Strain localization due to crack–microcrack interactions: X-FEM for a multifield approach

A survey of the extended finite element

“…Patzák and Jirásek [128] introduced macrocracks in the zones of highly localized damage developing in the simulation with smeared damage and plasticity models. Mariano and Stazi [129] analyzed the interaction between a macrocrack and a population of microcracks by adapting the extended finite element method to a multifield model of microcracked bodies.…”

Mixed Variational Formulations for Micro-cracked Continua in the Multifield Framework

“…The second example deals with the interactions of a macrocrack with the considered microcracked continuum. The problem was firstly tackled in [10], according to a variation of the well-known displacementbased X-FEM procedure [11]. The same example is also used to show some results on the convergence features of the proposed mixed schemes.…”

“…The constitutive relationships described in Section 1.2. allow to link their gradients to the relevant macro and micro stress fields. The numerical investigations discussed in [9] and [10] refer to displacement-based finite element methods that adopt post-processing techniques to derive the stress fields. The present paper proposes instead ad hoc mixed formulations that interpolate independently stresses and displacements.…”

Geometry of interactions in complex bodies

“…Moreover, by comparing the additive decomposition of F tot with the multiplicative one, namely F (m) F, we realize that F (m) = I+∇d (X) F −1 = I + gradd a . The direct (perhaps naive) description of the kinematics of microcracked bodies just sketched here follows [21], [26]; however, it can be obtained by using the procedure involving the limit of bodies described in [12], [13].…”

“…Sometimes the material substructure is a perfectly identifiable Lagrangian system as in the case of nematic liquid crystals [14], [20], [11] (in which the nematic molecules can be separated from the melt), sometimes it does not as in granular gases [6] and microcracked bodies [26]. In granular gases, e.g., a material element collects a family of sparse granules with peculiar velocities, so that the order parameter may be an element of a suitable Grassmanian of the tangent bundle of some finite-dimensional manifold, while for microcracked bodies each microcrack can be considered either as a sharp planar defect not interpenetrated by interatomic bonds or as an elliptic void, so it does not exist per se and has a volatile substance determined just by the surrounding matter.…”