Wave propagation in functionally graded materials by modified smoothed particle hydrodynamics (MSPH) method

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Abstract

We use the modified smoothed particle hydrodynamics (MSPH) method to study the propagation of elastic waves in functionally graded materials. An artificial viscosity is added to the hydrostatic pressure to control oscillations in the shock wave. Computed results agree well with the analytical solution of the problem. It is shown that, for the same placement of particles/nodes the MSPH method gives better results than the finite element method when the initial smoothing length in the MSPH method is 1.1 times the distance between two adjacent particles. Effects of the artificial viscosity are also examined, and the optimum value of the linear artificial viscosity that minimizes the relative error in computed stresses is found.

Introduction

Functionally graded structures are inhomogeneous bodies usually comprised of two constituents whose volume fractions vary continuously throughout the body so as to attain a specific variation of material moduli that minimizes a critical design variable, e.g. the maximum principal tensile stress or the maximum strain energy density, or the fundamental frequency of vibration. Lekhnitskii’s [1] book has solutions to many linear elastic problems for inhomogeneous materials. Batra’s [2] paper has an explicit solution and numerical results obtained by the finite element method (FEM) for the radial expansion of a circular Mooney–Rivlin cylinder with material moduli depending upon the radial coordinate. Interest in functionally graded materials (FGMs) seemed to originate with the International Symposium on FGMs [3]. Since the symposium, the literature on FGMs has exploded making a comprehensive review in the Section “Introduction” of a paper nearly impossible.

An advantage of FGMs is that the continuous variation of material properties eliminates sharp jumps in stresses that are likely to occur at interfaces between two distinct materials, as for example in laminated composites. This reduces the likelihood of two layers separating from each other. Also, under dynamic loading, there will be no wave reflections and refractions at such interfaces since material properties and the acoustic impedance vary continuously; this reduces wave dispersions. However, in the traditional FEM, material properties are evaluated at Gauss or quadrature points, and stresses are generally discontinuous across interelement boundaries unless one uses a mixed formulation and takes stresses also as unknowns; e.g. see [4]. Of course, these additional unknowns increase the number of variables and hence the number of equations to be solved. Even when values of material properties at nodes are given as a part of the input data and values at quadrature points interpolated from these nodal values, stresses across interelement boundaries are discontinuous. Batra and Love [5] by using the traditional FEM to analyze transient plane strain deformations of a FG linear elastic body have shown that with a fine mesh the computed wave speed, wave profile and the axial stress at the wave front match very well with their values derived from the analytical solution of Chiu and Erdogan [6]. Here we use a meshless method, namely the modified smoothed particle hydrodynamics (MSPH) method [7], to study a plane strain elastodynamic problem and delineate the effect of various parameters on the accuracy of the computed solution.

The Lagrangian SPH method, due to Lucy [8] and Gingold and Monaghan [9], has been used by Libersky and Petschek [10] for analyzing finite deformations of an elastoplastic body. There are two weaknesses in the traditional SPH method: the boundary deficiency in the sense that basis functions do not satisfy consistency condition near the boundaries and the tensile instability. Chen et al. [11], [12] proposed the corrective smoothed-particle method (CSPM) that removed the tensile instability deficiency of the SPH but not the boundary deficiency. However, the MSPH method is devoid of these two shortcomings of the SPH, and has been used to analyze two-dimensional transient heat conduction, wave propagation in a homogeneous linear elastic isotropic bar and the localization of deformation into a narrow zone of intense plastic deformation in thermoviscoplastic materials [13]. Sigalotti et al. [14] have proposed a technique to convert the standard SPH into a working shock capturing scheme without relying on solutions to the Riemann problem. Their modification allows solving problems with large discontinuities without using an artificial viscosity.

Wave propagation in FG elastic structures has numerically been studied by several investigators, e.g. see [15], [16], [17], [18]; additional references may be found in [6]. Free and forced vibrations of FG plates by a meshless method have been analyzed, among others, in [19], [20], [21], [22], [23], [24], [25], [26] and wave propagation in a segmented bar comprised of two materials by Batra and coworkers [27]. Here we study wave propagation, under uniaxial strain conditions, in a FG plate whose material properties vary continuously in the direction of wave propagation.

The rest of the paper is organized as follows. We review the MSPH method in Section 2, and formulate the wave propagation problem in an isotropic elastic body in Section 3. In Section 4 we compare the presently computed numerical solution with the analytical solution of Chiu and Erdogan [6], and delineate the effect on the accuracy of the computed solution of linear artificial viscosity, quadratic artificial viscosity, number of particles, smoothing length and the kernel function employed in the MSPH method. Findings of this work are summarized in Section 5 entitled conclusions.

Section snippets

The MSPH method

The Taylor series expansion of a function f(x) about the point x(i)=(x1(i),x2(i),x3(i)) isf(ξ)=f(x(i))+fxα(i)ξα-xα(i)+122fxα(i)xβ(i)ξα-xα(i)ξβ-xβ(i)+,where a repeated index implies summation over the range of the index, an index enclosed in parentheses is not summed and Greek indices α, β and γ range over 1, 2 and 3. Multiplying both sides of Eq. (2.1) with a kernel function W(x  ξ, h) of compact support 2h (i.e., W(x  ξ, h) = 0 for ∣x  ξ  2h, ∣x  ξ = distance between points x and ξ), its first

Formulation of the problem

In rectangular Cartesian coordinates, the conservation equations of mass and linear momentum aredρdt=-ρU,ββ,dUαdt=1ρσ,βαβ,where U is the velocity vector, σ the Cauchy stress tensor and t the time. Here we study a mechanical problem, and do not consider the energy equation.

Writing the stress tensor asσαβ=-Pδαβ+Sαβ,we assume that the hydrostatic pressure P is proportional to the compression ratio. That is,P=Kρρ0-1,where K is the bulk modulus and ρ0 the initial mass density. The Jaumann rate of

Numerical results

Values of various parameters used to compute results areE0=226.9GPa,ν=0.33,ρ0=8900kg/m3,a=0.3,m=3,n=1,l=5mm,CL=0.2,CQ=4.0,σ0=1.0GPa,t0=3μs.

Five hundred uniformly distributed particles are located along the length of the plate in the y-direction, and the initial smoothing length, h, is set equal to 1.1Δ where Δ is the distance between two adjacent particles.

Fig. 2(a)–(d) compares at four different times, 2 μs, 4 μs, 7.11 μs and 12 μs, the distribution of the axial stress σ with that obtained from

Conclusions

It has been shown that for wave propagation in a functionally graded elastic material, the MSPH method gives results very close to those obtained analytically. The artificial viscosity is considered to diminish oscillations near the shock front. Because the material considered is inhomogeneous, the wave speed and the magnitude of the shock front change as the wave propagates in the plate.

By introducing a criterion to determine the error in the computed results, we have examined in detail the

Acknowledgments

This work was partially supported by the Office of Naval Research grants N00014-98-1-3000 and N00014-06-1-0567 with Dr. Y.D.S. Rajapakse as the program manager. Views expressed herein are those of authors, and not of the funding agency.

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