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.     

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.     

An interface crack in a functionally graded piezoelectric bi-layer under anti-plane shear impact

The transient response of an interface crack between two dissimilar functionally graded piezoelectric material (FGPM) layers under anti-plane shear impact loading is analyzed using the integral transform method. The properties of the FGPM layers vary continuously along the thickness, and the two layers are connected weak-discontinuously. Laplace transform and Fourier transform are used to reduce the problem to two sets of dual integral equations, which are then expressed to the Fredholm integral equations of the second kind. Numerical values on the dynamic energy release rate are presented for the FGPM to show the effects on the electric loading, variation and gradient of material properties, and thickness of layers. Following things are helpful to increase the resistance of transient fracture of interface crack in FGPMs: (a) increase of the material properties from the interface to the upper or lower free surface; (b) decrease of weak discontinuity at the interface; (c) increase of the gradient of material properties; (d) certain direction and magnitude of the electric loading; and (e) increase of the thickness of the FGPM layer.     

On the shear stress distribution between a functionally graded piezoelectric actuator and an elastic substrate and the reduction of its concentration

Recent advances in material processing technologies allow the production of piezoelectric materials with functionally graded material properties. We investigate the implications of functionally graded piezoelectric materials when used as actuators for structural control by examining the distribution of the actuating shear stress under a piezoelectric actuator of a functionally graded material (FGM) on an isotropic elastic half-space. It is shown that FGM materials can be used to adjust the shear stress distribution. In particular, the concentration near the edges of a conventional homogeneous piezoelectric actuator can be significantly reduced in an FGM actuator.     

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.     

Interface waves in functionally graded piezoelectric materials

We analyze the propagation of antiplane or shear-horizontal waves near the interface between two half-spaces of piezoelectric ceramics. The material properties vary in the direction perpendicular to the interface. Both electroded and unelectroded interfaces are considered. Transcendental equations that determine the dispersion relations of the waves are obtained. They reduce to a few known results in the literature as special cases. Different from similar waves in homogeneous materials, the waves obtained are dispersive. The equations for the dispersion relations are solved numerically. It is found that the wave speeds are sensitive to the variation of material properties. This suggests the possibility of manipulating the wave propagation behavior through proper design of materials.     

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.     

Dynamic stress from a subsurface cylindrical inclusion in a functionally graded material layer under anti-plane shear waves

The multiple scattering of shear waves and dynamic stress resulting from a subsurface cylindrical inclusion in a functionally graded material (FGM) layer bonded to homogeneous materials are investigated, and the analytical solution of this problem is derived. Image method is used to satisfy the traction free boundary condition of the FGM layer. The analytical solutions of wave fields around the actual and image inclusions are expressed by employing wave functions expansion method, and the expanded mode coefficients are determined by satisfying the continuous boundary conditions around the inclusions. Through the numerical solutions of dynamic stress concentration factors (DSCFs) around the inclusion, the effects of the position of the inclusion in the material layer, the properties of the inclusion, and the properties of the two phases of composites on the DSCFs are analyzed. Analyses show that when the cylindrical inclusion is stiffer than the two phases of FGMs, the dynamic stress around the inclusion increases greatly. When the distance between the surface of the structure and the inclusion is smaller, the effect of the properties of the inclusion becomes greater. When the cylindrical inclusion is softer than the two phases of FGMs, the maximum dynamic stress shows little difference; however, the variation of the distribution of the dynamic stress around the inclusion is greater.     

