Talk:Ogden hyperelastic model

Discussion of the Ogden Model of the Strain-Energy Function

The Ogden model of the strain energy function W is an empirical form of the Valanis-Landel separable function of the principal extension ratios λi such that:

                                            W = w(λ1) + w(λ2) + w(λ3), 

As pointed out by Valanis and Landel in their paper in the Journal of Applied Physics in 1967, Vol. 38, pp 2997-3002, titled "The Strain-Energy Function of a Hyperelastic Materials in Terms of the Extension Ratios, the form of w(λ) may be obtained trivially. This is done by fitting the curve w'(λ), where w'(λ) = dw/dλ, obtained from a 'strip bi-axial' test where an elastic membrane is extended in direction 1 while keeping the principal extension λ2 equal to unity. The fit is not unique since different algebraic expressions may fit this curve equally well. In fact Valanis and Landel, by precisely doing just that, found that the data of a wide range of experiments and range of deformation, 0.4< λ <3.2 could be described by the compact and mathematically elegant strain energy function

                                         W = 2μ Σ λi(logλi - 1), i = 1,2,3                                             (a)

or,

                                                w(λ) = 2μ λ(logλ - 1)                                                  (b)

                                                   w'(λ) = 2μlogλ                                                      (c)
             

The Valanis-Landel function is well known in the technical literature and has been quoted and read widely and I find it extraordinary, or perhaps careless or even remiss, that the authors of the present article have not included the Valanis- Landel seminal paper in their references.

Note the 'Ogden' function given above. How does one fit that function to:

                                                     w'(λ) = 2μlogλ 

which, to a great deal of accuracy, is the experimentally determined function w'(λ)? Why choose that function in particular when w'(λ) is given by such a simple form in Eq.(c)? Also if w'(λ) is not known analytically because the material involved is a new type of rubber, why choose the Ogden fitting function. Because after all, all empirical functions that fit the experimentally determined w'(λ) are just curve fits. It is the proponent's task to provide justification. ~ https://en.wikipedia.org/wiki/Special:Contributions/2602:306:32DF:2780:6113:B958:254C:5122

Wikipedia is always a work in progress. You are welcome to add this information to the article. Bbanerje (talk) 21:12, 2 November 2019 (UTC)