** Next:** Thermal strains (eigenstrains)
**Up:** Finite element theory
** Previous:** Electrical conductivity

The elasticity problem is set up similarly to the electrical conductivity problem, because the elastic energy stored also obeys a variational principle. The elastic energy stored is given by

We will use α and β exclusively as labels running over the six
components of the
strain **6**-vectors. The energy equation becomes

The idea of the finite element scheme, as for the electrical case, is to reduce
the energy equation
to a quadratic form containing the components of the elastic displacement
vector defined at the
nodes of the pixels. There is an elastic displacement defined at each node,
which has three
components in 3-D. We denote this by *u(m,*3). The *m* and *ib* integer labelling
system is the same
as before, but now all real variables have an extra label for the Cartesian
component being
considered:

*u _{mp}* =

=

= (ε

Each component of the displacement is linearly interpolated across the pixel
in exactly the
same way as the voltage was done in the electric problem. The *p*'th component
of the three-vector
at a point *(x,y,z)* in the pixel is then defined as

u) =
_{p}(x,y,zN)_{p,rq}(x,y,zu
_{rq} | (15) |

where *N*_{1,r2} = *N*_{2,r2} = *N*_{3,r3} = *N _{r}* given for the electrical problem, and

ε_{α}(x,y,z) = [L(_{α p}N_{ p,rq}x,y,z)] u_{rq} | (16) |

or | |

ε_{α}(x,y,z) = S_{α ,rq}(x,y,z
) u_{rq} | (17) |

where the components of *S _{α,rq}* can be found in
ELAS3D.F, in the subroutine FEMAT, where the matrix

**Table 6:**
Components of *L _{pq}*

Substituting into the energy expression above, in the Voigt notation, results in

Grouping the *S* and *C* matrices together, and performing the integral over the
pixel using the part that has *(x,y,z)* dependence, results in

is the stiffness matrix. It is also the same as the dynamical matrix arising
in lattice models of
elastic phenomena found in theoretical physics, where displacements at nodes
are connected by
various forces [6]. The first part of subroutine FEMAT computes the
stiffness matrix (*dk* in
the finite element programs) using Simpson's rule to perform the integration. As
in the electrical case, the integration is
exact, as there is no term to be integrated that is higher order than quadratic,
and Simpson's rule is
exact for quadratic functions.

Periodic boundary conditions result in exactly the same structure as in the
electrical case, but
with now the extra index for the Cartesian coordinates, and a different form
for the vector
δ* _{rp}* , expressed in terms of the six independent applied strains.
The components of
δ