Dynamic analysis of fixed cracks in composites by the extended finite element method

This paper is dedicated to simulation of dynamic analysis of fixed cracks in orthotropic media using an extended finite element method. This work is in fact an extension to dynamic problems of the recently developed orthotropic extended finite element method for fracture analysis of composites. In this method, the Heaviside and near-tip enrichment functions are used in the framework of the partition of unity for modeling crack discontinuity and crack-tip singularities within the classical finite element method. In this procedure, elements that include a crack are not required to conform to crack edges. Therefore, mesh generation can be performed without any need to comply to crack edges and the method is capable of modeling the crack propagation without any remeshing. To determine the fracture properties, mixed-mode dynamic stress intensity factors (DSIFs) are evaluated by means of domain separation integral (J^′-integral) method. Results of the proposed method are compared with other available analytical and computational results.     

Complete Tangent Stiffness for eXtended Finite Element Method by including crack growth parameters

The eXtended Finite Element Method (XFEM) is a useful tool for modeling the growth of discrete cracks in structures made of concrete and other quasi‐brittle and brittle materials. However, in a standard application of XFEM, the tangent stiffness is not complete. This is a result of not including the crack geometry parameters, such as the crack length and the crack direction directly in the virtual work formulation. For efficiency, it is essential to obtain a complete tangent stiffness. A new method in this work is presented to include an incremental form the crack growth parameters on equal terms with the degrees of freedom in the FEM‐equations. The complete tangential stiffness matrix is based on the virtual work together with the constitutive conditions at the crack tip. Introducing the crack growth parameters as direct unknowns, both equilibrium equations and the crack tip criterion can be handled within the same standard nonlinear iterations. This new solution strategy is believed to provide the modeling capabilities to deal with simultaneous growth of several cracks. A cohesive crack modeling is used. The method is applied to a partly cracked XFEM element of linear strain triangle type with the crack length as the unknown crack growth parameter. In this paper, two examples are given. The first example verifies the theory and the implementation. The second example is the benchmark test three point bending test, where the efficiency of the complete tangential behavior is shown. Copyright © 2013 John Wiley & Sons, Ltd.     

Three‐dimensional non‐planar crack growth by a coupled extended finite element and fast marching method

A numerical technique for non‐planar three‐dimensional linear elastic crack growth simulations is proposed. This technique couples the extended finite element method (X‐FEM) and the fast marching method (FMM). In crack modeling using X‐FEM, the framework of partition of unity is used to enrich the standard finite element approximation by a discontinuous function and the two‐dimensional asymptotic crack‐tip displacement fields. The initial crack geometry is represented by two level set functions, and subsequently signed distance functions are used to maintain the location of the crack and to compute the enrichment functions that appear in the displacement approximation. Crack modeling is performed without the need to mesh the crack, and crack propagation is simulated without remeshing. Crack growth is conducted using FMM; unlike a level set formulation for interface capturing, no iterations nor any time step restrictions are imposed in the FMM. Planar and non‐planar quasi‐static crack growth simulations are presented to demonstrate the robustness and versatility of the proposed technique. Copyright © 2008 John Wiley & Sons, Ltd.     

An adaptive multiscale method for crack propagation and crack coalescence

This work presents a new multiscale technique for the efficient simulation of crack propagation and crack coalescence of macrocracks and microcracks. The fully adaptive multiscale method is able to capture localization effect mesh independently. By modeling macrocracks and microcracks, the extended finite element method offers an accurate solution and captures cracks and their propagation without changing the mesh topology. Propagating and coaliting cracks of different length scales can therefore be easily modeled and updated during the computation process. Hence, the presented method is an efficient and accurate option for modeling cracks of different length scales. This is demonstrated in several numerical examples showing the interaction of microcracks and macrocracks. Copyright © 2012 John Wiley & Sons, Ltd.     

A mesh objective method for modeling crack propagation using the smeared crack approach

This paper presents a mesh objective method for modeling crack propagation in brittle materials using a conventional finite element formulation. The primary shortcoming of the smeared crack approach is its pathological sensitivity to the mesh orientation, which is manifested by shear locking and stress field misalignment around the crack tip. Such undesirable characteristics preclude the ability to model arbitrary crack propagation at an angle through the mesh. Several techniques are developed to address these shortcomings. First, to preclude shear locking, a modified failure constitutive model is developed, which projects out the spurious stress increments as the crack opens. If a crack exists in an element, a crack tracking algorithm is used to identify the neighboring elements most likely to show crack continuation. This algorithm also identifies a crossover element when a crack passes through adjacent sides of an element. Then, the characteristic element length used in the constitutive equation is changed with the objective of providing the correct failure energy per unit crack length, a procedure called crossover scaling. The examples provided demonstrate that the developed methods work collectively to provide a simple and efficient method for modeling failure in brittle materials without mesh bias.     

