Abstract This paper describes a mixed extended finite element (XFEM) formulation that can handle nearly incompressible material behaviour for fracture mechanics problems as well as for heterogeneous materials under small and finite elastic and elastoplastic deformations without showing locking behaviour. Typical applications are the simulation of cracks and their propagation in rubber materials, microstructure simulations of rubber and simulations of heterogeneities in J 2 and crystal plasticity. The developed finite element technique is an extension of the well established mixed Q1P0 formulation. Herein, enrichments as they are commonly used within the XFEM are considered to capture discontinuities within the displacement and/or strain field. The volumetric stress is discontinuous across a crack face or an interface between two materials. For that reason the piecewise constant ansatz used in the Q1P0 formulation for the volumetric stress needs to be enriched with suitable enrichment functions as well. Therefore, the developed mixed extended finite element formulation is named XQ1XP0. With several examples it is shown that the new formulation does not show locking effects even for almost incompressible materials and for significant plastic deformations and that it shows superior convergence behaviour compared to a standard formulation.

中文翻译：

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用于模拟几乎不可压缩材料和金属塑性中的裂纹和异质性的混合扩展有限元

摘要 本文描述了一种混合扩展有限元 (XFEM) 公式，该公式可以处理断裂力学问题以及在小而有限的弹性和弹塑性变形下的异质材料几乎不可压缩的材料行为，而不会显示锁定行为。典型应用是橡胶材料中裂纹及其扩展的模拟、橡胶的微观结构模拟以及J 2 和晶体塑性的不均匀性模拟。开发的有限元技术是完善的混合 Q1P0 公式的扩展。在此，XFEM 中通常使用的富集被认为是捕获位移和/或应变场内的不连续性。体积应力在裂纹面或两种材料之间的界面上是不连续的。因此，Q1P0 公式中用于体积应力的分段常数 ansatz 也需要用合适的富集函数来富集。因此，开发的混合扩展有限元公式命名为 XQ1XP0。几个例子表明，即使对于几乎不可压缩的材料和显着的塑性变形，新公式也没有显示出锁定效应，并且与标准公式相比，它显示出更好的收敛行为。