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 (209):

$$\boldsymbol{t_{(n)}}= f(r) \boldsymbol{n}.$$ (288)

For simplicity, in the face-to-face penalty 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):

$$\frac{\partial \boldsymbol{m} }{\partial \boldsymbol{u_p} }=\frac... ...bol{u_p} } = \frac{\partial \eta }{\partial \boldsymbol{u_p} } =\boldsymbol{0}.$$ (289)

and

$$\frac{\partial \boldsymbol{r} }{\partial \boldsymbol{u_p} }= \boldsymbol{I},$$ (290)

which leads to

$$\frac{\partial \boldsymbol{t_{(n)}}}{\partial \boldsymbol{u_p} } = \frac{\partial f}{\partial r} \boldsymbol{n} \otimes \boldsymbol{n}.$$ (291)

This is the normal contact contribution to Equation (287).