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1

On obtaining effective elasticity tensors with entries zeroing method

Gierlach B., Danek T.

Geology, Geophysics and Environment

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2018

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Vol. 44, no. 2

259--274

EN

The purpose of this paper is to propose a new method for obtaining tensors expressing certain symmetries, called effective elasticity tensors, and their optimal orientation. The generally anisotropic tensor being the result of in situ seismic measurements describes the elastic properties of a medium. It can be approximated with a tensor of a specific symmetry class. With a known symmetry class and orientation, one can better describe geological structure elements like layers and fissures. A method used to obtain effective tensor in the previous papers (i.e. Danek & Slawinski 2015) is based on minimizing the Frobenius norm between the measured and effective tensor of a chosen symmetry class in the same coordinate system. In this paper, we propose a new approach for obtaining the effective tensor with the assumption of a certain symmetry class. The entry zeroing method assumes the minimization of the target function, being the measure of similarity with the form of the effective tensor for the specific class. The optimization of orientation is made by means of the Particle Swarm Optimization (PSO) algorithm and transformations were parameterised with quaternions. To analyse the obtained results, the Monte-Carlo method was used. After thousands of runs of PSO optimization, values of quaternion parts and tensor entries were obtained. Then, thousands of realizations of generally anisotropic tensors described with normal distributions of entries were generated. Each of these tensors was the subject of separate PSO optimization, and the distributions of rotated tensor entries were obtained. The results obtained were compared with solutions of the method based on the Frobenius distances (Danek et al. 2013).

2

Finite element implementation of slightly compressible and incompressible first invariant-based hyperelasticity: theory, coding, exemplary problems

Suchocki C.

Journal of Theoretical and Applied Mechanics

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2017

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Vol. 55 nr 3

787--800

EN

The present study is concerned with the finite element (FE) implementation of slightly compressible hyperelastic material models. A class of constitutive equations is considered where the isochoric potential functions are based on the first invariant of the right Cauchy-Green (C-G) deformation tensor. Special attention is paid to the most recently developed model formulations. The incremental form of hyperelasticity and its numerical implementation into both commercial and non-commercial FE software are discussed. A Fortran 77 UMAT code is attached which allows for a simple implementation of arbitrary first invariant-based constitutive models into Abaqus and Salome-Meca FE packages. Several exemplary problems are considered.

3

A finite element implementation of Knowles stored-energy function: theory, coding and applications

Suchocki C.

Archive of Mechanical Engineering

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2011

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Vol. LVIII, nr 3

319-346

EN

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 performace of the developed code is examined using an exemplary problem.

PL

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.

4

Space of SO (3)-orbits of elasticity tensors

Bóna A., Bucataru I., Slawinski M. A.

Archives of Mechanics

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2008

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Vol. 60, nr 2

123-138

EN

We construct an eighteen-dimensional orbifold that is in a one-to-one correspondence with the space of SO (3)-orbits of elasticity tensors. This allows us to obtain a local parametrization of SO (3)-orbits of elasticity tensors by six SO (6)-invariant and twelve SO (3)-invariant parameters. This process unravels the structure of the space of the orbits of the elasticity tensors.

5

An energy-based yield criterion for solids of cubic elasticity and orthotropic limit state

Kowalczyk K., Ostrowska-Maciejewska J., Pęcherski R. B.

Archives of Mechanics

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2003

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Vol. 55, nr 5-6

431-448

EN

The aim of the paper is to formulate a particular case of the J. Rychlewski yield condition for anisotropic linear elastic solids with Hooke's law and the limit tensor representing elastic range in the Mises yield condition under the assumption that different symmetry of elasticity tensors and the limit tensor appears. The elasticity tensor C is assumed to have cubic symmetry. The yield condition is based on the concept of stored elastic energy density, the theory of proper elastic states and energy orthogonal stress states developed by J. Rychlewski [1-3]. Three possible specifications of energy-based yield condition for cubic crystals are considered: the criterion based on the total distortion energy, the criterion based on the energy accumulated in the three proper states pertinent to cubic symmetry and the energy based criterion for cubic symmetry in elastic range and orthotropic symmetry in the limit state. Physical motivation, comparison with available experimental results and possible applications in mechanics of anisotropic solids as well as in nanomechanics are discussed.