# A numerical boundary integral equation analysis for standard linear viscoelastic media made of functionally graded materials

- Hossein Ashrafi
^{1}Email author and - Mohammad Shariyat
^{2}

**9**:9

**DOI: **10.1186/s40712-014-0009-4

© Ashrafi et al.; licensee Springer 2014

**Received: **13 April 2014

**Accepted: **7 July 2014

**Published: **3 October 2014

## Abstract

### Background

Although structures made of functionally graded materials have been studied by many researchers, no research may be found in literature on boundary element analysis of the functionally graded viscoelastic structures.

### Methods

In the present paper, a 2D boundary element formulation capable of modeling time-dependent functionally graded materials (FGM) is presented. A numerical implementation of the Somigliana identity in terms of the displacements is developed to solve 2D problems of the exponentially graded viscoelasticity. The FGM concept can be applied to various materials, for structural and functional purposes. In this model, only Green functions of the nonhomogeneous elastostatic problems are needed with material properties that vary continuously along a given dimension.

### Results and Conclusions

Results reveal that the boundary element approach can successfully be employed for the present complicated problem for arbitrary time histories of the applied loads and arbitrary boundary conditions, without the need to use relaxation functions or mathematical transformations.

### Keywords

Boundary integral equation formulation Viscoelastic FGMs Nonhomogeneity Time-dependent materials## Background

In recent years, the functionally graded materials (FGMs), as a category of the composite materials, have generated a great deal of attention. An FGM is an advanced material whose composition changes gradually and results in corresponding changes in its properties (Suresh and Mortensen 1998). Because of the special features of the FGMs with potential applications to lots of engineering fields, they have attracted attention of numerous scientists and engineers in broad areas of research. Moreover, a variety of nonhomogeneous engineering media made of polymers, plastics, metals and alloys at elevated temperatures, composites, concrete, etc., exhibit significant rate and history dependencies. Appropriate simulation of these types of structures requires using appropriate viscoelastic models.

Boundary element method (BEM) has recently found considerable applications in solving lots of engineering problems, such as viscoelasticity, contact mechanics, elastoplasticity, thermoelasticity, fracture mechanics, elastodynamics, etc. (Aliabadi 2002). The viscoelastic media can effectively and accurately be treated by the BEM (Ashrafi and Farid 2009; Ashrafi et al. 2012). The BEM just requires the boundary data as input, and there is no need for discretizing the domain under consideration into elements. Adaptation of the BEM to nonhomogeneous media is a hard task; because determination of the fundamental solutions corresponding to the concentrated loads is difficult for such materials. The fundamental solutions for heat transfer problems in the nonhomogeneous media have been presented using BEM algorithms by some researchers (Shaw and Makris 1992; Clements 1998; Clements and Budhi 1999; Gray et al. 2003; Sutradhar and Paulino 2004; Kuo and Chen 2005). Also, the fundamental solutions of the FGMs for 2D and 3D elasticity problems have been recently developed in some other works (Chan et al. 2004; Criado et al. 2007, 2008). Using a Fourier transform technique, these functions have been derived for exponentially graded media. Although extensive works have been developed to effectively model the constitutive behavior of the FG structures, most of these researches have been limited to elastic behavior analysis (Gao et al. 2008; Wang and Qin 2012; Ashrafi et al. 2013a, 2013b).

## Methods

In this paper, the boundary element formulation is proposed based on the differential viscoelastic constitutive equations of nonhomogeneous SLS model. The resulting algorithm is capable of solving the quasistatic problems for exponentially graded viscoelastic materials with arbitrary boundary conditions and therefore, provides a reasonable model for certain realistic situations. Avoiding internal domain elements is one of the objectives of this paper, which results in numerical discretization of the boundary of the considered nonhomogeneous problem, only. Therefore, this work reduces the number of variables to be computed, which makes numerical treatment of the infinite and semi-infinite time-dependent problems easy.

### Constitutive equations

*σ*is the total stress and

*σ*

^{ e2 }and

*σ*

^{ K }are the elastic and the Kelvin viscoelastic parts of the SLS, respectively. The total strain component can also be divided into two parts as

*i*and

*j*. Also, the Kelvin viscoelastic stress components can be expressed as

*σ*

^{ v }and

*σ*

^{el}are the viscous and elastic parts of the stresses developed in the Kelvin unit of the SLS model, respectively.

