On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime

We consider a two-dimensional atomic mass spring system and show that in the small displacement regime the corresponding discrete energies can be related to a continuum Griffith energy functional in the sense of Gamma-convergence. We also analyze the continuum problem for a rectangular bar under tensile boundary conditions and find that depending on the boundary loading the minimizers are either homogeneous elastic deformations or configurations that are completely cracked generically along a crystallographic line. As applications we discuss cleavage properties of strained crystals and an effective continuum fracture energy for magnets

Phase-field modelling of fracture in single crystal plasticity

We propose a phase-field model for ductile fracture in a single crystal within the kinematically linear regime, by combining the theory of single crystal plasticity as formulated in Gurtin et al. (2010) and the phase-field formulation for ductile fracture proposed by Ambati et al. (2015) . The model introduces coupling between plasticity and fracture through the dependency of the so-called degradation function from a scalar global measure of the accumulated plastic strain on all slip systems. A viscous regularization is introduced both in the treatment of plasticity and in the phase-field evolution equation. Testing of the model on two examples for face centred cubic single crystals indicates that fracture is predicted to initiate and develop in the regions of the maximum accumulated plastic strain, which is in agreement with phenomenological observations. A rotation of the crystallographic unit cell is shown to affect the test results in terms of failure pattern and corresponding global and local response

A recursive-faulting model of distributed damage in confined brittle materials

We develop a model of distributed damage in brittle materials deforming in triaxial compression based on the explicit construction of special microstructures obtained by recursive faulting. The model aims to predict the effective or macroscopic behavior of the material from its elastic and fracture properties; and to predict the microstructures underlying the microscopic behavior. The model accounts for the elasticity of the matrix, fault nucleation and the cohesive and frictional behavior of the faults. We analyze the resulting quasistatic boundary value problem and determine the relaxation of the potential energy, which describes the macroscopic material behavior averaged over all possible fine-scale structures. Finally, we present numerical calculations of the dynamic multi-axial compression experiments on sintered aluminum nitride of Chen and Ravichandran [1994. Dynamic compressive behavior of ceramics under lateral confinement. J. Phys. IV 4, 177–182; 1996a. Static and dynamic compressive behavior of aluminum nitride under moderate confinement. J. Am. Soc. Ceramics 79(3), 579–584; 1996b. An experimental technique for imposing dynamic multiaxial compression with mechanical confinement. Exp. Mech. 36(2), 155–158; 2000. Failure mode transition in ceramics under dynamic multiaxial compression. Int. J. Fracture 101, 141–159]. The model correctly predicts the general trends regarding the observed damage patterns; and the brittle-to-ductile transition resulting under increasing confinement

3D numerical modelling of twisting cracks under bending and torsion of skew notched beams

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 e-ue-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.Peer ReviewedPostprint (author's final draft

A Griffith-Euler-Bernoulli theory for thin brittle beams derived from nonlinear models in variational fracture mechanics

We study a planar thin brittle beam subject to elastic deformations and cracks described in terms of a nonlinear Griffith energy functional acting on $SBV$ deformations of the beam. In particular we consider the case in which elastic bulk contributions due to finite bending of the beam are comparable to the surface energy which is necessary to completely break the beam into several large pieces. In the limit of vanishing aspect ratio we rigorously derive an effective Griffith-Euler-Bernoulli functional which acts on piecewise $W^{2,2}$ regular curves representing the midline of the beam. The elastic part of this functional is the classical Euler-Bernoulli functional for thin beams in the bending dominated regime in terms of the curve's curvature. In addition there also emerges a fracture term proportional to the number of discontinuities of the curve and its first derivative