Approximation of work calculation in theory of geotechnical plasticity

The expressions of generalized shear stress and generalized shear strain are introduced. After analyzing the two expressions, mistakes in calculating the increment of generalized shear stress and generalized shear strain are pointed out, and then the right calculations are given. Based on this, two expressions of work increment that one is described by mean stress, volumetric strain increment, generalized shear stress and generalized shear strain increment and the other is described by three-dimensional stresses and three-dimensional strain increments are compared. The conclusion is that the two expressions are not always consistent except in some particular conditions. An example of the particular conditions is that loading in constant Lode angle and also the Lode angle of the three-dimensional stresses is equal to that of the three-dimensional strain increments. The same inconsistency still exists in applications of plastic work increment, while unfortunately the equation of Lode angle of the three-dimensional stresses and the three-dimensional strain increments is not always satisfied for practical geomaterials. So in the theory of geotechnical plasticity, for the condition of loading in constant Lode angle, the plastic work increment expressed by mean stress, plastic volumetric strain increment, generalized shear stress and generalized plastic shear strain increment is just a kind of approximation.