An exact semi-analytical solution in the undrained cavity expansion can be obtained on the basis of the MCC model to research the cone penetration test and the pile driving. The field around the cavity can be divided into three zones: critical zone, plastic deformation zone and elastic deformation zone.In the elastic zone, an analytical solution for the distributions of stress and excess pore pressure is deduced according to the elastic theory. In the critical and plastic zone, a set of first-order nonlinear ordinary differential equations concerning stress can be obtained according to the associated flow rule and the lagrangian analysis method. The stressss and pore pore pressure can be solved as an initial value problem starting at the elastic-plastic boundary. The results show that the isotropic over consolidation ratio has a significant influence on the stresses and the excess pore pressure.The shear modulus vary significantly with the cavity radius and the isotropic over consolidation in the course of cavity expansion.