Geometry update driven by material forces for simulation of brittle crack growth in functionally graded materials

Functionally graded materials (FGMs) are advanced materials that possess continuously graded properties, such that the growth of cracks is strongly dependent on the gradation of the material. In this work a thermodynamic consistent framework for crack propagation in FGMs is presented, by applying a dissipation inequality to a time‐dependent migrating control volume. The direction of crack growth is obtained in terms of material forces as a result of the principle of maximum dissipation. In the numerical implementation a staggered algorithm—deformation update for fixed geometry followed by geometry update for fixed deformation—is employed within each time increment. The geometry update is a result of the incremental crack propagation, which is driven by material forces. The corresponding mesh is generated by combining Delaunay triangulation with local mesh refinement. Furthermore a Newton algorithm is proposed, taking into account mesh transfer of displacements for crack propagation in incremental elasticity. In two numerical examples brittle crack propagation in FGMs is investigated for various directions of strength gradation within the structures. Copyright © 2008 John Wiley & Sons, Ltd.     

Thermo-mechanical behavior of a viscoelastic FGMs coating containing an interface crack

In this paper the plane thermo-mechanical behavior of a crack in a viscoelastic functionally graded materials (FGMs) coating with arbitrary material properties bonded to a homogeneous substrate is studied. In order to avoid the complex forms that describe the viscoelastic properties of FGMs, a multi-layered model for the FGMs coating is developed. The compliance and thermal conductivity in the multi-layered model linearly vary in each layer. In this mixed boundary value problem, the system is reduced to singular integral equations and solved numerically with the Lobatto-Chebyshev collocation technique. Using the correspondence principle and Laplace transform, the problem of an interface crack between a homogeneous substrate and a viscoelastic FGMs is solved. Some numerical examples are given to demonstrate the accuracy, efficiency and versatility of the multi-layered model. The numerical results confirm that the fracture toughness of materials can be greatly improved by the graded variation of material parameters. It is also confirmed that the specific variation of material parameters greatly influences the fracture behavior of viscoelastic FGMs coating.     

K‐dominance of static crack tip in functionally gradient materials

K‐dominance of static crack tip in functionally gradient materials (FGMs) with a crack oriented along the direction of the elastic gradient is studied through coherent gradient sensing (CGS), digital speckle correlation method (DSCM) and finite element method (FEM). In the direction of crack propagation, the shear modulus has a linear variation with constant mass density and Poisson's ratio. First, the CGS and DSCM governing equations related to the measurements and the elastic solutions at mode I crack in FGMs are obtained in terms of the stress intensity factor, material constants and graded index. Secondly, two kinds of FGMs specimens and one homogenous specimen are prepared to observe the influences of the property variation on the K‐dominance. Then, CGS and DSCM experiments using three‐point‐bending of FGMs and homogenous beams are performed. Thirdly, based on the results of the experiments, the stress intensity factors of three kinds of specimens are calculated by CGS and DSCM. Meanwhile, the stress intensity factors are obtained by FEM. Finally, comparing the results from CGS, DSCM and FEM, the K‐dominance of mode‐I static crack tip in FGMs is discussed in detail.     

功能梯度压电带中偏心裂纹对SH波的散射问题

功能梯度材料在机械、光电、核能、生物工程领域的应用非常广泛.但由于生产技术及工作环境等方面的原因,功能梯度材料内部常常产生各种形式的裂纹并最终导致材料破坏,这将会给材料所处的整个系统带来巨大损失.因此研究功能梯度材料的断裂问题对于该种材料的设计,制备和合理、安全的应用具有极大的促进作用.本文在压电材料线性宏观理论下,研究了功能梯度压电带中偏心裂纹对SH波的散射问题.借助于积分变换方法,在电非渗透型边界条件的情况下,将所考虑的问题转化为奇异积分方程,运用Gauss-Chebyshev数值积分方法对奇异积分方程进行了数值求解,进而得到了裂纹尖端的应力和电位移强度因子.     

Crack growth simulation for arbitrarily shaped cracks based on the virtual crack closure technique

In this paper, crack growth simulation for arbitrarily shaped cracks was investigated based on the virtual crack closure technique. During simulations, the crack front was represented by an approximated zigzag line which had the same general shape as the given crack. For this approximated zigzag crack front, a modified approach was developed to determine the required nodal forces, virtually closed area and displacement opening. After the strain energy release rate G was calculated, crack growth was governed by the fracture criterion G/G _C = 1 at all the crack tip nodes. The important features of the proposed approach are that (i) a simple stationary finite element mesh can be used for arbitrarily shaped cracks and (ii) adaptive re-meshing technique is avoided in studying crack growth. Three cases having different initial crack shapes are presented to assess the validity of this approach and to evaluate the ease of use in tracking crack growth. Reasonable agreement between the present study and other approaches are obtained. The shape changes during crack propagation can also be tracked with ease.     

