Simulation of crack growth using cohesive crack method

We present an effective cohesive discrete crack method in the context of the Reproducing Kernel Particle Method (RKPM) in order to study fracture of concrete structures. The discrete crack approach is based on the visibility method and a simple node splitting scheme. We also present an effective implementation of the visibility method and an iteration free algorithm by including the cohesive force term directly into the stiffness equations. The crack is represented by straight-line segments and the cohesive zone model is employed to model the post-localization behavior of concrete. The method is applied to several examples involving mode I and mixed-mode fracture. These results are compared to experimental data and show good agreement.     

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。     

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.     

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)     

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.     

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)     

Implementation of a cohesive zone model for the investigation of the dynamic behavior of a rotating shaft with a transverse crack

The influence of a transverse crack on the vibration of a rotating shaft has been at the focus of attention of many researchers. The knowledge of the dynamic behavior of cracked shaft has helped in predicting the presence of a crack in a rotor. Here, the changing stiffness of the cracked shaft is investigated based on a cohesive zone model. This model is developed for mode-I plane strain and accounts for triaxiality of the stress state explicitly by using basic elastic-plastic constitutive relations. Then, the proposed numerical solution is compared to the switching crack model, which is based on linear elastic fracture mechanics. The cohesive zone model is implemented in finite element techniques to predict and to analyse the dynamic behavior of cracked rotor system. Timoshenko beam theory is used to model the discrete shaft under the effect of gravity, unbalance force and gyroscopic effect. The analysis includes the cohesive function for describing the breathing crack and the reduction of the second moment of area of the element at the location of the crack. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shape derivatives of energy and regularity of minimizers for shallow elastic shells with cohesive cracks

The paper addresses the analysis of a model for a thin shallow linear elastic shell with a smooth vertical through crack that is based on the Kirchhoff–Love shell theory and accounts for cohesive forces acting between the crack faces. We follow the basic idea behind the Barenblatt theory assuming that the density of the total energy spent by the cohesive forces is the sum of longitudinal and transverse contributions, each of which in general is nonconvex. In order to eliminate nonphysical interpenetration of the crack faces, an inequality constraint that involves both the normal component of the longitudinal displacements and the normal derivative of the transverse deflection of the crack faces is imposed. We first prove the existence of minimizers for the total energy and study in detail the Euler–Lagrange system for them. Then we derive the left and right Eulerian shape derivatives of the minimal value of the total energy by developing a fully variational technique. Finally we apply the developed technique coupled with a difference quotient argument to obtain higher differentiability results in Besov and Sobolev spaces for the minimizers.     

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)     

Cohesive zone model for a transverse breathing crack in a rotor

The breathing mechanism of a transversely cracked rotor and its influence on a rotor system that appears due to the shaft weight is studied. This breathing mechanism is based on experimental and simulation result for the crack shape reported in the literature. If the crack depth is small, the crack closure line is a straight line while for larger crack depths the crack closure becomes more curved. For both cases, a method is proposed for the evaluation of the stiffness losses in the cross section that contains the crack. This method is based on a cohesive zone model (CZM) instead of linear elastic fracture mechanics (LEFM) approach, because LEFM is valid only for the fully open crack and cannot be extended to other intermediate situations. As the crack is closed, the stress intensity factor (SIF) will not appear at the boundary between the closed cracked areas and the open cracked areas. The CZM is developed for mode-I plane strain conditions and accounts explicitly for triaxiality of the stress state by using constitutive relations. The proposed model gives more realistic results than models based on LEFM for the stiffness losses of the crack rotor system for a wide range of the crack depth. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Detonation-driven fracture in thin shell structures: Numerical studies

Detonation-driven fracture of thin structures is studied numerically by a 3D discrete crack meshfree method. These types of failure mechanisms play an important role in pipes and vessels. I therefore proposed a three-dimensional meshfree method and an efficient discrete crack model to describe crack propagation. The method is based on separation of particles similar to the visibility method but its implementation is more efficient. I assume here through-thickness cracks though the method can be extended to crack growth in arbitrary directions. The load is applied as travelling pressure wave obtained from pure fluid simulation in accordance with experimental measurements. Numerical results to experimental data show good agreement.     

Implementation of a cohesive zone model for crack closure analysis of rotors

The presence of a crack in a rotor introduces a local flexibility which affects its dynamic response. Moreover, the crack may open and close during the vibration period. The crack status is a function of time and also depends on the rotational speed and the vibration amplitude of the rotor. This nonlinear case is still a challenging research topic especially in the field of closing crack in the rotating shaft. A cohesive zone model is developed in order to analyze the stiffness of a crack in a rotating shaft. The proposed expression will be compared to three different crack models, namely, a breathing crack model, a switching crack model and an open crack model. Moreover, a cohesive law to predict and to analyse the stress at the crack tip is presented. The numerical model is implemented using a finite element formulation. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulating quasi-brittle failures including damage-induced softening based on the mechanism of stress redistribution

There have been extensive analytical and numerical studies on crack propagation and softening behavior in quasi-brittle materials, such as concrete, rock and bone, based on the smeared crack model or the discrete crack model, each of which has advantages and disadvantages. It is well known that stress redistribution has an important effect on cracking behavior and may be taken as the physics mechanism. Using the principle of superposition, the theoretical explanation is provided for stress redistribution during the progressive fracture process. Although this explanation deals with problems of linear elastic fracture mechanics, it is a clue of modeling softening behavior. Then, it is shown that the above stress redistribution based method satisfies the necessary condition that the specimen returns to the unstressed state whenever all the external forces are removed. Finally, a new model for simulating quasi-brittle softening is established via a kind of "sub-element" methodology. Numerical examples show the effectiveness of the above subelement method in modeling the behavior of block media with quasi-brittle block-block interfaces.     

