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Normal contact stiffness

The traction excerted by the master face on the slave face at a slave integration point p can be written analogous to Equation (166):

$\displaystyle \boldsymbol{t_{(n)}}= f(r) \boldsymbol{n}.$ (245)

For simplicity, in the face-to-face contact formulation it is assumed that within an increment the location $ (\xi _{m_k}, \eta _{m_k})$ of the projection of the slave integration points on the master face and the local Jacobian on the master face do not change. Consequently (cf. the section 6.7.5):

$\displaystyle \frac{\partial \boldsymbol{m} }{\partial \boldsymbol{u_p} }=\frac...
...bol{u_p} } = \frac{\partial \eta }{\partial \boldsymbol{u_p} } =\boldsymbol{0}.$ (246)


$\displaystyle \frac{\partial \boldsymbol{r} }{\partial \boldsymbol{u_p} }= \boldsymbol{I},$ (247)

which leads to

$\displaystyle \frac{\partial \boldsymbol{t_{(n)}}}{\partial \boldsymbol{u_p} } = \frac{\partial f}{\partial r} \boldsymbol{n} \otimes \boldsymbol{n}.$ (248)

This is the normal contact contribution to Equation (244).

guido dhondt 2018-12-15