PSI - Issue 38

2

S. Spanke et al. / Procedia Structural Integrity 38 (2022) 220–229 Author name / Structural Integrity Procedia 00 (2021) 000±000

221

The investigated material is a CFRP composite made of an epoxy resin from the manufacturer Hexion with the trade name RIM935. The used reinforcing fiber is a high tensile fiber from the company Zoletek (PX35 50K).

In the first part of this paper, the theory of the homogenization method is briefly introduced and the model setup is explained. Subsequently, the calculated homogenized e ff ective material parameters are compared with experimental results and evaluated. Finally, the main findings are summarized and discussed.

Nomenclature

σ i j Macroscopic stress tensor σ i j Microscopic stress tensor

δε i j Variation macroscopic strain tensor δε i j Variation microscopic strain tensor t i Chauchy’s stress δ w i Variation fluctuation I i j Unity tensor x j Unit vector u i Displacement L x RVE length in x direction L y RVE length in y direction L z RVE length in z direction

2. Theory and model building

The term representative volume was firstly introduced by Hill (1963). The current literature refer to it as represen tative volume elements (RVE), which represent the microstructure in a limited space. In general, the microstructure can be of any complexity, but the dimensions of an RVE model must be large enough to represent the microscopic behavior under investigation. In this work, the microstructure consists of fiber, matrix system and interface, that are defined as independent material parts.

2.1. Theory

The formation of volume integrals of heterogeneous microscopic quantities gives rise to averaged homogeneous macroscopic quantities:

1 | RVE | RVE 1 | RVE | RVE

σ i j =

σ i j dv

(1)

δε i j =

δε i j dv

(2)

For the numerical determination of the homogenized e ff ective material parameters of the unidirectional layer, it is nec essary to find boundary conditions for the RVE system that are compatible with the Hill-macro-homogenity condition Hill (1963). In the following, these compatible boundary conditions are developed based on the virtual work.

1 | RVE | RVE

σ i j δε i j =

σ i j δε i j dv

(3)

The local equilibrium position must be satisfied for any RVE subareas.

σ i j , j = 0

(4)

The following identity is introduced for σ i j δε i j : σ i j δ u i , j = σ i j , j δ u i + σ i j δ u i , j = σ i j δε i j

(5)