A variationally-consistent hybrid equilibrium element formulation for linear poroelasticity

A poroelastic medium is defined as a continuous system in which the mechanical response arises from the interaction between a deformable elastic solid skeleton and a pressurised fluid, fully saturating the interconnected porous network. The coupled theory of Poromechanics is effectively employed to solve a broad class of problems spanning various fields, ranging from its original application in Geomechanics to addressing contemporary challenges in tissue Biomechanics. Besides the seminal theory introduced by Biot, several variational multi-field principles have been proposed, leading to various finite element formulations. Finite Element approaches are mostly based on a two-field discretisation involving the coupled solid displacement and interstitial fluid pressure as primary variables. However, this choice of the two-primary variables leads to a saddle-point problem, posing a challenge in achieving accurate solutions and thus making the use of stabilisation methods rather mandatory. This study proposes a novel variationally consistent Hybrid Equilibrium Element formulation, based on the complementary strain energy function. A two-field minimisation principle is formulated, with total Cauchy stress and interstitial fluid pressure serving as primary variables. The proposed variational principle provides a variationally consistent framework for designing inherently self-equilibrated Hybrid Equilibrium Elements, where the solid displacement field emerges as a Lagrangian variable to enforce a co-diffusivity constraint at element sides. The fulfilment of the inf-sup condition, typically challenging in developing stable and computationally efficient finite element formulations, is no longer required, a minimisation principle having been adopted, rather than the classical saddle-point principle. The proposed Hybrid Equilibrium Element approach offers a statically admissible solution even in the proximity of singularities or high space gradient stress distribution, providing a more comprehensive assessment of stress distributions and concentrations. The convexity of the new variational principle has been established, ensuring solution uniqueness. The formulation is based on a 2-D triangular Hybrid Equilibrium Element employed to solve plane strain problems. The element has been implemented in the Open source Finite Element Analysis Program (FEAP) and its performance evaluated against classical benchmark problems for poroelasticity, with the emphasis on stress accuracy. The numerical results obtained for the quasi-static analysis of poroelastic systems show the advantages of the proposed approach, in which accurate stress distributions are displayed, even adopting quite coarse meshes. Remarkably, in the proposed approach no stabilisation technique is required.