Numerical analysis of 2-D crack propagation problems using the numerical manifold method

The numerical manifold method is a cover-based method using mathematical covers that are independent of the physical domain. As the unknowns are defined on individual physical covers, the numerical manifold method is very suitable for modeling discontinuities. This paper focuses on modeling complex crack propagation problems containing multiple or branched cracks. The displacement discontinuity across crack surface is modeled by independent cover functions over different physical covers, while additional functions, extracted from the asymptotic near tip field, are incorporated into cover functions of singular physical covers to reflect the stress singularity around the crack tips. In evaluating the element matrices, Gaussian quadrature is used over the sub-triangles of the element, replacing the simplex integration over the whole element. First, the method is validated by evaluating the fracture parameters in two examples involving stationary cracks. The results show good agreement with the reference solutions available. Next, three crack propagation problems involving multiple and branched cracks are simulated. It is found that when the crack growth increment is taken to be 0.5h≤da≤0.75h, the crack growth paths converge consistently and are satisfactory.     

几何非线性扩展有限元法及其断裂力学应用

该文将扩展有限元方法应用到几何非线性及断裂力学问题中,并研制开发了扩展有限元Fortran程序。扩展有限元法其计算网格与不连续面相互独立,因此模拟移动的不连续面时无需对网格进行重新剖分。该文推导了几何非线性扩展有限元法的公式,在常规有限元位移模式中,基于单位分解的思想加进一个阶跃函数和二维渐近裂尖位移场,反映裂纹处位移的不连续性,并用2个水平集函数表示裂纹;采用拉格朗日描述方程建立了有限变形几何非线性扩展有限元方程;采用多点位移外推法计算裂纹应力强度因子并通过最小二乘法拟合得到更精确的结果。最后给出的大变形算例表明该文提出的几何非线性的断裂力学扩展有限元方法和相应的计算机程序是合理可行的,而且对于含裂纹及裂纹扩展的问题,扩展有限元法优于传统的有限元法。     

A geometrically non-linear three-dimensional cohesive crack method for reinforced concrete structures

A three-dimensional meshfree method for modeling arbitrary crack initiation and crack growth in reinforced concrete structure is presented. This meshfree method is based on a partition of unity concept and formulated for geometrically non-linear problems. The crack kinematics are obtained by enriching the solution space in order to capture the correct crack kinematics. A cohesive zone model is used after crack initiation. The reinforcement modeled by truss or beam elements is connected by a bond model to the concrete. We applied the method to model the fracture of several reinforced concrete structures and compared the results to experimental data.     

Cohesive fracture modeling of crack growth in thick-section composites

This paper presents a combined method for modeling the mode-I and II crack growth behavior in thick-section fiber reinforced polymeric composites having a nonlinear material response. The experimental part of this study includes crack growth tests of a thick composite material system manufactured using the pultrusion process. It consists of alternating layers of E-glass unidirectional roving and continuous filament mats in a polymeric matrix. Integrated micromechanical and cohesive finite element (FE) models are used to simulate the crack growth response in eccentrically loaded single-edge-notch, (tension), ESE(T) and notched butterfly specimens. Micromechanical constitutive models for the mat and the roving layers are used to generate the effective nonlinear material behavior from the in situ fiber and matrix responses. The validity of the numerical modeling approach before the onset of crack growth is investigated using an infrared thermal method. Cohesive FE models are calibrated and used to simulate the complete crack growth behavior for different crack configurations. The proposed integrated framework of multi-scale material models with cohesive fracture models is shown to be an effective method for predicting the structural and material responses including failure load and crack growth in thick-section fiber reinforced polymeric composites.     

Complex variable method of computing Jk for bi-material interface cracks

A method using functions of a complex variable is developed for evaluation of J₁ and a modified J₂ integrals for bi-material interface cracks. This method, used in conjunction with the finite element method, would be useful in the prediction of stress intensity factors for cracks lying between the interface of two dissimilar materials. Since the direct evaluation of J₂ poses difficulties in modeling the singular behavior in the near vicinity around the crack tip for bi-material crack problems, it is modified by evaluating it around a contour path of small radius from the crack tip within the singularity dominated zone. It is shown that the stress intensity factors for a bi-material interface crack can be accurately evaluated using these j_k integrals.     

Numerical implementation of anisotropic damage mechanics

This paper describes implementation of anisotropic damage mechanics in the material point method. The approach was based on previously proposed, fourth‐rank anisotropic damage tenors. For implementation, it was convenient to recast the stress update using a new damage strain partitioning tensor. This new tensor simplifies numerical implementation (a detailed algorithm is provided) and clarifies the connection between cracking strain and an implied physical crack with crack opening displacements. By using 2 softening laws and 3 damage parameters corresponding to 1 normal and 2 shear cracking strains, damage evolution can be directly connected to mixed tensile and shear fracture mechanics. Several examples illustrate interesting properties of robust anisotropic damage mechanics such as modeling of necking, multiple cracking in coatings, and compression failure. Direct comparisons between explicit crack modeling and damage mechanics in the same material point method code show that damage mechanics can quantitatively reproduce many features of explicit crack modeling. A caveat is that strengths and energies assigned to damage mechanics materials must be changed from measured material properties to apparent properties before damage mechanics can agree with fracture mechanics.     

