PSI - Issue 28

Yakubu Kasimu Galadima et al. / Procedia Structural Integrity 28 (2020) 1094–1105 Author name / Structural Integrity Procedia 00 (2019) 000–000

1099

6

of the form:

0

0  

t

x

x V

 

(16)

i

ij j

j

In the Periodic Boundary Condition (PBC), the displacement field 0

i u over V  is of the form:

0 u x u     0 ij j i

0

x V

 

(17)

i

j

where 0 i u  is a displacement fluctuation that is assumed to be periodic over the RVE. That is, 0

i u  takes the same value

at two homologous points on opposite face of the RVE. 5. Numerical results

In this section, a numerical example of homogenization problem will be solved to validate the PD homogenization scheme to be used. Thereafter, a numerical problem will be solved to study the effect of inclusion shape on the effective properties of composites. 5.1. Model validation Before proceeding to achieve the objective of this work, a numerical problem will first be solved to validate the homogenization scheme. Consider a composite with long parallel cylindrical fibres as shown in Fig. 2a. The corresponding 2D RVE is depicted in Fig. 2b. The material parameters of the matrix and fibre are as follows: 200GPa fiber E  , 1/ 3 fiber   , 100GPa matrix E  , 1 / 3 matrix   . The problem is solved for fibre volume fraction 0.6 f  . The RVE is subjected to a PBC.

Fig.2. (a) Composite material, (b) RVE

This homogenization problem was solved using both Finite Element Method (FEM) and PD homogenization schemes. The deformed shape due to the application of loads on the RVE are shown in Fig. 3. The components of the effective stiffness tensor and hence the effective elastic modulus as obtained from the FEM and PD homogenization are shown in Table 1.