The dynamic response of an edge crack in a functionally graded orthotropic strip

The dynamic response of a functionally graded orthotropic strip with an edge crack perpendicular to the boundaries is studied. The material properties are assumed to vary continuously along the thickness direction. Laplace and Fourier transforms are applied to reduce the problem to a singular integral equation. Numerical results are presented to illustrate the influences of parameters such as the nonhomogeneity constant and geometry parameters on the dynamic stress intensity factors (SIFs).     

Transient thermal shock fracture analysis of functionally graded piezoelectric materials by the extended finite element method

Transient thermal dynamic analysis of stationary cracks in functionally graded piezoelectric materials (FGPMs) based on the extended finite element method (X-FEM) is presented. Both heating and cooling shocks are considered. The material properties are supposed to vary exponentially along specific direction while the crack-faces are assumed to be adiabatic and electrically impermeable. A dynamic X-FEM model is developed in which both Crank–Nicolson and Newmark time integration methods are used for calculating transient responses of thermal and electromechanical fields respectively. The generalized dynamic intensity factors for the thermal stresses and electrical displacements are extracted by using the interaction integral. The accuracy of the developed approach is verified numerically by comparing the calculated results with reference solutions. Numerical examples with mixed-mode crack problems are analyzed. The effects of the crack-length, poling direction, material gradation, etc. on the dynamic intensity factors are investigated. It shows that the transient dynamic crack behaviors under the cooling shock differ from those under the heating shock. The influence of the thermal shock loading on the dynamic intensity factors is significant.     

Cylindrical interface crack in a hollow layered functionally graded cylinder under static torsion

A hollow functionally graded composite cylinder under static torsion, which consists of an inner and outer elastic circular tube with a cylindrical interface crack, is studied in this work. By utilizing Fourier integral transform method, the mixed boundary value problem is reduced to a Cauchy singular integral equation, from which the numerical results of the stress intensity factor are obtained by the Lobatto–Chebyshev quadrature technique. Numerical results demonstrate the coupled effects of geometrical, physical, and functionally graded parameters on the interfacial fracture behavior.     

Mechanical modeling and transient anti-plane fracture analysis for the graded inter-diffusion regions in a bonded structure

An inter-diffusion interface model (IDIM) is put forward for a bonded structure. Laplace and Fourier integral transforms are applied to reduce the transient anti-plane fracture problem of the structure as a Cauchy singular integral equation. Lobatto-Chebyshev collocation method and Laplace numerical inversion transform are employed to evaluate transient stress intensity factors (TSIFs). The effects of geometrical and physical parameters on TSIFs are studied. Specially discussed are the influences of the weak/micro-discontinuity of the interfaces. Comparison between IDIM and the graded interlayer model indicates that if the inter-diffusion between the two original materials is prominent, the former should be applied instead of the latter in fracture analyses of bonded structures.     

Thermal fracture of a functionally graded/homogeneous bimaterial with system of cracks

The thermal fracture of a bimaterial consisting of a homogeneous material and a functionally graded material (FGM) with a system of internal cracks and an interface crack is investigated. The bimaterial is subjected to a heat flux. The thermal properties of FGM are assumed to be continues functions of the thickness coordinate, while the elastic properties are constants. The method of the solution is based on the singular integral equations. For a special case where the interface crack is much larger than the internal cracks in the FGM the asymptotic analytical solution of the problem is obtained as series in a small parameter (the ratio between sizes of the internal and interface crack) and the thermal stress intensity factors (TSIFs) are derived as functions of geometry of the problem and material characteristics. A parametric analysis of the effects of the location and orientation of the cracks and of the inhomogeneity parameter of FGM’s thermal conductivity on the TSIFs is performed. The results are applicable to such kinds FGMs as ceramic/ceramic FGMs, e.g., TiC/SiC, MoSi₂/Al₂O₃ and MoSi₂/SiC, and also some ceramic/metal FGMs.     

