PREDICTION OF ELASTIC CONSTANTS AND THERMAL EXPANSION
Finite Element Methods
Isostrain or isostress assumptions are unnecessary in codes based on finite
element formulations, since these codes compute internal stress distributions in detail,
using arbitrarily many degrees of freedom. The stiffness matrix is built up by imposing
homogeneous displacement boundary conditions and either integrating the internal
stresses to obtain the average stress or integrating the boundary tractions. The average
stress can be related to the boundary tractions via the average stress theorem (Eq. 5.7).
Building the complete stiffness matrix requires solving for six independent sets of
boundary conditions. Because of the potential lack of symmetry of a general unit cell, the
stiffness matrix can be fully populated (generally anisotropic). Although it can be proven
that the stiffness matrix for any heterogeneous unit cell must be symmetric, finite element
calculations can yield a nonsymmetric matrix due to the approximations involved. A
symmetric result, C
sym
, is usually created by the averaging operation
, (5.29)
where the superscript T denotes transpose.
There is no requirement for a finite element mesh to map directly to the unit cell's
internal geometry. For example, the mesh may be a regular array of cuboidal elements,
independent of the yarn paths. When the element stiffness matrix is generated, the code
must determine the local material stiffness at each Gaussian integration point. This gives
rise to so-called heterogeneous elements, which greatly simplify the meshing problem.
However, the stresses in heterogeneous elements may converge slowly with respect to
mesh density.
Finite element methods are used to solve unit cell formulations in the codes
SAWC, µTEX-10, and µTEX-20.
Laminate Analysis: Quasi-Laminar Textile Composites
The properties of quasi-laminar textile composites are usually and most
conveniently represented by the conventional A, B, and D matrices of classical lamination
theory. The matrix A defines the coupling between in-plane strains and in-plane force
resultants; and B and D the coupling between in-plane strains and bending moments or
bending curvatures and in-plane force resultants [5.1]:
