- 1.1 Introduction
- 1.2 Scope of Treatment
- 1.3 Analysis and Design
- 1.4 Conditions of Equilibrium
- 1.5 Definition and Components of Stress
- 1.6 Internal Force-Resultant and Stress Relations
- 1.7 Stresses on Inclined Sections
- 1.8 Variation of Stress Within a Body
- 1.9 Plane-Stress Transformation
- 1.10 Principal Stresses and Maximum in-plane Shear Stress
- 1.11 Mohr's Circle for Two-Dimensional Stress
- 1.12 Three-Dimensional Stress Transformation
- 1.13 Principal Stresses in Three Dimensions
- 1.14 Normal and Shear Stresses on an Oblique Plane
- 1.15 Mohr's Circles in Three Dimensions
- 1.16 Boundary Conditions in Terms of Surface Forces
- 1.17 Indicial Notation
- References
- Problems
1.17 Indicial Notation
A system of symbols, called indicial notation, index notation, also known as tensor notation, to represent components of force, stress, displacement, and strain is used throughout this text. Note that a particular class of tensor, a vector, requires only a single subscript to describe each of its components. Often the components of a tensor require more than a single subscript for definition. For example, second-order or second-rank tensors, such as those of stress or inertia, require double subscripting: t _{ij} , I _{ ij }. Quantities such as temperature and mass are scalars, classified as tensors of zero rank.
Tensor or indicial notation, here briefly explored, offers the advantage of succinct representation of lengthy equations through the minimization of symbols. In addition, physical laws expressed in tensor form are independent of the choice of coordinate system, and therefore similarities in seemingly different physical systems are often made more apparent. That is, indicial notation generally provides insight and understanding not readily apparent to the relative newcomer to the field. It results in a saving of space and serves as an aid in nonnumerical computation.
The displacement components u, v, and w, for instance, are written u _{1}, u _{2}, u _{3} (or u_{x} , u_{y} , u_{z} ) and collectively as u_{i} , with the understanding that the subscript i can be 1, 2, and 3 (or x, y, z). Similarly, the coordinates themselves are represented by x _{1}, x _{2}, x _{3}, or simply x_{i} (i = 1, 2, 3), and x_{x}, x_{y}, x_{z} , or x_{i} (i = x, y, z). Many equations of elasticity become unwieldy when written in full, unabbreviated term; see, for example, Eqs. (1.28). As the complexity of the situation described increases, so does that of the formulations, tending to obscure the fundamentals in a mass of symbols. For this reason, the more compact indicial notation is sometimes found in publications.
Two simple conventions enable us to write most equations developed in this text in indicial notation. These conventions, relative to range and summation, are as follows:
- Range convention: When a lowercase alphabetic subscript is unrepeated, it takes on all values indicated.
- Summation convention: When a lowercase alphabetic subscript is repeated in a term, then summation over the range of that subscript is indicated, making unnecessary the use of the summation symbol.
The introduction of the summation convention is attributable to A. Einstein (1879–1955). This notation, in conjunction with the tensor concept, has far-reaching consequences not restricted to its notational convenience [Refs. 1.14 and 1.15].