Propagation of an anti-plane moving crack in a functionally graded piezoelectric strip

Summary The propagation of an anti-plane moving crack in a functionally graded piezoelectric strip (FGPS) is studied in this paper. The governing equations for the proposed analysis are solved using Fourier cosine transform. The mixed boundary value problems of the anti-plane moving crack, which is assumed to be either impermeable or permeable, are formulated as dual integral equations. By appropriate transformations, the dual integral equations are reduced to Fredholm integral equations of the second kind. For the impermeable crack, the stress intensity factor (SIF) of the crack in the FGPS depends on both the mechanical and electric loading, whereas, the SIF for the permeable crack depends only on the mechanical loading. The results obtained show that the gradient parameter of the FGPS and the velocity of the crack have significant influence on the dynamic SIF.Support from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7081/00E) is acknowledged. Support from the National Natural Science Foundation of China (Project No. 10072041) is also acknowledged.     

Dynamic stress intensity factors of multiple cracks in a functionally graded orthotropic half-plane

In the present paper dynamic stress intensity factor and strain energy density factor of multiple cracks in the functionally graded orthotropic half-plane under time-harmonic loading are investigated. By utilizing the Fourier transformation technique the stress fields are obtained for a functionally graded orthotropic half-plane containing a Volterra screw dislocation. The variations of the material properties are assumed to be exponential forms which the equilibrium has an analytical solution. The dislocation solution is utilized to formulate integral equation for the half-plane weakened by multiple smooth cracks under anti-plane deformation. The integral equations are of Cauchy singular type at the location of dislocation which are solved numerically to obtain the dislocation density on the faces of the cracks. The dislocation densities are employed to determined stress intensity factor and strain energy density factors (SEDFs) for multiple smooth cracks under anti-plane deformation. Numerical examples are provided to show the effects of material properties and the crack configuration on the dynamic stress intensity factors and SEDFs of the functionally graded orthotropic half-plane with multiple curved cracks.     

Torsional impact of transversely isotropic solid with functionally graded shear moduli and a penny-shaped crack

The torsional impact response of a penny-shaped crack in an unbounded transversely isotropic solid is considered. The shear moduli are assumed to be functionally graded such that the mathematics is tractable. Laplace transform and Hankel transform are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress fields are obtained. Investigated are the influence of material nonhomogeneity and orthotropy on the dynamic stress intensity factor. The peak value of the dynamic stress intensity factor can be suppressed by increasing the shear moduli's gradient and/or increasing the shear modulus in a direction perpendicular to the crack surface.     

Exact solution and transient behavior for torsional vibration of functionally graded finite hollow cylinders

Exact solutions are obtained for transient torsio- nal responses of a finitely long, functionally graded hollow cylinder under three different end conditions, i.e. free-free, free-fixed and fixed-fixed. The cylinder with its external surface fixed is subjected to a dynamic shearing stress at the internal surface. The material properties are assumed to vary in the radial direction in a power law form, while keep invariant in the axial direction. With expansion in the axial direction in terms of trigonometric series, the governing equations for the unknown functions about the radial coordinate r and time t are deduced. By applying the variable substitution technique, the superposition method and the separation of variables consecutively, series-form solutions of the equations are obtained. Natural frequencies and the transient torsional responses are finally discussed for a functionally graded finite hollow cylinder.     

功能梯度材料的平面断裂力学分析

针对材料参数在厚度方向可能按任意连续变化的梯度材料，给出了一个新的分层模型，利用该模型求解了面内加载下梯度界面层和涂层中的界面裂纹问题，借助Fburier积分技术和传递矩阵方法，将该问题化为一个Cauchy型奇异积分方程，通过数值求解，得到感兴趣的应力强度因子，对不同形式的杨氏模量和泊松比，计算了界面裂纹应力强度因子，结果表明泊松比的变化形式对应力强度因子影响不大，可当作常数处理，而杨氏模量的影响则很大。     

The transient piezothermoelastic problem of a thick functionally graded thermopiezoelectric strip due to nonuniform heat supply

This paper is concerned with the theoretical treatment of the transient piezothermoelastic problem involving a thick functionally graded thermopiezoelectric strip due to nonuniform heat supply in the width direction. The thermal, thermoelastic and piezoelectric constants of the strip are assumed to vary exponentially in the thickness direction. The transient two-dimensional temperature is analyzed by the methods of Laplace and finite sine transformations. We obtain the exact solution for a simply supported strip under the state of plane strain. Some numerical results for the temperature change, the displacement, the stress and electric potential distributions are presented in figures and table. Furthermore, the influence of the nonhomogeneity of the material and that of the electric boundary conditions are investigated.     

