Keywords: failure and fracture, finite elements, strong discontinuities.

The lack of smoothness of the solutions involved when modeling failure in solids leads to difficult and challenging problems as it regards both its theoretical modeling and its numerical resolution. This non-smoothness is reflected by the formation of narrow zones where the strain is localized (e.g. shear bands) and its limit case consisting of discontinuous displacements (e.g. cracks), the so-called strong discontinuities. It is precisely a multi-scale treatment of these solutions that allows its efficient inclusion in general continuum problem and their sharp and efficient numerical resolution through finite elements with embedded discontinuities. The multi-scale framework developed in this work consists of the standard continuum mechanical problem as the large-scale problem (possibly considering other physical effects like thermal or porous fluid flow couplings and/or based on classical structural theories of beams, plates and shells rather than the continuum), with the discontinuities introduced locally in the small scales with a localized model (e.g. cohesive law) capturing the localized dissipative effects that lead the system to failure. These ideas lead naturally to the enhancement of the finite elements crossed by the discontinuity (e.g. crack) by local fields that are eliminated at the element level through their static condensation, thus resulting in computationally very efficient numerical techniques that can be easily incorporated in an existing general finite element code.

We have recently proposed a general strategy for the development of these finite element enhancements, namely, rather that trying to develop local interpolations of discontinuous displacement fields at the element level, the strains are enhanced through the proper modes capturing the separation of the element by the discontinuity. This strategy automatically considers the kinematics of the underlying element (triangular or quadrilateral, basic displacement, mixed or assumed strain formulations), hence leading to enhanced elements avoiding the so-called stress locking by which spurious transfers of stresses lead to an over-stiff resolution of the kinematics of the discontinuity. Following these ideas we have already developed new plane quadrilateral finite elements capturing single discontinuities in the infinitesimal and the finite deformation ranges [1,2], as well as several discontinuities branches as needed in the modeling of crack branching in dynamic fracture [3,4]. We focus in this contribution on the development of new three-dimensional brick elements, constructing and analyzing the separation modes in this general three-dimensional setting. We discuss the details of the propagation and representation of the discontinuity surfaces as well as the numerical integration and other implementation issues that appear in this setting. Several representative numerical simulations are presented to illustrate the use and performance of the newly developed finite element methods.

1

C. Linder, F. Armero, "Finite Elements with Embedded Strong Discontinuities for the Modeling of Failure in Solids", Int. J. Num. Meth. Engr., 72, 1391-1433, 2007. doi:10.1002/nme.2042

2

F. Armero, C. Linder, "New Finite Elements with Embedded Strong Discontinuities in the Finite Deformation Range", Comp. Meth. Appl. Mech. Engr., 197, 3138-3170, 2008. doi:10.1016/j.cma.2008.02.021

3

F. Armero, C. Linder, "Numerical Simulation of Dynamic Fracture Using Finite Elements with Embedded Discontinuities", Int. J. Fracture, 160, 119-141, 2009. doi:10.1007/s10704-009-9413-9

4

C. Linder, F. Armero, "Finite Elements with Embedded Branching?", Finite Elements in Analysis and Design, 45, 280-293, 2009. doi:10.1016/j.finel.2008.10.012

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