Boundary Element Methods for Functionally Graded Materials Abstract

IABEM 2002, International Association for Boundary Element Methods, UT Austin, TX, USA, May 28-30, 2002

Boundary Element Methods for Functionally Graded Materials Glaucio H. Paulino1∗ , Alok Sutradhar1 and L. J. Gray2 1

Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Newmark Laboratory, 205 North Mathews Avenue, Urbana, IL 61801, U.S.A. 2

Computer Science and Mathematics Division, Oak Ridge National Laboratory, Bldg 6012, Oak Ridge, TN 37831, U.S.A.

Abstract Functionally graded materials (FGMs) possess a smooth variation of material properties due to continuous change in microstructural details. For example, the material gradation may change gradually from a pure ceramic to a pure metal. This work focuses on potential (both steady state and transient) and elasticity problems for inhomogeneous materials. The Green’s function(GF) for these materials (e.g. exponentially graded) are expressed as the GF for the homogeneous material plus additional terms due to material gradation. The numerical implementations are performed using a Galerkin (rather than collocation) approximation. A number of examples have been carried out. The results of some specific test problems agree within plotting accuracy with available analytical solutions.

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INTRODUCTION

Functionally graded materials (FGMs) represent the next generation of high performance material systems [17]. Example applications include electronic components, thermal protective systems for reusable spacecrafts, and blast protection for critical structures. In an ideal FGM, the material properties may vary smoothly in one dimension (e.g., are constant in (x, y) but vary with z). A smooth transition region between a pure metal and a pure ceramic may result in a multifunctional material that combines the desirable high temperature properties and thermal resistance of a ceramic, with the fracture toughness and strength of a metal (See Figure 1 ). Comprehensive reviews of current FGM research may be found in the articles by Hirai [9], Markworth et al. [15], Paulino et al. [17] and the book by Suresh and Mortensen [23]. Computational analysis can be an effective method for designing FGM, and for understanding FGM behavior. For homogeneous media, boundary integral equation methods [3] have been used extensively. However, the reformulation in terms of integral equations relies upon having, as either a closed form or a computable expression, a fundamental solution or Green’s function ∗

Corresponding Author, Phone (217)333-3817, Fax (217)265-8041, e-mail: [email protected]

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Figure 1: Transition in a CrNi/PSZ FGM from CrNi alloy to zirconia (PSZ) [10]. (GF) of the partial differential equation. To a large extent, application of the boundary integral technique has therefore been limited to homogeneous, or piece-wise homogeneous, media. The fundamental solutions traditionally employed in boundary integral analysis for homogeneous materials are ‘free-space’ GFs: they satisfy the appropriate differential equation everywhere in space, except at the site where a point load driving force is applied. Derivations for some of the basic GFs can be found in references [2, 3]. Extensions to the case of nonhomogeneous materials are reported on references [1, 6, 20, 21, 22]. Two-dimensional GF results have appeared in conjunction with a convective heat transfer problem in a homogeneous material [14]. Steady state heat conduction with an arbitrary spatially varying conductivity has recently been investigated [7, 11] and has generated some debate in the literature [4, 19]. A GF for a special type of elastodynamics problem was obtained by Vrettos [25]. In this work, the free-space GF for graded materials, in which the thermal conductivity varies exponentially in one coordinate [8], is derived for two and three-dimensional cases. The boundary integral equation formulation is carried using a Galerkin approximation in a three-dimensional setup. The work is further extended to transient diffusion analysis by deriving the GF for the diffusion equation with an exponentially varying heat conductance and heat capacitance [24]. The numerical implementation for transient analysis is done using the Laplace Transform Galerkin Boundary Element method. Of particular importance for engineering mechanics applications is understanding FGM fracture behavior. Crack propagation modeling is one area for which a boundary integral approach is advantageous. The two principal reasons are the ability to handle the crack tip singularity, and thus achieve accurate stress intensity factors, and the simpler remeshing task as the crack propagates. Although a number of studies have been performed for discretely layered FGMs with finite element methods, for example [13], less has been done for continuously varying FGMs [18]. The ability to treat the continuously graded FGM properly is important for fracture analysis studies: the variation of the crack tip stress field in a continously graded FGM is different from that in a discretely layered FGM, as the crack in a layered model will be interacting with an interface. Thus, fracture analysis is a primary application of the GF approach developed herein. Recently, Martin et al. [16] has obtained the GF for a three-dimensional exponentially graded

