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

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.

[1]  H. A. Anziz.  Homogénéisation de composites élastiques périodiques à fort contraste : Conception de métamatériaux de second gradient , 2018 .

[2]  Antonio Cazzani,et al.  Large deformations induced in planar pantographic sheets by loads applied on fibers: Experimental validation of a discrete Lagrangian model , 2016 .

[3]  Guy Bouchitté,et al.  Homogenization of Thin Structures by Two-Scale Method with Respect to Measures , 2001, SIAM J. Math. Anal..

[4]  Pierre Seppecher,et al.  A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium , 1997 .

[5]  Pierre Seppecher,et al.  Strain gradient and generalized continua obtained by homogenizing frame lattices , 2018, Mathematics and Mechanics of Complex Systems.

[6]  Francesco dell’Isola,et al.  How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: approach “à la D’Alembert” , 2012 .

[7]  M. Briane,et al.  Homogenization of two-dimensional elasticity problems with very stiff coefficients , 2007 .

[8]  Francesco dell’Isola,et al.  Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices , 2015 .

[9]  Francesco dell’Isola,et al.  Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models , 2016 .

[10]  Ivan Giorgio,et al.  Numerical identification procedure between a micro-Cauchy model and a macro-second gradient model for planar pantographic structures , 2016 .

[11]  P. Seppecher,et al.  HOMOGENIZATION OF PERIODIC GRAPH-BASED ELASTIC STRUCTURES , 2018 .

[12]  Francesco dell’Isola,et al.  The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power , 1995 .

[13]  Barbara Gambin,et al.  Higher-Order Terms in the Homogenized Stress-Strain Relation of Periodic Elastic Media , 1989 .

[14]  G. Allaire.  Homogenization and two-scale convergence , 1992 .

[15]  Alessandro Della Corte,et al.  Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof , 2015 .

[16]  V. Zhikov,et al.  Homogenization of Differential Operators and Integral Functionals , 1994 .

[17]  A. Della Corte,et al.  The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[18]  Andrea Braides.  Γ-convergence for beginners , 2002 .

[19]  Pierre Seppecher,et al.  Linear elastic trusses leading to continua with exotic mechanical interactions , 2011 .

[20]  Pierre Seppecher,et al.  Truss Modular Beams with Deformation Energy Depending on Higher Displacement Gradients , 2003 .

[21]  Guy Bouchitté,et al.  Homogenization of a soft elastic material reinforced by fibers , 2002 .

[22]  Claude Boutin,et al.  Microstructural effects in elastic composites , 1996 .

[23]  Francesco dell’Isola,et al.  Linear pantographic sheets: Asymptotic micro-macro models identification , 2017 .

[24]  G. Nguetseng.  A general convergence result for a functional related to the theory of homogenization , 1989 .

[25]  P. Seppecher,et al.  Edge Contact Forces and Quasi-Balanced Power , 1997, 1007.1450.

[26]  O. Oleinik,et al.  Mathematical Problems in Elasticity and Homogenization , 2012 .

[27]  Cristian Barbarosie,et al.  Optimization of Bodies with Locally Periodic Microstructure , 2012 .

[28]  F. dell’Isola,et al.  Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium , 2016, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[29]  Andrea Braides.  Gamma-Convergence for Beginners , 2002 .

[30]  Cristian Barbarosie,et al.  Optimization of bodies with locally periodic microstructure by varying the periodicity pattern , 2014, Networks Heterog. Media.

[31]  P. Donato,et al.  An introduction to homogenization , 2000 .

[32]  Ivan Giorgio,et al.  Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model , 2017 .

[33]  G. D. Maso,et al.  An Introduction to-convergence , 1993 .

[34]  Andrey L. Piatnitski,et al.  Homogenization: Methods and Applications , 2007 .

[35]  Luca Placidi,et al.  A review on 2D models for the description of pantographic fabrics , 2016 .

[36]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .