Linear elasticity

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

Physically-Based Approach to the Mechanics of Strong Non-Local Linear Elasticity Theory

Abstract: In this paper the physically-based approach to non-local elasticity theory is introduced. It is formulated by reverting the continuum to an ensemble of interacting volume elements. Interactions between adjacent elements are classical contact forces while long-range interactions between non-adjacent elements are modelled as distance-decaying central body forces. The latter are proportional to the relative displacements rather than to the strain field as in the Eringen model and subsequent developments. At the limit the displacement field is found to be governed by an integro-differential equation, solved by a simple discretization procedure suggested by the underlying mechanical model itself, with corresponding static boundary conditions enforced in a quite simple form. It is then shown that the constitutive law of the proposed model coalesces with the Eringen constitutive law for an unbounded domain under suitable assumptions, whereas it remains substantially different for a bounded domain. Thermodynamic consistency of the model also has been investigated in detail and some numerical applications are presented for different parameters and different functional forms for the decay of the long range forces. For simplicity, the problem is formulated for a 1D continuum while the general formulation for a 3D elastic solid has been reported in the appendix.

Fracture initiation at sharp notches under mode I, mode II, and mild mixed mode loading

Abstract: In the context of linear elasticity, a stress singularity of the type Knrδ(δ<0) may exist at sharp re-entrant corners, with an intensity Kn In general the order of the stress singularity δ and the stress intensity differ for symmetric (mode I) and antisymmetric (mode II) loading Under general mixed-mode loadings, the magnitudes of the mode I and II intensities fully characterize the stress state in the region of the corner A failure criterion based on critical values of these intensities may be appropriate in situations where the region around the corner dominated by the singular fields is large compared to intrinsic flaw sizes, inelastic zones, and fracture process zone sizes We determined the mode I and II stress intensities for notched mode I tensile specimens and notched mode II flexure specimens using a combination of the Williams (1952) asymptotic method, dimensional considerations, and detailed finite element analysis We carried out a companion experimental study to extract critical values of the mode I and II stress intensities for a series of notched polymethyl methacrylate (PMMA) tensile and flexure specimens with notch angles of 90- The data show that excellent failure correlation is obtained, in both mode I and II loading, through the use of a single parameter, the critical stress intensity We then analyzed and tested a series of T-shaped structures containing 90- corners The applied tensile loading results in mixed-mode loading of the 90- corners Failure of the specimens is brittle and can be well-correlated with a critical mode I stress intensity criterion using the results of the notched mode I tensile tests This is attributed to large difference in the strength of the stress singularities in modes I and II: δ= -04555 and -00915 for modes I and II for a 90- notch As a result, the mode I loading dominates the failure process for the 90- corner in the T-structure

Equilibrium finite elements for the linear elastic problem

Abstract: We consider a class of equilibrium finite element methods for elasticity problems. The approximate stresses satisfy the equilibrium equations but the symmetry of the stress tensor is relaxed. Optimal error bounds for the stresses and numerical examples are given.

Scale{dependent homogenization of random composites as micropolar continua

Abstract: A multitude of composite materials ranging from polycrystals to rocks, concrete, and masonry overwhelmingly display random morphologies. While it is known that a Cosserat (micropolar) medium model of such materials is superior to a Cauchy model, the size of the Representative Volume Element (RVE) of the effective homogeneous Cosserat continuum has so far been unknown. Moreover, the determination of RVE properties has always been based on the periodic cell concept. This study presents a homogenization procedure for disordered Cosserat-type materials without assuming any spatial periodicity of the microstructures. The setting is one of linear elasticity of statistically homogeneous and ergodic two-phase (matrix-inclusion) random microstructures. The homogenization is carried out according to a generalized Hill–Mandel type condition applied on mesoscales, accounting for non-symmetric strain and stress as well as couple-stress and curvature tensors. In the setting of a two-dimensional elastic medium made of a base matrix and a random distribution of disk-shaped inclusions of given density, using Dirichlet-type and Neumann-type loadings, two hierarchies of scale-dependent bounds on classical and micropolar elastic moduli are obtained. The characteristic length scales of approximating micropolar continua are then determined. Two material cases of inclusions, either stiffer or softer than the matrix, are studied and it is found that, independent of the contrast in moduli, the RVE size for the bending micropolar moduli is smaller than that obtained for the classical moduli. The results point to the need of accounting for: spatial randomness of the medium, the presence of inclusions intersecting the edges of test windows, and the importance of additional degrees of freedom of the Cosserat continuum.

Two-dimensional contact analysis of elastic graded materials

Abstract: The present paper examines two-dimensional contact of a rigid cylinder on an elastic graded substrate. The normal, sliding and rolling type of contact are addressed. The effect of adhesion in frictionless contact is also examined. The elastically graded substrate is modeled to be locally isotropic with constant Poisson ratio and elastic modulus that varies with depth, y, according to a power law, E=E 0 y k ; 0≤k , y≥0 . Such variation covers a fairly broad class of graded materials. Exact results are derived within the context of small deformation linear elasticity. The results show that the power law elastic gradation can be very advantageous in the design of strong and wear resistance sliding surfaces.