Abstract
Recent interest in developing new constitutive models for the behavior of elastic materials has led to a new nonlinear theory of elasticity, wherein the linearized strain bears a nonlinear relationship with the Cauchy stress. An important advantage of such a nonlinear constitutive model is to predict the crack-tip fields more realistically by bounding the strain under the unbounded stress. In this paper, we employ such a nonlinear relationship to characterize the circular crack-tip fields in a strain-limiting elastic solid with a penny-shaped crack in the center. The behavior of the bulk material is defined through a nonlinear relationship between the stress and the strain. The mathematical model studied is bounded, Lipschitz continuous, coercive, and more importantly, monotone. The equilibrium equation for the 3-D body, coupled with a special choice of the response relation, yields a second-order, quasi-linear, partial-differential-equation system. The numerical solution of such a highly nonlinear system is obtained by using a conforming, bilinear, Galerkin-type finite-element technique through a software. In stark contrast with the theory of linearized elasticity, the growth of the near-tip strain is far less than that of the stress. Results of the stress intensity factor and the strain energy density are comparable with those of the linear-elasticity model. Our work demonstrates that such nonlinear strain-limiting theory can bound the strain in the crack tips, and can be used to study stress or strain-energy-density based fracture models.
Published Version
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