X. Han

TRANSIENT WAVES IN A FUNCTIONALLY GRADED CYLINDER

A hybrid numerical method (HNM) is presented for analyzing transient waves in a cylinder made of functionally graded material (FGM). In the HNM, the FGM cylinder is divided into N cylindrical elements with three-nodal line in the wall thickness. The elemental material properties are assumed to vary linearly in the thickness direction to better model the spatial variation of material properties of FGM. The Hamilton variational principle is used to develop governing equations of the cylinder. The displacement responses are determined by employing the Fourier transformations together with the modal analysis. The HNM is applied to analyze a number of FGM cylinders, and its efficiency is demonstrated.

Stress waves in functionally gradient materials and its use for material characterization

A method is presented to investigate elastic waves in functionally gradient material (FGM) plates excited by plane pressure wavelets. The FGM plate was first divided into linearly inhomogeneous elements (LIEs). A general solution for the equation of motion governing the LIE was derived. The general solution was then used together with the boundary and continuity conditions to obtain the displacement and stress in the frequency domain for an arbitrary FGM plate. The response of the plate to a pressure wavelet was obtained using Fourier transform techniques. Results obtained by the present method are compared with an existing method using homogeneous layer elements. Relationships between the surface displacement response and the material mechanical properties of FGM plates were also obtained. These relationships may be used for the material characterization of FGM plates.

Elastic waves in a functionally graded piezoelectric cylinder

An analytical–numerical method is presented for analyzing dispersion and characteristic surface of waves in a circular cylinder composed of functionally graded piezoelectric material (FGPM). In this method, the FGPM cylinder is divided into a number of annular elements with three-nodal lines in the wall thickness. The elemental mechanical as well as electrical properties are assumed to vary linearly in the thickness direction so as to better model the spatial variation of the mechanical and electrical properties of FGPM. The associated frequency dispersion equation is developed and the phase velocity and slowness as well as the group velocity and slowness are established in terms of the Rayleigh quotient. Six characteristic wave surfaces are introduced to visualize the effects of anisotropy and piezoelectricity on wave propagation. The calculation examples provide a full understanding of the complex phenomena of elastic waves in FGPM cylinders.

A combined genetic algorithm and nonlinear least squares method for material characterization using elastic waves

Abstract The inverse problem of material characterization is formulated as a parameter identification problem in which a set of parameters corresponding to the material property can be found by minimizing error functions formulated using the measured displacement response and the one computed by a forward solver based on projected candidates of parameters. A hybrid numerical method is employed as the forward solver to calculate the dynamic displacement response on the surface of the composite plate for given material property. A combined method is used as the inverse operator to determine the material property of composite plate. In this method, genetic algorithm is first used to select a set of better solutions close to the optima; then the nonlinear least squares method is applied using these better solutions as the initial guesses. Finally, the identification results can be determined from the solutions of nonlinear least squares method by comparing their corresponding error function values. Actual material characterizations of composites demonstrate the higher efficiency of the present method.

Material characterization of functionally graded material by means of elastic waves and a progressive-learning neural network

Abstract In this paper, a procedure is suggested for characterizing the material properties of functionally graded material (FGM) plate by the use of a modified hybrid numerical method (HNM) and a neural network (NN). The modified HNM is used to calculate the displacement responses of FGM plate to an incident wave for a known material property. The NN model is trained by using the results from the modified HNM. Once trained by, the NN model can be used for on-line characterization of material properties if the dynamic displacement responses on the surface of the FGM plate can be obtained. The material property so characterized is then used in the modified HNM to calculate the displacement responses. The NN model would go through a progressive retraining process until the calculated displacement responses obtained by using the characterized result is sufficiently close to the actual responses. This procedure is examined for two sets of material properties of a SiC–C FGM plate. It is found that the present procedure is very robust for determining material property distributions in the thickness direction of FGM plates.

