Abstract

The paper addresses the problem of tensile and mixed‐mode cracking within the so‐called smeared crack approach. Because lack of point‐wise convergence on stresses is deemed as the main difficulty to be overcome in the discrete problem, a (stabilized) mixed formulation with continuous linear strain and displacement interpolations is used. The necessary convergence rate can be proved for such a formulation, at least in the linear problem. Two standard local isotropic Rankine damage models with strain‐softening, differing in the definition of the damage criteria, are used as discrete constitutive model. Numerical examples demonstrate the application of the proposed formulation using linear triangular P1P1 and bilinear quadrilateral Q1Q1 mixed elements. The results obtained do not suffer from spurious mesh‐bias dependence without the use of auxiliary tracking techniques.

Abstract

This paper presents an explicit mixed finite element formulation to address compressible and quasi-incompressible problems in elasticity and plasticity. This implies that the numerical solution only involves diagonal systems of equations. The formulation uses independent and equal interpolation of displacements and strains, stabilized by variational subscales. A displacement sub-scale is introduced in order to stabilize the mean-stress field. Compared to the standard irreducible formulation, the proposed mixed formulation yields improved strain and stress fields. The paper investigates the effect of this enhancement on the accuracy in problems involving strain softening and localization leading to failure, using low order finite elements with linear continuous strain and displacement fields (P1P1 triangles in 2D and tetrahedra in 3D) in conjunction with associative frictional Mohr–Coulomb and Drucker–Prager plastic models. The performance of the strain/displacement formulation under compressible and nearly incompressible deformation patterns is assessed and compared to analytical solutions for plane stress and plane strain situations. Benchmark numerical examples show the capacity of the mixed formulation to predict correctly failure mechanisms with localized patterns of strain, virtually free from any dependence of the mesh directional bias. No auxiliary crack tracking technique is necessary.