Stabilized formulation for phase‐field fracture in nearly incompressible hyperelasticity
Stabilized formulation for phase‐field fracture in nearly incompressible hyperelasticity
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DOI：
10.1002/nme.7050
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发表时间：
2022-05
影响因子：
2.9
通讯作者：
Ida Ang;N. Bouklas;Bin Li
中科院分区：
文献类型：
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作者：
Ida Ang;N. Bouklas;Bin Li
本文给出了超弹性材料在不可压缩极限附近相场断裂的稳定公式。在这个极限下，传统的混合位移和压力公式必须满足解的稳定性的inf-sup条件。混合配方与损伤场的结合可以抑制裂纹的张开，因为体积变化会受到严重的惩罚，从而有效地产生压泡。为了克服这一瓶颈，我们利用了扰动拉格朗日公式的混合公式，该公式在未损伤的材料中强制不可压缩约束，并减少在损伤材料中的压力效应。基于欧拉-拉格朗日方程的残差与作用于权空间的微分算子相结合的网格相关稳定化技术，允许对弹性子问题的所有场变量进行线性内插。用三个有限变形的算例进行了验证：平面应力纯剪切试验、平面应力二维几何模型和三维缺口试件。在最后一个例子中，我们将混合公式与附加应变能分解相结合，以说明拉伸和压缩中的不同行为。结果表明，计算结果与裂纹尖端张开位移的解析解非常吻合，且在不可压缩极限下表现良好。
This work presents a stabilized formulation for phase‐field fracture of hyperelastic materials near the limit of incompressibility. At this limit, traditional mixed displacement and pressure formulations must satisfy the inf‐sup condition for solution stability. The mixed formulation coupled with the damage field can lead to an inhibition of crack opening as volumetric changes are severely penalized effectively creating a pressure‐bubble. To overcome this bottleneck, we utilize a mixed formulation with a perturbed Lagrangian formulation which enforces the incompressibility constraint in the undamaged material and reduces the pressure effect in the damaged material. A mesh‐dependent stabilization technique based on the residuals of the Euler–Lagrange equations multiplied with a differential operator acting on the weight space is used, allowing for linear interpolation of all field variables of the elastic subproblem. This formulation was validated with three examples at finite deformations: a plane‐stress pure‐shear test, a two‐dimensional geometry in plane‐stress, and a three‐dimensional notched sample. In the last example, we incorporate a hybrid formulation with an additive strain energy decomposition to account for different behaviors in tension and compression. The results show close agreement with analytical solutions for crack tip opening displacements and performs well at the limit of incompressibility.