A strong discontinuity based adaptive refinement approach for the modeling of crack branching

In the current work, the physical phenomena of dynamic fracture of brittle materials involving crack growth, acceleration and consequent branching is simulated. The numerical modeling is based on the approach where the failure in the form of cracks or shear bands is modeled by a jump in the displacement field, the so called ‘strong discontinuity’. The finite element method is employed with this strong discontinuity approach where each finite element is capable of developing a strong discontinuity locally embedded into it. The focus in this work is on branching phenomena which is modeled by an adaptive refinement method by solving a new sub-boundary value problem represented by a finite element at the growing crack tip. The sub-boundary value problem is subjected to a certain kinematic constraint on the boundary in the form of a linear deformation constraint. An accurate resolution of the state of material at the branching crack tip is achieved which results in realistic dynamic fracture simulations. A comparison of resulting numerical simulations is provided with the experiment of dynamic fracture from the literature. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Influence of Multiple Micro Cracks on the Damage Behavior of a Macro-Crack Tip北大核心CSCD

The solution of an infinite plane containing a macro crack and a cluster of micro cracks under uniaxial tensile load was presented based on Muskhelishvili’s complex function method and the stepwise recursive method. The stress field and stress intensity factor K were obtained. Combined with the damage mechanics, damage parameter D of the macro-crack tip and the micro-crack tip under uniaxial tension was redefined, and the influence of different damage zone forms on the damage of the crack tip was analyzed. The results show that, both the chain-distribution and the reverse-chain-distribution micro cracks have an amplifying effect on the macro crack growth, and the damage parameter increases with the decrease of the inclination angle of the micro crack and the reduction of the distance between the macro crack and the micro cracks. For a relatively small inclination angle of the micro crack, the damage parameters of the macro crack and the micro crack heightens, and the damage parameter of the macro crack increases with the micro-crack length. For evenly distributed micro cracks in the continuous damage zone, the micro cracks have an amplifying effect on the macro-crack growth, and the damage parameter of the macro crack increases with the micro-crack number. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

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.     

平面弹性裂纹分析的一种有效边界元方法

提出了一种简单而有效的平面弹性裂纹应力强度因子的边界元计算方法．该方法由Crouch与Starfield建立的常位移不连续单元和闫相桥最近提出的裂尖位移不连续单元构成A·D2在该边界元方法的实施过程中,左、右裂尖位移不连续单元分别置于裂纹的左、右裂尖处,而常位移不连续单元则分布于除了裂尖位移不连续单元占据的位置之外的整个裂纹面及其它边界．算例(如单向拉伸无限大板中心裂纹、单向拉伸无限大板中圆孔与裂纹的作用)说明平面弹性裂纹应力强度因子的边界元计算方法是非常有效的．此外,还对双轴载荷作用下有限大板中方孔分支裂纹进行了分析．这一数值结果说明平面弹性裂纹应力强度因子的边界元计算方法对有限体中复杂裂纹的有效性,可以揭示双轴载荷及裂纹体几何对应力强度因子的影响．     

Configurational-Force-Based Adaptive FE Solver for a Phase Field Model of Fracture

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by diffusive crack modeling, based on the introduction of a crack phase field as outlined in [1, 2]. Following these formulations, we outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids, develop incremental variational principles and, as an extension to [1, 2], consider their numerical implementations by an efficient h-adaptive finite element method. A key problem of the phase field formulation is the mesh density, which is required for the resolution of the diffusive crack patterns. To this end, we embed the computational framework into an adaptive mesh refinement strategy that resolves the fracture process zones. We construct a configurational-force-based framework for h-adaptive finite element discretizations of the gradient-type diffusive fracture model. We develop a staggered computational scheme for the solution of the coupled balances in physical and material space. The balance in the material space is then used to set up indicators for the quality of the finite element mesh and accounts for a subsequent h-type mesh refinement. The capability of the proposed method is demonstrated by means of a numerical example. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Phase Field Modeling of Brittle and Ductile Fracture

The phase field modeling of brittle fracture was a topic of intense research in the last few years and is now well-established. We refer to the work [1-3], where a thermodynamically consistent framework was developed. The main advantage is that the phase-field-type diffusive crack approach is a smooth continuum formulation which avoids the modeling of discontinuities and can be implemented in a straightforward manner by multi-field finite element methods. Therefore complex crack patterns including branching can be resolved easily. In this paper, we extend the recently outlined phase field model of brittle crack propagation [1-3] towards the analysis of ductile fracture in elastic-plastic solids. In particular, we propose a formulation that is able to predict the brittle-to-ductile failure mode transition under dynamic loading that was first observed in experiments by Kalthoff and Winkler [4]. To this end, we outline a new thermodynamically consistent framework for phase field models of crack propagation in ductile elastic-plastic solids under dynamic loading, develop an incremental variational principle and consider its robust numerical implementation by a multi-field finite element method. The performance of the proposed phase field formulation of fracture is demonstrated by means of the numerical simulation of the classical Kalthoff-Winkler experiment that shows the dynamic failure mode transition. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the calculation of energy release rate for curved cracks of rubbery materials

The energy release rate G for 2-D rubbery material problems with curved cracks is calculated with finite element solutions in this paper. Two approaches, a generalized domain integral method and the virtual crack extension method, are investigated. The generalized domain integral method is demonstrated to be “patch independent” and therefore a complicated finite element mesh adjacent to the crack tip area is not required.     

