3 - Stress–Strain Relations

Summary

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.