3.3.2.3 Tensor Transformations

Using the transformation operator $\hat{U}(\alpha,\beta)$ the strain tensor elements are transformed as follows:

$$\varepsilon^{'}_{\alpha\beta}=U_{i\alpha}U_{j\beta}\varepsilon_{ij}.$$ (3.46)

Therefore the main task is to determine the elements of the strain tensor in the interface coordinate system. The strain tensor elements in the principle coordinate system are then obtained as:

$$\varepsilon_{\alpha\beta}=U_{\alpha i}U_{\beta j}\varepsilon^{'}_{ij}.$$ (3.47)

The elastic stiffness tensor is transformed analogously:

$$c_{\alpha\beta\delta\gamma}^{'}=U_{i\alpha}U_{j\beta}U_{k\delta}U_{l\gamma}c_{ijkl}$$ (3.48)

S. Smirnov: