A hybrid finite element-scaled boundary finite element method for crack propagation modelling

This study develops a novel hybrid method that combines the finite element method (FEM) and the scaled boundary finite element method (SBFEM) for crack propagation modelling in brittle and quasi-brittle materials. A very simple yet flexible local remeshing procedure, solely based on the FE mesh, is used to accommodate crack propagation. The crack-tip FE mesh is then replaced by a SBFEM rosette. This enables direct extraction of accurate stress intensity factors (SIFs) from the semi-analytical displacement or stress solutions of the SBFEM, which are then used to evaluate the crack propagation criterion. The fracture process zones are modelled using nonlinear cohesive interface elements that are automatically inserted into the FE mesh as the cracks propagate. Both the FEM’s flexibility in remeshing multiple cracks and the SBFEM’s high accuracy in calculating SIFs are exploited. The efficiency of the hybrid method in calculating SIFs is first demonstrated in two problems with stationary cracks. Nonlinear cohesive crack propagation in three notched concrete beams is then modelled. The results compare well with experimental and numerical results available in the literature.     

Fatigue-life estimation of functionally graded materials using XFEM

The present work deals with the fatigue crack growth simulation of alloy/ceramic functionally graded materials (FGMs) using extended finite element method (XFEM). Various cases of FGM containing multiple inhomogeneities/discontinuities along with either a major edge or a center crack are taken for the purpose of simulation. The fatigue life of the FGM plate is calculated using Paris law of fatigue crack growth under cyclic loading. The effect of multiple inhomogeneities/discontinuities (minor cracks, holes/voids, and inclusions) on the fatigue life of cracked FGM plate is studied in detail. These simulations show that the presence of inhomogeneities/discontinuities in the domain significantly influences the fatigue life of the components.     

Application of an interface failure model to predict fatigue crack growth in an implanted metallic femoral stem

A novel computational modelling technique has been developed for the prediction of crack growth in load bearing orthopaedic alloys subjected to fatigue loading. Elastic-plastic fracture mechanics has been used to define a three-dimensional fracture model, which explicitly models the opening, sliding and tearing process. This model consists of 3D nonlinear spring elements implemented in conjunction with a brittle material failure function, which is defined by the fracture energy for each nonlinear spring element. Thus, the fracture energy criterion is implicit in the brittle material failure function to search for crack initiation and crack development automatically. A degradation function is employed to reduce interfacial fracture properties corresponding to the number of cycles; thus fatigue lifetime can be predicted. Unlike other failure modelling methods, this model predicts the failure load, crack path and residual stiffness directly without assuming any pre-flaw condition. As an example, fatigue of a cobalt based alloy (CoCrMo) femoral stem is simulated. Experimental fatigue data was obtained from four point bending tests. The finite element model simulated a fully embedded implant with a constant point load. Comparison between the model and mechanical test results showed good agreement in fatigue crack growth rate.     

Fluid–structure–soil interaction effects on the free vibrations of functionally graded sandwich plates

This study investigates the effects of fluid–structure and soil–structure interaction on the free vibration response of functionally graded sandwich plates. To this aim, an exemplary problem is analyzed, whereas a metal/ceramic sandwich plate is placed at the bottom of a tank filled in with fluid. Two cases are considered: (i) soft core, i.e., a sandwich plate with metal core and ceramic skins, and (ii) hard core, i.e., a sandwich plate with ceramic core and metal skins. In both cases, the skins are modelled as suitable functionally graded materials (FGMs). The soil is modelled as a Pasternak foundation. The free vibration analysis is carried out according to the extended higher order sandwich plate theory (EHSAPT). The fluid is assumed to be inviscid, incompressible, and irrotational. Hamilton’s principle is exploited to deduce the governing equations and the corresponding boundary conditions. The Rayleigh–Ritz method with two-variable orthogonal polynomials is used to compute the natural frequencies of the sandwich plate. The adopted approach is first validated through comparison with results published in the literature. Then, the effects are studied of several parameters on the dynamic response of the system.

     

A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics

We introduce a novel enriched Boundary Element Method (BEM) and Dual Boundary Element Method (DBEM) approach for accurate evaluation of Stress Intensity Factors (SIFs) in crack problems. The formulation makes use of the Partition of Unity Method (PUM) such that functions obtained from a priori knowledge of the solution space can be incorporated in the element formulation. An enrichment strategy is described, in which boundary integral equations formed at additional collocation points are used to provide auxiliary equations in order to accommodate the extra introduced unknowns. In addition, an efficient numerical quadrature method is outlined for the evaluation of strongly singular and hypersingular enriched boundary integrals. Finally, results are shown for mixed mode crack problems; these illustrate that the introduction of PUM enrichment provides for an improvement in accuracy of approximately one order of magnitude in comparison to the conventional unenriched DBEM.     

