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

2020 ◽  
Vol 136 ◽  
pp. 103730 ◽  
Author(s):  
Claire Lestringant ◽  
Basile Audoly
Keyword(s):  
Strain Gradient ◽  
Elastic Structures ◽  
Derivation Method ◽  
Systematic Derivation ◽  
Gradient Models

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

2021 ◽  
Vol 477 (2246) ◽  
pp. 20200620
Author(s):  
Houssam Abdoul-Anziz ◽  
Lukáš Jakabčin ◽  
Pierre Seppecher
Keyword(s):  
Strain Gradient ◽  
Periodic Lattice ◽  
Minimizing Sequences ◽  
Elastic Structures ◽  
Generalized Continua ◽  
Minimization Problems ◽  
Weak Part ◽  
Periodic Graph ◽  
The Matrix ◽  
Gradient Models

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).

Aifantis versus Lam strain gradient models of Bishop elastic rods

2019 ◽  
Vol 230 (8) ◽  
pp. 2799-2812 ◽  
Author(s):  
R. Barretta ◽  
S. Ali Faghidian ◽  
F. Marotti de Sciarra
Keyword(s):  
Strain Gradient ◽  
Elastic Rods ◽  
Gradient Models

scholarly journals Homogenization of frame lattices leading to second gradient models coupling classical strain and strain-gradient terms

2019 ◽  
Vol 24 (12) ◽  
pp. 3976-3999 ◽  
Author(s):  
Houssam Abdoul-Anziz ◽  
Pierre Seppecher ◽  
Cédric Bellis
Keyword(s):  
Displacement Field ◽  
Strain Gradient ◽  
Periodic Structures ◽  
Three Dimensional ◽  
Discrete Systems ◽  
Effective Energy ◽  
Classical Strain ◽  
Second Gradient ◽  
Second Gradient Models ◽  
Gradient Models

We determine in the framework of static linear elasticity the homogenized behavior of three-dimensional periodic structures made of welded elastic bars. It has been shown that such structures can be modeled as discrete systems of nodes linked by extensional, flexural/torsional interactions corresponding to frame lattices and that the corresponding homogenized models can be strain-gradient models, i.e., models whose effective elastic energy involves components of the first and the second gradients of the displacement field. However, in the existing models, there is no coupling between the classical strain and the strain-gradient terms in the expression of the effective energy. In the present article, under some assumptions on the positions of the nodes of the unit cell, we show that classical strain and strain-gradient strain terms can be coupled. In order to illustrate this coupling we compute the homogenized energy of a particular structure that we call asymmetrical pantographic structure.

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

2017 ◽  
Vol 25 (7) ◽  
pp. 1429-1449 ◽  
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.

Sign in / Sign up

Export Citation Format

Share Document