Abstract Asymptotic solutions are presented for a stationary crack normal to the boundary between two elastically mismatched solids such that the crack tip is located at the interface. The second-order term in the elastic asymptotic expansion was determined as a function of elastic mismatch for a thin cracked film on a substrate and for a thin cracked lamina between two substrates. Elastic–plastic analysis was performed using both modified boundary layer formulations and full field analyses. Analytic and numerical solutions in small strain yielding identify elastic mismatch and the T-stress as the determinants of crack tip constraint. The effect of constraint on the competition between interface failure and penetration is discussed.