In this paper, we tackle a persistent numerical instability within the total Lagrangian smoothed particle hydrodynamics (TLSPH) solid dynamics. Specifically, we address the hourglass modes that may grow and eventually deteriorate the reliability of simulation, particularly in the scenarios characterized by large deformations. We propose a generalized essentially non-hourglass formulation based on volumetric-deviatoric stress decomposition, offering a general solution for elasticity, plasticity, anisotropy, and other material models. Comparing the standard SPH formulation with the original non-nested Laplacian operator applied in our previous work \cite{wu2023essentially} to handle the hourglass issues in standard elasticity, we introduce a correction for the discretization of shear stress that relies on the discrepancy produced by a tracing-back prediction of the initial inter-particle direction from the current deformation gradient. The present formulation, when applied to standard elastic materials, is able to recover the original Laplacian operator. Due to the dimensionless nature of the correction, this formulation handles complex material models in a very straightforward way. Furthermore, a magnitude limiter is introduced to minimize the correction in domains where the discrepancy is less pronounced. The present formulation is validated, with a single set of modeling parameters, through a series of benchmark cases, confirming good stability and accuracy across elastic, plastic, and anisotropic materials. To showcase its potential, the formulation is employed to simulate a complex problem involving viscous plastic Oobleck material, contacts, and very large deformation.

翻译：本文致力于解决总拉格朗日光滑粒子流体动力学（TLSPH）固体动力学中持续存在的数值不稳定性问题。具体而言，我们针对可能增长并最终降低模拟可靠性的沙漏模式展开研究，尤其是在大变形场景中。基于体积-偏量应力分解，我们提出了一种广义本质上无沙漏格式，为弹性、塑性、各向异性及其他材料模型提供了通用解决方案。通过将标准SPH格式与我们先前工作中用于处理标准弹性沙漏问题的原始非嵌套拉普拉斯算子进行对比，我们引入了一种基于初始粒子间方向从当前变形梯度回溯预测所得差异的剪切应力离散修正方法。该格式应用于标准弹性材料时能够恢复原始拉普拉斯算子。由于修正项的无量纲特性，该格式能以极为直接的方式处理复杂材料模型。此外，我们引入幅值限制器以在差异不显著的域中最小化修正量。通过一系列基准算例（采用单一建模参数集）验证了该格式在弹性、塑性及各向异性材料中均具有良好的稳定性和精度。为展示其潜力，该格式被用于模拟涉及粘塑性Oobleck材料、接触及极大变形的复杂问题。