Elasticity theory. IV. The stress–strain relation

The stress–strain relation is analyzed in terms of the gyration tensor where the sum is over all N atomic groups comprising the elastomeric material. All groups are taken for convenience to have the same mass. This tensor is linear in the strain, and furthermore it may be evaluated from a configuration integral. The definition of the strain by use of S^αβ gives new insight into the connection between statistical and continuum mechanics, and further clarifies the significance of “affine deformation.” Stress components are obtained as derivatives of the (Helmholtz) free energy A with respect to the invariants of S^αβ. The first invariant is just the radius of gyration S² of the network, and the dependence of A on S² alone provides a basis for analysis of stress–strain isotherms.