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Archive of Mechanical Engineering

The Journal of Committee on Machine Building of Polish Academy of Sciences

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A Finite Element Implementation of Knowles Stored-Energy Function: Theory, Coding and Applications

Cyprian Suchocki
  • Institute of Mechanics and Printing, Warsaw University of Technology, ul. Narbutta 85, 02-524 Warszawa, Poland
Published Online: 2011-10-29 | DOI: https://doi.org/10.2478/v10180-011-0021-7

A Finite Element Implementation of Knowles Stored-Energy Function: Theory, Coding and Applications

This paper contains the full way of implementing a user-defined hyperelastic constitutive model into the finite element method (FEM) through defining an appropriate elasticity tensor. The Knowles stored-energy potential has been chosen to illustrate the implementation, as this particular potential function proved to be very effective in modeling nonlinear elasticity within moderate deformations. Thus, the Knowles stored-energy potential allows for appropriate modeling of thermoplastics, resins, polymeric composites and living tissues, such as bone for example. The decoupling of volumetric and isochoric behavior within a hyperelastic constitutive equation has been extensively discussed. An analytical elasticity tensor, corresponding to the Knowles stored-energy potential, has been derived. To the best of author's knowledge, this tensor has not been presented in the literature yet. The way of deriving analytical elasticity tensors for hyperelastic materials has been discussed in detail. The analytical elasticity tensor may be further used to develop visco-hyperelastic, nonlinear viscoelastic or viscoplastic constitutive models. A FORTRAN 77 code has been written in order to implement the Knowles hyperelastic model into a FEM system. The performance of the developed code is examined using an exemplary problem.

Wprowadzenie funkcji energii potencjalnej typu Knowlesa do systemu metody elementów skończonych: teoria, kodowanie i zastosowania

Praca przedstawia pełną drogę wprowadzania do systemu metody elementów skończonych (MES) równania konstytutywnego hipersprężystości zdefiniowanego przez użytkownika przy użyciu odpowiedniego tensora sztywności. Aby zilustrować metodykę wprowadzania równania konstytutywnego do MES posłużono się modelem materiału hipersprężystego typu Knowlesa, gdyż model ten dobrze opisuje nieliniową sprężystość w zakresie średnich deformacji. Stąd model Knowlesa pozwala na poprawny opis własności mechanicznych polimerów termoplastycznych, żywic, kompozytów polimerowych i niektórych tkanek biologicznych, jak np. tkanka kostna. Przedstawiono podział równania konstytutywnego na część izochoryczną i objętościową. Wyprowadzono analitycznie tensor sztywności odpowiadający modelowi Knowlesa. Tensor ten nie był dotąd prezentowany w literaturze. Omówiono szczegółowo sposób wyprowadzania analitycznych tensorów sztywności dla materiałów hipersprężystych. Wyznaczony tensor sztywności może dalej posłużyć do budowy równań konstytutywnych nieliniowej lepkosprężystości lub lepkoplastyczności. W celu wprowadzenia modelu do systemu MES napisany został program w języku FORTRAN 77. W pracy przedstawiono wyniki z prostej symulacji MES wykonanej z wykorzystaniem napisanego programu.

Keywords: elasticity tensor; tangent modulus tensor; material Jacobian; hyperelasticity; stored-energy potential; constitutive equation; finite element method

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Published Online: 2011-10-29

Published in Print: 2011-01-01


Citation Information: Archive of Mechanical Engineering. Volume 58, Issue 3, Pages 319–346, ISSN (Print) 0004-0738, DOI: https://doi.org/10.2478/v10180-011-0021-7, October 2011

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