Elastodynamic analysis of a crack at an arbitrary angle to the graded interfacial zone in bonded half-planes under antiplane shear impact

A solution is provided for the elastodynamic problem of a crack at an arbitrary angle to the graded interfacial zone in bonded media under the action of antiplane shear impact. The interfacial zone is modeled by a nonhomogeneous interlayer with the spatially varying shear modulus and mass density in terms of power functions between the two dissimilar, homogeneous half-planes. Based on the use of Laplace and Fourier integral transforms and the coordinate transformations of basic field variables, formulation of the transient crack problem is reduced to solving a Cauchy-type singular integral equation in the Laplace transform domain. The crack-tip response in the physical domain is recovered via the inverse Laplace transform and the values of dynamic mode III stress intensity factors are obtained as a function of time. A comprehensive parametric study is then presented of the effects of crack obliquity on the overshoot behavior of the transient crack-tip response, by plotting the peak values of the dynamic stress intensity factors versus the crack orientation angle for various material and geometric combinations of the bonded system.     

Bending of functionally graded cantilever beam with power-law non-linearity subjected to an end force

A realistic beam structure often exhibits material and geometrical non-linearity, in particular for those made of metals. The mechanical behaviors of a non-linear functionally graded-material (FGM) cantilever beam subjected to an end force are investigated by using large and small deformation theories. Young's modulus is assumed to be depth-dependent. For an FGM beam of power-law hardening, the location of the neutral axis is determined. The effects of depth-dependent Young's modulus and non-linearity parameter on the deflections and rotations of the FGM beams are analyzed. Our results show that different gradient indexes may change the bending stiffness of the beam so that an FGM beam may bear larger applied load than a homogeneous beam when choosing appropriate gradients. Moreover, the bending stress distribution in an FGM beam is completely different from that in a homogeneous beam. The bending stress arrives at the maximum tensile stress at an internal position rather than at the surface. Obtained results are useful in safety design of linear and non-linear beams.     

Analytical solution for functionally graded anisotropic cantilever beam subjected to linearly distributed load

The bending problem of a functionally graded anisotropic cantilever beam subjected to a linearly distributed load is investigated.The analysis is based on the exact elasticity equations for the plane stress problem.The stress function is introduced and assumed in the form of a polynomial of the longitudinal coordinate.The expressions for stress components are then educed from the stress function by simple differentiation. The stress function is determined from the compatibility equation as well as the bound- ary conditions by a skilful deduction.The analytical solution is compared with FEM calculation,indicating a good agreement.     

Interfacial fracture analysis of bonded dissimilar strips with a functionally graded interlayer under antiplane deformation

This paper provides the solution to the problem of dissimilar, homogeneous semi-infinite strips bonded through a functionally graded interlayer and weakened by an embedded or edge interfacial crack. The bonded system is assumed to be under antiplane deformation, subjected to either traction-free or clamped boundary conditions along its bounding planes. Based on the Fourier integral transform, the problem is formulated in terms of a singular integral equation which has a simple Cauchy kernel for the embedded crack and a generalized Cauchy kernel for the edge crack. In the numerical results, the effects of geometric and material parameters of the bonded system on the crack-tip stress intensity factors are presented in order to quantify the interfacial fracture behavior in the presence of the graded interlayer.     

