Hill’s Concentration Factors Approach

The Hill’s concentration factors approach is based on the concept similar to the standard mechanics approach. In this approach, a composite with two elastic phases is considered. These phases are fibre and matrix.

The average stress in composite is given from Equation (7.74). In this equation, the stresses in individual phases are used to give following equation

(7.99)

where  and  are the local stresses in fibre and matrix, respectively. Now let us define the volume averaged stress in fibre as

(7.100)

and volume averaged stress in matrix as

(7.101)

Putting these two definitions in Equation (7.99) and adjusting the  and  terms properly, we get the average stress in composite as

(7.102)

Similarly, we define the volume averaged strains in fibre and matrix as

(7.103)

Note: The average stresses in Equation (7.100) and (7.101) and average strains in Equation (7.103) are also known as phase averaged stresses and phase averaged strains, respectively.

Putting these definitions for the definition of average strain in composite, we get

(7.104)

Now let us derive the average stress in fibre and matrix using the pointwise constitutive equation for fibre and matrix in Equations (7.100) and (7.101) as

(7.105)