Transient temperature response in functionally gradient materials

The temperature response in functionally gradient materials (FGM), subjected to pulseor stepwise heating at the front surface, is evaluated. Applicability of the approximate solution for the temperature response is investigated by comparing it with an exact analytical solution for the FGM in which thermophysical properties have certain profiles. When the FGM is composed of conventional solid materials, appropriateness of the approximate solution for the FGM is demonstrated as far as the temperature response near the rear surface is concerned. The approximate solution is also compared with the solution for the multilayered material. It is shown that an eight-layered material can be regarded as an FGM, as far as the temperature response at the rear surface is concerned, and that the approximate solution can predict the temperature response within 6% error. Because of its simplicity and fair degree of agreement, the approximate solution is anticipated to be used not only for qualitative but also for quantitative prediction of the temperature response near the rear surface of the FGM in engineering applications.     

Cu/ABS复合导电梯度功能材料的制备和性能

借助HAAKE转矩流变仪,采用熔融共混法制备了一系列Cu粉含量不同的Cu/丙烯腈-丁二烯-苯乙烯共聚物(ABS)复合材料,采用多层叠压法制备了内部Cu粉含量逐渐变化的板状Cu/ABS复合导电梯度功能材料,并对其结构和性能进行了表征。结果表明,所制备导电梯度功能材料的性能与均质复合材料存在很大差异。电性能测试结果显示,随着Cu粉含量沿板材厚度方向的梯度增加,其导电性能发生逾渗转变,体积电阻率从一侧的1016Ω.cm降低到另一侧的105Ω.cm。弯曲性能测试表明,富含ABS树脂的一侧表现出较高的弯曲强度,仅比纯ABS的弯曲强度低6%;而富含填料的另一侧则表现出较高的模量,比纯ABS高约20%。实验证明,将导电复合材料做成梯度结构可以兼顾材料的导电性能和力学性能。     

Geometrically non‐linear thermoelastic analysis of functionally graded shells using finite element method

A finite element formulation governing the geometrically non‐linear thermoelastic behaviour of plates and shells made of functionally graded materials is derived in this paper using the updated Lagrangian approach. Derivation of the formulation is based on rewriting the Green–Lagrange strain as well as the 2nd Piola–Kirchhoff stress as two second‐order functions in terms of a through‐the‐thickness parameter. Material properties are assumed to vary through the thickness according to the commonly used power law distribution of the volume fraction of the constituents. Within a non‐linear finite element analysis framework, the main focus of the paper is the proposal of a formulation to account for non‐linear stress distribution in FG plates and shells, particularly, near the inner and outer surfaces for small and large values of the grading index parameter. The non‐linear heat transfer equation is also solved for thermal distribution through the thickness by the Rayleigh–Ritz method. Advantages of the proposed approach are assessed and comparisons with available solutions are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

The extended finite element method with new crack‐tip enrichment functions for an interface crack between two dissimilar piezoelectric materials

This paper studies the static fracture problems of an interface crack in linear piezoelectric bimaterial by means of the extended finite element method (X‐FEM) with new crack‐tip enrichment functions. In the X‐FEM, crack modeling is facilitated by adding a discontinuous function and crack‐tip asymptotic functions to the classical finite element approximation within the framework of the partition of unity. In this work, the coupled effects of an elastic field and an electric field in piezoelectricity are considered. Corresponding to the two classes of singularities of the aforementioned interface crack problem, namely, ? class and κ class, two classes of crack‐tip enrichment functions are newly derived, and the former that exhibits oscillating feature at the crack tip is numerically investigated. Computation of the fracture parameter, i.e., the J‐integral, using the domain form of the contour integral, is presented. Excellent accuracy of the proposed formulation is demonstrated on benchmark interface crack problems through comparisons with analytical solutions and numerical results obtained by the classical FEM. Moreover, it is shown that the geometrical enrichment combining the mesh with local refinement is substantially better in terms of accuracy and efficiency. Copyright © 2015 John Wiley & Sons, Ltd.     

Two-dimensional steady heat conduction in functionally gradient materials by triple-reciprocity boundary element method

Homogeneous heat conduction can be easily solved by means of the boundary element method. However, domain integrals are generally necessary to solve the heat conduction problem in the functionally gradient materials. This paper shows that the two-dimensional heat conduction problem in the functionally gradient materials can be solved approximately without a domain integral by the triple-reciprocity boundary element method. In this method, the distribution of domain effects is interpolated by integral equations. A new computer program is developed and applied to several problems.     

