Energy functions for rubber from microscopic potentials

Abstract

The finite deformation theory of rubber and related materials is based on energy functions that describe the macroscopic response of these materials under deformation. Energy functions and elastic constants are here derived from a simple microscopic ͑ball-and-spring͒ model. Exact uniaxial force-extension relationships are given for Hooke's Law and for the thermodynamic entropy-based microscopic model using the Gaussian and the inverse Langevin statistical approximations. Methods are given for finding the energy functions as expansions of tensor invariants of deformation, with exact solutions for functions that can be expressed as expansions in even powers of the extension. Comparison with experiment shows good agreement with the neo-Hookean energy function and we show how this derives directly from the simple Gaussian statistical model with a small modification.

Constrained Swelling of Polymer Networks: Characterization of Vapor-Deposited Cross-Linked Polymer Thin Films

“…Usually the chain deformation λ is taken relative to the unperturbed state ( r / n 1/2 l ), so that λ rel = λn –1/2 . The Kuhn–Grün function has been a basis of various models such as the three-chain model, eight-chain model, and various series expansions (cf., e.g., refs − ). The semiempirical function developed by Gent et al − is based on the logarithm of the first invariant of the Cauchy–Green strain vector, J 1 = λ x 2 + λ y 2 + λ z 2 – 3, as a way of inclusion of higher powers of the first strain invariant.…”

Experimental study of the influence of structural planes on the mechanical properties of sandstone specimens under cyclic dynamic disturbance

“…When ε D increases a certain extent, the increasing rate has a sudden increase, leading to the failure of the samples. The fitting curve of ε D and n is shown in Figure 13, which shows that the evolution law of ε D and n around Δ σ ′ conforms to the negative exponential function 66,67 and the inverse function expression of the Langevin function 68‐71 :where the A , B , a , b , and c are undetermined parameters.…”

Siewert solutions of transcendental equations, generalized Lambert functions and physical applications

“…The Langevin function and its inverse are relevant not only for magnetism, but also for other domains of physics with important practical applications, as polymers (polymer deformation and flow) [25], [26], [27], [28] or solar energy conversion (daily clearness index) [30], [29]. Researchers in these fields proposed a large number of useful analytical approximations for L (x) and L −1 (x).…”