Linear elasticity of polymers

In other applications where large-strain representation is necessary, logarithmic strains are often used. These are defined as 

There are other more formal representations of finite strain in applied mechanics, which we do not present here. They can be found in Green and Zerna (1954). 

The most prevalent and widely developed constitutive connections of polymers between strain and stress are dealt with in linear elasticity by applying the generalized form of Hooke s law which is presented in Chapter 4 for anisotropic solids of different symmetry classes starting with orthotropic solids and progressing up to isotropic solids. Here and in the following chapters we shall develop only the connection for isotropic solids, which is the most useful one and most often is quite sufficient in development of concepts. 

For isotropic elastic solids there are only two independent elastic constants, or compliances. While Young s modulus E and the shear modulus // are the most widely used, we shall choose as the two physically independent pair of moduli the shear modulus /i and the bulk modulus K, where the first gauges the shear response and the second the bulk or volumetric response. However, in stating the linear elastic response in the equations below we still choose the more compact pair of E and //. Thus, for the six strain elements we have 

In the physically separate and distinct representation of shear and volumetric response we introduce the concepts of the deviatoric stress and the deviatoric strain which are free of volumetric response and represent only the shear response. These are defined as  

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