Interface cracking between functionally graded coatings and a substrate under antiplane shear

Coating technology plays a significant role in a number of applications such as high temperatures, corrosion, oxidation, wear, and interface. In this paper, we investigate the interface cracking between ceramic and/or functionally graded coatings (FGM coatings) and a substrate under antiplane shear. Four coating models are considered, namely single layered homogeneous coating, double layered piece-wise homogeneous coating, single layered FGM coating and double layered coating with an FGM bottom coat. Mode III stress intensity factors (SIFs) are calculated for the different coating models. In the case of μ_L > μ₀ where μ₀ is the shear modulus of the substrate and μ_L the shear modulus of the material at the surface of the coating, it is found that the single layered FGM coating reduces SIF slightly, whereas the coating system with a top homogeneous layer and a thin FGM bottom layer reduces SIF significantly. In the case of μ_L < μ₀ the SIF is found to be larger for the FGM coatings than for the homogeneous coatings. The FGM coating, however, may still be superior to homogeneous coatings in this case as FGM coatings usually provide better bonding strength between the coating and substrate. Finally, the applicability of the SIF concept in the fracture of FGM coatings is discussed. Large modulus gradients in thin coatings may seriously restrict the application of SIFs as the SIF-dominant zone may fall into the crack tip nonlinear deformation and damage zone. The same argument exists for some interphase models in interface crack solutions.     

Inverse problems of material distributions for prescribed apparent fracture toughness in FGM coatings around a circular hole in infinite elastic media

This study is concerned with the inverse problem of calculating material distributions intending to realize prescribed apparent fracture toughness in functionally graded material (FGM) coatings around a circular hole in infinite elastic media. The incompatible eigenstrain induced in the FGM coatings after cooling from the sintering temperature, due to mismatch in the coefficients of thermal expansion, is taken into consideration. An approximation method of determining stress intensity factors is introduced for a crack in the FGM coatings in which the FGM coatings are homogenized simulating the nonhomogeneous material properties by a distribution of equivalent eigenstrain. A radial edge crack emanating from the circular hole in the homogenized coatings is considered for the case of a uniform pressure applied to the surfaces of the hole and the crack. The stress intensity factors determined for the crack in the homogenized coatings represent the approximate values of the stress intensity factors for the same crack in the FGM coatings, and are used in the inverse problem of calculating material distributions in the FGM coatings intending to realize prescribed apparent fracture toughness in the coatings. Numerical results are obtained for a TiC/Al₂O₃ FGM coating, which reveal that the apparent fracture toughness in FGM coatings around a circular hole in infinite elastic media can be controlled within possible limits by choosing an appropriate material distribution profile in the coatings.     

Cracking in coating-substrate composites with multi-layered and FGM coatings

This investigation evaluates, by finite element method, the stress intensity factors (SIF) of cracked multi-layered and functionally graded material (FGM) coatings of a coating-substrate composite, due to the action of uniform normal stress on the crack surfaces. The substrate is assumed to be homogeneous material, while the coating consists of multi-layered media or sigmoid FGMs. The sigmoid FGM is a kind of FGM in which the material properties of the coating are governed by two power-law functions of volume fractions such that the functions of the material property represent sigmoid distributions in the thickness direction, simply called S-FGM in this paper. For the multi-layered coatings, one, two, and four-layered homogeneous coatings with stepwise changing volume fractions are considered. The primary problem addressed herein is the appearance of a crack in the coating surface and its expansion into the substrate along the direction perpendicular to the interface between the coating and the substrate. The results show that if the coating is stiffer than the substrate, a crack in a one-layered coating is much more susceptible to propagation into the substrate than a crack in the two- or four-layered coating. But crack growth can be effectively averted by using an S-FGM coating. However, if the coating is softer than the substrate, the S-FGM coating behaves like a bridge to connect the soft coating and the stiff substrate, and facilitates the expansion of the crack expanding into the substrate. Whereas the one-layered coating can more effectively prevent the crack from propagating into the substrate than can the two- or four-layered coating. The investigation also indicates that the material gradations of S-FGMs influence SIFs obviously only when the crack tip is inside the coating that is stiffer than the substrate. As the crack extends through the coating and into the substrate, the material gradation of the S-FGM coating and the material mismatch of the multi-layered coating slightly bear on the values of SIF.     