Optimal design of compliant mechanisms using functionally graded materials

This research applies topology optimization to create feasible functionally graded compliant mechanism designs with the aim of improving structural performance compared to traditional homogeneous compliant mechanism designs. Converged functionally graded designs will also be compared with two-material compliant mechanism designs. Structural performance is assessed with respect to mechanical/geometric advantage and stress distributions. Two design problems are presented – a gripper and a mechanical inverter. A novel modified solid isotropic material with penalization (SIMP) method is introduced for representing local element material properties in functionally graded structures. The method of moving asymptotes (MMA) is used in conjunction with adjoint sensitivity analysis to find the optimal distribution of material properties. Geometric non-linear analysis is used to solve the mechanics problem based on the Neo-Hookean model for hyperelastic materials. Functionally graded materials (FGMs) have material properties that vary based on spatial position. Here, FGMs are implemented using two different resource constraints – one on the mechanism’s volume and the other on the integral of the Young’s modulus distribution throughout the design domain. Tensile tests are performed to obtain the material properties used in the analysis. Results suggest that FGMs can achieve the desired improvements in mechanical/geometric advantage when compared to both homogeneous and two-material mechanisms.     

Topology optimization considering fracture mechanics behaviors at specified locations

As a typical form of material imperfection, cracks generally cannot be avoided and are critical for load bearing capability and integrity of engineering structures. This paper presents a topology optimization method for generating structural layouts that are insensitive/sensitive as required to initial cracks at specified locations. Based on the linear elastic fracture mechanics model (LEFM), the stress intensity of initial cracks in the structure is analyzed by using singularity finite elements positioned at the crack tip to describe the near-tip stress field. In the topology optimization formulation, the J integral, as a criterion for predicting crack opening under certain loading and boundary conditions, is introduced into the objective function to be minimized or maximized. In this context, the adjoint variable sensitivity analysis scheme is derived, which enables the optimization problem to be solved with a gradient-based algorithm. Numerical examples are given to demonstrate effectiveness of the proposed method on generating structures with desired overall stiffness and fracture strength property. This method provides an applicable framework incorporating linear fracture mechanics criteria into topology optimization for conceptual design of crack insensitive or easily detachable structures for particular applications.     

A finite element analysis of the mixed mode bi-material fracture mechanics problem

Two aspects of the mixed mode bi-material fracture mechanics problem are investigated using finite elements. The stress intensity factors for an inclined crack at various distances from a bi-material interface are established as a function of inclination for two material pair combinations. The probable angle of crack extension is established for this problem using the maximum hoop stress criterion. The inclined terminal crack problem is studied using variable power singular elements at the interface. Crack tip stress distributions and probable angle of crack extension are presented as functions of crack inclination and material pair combinations. Crack tip stress distributions assuming an interfacial debonding criterion are also presented as functions of crack inclination and material pair combinations.     

Optimal tailoring of 2D volume-fraction distributions for heat-resisting functionally graded materials using FDM

Thermoelastic behavior of heat resisting functionally graded materials (FGMs), under given thermal loading and boundary conditions, is definitely characterized by the spatial distribution of volume fractions of constituent particles. Hence, the determination of volume-fraction distribution becomes a crucial step in tailoring an optimal heat-resisting FGM. In this paper, we address a two-dimensional volume-fraction optimization procedure for relaxing the effective thermal stress distribution. In order for the optimization efficiency and the volume-fraction continuity, we approximate the volume-fraction field with bilinear elements of n-times larger size than for the thermoelastic analysis. As well, the refined material-property estimate is employed to assure the design quality. Numerical experiments illustrating our theoretical work are also presented.     

FEM analysis of concrete structures subjected to mode-I and mixed-mode loading conditions

In this study, the finite element method (FEM) for a body containing displacement discontinuity is used for the investigation of tensile fracture behavior under mode-I and mixed-mode loading conditions in concrete structures. A mechanical model for the tensile fracture behavior is reduced to a mathematical problem, and the analysis method is proposed. With the aid of this method, several factors which govern tensile fracture are examined, such as the unloading path in the tension-softening behavior and the transmission of shear stresses across crack surfaces. A plain concrete beam without a notch is analyzed by first neglecting and then taking into account the unloading path in the tension-softening behavior to demonstrate the phenomenon of cracking localization in mode-I crack growth. Pullout test specimens of practical significance are analyzed in order to study the crack growth phenomenon under mixed-mode loading conditions. Cases with and without lateral confinement are considered and the results obtained from the present analysis are compared with those obtained from available experimental data. A simple model for shear transfer across crack surfaces is established. By incorporating this model in the program, a pullout test specimen with lateral confinement is analyzed to examine the influence of shear transfer across crack surfaces on cracking localization.     

