Abstract: Based on the theory and method for variational problems of variable boundary, the second variation of the energy functional for crack extension is obtained. If the second variation vanishes, the critical condition and the critical velocity for crack extension are got. Then the formula of life prediction can be applied to various conditions of solid system, such as elasticity, nonlinear elasticity, elasto-plasticity and creep condition, which is of great importance theoretically adn of great value practically.