Dynamic response of a crack in a functionally graded interface of two dissimilar piezoelectric half-planes

Summary  In this paper, the dynamic anti-plane crack problem of two dissimilar homogeneous piezoelectric materials bonded through a functionally graded interfacial region is considered. Integral transforms are employed to reduce the problem to Cauchy singular integral equations. Numerical results illustrate the effect of the loading combination parameter λ, material property distribution and crack configuration on the dynamic stress and electric displacement intensity factors. It is found that the presence of the dynamic electric field could impede of enhance the crack propagation depending on the time elapsed and the direction of applied electric impact. Received 4 December 2001; accepted for publication 9 July 2002 This work is supported by the National Natural Science Foundation of China through Grant No. 10132010.     

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).     

Exact analytical solution for free vibration of functionally graded micro/nanoplates via three-dimensional nonlocal elasticity

Using three-dimensional (3-D) nonlocal elasticity theory of Eringen, this paper presents closed-form solutions for in-plane and out-of-plane free vibration of simply supported functionally graded (FG) rectangular micro/nanoplates. Elasticity modulus and mass density of FG material are assumed to vary exponentially through the thickness of micro/nanoplate, whereas Poisson's ratio is considered to be constant. By employing appropriate displacement fields for the in-plane and out-of-plane modes that satisfy boundary conditions of the plate, ordinary differential equations of free vibration are obtained. Boundary conditions on the lateral surfaces are imposed on the analytical solutions of the equations to yield the natural frequencies of FG micro/nanoplate. The natural frequencies of FG micro/nanoplate are obtained for different values of nonlocal parameter and gradient index of material properties. The results of this investigation can be used as a benchmark for the future numerical, semi-analytical and analytical studies on the free vibration of FG micro/nanoplates.     

Stochastic heat conduction analysis of a functionally graded annular disc with spatially random heat transfer coefficients

The mean and variance of the temperature are analytically obtained in a functionally graded annular disc with spatially random heat transfer coefficients (HTCs) on the upper and lower surfaces. This annular disc has arbitrary variations in the HTCs (i.e., arbitrary thermal interaction with the surroundings) and gradient material composition only along the radial direction and is subjected to deterministic axisymmetrical heating at the lateral surfaces. The stochastic temperature field is analysed by considering the annular disc to be multilayered with spatially constant material properties and spatially constant but random HTCs in each layer. A type of integral transform method and a perturbation method are employed in order to obtain the analytical solutions for the statistics. The correlation coefficients of the random HTCs are expressed in the form of a linear function with respect to the radial distance as a non-homogeneous random field of discrete space. Numerical calculations are performed for functionally graded annular discs composed of stainless steel and ceramic, which comprise two types of material composition distributions. The effects of the magnitude of the means of HTCs, volume fraction distributions of the constitutive materials and correlation strengths of the HTCs on the standard deviation of the temperature are discussed.     

Large deflections of tapered functionally graded beams subjected to end forces

The large deflections of tapered functionally graded beams subjected to end forces are studied by using the finite element method. The material properties of the beams are assumed to vary through the thickness direction according to a power law distribution. A first order shear deformable beam element employed the exact polynomials to interpolate the transverse displacement and rotation, is formulated in the context of the co-rotational approach. The large deflection response of the beams is computed by using the arc-length control algorithm in combination with the Newton–Raphson iterative method. The numerical results show that the formulated element is capable to assess accurately the response of the beams by using just several elements. A parametric study is given to examine the influence of the material non-homogeneity, taper ratio as well as the aspect ratio on the large deflection behaviour of the beams.     

Long-wave asymptotic theories: The connection between functionally graded waveguides and periodic media

This article explores the deep connections that exist between the mathematical representations of dynamic phenomena in functionally graded waveguides and those in periodic media. These connections are at their most obvious for low-frequency and long-wave asymptotics where well established theories hold. However, there is also a complementary limit of high-frequency long-wave asymptotics corresponding to various features that arise near cut-off frequencies in waveguides, including trapped modes. Simultaneously, periodic media exhibit standing wave frequencies, and the long-wave asymptotics near these frequencies characterise localised defect modes along with other high-frequency phenomena. The physics associated with waveguides and periodic media are, at first sight, apparently quite different, however the final equations that distill the essential physics are virtually identical. The connection is illustrated by the comparative study of a periodic string and a functionally graded acoustic waveguide.     

Effects of initial stress on guided waves in orthotropic functionally graded plates

Because of the limitation of the manufacturing technology, initial stress in functionally graded materials (FGM) and structures is inevitable. Based on the theory of “Mechanics of Incremental Deformations”, the guided wave propagation in FGM plates under gravity, homogeneous initial stress in the thickness direction and inhomogeneous initial stress in the wave propagation direction is investigated. The Legendre polynomial series method is used to solve the coupled wave equations with variable coefficients. The convergence of the polynomial series method is discussed through the numerical examples. The effects of the initial stress on the Lamb-like wave and on the SH wave are investigated respectively and the numerical results show they are quite distinct. The effect of the gravity on the wave propagation can be ignored. The effects of the initial stress in the thickness direction are very different from those of the initial stress in the wave propagation direction, both on the dispersion curves and on the displacement and stress distributions.     

Single-step laser deposition of functionally graded coating by dual ‘wire-powder’ or ‘powder-powder’ feeding—A comparative study

The creation of iron-copper (Fe-Cu) alloys has practical application in improving the surface heat conduction and corrosion resistance of, for example, conformal cooling channels in steel moulds, but is difficult to achieve because the elements have got low inter-solubility and are prone to solidification cracking. Previous work by these authors has reported a method to produce a graded iron-nickel-copper coating in a single-step by direct diode laser deposition (DLD) of nickel wire and copper powder as a combined feedstock. This work investigates whether dual powder feeds can be used in that process to afford greater geometric flexibility and compares attributes of the ‘nickel wire and copper powder’ and ‘nickel powder and copper powder’ processes for deposition on a H13 tool steel substrate.In wire-powder deposition, a higher temperature developed in the melt pool causing a clad with a smooth gradient structure. The nickel powder in powder-powder deposition did not impart much heat into the melt pool so the melt pool solidified with sharp composition boundaries due to single metal melting in some parts. In wire-powder experiments, a graded structure was obtained by varying the flow rates of wire and powder. However, a graded structure was not realised in powder-powder experiments by varying either the feed or the directions. Reasons for the differences and flow patterns in the melt pools and their effect on final part properties of parts produced are discussed.     

Approximation of functionally graded plates with non-conforming finite elements

In this paper rectangular plates made of functionally graded materials (FGMs) are studied. A two-constituent material distribution through the thickness is considered, varying with a simple power rule of mixture. The equations governing the FGM plates are determined using a variational formulation arising from the Reissner–Mindlin theory. To approximate the problem a simple locking-free Discontinuous Galerkin finite element of non-conforming type is used, choosing a piecewise linear non-conforming approximation for both rotations and transversal displacement. Several numerical simulations are carried out in order to show the capability of the proposed element to capture the properties of plates of various gradings, subjected to thermo-mechanical loads.     

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.