Second gradient models as a particular case of microstructured models: a large strain finite elements analysis (original) (raw)
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A finite deformation second gradient theory of plasticity
Comptes Rendus De L Academie Des Sciences Serie Ii Fascicule B-mecanique, 2001
Extending the previous work by Chambon et al. [2] to the finite deformation regime, a local second gradient theory of plasticity for isotropic materials with microstructure is developed based on the multiplicative decomposition of the deformation gradient, the additive decomposition of the second deformation gradient and the principle of maximum dissipation. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Finite elements for elasticity with microstructure and gradient elasticity
International Journal for Numerical Methods in Engineering, 2008
We present a general finite element discretization of Mindlin's elasticity with microstructure. A total of 12 isoparametric elements are developed and presented, six for plane strain conditions and six for the general case of three-dimensional deformation. All elements interpolate both the displacement and microdeformation fields. The minimum order of integration is determined for each element, and they are all shown to pass the single-element test and the patch test. Numerical results for the benchmark problem of one-dimensional deformation show good convergence to the closed-form solution. The behaviour of all elements is also examined at the limiting case of vanishing relative deformation, where elasticity with microstructure degenerates to gradient elasticity. An appropriate parameter selection that enforces this degeneration in an approximate manner is presented, and numerical results are shown to provide good approximation to the respective displacements and strains of a gradient elastic solid. Copyright © 2007 John Wiley & Sons, Ltd.
A direct finite element implementation of the gradient-dependent theory
International Journal for Numerical Methods in Engineering, 2005
The enhanced non-local gradient-dependent theories formulate a constitutive framework on the continuum level that is used to bridge the gap between the micromechanical theories and the classical (local) continuum. They are successful in explaining the size effects encountered at the micron scale and in preserving the well-posedeness of the (I)BVP governing the solution of material instability triggering strain localization. This is due to the incorporation of an intrinsic material length scale parameter in the constitutive description. However, the numerical implementation of these theories is not a direct task because of the higher order of the governing equations. In this paper a direct computational algorithm for the gradient approach is proposed. This algorithm can be implemented in the existing finite element codes without numerous modifications as compared to the current numerical approaches (Int. A predictor-corrector scheme is proposed for the solution of the non-linear algebraic problem from the FEM. The expressions of the continuum and consistent tangent matrices are provided. The method is validated by conducting various numerical tests. As a result, pathological mesh dependence as obtained in finite element computations with conventional continuum models is no longer encountered. of numerical techniques where the length scale parameter is used as a localization limiter; i.e. as a means of preserving the well-posedeness and discretization sensitivity in (initial) boundary value problems [(I)BVP] for strain-softening ductile behaviour and damage-softening brittle behaviour. Another research direction is the need for the development of continuum micromechanical-based theories where the length scale parameter is used to capture the size dependence at the micro-scale [1-10].
The current trend towards miniaturization in the microelectronics industry has pushed for the development of theories intended to explain the behavior of materials at small scales. In the particular case of metals, a class of available non-classical continuum mechanics theories has been recently employed in order to explain the wide range of observed behavior at the micron scale. The practical use of the proposed theories remains limited due to issues in its numerical implementation. First, in displacement-based finite element formulations the need appears for higher orders of continuity in the interpolation shape functions in order to maintain the convergence rate upon mesh refinement. This limitation places strong restrictions in the geometries of the available elements. Second, the available inelastic constitutive models for small scale applications have been cast into deformation theory formulations limiting the set of problems to those exhibiting proportional loading only. In this article two contributions are made for the particular case of a Cosserat couple stress continuum. First it describes a numerical scheme based on a penalty function/reduced integration approach that allows for the proper treatment of the higher order terms present in Cosserat like theories. This scheme results in 1 PhD in computational mechanics, jgomezc1@eafit.edu.co, associate professor, applied mechanics group, EAFIT University, Medellín-Colombia.
On nonlinear dilatational strain gradient elasticity
Continuum Mechanics and Thermodynamics, 2021
We call nonlinear dilatational strain gradient elasticity the theory in which the specific class of dilatational second gradient continua is considered: those whose deformation energy depends, in an objective way, on the gradient of placement and on the gradient of the determinant of the gradient of placement. It is an interesting particular case of complete Toupin–Mindlin nonlinear strain gradient elasticity: indeed, in it, the only second gradient effects are due to the inhomogeneous dilatation state of the considered deformable body. The dilatational second gradient continua are strictly related to other generalized models with scalar (one-dimensional) microstructure as those considered in poroelasticity. They could be also regarded to be the result of a kind of “solidification” of the strain gradient fluids known as Korteweg or Cahn–Hilliard fluids. Using the variational approach we derive, for dilatational second gradient continua the Euler–Lagrange equilibrium conditions in bo...
