Wave propagation and non-local effects in periodic frame materials: Generalized continuum mechanics

Abstract

Through the analysis of the wave propagation in infinite two-dimensional periodic frame materials, this paper illustrates the complexity of their dynamic behavior. Assuming the frame size is small compared to the wavelength, the homogenization method of periodic discrete media coupled with normalization is used to identify the macroscopic behavior at the leading order. The method is applied on a frame material with the vertical elements stiffer than the horizontal elements. Such a material is highly anisotropic and presents a large contrast between the rigidities of the possible mechanisms. Thus the waves associated with different kinematics appear in different frequency ranges. Moreover, the stiffer elements can deform in bending at the macroscopic scale. The equivalent continuum is a second-grade medium at the leading order and shear waves can be dispersive. A criterion is proposed to easily determine when this bending effect has to be taken into account.

Keywords

Reticulated materials second-grade media meta-materials homogenization dynamics wave propagation generalized media local resonance

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References

Bažant

Christensen

. Analogy between micropolar continuum and grid frameworks under initial stress. Int J Solids Struct 1972; 8(3): 327–346.

Noor

. Continuum modeling for repetitive lattice structures. Appl Mech Rev 1988; 41(7): 285–296.

Kumar

McDowell

. Generalized continuum modeling of 2-D periodic cellular solids. Int J Solids Struct 2004; 41(26): 7399–7422.

Cosserat

. Théorie des corps déformables (Theory of deformable bodies). Paris: A. Hermann et fils, 1909.

Eringen

. Linear theory of micropolar elasticity. J Math Mech 1966; 15: 909–923.

Eringen

Suhubi

. Nonlinear theory of simple micro-elastic solids—I. Int J Eng Sci 1964; 2(2): 189–203.

Suhubi

Eringen

. Nonlinear theory of micro-elastic solids—II. Int J Eng Sci 1964; 2(4): 389–404.

Mindlin

. Micro-structure in linear elasticity. Arch Ration Mech An 1964; 16(1): 51–78.

Toupin

. Elastic materials with couple-stresses. Arch Ration Mech An 1962; 11: 385–414.

10.

Mindlin

Eshel

. On first strain-gradient theories in linear elasticity. Int J Solids Struct 1968; 4(1): 109–124.

11.

Alibert

J-J

Seppecher

dell’Isola

. Truss modular beams with deformation energy depending on higher displacement gradients. Math Mech Sol 2003; 8(1): 51–73.

12.

Germain

. The method of virtual power in continuum mechanics. Part 2: Microstructure. Siam J Appl Math 1973; 25(3): 556–575.

13.

Maugin

. A historical perspective of generalized continuum mechanics. In Altenbach

Maugin

Erofeev

(eds) Mechanics of Generalized Continua (Advanced Structured Materials, Vol. 7). Berlin, Heidelberg: Springer-Verlag, 2011, pp.3–19.

14.

Tekoglu

Onck

. Size effects in two-dimensional Voronoi foams: A comparison between generalized continua and discrete models. J Mech Phys Solids 2008; 56(12): 3541–3564.

15.

Pradel

. Homogénéisation des milieux discrets périodiques orientés. Une application aux mousses (Homogenization of discrete periodic oriented media. An applications to foams). PhD Thesis. École Nationale des Ponts et Chaussées, 1998.

16.

Martinsson

Babuška

. Homogenization of materials with periodic truss or frame micro-structures. Math Mod Meth Appl S 2007; 17(5): 805–832.

17.

Chesnais

Boutin

Hans

. Effects of the local resonance on the wave propagation in periodic frame structures: Generalized Newtonian mechanics. J Acoust Soc Am 2012; 132(4, Part 2): 2873–2886.

18.

Pideri

Seppecher

. A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Continuum Mech Therm 1997; 9(5): 241–257.

19.

Sudret

De Buhan

. A multiphase model for materials reinforced by linear inclusions. Comptes Rendus de l’Académie des Sciences—Series IIB 1999; 327(1): 7–12.

20.

Boutin

Soubestre

. Generalized inner bending continua for linear fiber reinforced materials. Int J Solids Struct 2011; 48(3–4): 517–534.

21.

Boutin

Soubestre

Dietz

Taylor

. Experimental evidence of the high-gradient behaviour of fiber reinforced materials. Eur J Mech A—Solid 2013; 42(0): 280–298.

22.

Ferretti

Madeo

dell’Isola

Boisse

. Modeling the onset of shear boundary layers in fibrous composite reinforcements by second-gradient theory. ZAMP—J Appl Math Phys 2013, published online.

23.