An Adaptive Finite Element Approach for Brittle Fracture

The extension of the finite element method to take discrete fracture and failure modes into account is a current field of research. In recent times, first results in terms of cohesive element formulations have been introduced into commercial applications. Such element formulations are able to cover the discrete behaviour of interfaces between different materials or the mechanical processes of thin layers. These approaches are not suitable for simulations with unknown crack paths in homogeneous materials, due to the initial elastic phase of the material formulation and the necessity to define potential crack paths a priori. The presented strategy starts with an unextended model and modifies the structure during the computations in terms of an adaptive procedure. The idea is to generate additional elements, based on the cohesive element formulation, to approximate arbitrary crack paths. For this purpose, a failure criterion is introduced. For nodes where the limiting value is reached, cohesive elements are introduced between the volume element boundaries of accordingly facets and corresponding nodes are duplicated. Necessary modifications for this application on system level as well as the element and the material formulation are introduced. By means of some numerical examples, the functionality of the presented procedure is demonstrated. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase field simulation of thermomechanical fracture

In Francfort and Marigo's variational free-discontinuity formulation of brittle fracture [1] cracking is regarded as an energy minimization process, where the total energy is minimized with respect to any admissible crack set and displacement field. No additional criterion is needed to determine crack paths, branching of cracks and crack initiations. However, a direct discretization of the model is faced with significant technical problems, as it involves minimizations in a set of possibly discontinuous functions. A regularized version of the model has been introduced by Bourdin [2] and based on this, we use the concept of a continuum phase field model to simulate cracking processes. Cracks are indicated by the order parameter of the phase field model and cracking can be regarded as a phase transition problem. Additionally, introducing the heat equation into the model, it is capable to also take account of crack propagation due to thermal stresses. In the numerical implementation, crack parameter as well as temperature are treated as additional degrees of freedom and the coupled field equations are solved using the finite element method together with an implicit time integration scheme. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Subcritical and critical states of a crack with failure zones

The problem on the stress–strain state near a mode I crack in an infinite plate is solved in the frame of a cohesive zone model. The complex variable method of Muskhelishvili is used to obtain the crack opening displacements caused by the cohesive traction, which models the failure zone at the crack tip, as well as by the external load. The finite stress condition and logarithmic singularity of the derivative of the separation with respect to the coordinate at the tip of a physical crack are taken into account.The cohesive traction distribution is sought in a piecewise linear form, nodal values of which are being numerically chosen to satisfy the traction-separation law. According to this law, the cohesive traction is coupled with the corresponding separation and fracture toughness. The tips of the physical crack and cohesive zone (geometric variables) along with the discrete cohesive traction are used as the problem parameters determining the stress-strain state. If the crack length is included in the set, then the critical crack size can be found for the given loading intensity.The obtained determining system of equations is solved numerically. To find the initial point for a standard numerical algorithm, the asymptotic determining system is derived. In this system, the geometric variables can be easily eliminated, which make it possible to linearize the system.In the numerical examples, the one-parameter traction-separation laws are used. Influence of the shape parameters of the law on the critical crack size and the corresponding cohesive length is studied. The possibility of using asymptotic solutions for determining the critical parameters is analysed. It is established that the critical crack length slightly depends on the shape parameter, while the cohesive length shows a strong dependence on the shape of cohesive laws.     

A Variational–Based Formulation of Regularized Brittle Fracture

The paper performs a comparative study of variational-based brittle fracture with its gradient-type regularization, and outlines aspects of the numerical implementations of both approaches. The latter smoothes out sharp displacement discontinuities of cracks. On the side of discrete crack modeling, we propose a variational framework of configurational-force-driven crack propagation. The latter provides the basis for the computation of material nodal forces and drives the crack propagation in our proposed finite element framework with adaptive nodal doubling. Such a formulation is of limited applicability for the modeling of crack inititation in homogeneous bodies without defects and in situations with complex crack branching. This can be overcome by a regularized crack modeling. Here, an elliptic approximation of the crack surface term yields a regularized two field functional, where an additional scalar field approximates the set of discontinuities. We provide a simple interpretation of such a transition from the sharp crack to the regularized setting. It results in a smooth continuum-damage-type theory with a specific gradient-damage and hardening terms, depending on a length scale that represents the width of a zone that surrounds the crack. Such a variational framework is implemented by a coupled two-field finite element framework in a staightforward manner. We compare representative numerical results obtained by both methods for selected crack patterns and highlight the pro and contra of both meshes. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling quasi-static crack growth with the embedded finite element method on multiple levels