NiCr/ZrO2功能梯度复合材料中混合型准静态裂纹启裂的数字散斑相关方法实验研究

采用粉末冶金方法制备出NiCr/ZrO₂功能梯度材料FGMs。通过2种断裂试件研究了材料梯度对混合型断裂行为的影响(FGM-A试件,裂纹位于试件的弹性模量较大一侧;FGM-B试件,裂纹位于试件的弹性模量较小一侧)。对2种断裂试件在非对称载荷下进行准静态断裂实验,并利用数字散斑相关方法测得Ⅰ、Ⅱ型应力强度因子。结果表明：FGM-A的裂纹的开裂角小于FGM-B的开裂角;FGM-A的弹性梯度对静态裂纹有保护作用;弹性模量的梯度变化和裂尖局部材料的断裂韧性会影响混合型裂纹的启裂。     

Stress field at V-notch tip in polymer materials using digital gradient sensing

The digital gradient sensing (DGS) technique was used to study the stress field at the V-notch tip of polymer materials. First, DGS governing equations at the V-notch tip were deduced using the elastic singular stress fields. Then, theoretical angular deflections of light rays at the V-notch tip were simulated, and the effect of notch angle on the angular deflections was analyzed. Finally, the DGS experiments were conducted, and the angular deflection contours were obtained. The results show that the stress intensity factors at the V-notches extracted from the angular deflections agree well with the results calculated from the finite element method.     

3D crack analysis in functionally graded materials

Elastostatic crack analysis in three-dimensional, continuously non-homogeneous, isotropic and linear elastic functionally graded materials and structures is presented in this paper. A boundary-domain-integral equation formulation is applied for this purpose, which uses the elastostatic fundamental solutions for homogeneous, isotropic and linear elastic materials and involves a domain-integral due to the material’s non-homogeneity. To avoid displacement gradients in the domain-integral, normalized displacements are introduced. The domain-integral is transformed into boundary-integrals over the global boundary of the cracked solids by using the radial integration method. A meshless scheme is developed, which requires only the conventional boundary discretization and additional interior nodes instead of interior cells or meshes. Numerical examples for three-dimensional crack problems in continuously non-homogeneous, isotropic and linear elastic FGMs are presented and discussed, to show the effects of the material gradation on the crack-opening-displacements and the stress intensity factors.     

Estimation of the thermal diffusivity profile in functionally gradient materials with the stepwise heating method

Temperature response in a functionally gradient material (FGM) which is subjected to stepwise heating is investigated, to estimate the profile of the thermal diffusivity from the temperature response at the rear surface of the FGM. Emphasis is placed on a distribution parameter which gives the profiles of the thermophysical properties when an exact analytical solution exists for the temperature response in the FGM. An explicit expression to determine the distribution parameter is obtained as a function of the thermophysical properties at the rear surface. This explicit expression can represent the dependence of the temperature response on the thermophysical properties within 5% in comparison to the exact solution. It is expected that this identification can provide useful insight into the estimation of thermophysical properties in FGMs. The usefulness of this relation is also examined by comparing given and estimated profiles for the thermal diffusivity. Fair agreement is demonstrated as far as the trend and the approximate magnitude are concerned.     

梯度陶瓷喷嘴残余应力的有限元分析

为了提高喷嘴的抗冲蚀磨损能力,将梯度功能材料理论运用于喷嘴材料的设计中,改传统的均质喷嘴材料为非均质喷嘴材料,提出在梯度陶瓷喷嘴制备中将残余压应力引入喷嘴入口的设计目标.在组成分布指数一定的条件下,针对主要设计参数对梯度陶瓷喷嘴残余应力的影响进行有限元分析,探讨了梯度层厚度、临界梯度层材料组分差对SiC/(W,Ti)C单梯度陶瓷喷嘴残余热应力的影响规律,在组成分布指数取0.5时,优化SiC/(W,Ti)C梯度陶瓷喷嘴梯度层厚和临界梯度层材料组分差.结果表明,残余应力随梯度层厚h及临界梯度层SiC体积组分差的不同产生很大差异,合理设计梯度层厚h及临界梯度层SiC体积组分差可在喷嘴入口形成有效残余压应力,最佳梯度层厚为5mm,临界梯度层SiC组分差小于5%(体积分数).     

