2.1. Homogenization Process and Stiffness Prediction Method
The homogenization theory has been widely recognized to predict the stiffness of composites, and its concept is shown in
Figure 1. In order to solve the difficulty of large-scale macro-micro joint computation, the concept of microelement is introduced. A microelement represents a point in the composite structure, and its average stress and strain relation is regarded as the effective constitutive relation of composite. In fact, the non-uniform element is replaced by a representative volume element (RVE) with the above average stress and strain relationship. The above concepts are applied to every point of the composite structure. Based on such way, the original problem of analyzing the non-uniform composite structure was converted to the problem of analyzing the properties of the macro structure after homogenization, which greatly simplified the original problem [
24,
25].
Meso-level composites often exhibit a high degree of heterogeneity, and even in the neighborhood
with a very small macroscopic position
, the field variables will vary greatly, as shown in
Figure 2.
Considering the periodicity of the composite structure, these field variables can be expressed as follows:
where,
is the macroscopic scale,
is the mesoscopic scale,
is the ratio of the two scales,
is the period of the periodic function, and
is the integer.
The displacement is expressed as a function of double-scale coordinates and expanded according to small parameters:
is a periodic function of
. According to constitutive equation and geometric equation in elastic mechanics, the above asymptotic expansion was substituted into the equilibrium equation, and the left side of the equation was arranged as an asymptotic expansion with respect to
. We set the coefficient of
as zero to obtain a series of governing equations, in which:
In Equation (4),
is a function of stress field. By simplifying the operation, the following equation was obtained:
The periodic function
can be expressed as the region of a single cell in the axial braided carbon/carbon composite.
is the boundary of the cell, and
is the vertex of the cell on the boundary. The homogenization coefficient of a single cell can be expressed as:
is the second term of the asymptotic expansion of
, which is called the mesoscopic stress field. The mesoscopic stress field includes the macroscopic parameters
and the mesoscopic parameters
, which reflect the fluctuation of stress in the scale of RVE.
Gauss theorem is used to treat the governing equation of the homogenization coefficient of Equation (11), and the equation is converted into a “weak” solution equivalent to the original problem as follows:
From Equation (12) can be found in the original control equation is equivalent to the three dimensional linear elastic problems in the form of the "weak" solution of equation, the difference is that the material within a plane distribution related to the material properties and boundary shape, the plane distribution of force on the internal boundary of different material in the RVE (), pointing to the internal material area, the direction of the component for .
The two terms in the integral sign of Equation (12) were separated to obtain the deformation of the homogenization coefficient:
From Equation (13), it can be seen that in the boundary force method, the homogenization coefficient is equal to the volume average of the stress of the three-dimensional linear elastic problem on the RVE plus the volume average of the elastic tensor on the RVE [
16,
17,
18]. The homogenization coefficient is the effective elastic property of macroscopic structure.
2.2. A Method for Predicting CTE of Composites Based on Energy Method
The CTE prediction method based on the energy method is more convenient to calculate than the simple homogenization method. Therefore, the following method combined with FEM was used to calculate the CTE of materials in this paper [
19,
22].
The basic idea of the energy method is that based on the relationship between microstructure and homogeneous equivalent body, the energy expression of the equivalent property of composite materials can be obtained by deducing the relationship between the equivalent property of composite materials and the deformation energy of microstructure [
16,
17].
For the microstructure shown in
Figure 1, when the temperature rise value is
, the average stress and average strain of the microstructure have the following relationship:
where
,
are the stiffness matrix and CTE matrix of axial braided C/C composites.
The deformation energy of the microstructure of the material is shown in Equation (15):
For working condition 1 (
) as shown in
Figure 3a, the microstructure deformation energy caused by
temperature change is the same as that of working condition 4 without temperature load as shown in
Figure 3b. The displacement load
in working condition 4 is equal to the displacement load under working condition 1. The force
is the interface force caused by the difference of thermal stress between the two loads. The expressions of the two are Equations (16) and (17).
For the microstructure of working condition 1, the variable properties of its homogeneous equivalent are equal to the difference between the variable properties of working condition 1 and that of working condition 6, as shown in Equation (18):
Similar to Equation (18), the variable properties of the homogeneous equivalent of condition 2 (
) and condition 3 (
) can be expressed as Equations (19) and (20):
The FEM equation of the microstructure under operating conditions 4, 5 and 6 can be expressed as Equation (20).
where,
is the unknown node displacement vector under condition
, and
is the known node displacement vector.
is the known node force vector under condition
, and
is the unknown node force vector. To solve Equation (20), the following Equation (21) can be obtained.
The microstructural variable performance of working condition
can be expressed in the following form:
The calculation method of CTE of microstructure can be obtained by solving Equation (22) as shown in Equation (23):
In the equation .