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Elastic moduli

 

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

img134.gif

where the strain tensor εpq and elastic moduli tensor Cpqrs are in full tensorial form, p,q,r,s = 1, 2, or 3, and the integral is over the volume of a single pixel. The total energy is obtained by summing over all pixels. Since the strain tensor is symmetric, a simpler notation is usually used, the Voigt notation, where the strain is taken to be a 6-vector containing the six independent strains (εxx, εyy, εzz , εxz, εyz, εxy), and then Cpqrs is written as Cαβ, where

img140.gif

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

img143.gif

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:


ump = p'th component of displacement at m'th node
Cαβ = elastic moduli tensor (Voigt notation) for a pixel
img147.gif = ( Exx, Eyy, Ezz, Exz, Eyz, Exy) = overall elastic strains applied to system
= (εxx, εyy, εzz, εxz, εyz, εxy) = local strains at a point (x,y,z) in a pixel
Drp,sq = stiffness matrix in a pixel
Np,rq = shape matrix for cubic pixel

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
up(x,y,z) = Np,rq(x,y,z)urq (15)

where N1,r2 = N2,r2 = N3,r3 = Nr given for the electrical problem, and urq is the q'th component of the displacement on the r'th node, r = 1,8. Clearly, in this structure, we have N1,r2 = N1,r3 = N2,r1 = N2,r3 = N3,r1 = N3,r2 = 0. N is a 3 x (8,3) matrix. To construct the 6-vector strain from this, we need to multiply by (or operate with) a matrix of derivatives, Lpq, that is 6 x 3. The components Lpq are given in Table 6 below. This results in

εα(x,y,z) = [Lα pN p,rq(x,y,z)] urq(16)
or
εα(x,y,z) = Sα ,rq(x,y,z ) urq(17)

where the components of Sα,rq can be found in ELAS3D.F, in the subroutine FEMAT, where the matrix es(n1, n2, n3) is equivalent to Sα,rq. If eq. (17) is integrated over a pixel, it gives the average strain in the pixel, in terms of the nodal displacements. When first multiplied by Cαβ and then integrated, the average stress in the pixel results. When the solution displacements to the problem are obtained, these can be used to compute the average stress and strain in the system, which define the effective quantities and can also give insight into local stress and strain fields around microstructural features.


  img171.gif


Table 6: Components of Lpq

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

img172.gif

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

img174.gif

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 δrp , expressed in terms of the six independent applied strains. The components of δrp are given in Table 7. The periodic boundary conditions result in a term linear in the displacements, denoted b · U, as well as a constant term C that is quadratic in the applied strains. The gradient of the energy is the same as in the electric case, but with the extra index for the Cartesian coordinates of the displacement. Note that the terms of b, by definition, are linear in the applied strains (see 2.2). This is important for the development of the eigenstrain or thermal strain case, discussed next.

  

tab7b.gif


Table 7: δrp vectors for the 8 pixel corners


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