To analyze the mechanical properties of the network, we now consider a general affine deformation , which deforms a point in the network from to according to the relation . This defines the Cauchy-Green deformation tensor :
The elastic free energy per network strand of the distorted state is then given by the quenched average over the distribution of strands , :
where indicates the average with respect to the probability distribution function before deformation, . is simply the unit of thermal energy. This implies the following average:
However, this average over a non-Gaussian distribution is inconvenient, and we choose to rewrite Eq.(8) for as:
which is a Gaussian distribution with a non-trivial multiplicative factor. Thus the quenched average in Eq.(12) becomes a Gaussian average of a polynomial in , and Wick's theorem can again be employed after substituting to give the rubber-elastic contribution to the free energy density :
where we have dropped the irrelevant constant, and is the average number of chain strands per unit volume. Note that the terms start at order 1/N, the terms start at order and so on.
Equation (14) gives the full tensorial free energy with the finite extensibility effect, and it can be viewed as an extension of the previous work on uniaxial deformations . Our original motivation for such a tensorial formulation is partly that finite extensibility effects are likely to be qualitatively important in nematic elastomers which, being anisotropic, makes the tensorial formulation crucial. Whilst the results shown here is still being generalised for the nematic elastomer problem, it has some interesting conclusions on rubber elasticity as we show below.
The deformation tensor for a homogeneous, isotropic and elastic material can be diagonalised, leaving three principal strains along a set of orthogonal axes. If we further assume the imcompressibility, which imposes , the deformation is characterised by the two remaining invariants of the deformation tensor :
Thus biaxial deformation is the most general deformation, and uniaxial deformation is only a special case. The most strict test of a physical model is therefore obtained under the former . Rewriting eq.(14) in terms of , we have:
where . The derivatives are:
Experimentally, Kawabata and coworkers [24, 25, 26] have studied the behaviour of rubbers under multiaxial deformation. Special attention was paid to small deformations, at which experimental error was prevalent. While their results can be well fitted by some theoretical models at large strains, no previously available model can explain the negative observed at low strains . We can see from eq.(18) that, affine assumption with finite extensibility can clearly gives rise to a negative at order 1/N or higher. If that is really the case, then the strain at which experimental value of turns positive (at the order of ) is also the strain at which the affine assumption breaks down at the 1/N order. This seems reasonable as mechanisms like sliplinks may not contribute at very low strains, and affine deformation stays valid. Due to the breakdown of affine assumption at slightly large strains and experimental errors at low strains, quantitative comparison of eq.(18) with experiments is difficult at this stage. Comparison with computer simulations in Ref., is not possible due to differences in adopted models and the error in the inverse Langevin approximation discussed earlier. But the message of this paper is still clear, namely that affine deformation coupled with finite extensible chains can account for the negative , observed experimentally. A more successful model may be possible if it includes this physical ingredient.
Finally, we derive an expansion expression for small strains. We write , and the constant volume condition requires , which leads to:
via the Cayley-Hamilton theorem. Here in writing Eq.(19), we have assumed the matrix to be symmetric. It is an assumption without loss of generality since the anti-symmetric part of the matrix gives rise to a net rotation, which leaves the system invariant . Thus the above elastic energy expression, Eq.(14), can be rewritten as (again dropping the constant term):
This now becomes an expansion in both and 1/N. As expected, we indeed see the 1/N deviations away from the usual Gaussian behaviour, which is simply . Cubic, quartic and even higher order (in strain) corrections are also present, but are smaller by increasing powers of 1/N. With N being typically of the order 20 for rubber and possibly less for liquid crystalline elastomers, the correction can be quite significant even at small deformations.