![](media/image1.jpeg){width="1.4791666666666667in"
height="0.4791666666666667in"} **Advanced Scientific Computing
Research**

**FY 2003 Accomplishment**

**Boundary Integral Analysis for Functionally Graded Materials**

## L. J. Gray ^\*^, Oak Ridge National Laboratory

#### T. Kaplan, Oak Ridge National Laboratory

Summary

***Fundamental solutions (Green\'s functions) to the partial
differential equations in a functionally graded material (FGM) have been
derived. Based upon these exact solutions, effective numerical methods
for solving boundary integral equations can be applied to important
engineering problems, e.g., fracture analysis, and material design
optimization***.

Functionally graded materials (FGMs) are a class of relatively new and
promising materials that have emerged from the need to optimize material
performance.  In a homogeneous material the properties are constant,
whereas in an FGM, the material properties vary continuously with
position, usually in one coordinate direction.

As an example, a material that transitions smoothly from a pure metal to
a pure ceramic would combine the high temperature and compressive
strength properties of the ceramic with the fracture toughness and
thermal conductivity of the metal.  Compared to a layered system, e.g.,
a ceramic coating on a metal substrate, an FGM avoids the discontinuity
in material properties across the interface. A sharp interface can lead
to stress concentrations that eventually produce material failure.  FGMs
have already been employed in many important areas, e.g., thermal
barrier coatings for aerospace applications, graded refractive index
materials for optical applications, and implants for bio-medical
applications.

![](media/image2.jpeg){width="3.0097222222222224in"
height="3.6756944444444444in"}

*Figure 1. The FGM Advantage.*

As analytical solutions for non-homogeneous materials are rare, there is
a need for reliable and efficient numerical methods for solving problems
in FGMs.  In particular, a critical engineering need, and a primary goal
of this work, is to understand failure in FGMs, i.e., crack propagation
studies.  For homogeneous materials, integral equation methods have two
significant advantages for these analyses. First, the re-meshing
required as the crack grows is only at the crack tip, and is therefore
(compared to volume methods such as finite elements (FEM)) relatively
trivial.  Second, unlike FEM, the boundary-only approach avoids a direct
confrontation with the stress singularity ahead of the crack.  As a
consequence, an appropriate crack tip element \[1\] produces highly
accurate stress intensity factors (which determine crack growth), even
with seemingly coarse meshes. Thus, the overall computation time,
including mesh generation, for a boundary integral analysis is likely to
be significantly less than for an FEM simulation having corresponding
accuracy.

Integral equation methods, however, are based upon knowing the
**Green\'s function**, and have therefore been limited almost
exclusively to homogeneous materials.  The Green\'s function is an exact
solution of the partial differential equations, one having an impulse
forcing term at one point, and are known only for the simplest
equations.

We have successfully derived the Green\'s function for an FGM wherein
the material properties vary exponentially through the solid.  The
Green\'s function has been obtained in closed form for the Laplace
equation (heat conduction) \[2,3\], and in a readily computable form for
elasticity \[4,5\] in both two and three dimensions.  With these
Green\'s function, an FGM analysis can be formulated in terms of
boundary integral equations as simply as for homogeneous media. Note
that the advantages of a boundary integral formulation for fracture
analysis are amplified in the FGM setting.  For an FGM, the FEM volume
mesh must be fine enough to represent the exponential grading in the
material; this adds to the computation cost and further complicates
automatic re-meshing schemes.  In the boundary integral approach, the
material gradation in the volume is exactly encapsulated in the Green\'s
function.

The ability to perform computational analyses is highly important for
FGM research and development, as the manufacturing and experimental
testing of these materials is in its infancy and is not simple nor
inexpensive.  In particular, parameter design optimization and crack
propagation are two areas of major importance for which numerical
simulations can be highly useful.

\[1\] L. J. Gray, A.-V. Phan, G. H. Paulino, and T. Kaplan, \"Improved
Quarter-Point Crack Tip Element,\" *Engineering Fracture Mechanics*
**70**, pp. 269-283 (2003).

# \[2\] L. J. Gray, T. Kaplan, J. D. Richardson, and G. H. Paulino, "Green's Function and Boundary Integral Analysis for Exponentially Graded Materials: Heat Conduction," ASME Journal of Applied Mechanics (in press).

\[3\] A. Sutradhar, G. H. Paulino, and L. J. Gray, "Symmetric Galerkin
boundary element method for heat conduction in functionally graded
materials," *Int. J. Num. Meth. Eng*. (submitted)

\[4\] P. A. Martin, J. D. Richardson, L. J. Gray, and J. Berger,
"Green\'s functions for an Exponentially Graded Elastic Material,"
*Proc. Royal Soc*. **458**, 1931-1948 (2002)

\[5\] Y.-S. Chan, L. J. Gray, T. Kaplan, and G. H. Paulino, \"Green\'s
Function for a Two-Dimensional Exponentially-Graded Elastic Medium,\"
*Proc. Royal Soc.* (submitted)

**For further information on this subject contact:**

Dr.Chuck Romine, Program Manager

Mathematical, Information, and Computational

Sciences Division

Office of Advanced Scientific Computing Research

Phone: 301-903-5152

romine@er.doe.gov