A linearised consistent mixed displacement-pressure formulation for hyperelasticity

Abstract: We propose a novel mixed displacement-pressure formulation based on an energy functional that takes into account the relation between the pressure and the volumetric energy function. We demonstrate that the proposed two-field mixed displacement-pressure formulation is not only applicable for nearly and truly incompressible cases but also is consistent in the compressible regime. Furthermore, we prove with analytical derivation and numerical results that the proposed two-field formulation is a simplified and efficient alternative for the three-field displacement-pressure-Jacobian formulation for hyperelastic materials whose strain energy density functions are decomposed into deviatoric and volumetric parts.

• The paper introduces a unified two-field formulation that bridges compressible and incompressible hyperelastic simulations.
• It simplifies computations by removing the Jacobian field and consistently yielding symmetric matrix systems for enhanced efficiency.
• Numerical experiments validate its robustness across varying Poisson's ratios, highlighting its potential for biomedical and synthetic material applications.

A Mixed Displacement-Pressure Formulation for Hyperelasticity

The paper presents a novel approach in computational mechanics, specifically tackling the challenges associated with modeling hyperelastic materials, which are pivotal in simulating the behavior of rubber-like polymers and soft tissues. These materials exhibit a range of deformation behaviors, spanning from compressible to truly incompressible characteristics, necessitating sophisticated numerical methods for accurate simulation.

The focus of this work is a mixed displacement-pressure formulation based on an energy functional that accounts for the interplay between pressure and volumetric energy. This formulation extends its applicability across the spectrum of material compressibility: from compressible, nearly incompressible, to truly incompressible materials.

Key Contributions

1. Unified Framework: The proposed two-field mixed displacement-pressure formulation provides a unified framework applicable to both incompressible and compressible regimes, overcoming the limitations of existing displacement-pressure models that lose accuracy in the compressible domain.
2. Efficiency Over the Three-Field Formulation: This two-field approach is posited as an efficient alternative to the three-field displacement-pressure-Jacobian formulation, especially for hyperelastic materials where strain energy is decomposed into deviatoric and volumetric parts. The elimination of the Jacobian field variable simplifies computation without loss of accuracy.
3. Symmetric Matrix Systems: The formulation consistently results in symmetric matrix systems, irrespective of the specific volumetric energy function employed. This feature can lead to performance benefits, as symmetrical matrix systems are computationally less intensive to solve compared to their asymmetric counterparts.
4. Avoidance of Complementary Energy Functions: Unlike some advanced formulations requiring complementary energy functions for obtaining symmetric systems, this formulation avoids such complexities, easing implementation and broadening applicability across different types of hyperelastic models.

Numerical Validation

The paper rigorously demonstrates the formulation's accuracy and robustness through numerical experiments. Notably, the cylindrical bar example shows that the proposed formulation maintains fidelity across various Poisson's ratios, effectively bridging the compressible and nearly incompressible behaviors. Comparison against traditional methods using higher-order elements highlights its computational efficiency and accuracy.

Implications

This formulation has significant implications in the field of computational solid mechanics, where simulating nearly incompressible materials has often posed challenges. By providing a method that does not rely on the perturbation of the Lagrangian or complex dual potentials, it facilitates more straightforward modeling of materials in critical applications, such as in biomedical engineering and the design of advanced synthetic materials.

Speculation on Future Developments

This formulation might pave the way for further advancements in computational methods for multiphysics problems involving hyperelasticity, such as coupled thermo-mechanical and electro-mechanical simulations. The ability to handle complex loading and boundary conditions while maintaining computational efficiency could spur innovations in adaptive mesh refinement techniques and real-time simulations in structural analysis.

In summary, this paper offers a significant step toward versatile and computationally efficient finite element formulations for hyperelastic materials, accommodating a wide range of mechanical behaviors with potential applications across various engineering disciplines.