In this study, linear plane wave propagation in pantographic lattices is investigated using a two-dimensional (2D) second-gradient continuum model. Primarily, we aim to investigate wave transmission and reflection phenomena occurring when a pantographic lattice is attached to a material modeled by classical first-gradient theory. From a practical point of view, this would be a possible application for pantographic lattices due to their promising mechanical behavior. From a theoretical point of view, this investigation let us to elucidate wave propagation between second- and first-gradient continua. The adapted second-gradient model is first linearized, and the dispersion relations are obtained. Boundary and jump conditions at the discontinuity surface are identified accordingly. It is shown how elastic constants may affect the wave propagation between the two continua and can be tuned to control such wave propagation phenomena.
dell’IsolaFAndreausUPlacidiL. At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola. Math Mech Solids2015; 20: 887–928.
2.
dell’IsolaFCorteADGiorgioI. Higher-gradient continua: the legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives. Math Mech Solids2017; 22: 852–872.
3.
BarchiesiECiallellaAScerratoD. A partial report on the controversies about the principle of virtualwork: from Archytas of Tarentum to Lagrange, Piola, Mindlin and Toupin. In: dell’IsolaFEugsterSRSpagnuoloM, et al. (eds) Evaluation of scientific sources in mechanics. Berlin: Springer, 2022; pp. 341–377.
4.
dell’IsolaFEugsterSRFedeleR, et al. Second-gradient continua: from Lagrangian to Eulerian and back. Math Mech Solids2022; 27: 2715–2750.
5.
GolaszewskiMGrygorukRGiorgioI, et al. Metamaterials with relative displacements in their microstructure: technological challenges in 3D printing, experiments and numerical predictions. Continuum Mech Therm2019; 31: 1015–1034.
6.
dell’IsolaFSeppecherPAlibertJJ, et al. Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Continuum Mech Therm2019; 31: 851–884.
7.
dell’IsolaFSeppecherPSpagnuoloM, et al. Advances in pantographic structures: design, manufacturing, models, experiments and image analyses. Continuum Mech Therm2019; 31: 1231–1282.
8.
JuritzaAYangHGanzoschG. Qualitative investigations of experiments performed on 3D-FDM-printed pantographic structures made out of PLA. In: New achievements in continuum mechanics and thermodynamics. Berlin: Springer, 2019, pp. 197–209.
9.
PlacidiLMisraABarchiesiE. Simulation results for damage with evolving microstructure and growing strain gradient moduli. Continuum Mech Therm2019; 31: 1143–1163.
10.
PlacidiLBarchiesiEMisraA, et al. Micromechanics-based elasto-plastic–damage energy formulation for strain gradient solids with granular microstructure. Continuum Mech Therm2021; 33: 2213–2241.
11.
TimofeevDBarchiesiEMisraA, et al. Hemivariational continuum approach for granular solids with damage-induced anisotropy evolution. Math Mech Solids2021; 26: 738–770.
12.
YangHTimofeevDAbaliBE, et al. Verification of strain gradient elasticity computation by analytical solutions. Z Angew Math Mech2021; 101: e202100023.
13.
DesmoratBSpagnuoloMTurcoE. Stiffness optimization in nonlinear pantographic structures. Math Mech Solids2020; 25: 2252–2262.
14.
CiallellaAPasqualiDGołaszewskiM, et al. A rate-independent internal friction to describe the hysteretic behavior of pantographic structures under cyclic loads. Mech Res Commun2021: 116103761.
15.
SpagnuoloMAndreausUMisraA, et al. Mesoscale modeling and experimental analyses for pantographic cells: effect of hinge deformation. Mech Mater2021; 160: 103924.
16.
SpagnuoloMYildizdagMEAndreausU, et al. Are higher-gradient models also capable of predicting mechanical behavior in the case of wide-knit pantographic structures?Math Mech Solids2021; 26: 18–29.
17.
BarchiesiEEugsterSRdell’isolaF, et al. Large in-plane elastic deformations of bipantographic fabrics: asymptotic homogenization and experimental validation. Math Mech Solids2020; 25: 739–767.
18.
BarchiesiEYangHTranC, et al. Computation of brittle fracture propagation in strain gradient materials by the FEniCS library. Math Mech Solids2021; 26: 325–340.
