THE THEORY OF SHEAR-STRESS AND SHEAR STRAIN ON PLANES INCLINED TO THE PRINCIPAL DIRECTIONS

Given the magnitudes and orientations of the principal stresses, the normal and shear components of stress acting on a plane of arbitrary orientation are determined either graphically or with the aid of a novel type of vector product. The normal stress σn due to the action of a stress vector σ on a plane of pole n is the star product n*σ, which is defined as a vector oriented parallel to n with magnitude equal to the dot product n·σ. The shear stress vector σ3, is given by the vector difference σ -σn. In the case of a deformation, given the magnitudes and orientations of the principal stretches, the shear strain of an arbitrary line is determined by a related, but significantly different, procedure. © 1990.