Preparation and characterization of bioactive monolayer and functionally graded coatings

Hydroxyapatite powders were made by reacting orthophosphoric acid with calcium hydroxide and dense bioactive coatings were subsequently produced by the plasma spray technique. Three types of hydroxyapatite (flame spheroidized) monolayer coatings and three types of functionally graded coatings were manufactured. It was found that average microhardness values of monolayer coatings decreased as the indentation load increased. The relationship between indentation load and indent diagonal length observed Meyer's law. Microhardness and fracture toughness of coatings were affected by characteristics of feedstock powders for plasma spraying. The indentation fracture toughness of coatings could be significantly increased by incorporating a toughening phase. ©1999 Kluwer Academic Publishers     

Analytical solutions for an antiplane problem of two dissimilar functionally graded magnetoelectroelastic half-planes

In this study, the two-dimensional analysis of two bonded dissimilar half-planes for functionally graded magnetoelectroelastic materials subjected to generalized line forces and screw dislocations is presented. The variations of the material properties are assumed to be exponential forms. By using the Fourier-transform technique, Green’s function for the transversely isotropic magnetoelectroelastic nonhomogeneous bimaterial is obtained from the jump and interface continuity conditions. Numerical results for the full-field distributions of the stresses, electric fields, and magnetic fields with different functionally graded parameters are presented. It is noted that the elastic, electric, and magnetic fields are all continuous at the interface in the nonhomogeneous bimaterial if the material constants at the interface are continuous. Furthermore, this functionally graded bimaterial has identical contour slopes for the generalized stresses across the interface.     

A novel technique for processing functionally graded HA coatings

Hydroxyapatite (HA) films were deposited using dual ion beam sputtering. Deposition was carried out with an in situ heat treatment at three temperature settings during deposition. X-ray diffraction of the films at the surface revealed that the deposited film is composed of hydroxyapatite crystalline and amorphous phases. Cross-sectional transmission electron microscopy analysis displayed that the films have a graded crystal structure with the crystalline layer near the substrate and the amorphous layer at the top surface. Compositional analysis was performed using SEM-EDX at the top surface as well as STEM-EDX at the cross-section of the film. The average calcium to phosphorous ratio at the surface is 1.46, obtained by SEM-EDX. The Ca/P ratios in the crystalline and amorphous layers of the film are 1.6 to 1.7, close to the ratio of 1.67 for HA.     

Design of functionally graded carbon coatings against contact damage

Three different functionally graded amorphous carbon (a-C) thin films were deposited on to aluminium substrates using a closed-field unbalanced magnetron sputtering ion plating method. The closed-field configuration prohibits the loss of secondary electrons and consequently enhances the plasma density significantly. The functional gradient of the a-C films was achieved by varying the bias voltage linearly during deposition. Three graded a-C systems possessing different variations in Young's modulus were deposited with the highest Young's modulus at the (i) top surface, (ii) interface or (iii) middle of the film. Of the three systems investigated, the one with the highest Young's modulus at the middle of the film thickness was found to exhibit significantly lower levels of cracking at higher indentation depths. Finite element models that included an embedded ring crack controlled by cohesive zone elements were developed to clarify the effect of ring cracks on the deformation of the films. This study provides guidance for the design of functionally graded coatings against contact damage.     

Buckling of functionally graded circular columns including shear deformation

An analysis of the stability of circular cylindrical columns/beams composed of functionally graded materials is made where shear deformation is taken into account. In this study, a new approach is carried out. Different from the assumption of uniform shear stress at the cross-section adopted in the Timoshenko beam theory, proposed model provides a new approach for treating the problem. Based on the traction-free surface condition, two coupled governing equations for the deflection and rotation are derived, and a single governing equation is further obtained. A comparison of buckling loads derived from the proposed circular column model and the Timoshenko and Euler–Bernoulli theories of beams is made. Moreover, the effects of radial gradient on buckling loads of elastic columns with circular cross-section made of functionally graded materials are elucidated. Finally, the stability of double-walled carbon nanotubes is considered and critical stress is determined and compared with existing results. The results obtained by the proposed model show very good agreement with the results of the Timoshenko beam theory or Reddy–Bickford beam theory.     

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.     