Self-consistent elastoplastic stress solutions for functionally graded material systems subjected to thermal transients

In this work, a self-consistent constitutive framework is proposed to describe the behaviour of a generic three-layered system containing a functionally graded material (FGM) layer subjected to thermal loading. Analytical and semi-analytical solutions are obtained to describe the thermo-elastic and thermo-elastoplastic behaviour of a three-layered system consisting of a metallic and a ceramic layer joined together by an FGM layer of arbitrary composition profile. Solutions for the stress distributions in a generic FGM system subjected to arbitrary temperature transient conditions are presented. The homogenisation of the local elastoplastic FGM behaviour in terms of the properties of its individual phases is performed using a self-consistent approach. In this work, power-law strain hardening behaviour is assumed for the FGM metallic phase. The stress distributions within the FGM systems are compared with accurate numerical solutions obtained from finite element analyses and good agreement is found throughout. Solutions are also given for the critical temperature transients required for the onset of plastic deformation within the three-layered systems.     

Non-axisymmetric thermal stress of a functionally graded coated circular inclusion in an infinite matrix

This paper is to study the non-axisymmetric two-dimensional problem of thermal stresses in an infinite matrix with a functionally graded coated circular inclusion based on complex variable method. With using the method of piece-wise homogeneous layers, the general solution for the functionally graded coating having radial arbitrary elastic properties is derived when the matrix is subjected to uniform heat flux at infinity, and then numerical results are presented for several special examples. It is found that the existence of the functionally graded coating can change interfacial thermal stresses, and choosing proper change ways of the radial elastic properties in the coating can obviously reduce the thermal stresses.     

A moving mode-III crack in functionally graded piezoelectric material: permeable problem

In this paper a moving mode-III crack in functionally graded piezoelectric materials (FGPM) is studied. The crack surfaces are assumed to be permeable. The governing equations for FGPM are solved by means of Fourier cosine transform. The mathematical formulation for the permeable crack condition is derived as a set of dual integral equations, which, in turn, are reduced to a Fredholm integral equation of the second kind. The results obtained indicate that the stress intensity factor of moving crack in FGPM depends only on the mechanical loading. The gradient parameter of the FGPM and the moving velocity of the crack do have significant influence on the dynamic stress intensity factor.     

Dynamic response for functionally graded materials with penny-shaped cracks

This paper provides a method for studying the penny-shaped cracks configuration in functionally graded material(FGM) structures subjected to dynamic or steady loading. It is assumed that the FGMs are transversely isotropic and all the material properties only depend on the axial coordinatez. In the analysis, the elastic region is treated as a number of layers. The material properties are taken to be constants for each layer. By utilizing the Laplace transform and Hankel transform technique, the general solutions for the layers are derived. The dual integral equations are then obtained by introducing the mechanical boundary and layer interface conditions via the flexibility/stiffness matrix approach. The stress intensity factors are computed by solving dual integral equations numerically in Laplace transform domain. The solution in time domain is obtained by utilizing numerical Laplace inverse. The main advantage of the present model is its ability for treating multiple crack configurations in FGMs with arbitrarily distributed and continuously varied material properties by dividing the FGMs into a number of layers with the properties of each layer slightly different from one another. This work was supported by Failure Mechanics Laboratory of State Education Commission and the Post-doctor Research Fund of China.     

Effects of the weak/micro-discontinuity of interface on the fracture behavior of a functionally graded coating with an inclined crack

The mechanical model was established for the anti-plane fracture problem of a functionally graded coating–substrate system with a coating crack inclined to the weak/micro-discontinuous interface. The Cauchy singular integral equation for the crack was derived using Fourier integral transform, and the Lobatto–Chebyshev collocation method put up by Erdogan and Gupta was used to get its numerical solution. Finally, the effects of the weak/micro-discontinuity of the interface on SIFs were analyzed, the “affected regions” corresponding to the two crack tips have been obtained and their engineering significance was discussed. It was indicated that, for the crack tip in the corresponding “affected region”, to reduce the weak-discontinuity of the interface and to make the interface micro-discontinuous are the two effective ways to reduce the SIF, and the latter way always has more remarkable SIF-reduction effect. For the crack tip outside the “affected region”, its SIF is mainly influenced by material stiffness, and to prevent such a tip from growing toward the interface “softer coating and stiffer substrate” is a more advantageous combination than “stiffer coating and softer substrate”.