*C*

_{ ijkm }are the nonhomogeneous isotropic elasticity tensor, which is a function of the coordinate vector x. The most general form of the fourth-order isotropic tensor

*C*

_{ ijkm }can be shown to have the following form

*λ*and

*μ*are Lame’s constants, given by

*E*and

*ν*are Young's modulus and Poisson's ratio, respectively. The exponential material gradation of Lame's constants in the

*x-*direction is chosen as

*γ*is the gradient parameter. A material with exponentially gradation has been widely used in the literature, since such a graded composition represents a justifiable model for certain real situations of the time-dependent nonhomogeneous materials (Jin 2006).

*K*

_{ ijmn }represents the nonhomogeneous, isotropic viscosity characteristic tensor of the material that is a function of the spatial variable

*x*and can be defined as

*β*

_{ μ }and

*β*

_{ λ }are the hydrostatic and the deviatoric viscosity coefficients of the model, respectively.

*β*

_{ μ }=

*β*

_{ λ }. Thus, the general viscoelastic equation of the SLS model can be obtained as

### Boundary integral equations

*B*

_{ j }is a body force acting in the

*j*direction and

*ψ*

_{ i }is the fundamental solution which represents effect of a unit concentrated load applied at a point of an infinite domain. After using the fundamental solution in the corresponding Green's function, one can reduce it to the BEM. The direct integral equation requires a displacement fundamental solution. The fundamental solution for 2D elastic problems in exponentially graded structures has been recently derived as (Chan et al. 2004)

*K*

_{0}and

*K*

_{1}are the modified Bessel functions.

*∂D*is the boundary of the problem and

*n*

_{ j }is the outward normal vector; we have

*e*

_{ i }corresponds to a unit positive force in the

*i*direction applied at

*z*point, and Δ (

*x, z*) is the Dirac delta function, in wherein

*z*and

*x*represent the field and the source points, respectively. Equation 18, by performing some mathematical manipulations, can be rewritten as

*Є*

_{ ji }) is exactly the well-known one of the in elastostatic formulations. Equation 20 may be considered as the integral constitutive equation of time-dependent functionally graded structures modeled properly by the SLS model.

The domain integral of the body forces can easily be transformed into an equivalent boundary integral equation, which results in equation in terms of the boundary values only; however, for more simplicity, we neglect it. In a similar manner, the boundary integral equations of the stresses can be derived.

*∂D*into

*N*

_{e}elements; so that after choosing identical numbers of the source points and the nodes, and calculating all integrals, the BEM discretization equations may be derived in a matrix form as follows

It should be noted in the, for solving the above time-dependent differential equation, it was necessary to approximate the displacement and the traction rates in the time domain by a time marching treatment. Finally, the presented algorithm has been cast into a unique program and then solved by the MATLAB software.

## Results and discussion

For evaluating the accuracy and the efficiency of the proposed approach, a numerical viscoelastic problem wherein the material properties are assumed to be exponential functions of the Cartesian coordinate *x*, is considered.

*x*-direction and two circular cutouts as shown in Figure 1 is considered. The considered plate is subjected to a uniform traction

*P*in the

*x*-direction at the edge

*x = a*, where

The opposite end of the considered plate is fixed, while its other boundaries are free of tractions. It is assumed that the plate is sufficiently thin such that a plane stress condition holds. In this problem, Boltzmann's solid model is employed to formulate the differential constitutive equations of the heterogeneous viscoelastic structures.

This problem is solved using the proposed method, adopting the geometric and material parameters as *a =* 2 m, *b =* 1 m, *c =* 1 m, *γ =* 0.25, 1, 2 and 3, *λ*_{
0
} *=* 5 GPa, *μ*_{
0
} *=* 350 MPa, *υ =* 0.4 and *β* = 24. The diameter of the circular cutouts is *d =* 0.35 m. Due to using the present-graded boundary integral equation approach, only about 54 boundary elements have been used for the boundary discretization. Using the traditional finite element procedures necessitates using thousands of elements and nodal points.

*x = 0.75*,

*y = 0.5*) is depicted in Figure 2 for exponential material properties variations and various material gradation exponents.

*x =*2,

*y*= 0) of the heterogeneous viscoelastic plate with exponential material properties variations is shown in Figure 7. These results are given for different material gradation exponents.

## Conclusions

In the present paper, a new numerical formulation is presented for accomplishment of the simplified viscoelastic analysis of the functionally graded media by the BEM. This approach avoids using relaxation functions or mathematical transformations, and it is capable of solving the quasistatic viscoelastic problems with arbitrary load time dependence and arbitrary boundary conditions. A numerical example, as an application, was provided to evaluate this boundary element formulation. Only the displacement and traction fundamental solutions of the FG elastostatic structures are needed in the present formulation. This computational system was easily solved by adopting a linear time approximation for the displacement and traction rates. Other advantage of the presented approach is that the mathematical integral representation needs only the boundary data. Numerical discretization was done without any domain approximations, and the integral equations were applied only on the boundaries.