The interface crack problem under a concentrated load for a functionally graded coating–substrate composite system

The plane crack problem for a functionally graded coating–substrate system under a concentrated load is studied in this paper. The medium consists of a functionally graded coating bonded to a homogeneous substrate of finite thickness, containing an interface crack of finite length. With use of the integration transform and differential factor methods, the displacement form can be obtained. By introducing auxiliary functions, the present problem can be turned into solving a group of singular integral equations. The mixed-mode stress intensity factors (SIFs) and strain energy release rates (SERRs) are obtained. The influences of the parameters such as the load location, nonhomogeneity constants and the geometry parameters on the SIFs and SERRs are studied.     

Design of functionally graded thermal barrier coatings based on a nonlinear micromechanical approach

This study presents the methodology of analysis and design of functionally graded thermal barrier coatings (FG TBCs) consisting of ceramics and metals based on a mean-field micromechanical approach taking into account the time-independent and dependent inelastic deformation, such as plasticity of metals, creep of metals and ceramics, and diffusional mass flow at the ceramic/metal interface. Ni–ZrO₂ systems were numerically studied and the effect of the compositional gradation patterns and time-dependent inelastic deformation on the micro- and macro-stress states in the FG TBCs were examined. The suitable compositional gradation patterns were proposed for typical thermo-mechanical boundary conditions. It was shown that the creep deformation near the ceramic surface can introduce large tensile stresses, potentially leading to thermal fracture, which is possibly reduced by controlling the ability of creep of the ceramics.     

Elastic response of functionally graded circular plates under a drop-weight

The three-dimensional impact behaviours of functionally graded (FG) circular plates were studied under a drop-weight. The functionally graded circular plate was composed of ceramic (SiC) and metal (Al) phases and the through-thickness mechanical properties through the region between the metal and ceramic layers vary continuously according to a power-law distribution of the volume fraction of the ceramic. The through-thickness material properties of the FG circular plate were determined using the Mori–Tanaka scheme. The effects of layer number and compositional gradient exponent as well as impactor velocity and plate radius on the elastic impact response of the FG circular plates were investigated. The compositional gradient exponent, impactor velocity and plate radius played an important role on the impact response of the FG circular plates, whereas the layer number through the plate thickness had a minor effect. In addition, the failure strains in all layers were determined using Tamura–Tomota–Ozowa (TTO) model in order to predict the damage regions in each layer through the thickness of FG circular plates.     

Theoretical analysis of a functionally graded shape memory alloy plate under graded temperature loading

As a new kind of functional material, functionally graded shape memory alloy (FG-SMA) possesses the excellent properties of both shape memory alloy (SMA) and functionally graded material (FGM). By combining the heat conduction theory with the theory of the mechanics of composite materials, a macro constitutive model, which can be used to describe the mechanical behavior of FG-SMA under graded temperature loading, is established. With this macro constitutive model, the thermo-mechanical properties of an FG-SMA plate, which is composed by ceramic and SMA and subjected to different surface temperature loads, are investigated. The theoretical results show good agreement with the existing data, which indicates that the macro constitutive model provided here is valid. The obtained results show that the martensite transformation does not always happen first at the top or bottom of the plate, and it is dependent on the surface temperatures. It can also be found that the stress decreases markedly due to the martensite transformation. This research can provide a basis for the design and in-depth investigation of FG-SMA material.     

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.     

The time-dependent interaction between inclusions and a cracked matrix subjected to antiplane shear

Summary. The time-dependent interaction between multiple circular inclusions and a cracked matrix in the antiplane viscoelastic problem is discussed in this paper. The fundamental elastic solution is obtained as a rapidly convergent series in terms of complex potentials via successive iterations of Möbius transformation in order to satisfy continuity conditions on multiple interfaces. Based on the correspondence principle, the Laplace transformed viscoelastic solution is then directly determined from the corresponding elastic one. In association with the singular integral technique, the time-dependent mode-III stress intensity factor of the crack tip can be solved numerically in a straightforward manner. Finally, some typical examples of an arbitrary crack lying in a matrix with various material properties under various loading types are also discussed. The results show that, depending on the relative locations and material properties of inclusions, the evolution of the stress intensity factor (SIF) may increase or decrease with time.