A gradient approach to localization of deformation. I. Hyperelastic materials
Journal of Elasticity, 1986
By utilizing methods recently developed in the theory of fluid interfaces, we provide a new framework for considering the localization of deformation and illustrate it for the case of hyperelastic materials. The approach overcomes one of the major shortcomings in constitutive equations for solids admitting localization of deformation at finite strains, i.e. their inability to provide physically acceptable solutions to boundary value problems in the post-localization range due to loss of ellipticity of the governing equations. Specifically, strain-induced localized deformation patterns are accounted for by adding a second deformation gradient-dependent term to the expression for the strain energy density. The modified strain energy function leads to equilibrium equations which remain always elliptic. Explicit solutions of these equations can be found for certain classes of deformations. They suggest not only the direction but also the width of the deformation bands providing for the first time a predictive unifying method for the study of pre-and post-localization behavior. The results derived here are a three-dimensional extension of certain one-dimensional findings reported earlier by the second author for the problem of simple shear.
International Journal of Solids and Structures
The stress-gradient theory has a third order tensor as kinematic degree of freedom, which is workconjugate to the stress gradient. This tensor was called micro-displacements just for dimensional reasons. Consequently, this theory requires a constitutive relation between stress gradient and micro-displacements, in addition to the conventional stress-strain relation. The formulation of such a constitutive relation and identification of the parameters therein is difficult without an interpretation of the micro-displacement tensor. The present contribution presents an homogenization concept from a Cauchy continuum at the micro-scale towards a stress-gradient continuum at the macro-scale. Conventional static boundary conditions at the volume element are interpreted as a Taylor series whose next term involves the stress gradient. A generalized Hill-Mandel lemma shows that the micro-displacements can be identified with the deviatoric part of the first moment of the microscopic strain field. Kinematic and periodic boundary conditions are provided as alternative to the static ones. The homogenization approach is used to compute the stress-gradient properties of an elastic porous material. The predicted negative size effect under uni-axial loading is compared with respective experimental results for foams and direct numerical simulations from literature.
Theory and numerics of higher gradient inelastic material behavior
The work presented in this thesis has been carried out during the period 1999-2003 at the Chair of Applied Mechanics at the University of Kaiserslautern. The financial support of the DFG (Deutsche Forschungsgemeinschaft) within the project 'Theorie und Numerik von Mono-und Polykristallplastizität unter Berücksichtigung höherer Gradienten' (STE-544/7-1-3) is gratefully acknowledged. In the first place, I would like to thank Professor Paul Steinmann for his constant support and guidance, his never ending patience, inspiring comments and the time he put into my thesis. He motivated me to come to Kaiserslautern and take up this PhD-project and I owe him a great deal for making these last four years a successful and enjoyable stage of my life. My sincere thanks go to Professor Bob Svendsen and Professor Ahmed Benallal, who spontaneously agreed to become correferees for my thesis. Many scientific discussions and valuable remarks encouraged and helped me getting a deeper insight into the subject of my work, and at some stages even more enlightening were the insights they provided to the overall picture. Special thanks go to Professor Erwin Stein who introduced me to the fascinating world of mechanics, in particular computational mechanics, already during my study at the university in Hanover. Furthermore, I would like to thank Professor Kaspar Willam for his hospitality and support during my stay at the Colorado University Boulder, where I carried out my diploma thesis. The open-minded atmosphere I encountered there raised my interest to carry on with scientific research. I appreciated the pleasant working climate at the Chair of Applied Mechanics at the University of Kaiserslautern, which is undoubtly due to my colleagues. In particular, I like to thank my roommates during the period of this work: Thomas Svedberg, Ellen Kuhl and Bernd Kleuter, who made our officelife a great deal more enjoyable. Last, but not least, I would like to