Dispersion of waves and characteristic wave surfaces in functionally graded piezoelectric plates

An inhomogeneous layer element method is presented to analyze the dispersion of waves and characteristic wave surfaces in plates of functionally graded piezoelectric material (FGPM). In this method, the FGPM plate is divided into a number of layered elements. The elemental elastic and electric properties are assumed as linear functions of the thickness to adopt the variety of the material property of FGPM. The Hamilton principle is applied to determine the governing equations. The phase velocity surface, phase slowness surface, phase wave surface, group velocity surface, group slowness surface, and group wave surface for FGPM plate are formulated using Rayleigh quotient and the orthogonality condition of the eigenvectors. These six surfaces are then used to illustrate the characteristics of waves in FGPM plates. Numerical examples are presented using the present formulations to analyze dispersions and characteristics of waves in FGPM plates.

Effects of SH waves in a functionally graded plate

Abstract A computational method is presented to investigate SH waves in functionally graded material (FGM) plates. The FGM plate is first divided into quadratic layer elements (QLEs), in which the material properties are assumed as a quadratic function in the thickness direction. A general solution for the equation of motion governing the QLE has been derived. The general solution is then used together with the boundary and continuity conditions to obtain the displacement and stress in the wave number domain for an arbitrary FGM plate. The displacements and stresses in the frequency domain and time domain are obtained using inverse Fourier integration. Furthermore, a simple integral technique is also proposed for evaluating modified Bessel functions with complex valued order. Numerical examples are presented to demonstrate this numerical technique for SH waves propagating in FGM plates.

An inverse procedure for determination of material constants of composite laminates using elastic waves

A procedure is proposed to determine the material constants of composite laminates from dynamic displacement responses (wave response) obtained at only one receiving point of laminate surfaces. A hybrid numerical method is used for forward computation that relates the material constants to the displacement responses. A uniform micro-genetic algorithm is adopted to develop an inverse procedure and its effectiveness to the present type of applications is shown. The robustness of the procedure to the effect of measurement noise is investigated by adding Gauss noise to the input displacement response. Numerical simulations are performed on laminates with different composite materials, layer orientations, layer number and thickness, the satisfactory identification results for all cases demonstrate the effectiveness of the present procedure, as well as its robustness to the noises effects.

Material characterization of FGM plates using elastic waves and an inverse procedure

A computational inverse procedure is presented for characterization of the material properties of functionally graded materials (FGMs) using the surface displacement response of the plate. Amodified hybrid numerical method is first developed combining some existing techniques to compute the wave field in an FGM plate for given material properties and their variation in the thickness direction. The modified HNM allows a linear variation of material properties in the element in the thickness direction. This is to reduce the number of elements needed to model the material variation of FGM. The modified HNM is proven more efficient than the original HNM. The high efficiency paves the way for inverse procedure. The reconstruction of the material properties of FGM plates is performed using an inverse procedure of nonlinear least squares method. Numerical examples are presented to demonstrate the present procedure for material characterization of FGM plates. The present procedure can recover accurately the material properties of an FGM with an initial guess of up to 40% off from their true values.

A computational inverse technique for material characterization of a functionally graded cylinder using a progressive neural network

Abstract A computational inverse technique of neural network (NN) by means of elastic waves to material characterization of functionally graded material (FGM) cylinder is presented. The displacement responses on the outer surface are used as the inputs for the NN model. The outputs of the NN are the material property of FGM cylinder. The analytical–numerical method is used as the forward solver to calculate the displacement responses of FGM cylinder to an incident wave for the known material property. The NN model is trained using the results from the forward solver. Once trained by, the NN model can be used for on-line characterization of material property if the dynamic displacement responses on the outer surface of the cylinder can be obtained. The characterized material property is then used to calculate the displacement responses. The NN model would go through a progressive retraining process until the calculated displacement responses using the characterized result are sufficiently close to the actual responses. This procedure is examined for material characterization of an actual FGM cylinder composed of stainless steel and silicon nitride. It is found that the present procedure is very robust for determining the material property distribution in the thickness direction of FGM cylinders.