Formulation and Application of the Initially Rigid Cohesive Finite Element Method

The presented procedure for cohesive crack propagation is based on an adaptive finite element (FE) implementation, which enables the introduction of cohesive surfaces in dependence on the current crack state. In contrast to already existing formulations, the focus of the present model lies on failure processes that can be described at quasi-static conditions within an implicit framework. Furthermore, an extension for mesh independent crack propagation in terms of an additional mesh adaptive formulation is presented. By the evaluation of the failure criterion considering the preferred crack direction, a new crack tip coordinate is computed and the discretization is accordingly adjusted. The remaining mesh is modified for the new boundary representation. The application of the proposed method is shown by the numerical investigation of a concrete fracture specimen from an experimental research project. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite Element Modelling of Cracks Based on the Partition of Unity Method

In this paper a finite element model for the analysis of brittle materials in the post cracking regime is presented. The model allows the representation of failure zones several times smaller than the structure itself using relatively coarse finite element meshes. The formulation is based on the partition of unity method. Discontinuous shape functions are used to enrich the continuous approximation of the displacement field where a crack has opened [2]. The magnitude of the displacement jump is determined by extra degrees of freedom at existing nodes. The crack path is completely independent of the structure of the mesh and is continuous across element boundaries. To model inelastic deformations around the crack tip a cohesive crack model is used. A representative numerical example illustrates the performance of the proposed model.     

A generalized cohesive element technique for arbitrary crack motion

A computational method for arbitrary crack motion through a finite element mesh, termed as the generalized cohesive element technique, is presented. In this method, an element with an internal discontinuity is replaced by two superimposed elements with a combination of original and imaginary nodes. Conventional cohesive zone modeling, limited to crack propagation along the edges of the elements, is extended to incorporate the intra-element mixed-mode crack propagation. Proposed numerical technique has been shown to be quite accurate, robust and mesh insensitive provided the cohesive zone ahead of the crack tip is resolved adequately. A series of numerical examples is presented to demonstrate the validity and applicability of the proposed method.     

Analysis of concrete fracture using a novel cohesive crack method

Numerical analysis of fracture in concrete is studied with a simplified discrete crack method. The discrete crack method is a meshless method in which the crack is modeled by discrete cohesive crack segments passing through the nodes. The cohesive crack segments govern the non-linear response of concrete in tension softening and introduce anisotropy in the material model. The advantage of the presented discrete crack method over other discrete crack method is its simplicity and applicability to many cracks. In contrast to most other discrete crack methods, no representation of the crack surface is needed. On the other hand, the accuracy of discrete crack methods is maintained. This is demonstrated through several examples.     

A CQM-based BEM for transient heat conduction problems in homogeneous materials and FGMs

A method for numerical solution of time-domain boundary integral formulations of transient problems governed by the heat equation is presented. The heat conduction problem is analyzed considering homogeneous and non-homogeneous media. In the case of the non-homogeneous media, the conductor material is assumed to be a functionally graded material, i.e., the material properties vary spatially according to known smooth functions. For some specific spatial variations of the material properties, the fundamental solution and the boundary integral equation of the problem are obtained thanks to a change of variables that transforms the original problem to the standard heat conduction problem for homogeneous materials. For the treatment of time-dependent terms, the convolution quadrature method is adopted to approximate numerically the integral equation of the time-domain boundary element method. In the case that the responses are required at a large number of interior points, the convolution performed to calculate them is very time consuming. It is shown that the discrete convolution of the proposed formulation can be computed by means of the fast Fourier transform technique, which considerably reduces the computational complexity. Results for some transient heat conduction examples are presented to validate the numerical techniques studied.     

