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.

众所周知，线性化弹性理论在应用于某些简单的断裂模型时承认奇异应变在逻辑上不一致的解，而该理论是在无穷小位移梯度的渐近极限中对有限弹性的一阶近似。同时，应变限制模型是非线性隐式本构关系的一个特殊子类，可预测整个材料体中的均匀有界应变，包括应变集中器（如裂纹尖端或凹角）处。这种非线性近似在柯西或格林弹性的标准线性化过程中是不可能的。在这项工作中，我们检查了几个边界值问题的有限元离散化，以研究实体中的应力-应变状态，其响应由非线性应变限制弹性理论描述。分析了反平面剪切中的缺口、定向裂纹和界面裂纹问题。数值结果表明线性化应变保持低于可以固定的值因此，先验保证了非线性模型的有效性。此外，我们在每个示例中都发现裂纹尖端附近的高应力值，从而表明裂纹尖端充当固定裂纹的单一能量汇。我们还计算了本研究中的应力强度因子 (SIF)。非线性应变限制模型中 SIF 的计算值对应于经典线性模型的计算值，从而为可能的局部断裂准则提供了原则。线性化应变与应力呈非线性关系的应变限制理论框架可以为开发具有物理意义的模型以研究弹性固体中的裂纹演化提供合理的基础。