Starting from a general description of model gels and key experimental variables, thermodynamics and swelling equilibria are described. Based on the fundamentals, behaviors of gels under tension, shear, and temperature variation are explained using a combination of theory and experiments. Phase transitions of gels, where volume changes of several orders of magnitude are of common occurrence, are presented in details to enable researchers to design new hydrogels for their intended industrial purpose.

The generalized Hooke's law is introduced, which represents six linear relations between the stress and strain components in the case of small elastic deformations. For isotropic materials, only two independent elastic constants appear in these stress–strain relations. Each longitudinal strain component depends linearly on the three orthogonal components of the normal stress; the relationship involves two constants: Young's modulus of elasticity and Poisson's coefficient of lateral contraction. Each shear strain component is proportional to the corresponding shear stress component; the shear modulus relates the two. The volumetric strain is proportional to the mean normal stress, with the elastic bulk modulus relating the two. The inverted form of the generalized Hooke's law is derived, which expresses the stress components as a linear combination of strain components. Lamé elastic constants appear in these relations. The Duhamel–Neumann law of linear thermoelasticity is formulated, which incorporates the effects of temperature on stresses and strains. The Beltrami–Michell compatibility equations with and without temperature effects are derived.

In the previous chapter, some background was provided about types of reinforcement and their distribution within different matrices. Attention is now turned to predicting the behaviour of the resulting composites. The prime concern is with mechanical properties. The reinforcement is usually designed to enhance the stiffness and strength of the matrix. The details of this enhancement can be rather complex. The simplest starting point is the elastic behaviour of a composite with aligned long (continuous) fibres. This arrangement creates high stiffness (and strength) in the fibre direction. However, it is also important to understand the behaviour when loaded in other directions, so the treatment also covers transverse loading. In this chapter, and in the following one, perfect bonding is assumed at the fibre/matrix interface. Details concerning this region, and consequences of imperfect bonding, are considered in Chapter 6.