Caillerie

Trompette

Verna

. Homogenisation of periodic trusses. In: IASS symposium, 10 years of progress in shell and spatial structures, Madrid, 1989.

24.

Tollenaere

Caillerie

. Continuous modeling of lattice structures by homogenization. Adv Eng Softw 1998; 29(7–9): 699–705.

25.

Moreau

Caillerie

. Continuum modeling of lattice structures in large displacement applications to buckling analysis. Comput Struct 1998; 68(1–3): 181–189.

26.

Caillerie

Mourad

Raoult

. Cell-to-muscle homogenization. Application to a constitutive law for the myocardium. ESAIM—Math Model Num 2003; 37(4): 681–698.

27.

Caillerie

Mourad

Raoult

. Discrete homogenization in graphene sheet modeling. J Elasticity 2006; 84(1): 33–68.

28.

Boutin

Hans

. Homogenisation of periodic discrete medium: Application to dynamics of framed structures. Comput Geotech 2003; 30(4): 303–320.

29.

Hans

Boutin

. Dynamics of discrete framed structures: A unified homogenized description. J Mech Mater Struct 2008; 3(9): 1709–1739.

30.

dell’Isola

Seppecher

. The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power. Comptes Rendus de l’Académie des Sciences—Series IIB 1995; 321(8): 1–26.

31.

dell’Isola

Madeo

Placidi

. Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua. ZAMM—J Appl Math Mech 2012; 92(1): 52–71.

32.

Placidi

Rosi

Giorgio

Madeo

. Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials. Math Mech Sol 2013; 18(7): 1–24.

33.

Bensoussan

Lions

J-L

Papanicolaou

. Asymptotic Analysis for Periodic Structures (Studies in Mathematics and Its Applications, Vol. 5). Amsterdam: North Holland, 1978.

34.

Sanchez-Palencia

. Non-Homogeneous Media and Vibration Theory (Lecture Notes in Physics, Vol. 127). Berlin: Springer-Verlag, 1980.

35.

Chiheb

Cioranescu

El Janati

Panasenko

. Reinforced reticulated structures in elasticity. Cr Acad Sci I—Math 1998; 326(7): 897–902.

36.

Cioranescu

Saint Jean Paulin

. Homogenization of Reticulated Structures. (Applied Mathematical Sciences, Vol. 136). New York: Springer-Verlag, 1999.

37.

Caillerie

. General study of a kind of singular stiff problems. Cr Acad Sci I—Math 1996; 323(7): 835–840.

38.

Caillerie

Sanchez-Palencia

. Elastic thin shells—Asymptotic theory in the anisotropic and heterogeneous cases. Math Mod Meth Appl S 1995; 5(4): 473–496.

39.

Soubestre

Boutin

. Non-local dynamic behavior of linear fiber reinforced materials. Mech Mater 2012; 55: 16–32.

40.

Auriault

J-L

Boutin

. Long wavelength inner-resonance cut-off frequencies in elastic composite materials. Int J Solids Struct 2012; 49(23–24): 3269–3281.

41.

Boutin

Auriault

J-L

. Rayleigh scattering in elastic composite materials. Int J Eng Sci 1993; 31(12): 1669–1689.

42.

Phani

Woodhouse

Fleck

. Wave propagation in two-dimensional periodic lattices. J Acoust Soc Am 2006; 119(4): 1995–2005.

43.

Boutin

Auriault

J-L

. Dynamic behaviour of porous media saturated by a viscoelastic fluid. Application to bituminous concretes. Int J Eng Sci 1990; 28(11): 1157–1181.

44.

dell’Isola

Sciarra

Vidoli

. Generalized Hooke’s law for isotropic second gradient materials. P Roy Soc A—Math Phy 2009; 465(2107): 2177–2196.

45.

Moustaghfir

Daya

Braikat

Damil

Potier-Ferry

. Evaluation of continuous modelings for the modulated vibration modes of long repetitive structures. Int J Solids Struct 2007; 44(21): 7061–7072.

46.

Craster

Kaplunov

Pichugin

. High-frequency homogenization for periodic media. P Roy Soc A—Math Phy 2010; 466(2120): 2341–2362.

47.

Craster

Kaplunov

Postnova

. High-frequency asymptotics, homogenisation and localisation for lattices. Q J Mech Appl Math 2010; 63(4): 497–519.

48.

Manevitch

Oshmyan

. An asymptotic study of the linear vibrations of a stretched beam with concentrated masses and discrete elastic supports. J Sound Vib 1999; 223(5): 679–691.

49.

Boutin

Rallu

Hans

. Large scale modulation of high frequency acoustic waves in periodic porous media. J Acoust Soc Am 2012; 132(6): 3622–3636.