Meshless Animation of Fracturing Solids

We present a new meshless animation framework for elastic and plastic materials that fracture. Central to our method is a highly dynamic surface and volume sampling method that supports arbitrary crack initiation, propagation, and termination, while avoiding many of the stability problems of traditional mesh-based techniques. We explicitly model advancing crack fronts and associated fracture surfaces embedded in the simulation volume. When cutting through the material, crack fronts directly affect the coupling between simulation nodes, requiring a dynamic adaptation of the nodal shape functions. We show how local visibility tests and dynamic caching lead to an efficient implementation of these effects based on point collocation. Complex fracture patterns of interacting and branching cracks are handled using a small set of topological operations for splitting, merging, and terminating crack fronts. This allows continuous propagation of cracks with highly detailed fracture surfaces, independent of the spatial resolution of the simulation nodes, and provides effective mechanisms for controlling fracture paths. We demonstrate our method for a wide range of materials, from stiff elastic to highly plastic objects that exhibit brittle and/or ductile fracture.     

A robust and consistent remeshing-transfer operator for ductile fracture simulations

This paper addresses the numerical simulation of quasi-static ductile fracture. The main focus is on numerical and stability aspects related to discrete crack propagation. Crack initiation and propagation are taken into account, both driven by the evolution of a discretely coupled damage variable. Discrete ductile failure is embedded in a geometrically nonlinear hyperelasto-plastic model, triggered by an appropriate criterion that has been evaluated for tensile and shear failure. A crack direction criterion is proposed, which is validated for both failure cases and which is capable of capturing the experimentally observed abrupt tensile–shear transition. In a large strain finite element context, remeshing enables to trace the crack geometry as well as to preserve an adequate element shape. Stability of the computations is an important issue during crack propagation that can be compromised by two factors, i.e. large stress redistributions during the crack opening and the transfer of variables between meshes. A numerical procedure is developed that renders crack propagation considerably more robust, independently of the mesh fineness and crack discretisation. A consistent transfer algorithm and a crack relaxation method are proposed and implemented for this purpose. Finally, illustrative simulations are compared with published experimental results to highlight the features mentioned.     

Boundary element analysis of three-dimensional mixed-mode cracks via the interaction integral

A three-dimensional boundary element method (BEM) implementation of the interaction integral methodology for the numerical analysis of mixed-mode three-dimensional cracks is presented in this paper. The interaction integral is evaluated from a domain representation naturally compatible with the BEM, since stresses, strains and derivatives of displacements at internal points can be evaluated using their appropriate boundary integral equations. Special emphasis is put in the selection of the auxiliary function that represents the virtual crack advance in the domain integral. This is found to be a key feature to obtain reliable results at the intersection of crack fronts with free surfaces. Several examples are analysed to demonstrate the efficiency and accuracy of the implementation.     

Automatic LEFM crack propagation method based on local Lepp–Delaunay mesh refinement

A numerical method for 2D LEFM crack propagation simulation is presented. This uses a Lepp–Delaunay based mesh refinement algorithm for triangular meshes which allows both the generation of the initial mesh and the local modification of the current mesh as the crack propagates. For any triangle t, Lepp(t) (Longest Edge Propagation Path of t) is a finite, ordered list of increasing longest edge neighbor triangles, that allows to find a pair of triangles over which mesh refinement operations are easily and locally performed. This is particularly useful for fracture mechanics analysis, where high gradients of element size are needed. The crack propagation is simulated by using a finite element model for each crack propagation step, then the mesh near the crack tip is modified to take into account the crack advance. Stress intensify factors are calculated using the displacement extrapolation technique while the crack propagation angle is calculated using the maximum circumferential stress method. Empirical testing shows that the behavior of the method is in complete agreement with experimental results reported in the literature. Good results are obtained in terms of accuracy and mesh element size across the geometry during the process.     

