The Mori-Tanaka method is one of the most popular mean-field methods for predicting the elastic properties of composites with unidirectional inclusions. Following the formalism developed by [Benveniste, (1987)] (see also [Tucker, C. L. et al. (1999)]), the Mori-Tanaka method provide the following estimate of the stiffness tensor of a unidirectional composite:

(3–1)

The strain concentration tensor in the inclusions is obtained from the Eshelby tensor

(3–2)

(3–3)

The Eshelby tensor only depends on the inclusion shape, its orientation, and on the matrix elastic constants. Expressions of the Eshelby tensor for the elastostatics problem for different type of inclusions can be found for example in [Parnell, (2016)]. Material Designer supports ellipsoidal inclusion types.

When the fiber in the composites are not fully aligned, an orientation averaging procedure is carried out to predict the elastic properties of the composite. Following the approach of [Advani et al, (1987)] the properties of the composite are taken as an average of unidirectional composite properties over all directions, weighted by the orientation distribution function :

(3–4)

Here is the stiffness tensor of the composite with misaligned fiber orientations, is the stiffness tensor of a unidirectional composite with inclusions aligned along the direction and 𝛀 is the unit sphere.

Assuming the inclusions to be transversely isotropic along and the matrix to be isotropic, the unidirectional composite is transversely isotropic along and the orientations averaged stiffness tensor can be written as [Advani et al, (1987)]

(3–5)

where

(3–6)

(3–7)

denote the second and fourth-order orientation tensors, respectively. The constants B₁,…,B₅ are related to the five independent components of the transversely isotropic elasticity tensor (Equation 3–1) of the unidirectional composite.

The spatial distribution of the second order orientation tensor is commonly provided by injection molding software, while the fourth-order orientation tensor is usually unknown. Based on the second order orientation tensor, Material Designer estimates the fourth order orientation tensor using a closure approximation. Both the smooth and fitted variants of the orthotropic closure approximation proposed in [Cintra et al., (1995)] are available. In addition, a combination of these two is provided: while being based on the fitted version, it reduces to the smooth variant in case of isotropic or transversely isotropic orientation configurations.