Non-linear effects in seismic waves in high-energy earthquakes: A two-dimensional analysis for non-homogeneous isotropic media with a view towards the study of the 2009 L’Aquila earthquake
Restricted accessResearch articleFirst published online February, 2026
Non-linear effects in seismic waves in high-energy earthquakes: A two-dimensional analysis for non-homogeneous isotropic media with a view towards the study of the 2009 L’Aquila earthquake
It is usually accepted in geophysics (and in civil engineering) that linear models can be used for describing an earthquake and the consequent seismic waves’ propagation. However, the large deformation experienced by the soil in these situations suggests that this paradigm requires more critical consideration. In fact, we claim that, in the vicinity of some discontinuities (that are common in all the geophysical applications of continuum models), the corresponding strain concentrations make the hypothesis of small deformation to be inadequate. In this paper, we verify the inappropriateness of the linear paradigm in a simple but reasonable case, with a view to a future application of this study to the effects of the 2009 L’Aquila earthquake. To this aim, we start with an analysis which is restricted to a two-dimensional body (i) with an inhomogeneity that resembles the Aterno River Valley, central Italy and (ii) with a non-linearity that is the most simple one, choosing the strain energy to be quadratic in the non-linear measures of deformation. More precisely, we consider a 2D piecewise homogeneous domain and a material that is viscoelastic isotropic and geometrically non-linear. We apply, to the bottom of such a domain, a seismic excitation and calculate the differences in the response between the linear and the geometrically non-linear cases. Using a suitably designed numerical model, we prove that, as conjectured, these differences not only originate near the pre-defined geometrical discontinuities but also propagate throughout the rest of the domain. Moreover, we find numerical predictions of the frequency ratios and ground acceleration time dependence and amplitude that produce, in the case of non-linear models, predictions which are closer to experimental evidence than those obtained using the corresponding linear model.
EremeyevVAAlzahraniFSCazzaniA, et al. On existence and uniqueness of weak solutions for linear pantographic beam lattices models. Continuum Mech Therm2019; 31(6): 1843–1861.
2.
CiallellaAPasqualiDD’AnnibaleF, et al. Shear rupture mechanism and dissipation phenomena in bias-extension test of pantographic sheets: numerical modeling and experiments. Math Mech Solids2022; 27(10): 2170–2188.
3.
LaudatoMBarchiesiE. Non-linear dynamics of pantographic fabrics: modelling and numerical study. In: SumbatyanMA (ed.) Wave dynamics, mechanics and physics of microstructured metamaterials: Theoretical and experimental methods. Cham: Springer, 2019, pp. 241–254.
4.
dell’isolaFCorteADBattistaA, et al. Extensible beam models in large deformation under distributed loading: a numerical study on multiplicity of solutions. In: AltenbachHMüllerWHAbaliBE (eds) Higher gradient materials and related generalized continua. Cham: Springer, 2019, pp. 19–41.
5.
EremeyevVZubovL. On constitutive inequalities in nonlinear theory of elastic shells. J Appl Math Mech2007; 87(2): 94–101.
6.
AbaliBEMüllerWHEremeyevVA. Strain gradient elasticity with geometric nonlinearities and its computational evaluation. Mech Adv Mater Modern Process2015; 1(1): 1–11.
EremeyevVAPietraszkiewiczW. The nonlinear theory of elastic shells with phase transitions. J Elasticity2004; 74(1): 67–86.
9.
EugsterSR. Numerical analysis of nonlinear wave propagation in a pantographic sheet. Math Mech Complex Syst2022; 9(3): 293–310.
10.
TurcoEBarchiesiECiallellaA, et al. Nonlinear waves in pantographic beams induced by transverse impulses. Wave Motion2022; 115: 103064.
11.
EremeyevVALebedevLPCloudMJ. Acceleration waves in the nonlinear micromorphic continuum. Mech Res Commun2018; 93: 70–74.
12.
Abd-allaAAlshaikhFGiorgioI, et al. A mathematical model for longitudinal wave propagation in a magnetoelastic hollow circular cylinder of anisotropic material under the influence of initial hydrostatic stress. Math Mech Solids2016; 21(1): 104–118.
13.
