Linear Elastic Response of Materials

Under more complex states of loading, Hooke s law required elaboration. The generalization of Hooke s law is the assertion that an arbitrary component of the 

What this equation tells us is that a particular state of stress is nothing more than a linear combination (albeit perhaps a tedious one) of the entirety of components of the strain tensor. The tensor Cijn is known as the elastic modulus tensor or stiffness and for a linear elastic material provides nearly a complete description of the material properties related to deformation under mechanical loads. Eqn (2.52) is our first example of a constitutive equation and, as claimed earlier, provides an explicit statement of material response that allows for the emergence of material specificity in the equations of continuum dynamics as embodied in eqn (2.32). In particular, if we substitute the constitutive statement of eqn (2.52) into eqn (2.32) for the equilibrium case in which there are no accelerations, the resulting equilibrium equations for a linear elastic medium are given by 

As usual, we have invoked the summation convention and in addition have assumed that the material properties are homogeneous. For an isotropic linear elastic solid, the constitutive equation relating the stresses and strains is given by 

Note that these equations are a special case of the equilibrium equations revealed in eqn (2.53) in the constitutive context of an isotropic linear elastic solid. 

As we will see in coming chapters, the logic that rests behind this model is a 

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