Mass lumping strategies for X‐FEM explicit dynamics: Application to crack propagation

This paper deals with the numerical modelling of cracks in the dynamic case using the extended finite element method. More precisely, we are interested in explicit algorithms. We prove that by using a specific lumping technique, the critical time step is exactly the same as if no crack were present. This somewhat improves a previous result for which the critical time step was reduced by a factor of square root of 2 from the case with no crack. The new lumping technique is obtained by using a lumping strategy initially developed to handle elements containing voids. To be precise, the results obtained are valid only when the crack is modelled by the Heaviside enrichment. Note also that the resulting lumped matrix is block diagonal (blocks of size 2 × 2). For constant strain elements (linear simplex elements) the critical time step is not modified when the element is cut. Thanks to the lumped mass matrix, the critical time step never tends to zero. Moreover, the lumping techniques conserve kinetic energy for rigid motions. In addition, tensile stress waves do not propagate through the discontinuity. Hence, the lumping techniques create neither error on kinetic energy conservation for rigid motions nor wave propagation through the crack. Both these techniques will be used in a numerical experiment. Copyright © 2007 John Wiley & Sons, Ltd.     

Estimation of the fracture front propagation time in structural components

We discuss special features of the fracture front propagation in structural components (variable-thickness disks and plates, internally pressurized thick-walled vessels). The upper and lower bounds of the relative time of the front propagation have been estimated. Some recommendations regarding the application of the results obtained are provided. __________ Translated from Problemy Prochnosti, No. 6, pp. 13–24, November–December, 2007.     

POST-COLLAPSE BIFURCATION ANALYSIS OF SHELLS OF REVOLUTION BY THE ACCUMULATED ARC-LENGTH METHOD

Shells of revolution subject to axisymmetric loads often fail by non-symmetric bifurcation buckling after non-linear axisymmetric deformations. A number of computer programmes have been developed in the past decades for these problems, but none of them is capable of bifurcation analysis on the descending branch of the primary load–deflection path following axisymmetric collapse/snap-through. This paper presents the first finite element formulation of post-collapse bifurcation analysis of axisymmetric shells in which a modified arc-length method, the accumulated arc-length method, is developed to effect a new automatic bifurcation solution procedure. Numerical examples are presented to demonstrate the validity and capability of the formulation as well as the practical importance of post-collapse bifurcation analysis. The accumulated arc-length method proposed here can also be applied to the post-collapse bifurcation analysis of other structural forms. © 1997 by John Wiley & Sons, Ltd.     

Numerical models of shear fracture propagation

Two-dimensional numerical models using the displacement discontinuity method and the PATH algorithm show that shear fractures can propagate in three different patterns, depending on the lateral normal stress (LNS). Under 0 or very small LNS they propagate by open wing cracks, as shown in many previous results. Under increasing LNS, the extension consists first of a short wing crack followed by a succession of small closed and open segments forming a stair-step pattern. As the LNS increases, the length of the closed segments increases, thus changing the general orientation of the extension. Under sufficiently high LNS the transition is complete, and propagation takes place as a closed shear fracture. These results provide a remarkable confirmation of the theoretical predictions made by Cherepanov [J. Appl. Math. Mech. (English transl. of Prikl. Mate. Mekh.) 30 (1966) 96]. They also explain the difference between the fracture parameters in tension and in shear.     

Fully-automatic modelling of cohesive crack growth using a finite element-scaled boundary finite element coupled method

This study develops a method coupling the finite element method (FEM) and the scaled boundary finite element method (SBFEM) for fully-automatic modelling of cohesive crack growth in quasi-brittle materials. The simple linear elastic fracture mechanics (LEFM)-based remeshing procedure developed previously is augmented by inserting nonlinear interface finite elements automatically. The constitutive law of these elements is modelled by the cohesive/fictitious crack model to simulate the fracture process zone, while the elastic bulk material is modelled by the SBFEM. The resultant nonlinear equation system is solved by a local arc-length controlled solver. The crack is assumed to grow when the mode-I stress intensity factor K_I vanishes in the direction determined by LEFM criteria. Other salient algorithms associated with the SBFEM, such as mapping state variables after remeshing and calculating K_I using a “shadow subdomain”, are also described. Two concrete beams subjected to mode-I and mixed-mode fracture respectively are modelled to validate the new method. The results show that this SBFEM-FEM coupled method is capable of fully-automatically predicting both satisfactory crack trajectories and accurate load-displacement relations with a small number of degrees of freedom, even for problems with strong snap-back. Parametric studies were carried out on the crack incremental length, the concrete tensile strength, and the mode-I and mode-II fracture energies. It is found that the K_I ? 0 criterion is objective with respect to the crack incremental length.     