Thermal shock analysis for a functionally graded plate with a surface crack

Summary The fracture behavior of a functionally graded material (FGM) plate subjected to a thermal shock is studied. A surface crack is considered. The thermomechanical properties of the FGM plate are assumed to vary along the thickness direction. By using a perturbation method, the transient temperature field is solved. Then the transient thermal stresses and the corresponding thermal stress intensity factor (TSIF) are obtained. The transient thermal stresses and TSIF in an FGM ceramic/metal (ZrO₂/Ti-6Al-4V) plate are shown in figures. Dedicated to Professor Franz Ziegler on the occasion of his 70th birthday     

Stress intensity factors of two diametrically opposed edge cracks in a thick-walled functionally graded material cylinder

A method is developed to evaluate stress intensity factors for two diametrically-opposed edge cracks emanating from the inner surface of a thick-walled functionally graded material (FGM) cylinder. The crack and the cylinder inner surfaces are subjected to an internal pressure. The thermal eigenstrain induced in the cylinder material due to nonuniform coefficient of thermal expansion after cooling from the sintering temperature is taken into account. First, the FGM cylinder is homogenized by simulating its nonhomogeneous material properties by an equivalent eigenstrain, whereby the problem is reduced to the solution of a cracked homogenized cylinder with an induced thermal and an equivalent eigenstrains and under an internal pressure. Then, representing the cracks by a continuous distribution of edge dislocations and using their complex potential functions, generalized formulations are developed to calculate stress intensity factors for the cracks in the homogenized cylinder. The stress intensity factors calculated for the cracks in homogenized cylinder represents the stress intensity factors for the same cracks in the FGM cylinder. The application of the formulations are demonstrated for a thick-walled TiC/Al₂O₃ FGM cylinder and some numerical results of stress intensity factors are presented for different profiles of material distribution in the FGM cylinder.     

Periodic cracking of functionally graded coatings

The antiplane elasticity problem for a functionally graded coating bonded to a homogeneous half space is considered. The coating is assumed to contain periodic cracks perpendicular to the surface. The problem is formulated in terms of an integral equation with strongly singular kernels. Three dimensionless parameters representing the crack depth, the material nonhomogeneity and the crack periodicity are identified. In addition to the mode III stress intensity factor calculated by varying these three parameters, the results presented include a qualitative discussion of the question of fracture instability, the effect of periodic cracking on the relaxation of stresses on the coating surface, and the comparison of the total strain energy released as a result of surface cracking with that assumed in a simple stress relaxation model.     

Stress intensity factors for an interface crack between a functionally graded coating and a homogeneous substrate

In this paper we consider the problem of a functionally graded coating bonded to a homogeneous substrate with a partially insulated interface crack between the two materials subject to both thermal and mechanical loading. The problem is solved under the assumption of plane strain and generalized plane stress conditions. The heat conduction and the plane elasticity equations are converted analytically into singular integral equations which are solved numerically to yield the temperature and the displacement fields in the medium as well as the crack tip stress intensity factors. A crack-closure algorithm recently developed by the authors is applied to handle the problem of having negative mode I stress intensity factors. The Finite Element Method was additionally used to model the crack problem and to compute the crack-tip stress intensity factors. The main objective of the paper is to study the effect of the material nonhomogeneity parameters, partial insulation of the crack surfaces and crack-closure on the crack tip stress intensity factors for the purpose of gaining better understanding of the thermo-mechanical behavior of graded coatings.     

Interaction integrals for thermal fracture of functionally graded piezoelectric materials

This paper presents domain form of the interaction integrals based on three independent formulations for computation of the stress intensity factors and electric displacement intensity factor for cracks in functionally graded piezoelectric materials subjected to steady-state thermal loading. Each of the formulation differs in the way auxiliary fields are imposed in the evaluation of interaction integral and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appear in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded piezoelectric medium.     

Three-dimensional analysis of a functionally graded coating/substrate system of finite thickness

The concept of functionally graded material (FGM) is currently actively explored in coating design for the purpose of eliminating the mismatch of thermomechanical properties at the interfaces and thus increasing the resistance of coatings to functional failure. In the present paper, three-dimensional elastic deformation of a functionally graded coating/substrate system of finite thickness subjected to mechanical loading is investigated. A comparative study of FGM versus homogeneous coating is conducted to examine the effect of the coating type on stress and displacement fields in the system.     

