Abstract Strong anisotropies and/or near-incompressibility properties introduce internal constraints in material deformation. Numerical simulations comprising such a constrained behaviour show an overstiff structural response, referred to as element locking. Implementations based on mixed variational methods can heal locking but available solutions in the state-of-the-art are still non-optimal for anisotropic materials. This paper addresses this issue, by proposing a novel decomposition of anisotropic deformation modes on the basis of kinematic and energy requirements. Theoretical results exploit the Walpole’s formalism. The proposed kinematic split allows to introduce a new class of variational principles, referred to as energetically decoupled, for nearly-constrained transversely isotropic materials in linear elasticity. Low-order mixed finite element models are thus derived for treating near-inextensibility and/or near-incompressibility. Two-dimensional benchmark tests reproducing pure-bending and Cook’s membrane problems are conducted. Numerical results show that the accuracy of energetically decoupled formulations is high and robust with respect to variations of material properties, while the accuracy of non-energetically decoupled formulations is more sensitive.

中文翻译：

---

近乎约束的横向各向同性线性弹性：混合有限元公式的能量一致各向异性变形模式

摘要 强各向异性和/或近乎不可压缩性在材料变形中引入了内部约束。包含这种受约束行为的数值模拟显示了过硬的结构响应，称为单元锁定。基于混合变分方法的实现可以修复锁定，但现有技术中的可用解决方案对于各向异性材料仍然不是最佳的。本文通过提出基于运动学和能量要求的各向异性变形模式的新分解来解决这个问题。理论结果利用了沃波尔的形式主义。提议的运动学分裂允许引入一类新的变分原理，称为能量解耦，用于线性弹性中几乎受约束的横向各向同性材料。因此导出低阶混合有限元模型用于处理近乎不可扩展性和/或近乎不可压缩性。进行了再现纯弯曲和库克膜问题的二维基准测试。数值结果表明，能量去耦公式的准确性对于材料特性的变化是高且稳健的，而非能量去耦公式的准确性更敏感。