Kinking In Elastoplastic Continua Under Plane Strain

Many studies have been devoted to kinking and the study of stress-induced phase transition, but the continuum models used are usually based on thermoelasticity, nonlinear elasticity or hyperelasticity. In this paper, we will provide a general analysis of kinking in elastoplastic continua under plane strain deformation by modeling kinking as a stress-induced phase transformation.Ericksen has studied on stress-induced phase transition for nonlinear elastic bars using Maxwell stress. Phase transformation in elastoplastic bars are investigated for both infinitesimal deformation and finite deformation by Zhang etc. It is proved that for any assumed strain-softening curve the algebraic sum of the areas enclosed by the straight line of Maxwell stress and the assumed curve is always equal to zero, which agrees with the result give by Ericksen for the analysis of phase transformations in nonlinearly elastic bars. Numerical examples demonstrate that the jump of strain across the interface between phases is very large when the ratio of plastic tangential modulus to elastic modulus is smaller. In this case it is necessary to analyze the phase transformation using the finite deformation theory to induce errors.In the present paper a general analysis of kinking in elastoplastic continua under plane strain deformation is presented by modeling kinking as a stress-induced phase transformation. The strain-energy function for nonlinear elastic materials is extended to obtain a stress-work function for elastoplastic continua. With the aid of the stress-work function, plane strain kinking under biaxial loading in isotropic elastoplastic continua is analyzed. The analysis is reduced to finding the minimum value of the loading at which the jump conditions of phase transformations have a unique, real, physically acceptable solution. The equations are solved numerically by homotopy continuation methods. A numerical procedure is illustrated and it is verified that kinking can definitely occur in elastoplastic continua with strain-softening behavior. The kinking stress, the stresses and strains inside and outside the kink-band, the kink band orientation angle and the kink angle, all that need to be determined for kinking, are determined. A description about the variation of these stresses and angles to different strain-induced coefficients and load ratios are also supplied. The results indicated that strain-induced coefficient has little influence on kinking stress, while a bigger influence on kink band orientation angle exists when the load ratio is greater than zero comparing with smaller than zero. Both of these two curves are monotony. It also demonstrates that the strain-induced coefficient has an obvious influence on the kink angle, and an upper limit exists around the load ratio equals to negative one. Results coincide with others indicate the method and the program's correctness. When wrong Maxwell relationship is used, the wrong calculation results point out it correctly.