PSI - Issue 38

S. Spanke et al. / Procedia Structural Integrity 38 (2022) 220–229

222

3

Author name / Structural Integrity Procedia 00 (2021) 000±000

The gradient of the variation δ u i , j can be transformed into the variation δε i j by observing the symmetry condition of a gradient:

1 2 u i , j

+ u j , i = δε i j

(6)

δ u i , j =

Using Gaussian integration and observing Cauchy’s fundamental theorem σ i j n j = t i , we obtain the following for equation (3):

1 | RVE | RVE

σ i j δ u i

1 | RVE | γ RVE

σ i j δε i j =

dv =

t i δ u i da

(7)

, j

With the introduction of a periodic variation of δ u i which is composed of a fluctuation δ w i and the macroscopic variation δε i j x j the following applies:

δ u i = δ w i + δε i j x j

(8)

Fig. 1. (a) Homogeneous body; (b) Inhomogeneous microstructure.

Substituting the periodic variation δ u i equation (8) into equation (7), it follows that:

t i δ w i + δε i j x j da

1 | RVE | γ RVE

σ i j δε i j =

(9)

With assumption of x j , i = I i j and σ i j x j , i

= σ i j , i x j + σ i j I i j = σ i j

(10)

results in the following identity for the macroscopic stress tensor:

1 | RVE | RVE

σ i j x j

1 | RVE | γ RVE

σ i j =

dv =

t j x j da

(11)

, i

Taking into account equation (11), the virtual work results in the following expression:

1 | RVE | γ RVE

σ i j δε i j = σ i j δε i j +

t i δ w i da

(12)

To satisfy Hill’s homogeneity condition, the boundary integral of the variation of the fluctuation δ w i must satisfy the following condition: γ RVE t i δ w i da = 0 (13) 2.2. Numerical definition of the periodic boundary conditions The numerical implementation of the periodic boundary conditions must satisfy equation (13) and is illustrated in the following using the unit cell. The structure of the unit cell is composed of lines vertices and faces and is oriented to Barbero (2008).