2. Constitutive Equation

A constitutive equation is a mathematical relation between two or more physical quantities. To define such mathematical relations, coefficients that are specific to a material, or to a composite material, known as material constants, are needed.As quoted above, Robert Hooke (see Sect. F.2 for a biography) was the first to establish a constitutive relation for elastic bodies by observing that the elongation of a coil spring, a spiral spring, a wire string, and also the bending of a straight piece of wood, are directly proportional to the weight attached to them (Hooke R, De Potentia Restitutivâ, or of spring, explaining the power of springy bodies. Martyn, London, 1678, 56pp). When this observation is used to establish linear relations between the stress and strain of an elastic body, it is called Hooke’s law, to honor the original contribution by Hooke.Since Hooke’s observation, it took almost 130 years, until 1807, that a material constant was explicitly defined by Thomas Young (Young T, A course of lectures on natural philosophy and the mechanical arts, vol 2. Joseph Johnson, London, 1807, 738pp) (see Sect. F.4 for a biography), and he stated:

The modulus of the elasticity of any substance is a column of the same substance, capable of producing a pressure on its base which is to the weight causing a certain degree of compression, as the length of the substance is to the diminution of its length.

In the above, Young first defined a weight of modulus of elasticity, which was dependent on the cross-sectional area of the column. When the weight of modulus is divided by the cross-sectional area to yield a height of modulus, then it becomes a material property only, which in present-day terminology is called the Young’s modulus. In fact, it was quite amazing that Young was able to determine the Young’s modulus of a steel to be 2. 9 × 10⁷ psi, which is the same as the present-day determined value, by finding the frequency of vibration of a tuning fork (Timoshenko SP, History of strength of materials. Dover, New York, 1983, 452pp; Young T, A course of lectures on natural philosophy and the mechanical arts, vol 2. Joseph Johnson, London, 1807, 738pp)!Other examples of constitutive equations include Newton’s law of viscosity, relating the rate of fluid shear strain to applied shear stress, through a material constant called viscosity, and Darcy’s law of porous medium, relating the fluid specific flux to the applied head gradient, through a constant called hydraulic conductivity, which is a combined property of the porous medium void space geometry and the fluid residing in it.These constitutive equations (laws) are often combined with other physical laws of more general nature, such as mass, momentum, and energy conservation, which are not specific to a material, to form a set of governing equations to predict the physical response of matters subject to disturbances (see Chap. 6).In this chapter, we shall construct the constitutive equations for poroelasticity. Particularly, we are interested in the deformation of the porous solid and the fluid, subjected to applied forces and pressure.