Analysis of a mode III crack problem in a functionally graded coating-substrate system with finite thickness

This paper presents an analysis of the static problem of model III crack of a functionally graded coating-substrate system with an internal crack perpendicular to the interface under antiplane shear loading when the coating layer and substrate have finite thickness. After the Fourier transform method is employed, the expressions of the displacement components can be obtained. Integral transforms are employed to reduce the problem to a singular integral equation that can be solved numerically. The influences of the nonhomogeneity constant, relative crack length and thickness ratio are quantitatively studied.     

Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading*

In this paper, the transient dynamic stress intensity factor (SIF) is determined for an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading. An anti-plane step loading is assumed to act suddenly on the surface of interface crack of finite length. The stress field incurred near the crack tip is analyzed. The integral transformation method and singular integral equation approach are used to get the solution. By virtue of the integral transformation method, the viscoelastic mixed boundary problem is reduced to a set of dual integral equations of crack open displacement function in the transformation domain. The dual integral equations can be further transformed into the first kind of Cauchy-type singular integral equation (SIE) by introduction of crack dislocation density function. A piecewise continuous function approach is adopted to get the numerical solution of SIE. Finally, numerical inverse integral transformation is performed and the dynamic SIF in transformation domain is recovered to that in time domain. The dynamic SIF during a small time-interval is evaluated, and the effects of the viscoelastic material parameters on dynamic SIF are analyzed.     

Antiplane crack analysis of a functionally graded material by a BIEM

Elastostatic analysis of an antiplane crack in a functionally graded material (FGM) is performed by using a hypersingular boundary integral equation method (BIEM). An exponential law is applied to describe the spatial variation of the shear modulus of the FGM. A Galerkin method is applied for the numerical solution of the hypersingular traction BIE. Both unidirectional and bidirectional material gradations are investigated. Stress intensity factors for an infinite and linear elastic FGM containing a finite crack subjected to an antiplane crack-face loading are presented and discussed. The influences of the material gradients and the crack orientation on the stress intensity factors are analyzed.     

Fracture dynamics analysis of mode III interface moving crack on functionally graded coating

The interface moving crack between the functionally graded coating and infinite substrate structure with free boundary is investigated in this paper. By application of the interface bonding conditions of the two media, all the quantities have been represented by means of a single unknown function. With the help of the exponent model of the shear modulus and density, the dual integral equation of moving crack problem is obtained by Fourier transform. The displacement is expanded into series form using Jacob Polynomial, and then the semi‐analytic solution of dynamic stress intensity factor is derived by Schmidt method. Dynamic stress intensity factor is influenced by those parameters such as crack velocity, graded parameter and coating height.     

Steady state thermoelastic contact problem in a functionally graded material

This paper is concerned with the stationary plane contact of a functionally graded heat conducting punch and a rigid insulated half-space. The frictional heat generation inside the contact region due to sliding of the punch over the half-space surface and the heat radiation outside the contact region are taken into account. Elastic coefficient μ, thermal expansion coefficient α_t and coefficient of thermal conductivity k are assumed to vary along the normal to the plane of contact. With the help of Fourier integral transform the problem is reduced to a system of two singular integral equations. The equations are solved numerically. The effects of nonhomogeneity parameters in FGMs and thermal effect are discussed and shown graphically.     

Anti-plane problem of periodic interface cracks in a functionally graded coating-substrate structure

In this paper, the interface cracking between a functionally graded material (FGM) and an elastic substrate is analyzed under antiplane shear loads. Two crack configurations are considered, namely a FGM bonded to an elastic substrate containing a single crack and a periodic array of interface cracks, respectively. Standard integral-transform techniques are employed to reduce the single crack problem to the solution of an integral equation with a Cauchy-type singular kernel. However, for the periodic cracks problem, application of finite Fourier transform techniques reduces the solution of the mixed-boundary value problem for a typical strip to triple series equations, then to a singular integral equation with a Hilbert-type singular kernel. The resulting singular integral equation is solved numerically. The results for the cases of single crack and periodic cracks are presented and compared. Effects of crack spacing, material properties and FGM nonhomogeneity on stress intensity factors are investigated in detail.     

Simulation of crack paths in functionally graded materials

One of the main interests of fracture mechanics in functionally graded materials is the influence of such an inhomogeneity on crack propagation processes. Using the Griffith’ energy principle, the change of energy has to be calculated, if the crack starts to propagate. In homogeneous linear-elastic structures (asymptotically precise) formulas for the energy release rate are known, but a direct transfer of these methods to functionally graded materials can lead to very inaccurate results. Moreover, the influence of the inhomogeneity on the crack path cannot be seen. Here, a simple model for functionally graded materials is introduced. For this model, a formula for the change of potential energy is derived, giving detailed information on the effect of the gradation on crack propagation.     

Analysis of the driving force for a generally oriented crack in a functionally graded strip sandwiched between two homogeneous half planes

The driving forces for a generally oriented crack problem embedded in a Functionally Graded strip sandwiched between two half plane are analyzed using singular integral equations with Cauchy kernels, and integrated using Lobatto-Chebyshev collocation. Mixed-mode Stress Intensity Factors (SIF) and Strain Energy Release Rates (SERR) are calculated. The Stress Intensity Factors are compared for accuracy with previously published results. Parametric studies are conducted for various non-homogeneity ratios, crack lengths, crack orientation and thickness of the strip. It is shown that the SERR is more complete and should be used for crack propagation analysis.     

Analysis of an interface crack for a functionally graded strip sandwiched between two homogeneous layers of finite thickness

The interface crack problem for a composite layer that consists of a homogeneous substrate, coating and a nonhomogeneous functionally graded interphase was formulated for singular integral equations with Cauchy kernels, which were integrated using the Lobatto–Chebyshev collocation technique. Mixed-Mode Stress Intensity Factors (SIFs) and Strain Energy Release Rates were calculated. The SIFs were compared for accuracy with relevant results previously published. The parametric studies were conducted for the various thickness of each layer and for various nonhomogeneity ratios. Particular application to the Zirconia thermal barrier on steel substrate is demonstrated.     

Thermal resistance inside an interface crack in the functionally graded sandwich structures

This article proposes a thermal facture model of sandwich structures occupying a functionally graded interlayer with thermal resistance inside the crack region introduced. The crack surfaces are partially thermally insulated with thermally insulated crack and thermally conductive crack as limiting cases. A system of singular integral equations in thermo‐elastic field is reduced and solved numerically by using the collocation methods with higher asymptotic terms in order to improve the convergence and accuracy. For a special case, exact solution is derived to validate the theoretical analysis and numerical computation. Numerical analyses are conducted to reveal the influences of the graded interlayer size, graded parameter and dimensionless thermal resistance on the temperatures along the crack plane and the stress intensity factors.     

The surface crack problem for a layered plate with a functionally graded nonhomogeneous interface

The plane elasticity solution is presented in this paper for the crack problem of a layered plate. A functionally graded interfacial region is assumed to exist as a distinct nonhomogeneous transitional layer with the exponentially varying elastic property between the dissimilar homogeneous surface layer and the substrate. The surface layer contains a crack perpendicular to the boundaries. The Fourier transform technique is used to formulate the problem in terms of a singular integral equation. The main results presented are the variations of stress intensity factors as functions of geometric and material parameters of the layered plate.     

A double receding contact axisymmetric problem between a functionally graded layer and a homogeneous substrate

In this paper, the axisymmetric problem of a frictionless double receding contact between a rigid stamp of axisymmetric profile, an elastic functionally graded layer and a homogeneous half space is considered. The graded layer is modelled as a nonhomogeneous medium with an isotropic stress-strain law. Assuming the double contact between the bodies to be frictionless, only compressive normal tractions can be transmitted in each contact area while the rest of the surface is free of tractions. Using an appropriate integral transform, the axisymmetric elasticity equations are converted analytically into a system of singular integral equations where the unknowns are the pressures and the radii of the receding contact area in the two contact zones. The global equilibrium conditions are supplemented to solve the problem. The singular integral equations are solved numerically using orthogonal Chebyshev polynomials. An iterative scheme based on the Newton-Raphson method is employed to obtain the receding contact radii and pressures that satisfy the equilibrium conditions. The main objectives of the paper are to study the effect of the nonhomogeneity parameter, the thickness of the graded layer and the magnitude of the applied load on the contact pressures, the radii of the receding contact zones and the indentation for the case of a spherical rigid punch.     