## Declarations

## Authors’ Affiliations

## References

- Aliabadi, MH. (2002).
*The boundary element method: applications in solids and structures*. New York: Wiley.Google Scholar - Ashrafi, H, Asemi, K, Shariyat, M, & Salehi, M. (2013a). Two–dimensional modeling of heterogeneous structures using graded finite element and boundary element methods.
*Meccanica, 48*, 663–680.MathSciNetView ArticleGoogle Scholar - Ashrafi, H, Asemi, K, & Shariyat, M. (2013b). A three-dimensional boundary element stress and bending analysis of transversely/longitudinally graded plates with circular cutouts under biaxial loading.
*European Journal of Mechanics A/Solids, 42*, 344–357.MathSciNetView ArticleGoogle Scholar - Ashrafi, H, Bahadori, MR, & Shariyat, M. (2012). Modeling of viscoelastic solid polymers using a boundary element formulation with considering a body load.
*Advanced Materials Research, 463*, 499–504.View ArticleGoogle Scholar - Ashrafi, H, & Farid, M. (2009). A mathematical boundary integral equation analysis of standard viscoelastic solid polymers.
*Computational Mathematics and Modeling, 20*, 397–415.MATHMathSciNetView ArticleGoogle Scholar - Chan, YS, Gray, LJ, Kaplan, T, & Paulino, GH. (2004). Green’s function for a two–dimensional exponentially graded elastic medium.
*Proceedings of the Royal Society A, 460*, 1689–1706.MATHMathSciNetView ArticleGoogle Scholar - Clements, DL. (1998). Fundamental solutions for second order linear elliptic partial differential equations.
*Computational Mechanics, 22*, 26–31.MATHMathSciNetView ArticleGoogle Scholar - Clements, DL, & Budhi, WS. (1999). A boundary element method of the solution of a class of steady-state problems for anisotropic media.
*Journal of Heat Transfer, 121*, 462–465.View ArticleGoogle Scholar - Criado, R, Gray, LJ, Mantic, V, & Paris, F. (2008). Green’s function evaluation for three–dimensional exponentially graded elasticity.
*International Journal for Numerical Methods in Engineering, 74*, 1560–1591.MATHMathSciNetView ArticleGoogle Scholar - Criado, R, Ortiz, JE, & Mantic, V. (2007). Boundary element analysis of three–dimensional exponentially graded isotropic elastic solids.
*Computer Methods in Applied Mechanics and Engineering, 22*, 151–164.MATHMathSciNetGoogle Scholar - Gao, XW, Zhang, C, Sladek, J, & Sladek, V. (2008). Fracture analysis of functionally graded materials by a BEM.
*Composites Science and Technology, 68*, 1209–1215.View ArticleGoogle Scholar - Gray, LJ, Kaplan, T, Richardson, JD, & Paulino, GH. (2003). Green’s function and boundary integral analysis for exponentially graded materials: heat conduction.
*Journal of Applied Mechanics, 40*, 543–549.View ArticleGoogle Scholar - Jin, Z-H. (2006). Some notes on the linear viscoelasticity of functionally graded materials.
*Mathematics and Mechanics of Solids, 11*, 216–224.MATHMathSciNetView ArticleGoogle Scholar - Kuo, HY, & Chen, T. (2005). Steady and transient Green’s functions for anisotropic conduction in an exponentially graded solid.
*International Journal of Solids and Structures, 42*, 1111–1128.MATHMathSciNetView ArticleGoogle Scholar - Mase, GT, & Mase, GE. (1999).
*Continuum mechanics for engineers*. New York: CRC Press.MATHGoogle Scholar - Shaw, RP, & Makris, N. (1992). Green’s function for Helmholtz and Laplace equations in heterogeneous media.
*Engineering Analysis with Boundary Elements, 10*, 179–183.View ArticleGoogle Scholar - Suresh, S, & Mortensen, A. (1998).
*Functionally graded materials*. London: Institute of Materials, IOM Communications.Google Scholar - Sutradhar, A, & Paulino, GH. (2004). The simple boundary element method for transient heat conduction in functionally graded materials.
*Computer Methods in Applied Mechanics and Engineering, 193*, 4511–4539.MATHView ArticleGoogle Scholar - Sutradhar, A, Paulino, GH, & Gray, LJ. (2008).
*Symmetric Galerkin boundary element method*. New York: Springer.MATHGoogle Scholar - Wang, H, & Qin, Q-H. (2012). Boundary integral based graded element for elastic analysis of 2D functionally graded plates.
*European Journal of Mechanics A/Solids, 33*, 12–23.MathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.