The asymptotic solution of a contact problem with a half-unknown boundary of the contact region. (Q2761303)

A problem is considered on impressing a die with a base in the form of elliptical paraboloid NEWLINE\[NEWLINE x_3 = -\Phi_\varepsilon(x_1,x_2),\quad \Phi_\varepsilon(x_1,x_2) = A[x_1^2+x_2^2 + \varepsilon(x_1^2-x_2^2)], NEWLINE\]NEWLINE where \(\,\varepsilon>0\,\) is a small parameter, into the elastic half-space \(\,x_3>0\,\) without friction. Under some assertions a solution is constructed by joining the asymptotic expansions. The asymptotics of the contact area boundary is written down in the explicit form.