A way to determine the positive direction of the shear force on the elemental area

Abstract

This study is devoted to amendment to some concepts related to the construction of Mohr's circles on the plane of variables “normal and tangential components of the stress vector on the elemental area”. As the tangential component is positive by definition (as a square root), we have to talk only about semicircles instead of Mohr's circles. To introduce negative values, we bring in the concept of the positive direction of the shear force connected with the projection on the first principal direction of the stress tensor. The considered approach allows us to determine the direction of the shear force (positive/negative) relatively to the principal axes of the stress tensor on any elemental area with known values of the principal stresses. The same approach is applied to the vector of deformations on the elemental area. To represent the application of these two vectors on the elemental area, we consider the work done by the forces acting (in the form of the Cauchy vector of stresses) on changes in the vector of strains. It is also shown that this work, even in the case of elasticity, does not always lead to an unambiguous result. It does not depend on the loading path only on octahedral elemental areas. The foregoing does not negate the existence of the elasticity potential as a whole (non-potency on one elemental area is annulled by the same non-potency on the other one). All this is important when, based on a set of slip areas, physical theories of plasticity and destruction (slip theories) are constructed.