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Next: Summary Up: Rubber Elasticity with Finite Previous: Single Chain Statistics

Network Average

To analyze the mechanical properties of the network, we now consider a general affine deformation tex2html_wrap_inline771 , which deforms a point in the network from tex2html_wrap_inline773 to tex2html_wrap_inline717 according to the relation tex2html_wrap_inline777 . This defines the Cauchy-Green deformation tensor tex2html_wrap_inline771 :

equation182

The elastic free energy per network strand of the distorted state is then given by the quenched average over the distribution of strands tex2html_wrap_inline781 , [8]:

  equation188

where tex2html_wrap_inline783 indicates the average with respect to the probability distribution function before deformation, tex2html_wrap_inline781 . tex2html_wrap_inline787 is simply the unit of thermal energy. This implies the following average:

  eqnarray503

However, this average over a non-Gaussian distribution is inconvenient, and we choose to rewrite Eq.(8) for tex2html_wrap_inline781 as:

eqnarray505

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 tex2html_wrap_inline717 , and Wick's theorem can again be employed after substituting tex2html_wrap_inline793 to give the rubber-elastic contribution to the free energy density tex2html_wrap_inline795 :

  eqnarray507

where we have dropped the irrelevant constant, and tex2html_wrap_inline797 is the average number of chain strands per unit volume. Note that the tex2html_wrap_inline799 terms start at order 1/N, the tex2html_wrap_inline803 terms start at order tex2html_wrap_inline805 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 [9]. 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 tex2html_wrap_inline771 for a homogeneous, isotropic and elastic material can be diagonalised, leaving three principal strains tex2html_wrap_inline809 along a set of orthogonal axes. If we further assume the imcompressibility, which imposes tex2html_wrap_inline811 , the deformation is characterised by the two remaining invariants of the deformation tensor [23]:

eqnarray511

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 [17]. Rewriting eq.(14) in terms of tex2html_wrap_inline813 , we have:

equation513

where tex2html_wrap_inline815 . The derivatives tex2html_wrap_inline817 are:

  eqnarray515

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 tex2html_wrap_inline697 observed at low strains [17]. We can see from eq.(18) that, affine assumption with finite extensibility can clearly gives rise to a negative tex2html_wrap_inline697 at order 1/N or higher. If that is really the case, then the strain at which experimental value of tex2html_wrap_inline697 turns positive (at the order of tex2html_wrap_inline827 ) 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.[20], 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 tex2html_wrap_inline697 , 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 tex2html_wrap_inline833 , and the constant volume condition requires tex2html_wrap_inline835 , which leads to:

  equation517

via the Cayley-Hamilton theorem. Here in writing Eq.(19), we have assumed the matrix tex2html_wrap_inline837 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 gif. Thus the above elastic energy expression, Eq.(14), can be rewritten as (again dropping the constant term):

  eqnarray521

This now becomes an expansion in both tex2html_wrap_inline837 and 1/N. As expected, we indeed see the 1/N deviations away from the usual Gaussian behaviour, which is simply tex2html_wrap_inline847 . 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.


next up previous
Next: Summary Up: Rubber Elasticity with Finite Previous: Single Chain Statistics

Yong Mao
Wed Apr 30 14:46:27 BST 1997