A Finite Element Approach for the Simulation of Quasi-Brittle Fracture

In the context of a strong discontinuity approach, we propose a finite element formulation with an embedded displacement discontinuity. The basic assumption of the proposed approach is the additive split of the total displacement field in a continuous and a discontinuous part. An arbitrary crack splits the linear triangular finite element into two parts, namely a triangular and a quadrilateral part. The discontinuous part of the displacement field in the quadrilateral portion is approximated using linear shape functions. For these purposes, the quadrilateral portion is divided into two triangular parts which is in this way similar to the approach proposed in [5]. In contrast, the discretisation is different compared to formulations proposed in [1] and [3], where the discontinuous part of the displacement field is approximated using bilinear shape functions. The basic theory of the underlying finite element formulation and a cohesive interface model to simulate brittle fracture are presented. By means of representative numerical examples differences and similarities of the present formulation and the formulations proposed in [1] and [3] are highlighted. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of material failure by combining the advantages of embedded strong discontinuity approaches and classical interface elements

A finite element formulation within the framework of the Strong Discontinuity Approach suitable for the simulation of crack growth is presented. The formulation allows for intersecting discontinuities and similarly to classical interface elements, the cracks are introduced parallel to the element facets. However and in contrast to interface elements, the discontinuities are directly embedded in finite elements, based on the Enhanced Assumed Strain concept. It is shown that a realistic prediction of the mechanical response requires the consideration of more than one crack within each finite element. The proposed formulation is suitable to overcome locking effects and it automatically fulfills crack path continuity. The approach is strictly local yielding an efficient numerical formulation. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of microscopic failure in heterogeneous materials

The proper modeling of state-of-the-art engineering materials requires a profound understanding of the nonlinear macroscopic material behavior. Especially for heterogeneous materials the effective macroscopic response is amongst others driven by damage effects and the inelastic material behavior of the individual constituents [1]. Since the macroscopic length scale of such materials is significantly larger than the fine-scale structure, a direct modeling of the local structure in a component model is not convenient. Multiscale techniques can be used to predict the effective material behavior. To this end, the authors developed a modeling technique based on representative volume elements (RVE) to predict the effective material behavior on different length scales. The extended finite element method (XFEM) is used to model discontinuities within the material structure independent of the underlying FE mesh. A dual enrichment strategy allows for the combined modeling of kinks (material interfaces) and jumps (cracks) within the displacement field [2]. The gradual degradation of the interface is thereby controlled by a cohesive zone model. In addition to interface failure, a non-local strain driven continuum damage model has been formulated to efficiently detect localization zones within the material phases. An integral formulation introduces a characteristic length scale and assures the convergence of the approach upon mesh refinement [3]. The proposed method allows for an efficient modeling of substantial failure mechanisms within a heterogeneous structure without the need of remeshing or element substitution. Due to the generality of the approach it can be used on different length scales. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

半无穷大裂纹端部粘聚力分析

准脆性材料裂纹端部断裂过程区粘聚力是导致非线性断裂特性的重要原因,根据准脆性材料的断裂特性,对存在粘聚力分布的半无穷大裂纹力学分析模型,由变形叠加原理得到以该粘聚应力分布为未知函数的积分方程,通过对积分方程的分析推证,得到了该分布函数解的数学结构和级数型表达式;提出了由实际裂纹张开位移,确定裂纹端部粘聚力分布函数的两种方法:其一由连续的裂纹张开位移通过积分变换求解未知函数级数展开项的系数,其二是由离散的裂纹张开位移数据通过最小二乘法确定该函数;推导出了相应方法求解未知量的代数方程,并且给出了适当的算例和讨论。     

