Due to the high brittleness, cracks, holes, and other defects that are easily generated in quasi-crystal structures can affect safe applications in serious cases. For guided wave non-destructive testing, the propagation of Lamb and SH waves in functionally graded one-dimensional hexagonal quasi-crystal plates are investigated. Governing equations of wave motion in the context of Bak’s model are deduced and solved by the Legendre orthogonal polynomial method. Dispersion curves, phonon and phason displacement, and stress distributions are illustrated. The convergence of the present method applied to functionally graded quasi-crystal plates is verified. Moreover, the influences of the phonon-phason coupling effect and graded fields on wave characteristics are analyzed. Some new results are obtained: angular frequencies of phason modes always decrease as phonon-phason coupling coefficients, Ri, increase; and phonon and phason displacements of Lamb and SH waves at high frequencies are mainly distributed in the region that contains more quasi-crystal material with a smaller elasticity modulus and less rigidity. The obtained results establish the theoretical foundation of guided wave non-destructive testing for functionally graded quasi-crystal plates.

The elastoplastic behavior of a Functionally Graded Material (FGM) simply supported beam consisting of elastic material A and elastoplastic material B under uniformly distributed load is investigated. A power function is used to describe the volume fractions of the constituent materials, and the average stress of the FGM beam is obtained by using the averaging method. This method can avoid the assumption of the varying properties of the whole material, and can consider the different Possion’s ratios of the different constituent materials. What’s more, only the elastoplastic material B in the FGM beam will yield, and the yield function is determined by the stress of material B only, rather than the average stress of the whole material. The method used in this work is more closer to the real material than the method by assuming the variation of the whole properties of FGM. The theoretical results show a good agreement with the finite element results, which indicates that the method provided in this work is valid. With this method, the variation of the elastic and plastic areas, the stress distribution on the cross section, variation of the curvature and neutral layer, and the residual stress distribution of the FGM beam are discussed through numerical results. This work can provide a new way for the design and in-depth investigation of FGM material.

Propagation of harmonic Lamb waves in plates made of functionally graded materials (FGM) with transverse inhomogeneity is studied by combination of the Cauchy six-dimensional formalism and matrix exponential mapping. For arbitrary transverse inhomogeneity a closed form implicit solution for dispersion equation is derived and analyzed. Both the dispersion equation and the corresponding solution resemble ones obtained for stratified media. The dispersion equation and the corresponding solution are applicable to media with arbitrary elastic (monoclinic) anisotropy.

A theoretical model for the Functionally Graded Shape Memory Alloy (FG-SMA) cylinders subjected to internal pressure is investigated. The gradient properties in this work are embodied in the Young’s modulus and Poisson’s ratio gradient through the thickness of the cylinder. The critical transformation stresses and maximum formation strain are all assumed to be constant. Combining the elasticity and exponential function of the Young’s modulus and Poisson’s ratio with the different gradient parameters, the elastic stress distributions and displacement distributions for the FG-SMA cylinder under the internal pressure are obtained, respectively. To get the theoretical solution, the Tresca yield function and the ideal elastic–plastic constitutive model are selected for the shape memory alloy to illustrate the phase transformation. The relationships between the internal pressure and total strain at the internal radius with different gradient parameters are then given, and the results show that the total strains are greatly influenced by the different parameters.

Functionally graded material (FGM) has some particular characteristics due to the gradual variation of physical properties. The study on mechanical behavior of FGM is of great research value. In this work, a large scale FGM which filled with small glass spheres has been prepared by gravity assisted casting technique. The elastic material constants in static condition are measured. One optical experimental method, coherent gradient sensing (CGS), is introduced to study the mechanical behavior of FGM which has variation of material property in power-law. The governing equations of CGS which is used to represent the optics-mechanics relation of the singular field near the point of the outside force are derived based on the power-law asymptotic expansion. The experimental result shows this CGS method as a nondestructive methodology can be used to detect the damage in FGM with high accuracy.

Based on the Timoshenko beam model, the nonlinear vibration of microbeams made of functionally graded (FG) materials is investigated under different boundary conditions. To consider small scale effects, the model is developed based on the most general form of strain gradient elasticity. The nonlinear governing equations and boundary conditions are derived via Hamilton's principle and then discretized using the generalized differential quadrature technique. A pseudo-Galerkin approach is used to reduce the set of discretized governing equations into a time-varying set of ordinary differential equations of Duffing-type. The harmonic balance method in conjunction with the Newton-Raphson method is also applied so as to solve the problem in time domain. The effects of boundary conditions, length scale parameters, material gradient index and geometrical parameters are studied. It is found that the importance of the small length scale is affected by the type of boundary conditions and vibration mode. Also, it is revealed that the classical theory tends to underestimate the vibration amplitude and linear frequency of FG microbeams.

In this paper, a new displacement based high-order shear deformation theory is introduced for the static response of functionally graded plate. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory presented is variationally consistent, has strong similarity with classical plate theory in many aspects, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. Numerical illustrations concerned flexural behavior of FG plates with Metal-Ceramic composition. Parametric studies are performed for varying ceramic volume fraction, volume fraction profiles, aspect ratios and length to thickness ratios. The validity of the present theory is investigated by comparing some of the present results with those of the classical, the first-order and the other higher-order theories. It can be concluded that the proposed theory is accurate and simple in solving the static behavior of functionally graded plates.

This paper is concerned with two-dimensional (r, θ) thermoelastic dynamic responses of a long functionally graded hollow cylinder subjected to asysmmetrical thermal and mechanical loads. The material properties, except the Poisson's ratio, are assumed to be temperature independent and vary exponentially and continuously in the radial direction. By means of finite difference method and Newmark method, the motion governing equations of the long FGM hollow cylinder are solved. Comparisons between this paper's results and the corresponding analytical results validate the proposed solution. In addition, the effects of the volume fraction, temperature boundary conditions on the hollow cylinder's deformations and stresses distributions are examined, and many other valuable thermoelastic dynamic characteristics are revealed.

This paper is concerned with an axially symmetric frictionless contact between an elastically transversely isptropic functionally graded half-space and a rigid base that has a small axisymmetric surface recess. The graded half-space is modeled as a nonhomogeneous medium. We reduce the problem to solving Fredholm integral equations, solve these equations numerically and establish a relationship between the applied pressure and gap radius. The effects of anisotropy and nonhomogeneity parameter of the graded half-space on the normal pressure as well as on the critical pressure have been shown graphically.