We propose a one-dimensional, nonconvex elastic constitutive model with higher gradients that can predict spontaneous fracture at a critical load via a bifurcation analysis. It overcomes the problem of discontinuous deformations without additional field variables, such as damage or phase-field variables, and without a priori specified surface energy. Our main tool is the inverse deformation, which can be extended to be a piecewise smooth mapping even when the original deformation has discontinuities describing cracks opening. We exploit this via the inverse-deformation formulation of finite elasticity due to Shield and Carlson, including higher gradients in the energy. The problem is amenable to a rigorous global bifurcation analysis in the presence of a unilateral constraint. Fracture under hard loading occurs on a bifurcating solution branch at a critical applied stretch level, with a sudden drop to zero stress. Fractured solutions are found to have sharply defined cracks with surface energy arising from higher gradient effects.

我们提出了具有较高梯度的一维非凸弹性本构模型，该模型可以通过分叉分析来预测临界载荷下的自发断裂。它克服了不连续变形的问题，而没有附加的场变量，例如损伤或相场变量，并且没有先验的指定表面能。我们的主要工具是逆向变形，即使原始变形具有不连续性来描述裂纹的开口，也可以将其扩展为分段平滑映射。由于Shield和Carlson的作用，我们通过有限弹性的反变形公式来利用这一点，其中包括较高的能量梯度。在存在单边约束的情况下，该问题适合进行严格的全局分叉分析。硬负荷下的断裂会在分叉的溶液分支上以临界施加拉伸水平发生，并突然降至零应力。发现破裂的溶液具有尖锐的裂纹，其表面能来自较高的梯度效应。