Abstract A crack model is introduced in which the non-linear force-deformation relation, just ahead of the crack tip, is taken into account. The analysis of the crack model leads to an integral equation of the Fredholm type, which is solved numerically by approximating it by a finite system of linear equations. The distributions of stress and deformation are calculated for different loads increasing from zero to the maximum value of stable equilibrium. The conditions, for which instability is reached, are also determined by investigating the eigenvalues of the system. Comparison of the results with existing theories is made.