求解层间界面反平面剪切破坏的剪切梁模型(Ⅰ)——基本特性

在反平面剪切载荷及侧压力共同作用下引起的裂纹及裂纹扩展导致的层间界面失效,是岩土工程层间界面及砌体结构中界面层上典型的失效方式.运用弹性力学和断裂力学的理论原理,提出了能够反映上述层间界面断裂失效问题力学特性的剪切梁模型.文中采用具有应力软化特性的“粘性裂纹”(内聚力裂纹)模型来表述层间裂纹前方损伤过程区的本构行为.对通过粘性层结合在一起的两个弹性板,在反平面剪切载荷及侧压力共同作用下的力学行为作了解析分析计算,研究了层间界面裂纹扩展规律.     

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.     

Free vibration analysis of cracked beams by using rigid segment method

This paper deals with the analysis of influence of crack parameters to the modal characteristics of beams at various boundary conditions by using rigid segment method. The beam was discretized by a number of rigid segments which were connected by elastic joints with three degrees of freedom, while the crack was described by cracked element based on fracture mechanics. This model allows detection of coupling between the axial and transverse vibrations under the special boundary conditions. The proposed approach covers both the Euler–Bernoulli and Timoshenko beam model. The efficiency of the method was shown through the few numerical examples.     

混凝土断裂力学虚拟裂缝模型的半解析有限元法

利用平面扇形域哈密顿体系的方程,通过分离变量法及共轭辛本征函数向量展开法,以解析的方法推导出基于混凝土断裂力学中虚拟裂缝模型的平面裂纹解析元列式．将该解析元与有限元相结合,构成半解析的有限元法,可求解任意几何形状和荷载混凝土平面裂纹的虚拟裂缝模型计算问题．数值计算结果表明方法对该类问题的求解是十分有效的,并有较高的精度．     

Interlaminar Crack Propagation Analysis of ENF Specimens Based on the Cohesive Zone Model北大核心CSCD

Based on the classical laminated plate theory and the cohesive zone model, a theoretical model for general delamination cracked laminates was established for crack propagation of pure mode Ⅱ ENF specimens. Compared with the conventional beam theory, the proposed model fully considered the softening process of the cohesive zone and introduced the nonlinear behavior of ENF specimens before failure. The predicted failure load is smaller than that under the beam theory and closer to the experimental data in literatures. Compared with the beam theory with only fracture toughness considered, the proposed model can simultaneously analyze the influences of the interface strength, the fracture toughness and the initial interface stiffness on the load-displacement curves in ENF tests. The results show that, the interface strength mainly affects the mechanical behavior of specimens before failure, but has no influence on crack propagation. The fracture toughness is the main parameter affecting crack propagation, and the initial interface stiffness only affects the linear elastic loading stage. The cohesive zone length increases with the fracture toughness and decreases with the interface strength. The effect of the interface strength on the cohesive zone length is more obvious than that of the fracture toughness. When the adhesive zone tip reaches the half length of the specimen, the adhesive zone length will decrease to a certain extent. Copyright ©2022 Applied Mathematics and Mechanics. All rights reserved.     

幂律非线性粘弹性材料中的裂纹扩展*

对蠕变不可压幂律非线性粘弹性材料中裂纹的蠕变扩展进行了分析,为描述银纹带中的力学行为,假设在裂纹尖端邻域中断裂过程区中分布着阻抗裂纹张开的粘聚应力б_f,.通过对均匀应力参考状态平凡解的摄动,将非线性粘弹性问题化成线性问题处理.对于幂指数.n≌1的弱非线性情况得到了应力与位移表达式.提出断裂过程区局域能量判据,导出了裂纹孕育时间t*与蠕变扩展率a的预测公式.     

Solutions and continuum limits to nonlocal discrete sine-Gordon equations: Bilinearization reduction method

In this paper, we investigate local and nonlocal reductions of a discrete negative order Ablowitz–Kaup–Newell–Segur equation. By the bilinearization reduction method, we construct exact solutions in double Casoratian form to the reduced nonlocal discrete sine-Gordon equations. Then, nonlocal semidiscrete sine-Gordon equations and their solutions are obtained through the continuum limits. The dynamics of soliton solutions are analyzed and illustrated by asymptotic analysis. The research ideas and methods in this paper can be generalized to other nonlocal discrete integrable systems.