AbaliBEBarchiesiE. Additive manufacturing introduced substructure and computational determination of metamaterials parameters by means of the asymptotic homogenization. Continuum Mech Therm2021; 33(4): 993–1009.
14.
CefisNFedeleRBeghiMG. An integrated methodology to estimate the effective elastic parameters of amorphous tio2 nanostructured films, combining sem images, finite element simulations and homogenization techniques. Mech Res Commun2023; 131: 104153.
15.
FedeleRMaierG. Flat-jack tests and inverse analysis for the identification of stress states and elastic properties in concrete dams. Meccanica2007; 42(4): 387–402.
16.
FedeleRSessaSValorosoN. Image correlation-based identification of fracture parameters for structural adhesives. Tech Mech-Eur J Eng Mech2019; 32(2–5): 195–204.
17.
DarbanHLucianoRCaporaleA, et al. Higher modes of buckling in shear deformable nanobeams. Int J Eng Sci2020; 154: 103338.
18.
BormannPBaumbachMBockG, et al. Seismic sources and source parameters. In: BormannP (ed.) New manual of seismological observatory practice (NMSOP-1). Potsdam: Deutsches GeoForschungsZentrum GFZ, 2009, pp. 1–94.
19.
TurcoEBarchiesiE. Kinematically-triggered nonlinear vibrations of Hencky-type pantographic sheets. Math Mech Complex Syst2021; 9: 311–335.
20.
LucianoRDarbanHBartolomeoC, et al. Free flexural vibrations of nanobeams with non-classical boundary conditions using stress-driven nonlocal model. Mech Res Commun2020; 107: 103536.
21.
MancusiGFabbrocinoFFeoL, et al. Size effect and dynamic properties of 2D lattice materials. Compos B Eng2017; 112: 235–242.
22.
FunariMFSpadeaSFabbrocinoF, et al. A moving interface finite element formulation to predict dynamic edge debonding in FRP-strengthened concrete beams in service conditions. Fibers2020; 8(6): 42.
23.
TurcoEBarchiesiE. Kinematically triggered nonlinear vibrations of Hencky-type pantographic sheets. Math Mech Complex Syst2022; 9(3): 311–335.
PlacidiLBarchiesiEDell’isolaF, et al. On a hemi-variational formulation for a 2D elasto-plastic-damage strain gradient solid with granular microstructure. Math Eng2022; 5: 1–24.
26.
ScrofaniABarchiesiEChiaiaB, et al. Fluid diffusion related aging effect in a concrete dam modeled as a Timoshenko beam. Math Mech Complex Syst2023; 11(2): 313–334.
27.
ChiaiaBBiagiVDPlacidiL. A damaged non-homogeneous Timoshenko beam model for a dam subjected to aging effects. Math Mech Solids2021; 26(5): 694–707.
28.
LucaGDScarpaRFilippiL, et al. A detailed analysis of two seismic sequences in Abruzzo, Central Apennines, Italy. J Seismol2000; 4(1): 1–21.
29.
ScognamiglioLTintiEQuintilianiM. Time Domain Moment Tensor (TDMT) [Data set]. Istituto Nazionale di Geofisica e Vulcanologia (INGV), 2006. DOI: 10.13127/TDMT.
30.
ChiarabbaCAmatoAAnselmiM, et al. The 2009 L’Aquila (Central Italy) mw6.3 earthquake: main shock and aftershocks. Geophys Res Lett2009; 36(18): 1–6.
31.
RovidaALocatiMCamassiR, et al. Catalogo Parametrico dei Terremoti Italiani (CPTI15), versione 4.0 [Data set]. Istituto Nazionale di Geofisica e Vulcanologia (INGV), 2022. DOI: 10.13127/cpti/cpti15.4.
32.
RovidaALocatiMCamassiR, et al. The Italian earthquake catalogue CPTI15. B Earthq Eng2020; 18(7): 2953–2984.
33.
LucaGDMarcucciSMilanaG, et al. Evidence of low-frequency amplification in the city of L’Aquila, Central Italy, through a multidisciplinary approach including strong- and weak-motion data, ambient noise, and numerical modelling. Bull Seismol Soc Am2005; 95(4): 1469–1481.
