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Abstract 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.
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Locking treatment of penalty-based gradient-enhanced damage formulation for failure of compressible and nearly incompressible hyperelastic materials
Abstract Soft materials are of major interest for biomechanics applications due to their high deformability and susceptibility to experience damage events under different loading scenarios. The present study is concerned with modelling damage evolution processes in these nonlinear materials whose structural responses are prone to locking when low-order kinematic interpolation is employed in the context of nonlinear Finite Element schemes. For this reason, a pair of gradient-enhanced continuum damage schemes are proposed with the aim of tackling mechanical failure problems in applications that exhibit shear and volumetric locking. In particular, we present the consistent formulation and the assessment of the corresponding performance of (i) a mixed displacement-enhanced assumed strain employing a total Lagrangian formulation, and (ii) a three-field mixed displacement-pressure-Jacobian formulation. The novel and formulations are consistently derived and numerically implemented, providing a satisfactory agreement with respect to built-in elements handling the treatment of shear and volumetric locking, respectively, in conjunction to the modelling damage phenomena via the use of a penalty-based gradient-enhanced formulation. This performance is examined via several numerical applications. Furthermore, the final example justifies the need for a formulation combining both mixed FE approaches to simulate problems encompassing both locking issues (shear and volumetric locking), which can be performed using a combination of the and herein proposed.
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- Award ID(s):
- 2119716
- PAR ID:
- 10403306
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Computational Mechanics
- Volume:
- 72
- Issue:
- 4
- ISSN:
- 0178-7675
- Format(s):
- Medium: X Size: p. 635-662
- Size(s):
- p. 635-662
- Sponsoring Org:
- National Science Foundation
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