Generalized continua and non‐homogeneous boundary conditions in homogenisation methods

Forest, Samuel; Trinh, Duy Khanh

doi:10.1002/zamm.201000109

Cited by 166 publications

(138 citation statements)

References 52 publications

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“…It coincides with the mean value of the second gradient of the local displacement field inside the unit cell in the absence of holes, cracks nor rigid particles [12]. The conjugate quantity in the work of internal forces is the third rank tensor of double stress or hyper -stress tensor M ijk .…”

Section: Second Gradient Overall Modeling Of Compositessupporting

confidence: 53%

“…Such polynomial developments represent an alternative to multiscale asymptotic expansions to derive effective higher order properties, with the advantage that they can be used in a straightforward manner, irrespective of the local linear or nonlinear behaviour of the composite material. The complete Ansatz polynomial, introduced for instance in [12], is…”

Section: Introductionmentioning

confidence: 99%

“…This is no longer the case when higher order terms are included. The fact that v can generally not be expected to be periodic was first discussed in [29] and proved in [12,26] by determining a representative volume element size for quadratic and cubic boundary conditions. This methodology was recently applied to a simple two -phase periodic microstructure in [25] and shown to deliver better description of the displacement field in a loaded composite structure than the usual Cauchy approach.…”

Section: Introductionmentioning

confidence: 99%

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Generalized continuum overall modelling of periodic composite structures

Trinh

Forest

2011

Vietnam J. Mech.

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Abstract. Classical homogenization methods fail to reproduce the overall response of composite structures when macroscopic strain gradients become significant. Generalized continuum models like Cosserat, strain gradient and micromorphic media, can be used to enhance the overall description of heterogeneous materials when the hypothesis of scale separation is not fulfilled. We show in the present work how the higher order elasticity moduli can be identified from suitable loading conditions applied to the unit cell of a periodic composite. The obtained homogeneous substitution generalized continuum is used then to predict the response of a composite structure subjected to various loading conditions. Reference finite element computations are performed on the structure taking all the heterogeneities into account. The overall substitution medium is shown to provide improved predictions compared to standard homogenization. In particular the additional boundary conditions required by generalized continua makes it possible to better represent the clamping conditions on the real structure.

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Section: Second Gradient Overall Modeling Of Compositessupporting

confidence: 53%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalized continuum overall modelling of periodic composite structures

Trinh

Forest

2011

Vietnam J. Mech.

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“…A computational strategy is introduced to unambiguously determine this fluctuation. The approach is well-suited for elastic as well as elastoplastic composites, see [25].…”

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confidence: 99%

Second Gradient and Generalized Continua. A workshop held on 12–16 March 2012 in Cisterna di Latina

Dell’Isola

Forest

2014

Z Angew Math Mech

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This editorial intends to complement the scientific content of the papers which follow in this special issue. The workshop "Second Gradient and Generalized Continua" held on 12-16 March 2012 at Palazzo Caetani in Cisterna di Latina, Italy was organized by Francesco dell'Isola and Samuel Forest. Very interesting, exciting and creative discussions started nearly immediately during the seminars and the round tables: the spirit of all participants was very productive and open minded and nobody felt necessary to hide his views, also when they were in contrast with those of other participants, so that a truly frank and direct scientific discussion was possible. The video-recorded seminars are available at the website http://memocs.univaq.it/?page id=2858, http://memocs.univaq.it/?page id=3071.In the following paragraphs we will reproduce, following the seminar order, only the abstracts of those presentations which did not produce a paper published in the present issue of ZAMM.Francesco dell'Isola: "Contact interactions in N-th gradient continua". Cauchy format for continuum mechanics is not general enough to include the description of some important phenomena occurring in a large class of "microscopically" inhomogeneous bodies. This circumstance has been recognized already by E. and F. Cosserat and later by Mindlin, Rivlin, Toupin, and Green. Particularly detailed is the development of second gradient model due to Paul Germain, which is the simplest continuum model which cannot be included in the Cauchy format. Actually Cauchy Continuum model has a wide but limited scope of predictivity: therefore the logical possibility of generalizing it truly corresponds to the need of describing some new and interesting phenomena, which include, but are not limited to, the formation of many different kinds of boundary layers. The class of generalized (N -th gradient) continua which is presented here is constituted by those models in which deformation energy may depend on N + 1 gradients of displacement field. The precise definition of N -th gradient continua and the nature of contact interactions which may arise in them can be mathematically formulated and characterized only by using suitable concepts from the theory of distribution and differential geometry, but it is definitively not so difficult to generalize -in the considered context -the Cauchy concept of stress state. The mathematical abstraction required does not seem inappropriate as it actually supplies a useful description of some phenomena of relevance in the mechanics of growing tissues and opens interesting perspectives in the design of artificial materials showing "exotic" mechanical behavior. More details about the presented ideas can be found in [17,18].Pierre Seppecher: "Linear elastic trusses leading to continua with exotic mechanical interactions". When studying the statics of a truss made of a discrete network of nodes joined by linear elastic springs, the kernel of the quadratic potential energy (space of floppy modes) plays an essential role. In particular w...

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“…The anisotropic case involving a large number of additional parameters is much more challenging from the identification perspective. Extended homogenization procedures can be used to identify the whole set of parameters from the consideration of an underlying periodic microstructure, as reviewed in [68][69][70].…”

Section: (B) Scalar Microstrainmentioning

confidence: 99%

Nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage

Forest¹

2016

Proc. R. Soc. A.

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International audienceThe construction of regularization operators presented in this work is based on the introduction of strain or damage micromorphic degrees of freedom in addition to the displacement vector and of their gradients into the Helmholtz free energy function of the constitutive material model. The combination of a new balance equation for generalized stresses and of the micromorphic constitutive equations generates the regularization operator. Within the small strain framework, the choice of a quadratic potential w.r.t. the gradient term provides the widely used Helmholtz operator whose regularization properties are well known: smoothing of discontinuities at interfaces and boundary layers in hardening materials, and finite width localization bands in softening materials. The objective is to review and propose nonlinear extensions of micromorphic and strain/damage gradient models along two lines: the first one introducing nonlinear relations between generalized stresses and strains; the second one envisaging several classes of finite deformation model formulations. The generic approach is applicable to a large class of elastoviscoplastic and damage models including anisothermal and multiphysics coupling. Two standard procedures of extension of classical constitutive laws to large strains are combined with the micromorphic approach: additive split of some Lagrangian strain measure or choice of a local objective rotating frame. Three distinct operators are finally derived using the multiplicative decomposition of the deformation gradient. A new feature is that a free energy function depending solely on variables defined in the intermediate isoclinic configuration leads to the existence of additional kinematic hardening induced by the gradient of a scalar micromorphic variable

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Generalized continua and non‐homogeneous boundary conditions in homogenisation methods

Cited by 166 publications

References 52 publications

Generalized continuum overall modelling of periodic composite structures

Generalized continuum overall modelling of periodic composite structures

Second Gradient and Generalized Continua. A workshop held on 12–16 March 2012 in Cisterna di Latina

Nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage

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