## 3D Lame Equation

# Background

Let be a bounded, simply or multiply connected, domain in with Lipschitz boundary occupying a homogeneous isotropic elastic solid that is characterized by the Lamé parameters and In linear elasticity, the Lamé equation which describes the static equilibrium of a deformable body in terms of the displacement vector is defined as

where represents a body force prescribed on To deal with the Lamé equation, let where be the unit normal to directed towards the *exterior* of and let be the traction vector on associated with the displacement where the stress tensoris introduced as

with denoting the symmetric second-order identity tensor on and the superscript "" representing the transpose symbol. A solution of the Lamé equation admits an integral representation, known as the Somigliana identity, expressed as

where the kernels and are given respectively as

and

Here the elastic constant is called the Poisson ratio of the solid and is the usual Euclidean norm in defined as . Moreover, it was established in [2] that the Newton potential admits a boundary representation as

where

with denoting an extension of the body force into any ball centered at and containing . In particular, a continuation of the body force can be specified as

This representation of the elastic Newton potential in term of surface integral allows a numerical solution of the Lamé equation that does not require a volume-fitted mesh.

# Solution via a boundary element method

To approximately solve the Lamé equation via a Boundary Element Method (BEM), the surface is usually discretized into *flat* triangles using a mesh generation software (e.g. CUBIT).

With reference to [1], the vector fields and are assumed to have a polynomial variation over each triangle (boundary element)

## Approaches:

Galerkin BEM

# References

- [1] S. Nintcheu Fata.

Explicit expressions for three-dimensional boundary integrals in linear elasticity.

*J. Comput. Appl. Math.*, 235(15):4480-4495, 2011. - [2] S. Nintcheu Fata.

Boundary integral approximation of volume potentials in three-dimensional linear elasticity.

*J. Comput. Appl. Math.*, 242(1):275-284, 2013.