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.

The development of slip lines, due to strain localization, is a common cause for failure of soil in many circumstances investigated in geotechnical engineering. Through the use of numerical methods -like finite elements-many practitioners are able to take into account complex geometrical and physical conditions in their analyses. However, when dealing with shear bands, standard finite elements display lack of precision, mesh dependency and locking. This paper introduces a (stabilized) mixed finite element formulation with continuous linear strain and displacement interpolations. Von Mises and Drucker-Prager local plasticity models with strain softening are considered as constitutive law. This innovative formulation succeeds in overcoming the limitations of the standard formulation and provides accurate results within the vicinity of the shear bands, specifically without suffering from mesh dependency. Finally, 2D and 3D numerical examples demonstrate the accuracy and robustness in the computation of localization bands, without the introduction of additional tracking techniques as usually required by other methods.

This paper presents the 2D and 3D numerical analysis of pullout tests on steel anchorages in concrete blocks using standard and mixed finite elements. A novel (stabilized) mixed formulation in the variables of total strain ε and displacements u is introduced to overcome the intrinsic deficiencies of the standard displacement-based one in the context of localization of strains, such as mesh dependency. The quasi-brittle behaviour of concrete is described through an elastoplastic constitutive law with a local Rankine yielding criterion. The proposed formulation is shown to be a reliable and accurate tool, sensitive to the physical parameters of the pullout tests, but objective with respect to the adopted FE mesh. Furthermore, the mixed ε/u finite element is able to capture the correct failure mechanism with relatively coarse discretizations. At the same time, the spurious behavior of the standard formulation is not alleviated by mesh-refinement.

The testing of mode III and mixed mode failure is every so often encountered in the dedicated literature of mechanical characterization of brittle and quasi-brittle materials. In this work, the application of the mixed strain displacement ε − u finite element formulation to three examples involving skew notched beams is presented. The use of this FE technology is effective in problems involving localization of strains in softening materials.The objectives of the paper are: (i) to test the mixed formulation in mode III and mixed mode failure and (ii) to present an enhancement in terms of computational time given by the kinematic compatibility between irreducible displacement-based and the mixed strain-displacement elements.Three tests of skew-notched beams are presented: firstly, a three point bending test of a PolyMethyl MethaAcrylate beam; secondly, a torsion test of a plain concrete prismatic beam with square base; finally, a torsion test of a cylindrical beam made of plain concrete as well. To describe the mechanical behavior of the material in the inelastic range, Rankine and Drucker-Prager failure criteria are used in both plasticity and isotropic continuum damage formats.The proposed mixed formulation is capable of yielding results close to the experimental ones in terms of fracture surface, peak load and global loss of carrying capability. In addition, the symmetric secant formulation and the compatibility condition between the standard irreducible method and the strain-displacement one is exploited, resulting in a significant speedup of the computational procedure.

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