Constitutive Equations: General Principles

The general principles of continuous media apply to a large variety of materials. Over the last half-century the development and production of new materials, initially linked to oil derivatives like polymers, but further on to composites, bio-materials, food and drugs, etc. launched the need to describe mathematically the mechanical behavior of those products. The principles of writing relevant constitutive equations were elaborated step by step by generalization of the concepts of mechanics to continua and by a constant interplay between theory and experiments. This lengthy process gave rise to the first nonlinear models that constituted the cornerstone for the development of numerical simulations. The theory of constitutive equations elaborates relations linking the stress tensor to the motion. These constitutive relationships quantify the mechanical behavior of these materials. In this monograph this concept of constitutive equations will also be used, but also extended and adapted to represent the behavior of a turbulent flow. Rivlin (Q Appl Math 15:212–215, 1957) suggested such an analogy between a non-Newtonian fluid and turbulent Newtonian flow over a half-century ago. The analogy was primarily based on the appearance of secondary motions in both the laminar flow of a non-Newtonian fluid and the turbulent flow of a Newtonian fluid in a pipe with elliptical cross-section; whereas, for the laminar flow of a Newtonian fluid the flow is rectilinear. Such behavior is induced through the appearance of normal stress effects, that is, normal stresses associated with the extra-stress of the non-Newtonian fluid in a laminar flow and the turbulent stress of the Newtonian fluid in a turbulent flow.