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elastic medium, and similar work has also been done for two-dimensional problems [5]. The remainder of this paper is organized as follows. The FGM GFs for different problems are presented in Section 2. Section 3 discusses some test results from the Galerkin numerical implementation of the boundary integral formulation, and Section 4 contains some concluding remarks.

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GREEN’S FUNCTION APPROACH The Green’s function for an FGM can be written as a function of GF GM = Ghomogeneous + Ggraded term

(1)

where Ghomogeneous refers to the two-point GF for the corresponding homogeneous material and Ggraded term refers to the additional terms due to property gradation. Here the concept illustrated by Eq. (1) is applied to exponentially graded materials.

2.1

Steady state heat conduction

Steady state isotropic heat conduction in a solid is governed by the equation ∇ • (k∇φ) = 0 .

(2)

Here φ = φ(x, y, z) is the temperature function, and we assume that the FGM is defined by the thermal conductivity k(x, y, z) = k(z) = k0 e2βz , (3) where β is the (dimensionless) nonhomogeneity parameter. The free-space GF for this problem is eβ(r+Rz ) 4πr

(4)

1 eβ(r+Rz ) − 1 + 4πr 4πr

(5)

G(P, Q) = which can be rewritten in the form of Eq. (1) as G(P, Q) =

where Rz = zQ − zP , and r is the norm of R, i.e. r = kRk = kQ − Pk. The first term of Eq. (5) is the GF for Laplace equation in homogeneous media, i.e. Ghomogeneous . The second term is bounded and is referred as Ggraded term . When β tends to zero (material is homogeneous), this graded term vanishes. This approach makes the boundary integral implementation simpler and more useful. In two dimensions the GF becomes [8] i g(xQ , zQ ; xP , zP ) = eβRz H01 (iβr) . 4 3

(6)


where H01 is the zero-order first kind Hankel function, known to be the solution of the Helmholtz equation in two dimensions. The governing boundary integral equation for the FGM problem is ³ ² Z ∂ φ(P ) + G(P, Q) − 2βnz G(P, Q) dQ = φ(Q) ∂n Σ Z ∂ (7) G(P, Q) φ(Q) dQ , ∂n Σ which differs in form from the usual integral statements by the presence of the additional term (containing the β factor) multiplying φ(Q).

2.2

Transient Analysis

The transient diffusion equation is given by ∇ · (k∇φ) = c

∂φ ∂t

(8)

where φ = φ(x, y, z; t) is the temperature function, c is the specific heat and k is the thermal conductivity. We assume that the thermal conductivity varies exponentially in one cartesian coordinate according to Eq. (3). The specific heat is also graded with the same functional variation as the conductivity, i.e. c(x, y, z) = c0 e2βz . (9) Therefore k/c is a constant. The 3D fundamental solution to the FGM diffusion equation is derived as r2 1 −β(z−zp )−β 2 ατ − 4ατ (10) G(P, Q) = 3 e (4πατ ) 2 where α = k0 /c0 and τ = tF − t. The function G represents the temperature field at time tF produced by an instantaneous source of heat at point P and time t. The transient problem is solved using the transform-space approach. In this process the Green’s function is derived for the Laplace transform space as G(P, Q, s) =

1 βz −√β 2 +(s/α) r e e , 4πr

(11)

where s is the transform variable. The details of the corresponding Galerkin BEM formulation and implementation can be found in Sutradhar et al. [24].

