Strain localization and quasi-brittle failure in frictional-cohesive materials is still an open and challenging problem in computational mechanics. Owing to its complexity and the significant implications on numerous engineering problems, a considerable effort has been devoted to the development of theories and techniques capable of dealing with this topic. The introduction of numerical methods in the 70's provided a way to compute solutions, even if approximated. The Finite Element Method is able to describe a large number of geometries, engineering problems and various boundary conditions and, for this reason, the displacement-based formulation represents the preferred choice in the mechanical analysis of solids. Moreover, assuming the displacement jump created by a crack to be smeared across an element band, the calculation of the onset and the evolution of a fracture can be performed. However, standard finite elements are well-known to behave poorly in the case of strain localization of softening materials. Indeed, the irreducible formulation is mesh-biased and the resulting fracture direction is frequently incorrect. Plasticity constitutive models are largely affected by this issue, being directional by their very nature. In addition, when dealing with isochoric conditions, locking of the stresses provokes spurious pressure oscillations, that spoil the numerical solution. Both problems can be shown not to be related to the mathematical statement of the continuous problem but to its discrete (FEM) counterpart.

In this work, a novel mixed ε-u strain-displacement finite element method for strain localization and failure in plasticity is presented. Thanks to the independent interpolation of the strain and displacement fields, it is characterized by enhanced kinematic properties which result in an improvement in the accuracy of stresses and deformations. Moreover, it is proved that the numerical quandaries typical of the irreducible formulation are alleviated with the introduction of this FE technology. The ε-u FEM is applied to 2D and 3D problems aimed at benchmarking its numerical capabilities as well as proving high-fidelity predictions and simulations of experimental results.

Firstly, failure under Mode I (opening) loading is considered, using a Rankine failure criterion to describe the mechanical behavior of materials, such as concrete, which exhibit cracking under tensile load. Secondly, failure under Mode II (shearing) loading is studied, employing the J2 von Mises and the Drucker-Prager failure criteria for incompressible and compressible plasticity cases. Thirdly, failure under Mode III (tearing) and Mixed Mode loading is discussed. To study the complex stress state arising in torsional and skew-symmetrical bending cases, Rankine and Drucker-Prager failure criteria are developed in both plasticity and isotropic continuum damage models. Finally, the formulation is applied to crack propagation in weak snowpack layers, which is the main cause for the initiation of snow avalanches.

From the results, three main conclusions emerge: (i) the mixed ε-u finite element method proposed is capable of overcoming many of the challenges posed by strain localization in solids, providing reliable and accurate solutions; (ii) the smeared crack approach is able to describe effectively the creation and propagation of fracture surfaces in Mode I, Mode II, Mode III and Mixed Mode loading; (iii) the improvement of the kinematic description, with continuity of displacements and strains, is considered a key factor to empower the numerical solution.

The ε-u finite elements share numerous aspects with the standard displacement-based ones, in terms of implementation of constitutive laws, initial set of data and geometrical discretization. However, the proposed mixed formulation is superior in predicting peak loads, strain localization patterns and failure mechanisms, demonstrating its generality and its possibilities in the engineering practice.

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