The meaning of the stress tensor is elucidated and its representation with the Mohr circle is presented.

The distribution of the external forces acting on a body affects both the internal and external deformation of the body. The internal deformations in particular depend on how the forces are distributed throughout the body. Stress is a key concept that gives us a way to characterize those internal force distributions. This chapter will discuss in depth the stress concept, including stress transformations, principal stresses, states of stress, and Mohr's circle. MATLAB® will be used as the principal tool for calculations.

The analysis of normal and shear stresses over differently oriented surface elements through a considered material point is presented. The Cauchy relation for traction vectors is introduced, which leads to the concept of a stress tensor. The analysis is presented of one-, two-, and three-dimensional states of stress, the principal stresses (maximum and minimum normal stresses), the maximum shear stress, and the deviatoric and spherical parts of the stress tensor.The equations of equilibrium are derived and the corresponding boundary conditions are formulated.

At any point in a glacier, there are three normal and six shear stresses.Coordinate axes can be chosen so that the shear stresses vanish. The remaining normal stresses are known as the principal stresses. Certain combinations of the stresses do not vary with the orientation of the coordinate axes. These are known as invariants of the stress tensor. The second invariant is one half the sum of squares of all nine stresses in the tensor. This stress is used in the common flow law for ice, so the deformation rate depends on all the stresses acting, not just on those acting in the direction of the deformation.Balancing forces on an element of ice at a point leads to an equation for the conservation of linear momentum. The strain along a line is defined as the change in length per unit length. There are also three normal and six shear strain rates. Again, axes can be chosen so that the shear strain rates disappear.The remaining normal strain rates are called the principal strain rates.In an isotropic material the principal axes of stress and strain rate coincide. Ice is commonly assumed to be isotropic for purposes of theoretical calculations, although this is clearly not true.

In the previous chapter, it was shown that an aligned composite is usually stiff along the fibre axis, but much more compliant in the transverse directions. Sometimes, this is all that is required. For example, in a slender beam, such as a fishing rod, the loading is often predominantly axial and transverse or shear stiffness are not important. However, there are many applications in which loading is distributed within a plane: these range from panels of various types to cylindrical pressure vessels. Equal stiffness in all directions within a plane can be produced using a planar random assembly of fibres. This is the basis of chopped-strand mat. However, demanding applications require material with higher fibre volume fractions than can readily be achieved in a planar random (or woven) array. The approach adopted is to stack and bond together a sequence of thin ‘plies’ or ‘laminae’, each composed of long fibres aligned in a single direction, into a laminate. It is important to be able to predict how such a construction responds to an applied load. In this chapter, attention is concentrated on the stress distributions that are created and the elastic deformations that result. This involves consideration of how a single lamina deforms on loading at an arbitrary angle to the fibre direction. A summary is given first of some matrix algebra and analysis tools used in elasticity theory.