# OOF: Finite Element Analysis of Microstructures

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Background Material: Basic Anisotropic Thermoelasticity

In the current version of *OOF*, all elements are triangles consisting
of three *nodes* located at the triangle vertices. The nodes may
be located at the boundary of the mesh, or they can be shared with
other elements. The elements share information with each other (i.e.,
how much the corners have been displaced) through their nodes.

The current version of *OOF* is thermoelastic, and so the pertinent
field quantities are the displacement (vector) field and the
uniform temperature change . The displacement field imparts
a strain (tensor) field through the rotation-invariant values of
spatial derivatives of the displacement field:
-which is a long-winded way of saying
the nodal displacements impose a strain, , on the
elements through a predictable formula.

The temperature is given as a difference . This difference is relative to the temperature at which the initial mesh is stress free. Each element undergoes a uniform strain, , in the stress free state (i.e., if the nodes are completely unrestrained or the stress tensor ). The second rank tensor is the (symmetric) thermal expansion tensor. It has additional symmetries according to the point group of the underlying material. See Nye [2] for a lucid discussion of how material symmetry is expressed in tensors representing linear constitutive behavior.

The stress is a second rank tensor which represents the
the total *force* transmitted in the -direction through
a planar region with its normal parallel to the
-direction in the limit that the area of the region
shrinks to a point. Torque balance requires that
in linear elasticity.

In linear elasticity, the stress is related to the strains implied by
the displacement field and the stress-free strain^{4.1}
through the linear relation:

The fourth-rank tensor is the

*stiffness tensor*. The symmetry of both and imply that , so it can have at most 36 independent components. An additional requirement, that the elastic energy density be frame invariant, imposes the symmetry and reduces the number of independent components to 21.

It is convenient to take advantage of the symmetries to represent the
stiffness tensor as a symmetric matrix. This can lead to
*silly mistakes* since the matrix does not rotate in the usual
straight-forward way. Fortunately, *OOF* handles the rotations for
you (see Section 4.3). To convert the fourth-rank
tensor into a two dimensional matrix, we adopt the convention in Nye
[2] and replace the six distinct pairs with a single
integer as follows:

Four Index Tensor Indices | 11 | 22 | 33 | 23 or 32 | 31 or 13 | 12 or 21 |

Two Index Matrix Indices | 1 | 2 | 3 | 4 | 5 | 6 |

In *OOF*, the stiffness components are entered using the matrix
notation. The symmetry of each particular material places additional
constraints on the stiffness matrices. These additional symmetries
are illustrated in Figures 4.2,
4.3,
4.4, and 4.5 (following Nye [2]).

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