J. Mosler

On the modeling of highly localized deformations induced by material failure: The strong discontinuity approach

SummaryNumerical analyses of large engineering structures undergoing highly localized deformations induced by material failure such as cracking in concrete or shear bands in soils still represent a challenge to the scientific community. In this paper, an efficient concept suitable for the analysis of those problems is presented. More precisely, an overview of the Strong Discontinuity Approach (SDA) is given. This specific approach is characterized by the incorporation of strong discontinuities, i.e. discontinuous displacement fields, into standard displacement-based finite elements by means of the Enhanced Assumed Strain (EAS) concept. The fundamentals of the SDA are illustrated and compared to those of other models based on discontinuous deformation mappings. The main part of this contribution deals with the numerical implementation of the SDA. Besides the original finite element formulation of the SDA, two more recently proposed algorithmic frameworks which avoid the use of the static condensation technique are presented. Both models result in a set of equations formally identical to that known from classical plasticity theory and, consequently, it can be solved by applying the return-mapping algorithm. Several recently suggested extensions of the SDA such as rotating surfaces of discontinuous displacements and intersecting discontinuities are discussed and investigated by means of finite element analyses. The applicability of the SDA as well as its numerical performance is illustrated by means of fully three-dimensional ultimate load analyses.

Efficient computation of the elastography inverse problem by combining variational mesh adaption and a clustering technique

This paper is concerned with an efficient implementation suitable for the elastography inverse problem. More precisely, the novel algorithm allows us to compute the unknown stiffness distribution in soft tissue by means of the measured displacement field by considerably reducing the numerical cost compared to previous approaches. This is realized by combining and further elaborating variational mesh adaption with a clustering technique similar to those known from digital image compression. Within the variational mesh adaption, the underlying finite element discretization is only locally refined if this leads to a considerable improvement of the numerical solution. Additionally, the numerical complexity is reduced by the aforementioned clustering technique, in which the parameters describing the stiffness of the respective soft tissue are sorted according to a predefined number of intervals. By doing so, the number of unknowns associated with the elastography inverse problem can be chosen explicitly. A positive side effect of this method is the reduction of artificial noise in the data (smoothing of the solution). The performance and the rate of convergence of the resulting numerical formulation are critically analyzed by numerical examples.

Aspects of interface elasticity theory

Interfaces significantly influence the overall material response especially when the area-to-volume ratio is large, for instance in nanocrystalline solids. A well-established and frequently applied framework suitable for modeling interfaces dates back to the pioneering work by Gurtin and Murdoch on surface elasticity theory and its generalization to interface elasticity theory. In this contribution, interface elasticity theory is revisited and different aspects of this theory are carefully examined. Two alternative formulations based on stress vectors and stress tensors are given to unify various existing approaches in this context. Focus is on the hyper-elastic mechanical behavior of such interfaces. Interface elasticity theory at finite deformation is critically reanalyzed and several subtle conclusions are highlighted. Finally, a consistent linearized interface elasticity theory is established. We propose an energetically consistent interface linear elasticity theory together with its appropriate stress measures.

A Variationally Consistent Approach for Crack Propagation Based on Configurational Forces

This paper is concerned with a variationally consistent approach suitable for the analysis of cracking in brittle materials. In line with the pioneering works by Griffith, it is assumed that a crack propagates, if this is energetically favorable. However, in order to bypass the well-known defects of Griffiths original idea such as the requirement of a pre-existing crack, a modified energy-based criterion is proposed. In contrast to Griffith and similar to Francfort and Marigo, the novel cracking model is based on a finite crack extension. More precisely, new crack surfaces form, if this leads to a reduction in energy within a finite (but not global) neighborhood. The features of the advocated model are critically analyzed.

An efficient algorithm for the inverse problem in elasticity imaging by means of variational r-adaption

A novel finite element formulation suitable for computing efficiently the stiffness distribution in soft biological tissue is presented in this paper. For that purpose, the inverse problem of finite strain hyperelasticity is considered and solved iteratively. In line with Arnold et al (2010 Phys. Med. Biol. 55 2035), the computing time is effectively reduced by using adaptive finite element methods. In sharp contrast to previous approaches, the novel mesh adaption relies on an r-adaption (re-allocation of the nodes within the finite element triangulation). This method allows the detection of material interfaces between healthy and diseased tissue in a very effective manner. The evolution of the nodal positions is canonically driven by the same minimization principle characterizing the inverse problem of hyperelasticity. Consequently, the proposed mesh adaption is variationally consistent. Furthermore, it guarantees that the quality of the numerical solution is improved. Since the proposed r-adaption requires only a relatively coarse triangulation for detecting material interfaces, the underlying finite element spaces are usually not rich enough for predicting the deformation field sufficiently accurately (the forward problem). For this reason, the novel variational r-refinement is combined with the variational h-adaption (Arnold et al 2010) to obtain a variational hr-refinement algorithm. The resulting approach captures material interfaces well (by using r-adaption) and predicts a deformation field in good agreement with that observed experimentally (by using h-adaption).

On the Implementation of Variational Constitutive Updates at Finite Strains

In this paper an efficient, variationally consistent, algorithmic formulation for rate-independent dissipative solids at finite strain is presented. Focusing on finite strain plasticity theory and adopting the formalism of standard dissipative solids, the considered class of constitutive models can be defined by means of only two potentials being the Helmholtz energy and the yield function (or equivalently, a dissipation functional). More importantly, by assuming associative evolution equations, these potentials allow to recast finite strain plasticity into an equivalent, variationally consistent minimization problem, cf. [1’4]. Based on this physically sound theoretical approach, a novel numerical implementation is discussed. Analogously to the theoretical part, it is variationally consistent, i.e., all unknown variables follow naturally from minimizing the energy of the respective system. Extending previously published works on such methods, the advocated numerical scheme does not rely on any material symmetry regarding the elastic and the plastic response and covers isotropic, kinematic and combined hardening, cf. [5, 6].

Prediction of macroscopic material failure based on microscopic cohesive laws

A three-dimensional finite element formulation is applied to the process of determination of macroscopic material properties based on constitutive relationships characterising a microscale. More specifically, a macroscopic failure criterion is computed numerically. The adopted finite element model captures the localised fully nonlinear kinematics associated with the failure on the microscale by means of the Strong Discontinuity Approach (SDA). In contrast to classical continuum mechanics, the deformation gradient is additively decomposed into a conforming part corresponding to a smooth deformation mapping and an enhanced part reflecting the final failure kinematics of the microscale. The implementation of the Enhanced-Assumed-Strain (EAS) concept leads to the elimination of the additional degrees of freedom (displacement jump) on the material point level. More precisely, the applied numerical implementation is similar to that of standard (finite) plasticity. The model does not require any assumption regarding neither the type of the finite elements, nor the constitutive behaviour. Any traction-separation law, connecting the displacement jump to the traction vector, can be chosen. Based on the proposed finite element formulation, microscopic material properties (traction-separation laws) are then used for the computation of the macroscopic material failure. The applicability of the presented numerical model is demonstrated by means of rather academic examples.