The current work presents the multilevel approach of the embedded finite element method which is obtained by combining features of the method of domain decomposition with those of the standard embedded finite element method. The conventional requirement of fine mesh in a possible failure zone is rendered unnecessary with the new approach thereby reducing the computational expense. In addition, it is also possible to stop a propagating crack-tip in the middle of a finite element. In this approach, the finite elements at the failure-prone zone where cracks or shear bands, referred to as strong discontinuities which represent jumps in the displacement field, can form and propagate based on some failure criterion are treated as separate sub-boundary value problems which are adaptively discretized during the run time into a number of sub-elements and subjected to a kinematic constraint on their boundary. Each sub-element becomes equally capable of developing a strong discontinuity depending upon its state of stress. A linear displacement based constraint is applied initially which is modified accordingly as soon as a strong discontinuity propagates through the boundary of the main finite element. At the local equilibrium, the coupling between the quantities at two different levels of discretization is obtained by matching the virtual energies due to admissible variations of the main finite element and its constituent sub-elements. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Configurational-Force-Driven Crack Propagation and Adaptive Mesh Refinement in Finite Inelasticity

We introduce a consistent variational framework for inelasticity at finite strains, yielding dual balances in physical and material space as the Euler equations. The formulation is employed for the simultaneous usage of configurational forces as both driving forces for crack propagation as well as h-adaptive mesh refinement. The theoretical basis builds upon a global balance of internal and external power, where the mechanical response is exclusively governed by two scalar functions, the free energy function and a dissipation potential. The resulting variational structure is exploited in the context of fracture mechanics and yields evolution equations for internal variables. In the discrete setting, we present a geometry model fully separated from the finite element mesh structure that represents structural changes of the material configuration due to crack propagation. Advanced meshing algorithms provide an optimal discretization at the crack tip. Local and global criteria are obtained via error estimators based on configurational forces being interpreted as indicators of an energetic misfit due to an insufficient discretization. The numerical handling is decomposed into a staggered algorithm scheme for the dual set of equilibrium equations in material and physical space and efficient mesh generation tools. Exemplary numerical examples are considered to illustrate the method and to underline the effects of inelastic material behaviour in the presented context. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

快传播裂缝尖周围的温度场

裂缝进入快传播时,裂缝尖周围的温度升高是一个十分重要的实际问题,它不仅取决于一些材料常数,也取决于传播速度和热源的密度分布.本文讨论了裂缝尖周围塑性区形状以及热源密度,提出了一个温度场模型.对PMMA材料进行了数值计算,并将结果与其它理论和实验结果作了比较.     

Crack identification of the sprocket wheel in sintering machine based on WFEM

In order to identify the crack of the sprocket wheel correctly, the wavelet finite element method is studied in depth. Firstly, the progress of study on the wavelet finite element method is summarized, and then the basic property of wavelet analysis is analyzed, and then the wavelet finite element theory model of sprocket wheel in sintering machine is studied, and the Daubechies wavelet plate element and isoparametric plate element of crack tip are established, and then the theory of constructing crack identification database of the crack for sprocket wheel is studied, and finally the effective of this method is verified by identify the sprocket wheels with five kinds of cracks based on the vibration test, and results showed that this method can identify the crack of the sprocket wheel correctly.     

The effect of microdamage on fatigue-crack growth

The theory of fatigue-crack growth, based on a synthesis of fracture mechanics and continuum mechanics of microdamage accumulation, is applied to the problem of crack growth under cyclic loading, taking into account the plastic deformations in the tip zones. The model of a thin plastic zone, which is a region of considerable plastic deformations at the crack tip, is supplemented by taking into account the effect of microdamage on the value of the specific fracture work and the limit stresses in the tip zone. Governing equations which describe fatigue-crack growth taking these factors into account are derived. The effect of the material characteristics and the load parameters on the growth rate and the distribution of microdamage in the tip zone and on its extensions is investigated by a computational experiment. Particular attention is given to the initial stage when crack growth may occur abruptly and the growth rate depends substantially on the initial conditions     

Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations

Summary The Lagrange-Galerkin method is a numerical technique for solving convection — dominated diffusion problems, based on combining a special discretisation of the Lagrangian material derivative along particle trajectories with a Galerkin finite element method. We present optimal error estimates for the Lagrange-Galerkin mixed finite element approximation of the Navier-Stokes equations in a velocity/pressure formulation. The method is shown to be nonlinearly stable.     

裂纹尖端场弹塑性分析的加权残数法及塑性应力强度因子的计算

本文以幂强化材料,平面应变情形为例,系统地提出了裂纹尖端场弹塑性分析的加权残数法,并根据此法,得出了裂纹尖端场的解析式弹塑性近似解.在此基础上.对整个裂纹区域,构造了弹塑性解叠加非线性有限元计算塑性应力强度因子的方法,从而为裂纹尖端场和整个裂纹体的分析和计算,提供了一个方法.