21.5.2 Three-dimensional Interface Behavior

The two-dimensional interface model is extended to a three-dimensional (see Van Zijl [106] ), which enables the description of delamination (tension cut-off) and relative shear-slipping of two planes (Coulomb friction). No three-dimensional compression cap is implemented in DIANA-9.3. Now the generalized stress and strain vectors are:

$$\boldsymbol\sigma \left\{\vphantom{ \negthickspace \begin{array}{c} \sigma \\ \tau_{s} \\ \tau_{t} \end{array} \negthickspace }\right. \begin{array}{c} \sigma \\ \tau_{s} \\ \tau_{t} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \sigma \\ \tau_{s} \\ \tau_{t} \end{array} \negthickspace }\right\} \boldsymbol\varepsilon \left\{\vphantom{ \negthickspace \begin{array}{c} u \\ v \\ w \end{array} \negthickspace }\right. \begin{array}{c} u \\ v \\ w \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} u \\ v \\ w \end{array} \negthickspace }\right\}$$ (21.62)

where the shear stresses $\tau_{{s}}^{}$ and $\tau_{{t}}^{}$ act in the local plane of the interface, v and w are the relative shearing displacements in the interface plane and $\sigma$ and u the stress and relative displacement respectively normal to the plane. The stiffness matrix is defined as

$$\left[\vphantom{ \; k_{n} \; k_{s} \; k_{t} \; }\right. \left.\vphantom{ \; k_{n} \; k_{s} \; k_{t} \; }\right]$$ (21.63)

Figure 21.18 shows the three-dimensional interface material law.

Figure 21.18: Three-dimensional interface yield function

Apart from the added stress and strain component, the two-dimensional tension criterion f₂ of (21.51) remains unchanged. For the Coulomb friction part the yield function becomes

$$\sqrt{{ \tau_{s}^{2} + \tau_{t}^{2} }} \sigma \Phi$$ (21.64)

As for the two-dimensional case adhesion softening and friction softening are modeled as described by (21.38) and (21.39). A non-associated plastic potential is chosen, giving the flow rule

$$\Delta \boldsymbol\varepsilon \Delta \lambda {\frac{{ \partial g }}{{ \partial \boldsymbol{\sigma } }}} \Delta \lambda \left\{\vphantom{ \negthickspace \begin{array}{c} \Psi \\ [\bigs... ...} }{ \sqrt{ \tau_{s}^{2} + \tau_{t}^{2} } } \end{array} \negthickspace }\right. \begin{array}{c} \Psi \\ [\bigskipamount] \dfrac{ \tau_{s} }{ \s... ...\ [4ex] \dfrac{ \tau_{t} }{ \sqrt{ \tau_{s}^{2} + \tau_{t}^{2} } } \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \Psi \\ [\bigsk... ... }{ \sqrt{ \tau_{s}^{2} + \tau_{t}^{2} } } \end{array} \negthickspace }\right\}$$ (21.65)

with the mobilized dilatancy $\Psi$ defined as before by (21.49). However, now the strain softening is governed by the equivalent shear displacement

$$\Delta \kappa \sqrt{{ \left( \Delta v_{\mathrm{p}} \right)^{2} + \left( \Delta w_{\mathrm{p}} \right)^{2} }} \Delta \lambda$$ (21.66)