Response and control of functionally graded laminated piezoelectric shells under thermal shock and moving loadings

Based on the first-order shear deformation theory (FSDT), approximate solution for FG (functionally graded) laminated piezoelectric cylindrical shells under thermal shock and moving mechanical loads is given utilizing Hamilton’s principle. The thin piezoelectric layers embedded on inner and outer surfaces of the functionally graded layer are acted as distributed sensor and actuator to control dynamic characteristics of the FG laminated cylindrical shells. Here, the modal analysis technique and Newmark’s integration method are used to calculate the dynamic response of FG laminated cylindrical shells. Constant-gain negative velocity feedback approach is used for active vibration control. The active vibration control to a single moving concentrated loading, thermal shock loading and a continuous stream of moving concentrated loadings is, respectively, investigated. Results indicate that the control gain and velocity of moving loadings have significant effects on the dynamic response and resonance of the system.     

Griffith crack moving along the interface of two dissimilar piezoelectric materials

The problem of an anti-plane Griffith crack moving along the interface of dissimilar piezoelectric materials is solved by using the integral transform technique. It is shown from the result that the intensity factors of anti-plane stress and electric displacement are dependent on the speed of the Griffith crack as well as the material coefficients. When the two piezoelectric materials are identical, the present result will reduce to the result for the problem of an anti-plane moving Griffith crack in homogeneous piezoelectric materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

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.     

Crack problem for a functionally graded layer on an elastic foundation

In this paper internal and edge crack problems for an FGM layer attached to an elastic foundation are considered. This model can be used to simulate circumferential crack problem for a thin walled cylinder. It is assumed that the mechanical properties of the layer are varying in thickness direction. Crack is assumed to be perpendicular to the surfaces. For this geometry stress intensity factors are calculated for a number of different crack surface tractions. By using the calculated stress intensity factors and the principle of superposition it is possible to obtain solutions for physically meaningful cases such as fixed grip constant strain loading, membrane loading and bending. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dynamic fracture behavior of an internal interfacial crack between two dissimilar magneto-electro-elastic plates

In this paper, the anti-plane problem for an interfacial crack between two dissimilar magneto-electro-elastic plates subjected to anti-plane mechanical and in-plane magneto-electrical impact loadings is investigated. Four kinds of crack surface conditions are adopted: magneto-electrically impermeable (Case 1), magnetically impermeable and electrically permeable (Case 2), magnetically permeable and electrically impermeable (Case 3), and magneto-electrically permeable (Case 4). The position of the interfacial crack is arbitrary. The Laplace transform and finite Fourier transform techniques are employed to reduce the mixed boundary-value problem to triple trigonometric series equations in the Laplace transform domain. Then the dislocation density functions and proper replacements of the variables are introduced to reduce the series equations to a standard Cauchy singular integral equation of the first kind. The resulting integral equation together with the corresponding single-valued condition is approximated as a system of linear algebra equations, which can easily be solved. Field intensity factors and energy release rates are determined and discussed. The effects of loading combination parameters on dynamic energy release rate are plotted for Cases 1-3. On the other hand, since the magneto-electrically permeable condition is perhaps more physically reasonable for type III crack, the effect of the crack configuration on the dynamic fracture behavior of the crack tips is studied in detail for Case 4. The results could be useful for the design of multilayered magneto-electro-elastic structures and devices.     

Dynamic behaviour of edge-cracked shear deformable functionally graded beams on an elastic foundation under a moving load

This paper studies the dynamic response of functionally graded beams with an open edge crack resting on an elastic foundation subjected to a transverse load moving at a constant speed. It is assumed that the material properties follow an exponential variation through the thickness direction. Theoretical formulations are based on Timoshenko beam theory to account for the transverse shear deformation. The cracked beam is modeled as an assembly of two sub-beams connected through a linear rotational spring. The governing equations of motion are derived by using Hamilton’s principle and transformed into a set of dynamic equations through Galerkin’s procedure. The natural frequencies and dynamic response with different end supports are obtained. Numerical results are presented to investigate the influences of crack location, crack depth, material property gradient, slenderness ratio, foundation stiffness parameters, velocity of the moving load and boundary conditions on both free vibration and dynamic response of cracked functionally graded beams.     

A penny-shaped crack in a functionally graded piezoelectric strip under thermal loading

The mixed-mode thermoelectromechanical fracture problem for a functionally graded piezoelectric material (FGPM) strip with a penny-shaped crack is considered. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under thermal loading. The crack faces are supposed to be insulated thermally and electrically. The thermal and electromechanical problems are reduced to singular integral equations and solved numerically. The stress and electric displacement intensity factors are presented for different crack size, crack position and material nonhomogeneity.