Consider the two bodies shown acted on by uniformly distributed pressure $p.$ Perform contact analysis and calculate the pressure distribution over the contact plane.

$E=2\times10^5\text{ MPa},$ $\nu=0.3.$

None.

$p=-10\text{ MPa}.$

• Pentahedra, CUBE55K1

The contact search is performed at external Gauss integration points accessed by the FE algorithm. Contact constraints are enforced by the penalty method. Thus, the unknown contact pressure $p_\mathtt{IG}$ is approximated by the penalty function $$p_\mathtt{IG}=\xi\pi_\mathtt{IG},$$ where $\xi$ denotes the value of the penalty parameter and $\pi_\mathtt{IG}$ is the penetration determined at the Gauss integration point IG. The essential part of the analysis is a good choice of the penalty parameter $\xi,$ whose value can be estimated from the stiffness of a cubic element $$\xi=f_s\frac{K}{\sqrt{\det\mathbf{J}^S}},$$ where $f_s$ is the inverse of relative displacement error, $K$ the bulk modulus and $\mathbf{J}^S$ the surface Jacobian. For example, if we require the relative displacement error to be $0.01$ (i.e., the ratio of penetration to the displacement of the contact plane should not exceed $0.01$), we set the penalty parameter to $\xi=10^{13}~\text{N/m}^3.$