19.
dell’IsolaFGiorgioIPawlikowskiM, et al. Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc R Soc A Math Phys Eng Sci2016; 472: 20150790.
20.
GiorgioIRizziNTurcoE. Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proc R Soc A Math Phys Eng Sci2017; 473: 20170636.
21.
SpagnuoloMBarczKPfaffA, et al. Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mech Res Commun2017; 83: 47–52.
22.
TurcoECazzaniARizziNL, et al. Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Z Angew Math Phys2016; 67: 1–28.
23.
TurcoEBarchiesiEdell’IsolaF. In-plane dynamic buckling of duoskelion beam-like structures: discrete modeling and numerical results. Math Mech Solids2021; 27: 1164–1184.
24.
TurcoE. Stepwise analysis of pantographic beams subjected to impulsive loads. Math Mech Solids2021; 26: 62–79.
25.
TurcoEMisraASarikayaR, et al. Quantitative analysis of deformation mechanisms in pantographic substructures: experiments and modeling. Continuum Mech Therm2019; 31: 209–223.
26.
YildizdagMEBarchiesiEdell’IsolaF. Three-point bending test of pantographic blocks: numerical and experimental investigation. Math Mech Solids2020; 25: 1965–1978.
27.
VangelatosZYildizdagMEGiorgioI, et al. Investigating the mechanical response of microscale pantographic structures fabricated by multiphoton lithography. Extreme Mech Lett2021; 43: 101202.
28.
SpagnuoloMYildizdagMEPinelliX, et al. Out-of-plane deformation reduction via inelastic hinges in fibrous metamaterials and simplified damage approach. Math Mech Solids2022; 27: 1011–1031.
29.
AydinGYildizdagMEAbaliBE. Strain-gradient modeling and computation of 3-D printed metamaterials for verifying constitutive parameters determined by asymptotic homogenization. In: Theoretical analyses, computations, and experiments of multiscale materials. Berlin: Springer, 2022, pp. 343–357.
30.
RahaliYGiorgioIGanghofferJ, et al. Homogenization à la piola produces second gradient continuum models for linear pantographic lattices. Int J Eng Sci2015; 97: 148–172.
31.
BoutinCGiorgioIPlacidiL, et al. Linear pantographic sheets: asymptotic micro-macro models identification. Math Mech Complex Syst2017; 5: 127–162.
32.
BarchiesiEHarschJGanzoschG, et al. Discrete versus homogenized continuum modeling in finite deformation bias extension test of bi-pantographic fabrics. Continuum Mech Therm. Epub ahead of print 14August2020. DOI: 10.1007/s00161-020-00917-w.
33.
BarchiesiEPlacidiL. A review on models for the 3D statics and 2D dynamics of pantographic fabrics. In: SumbatyanM (ed.) Wave dynamics and composite mechanics for microstructured materials and metamaterials. Berlin: Springer, 2017, pp. 239–258.
34.
BarchiesiELaudatoMDi CosmoF. Wave dispersion in non-linear pantographic beams. Mech Res Commun2018; 94: 128–132.
35.
LaudatoMManzariLBarchiesiE, et al. First experimental observation of the dynamical behavior of a pantographic metamaterial. Mech Res Commun2018; 94: 125–127.
36.
LaudatoMManzariLGiorgioI, et al. Dynamics of pantographic sheet around the clamping region: experimental and numerical analysis. Math Mech Solids2021; 26: 1515–1537.
37.
ShekarchizadehNLaudatoMManzariL, et al. Parameter identification of a second-gradient model for the description of pantographic structures in dynamic regime. Z Angew Math Phys2021; 72: 1–24.
38.
TurcoEBarchiesiEdell’IsolaF. A numerical investigation on impulse-induced nonlinear longitudinal waves in pantographic beams. Math Mech Solids2022; 27: 22–48.
39.
dell’IsolaFMadeoAPlacidiL. Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua. Z Angew Math Mech2012; 92: 52–71.
40.
PlacidiLRosiGGiorgioI, et al. Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials. Math Mech Solids2014; 19: 555–578.
41.
EremeyevVAdell’IsolaFBoutinC, et al. Linear pantographic sheets: existence and uniqueness of weak solutions. J Elast2018; 132: 175–196.