34.
MilanaGAzzaraRMBertrandE, et al. The contribution of seismic data in microzonation studies for downtown L’Aquila. B Earthq Eng2011; 9(3): 741–759.
35.
PugliaRDitommasoRPacorF, et al. Frequency variation in site response as observed from strong motion data of the L’Aquila (2009) seismic sequence. B Earthq Eng2011; 9(3): 869–892.
36.
RagozzinoE. Nonlinear seismic response in the western L’Aquila basin (Italy): numerical FEM simulations vs. Ground motion records. Eng Geol2014; 174: 46–60.
37.
FerraroAGrassoSMaugeriM, et al. Seismic response analysis in the southern part of the historic centre of the city of L’Aquila (Italy). Soil Dyn Earthq Eng2016; 88: 256–264.
38.
GiulioGDGaudiosiICaraF, et al. Shear-wave velocity profile and seismic input derived from ambient vibration array measurements: the case study of downtown L’Aquila. Geophys J Int2014; 198: 848–866.
39.
TalliniMSardoLLSpadiM. Seismic site characterisation of red soil and soil-building resonance effects in L’Aquila downtown (Central Italy). Bull Eng Geol Environ2020; 79(8): 4021–4034.
40.
BergamaschiFCultreraGAzzaraR, et al. Evaluation of site effects in the Aterno river valley (Central Italy) from aftershocks of the 2009 L’Aquila earthquake. B Earthq Eng2011; 9: 697–715.
41.
CalderoniGRovelliAGiovambattistaRD. Large amplitude variations recorded by an on-fault seismological station during the L’Aquila earthquakes: evidence for a complex fault-induced site effect. Geophys Res Lett2010; 37: L24305.
42.
AkinciAMalagniniLSabettaF. Characteristics of the strong ground motions from the 6 April 2009 L’Aquila earthquake, Italy. Soil Dyn Earthq Eng2010; 30(5): 320–335.
43.
LanzoGPagliaroliA. Seismic site effects at near-fault strong-motion stations along the Aterno River Valley during the Mw=6.3 2009 L’Aquila earthquake. Soil Dyn Earthq Eng2012; 40: 1–14.
44.
BindiDPacorFLuziL, et al. The Mw 6.3, 2009 L’Aquila earthquake: source, path and site effects from spectral analysis of strong motion data. Geophys J Int2009; 179(3): 1573–1579.
45.
AmeriGBindiDPacorF, et al. The 2009 April 6, Mw 6.3, L’Aquila (Central Italy) earthquake: finite-fault effects on intensity data. Geophys J Int2011; 186(2): 837–851.
46.
UgurhanBAskanAAkinciA, et al. Strong-ground-motion simulation of the 6 April 2009 L’Aquila, Italy, earthquake. Bull Seismol Soc Am2012; 102(4): 1429–1445.
47.
FelicettaCRussoED’AmicoM, et al. Italian Accelerometric Archive v4.0. Istituto Nazionale di Geofisica e Vulcanologia, Dipartimento della Protezione Civile Nazionale, 2023. DOI: 10.13127/itaca.4.0.
48.
CelebiMBazzurroPChiaraluceL, et al. Recorded motions of the 6 April 2009 Mw 6.3 L’Aquila, Italy, earthquake and implications for building structural damage: overview. Earthq Spectra2010; 26(3): 651–684.
49.
StewartJPLanzoGPagliaroliA, et al. Ground motion recordings from the Mw 6.3 2009 L’Aquila earthquake in Italy and their engineering implications. Earthq Spectra2012; 28(1): 317–345.
50.
RupakhetyRSigbjörnssonR. A note on the L’Aquila earthquake of 6 April 2009: permanent ground displacements obtained from strong-motion accelerograms. Soil Dyn Earthq Eng2010; 30(4): 215–220.
51.
AugentiNParisiF. Learning from construction failures due to the 2009 L’Aquila, Italy, earthquake. J Perform Constr Fac2010; 24: 536–555.
52.
CucciLTertullianiA. Clues for a relation between rotational effects induced by the 2009 Mw 6.3 L’Aquila (Central Italy) earthquake and site and source effects. B Seismol Soc Am2011; 101: 1109–1120.
