Rubber elasticity stress tensor

The c is the elongation ratio in the jc-axis as compared to the isotropically swollen state with the same volume fraction. Here, the average stress tensor fly in Eq. (4.36) is diagonal and it follows a famous relation in rubber elasticity [1],... 

In Section I, the discussion dealt with the significant role of nonbonded interactions in the development of the full stress tensor, mean plus deviatoric, in rubber elasticity, in the important high reduced density regime p > 1. Here, we present some concepts and formulations that apply to this regime. 

The difference between nematic and isotropic elastomers is simply the molecular shape anisotropy induced by the LC order, as discussed in Sect. 2. The simplest approach to nematic rubber elasticity is an extension of classical molecular mbber elasticity using the so-called neo-classical Gaussian chain model [64] see also Warner and Terentjev [4] for a detailed presentation. Imagine an elastomer formed in the isotropic phase and characterized by a scalar step length Iq. After cooling down to a monodomain nematic state, the chains obtain an anisotropic shape described by the step lengths tensor Ig. For this case the stress-strain relation can be written as ... 

Another version of the tube model in rubber elasticity has been reviewed by Graessley. It uses the formalism of the classical paper of Doi and Edwards to calculate the stress-strain relationship. Again the basic property is the primitive path and correlation functions of the primitive path segments , which determines the relaxation of the primitive path. Static properties can be worked out from this dynamic consideration and in the static limit the Cartesian stress tensor can be written as... 

From the viewpoint of the mechanics of continua, the stress-strain relationship of a perfectly elastic material is fully described in terms of the strain energy density function W. In fact, this relationship is expressed as a linear combination erf the partial derivatives of W with respect to the three invariants of deformation tensor, /j, /2, and /3. It is the fundamental task for a phenomenologic study of elastic material to determine W as a function of these three independent variables either from molecular theory or by experiment. The present paper has reviewed approaches to this task from biaxial extension experiment and the related data. The results obtained so far demonstrate that the kinetic theory of polymer network does not describe actual behavior of rubber vulcanizates. In particular, contrary to the kinetic theory, the observed derivative bW/bI2 does not vanish. 

On the other hand, rubber can deform elastically up to extensions of as much as 7. As rubber came into use as an engineering material during World War n, a need arose to express Hooke s law for large deformations. Using the Finger deformation tensor, we can come up with the result quite easily. If the stress at any point is linearly proportional to deformation and if the material is isotropic (i.e., has the same proportionality in all directions), then the extra stress due to tteformation should be determined by a constant times the deformation. 

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