2.3

Elasticity

The material properties are given by explicit functions. For instance, for exponential material variation, the Lam´e moduli of the isotropic solid is given by λ(x) = λ0 exp(2β · x) and µ(x) = µ0 exp(2β · x), 4

(12)


where λ0 and µ0 are constants, and β is a given constant vector. We say that the solid is exponentially graded in the direction of β. Evidently, Poisson’s ratio is constant for such a solid. The Green’s function G can be written as ¨ © G(x; x0 ) = exp{−β · (x + x0 )} G0 (x; x0 ) + Gg (x; x0 ) , where Gij (x; x0 ) gives the i-th component of the displacement at x due to a point force acting in the j-th direction at x0 . Here G0 is the Kelvin solution, and the additional grading term Gg is bounded. It is given as a three-dimensional Fourier integral, and the main task is to evaluate this integral. It can be reduced to an explicit term, some single integrals of modified Bessel functions over a finite interval, and some double integrals of elementary functions over finite regions [16]. Similar expressions are also obtained for two dimensional elastic media as well [5].

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NUMERICAL EXAMPLES

Here an example of the steady state heat conduction in an FGM rotor and another example for transient analysis in an FGM cube are presented. The numerical implementation of the elasticity and crack problems are in progress.

3.1

FGM Rotor

This example is a rotor with eight mounting holes. Due to the eight-fold symmetry, only oneeighth of the rotor is modeled, as drawn in Figure 2. The grading direction for the rotor is parallel to its line of symmetry, which is taken as the z-axis, and the thermal conductivity for the rotor varies according to W (13) k(z) = 20e330z ◦ . mK The temperature is specified along the inner radius as Tinner = 20 + 1.25 × 106 (z − 0.01)2 and outer radius as Touter = 150 + 1.25 × 106 (z − 0.01)2 , also a uniform heat flux of 5 × 105 W/m2 is added on the bottom surface where z = 0. All other surfaces are insulated. The BEM solution is compared with an FEM solution obtained from a commercial package using ten-node tetrahedral elements to handle the geometric complexity of the rotor. The FEM mesh required 95, 880 nodes, whereas the BEM mesh employed 3, 252 nodes. The temperature distribution around the hole is shown in Figure 3. The angle θ is measured from a line passing through the line of symmetry for the geometry and the center line of the hole as shown in Figure 3. Though surface nodal positions in the two models were not coincident in general, the plot shows a strong agreement in the two solutions. To see the effects of the grading upon the solution, the corresponding results for the ungraded β = 0 (k(z) ≡ 20) rotor are also shown. The radial heat flux is plotted in Figure 4. The negative sign indicates that the flow of heat is toward the interior of the rotor. A limitation on the use of piece-wise constant conductivities in FEM models may be evident in the plot where the FEM nodal value at z = 0.01 seems to fall out of line with the other values on the curve. The behavior should be fully expected, however, given the local error associated with the piece-wise constant approximation. As should also be expected, the nodal flux values from the BEM solution seem to fall onto a single curve even 5


0.0075Dia

0.05R

0.03R

0.01

0.01

0.03827

Figure 2: Geometry of the functionally graded rotor.

o

Temperature ( C)

200.0

150.0

FEM (graded) BEM (graded) FEM (homogeneous) BEM (homogeneous)

100.0

50.0

0

p/4

π/2 θ

3π/4

π

Figure 3: Temperature distribution around the hole on the z = 0.01 surface.

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2

Radial Heat Flux (W/m )

1.0e+06 −1.0e+06 −3.0e+06 −5.0e+06 FEM BEM

−7.0e+06 −9.0e+06 0.000

0.002

0.004 0.006 z (m)

0.008

0.010

Figure 4: Radial heat flux along the inside corner.

