Displacements- Vectors and Tensors
A second basic concept or theme in solid mechanics, is the development of a general method of describing changes in physical quantities within an artificial coordinate system. As we shall see, this involves tensors of various orders.
A tensor is a physical quantity which, in its essence, remains unchanged when subject to any admissible transformation of the reference frame. The rules of tensor transformation can be expressed analytically (or graphically), but it is the unchanging aspects of a tensor that verify its existence and are the most interesting physically. Seldom in solid mechanics is a tensor confused with a matrix which is simply an operator. A tensor can be written in matrix form, and therefore the two can look alike on paper, but a matrix as an array of numbers has no physical meaning and the transformation of a matrix to a new reference frame is impossible. Matrix notation and matrix algebra can apply to tensors, but few matrices, as such, are found in the study of solid mechanics.
All of physics is a study of tensors of some order. Scalars such as temperature or pressure, where one invariant quantity (perhaps with a sign) describes them, are tensors of order zero while vectors such as force and acceleration are tensors of the first order. Stress, strain, and inertia are second-order tensors, sometimes called dyadics. Since the physical universe is described by tensors and the laws of physics are laws relating them, what we must do in mechanics is learn to deal with tensors whether we bother to call them that or not.
A transformation tensor is the next higher order than the tensor it trans-forms. A tensor of second order, therefore, changes a vector at some point into another vector while, as we shall see, it takes a fourth-order tensor to transform stress or strain.* A tensor field is simply the spacial (x, y, z) and/or time varia-tion of a tensor. It is this subject, scalar fields, vector fields, and second-order tensor fields that is the primary focus of solid mechanics. More specifically, the goal is to determine the vector field of displacement and second-order stress and strain fields in a 'structure,' perhaps as a function of time as well as position, for specific material properties (elastic, viscoelastic, plastic) due to loads on the boundary, body forces, imposed displacements, or temperature changes.