含任意方向裂纹功能梯度材料的应力分析研究

功能梯度材料是在航空航天领域的需求背景下发展起来的,但由于生产技术及工作环境等方面的原因,功能梯度材料内部常常产生各种形式的裂纹并最终导致材料破坏,因此研究含任意方向裂纹功能梯度材料的断裂问题具有重要意义。以含有任意方向裂纹的功能梯度材料为对象,运用积分变换方法,给出了相应材料平面问题的位移场的形式解。通过引入辅助函数并利用相关条件,可将问题转化为求解一组带有Cauchy核的奇异积分方程,继而采用Lobatto-Chebyshev方法对奇异积分方程进行数值求解。最后分析了裂纹方向、材料非均匀指数、载荷条件对混合型应力强度因子的影响。      

Fracture Mechanics Analysis Model for Functionally Graded Materials with Arbitrarily Distributed Properties

Functionally Graded Materials (FGMs) have been developed as super-resistant materials for propulsion systems and airframe of space-planes in order to decrease thermal stresses and to increase the effect of protection from heat. It has been experimentally observed that surface cracking in FGMs is the most common failure mode of a metal-ceramic FGM when it is subjected to a thermal shock. Therefore, it is very important to consider the thermally induced fracture behaviors of FGMs. In this paper, a functionally graded material strip containing an embedded or a surface crack perpendicular to its boundaries is considered. The graded region is treated as a large number of single layers, with each layer being homogeneous material. The problem is reduced to an integral equation and is solved numerically. Unlike most of the existing researches, which considered only certain assumed material distributions, the method developed in this paper can be used to investigate functionally graded materials with arbitrarily varied material properties.     

Investigation of fracture characterizations of functionally graded materials

In this paper, the mode I crack problem of functionally gradient materials (FGMs) with the gradient direction parallel to the crack is discussed, and the differences of stress distribution between the gradient materials and the homogeneous materials are analyzed. It is shown that a mode I crack problem of FGMs with the gradient direction parallel to the crack direction can become a mixed‐mode crack problem. In FGMs, the crack initiation angles are determined by the fracture toughness gradient, elastic modulus and crack mode. If the gradient coefficients are small, the crack initiation angles in FGMs are the same as those in homogeneous materials. If the elastic modulus gradient is large, the principal stress terms without the gradient coefficients can be ignored in obtaining the crack initiation angle. In this study, all the above results are generalized to the mixed‐mode crack problems with arbitrary angle between the gradient direction and the crack direction.     

Interaction integral procedures for 3-D curved cracks including surface tractions

This study examines a two-state interaction integral for the direct computation of mixed-mode stress intensity factors along curved cracks under remote mechanical loads and applied crack-face tractions. We investigate the accuracy of stress intensity factors computed along planar, curved cracks in homogeneous materials using a simplified interaction integral that omits terms to reflect specifically the effects of local crack-front curvature. We examine the significance of the crack-face traction term in the interaction integral, and demonstrate the benefit of a simple, exact numerical integration procedure to evaluate the integral for one class of three-dimensional elements. The work also discusses two approaches to compute auxiliary, interaction integral quantities along cracks discretized by linear and curved elements. Comparisons of numerical results with analytical solutions for stress intensity factors verify the accuracy of the proposed interaction integral procedures.     

Numerical investigation of dynamic crack branching under biaxial loading

Dynamic crack growth and branching of a running crack under various biaxial loading conditions in homogeneous and heterogeneous brittle or quasi-brittle materials is investigated numerically using RFPA2D (two-dimensional rock failure process analysis)-Dynamic program which is fully parallelized with OpenMP directives on Windows. Six 2D models were set up to examine the effect of biaxial dynamic loading and heterogeneity on crack growth. The numerical simulation vividly depicts the whole evolution of crack and captured the crack path and the angles between branches. The path of crack propagation for homogenous materials is straight trajectory while for heterogeneous materials is curved. Increasing the ratio of the loading stress in x-direction to the stress in y-direction, the macroscopic angles between branches become larger. Some parasitic small cracks are also observed in simulation. For heterogeneous brittle and quasi-brittle materials coalescence of the microcracks is the mechanism of dynamic crack growth and branching. The crack tip propagation velocity is determined by material properties and independent of loading conditions.     