Rupture in soft biological tissues modeled by a phase-field method

We present an application of the phase-field method of fracture to the simulation of artery rupture at large strains. To achieve this, the crack driving force function associated with the evolution of the crack phase-field is modified to account for the inherent anisotropy of the soft biological tissues. The phase-field methods present a promising and innovative approach to the thermodynamically consistent modeling of fracture. A key advantage lies in the prediction of the complex crack topologies where the cohesive zone approaches to fracture are known to suffer. A regularized crack surface functional is introduced that Γ-converges to a sharp crack topology for vanishing length scale parameter. The evaluation of the phase-field follows the minimization of this crack surface functional. The phase-field variable can be treated as a geometric quantity whose evolution is coupled to the anisotropic bulk response in a modular format in terms of a crack driving state function. A stress-based anisotropic failure criterion is introduced whose maximum value from the deformation history drives the irreversible crack phase-field. The formulation is verified by the finite element based simulation of a real arterial cross-section undergoing rupture in a two-dimensional setting when subjected to inflation pressure. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A combined nodal continuous-discontinuous finite element formulation for the Maxwell problem

Continuous Galerkin formulations are appealing due to their low computational cost, whereas discontinuous Galerkin formulation facilitate adaptative mesh refinement and are more accurate in regions with jumps of physical parameters. Since many electromagnetic problems involve materials with different physical properties, this last point is very important. For this reason, in this article we have developed a combined cG-dG formulation for Maxwell’s problem that allows arbitrary finite element spaces with functions continuous in patches of finite elements and discontinuous on the interfaces of these patches. In particular, the second formulation we propose comes from a novel continuous Galerkin formulation that reduces the amount of stabilization introduced in the numerical system. In all cases, we have performed stability and convergence analyses of the methods. The outcome of this work is a new approach that keeps the low CPU cost of recent nodal continuous formulations with the ability to deal with coefficient jumps and adaptivity of discontinuous ones. All these methods have been tested using a problem with singular solution and another one with different materials, in order to prove that in fact the resulting formulations can properly deal with these problems.     

Infinite elements in a poroelastodynamic FEM

Wave propagation phenomena in unbounded domains occur in many engineering applications, e.g., soil structure interactions. When simulating unbounded domains, infinite elements are a possible choice to describe the far field behavior, whereas the near field is described through conventional finite elements. Finite element formulations for porous materials in terms of Biot's theory [1] have been published, e.g., by Zienkiewicz [2]. For infinite elements, several approaches are described in [3,4]. Infinite elements are based on special shape functions to approximate the semi-infinite geometry as well as the Sommerfeld radiation condition, i.e., the waves decay with distance and are not reflected at infinity. If there is only one wave traveling in the media, a formulation in time-domain can be established. But in poroelastodynamics, there are three body waves and eventually also a Rayleigh wave. Unfortunately, the extension to more than one wave is not straight forward. Here, an infinite element is presented which can handle all wave types, as it is needed in poroelasticity. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mixed and hybrid formulations for the three‐dimensional magnetostatic problem

We propose mixed and hybrid formulations for the three‐dimensional magnetostatic problem. Such formulations are obtained by coupling finite element method inside the magnetic materials with a boundary element method. We present a formulation where the magnetic field is the state variable and the boundary approach uses a scalar Dirichlet‐Neumann map to describe the exterior domain. Also, we propose a second formulation where the magnetic induction is the state variable and a vectorial Dirichlet‐Neumann map is used to describe the outer field. Numerical discretizations with “edge” and “face” elements are proposed, and for each discrete problem we study an “inf‐sup” condition. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 85–104, 2002     

Application of the discrete element method to model cohesive materials

In this contribution we show how the discrete element method (DEM) can be applied to model cohesive materials which exhibit ductile behaviour by introducing connective elements that can bear a certain load. The modelled material is verified by simulating a orthogonal cutting process which is compared to experimental results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Material-Force-Based Refinement Indicators in Adaptive Strategies