200.0

FEM BEM (interior)

o

Temperature ( C)

150.0

100.0

50.0

0.0

0.030

0.035

0.040 0.045 Radial coordinate

0.050

Figure 5: Computed interior temperature values in the graded rotor. 7


in the region of the steepest conductivity gradient. This is not to say that BEM is necessarily better than FEM for graded analysis: the finite element method is not restricted to using the discontinuous piece-wise constant approximation available in existing packages. It is possible to incorporate continuous grading within individual elements, and codes having this capability are being developed [12, 17]. As a final test, Figure 5 displays a comparison between the FEM interior temperature values, and corresponding values computed from the BEM solution (in a post-processing calculation). The values are shown for a line of points on the mid-z (z = 0.005) plane in the radial direction, passing through the middle of the hole. Again, the BEM and FEM results agree quite well.

3.2

Transient analysis of an FGM cube with constant temperature on two planes

The problem of interest is shown in Figure 6. The cube initial temperature is zero. Then the top surface of the cube at [z = 1] is maintained at a temperature of T = 100 while the bottom face at [z = 0] is zero. The remaining four faces are insulated (zero normal flux). The boundary and initial conditions are φ(x, y, 0; t) = 0 φ(x, y, 1; t) = 100 φ(x, y, z; 0) = 0

(14)

The thermal conductivity and the specific heat are taken to be k(x, y, z) = k0 e2βz = 5e3z c(x, y, z) = c0 e2βz = 1e3z

(15) (16)

The analytical solution for temperature is derived, φ(x, y, z; t) = φs (x, y, z) + φt (x, y, z; t) ° ± ∞ nπz −βz − nL2 π2 2 +β 2 αt 1 − e−2βz X + e e Bn sin = T 1 − e−2βL n=1 L where L is the dimension of the cube (in the z-direction) and µ ´ 2T eβL 1 + e−2βL − nπ cos nπ βL sin nπ Bn = − 2 2 β L + n2 π 2 1 − e−2βL

(17)

(18)

The Galerkin BEM mesh has 1200 elements with 200 elements on each face. Numerical solutions for the temperature profile at different times are shown in Figure 7. Notice that the temperature variation matches the analytical solution.

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z Flux=0(back)

(1,1,1)

Temp=100(top)

Flux=0(front)

Flux=0(right)

y Flux=0(left)

Temp=0.0(bottom)

x

(0,0,0)

Figure 6: Geometry and boundary conditions of the FGM cube problem with constant temperature on two planes. The faces with prescribed temperature are shaded (Example 2).

100.0

Temperature

80.0

Analytical BEM (t=0.001) BEM (t=0.01) BEM (t=0.02) BEM (t=0.05) BEM (t=0.1)

60.0

40.0

20.0

0.0 0.0

0.2

0.4 0.6 z coordinate

0.8

1.0

Figure 7: Temperature profile in z direction at different time levels for the FGM cube problem with constant temperature on two planes (Example 2).

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4

CONCLUSIONS

Boundary element analysis with "augmented" Green’s functions can be successfully applied to FGMs and can maintain the boundary-only aspect of the method. The numerical results presented in this paper have shown that it is relatively simple to implement the FGM Green’s function in a standard boundary integral (Galerkin) approximation, and that accurate results are obtained. For graded materials, this offers the possibility of efficient and accurate solution of those types of problems for which a boundary integral analysis is particularly advantageous, such as shape optimization, moving boundaries, and small scale structures.

Acknowledgments Work at ORNL was supported by the Applied Mathematical Sciences Research Program of the Office of Mathematical, Information, and Computational Sciences, U.S. Department of Energy under contract DE-AC05-00OR22725 with UT-Battelle, LLC. A. Sutradhar acknowledges the support from the DOE Higher Education Research Experience (HERE) Program at Oak Ridge National Laboratory. G.H. Paulino acknowledges the support from the National Science Foundation (NSF) under grant No. CMS-0115954.

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