A mixed displacement-pressure-stress stabilized finite element formulation for a finite strain damage model

In this work, we describe a finite element formulation for the approximation of solid mechanics problems using a damage model under finite strain conditions. The balance equations are written in a total Lagrangian framework, employing the deviatoric component of the second Piola–Kirchhoff stress tensor, the displacement, and the pressure as primary variables. Introducing the pressure as a variable enables the treatment of incompressible materials, while incorporating the stress improves the stress approximation, which is crucial when nonlinear material laws depending on stress (or strain) are considered. In particular, we adopt the damage model proposed by Comellas et al. (International Journal for Numerical Methods in Engineering, Vol. 105, pp. 781–800, 2016), which generalizes previous isotropic damage models from infinitesimal strains to finite ones. This damage model is combined with a hyperelastic formulation for the reversible component of the deformation. The three-field formulation we consider was first introduced and analyzed for the Stokes problem by Codina (SIAM Journal on Numerical Analysis, Vol. 47, pp. 699–718, 2009). The interest of interpolating stress as an independent variable was highlighted in the work of Cervera et al. (Computer Methods in Applied Mechanics and Engineering, Vol. 199, pp. 2559–2570, 2010), and has since been successfully applied to numerous problems involving both linear and nonlinear constitutive behavior under the small strain assumption. More recently, Codina et al. (International Journal for Numerical Methods in Engineering, Vol. 125, e7540, 2024), extended the three-field formulation to geometrically nonlinear problems. The purpose of the present work is to combine these approaches, addressing problems that involve both nonlinear constitutive laws and geometrical nonlinearity with a mixed, three-field approach.