A material-force-based refinement indicator for adaptive finite element strategies for finite elasto-plasticity is proposed. Starting from the local format of the spatial balance of linear momentum, a dual material counterpart in terms of Eshelby's energy-momentum tensor is derived. For inelastic problems, this material balance law depends on the material gradient of the internal variables. In a global format the material balance equation coincides with an equilibrium condition of material forces. For a homogeneous body, this condition corresponds to vanishing discrete material nodal forces. However, due to insufficient discretization, spurious material forces occur at the interior nodes of the finite element mesh. These nodal forces are used as an indicator for mesh refinement. Assigning the ideas of elasticity, where material forces have a clear energetic meaning, the magnitude of the discrete nodal forces is used to define a relative global criterion governing the decision on mesh refinement. Following the same reasoning, in a second step a criterion on the element level is computed which governs the local h-adaptive refinement procedure. The mesh refinement is documented for a representative numerical example of finite elasto-plasticity. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stabilized Mixed Tetrahedrals with Volume and Area Bubble Functions at Large Deformations

In this contribution, stabilized mixed finite tetrahedral elements are presented in order to avoid volume locking and stress oscillations. Geometrically non-linear elastic problems are addressed. The mixed method of incompatible modes is considered. As a key idea, volume and area bubble functions are used for the method of incompatible modes [1], thus giving it the interpretation of a mixed finite element method with stabilization terms. Concerning non-linear problems these are non-linearly dependent on the current deformation state, however, linearly dependent stabilization terms are used. The approach becomes most attractive for the numerical implementation, since the use of quantities related to the previous Newton iteration step is completely avoided. The variational formulation for the standard two-field method, the method of incompatible modes in finite deformation problems is derived for a hyper elastic Neo-Hookean material. In the representative examples Cook's membrane problem and a block under central pressure illustrate the good performance of the presented approaches compared to existing finite element formulations. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite element-based approach to modeling heterogeneous objects

Advances in materials and rapid prototyping technology, makes developing rapid-prototype heterogeneous parts possible. This paper proposes a finite element-based approach for modeling heterogeneous parts to facilitate rapid prototyping. In this approach, traditional geometrical and topological data are blended with materials information in the CAD model. It builds on a CAD system to create parts’ geometry which is then subdivided into a set of tetrahedral finite elements. With this, the design problem of material distribution in the whole body can be handled at the finite element level. The material composition of each point inside a finite element is formulated in terms of four control nodes of the finite element and a blending function. Moreover, design rules for modeling heterogeneous parts are introduced in the paper, making the modeling solution more robust.     

A BE based finite element for crack propagation processes

A boundary element based finite macro element for the simulation of 3D crack propagation in the framework of linear elastic fracture mechanics is presented. While the major part of the numerical model is discretized with finite elements, a small domain containing the crack is meshed with boundary elements. By means of the Symmetric Galerkin BEM a stiffness formulation for the cracked BE domain is obtained which enables a direct FEM/BEM coupling. All necessary operations for the crack propagation are carried out within this boundary element based finite macro element and exploit the potential of the boundary integral formulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A phase field model for fracture

The variational formulation of brittle fracture as formulated for example by Francfort and Marigo in [1], where the total energy is minimized with respect to any admissible crack set and displacement field, allows the identification of crack paths, branching of preexisting cracks and even crack initiation without additional criteria. For its numerical treatment a continuous approximation of the model in the sense of Γ-convergence has been presented by Bourdin in [2]. In the regularized Francfort–Marigo model cracks are represented by an additional field variable (secondary variable) s∈[0,1] which is 0 if the material is cracked and 1 if it is undamaged. In this work, we reinterpret the crack variable as a phase field order parameter and address cracking as a phase transition problem. The crack growth is governed by the evolution equation of the order parameter which resembles the Ginzburg–Landau equation. The numerical treatment is done by finite elements combined with an implicit Euler scheme for the time integration. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient simulation of 3D fatigue crack growth based on boundary elements

The three-dimensional simulation of fatigue crack growth under the consideration of 3D effects is presented to follow complex crack paths as realistic as possible from the macroscopic point of view. This aim is mainly based on two aspects. Firstly, an accurate stress analysis is performed by the boundary element method in terms of the 3D Dual BEM. Secondly, a suitable 3D crack growth criterion based on experimental observations is utilized. Due to the non-linearity of crack growth the whole procedure is embedded in an advanced incremental crack growth algorithm. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)