stresses that are much larger than those obtained within the local theory. This observation has implications for defect nucleation in these structures. Furthermore, the new non-local model improves upon the Gaussian one by predicting a more realistic wave dispersion relationship, with essentially zero group velocity at the boundary of the Brillouin zone.