Elasticity of Rubbery Materials

The Elasticity of Rubbery Materials Results from the Sum of Chain Entropies 

Rubber and other pol nneric materials are elastic. Polymeric elastomers are covalently cross-linked networks of pol Tner chains. Here w e describe one of the simplest and earliest models for the retractive forces of polymeric materials, the affine network model. 

Suppose you have m chains, all of the same length N. The bond length is b. The chain ends are covalently cross-linked at junction points. Assume there arc no intermolecular interactions, and that the total elastic free energy of the material is the sum of the elastic free energies of each of the chains. For the undeformed material, indicated by subscript 0, the end of a chain is at (X(),yo,zo) and 

Summing over m independent chains gives the total free energy f ,  

In a macroscopic material, applying a force in one direction can cause forces and deformations in other directions. The forces per unit area in various directions are called stresses and the relative displacements are called strains. Stresses are derivatives of the free energy with respect to strains. If you deform a material by a factor a, where the x-axis, y- axis, and z-axis deformations depend on a, the free energy is a function F(Ax( ), Ay ( ),Az( )). The derivative can be written in terms of Al, A and A as 

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