Material Designer offers two methods for the analytical homogenization of the coefficients of thermal expansion of a unidirectional composite, namely Schapery and Mori-Tanaka methods.
Schapery's Method
Consider a unidirectional composite with inclusions transversely isotropic in the x direction. In the Schapery's method [Schapery, (1968)], the thermal expansion coefficient in the longitudinal direction is computed as
 | (3–8) |
and the thermal expansion coefficient in the transverse directions is computed as
 | (3–9) |
Here the Young moduli and Poisson ratios of the matrix and fiber constituents are denoted by E and ν.
Note that in the Schapery's method inclusions are modeled as infinitely long cylinders.
Mori-Tanaka Method
Following the formalism in [Lu P., (2013)], the thermal expansion coefficients predicted by the Mori-Tanaka method read
 | (3–10) |
where
 | (3–11) |
and 
is the Eshelby tensor introduced in the section above,
Homogenization of Unidirectional Composites.
In case of a composite with misaligned fiber orientations, the orientation averaged thermal expansion coefficients are given by [Camacho et al., (1990)]
 | (3–12) |
where
the constants R 1 and R 2 depend on the stiffness tensor and thermal expansion coefficients of the unidirectional composite
and 
denote the second order orientation and identity
tensors in Voigt notation, respectively;the orientations averaged stiffness tensor
is computed as described in the section above,
Orientations Averaging.