A boundary layer framework considering material gradation effects

This paper describes the development and application of a novel modified boundary layer (MBL) model for graded nonhomogeneous materials, e.g. functionally graded materials (FGMs). The proposed model is based on a middle-crack tension, M(T), specimen with traction boundary conditions applied to the top and lateral edges of the model. Finite element analyses are performed using WARP3D, a fracture mechanics research finite element code, which incorporates elements with graded elastic and plastic properties. Elastic crack-tip fields obtained from the proposed MBL model show excellent agreement with those obtained from full models of the cracked component for homogeneous and graded nonhomogeneous materials. The K-T dominance of FGMs is investigated by comparing the actual stress fields with the asymptotic stress fields (the Williams’ solution). The examples investigated in the present study consider a crack parallel to the material gradient. Results of the present study provide insight into the K-T dominance of FGMs and also show the range of applicability of the proposed MBL model. The MBL model is applied to analyze the elastic-plastic crack-tip response of a Ti/TiB FGM SE(T) specimen. The numerical results demonstrate that the proposed MBL model captures the effect of T-stress on plastic zone size and shape, constraint effects, etc. for such configurations.     

Crack analysis in unidirectionally and bidirectionally functionally graded materials

Mixed-mode crack analysis in unidirectionally and bidirectionally functionally graded materials is performed by using a boundary integral equation method. To make the analysis tractable, the Young's modulus of the functionally graded materials is assumed to be exponentially dependent on spatial variables, while the Poisson's ratio is assumed to be constant. The corresponding boundary value problem is formulated as a set of hypersingular traction boundary integral equations, which are solved numerically by using a Galerkin method. The present method is especially suited for straight cracks in infinite FGMs. Numerical results for the elastostatic stress intensity factors are presented and discussed. Special attention of the analysis is devoted to investigate the effects of the material gradients and the crack orientation on the elastostatic stress intensity factors.     

A multi-scale model for coupled heat conduction and deformations of viscoelastic functionally graded materials

An integrated micromechanical-structural framework is presented to analyze coupled heat conduction and deformations of functionally graded materials (FGM) having temperature and stress dependent viscoelastic constituents. A through-thickness continuous variation of the thermal and mechanical properties of the FGM is approximated as an assembly of homogeneous layers. Average thermo-mechanical properties in each homogeneous medium are computed using a simplified micromechanical model for particle reinforced composites. This micromechanical model consists of two isotropic constituents. The mechanical properties of each constituent are time–stress–temperature dependent. The thermal properties (coefficient of thermal expansion and thermal conductivity) of each constituent are allowed to vary with temperature. Sequentially coupled heat transfer and displacement analyses are performed, which allow analyzing stress/strain behaviors of FGM having time and temperature dependent material properties. The thermo-mechanical responses of the homogenized FGM obtained from micromechanical model are compared with experimental data and the results obtained from finite element (FE) analysis of FGMs having microstructural details. The present micromechanical-modeling approach is computationally efficient and shows good agreement with experiments in predicting time-dependent responses of FGMs. Our analysis forecasts a better design for creep resistant materials using particulate FGM composites.     

梯度复合材料裂纹扩展路径和起裂载荷的有限元分析

为了模拟功能梯度材料(FGM)在工程应用中可能会出现的断裂问题并计算相应的开裂载荷,通过编写用户自定义UEL子程序将梯度扩展单元嵌入到ABAQUS软件中模拟功能梯度材料的物理场,并编写交互能量积分后处理子程序计算裂纹尖端的混合模式应力强度因子(SIF),采用最大周向应力准则编写子程序计算裂纹的偏转角,并模拟了裂纹扩展路径,计算了裂纹的起裂载荷。讨论了材料梯度参数对裂纹扩展路径以及起裂载荷的影响规律。通过与均匀材料的对比,验证了功能梯度材料断裂性能的优越性。研究表明:外载平行于梯度方向时,垂直梯度方向的初始裂纹朝着等效弹性模量小的方向扩展,且偏转角在梯度指数线性时出现峰值,并随着组分弹性模量比的增加而变大;当外载和初始裂纹均平行于梯度方向时,材料等效弹性模量和断裂韧性的增加或者梯度指数的减小都导致起裂载荷变大。     

