By I. M. Ward

Ward and Sweeney, either affiliated with the IRC in Polymer technological know-how and know-how on the college of Leeds, united kingdom, introduce the mechanical habit of good polymers, including new fabric on mechanical relaxations and anisotopy, composites modeling, nonlinear viscoelasticity, and fracture of difficult polymers to this moment version. The obtainable technique of the e-book has been retained for this variation, with every one bankruptcy designed to be self-contained and the speculation and functions of the topic brought the place applicable. bankruptcy difficulties and mathematical appendices are incorporated. The publication is for college kids of fabrics, chemistry, physics, and engineering.

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E. U ¼ f (º1 , º2 , º3 ) (2:2) It would be unreasonable for a physical quantity such as energy to depend on the choice of axes. The use of principal extension ratios, with values independent of the axis set, goes some way to ensuring that this is not the case. However, the choice of subscripts 1, 2 and 3 is arbitrary, so the chosen form must be a symmetric function of º1 , º2 and º3 . e. for zero strain. A further requirement is that for small strains we should obtain Hooke’s law for simple tension and the equivalent equation for simple shear.

24) ˜sb ¼ À12 mkfº21 þ º22 þ º23 À 3g (3:27) We can now add the contribution of all the chains in the network (N per unit volume) and obtain the entropy change of the network ˜S, where ˜S ¼ N X ˜s ¼ À12 Nkf(º21 þ º22 þ º23 ) À 3g (3:28) 1 Assuming no change in internal energy on deformation, this gives the change in the Helmholtz free energy ˜A ¼ ÀT ˜S ¼ 12 NkT (º21 þ º22 þ º23 À 3) If we assume that the strain-energy function U is zero in the undeformed state this gives U ¼ ˜A ¼ 12 NkT (º21 þ º22 þ º23 À 3) (3:29a) MODIFICATIONS OF THE SIMPLE MOLECULAR THEORY 43 Consider simple elongation º in the x direction.

6) above, which was derived as the equivalent of Hooke’s law for ﬁnite deformations. It will be shown that the reason for this direct link between the behaviour at a molecular level and the mechanics of ﬁnite elasticity arises because of the applicability of the so-called ‘afﬁne deformation’ assumption. Afﬁne deformation in the molecular theory of rubber elasticity means that we can assume that the changes in the length and orientation of lines joining adjacent cross-links in the molecular network are identical to the changes in lines marked on the macroscopic rubber.