Nonlinear Behavior at Equilibrium

A full discussion of stress-strain relations at equilibrium for large deformations in rubbery cross-linked polymers is beyond the scope of this chapter there have been many investigations of uniaxial deformations (simple extension, compression,) torsion, and biaxial deformations, which have been critically reviewed elsewhere. A few comments will introduce the discussion of nonlinear viscoelastic behavior. 

The state of strain in large deformations is commonly described either by the principal extension ratios, Xi, X2, X3, deflned in the notation of Chapter 1 as X,- = 1 + Uilxi, with the coordinate axes suitably oriented, or by three strain invariants whose values are independent of the coordinate system. In simple extension, Xi = 1 + e, where e is the (practical) tensile strain U jx cf. equation 8 above), not to be confused with the e in equations 3,4 and 6. Most of this section is concerned with simple extension. 

The statistical theory of rubberlike elasticity predicts the following stress-strain relation in simple extension, often designate as neo-Hookean  

Thus the strain function (Xi) in equation 8 is (X — 1 /X )/3, which correctly reduces to e as Xi approaches unity, to give the correct deflnition of Young s modulus Ee in small strains. (Here ar is the stress calculated on the actual cross-section area sometimes in the literature the nominal or engineering stress is expressed on the basis of the initial cross-section, and is lower by a factor of 

Equation 13 does not hold closely above extensions of about 20%. For larger deformations the following formulation, known as the Mooney-Rivlin equation, is commonly used  

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