Finite element modelling of multiple cohesive discrete crack propagation in reinforced concrete beams

This paper presents a finite element (FE) model for fully automatic simulation of multiple discrete crack propagation in reinforced concrete (RC) beams. The discrete cracks are modelled based on the cohesive/fictitious crack concept using nonlinear interface elements with a bilinear tensile softening constitutive law. The model comprises an energy-based crack propagation criterion, a simple remeshing procedure to accommodate crack propagations, two state variable mapping methods to transfer structural responses from one FE mesh to another, and a local arc-length algorithm to solve system equations characterised by material softening. The bond-slip behaviour between reinforcing bars and surrounding concrete is modelled by a tension-softening element. An example RC beam with well-documented test data is simulated. The model is found capable of automatically modelling multiple crack propagation. The predicted cracking process and distributed crack pattern are in close agreement with experimental observations. The load-deflection relations are accurately predicted up to a point when compressive cracking becomes dominant. The effects of bond-slip modelling and the efficiency and effectiveness of the numerical algorithms, together with the limitations of the current model, are also discussed.     

Ductile fracture initiation and propagation modeling using damage plasticity theory

Ductile fracture is often considered as the consequences of the accumulation of plastic damage. This paper is concerned with the application of a recently developed damage plasticity theory incorporates the pressure sensitivity and the Lode angle dependence into a nonlinear damage rule and the material deterioration. The ductile damaging process is calculated through the so-called “cylindrical decomposition” method. The constitutive equations are discussed and numerically implemented. An experimental and numerical investigation for three-point bending tests is reported for aluminum alloy 2024-T351. Crack initiation and propagation in compact tension specimens are also studied numerically. These simulation results show good agreement with experiments. The present model can successfully predict slant fracture as well as the formation of shear lips.     

Elasto-plastic finite element analysis of a crack in an infinite plate

The elastic support method was recently developed to simulate the effects of unbounded solids in the finite element analysis of stresses and displacements. The method eliminates all the computational disadvantages encountered in the use of `infinite' elements or coupled finite element boundary element methods while retaining all the computational advantages of the finite element method. In this paper, the method is extended to the elasto-plastic analysis of fracture in infinite solids by using the load increment approach and including the effects of strain hardening. Numerical tests and parametric study are conducted by analysing a straight crack in an infinite plate. Present results for J integrals and plastified zones are compared, respectively, with analytical solutions and available results obtained by using the body force method. The agreement between the results is found to be very good even if the truncation boundary of the finite element model is located very close to the crack tip or the plastified zone.     

A model for short crack propagation in polycrystalline materials

A model for microstructurally short crack propagation in a grain structure of a polycrystalline material is developed. The crack propagation model is based on a crystal plasticity model and a microstructurally short crack propagation model in the spirit of the model by Navarro and de los Rios [A model for short fatigue crack propagation with an interpretation of the short-long crack transition. Fatigue Fract Eng Mater Struct 1987;10:169-86]. Numerical examples, where the combined crystal plasticity and crack propagation model is implemented in a model of a microstructure representing a duplex stainless steel, concludes the paper. Results showing how the misorientation of the crack- and slip-directions between two adjacent austenitic grains influences the crack propagation rate, as the crack propagates across their common grain boundary, are given.     

弧长法中初始荷载增量参数符号确定准则的改进

在结构非线性全过程跟踪分析中,弧长控制类方法由于其概念简单明了、计算方便可靠,目前已成为一种最主要的跟踪技术。在该方法的使用过程中,初始荷载增量参数符号的确定非常重要,它决定了当前跟踪分析是向前还是返回(Tracing-back)。本文在比较现有准则的基础上,提出了一种依据当前刚度参数进行判定的新准则,并以两个经典算例为例进行考证,其中一个算例同时存在跳跃(Snap-Through)及跳回(Snap-Back)现象且具有分枝路径。分析结果表明,该准则简单实用,在跟踪复杂失稳过程中是非常有效的,具有很好的适用性。     

An indirect boundary element method for three-dimensional dynamic fracture mechanics

Indirect boundary element methods (fictitious load and displacement discontinuity) have been developed for the analysis of three-dimensional elastostatic and elastodynamic fracture mechanics problems. A set of boundary integral equations for fictitious loads and displacement discontinuities have been derived. The stress intensity factors were obtained by the stress equivalent method for static loading. For dynamic loading the problem was studied in Laplace transform space where the numerical calculation procedure, for the stress intensity factor K_I(p), is the same: as that for the static problem. The Durbin inversion method for Laplace transforms was used to obtain the stress intensity factors in the time domain K_I(t). Results of this analysis are presented for a square bar, with either a rectangular or a circular crack, under static and dynamic loads.