A total-Lagrangian material point method (TLMPM) with a mixed formulation is proposed in this article for the mechanical analysis of incompressible soft materials coupling the mass growth and massive deformation. In this work, the growth-induced large deformation behavior is handled within the framework of the material point method (MPM) based on the multiplicative decomposition of deformation gradient. To overcome the volumetric locking caused by the incompressibility of soft materials, a weak-form equation for hydrostatic pressure is implemented into a mixed MPM framework. To deal with the massive deformation caused by growth, the total-Lagrangian formulation is further introduced into the mixed MPM. The TLMPM discretized via the B-spline basis functions is then developed in the proposed solution procedure. The efficiency and accuracy of the proposed method are demonstrated by several representative two- and three-dimensional numerical examples with large deformation such as the free growth of blocks and constrained growth of rings. The challenging problems including the volumetric growth of bilayers with complex geometry and the strain-driven growth of skin are further investigated using the proposed TLMPM to illustrate its ability for evaluating the growth phenomena and behaviors as observed in nature and engineering.

中文翻译：

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不可压缩软材料耦合生长与大变形的全拉格朗日材料点法

本文提出了一种具有混合公式的全拉格朗日材料点法 (TLMPM)，用于耦合质量增长和大变形的不可压缩软材料的力学分析。在这项工作中，生长引起的大变形行为是在基于变形梯度乘法分解的材料点法（MPM）的框架内处理的。为了克服软材料不可压缩性引起的体积锁定，在混合 MPM 框架中实现了静水压力的弱形式方程。为了处理由生长引起的大量变形，在混合 MPM 中进一步引入了全拉格朗日公式。然后在建议的求解过程中开发通过 B 样条基函数离散化的 TLMPM。通过几个具有大变形的代表性二维和三维数值例子，例如块的自由生长和环的约束生长，证明了所提出方法的效率和准确性。使用拟议的 TLMPM 进一步研究了具有复杂几何形状的双层的体积生长和皮肤的应变驱动生长等具有挑战性的问题，以说明其评估自然和工程中观察到的生长现象和行为的能力。