Cracking node method for dynamic fracture with finite elements

A new method for modeling discrete cracks based on the extended finite element method is described. In the method, the growth of the actual crack is tracked and approximated with contiguous discrete crack segments that lie on finite element nodes and span only two adjacent elements. The method can deal with complicated fracture patterns because it needs no explicit representation of the topology of the actual crack path. A set of effective rules for injection of crack segments is presented so that fracture behavior beginning from arbitrary crack nucleations to macroscopic crack propagation is seamlessly modeled. The effectiveness of the method is demonstrated with several dynamic fracture problems that involve complicated crack patterns such as fragmentation and crack branching. Copyright © 2008 John Wiley & Sons, Ltd.     

A phase field model of dynamic fracture: Robust field updates for the analysis of complex crack patterns

The numerical modeling of dynamic failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies and demands the formulation of additional branching criteria. This drawback can be overcome by a diffusive crack modeling, which is based on the introduction of a crack phase field. Following our recent works on quasi‐static modeling of phase‐field‐type brittle fracture, we propose in this paper a computational framework for diffusive fracture for dynamic problems that allows the simulation of complex evolving crack topologies. It is based on the introduction of a local history field that contains a maximum reference energy obtained in the deformation history, which may be considered as a measure of the maximum tensile strain in the history. This local variable drives the evolution of the crack phase field. Its introduction provides a very transparent representation of the balance equation that governs the diffusive crack topology. In particular, it allows for the construction of a very robust algorithmic treatment for elastodynamic problems of diffusive fracture. Here, we extend the recently proposed operator split scheme from quasi‐static to dynamic problems. In a typical time step, it successively updates the history field, the crack phase field, and finally the displacement field. We demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples, which show the evolution of complex crack patterns under dynamic loading. Copyright © 2012 John Wiley & Sons, Ltd.     

Modeling of fatigue crack propagation using dual boundary element method and Gaussian Monte Carlo method

This paper studies the modeling of fatigue crack propagation on a multiple crack site of a finite plate using deterministic and probabilistic methods. Stress intensity factor has been calculated by the combined deterministic approach of the dual boundary element method (DBEM) and the probabilistic approach of the Gaussian Monte Carlo method. The Gaussian Monte Carlo method has been incorporated to simulate the random process of the fatigue crack propagation. A finite plate of aluminum alloy 2024-T3 with a thickness of 1.6 mm and 14 holes is analyzed and the fatigue life of the plate is predicted by following a linear elastic law of fracture mechanics. The results of fatigue life predicted by DBEM-Monte Carlo method are in good agreement with experimental ones. The same approach is also applied to two other engineering applications of a gear tooth and a bracket.     

Bridging cell multiscale modeling of fatigue crack growth in fcc crystals

The previously developed bridging cell method for modeling coupled continuum/atomistic systems at finite temperature is used to model fatigue crack growth in single crystal nickel under two crystal orientations at different temperatures. The method is expanded to implement a temperature‐dependent embedded atom method potential for finite temperature simulations avoiding time‐scale restrictions associated with small timesteps. Results for the fatigue simulation were compared with respect to deformation behavior, stress distribution, and crack length. Results showed very different crack growth mechanisms between the two crystal orientations as well as reduced resistance to crack growth with increased temperature. Copyright © 2015 John Wiley & Sons, Ltd.     

A mesh-independent method for planar three-dimensional crack growth in finite domains

The discrete crack mechanics (DCM) method is a dislocation-based crack modeling technique where cracks are constructed using Volterra dislocation loops. The method allows for the natural introduction of displacement discontinuities, avoiding numerically expensive techniques. Mesh dependence in existing computational modeling of crack growth is eliminated by utilizing a superposition procedure. The elastic field of cracks in finite bodies is separated into two parts: the infinite-medium solution of discrete dislocations and an finite element method solution of a correction problem that satisfies external boundary conditions. In the DCM, a crack is represented by a dislocation array with a fixed outer loop determining the crack tip position encompassing additional concentric loops free to expand or contract. Solving for the equilibrium positions of the inner loops gives the crack shape and stress field. The equation of motion governing the crack tip is developed for quasi-static growth problems. Convergence and accuracy of the DCM method are verified with two- and three-dimensional problems with well-known solutions. Crack growth is simulated under load and displacement (rotation) control. In the latter case, a semicircular surface crack in a bent prismatic beam is shown to change shape as it propagates inward, stopping as the imposed rotation is accommodated.