A finite-element discretization of some boundary value problems for nonlinear strain-limiting elastic bodies

Abstract

It is well known that the linearized theory of elasticity admits the logically inconsistent solution of singular strains when applied to certain naive models of fracture while the theory is a first-order approximation to finite elasticity in the asymptotic limit of infinitesimal displacement gradient. Meanwhile, the strain-limiting models, a special subclass of nonlinear implicit constitutive relations, predict uniformly bounded strain in the whole material body including at the strain-concentrator such as a crack tip or reentrant corner. Such a nonlinear approximation cannot be possible within the standard linearization procedure of either Cauchy or Green elasticity. In this work, we examine a finite-element discretization for several boundary value problems to study the state of stress–strain in the solid body of which response is described by a nonlinear strain-limiting theory of elasticity. The problems of notches, oriented cracks, and an interface crack in anti-plane shear are analyzed. The numerical results indicate that the linearized strain remains below a value that can be fixed a priori, therefore, ensuring the validity of the nonlinear model. In addition, we find high stress values in the neighborhood of the crack tip in every example, thereby suggesting that the crack tip acts as a singular energy sink for a stationary crack. We also calculate the stress intensity factor (SIF) in this study. The computed value of SIF in the nonlinear strain-limiting model is corresponding to that of the classical linear model, and thereby providing a tenet for a possible local criterion for fracture. The framework of strain-limiting theories, within which the linearized strain bears a nonlinear relationship with the stress, can provide a rational basis for developing physically meaningful models to study a crack evolution in elastic solids.

Existence and Uniqueness of Global Weak Solutions to Strain-Limiting Viscoelasticity with Dirichlet Boundary Data

“…The authors study the timeindependent problem in three different crack geometries in the antiplane setting. The numerical results presented in [37] indicate that the linearized strain remains bounded a priori below a fixed value, while the value of the stress is able to be very high. Indeed, near the crack tip, the stress grows significantly faster than the strain.…”

“…Furthermore, there has been recent study of a finite-element discretization of problems based on strain-limiting elasticity in [37]. The authors study the timeindependent problem in three different crack geometries in the antiplane setting.…”

A finite element model for a coupled thermo-mechanical system: nonlinear strain-limiting thermoelastic body

“…For strains, we can see much smaller values along the radial line leading up to the crack-tip, as expected from the nonlinear strain-limiting model. Also note that compared to Figure 14 (b), Figure 16 (b) has a relatively sharper change in color for the legend (e.g., see the cyan-colored line in between the blue and the green) over the domain and in the vicinity of the tip, implying that the the contour lines for the strain near the tip along with the shape of crack for the strain-limiting is much different from the linear model (35).…”

“…where ω 0 , ω 1 , ω 2 are constants. Models of the type (33) or (35) are very important in their own rights, as these could be compelling candidates to describe the response of brittle materials such as rock, gum metal, ceramics, glass, and more specifically in the study of evolution of cracks and fractures in brittle bodies.…”

“…In particular, the results in (24-26; 30; 34; 35) predict crack-tip stress singularity similar to the classical LEFM model for static crack problems. This implies that the crack-tip is a singular energy sink hence one can utilize a local fracture criterion to study the evolution of cracks within the nonlinear strain-limiting relation (35)(36)(37). Thus, the objective of this study is to offer a reasonable rationale to use the nonlinear strain-limiting models and describe a reliable thermoelastic response particularly around the crack-tip.…”