Dynamic stress intensity factor of a functionally graded material under antiplane shear loading

Summary The dynamic response of a finite crack in an unbounded Functionally Graded Material (FGM) subjected to an antiplane shear loading is studied in this paper. The variation of the shear modulus of the functionally graded material is modeled by a quadratic increase along the direction perpendicular to the crack surface. The dynamic stress intensity factor is extracted from the asymptotic expansion of the stresses around the crack tip in the Laplace transform plane and obtained in the time domain by a numerical Laplace inversion technique. The influence of graded material property on the dynamic intensity factor is investigated. It is observed that the magnitude of dynamic stress intensity factor for a finite crack in such a functionally graded material is less than in the homogeneous material with a property identical to that of the FGM crack plane.     

Fatigue crack growth simulations of interfacial cracks in bi-layered FGMs using XFEM

An investigation of fatigue crack growth of interfacial cracks in bi-layered materials using the extended finite element method is presented. The bi-material consists of two layers of dissimilar materials. The bottom layer is made of aluminium alloy while the upper one is made of functionally graded material (FGM). The FGM layer consists of 100 % aluminium alloy on the left side and 100 % ceramic (alumina) on the right side. The gradation in material property of the FGM layer is assumed to be exponential from the alloy side to the ceramic side. The domain based interaction integral approach is extended to obtain the stress intensity factors for an interfacial crack under thermo-mechanical load. The edge and centre cracks are taken at the interface of bi-layered material. The fatigue life of the interface crack plate is obtained using the Paris law of fatigue crack growth under cyclic mode-I, mixed-mode and thermal loads. This study reveals that the crack propagates into the FGM layer under all types of loads.     

Transient thermal stress analysis of an edge crack in a functionally graded material

An edge crack in a strip of a functionally graded material (FGM) is studied under transient thermal loading conditions. The FGM is assumed having constant Young's modulus and Poisson's ratio, but the thermal properties of the material vary along the thickness direction of the strip. Thus the material is elastically homogeneous but thermally nonhomogeneous. This kind of FGMs include some ceramic/ceramic FGMs such as TiC/SiC, MoSi₂/Al₂O₃ and MoSi₂/SiC, and also some ceramic/metal FGMs such as zirconia/nickel and zirconia/steel. A multi-layered material model is used to solve the temperature field. By using the Laplace transform and an asymptotic analysis, an analytical first order temperature solution for short times is obtained. Thermal stress intensity factors (TSIFs) are calculated for a TiC/SiC FGM with various volume fraction profiles of the constituent materials. It is found that the TSIF could be reduced if the thermally shocked cracked edge of the FGM strip is pure TiC, whereas the TSIF is increased if the thermally shocked edge is pure SiC.     

A periodic array of cracks in functionally graded materials subjected to transient loading

A periodic array of cracks in an infinite functionally graded material under transient mechanical loading is investigated. In-plane normal (mode I) and shear (mode II) loading conditions are considered. For each individual loading mode, a singular integral equation is derived, in which the crack surface displacements are unknown functions. Numerical results are obtained to illustrate the variation of the stress intensity factors as a function of the crack periodicity for different values of material inhomogeneity, either at the transient state or steady state. The material inhomogeneity can increase or decrease the mode I and mode II stress intensity factors. Compared with the single crack solution, it is also shown that multiple cracking may decrease the mode I stress intensity factors, but enhance the mode II stress intensity factors significantly.     

Collinear cracks in a layered half – plane with a graded nonhomogeneous interfacial zone – Part II: Thermal shock response

In Part I of the paper, the problem of collinear cracks in a layered half-plane with a graded nonhomogeneous interfacial zone was investigated under mechanical loading and the cracking behavior was addressed by evaluating the stress intensity factors as functions of various geometric and material parameters. In Part II, the solution framework is extended to the problem of thermal shock on the basis of uncoupled, quasi-static thermoelasticity. The interfacial zone, in this case, is assumed to have the graded thermoelastic properties. Using the principle of superposition, a system of singular integral equations is solved subjected to equivalent crack surface tractions obtained form the transient thermoelasticity solution for a uncracked medium. Main results presented are the transient thermal stress intensity factors of collinear cracks to illustrate the parametric effects of geometric and material combinations of the layered medium with the thermoelastically graded interfacial zone. This revised version was published online in July 2006 with corrections to the Cover Date.     

Periodically distributed parallel cracks in a functionally graded piezoelectric (FGP) strip bonded to a FGP substrate under static electromechanical load

In this paper, the Fourier integral transform–singular integral equation method is presented for the problem of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a different functionally graded piezoelectric material. The properties of two materials, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The crack surface condition is assumed to be electrically impermeable or permeable. The mixed boundary value problem is reduced to a singular integral equation over crack by applying the Fourier transform and the singular integral equation is solved numerically by using the Lobatto–Chebyshev integration technique. The analytic expressions of the stress intensity factors and the electric displacement intensity factors are derived. The effects of the loading parameter λ, material constants and the geometry parameters on the stress intensity factor, the energy release ratio and the energy density factor are studied.     

