A stabilized mixed three-field formulation for stress accurate analysis including the incompressible limit in finite strain solid dynamics

In this work a new methodology for finite strain solid dynamics problems for stress accurate analysis including the incompressible limit is presented. In previous works, the authors have presented the stabilized mixed displacement/pressure formulation to deal with the incompressibility constraint in finite strain solid dynamics. To this end, the momentum equation is complemented with a constitutive law for the pressure which emerges from the deviatoric/volumetric decomposition of the strain energy function for any hyperelastic material model. The incompressible limit is attained automatically depending on the material bulk modulus. This work exploits the concept of mixed methods to formulate stable displacement/pressure/deviatoric stress finite elements. The final goal is to design a finite element technology able to tackle simultaneously problems which may involve incompressible behavior together with a high degree of accuracy of the stress field. The variational multi-scale stabilization technique and, in particular, the orthogonal subgrid scale method allows the use of equal-order interpolations. These stabilization procedures lead to discrete problems which are fully stable, free of volumetric locking, stress oscillations and pressure fluctuations. Numerical benchmarks show that the results obtained compare very favorably with those obtained with the corresponding stabilized mixed displacement/pressure formulation.

Automated summary

In this work, a new methodology is presented for accurately analyzing stress in finite strain solid dynamics problems, including incompressibility. The momentum equation is complemented with a constitutive law for pressure, which is derived from the deviatoric/volumetric decomposition of the strain energy function for any hyperelastic material model. The incompressible limit is automatically achieved depending on the material bulk modulus. This work utilizes mixed methods to formulate stable displacement/pressure/deviatoric stress finite elements, aiming to simultaneously handle problems involving incompressible behavior and a high degree of stress field accuracy. Numerical benchmarks show favorable results compared to the corresponding stabilized mixed displacement/pressure formulation.