Stress and strain

Abstract

Stress (force per unit area) and strain (change in length per unit length) are both symmetric second rank tensors like the dielectric constant, but they are not property tensors. Experimenters are at liberty to apply different types of forces to a specimen, therefore there is no reason that the stress tensors (and the resulting strain tensor) must conform to the crystal symmetry. Stress and strain tensors do not obey Neumann’s Law. They are sometimes called field tensors to distinguish them from property tensors like the dielectric constant. Property tensors are relationships between field tensors. For the same reason, electric and magnetic fields are first rank field tensors, as are magnetization and polarization. They do not obey the symmetry principles as first rank property tensors such as pyroelectricity or the magnetocaloric effect are required to do. In arbitrary coordinate systems, the state of stress in a specimen is described by nine components of the stress tensor: The first subscript refers to the direction of the force, the second to the normal to the face on which the force acts. To prevent translational motion, each force is balanced by an equal and opposite force on the reverse side of the specimen. Stress component X22 is a tensile stress in which both the force and the normal are along Z2, and X12 is a shear stress in which a force along Z1 acts on a face normal to Z2. For static equilibrium, the torques must be balanced, otherwise rotation occurs; this means that the stress tensor must be symmetric with X12 = X21, X13 = X31, and X23 = X32. Thus the stress state is specified by six independent components: three tensile stresses X11, X22, and X33, and three shear components X12, X13, and X23. For an arbitrary axial system (new axes) the general stress tensor can be rewritten as a 6 × 1 column matrix: The first three components in the column matrix are tensile stresses along Z'1, Z '2, Z '3, and the last three are shear stresses about Z '1, Z '2, Z '3. Both the tensor and matrix forms are widely used in the literature.