Thermomechanically coupled sensitivity analysis and design optimization of functionally graded materials

This paper presents a systematic numerical technique for performing sensitivity analysis of coupled thermomechanical problem of functionally graded materials (FGMs). General formulations are presented based on finite element model by using the direct method and the adjoint method. In the modeling of spatial variances of material properties, the graded finite element method is employed to conduct the heat transfer analysis and structural analysis and their sensitivity analysis. The design variables are the volume fractions of FGMs constituents and structural shape parameters. The design optimization model is then constructed and solved by the sequential linear programming (SLP). Numerical examples are presented to demonstrate the accuracy and the applicability of the present method.     

Shape optimization for 2-D mixed-mode fracture using Extended FEM (XFEM) and Level Set Method (LSM)

Weight and service life are often the two most important considerations in design of structural components. This research incorporates a novel crack propagation analysis technique into shape optimization framework to support design of 2-D structural components under mixed-mode fracture for: (1) maximum service life, subject to an upper limit on volume, and (2) minimum weight subject to specified minimum service life. In both cases, structural performance measures are selected as constraints and CAD dimensions are employed as shape design variables. Fracture parameters, such as crack growth rate and crack growth direction are computed using extended finite element method (XFEM) and level set method (LSM). XFEM employs special enrichment functions to incorporate the discontinuity of structural responses caused by the crack surfaces and crack tip fields into finite element approximation. The LSM utilizes level set functions to track the crack during the crack propagation analysis. As a result, this method does not require highly refined mesh around the crack tip nor re-mesh to conform to the geometric shape of the crack when it propagates, which makes the method extremely attractive for crack propagation analysis. An accurate and efficient semi-analytical design sensitivity analysis (DSA) method is developed for calculating gradients of fracture parameters. Two different approaches—a batch-mode, gradient-based, nonlinear algorithm and an interactive what-if analysis—are used for optimization. An engine connecting rod example is used to demonstrate the feasibility of the proposed method.     

Mixed-mode simulation for IEEE 802.11-operated WLANs

In this paper, we address the issue of integrating packet-level simulation with fluid-model-based simulation for IEEE 802.11-operated wireless LANs (WLANs), so as to combine the performance gain of the latter with the accuracy and packet-level detail afforded by the former. In mixed-mode simulation, foreground flows operate in the packet mode, while the other background flows are approximated into a collection of fluid chunks and simulated in the fluid mode. As these two types of flows influence each other at the point of interaction, e.g. the wireless channel in a WLAN, we derive the model of interaction at the wireless medium. We then implement mixed-mode simulation in ns-2 as well as MATLAB, and conduct a comprehensive simulation study to evaluate its performance with respect to the capability of keeping track of network dynamics (in terms of the TCP behavior), accuracy (in terms of the error discrepancy in throughput), and efficiency (in terms of the speed-up in carrying out simulation).Simulation results indicate that, for IEEE 802.11-operated WLANs, mixed-mode simulation significantly expedites the simulation, and yet provides the same level of packet details and accuracy as packet-level simulation does for flows of interest. Specifically, the error discrepancy incurred in mixed-mode simulation is within 2% of the maximum channel bandwidth, and reduces the execution time, in the best case, by two orders of magnitude.     

柴油机机体横隔板断裂失效分析

针对高速大功率柴油机机体横隔板断裂失效现象,应用断裂力学和有限元方法,提出了机体横隔板断裂力学有限元分析的基本方法,建立了合理的含有裂纹的二维有限元模型,进行了静载荷作用下两种不同加载方式的分析,得到了柴油机机体横隔板的应力强度因子K、J积分,并考察了横隔板在静载荷作用下的承载能力;采用3种复合型裂纹扩展准则,预测了裂纹扩展方向,预测结果与横隔板的实际断裂情况相吻合,证明了该方法的正确性,对今后柴油机机体设计有一定的借鉴价值.     

Continuum shape sensitivity analysis of a mixed-mode fracture in functionally graded materials

This paper presents two new methods for conducting a continuum shape sensitivity analysis of a crack in an isotropic, linear-elastic functionally graded material. These methods involve the material derivative concept from continuum mechanics, domain integral representation of interaction integrals, known as the M-integral, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed to calculate the sensitivity of stress–intensity factors. Since the governing variational equation is differentiated prior to the process of discretization, the resulting sensitivity equations are independent of approximate numerical techniques, such as the meshless method, finite element method, boundary element method, or others. Three numerical examples are presented to calculate the first-order derivative of the stress–intensity factors. The results show that first-order sensitivities of stress intensity factors obtained using the proposed method are in excellent agreement with the reference solutions obtained using the finite-difference method for the structural and crack geometries considered in this study.