Abstract
Optimized material parameters obtained from parameter identification for verification wrt a certain loading scenario are amenable to two deficiencies: Firstly, they may lack a general validity for different loading scenarios. Secondly, they may be prone to instability, such that a small perturbation of experimental data may ensue a large perturbation for the material parameters. This paper presents a framework for extension of hyperelastic models for rubber-like materials accounting for both deficiencies. To this end, an additive decomposition of the strain energy function is assumed into a sum of weighted strain mode related quantities. We propose a practical guide for model development accounting for the criteria of verification, validation and stability by means of the strain mode-dependent weighting functions and techniques of model reduction. The approach is successfully applied for 13 hyperelastic models with regard to the classical experimental data on vulcanized rubber published by Treloar (Trans Faraday Soc 40:59–70, 1944), showing both excellent fitting capabilties and stable material parameters.
Highlights
1.1 State of the art on hyperelasticityRubber-like materials or elastomers, respectively, consist of randomly oriented chain-like macromolecules with more or less closely connected entanglements or cross-links
The solution of the underlying least-squares problem for parameter-identification might render a satisfactory agreement between simulated and experimental data; it might be susceptible to instability, in the sense, that a small perturbation of experimental data may lead to a large deviation of the resulting parameter solution
This contribution presents a practical guide for model development accounting for the criteria of verification, validation and stability
Summary
Rubber-like materials or elastomers, respectively, consist of randomly oriented chain-like macromolecules with more or less closely connected entanglements or cross-links. Two main characteristics are their ability for large deformations subject to relatively small stresses and their retaining of the initial configuration after unloading without considerable permanent deformation This behavior is attributed to the network entropy as the orientation of chains alters with deformation. Numerous phenomenological and micro-mechanically motivated models have been proposed in the literature in order to capture the elastic and nearly incompressible mechanical of rubber-like materials The former can be classified into invariant-based and principal-stretch-based formulations, cf e.g. For the case of isotropy, an appropriate set of invariants dependent on the right Cauchy–Green tensor are selected, which are included as polynomials with sufficiently high orders into the strain energy function, known as Rivlin’s expansion, [43] Classical examples such as the Neo-Hooke material are of Mooney–Rivlin type, cf., e.g., [33,40,41,42]. Many of the above-mentioned models for rubber-like materials are well advanced from the mathematical point of view, [17], and the numerical point of view [31,45]
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