Homogenization of an elastic material reinforced by very strong fibres arranged along a periodic lattice

We provide in this paper homogenization results for the L 2 -topology leading to complete strain-gradient models and generalized continua. Actually, we extend to the L 2 -topology the results obtained in (Abdoul-Anziz & Seppecher, 2018 Homogenization of periodic graph-based elastic structures. Journal de l’Ecole polytechnique–Mathématiques 5 , 259–288) using a topology adapted to minimization problems set in varying domains. Contrary to (Abdoul-Anziz & Seppecher, 2018 Homogenization of periodic graph-based elastic structures. Journal de l’Ecole polytechnique–Mathématiques 5 , 259–288) we consider elastic lattices embedded in a soft elastic matrix. Thus our study is placed in the usual framework of homogenization. The contrast between the elastic stiffnesses of the matrix and the reinforcement zone is assumed to be very large. We prove that a suitable choice of the stiffness on the weak part ensures the compactness of minimizing sequences while the energy contained in the matrix disappears at the limit: the Γ-limit energies we obtain are identical to those obtained in (Abdoul-Anziz & Seppecher, 2018 Homogenization of periodic graph-based elastic structures. Journal de l’Ecole polytechnique–Mathématiques 5 , 259–288).

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Asymptotically exact strain-gradient models for nonlinear slender elastic structures: A systematic derivation method

Journal of the Mechanics and Physics of Solids ◽

10.1016/j.jmps.2019.103730 ◽

2020 ◽

Vol 136 ◽

pp. 103730 ◽

Cited By ~ 4

Author(s):

Claire Lestringant ◽

Basile Audoly

Keyword(s):

Strain Gradient ◽

Elastic Structures ◽

Derivation Method ◽

Systematic Derivation ◽

Gradient Models

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Finite-deformation second-order micromorphic theory and its relations to strain and stress gradient models

Mathematics and Mechanics of Solids ◽

10.1177/1081286517720844 ◽

2017 ◽

Vol 25 (7) ◽

pp. 1429-1449 ◽

Cited By ~ 13

Author(s):

Samuel Forest ◽

Karam Sab

Keyword(s):

Strain Gradient ◽

Finite Deformation ◽

Stress Gradient ◽

Higher Order ◽

Second Order ◽

Stress And Strain ◽

Dynamical Equations ◽

Generalized Continua ◽

Micromorphic Theory ◽

Gradient Models

Germain’s general micromorphic theory of order [Formula: see text] is extended to fully non-symmetric higher-order tensor degrees of freedom. An interpretation of the microdeformation kinematic variables as relaxed higher-order gradients of the displacement field is proposed. Dynamical balance laws and hyperelastic constitutive equations are derived within the finite deformation framework. Internal constraints are enforced to recover strain gradient theories of grade [Formula: see text]. An extension to finite deformations of a recently developed stress gradient continuum theory is then presented, together with its relation to the second-order micromorphic model. The linearization of the combination of stress and strain gradient models is then shown to deliver formulations related to Eringen’s and Aifantis’s well-known gradient models involving the Laplacians of stress and strain tensors. Finally, the structures of the dynamical equations are given for strain and stress gradient media, showing fundamental differences in the dynamical behaviour of these two classes of generalized continua.

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Cosserat Modeling of Size Effects in Crystalline Solids

MRS Proceedings ◽

10.1557/proc-653-z8.2 ◽

2000 ◽

Vol 653 ◽

Author(s):

Samuel Forest

Keyword(s):

Stress State ◽

Size Effects ◽

Elastic Inclusion ◽

Characteristic Size ◽

Infinite Matrix ◽

Crystalline Solids ◽

Generalized Continua ◽

Absolute Size ◽

Kink Bands ◽

The Matrix

AbstractThe mechanics of generalized continua provides an efficient way of introducing intrinsic length scales into continuum models of materials. A Cosserat framework is presented here to descrine the mechanical behavior of crystalline solids. The first application deals with the problem of the stress field at a crak tip in Cosserat single crystals. It is shown that the strain localization patterns developping at the crack tip differ from the classical picture : the Cosserat continuum acts as a bifurcation mode selector, whereby kink bands arising in the classical framework disappear in generalized single crystal plasticity. The problem of a Cosserat elastic inclusion embedded in an infinite matrix is then considered to show that the stress state inside the inclusion depends on its absolute size lc. Two saturation regimes are observed : when the size R of the inclusion is much larger than a characteristic size of the medium, the classical Eshelby solution is recovered. When R is much small than the inclusion, a much higher stress is reached (for an inclusion stiffer than the matrix) that does not depend on the size any more. There is a transition regime for which the stress state is not homogeneous inside the inclusion. Similar regimes are obtained in the study of grain size effects in polycrystalline aggregates of Cosserat grains.

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