Fiber Reinforcement

You can define the material properties of the matrix constituent of the composite using a different material behavior. The properties of the matrix are undefined in the fiber reinforcement material model. You also specify various parameters to define a semianalytical homogenization approach to compute the thermal and mechanical properties of the composite material.

The fiber reinforcement material option:

supports one or more fiber reinforcements
supports fiber behaviors whose mechanical response can be modeled using a linear elastic response only
supports specification of expansion, thermal conductivity, and density of each fiber reinforcement
requires you to provide the homogenization data, the number of fiber reinforcements, the fiber geometric and orientation data, and the material data for each reinforcement
uses formulations based on Eshelby's solution and
can predict volume-averaged microlevel solutions in each constituent of the composite

Mean-Field Homogenization

The mean-field strain and stress in each phase I is defined as

{\bar{σ}}_{I} = {〈 σ 〉}_{I}

{\bar{ϵ}}_{I} = {〈 ϵ 〉}_{I},

where

〈 f 〉 = \frac{1}{V} \int_{V}^{} f d V .

The averaged macro fields can be written as

〈 • 〉 = v_{f}^{M} {〈 • 〉}_{M} + \sum_{I = 1}^{N} v_{f}^{I} {〈 • 〉}_{I},

where $v_{f}^{M}$ is the volume fraction of the matrix phase and $v_{f}^{I}$ is the volume fraction of the inclusion phase. For the single-inclusion case, if all the constituents are linear elastic, the strain in the inclusion is related to the strain in the matrix through a concentration tensor $A$ .

{〈 ϵ 〉}_{I} = A {〈 ϵ 〉}_{M} .

The form of the concentration tensor defines the homogenization methods below.

Mori-Tanaka (MT)

The Mori-Tanaka formulation assumes that each inclusion behaves like an isolated inclusion and the strain in the matrix is considered as the far-field strain. Therefore, the concentration tensor can be written as

A = {[E : (C_{M}^{- 1} C_{I} - I) + I]}^{- 1} .

Voigt and Reuss Models

The simplest mean-field homogenization formulations are the Voigt and Reuss models. These models do not take into account the shape or the orientation of the inclusions. However, they provide the upper and lower bounds of the macro stiffness modulus and, therefore, can be used for validation. The Voigt model is also used in a two-step approach to model composites with multiple inclusions.

The Voigt model assumes uniform strain in the composite ${〈 ϵ 〉}_{I} = {〈 ϵ 〉}_{M}$ , which gives

A = I .

The Reuss model assumes uniform stress in the composite ${〈 σ 〉}_{I} = {〈 σ 〉}_{M}$ , which gives

A = C_{I}^{- 1} C_{M},

where $C_{M}$ is the stiffness of the matrix and $C_{I}$ is the stiffness of the inclusion.

Inverse Mori-Tanaka (Inverse MT)

The inverse Mori-Tanaka formulation assumes a high volume fraction of the inclusion and permutes the properties of the matrix and inclusion in a single inclusion problem, which gives

A = {[E : (C_{I}^{- 1} C_{M} - I) + I]}^{- 1} .

Balanced Model

The Mori-Tanaka and inverse Mori-Tanaka formulations give the upper and lower bounds of the composite stiffness for low and high volume fractions. The balanced formulation is the interpolative formulation, which can be written as

H_{1} = {[E : (C_{M}^{- 1} C_{I} - I) + I]}^{- 1},

H_{2} = {[E : (C_{I}^{- 1} C_{M} - I) + I]}^{- 1},

A = {[(1 - φ) H_{1}^{- 1} + φ H_{2}^{- 1}]}^{- 1},

where $φ$ is the smooth interpolation function proposed by Lielens (Lielens, 1999),

φ = \frac{1}{2} v_{f}^{I} (1 + v_{f}^{I}) .

Isotropization

Options are available to use the isotropic projection of the tangent matrix to compute different parts of the concentration tensor. For better predictions, Doghri et al. (Doghri, 2010) recommended the following:

If the general method is used, only Eshelby's tensor is computed with the isotropic projection.
If the spectral/modified spectral method is used, Hill's tensor given by $P = E : C_{M}^{- 1}$ is computed with the isotropic projection.

It is possible that some choices can lead to unphysical prediction of microlevel strain resulting in unstable material behavior in the composite. In addition, the tangent stiffness matrix for the composite might lose its symmetry after homogenization; therefore, you must use the unsymmetric solver to achieve convergence.

Input File Usage

Set isotropization to ALLISO to use the isotropic projection of the matrix stiffness to compute all parts of the concentration tensor by default.

Set isotropization to E-ISO to use the isotropic projection of the matrix stiffness to compute only Eshelby's tensor.

Set isotropization to P-ISO to use the isotropic projection of the matrix stiffness to compute Hill's tensor.

Including Fiber Direction

By default, the orientation of the inclusion or void phase is fixed and is aligned with the x-axis of the local coordinate system specified in the section definition. You can specify a different orientation by giving the three components of the orientation vector in the local coordinate system, as shown in Mean-Field Homogenization. The components of the orientation unit vector, $p$ , are

\begin{matrix} p_{1} & = & \sin θ \cos φ, \\ p_{2} & = & \sin θ \sin φ, \\ p_{3} & = & \cos θ . \end{matrix}