Crack Spacing Effect on the Brittle Fracture Characteristics of Semi-infinite Functionally Graded Materials with Periodic Edge Cracks

The effect of crack spacing on the brittle fracture characteristics of a semi-infinite functionally graded material (FGM) with periodic edge cracks is discussed. The incompatible eigenstrain induced in the material due to mismatch in the coefficients of thermal expansion is considered in the analysis. The nonhomogeneity of the material is simulated by an equivalent eigenstrain, whereby the problem is reduced to that of a cracked homogeneous material with incompatible and equivalent eigenstrains. A method is then formulated to calculate the stress intensity factor of periodic edge cracks in such a semi-infinite homogeneous medium and applied to calculate apparent fracture toughness of a semi-infinite FGM from its prescribed composition profile. Inverse calculation is also carried out to compute composition profile from prescribed apparent fracture toughness of the semi-infinite FGM. Numerical calculations are carried out for semi-infinite TiC/Al₂O₃ FGM and the results are shown in the figures.     

A periodic array of cracks in functional graded materials subjected to thermo-mechanical loading

A periodic array of cracks in an infinite functionally graded material under mechanical and/or thermal loading is investigated. Due to non-uniform heating or cooling, compressive stresses occur causing the crack surfaces to come into contact at a certain contact length. The mixed boundary value problem is reduced to a singular integral equation with the crack contact length as an additional unknown variable. Numerical results for stress intensity factors and the crack contact length are obtained as a function of crack spacing. Effect of the material non-homogeneity on the crack tip intensity factors is discussed. Some suggestions are made for the design of thermal resistive functionally graded materials.     

Thermoelastic interaction of two offset interfacial cracks in bonded dissimilar half-planes with a functionally graded interlayer

This paper deals with the thermoelasticity problem of bonded dissimilar half-planes with a functionally graded interlayer, weakened by a pair of two offset interfacial cracks. The material nonhomogeneity in the graded interlayer is represented by spatially varying thermoelastic moduli expressed in terms of exponential functions. The cracks are assumed to be thermally insulated disturbing a steady-state uniform heat flow, and the solution is obtained within the framework of linear plane thermoelasticity. The Fourier integral transform method is employed, and the formulation of the current nonisothermal crack problem is reduced to two sets of Cauchy-type singular integral equations for temperature and thermal stress fields in the bonded system. In the numerical results, parametric studies are conducted so that the variations in mixed-mode thermal stress intensity factors are presented as a function of offset crack distance for various geometric and material combinations of the dissimilar homogeneous media bonded through the thermoelastically graded interlayer, elaborating thermally induced singular interaction of the two neighboring interfacial cracks.     

Quantitative evaluation of the fatigue limit of a metal with an arbitrary crack under a stress controlled condition – Stress Ratio R=−1

The static crack problem of a functionally graded coating-substrate structure with an internal or edge crack perpendicular to the interface is investigated under an in-plane load. The structure is made up of a functionally graded coating with an internal or edge crack and a homogeneous substrate of finite thickness. The material properties are assumed to vary continuously from the coating to the substrate. By use of Fourier transform method, the mixed boundary value problem is reduced to a singular integral equation which can be solved numerically. During the analysis, a higher-order term is obtained in the asymptotic analysis of the singular kernel to improve the convergence efficiency of numerical integrals. The influences of material constants and the geometry parameters on the stress intensity factors (SIFs) are studied. In Part II of this paper, the transient response of the structure subjected to an in-plane impact is investigated.     

Determining the mixed mode stress intensity factors of surface cracks in functionally graded hollow cylinders

This work reports about an investigation on mixed mode stress intensity factors (SIFs) of three-dimensional (3D) surface cracks in hollow cylinders made-up of functionally graded material (FGM). A finite element implementation of the interaction energy integral in domain form is employed to extract the SIFs. In turn, surface cracks located at the inner and outer wall of the cylinders are considered, and the influence of exponentially varying Poisson’s ratio and Young’s modulus in radial direction on the SIFs is studied in detail. The computational results reported herein show that graded materials properties can significantly affect the magnitude and the distribution of SIFs along 3D crack fronts in FGM hollow cylinders.