Effective Axial Modulus :

The unit cell as shown in Figure 7.2 is used to compute the effective axial modulus . It should be noted that the thickness of the unit cell is not important in this computation. Further, the cross sectional shapes are not considered in this calculation. However, the cross sectional areas are important in this calculation. The thicknesses of the fibre and matrix constituents are same in the unit cell. Hence, the areas of the constituents represent the volume fractions of the constituents.

In the calculation of effective axial modulus, it is assumed that the axial strain in the composite is uniform such that the axial strains in the fibers and matrix are identical. This assumption is justified by the fact that the fibre and the matrix in the unit cell are perfectly bonded. Hence, the elongation in the axial direction of the fibre and matrix will also be identical. Thus, the strains in the fibre and matrix can be given as

(7.14)

where,  is the axial strain in the composite and  and  are the axial strains in fibre and matrix, respectively. Now, let  and  be the axial Young’s moduli of the fibre and matrix, respectively. We can give the axial stress in the fibre,  and matrix,  as

(7.15)

Using the above equation and the cross section areas of the respective constituent in the unit cell, we can calculate the forces in them as

(7.16)

 

 

 

The total axial force in the composite is sum of the axial forces in fibre and matrix. Thus, the total axial force in the composite substituting the expressions for axial strains in fibre and matrix from Equation (7.14) in above equation, can be given as

(7.17)

Now  be the average axial stress in composite. The total cross sectional area of the composite is . Thus, using the average axial stress and cross sectional area of the composite, the axial force is

(7.18)

Thus, combining Equation (7.17) and Equation (7.18) and rearranging, we get

(7.19)

Figure 7.3:  (a) Undeformed unit cell under  (b) and (c) deformed individual constituents of the unit cell

 

Let us define

(7.20)

Further, noting that the ratios  and  for same length of fibre and matrix represent the fibre and matrix volume fractions, respectively. Thus, combining Equations (7.19) and (7.20), we get

(7.21)

The above equation relates the axial modulus of the composite to the axial moduli of the fibre and matrix through their volume fractions. Thus, the effective axial modulus is a linear function of the fiber volume fraction. This equation is known as rule of mixtures equation. It should be noted that the effective properties are functions of the fiber volume fractions; hence it should always be quoted in reporting the effective properties of a composite.