标题： 正交各向异性断裂的计算洞察：非均匀载荷下应变限制材料中的裂纹尖端场 显示英文标题

标题： Computational Insights into Orthotropic Fracture: Crack-Tip Fields in Strain-Limiting Materials under Non-Uniform Loads

摘要： 一个有限元框架被提出用于分析横观各向同性、应变限制弹性材料中的裂纹尖端现象。 机械响应由一个代数非线性的本构模型表征，该模型将应力与线性化应变相关联。 裂纹顶端的非物理应变奇异性得到缓解，确保应变幅度是有限的。 该方法显著推进了边界值问题（BVP）的公式化，特别是对于一阶近似理论。 对于具有裂纹的横观各向同性弹性固体，从线动量平衡和非线性本构模型导出的控制平衡方程被简化为二阶、向量值、拟线性椭圆BVP。 该BVP通过一种稳健的数值方案求解，该方案结合了Picard型线性化与连续Galerkin有限元方法进行空间离散化。 给出了各种加载条件下的数值结果，包括均匀拉伸、非均匀斜率和抛物线加载，并考虑了两种不同的材料纤维取向。 结果表明，裂纹尖端应变的增长明显低于应力的增长。 然而，发现应变能密度集中在裂纹尖端，这与线弹性断裂力学原理一致。 所提出的框架为制定物理上有意义的严格BVP提供了稳健的基础，这对于在不同加载条件下研究各向异性、应变限制弹性固体中的基本过程（如裂纹扩展、损伤和成核）至关重要。 显示英文摘要

摘要： A finite element framework is presented for analyzing crack-tip phenomena in transversely isotropic, strain-limiting elastic materials. Mechanical response is characterized by an algebraically nonlinear constitutive model, relating stress to linearized strain. Non-physical strain singularities at the crack apex are mitigated, ensuring bounded strain magnitudes. This methodology significantly advances boundary value problem (BVP) formulation, especially for first-order approximate theories. For a transversely isotropic elastic solid with a crack, the governing equilibrium equation, derived from linear momentum balance and the nonlinear constitutive model, is reduced to a second-order, vector-valued, quasilinear elliptic BVP. This BVP is solved using a robust numerical scheme combining Picard-type linearization with a continuous Galerkin finite element method for spatial discretization. Numerical results are presented for various loading conditions, including uniform tension, non-uniform slope, and parabolic loading, with two distinct material fiber orientations. It is demonstrated that crack-tip strain growth is substantially lower than stress growth. Nevertheless, strain-energy density is found to be concentrated at the crack tip, consistent with linear elastic fracture mechanics principles. The proposed framework provides a robust basis for formulating physically meaningful, rigorous BVPs, critical for investigating fundamental processes like crack propagation, damage, and nucleation in anisotropic, strain-limiting elastic solids under diverse loading conditions.