A quasi‐static crack growth simulation based on the singular ES‐FEM

In the edge‐based smoothed finite element method (ES‐FEM), one needs only the assumed displacement values (not the derivatives) on the boundary of the edge‐based smoothing domains to compute the stiffness matrix of the system. Adopting this important feature, a five‐node crack‐tip element is employed in this paper to produce a proper stress singularity near the crack tip based on a basic mesh of linear triangular elements that can be generated automatically for problems with complicated geometries. The singular ES‐FEM is then formulated and used to simulate the crack propagation in various settings, using a largely coarse mesh with a few layers of fine mesh near the crack tip. The results demonstrate that the singular ES‐FEM is much more accurate than X‐FEM and the existing FEM. Moreover, the excellent agreement between numerical results and the reference observations shows that the singular ES‐FEM offers an efficient and high‐quality solution for crack propagation problems. Copyright © 2011 John Wiley & Sons, Ltd.     

hp‐Generalized FEM and crack surface representation for non‐planar 3‐D cracks

A high‐order generalized finite element method (GFEM) for non‐planar three‐dimensional crack surfaces is presented. Discontinuous p‐hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts. The GFEM is able to model a crack arbitrarily located within a finite element (FE) mesh and thus the proposed method allows fully automated fracture analysis using an existing FE discretization without cracks. We also propose a crack surface representation that is independent of the underlying GFEM discretization and controlled only by the physics of the problem. The representation preserves continuity of the crack surface while being able to represent non‐planar, non‐smooth, crack surfaces inside of elements of any size. The proposed representation also provides support for the implementation of accurate, robust, and computationally efficient numerical integration of the weak form over elements cut by the crack surface. Numerical simulations using the proposed GFEM show high convergence rates of extracted stress intensity factors along non‐planar curved crack fronts and the robustness of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite element evaluation of mixed mode stress intensity factors in functionally graded materials

This paper is directed towards finite element computation of fracture parameters in functionally graded material (FGM) assemblages of arbitrary geometry with stationary cracks. Graded finite elements are developed where the elastic moduli are smooth functions of spatial co‐ordinates which are integrated into the element stiffness matrix. In particular, stress intensity factors for mode I and mixed‐mode two‐dimensional problems are evaluated and compared through three different approaches tailored for FGMs: path‐independent J^*_k‐integral, modified crack‐closure integral method, and displacement correlation technique. The accuracy of these methods is discussed based on comparison with available theoretical, experimental or numerical solutions. Copyright © 2001 John Wiley & Sons, Ltd.     

Cohesive fracture modeling of elastic-plastic crack growth in functionally graded materials

This work investigates elastic-plastic crack growth in ceramic/metal functionally graded materials (FGMs). The study employs a phenomenological, cohesive zone model proposed by the authors and simulates crack growth by the gradual degradation of cohesive surfaces ahead of the crack front. The cohesive zone model uses six material-dependent parameters (the cohesive energy densities and the peak cohesive tractions of the ceramic and metal phases, respectively, and two cohesive gradation parameters) to describe the constitutive response of the material in the cohesive zone. A volume fraction based, elastic-plastic model (extension of the original Tamura-Tomota-Ozawa model) describes the elastic-plastic response of the bulk background material. The numerical analyses are performed using WARP3D, a fracture mechanics research finite element code, which incorporates solid elements with graded elastic and plastic properties and interface-cohesive elements coupled with the functionally graded cohesive zone model. Numerical values of volume fractions for the constituents specified at nodes of the finite element model set the spatial gradation of material properties with isoparametric interpolations inside interface elements and background solid elements to define pointwise material property values. The paper describes applications of the cohesive zone model and the computational scheme to analyze crack growth in a single-edge notch bend, SE(B), specimen made of a TiB/Ti FGM. Cohesive parameters are calibrated using the experimentally measured load versus average crack extension (across the thickness) responses of both Ti metal and TiB/Ti FGM SE(B) specimens. The numerical results show that with the calibrated cohesive gradation parameters for the TiB/Ti system, the load to cause crack extension in the FGM is much smaller than that for the metal. However, the crack initiation load for the TiB/Ti FGM with reduced cohesive gradation parameters (which may be achieved under different manufacturing conditions) could compare to that for the metal. Crack growth responses vary strongly with values of the exponent describing the volume fraction profile for the metal. The investigation also shows significant crack tunneling in the Ti metal SE(B) specimen. For the TiB/Ti FGM system, however, crack tunneling is pronounced only for a metal-rich specimen with relatively smaller cohesive gradation parameter for the metal.     

