Mixed stabilized finite element methods in nonlinear solid mechanics. Part III: Compressible and incompressible plasticity

Abstract

This paper presents the application of a stabilized mixed strain/displacement …nite element formulation for the solution of nonlinear solid mechanics problems involving compressible and incompressible plasticity. The variational multiscale stabilization introduced allows the use of equal order interpolations in a consistent way. Such formulation presents two advantages when compared to the standard, displacement based, irreducible formulation: (a) it provides enhanced rate of convergence for the strain (and stress) …eld and (b) it is able to deal with incompressible situations. The …rst advantage also applies to the comparison with the mixed pressure/displacement formulation. The paper investigates the e¤ect of the improved strain and stress …elds in problems involving strain softening and localization leading to failure, using low order …nite elements with continuous strain and displacement …elds (P 1P 1 triangles or tetrahedra and Q1Q1 quadrilaterals, hexahedra, and triangular prisms) in conjunction with an associative frictional Drucker-Prager plastic model. The performance of the strain/displacement formulation under compressible and nearly incompressible deformation patterns is assessed and compared to a previously proposed pressure/displacement formulation. 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.

Explicit mixed strain–displacement finite elements for compressible and quasi-incompressible elasticity and plasticity

“…Table 1 shows analytical values of the localization angle of locale θ loc for the MC and DP in plane stress and plane strain situations. Cervera et al [10] have verified these results numerically for J2 plasticity and the DP model using an implicit mixed strain/displacement formulation. …”

“…Let us assume that the body forces b can be described completely in V h , so that P ⊥ h (b) = 0 [10,17]. Using the orthogonal projection, the subscales ε and u in Eqs.…”

“…(17) may be split as ∇ · σ h = ∇p h + ∇ · s h , being s h the deviatoric stress tensor and p h the mean stress. In quasi-incompressible situations, it has been found to be effective to use only the mean stress gradient ∇p h in the approximation for the displacement subscale [10]:…”

“…The plastic strain is integrated in time using return mapping algorithms [46]. Alternatively, the stress can be expressed in secant form as [10]:…”

“…In these works, deviatoric strains are calculated by differentiation of the displacement field; hence, the rate of convergence for the angular distortions is the same as in the irreducible formulation. More recently, Cervera et al [12,13,14] introduced a mixed ε− u strain/ displacement FE formulation and have applied it to address problems of strain localization using compressible and incompressible plasticity models [2,10]. The objective of such formulation is to achieve a discrete scheme with enhanced stress accuracy.…”