Fracture problems,vibration, buckling,and bending analyses of functionally graded materials: A state-of-the-art review including smart FGMS

Composite materials fail under extreme working conditions, particularly at high temperature, due to delamination (separation of fibers from matrix). And therefore it is needed to switch over functionally graded materials (FGMs) which can sustain at high temperature conditions (250–2000°C). There is a need to analyze the fracture and fatigue characteristics of FGM structures and so through this review the emphasis is given on fracture analysis of FGM materials. It has been reported that a combination of extended finite element method and isogeometric analysis methodologies has been used for general mixed-mode crack propagation problems after the introduction of extended isogeometric analysis. Furthermore, recent computational advances have been in the form of multiscale simulations where the part of model is simulated by a finer modeling scale, which can represent details of the material behavior and the interacting effects of material constituents in the finest way. The review is also focused on new advances in analytical and numerical methods for the stress, vibration, and buckling analyses of FGMs. Emphasis has been primarily on to restrict 2D analysis with sorts of compromise in the accuracy of results. First shear deformation theory (FSDT) and third-order shear deformation theory have been extensively used among the various 2D plate theories. FSDT can help us in terms of getting reasonably accurate results with less computational afford. This paper also outlines review on carbon nanotubes (CNT) reinforced FGMs, functionally graded nanocomposites, functionally graded single-walled CNT, FG nanobeam as well as functionally graded piezoelectric materials. Future applications would be based on these smart materials which are supposed to serve us in adverse conditions. Of course, with rise and advent of promising nanotechnology and its potential impact on aerospace industry as well as on other areas, it becomes important to us to compile this review article.     

A simplified method for the evaluation of fatigue crack front shapes under mode I loading

Two-dimensional elastic or elasto-plastic models dominate the current fatigue crack growth assessment and life prediction procedures for plate components with through-the-thickness cracks. However, as demonstrated in many theoretical and experimental papers, the stress field near the crack tip is always three-dimensional and, as a result, the fatigue crack front is not straight. It is normally curved towards the plate faces. Over the past few years there were a number of very careful numerical studies focusing on the evaluation of fatigue crack front shapes. However, the application of the direct numerical techniques to fatigue phenomena is a very tedious and time consuming process and, sometimes, it is quite ambiguous. In the current paper we develop a simplified method for the evaluation of the front shapes of through-the-thickness fatigue cracks. Further, we validate the developed method against experimental results, investigate the influence of various parameters on the crack front shapes at stable (steady-state) propagation and analyse the differences in the results of fatigue crack growth evaluation obtained with two- and three-dimensional approaches.     

XFEM simulation of cracks, holes and inclusions in functionally graded materials

The present work aims at the numerical simulation of inhomogeneities/discontinuities (cracks, holes and inclusions) in functionally graded materials (FGMs) using extended finite element method (XFEM). A FGM with unidirectional gradation in material properties is modeled under plane strain condition. The domain contains a major crack either at the center or at the edge of the domain along with multiple minor discontinuities/flaws such as minor cracks and/or voids/inclusions distributed all over the domain. The effect of the variation in stress intensity factor (SIF) of the major crack due to the presence of the minor cracks and voids/inclusions is studied in detail. The simulations show that the presence of minor discontinuities significantly affects the values of SIFs.     

Cracks and re-entrant corners in functionally graded materials

Functionally graded materials (FGMs) are special composites in which the volume fractions of constituent materials vary gradually, giving continuously graded mechanical properties. The aim of this paper is the evaluation of the strength of structures composed by FGMs incorporating re-entrant corners - tending to the more common crack for vanishing corner angle. The end result is useful in engineering applications predicting the strength of the element corresponding to the unstable brittle crack propagation in such innovative materials. To show the general validity of the method, heterogeneous plates under tension and beam under bending containing re-entrant corners and by varying corner angle, depth and grading of the FGM are considered. Ad hoc performed numerical finite element simulations, by using the FRANC2D code, agree with the theoretical predictions.     

Dynamic stress intensity factors around two parallel cracks in a functionally graded layer bonded to dissimilar half-planes subjected to anti-plane incident harmonic stress waves

The time-harmonic problem of determining the stress field around two parallel cracks in functionally graded materials (FGMs) is studied. The Fourier transform technique is used to reduce the boundary conditions to four simultaneous integral equations which are then solved by expanding the differences of crack surface displacements in a series. The unknown coefficients in the series are obtained by the Schmidt method. Numerical calculations are carried out for dynamic stress intensity factors (DSIF) in FGMs.