The XFEM for high‐gradient solutions in convection‐dominated problems

Convection‐dominated problems typically involve solutions with high gradients near the domain boundaries (boundary layers) or inside the domain (shocks). The approximation of such solutions by means of the standard finite element method requires stabilization in order to avoid spurious oscillations. However, accurate results may still require a mesh refinement near the high gradients. Herein, we investigate the extended finite element method (XFEM) with a new enrichment scheme that enables highly accurate results without stabilization or mesh refinement. A set of regularized Heaviside functions is used for the enrichment in the vicinity of the high gradients. Different linear and non‐linear problems in one and two dimensions are considered and show the ability of the proposed enrichment to capture arbitrary high gradients in the solutions. Copyright © 2009 John Wiley & Sons, Ltd.     

Mixed mode asymptotic crack tip fields in orthotropic materials: Derivation and range of dominance

In the first part of this work the asymptotic stress field distribution surrounding a crack in a generally orthotropic solid (i.e. one in which the material and loading axes do not coincide) is derived. The resulting crack tip stress intensity factors are also related to the energy release rate of the cracked solid. In any physical situation, however, the range of dominance of these asymptotic fields will be limited, and will depend on specific geometry and loading parameters. Thus, the second part of this work deals with the range of dominance of the derived stress fields in edge cracked bending loaded fiber reinforced composite plates. The lower and upper limits of the range of dominance of the solution are respectively determined from three-dimensional and two-dimensional full field finite element solutions. The comparison of full field solutions with the asymptotic result provides information on the latter's range of dominance. In all cases the effects of mixed mode loading are considered.     

Three-dimensional static and dynamic analyses of the functionally graded cylinder bonded to the laminated plate under general loading

A graded finite element method code based on Rayleigh–Ritz energy formulation is developed and implemented to study the elastic behavior of a layered plate loaded by a solid isotropic cylinder and a functionally graded interlayer. The applied nonaxisymmetric loading to the inner cylinder induces a stress concentration in the flexible part of the joint. The effects of different thicknesses and power law exponents of functionally graded interlayer on the distribution of displacements and stresses are investigated, which verifies the ability of functionally graded material to control the stress and displacement waves. The time-dependent response of the structure is also obtained based on Newmark's time integration method.     

A super‐element for crack analysis in the time domain

A super‐element for the dynamic analysis of two‐dimensional crack problems is developed based on the scaled boundary finite‐element method. The boundary of the super‐element containing a crack tip is discretized with line elements. The governing partial differential equations formulated in the scaled boundary co‐ordinates are transformed to ordinary differential equations in the frequency domain by applying the Galerkin's weighted residual technique. The displacements in the radial direction from the crack tip to a point on the boundary are solved analytically without any a priori assumption. The scaled boundary finite‐element formulation leads to symmetric static stiffness and mass matrices. The super‐element can be coupled seamlessly with standard finite elements. The transient response is evaluated directly in the time domain using a standard time‐integration scheme. The stress field, including the singularity around the crack tip, is expressed semi‐analytically. The stress intensity factors are evaluated without directly addressing singular functions, as the limit in their definitions is performed analytically. Copyright © 2004 John Wiley & Sons, Ltd.     

A micro‐mechanical damage model based on gradient plasticity: algorithms and applications

As soon as material failure dominates a deformation process, the material increasingly displays strain softening and the finite element computation is significantly affected by the element size. Without remedying this effect in the constitutive model one cannot hope for a reliable prediction of the ductile material failure process. In the present paper, a micro‐mechanical damage model coupled to gradient‐dependent plasticity theory is presented and its finite element algorithm is discussed. By incorporating the Laplacian of plastic strain into the damage constitutive relationship, the known mesh‐dependence is overcome and computational results are uniquely correlated with the given material parameters. The implicit C¹ shape function is used and can be transformed to arbitrary quadrilateral elements. The introduced intrinsic material length parameter is able to predict size effects in material failure. Copyright © 2002 John Wiley & Sons, Ltd.