express my gratitude to my parents and my brother, Jörg, for their continuous encouragement and for always standing behind me. Especially, I thank Klaus Schmitt for his support-not to forget his sometimes funny but always valuable questions and comments during the reading and re-reading of the manuscript at various stages of its development. Kaiserslautern, in April 2003 Tina Liebe ext damaged part of the body and corresponding boundary (internal, external) H damage growth p volumetric part of Cauchy stress s deviatoric part of Cauchy stress In addition, the material may possess internal micro-defects that may initiate micro-pores or even microcracks. Further evolution then triggers the formation of macro-pores and macro-cracks or simply material deterioration. In the simplest case such damage process may also be modeled as isotropic with the help of a scalar internal damage variable accounting for micro-defects. In general, it would be cumbersome to account explicitly for each and every atom within the crystal lattice. Consequently, the ideal forum to derive a theoretically as well as computationally manageable formulation is based on phenomenological modeling of fields and fluxes related to the microstructure, in particular, dislocations and micro-defects, within the framework of continuum mechanics. This leads to complex coupled non-linear boundary value problems that can mainly be solved in an approximated manner with the help of numerical methods. Moreover, there are two competing mechanisms: material oped in CHAPTER 3. Thereby, not only the theoretical aspects corresponding to the gradient enhancement complemented with a loss of ellipticity analysis are envisioned but the numerical solution of the coupled problem is also investigated. For verification purposes different element formulations are compared. Chapter 3 is completed by a compact introduction of the geometrically non-linear case, which will be illustrated in a numerical example. Following the preliminary excursion in phenomenological gradient plasticity, the general concept is ap-Prinzipiell lässt sich inelastisches duktiles Materialverhalten anhand mikromechanischer Eigenschaften auf der Basis von Einkristallen beschreiben. Darüber hinaus lassen sich Mehrkristalle aus einer ungeordneten Zusammensetzung von Einkristallen bilden. Typischerweise führt die Betrachtung von Mehrkristallen auf komplexe Phänomene, wie z.B. Texturbildung, Ausbildung von Korngrenzen und damit verbundene Fragestellungen von Anisotropie u.v.m., die jedoch nicht Gegenstand dieser Dissertation sind. Stattdessen werde ich mich hier auf eine Untersuchung von Einkristallen beschränken. Als treibender Prozess für elastoplastische Deformationen lassen sich Versetzungen bzw. der Versetzungsfluss physikalisch motivieren. So findet man an Kristallwänden mikroskopische Gleitstufen, die von Versetzungen herrühren. Darüber hinaus können die Versetzungen bei der 'Durchwanderung' des Kristalles auf Hindernisse treffen, sich festsetzen und somit zur Materialverfestigung beitragen. Im einfachsten Falle einer homogenen plastischen Deformation genügt es sogenannte statistisch verteilte Versetzungen anzusetzen. Im Rahmen einer kontinuumsmechanischen Beschreibung ist es dabei ausreichend, eine isotrope skalare interne Verfestigungsvariable als charakteristische Verfestigungsgröße zu definieren. Weiterhin weisen Materialien in der Regel interne Mikrodefekte auf, die wiederum Mikroporen oder sogar Mikrorisse initiieren können. Dies kann sich bei entsprechender Belastung zur Formation von Makroporen und Makrorissen steigern oder ganz allgemein zu einer Materialschädigung führen. Im einfachsten Generell werden die zugrunde liegende Theorie und Numerik des behandelten inelastischen Materialverhaltens in dem jeweiligen Kapitel erörtert. Zu Demonstrationszwecken sind die meisten numerischen Berechnungsbeispiele eindimensional gehalten, weiterführende Beispiele basieren auf der Annahme eines ebenen Verzerrungszustandes. Die dazu erforderliche numerische Umsetzung der spezifischen Materialmodelle erfolgte im Rahmen des Finite Element Programms PHOENIX des Lehrstuhls für Technische Mechanik, siehe die dazugehörige Dokumentation. Zur Einführung in die Thematik werden in KAPITEL 1 die verschiedenen Konzepte zur Formulierung inelastischen Materialverhaltens unter Berücksichtigung der Mikrostruktur beleuchtet. Einerseits ist es möglich, die kinematische Beschreibung mit internen Freiheitsgraden zu erweitern. Andererseits können stattdessen interne Variablen und deren Gradienten eingeführt werden. Im ersten Fall lassen sich sogenannte Mikrospannungen definieren, die in die Gesamtenergiebilanz eingehen und zusätzlich Gleichgewichtsaussagen über die Mikrospannungen erfordern. Der andere Ansatz geht von einem zusätzlichen