Talk:Rubber elasticity

Arithmetic Problems

   This treatment suffers the difficulty of disagreeing with observation, namely that force (at constant temperature) =(Const)x r cannot change sign, as r, the end-to-end distance cannot be <0, while in practice rubbers are often strained in (linear) compression as well as in extension--e.g., tire treads, shoe soles, shock mounts.  The difficulty is not in the model but in the arithmetic.
   
  The treatment is based on an expression for the probability that the chain ends are r units apart as a function of r.  Apart from notation this expression is the same as Eq. 3.9 in Ref. 4 (Treloar, "The Physics of Rubber Elasticity")(where it I given as P(r), not P(r,n), as n is a constant).  Differentiation shows the expression to have a maximum (most probable value) at r = √(2nb2/3)--see Ref 4, Eq. 3.10 and Fig. 3.5(b).   As entropy S≃lnP, S likewise would have a maximum at the same value of r; and as force F≃-dS/dr, F will accordingly be positive or negative as r is greater than (extension) or less than (compression) its most probable value, as is observed in practice (Ref. 4, Fig. 5.6).

  In the wiki article, when ln(P(r,n)dr) is substituted for S, only the exponential factor is carried over; the quadratic factor, which would constitute a 2ln(r) term is dropped.  Inclusion of the entire expression would lead to the appropriate maximum in S and a F(r) function similar to Ref. 4, Fig. 5.6, in agreement with experiment.

  Although ref. 4, Fig. 3.5(b) shows the maximum of P, and hence S, and hence +&- F as in Fig. 5.6,  it comes up with (Eq. 3.22) essentially the same force expression as the wiki article and which likewise cannot change sign.  This seems to result from substituting an expression for the rectangular-coordinate volume element, dτ(=dxdydz) for the spherical-coordinate volume element (cf. Fig. 3.6) and regarding it as a constant; this drops the 2ln(r) term from the entropy expression (Eqs. 3.18 and 3.19), just as the wiki article did.
  Ref. 4 observe that its "result (3.19) shows the entropy to have its maximum value when the two ends of the chain are coincident (r=0)...".  Eq. 3.9 shows P(0) = 0, which would have the probability of the maximum entropy conformation to be zero.  This unlikely event, as well as the disagreement with observation, can be avoided by keeping track of the terms.

Debosley (talk) 21:10, 29 May 2014 (UTC)

Serious problems with this page:

I applaud the effort of D Hanson in updating this page with modern modeling. The problems I see are

1) lack of balance ... apparently Hanson's work is the only work that is not merely historical. References are to him, or before 1950. Really? Nobody else at all?

2) No distinction between general truths of rubber elasticity and the details found by Hanson for polyisoprene; e.g. no discussion of the ideal rubber

3) A serious error in that dealing with polyisoprene (?cis 1-4 ? it is not clear) the word 'crystal' never appears. It is generally agreed that the turn upwards in the stress-strain curve for this material at high strain and room temperature is caused by strain induced crystallization (SIC). In this article (and Hanson's papers) it is said to be caused by elastic deformation of the chain. This would happen in materials that do not crystallize but they generally fail before getting into this regime. It is commonplace that molecular modeling does not capture crystallization in temperature regimes where it is experimentally seen; the space and time limitations of even modern computers do not include the nucleation events. But this does not mean it does not happen. I do not understand why the author does not restrict his model to the regions of strain or temperature where crystals do not form.

4) The falling slope of the stress-strain curve at lower strains is associated with details of molecular motion. But at least some of it is simply due to using engineering stress when the sample is reducing in cross-section. Identifying deviations from neo-Hookean would be more relevant.

5) A special region 1a is described in molecular terms and vaguely indicated in fig 1 as below 100% strain. The references show it to be a specially stiff region at up to 5% strain. Is there any experimental evidence for this to exist? It seems a simple experiment to do.

I do not want to get into an editing war here, but I feel the current version does not serve our readers well. Clavipes (talk) 03:49, 21 September 2019 (UTC)