53.
TertullianiALeschiuttaIBordoniP, et al. Damage distribution in L’Aquila city (Central Italy) during the 6 April 2009 earthquake. Bull Seismol Soc Am2012; 102(4): 1543–1553.
54.
MannellaAMacerolaLMartinelliA, et al. The local seismic response and the effects of the 2016 Central Italy earthquake on the buildings of L’Aquila downtown. Boll Geofis Teor Appl2019; 60(2): 263–276.
55.
DouglasJMonfortDNegulescuC, et al. Limits on the potential accuracy of earthquake risk evaluations using the L’Aquila (Italy) earthquake as an example. Ann Geophys2015; 58: S0214.
56.
VannoliPBurratoPValensiseG. The seismotectonics of the Po Plain (Northern Italy): tectonic diversity in a blind faulting domain. Pure Appl Geophys2015; 172: 1105–1142.
57.
TertullianiAArcoraciLBerardiM, et al. An application of EMS98 in a medium-sized city: the case of L’Aquila (Central Italy) after the April 6, 2009 Mw 6.3 earthquake. B Earthq Eng2011; 9: 67–80.
58.
BlumettiAMMannaPDComerciV, et al. Paleoseismicity of the San Demetrio ne’ Vestini fault (L’Aquila Basin, Central Italy): implications for seismic hazard. Quatern Int2017; 451: 129–142.
59.
BrunoPPGVillaniFImprotaL. High-resolution seismic imaging of fault-controlled basins: a case study from the 2009 Mw 6.1 Central Italy earthquake. Tectonics2022; 41(4): e2022TC007207.
60.
CivicoRSapiaVGiulioGD, et al. Geometry and evolution of a fault-controlled quaternary basin by means of TDEM and single-station ambient vibration surveys: the example of the 2009 L’Aquila earthquake area, Central Italy. J Geophys Res Solid Earth2017; 122(3): 2236–2259.
61.
NocentiniMCosentinoDSpadiM, et al. Plio-Quaternary geology of the Paganica-San Demetrio-Castelnuovo Basin (Central Italy). J Maps2018; 14: 411–420.
62.
SpadiMTalliniMAlbanoM, et al. New insights on bedrock morphology and local seismic amplification of the castelnuovo village (L’Aquila Basin, Central Italy). Eng Geol2022; 297: 106506.
63.
TalliniMSpadiMCosentinoD, et al. High-resolution seismic reflection exploration for evaluating the seismic hazard in a Plio-Quaternary intermontane basin (L’Aquila downtown, Central Italy). Quatern Int2019; 532: 34–47.
64.
CultreraGMucciarelliMParolaiS. The L’Aquila earthquake—a view of site effects and building behavior from temporary networks. B Earthq Eng2011; 9(3): 691–695.
65.
MonacoPAmorosoS. Site effects from the building scale to the seismic microzonation scale: examples from the experience of L’Aquila. Procedia Engineer2016; 158: 517–522.
66.
NunziataCCostanzoM. Ground motion modeling for site effects at L’Aquila and middle Aterno river valley (Central Italy) for the Mw 6.3, 2009 earthquake. Soil Dyn Earthq Eng2014; 61–62: 107–123.
67.
NunziataCVaccariFPanzaGF. The Mw 6.3, 2009 LAquila earthquake: Linear and nonlinear site effects. In: MazzolaniFM (ed.) Urban Habitat Constructions Under Catastrophic Events: Proceedings of the Cost C26 Action Final Conference. London: Taylor & Francis Group, 2010.
68.
NunziataCCostanzoMVaccariF, et al. Evaluation of linear and nonlinear site effects for the MW 6.3, 2009 L’Aquila earthquake. Earthquake Research and AnalysiseNew Frontiers in Seismology2012; 5: 155e176.
69.
MagnoniFCasarottiEMicheliniA, et al. Spectral-element simulations of seismic waves generated by the 2009 L’Aquila earthquake. Bull Seismol Soc Am2014; 104(1): 73–94.
70.
SmerziniCVillaniM. Broadband numerical simulations in complex near-field geological configurations: the case of the 2009 Mw 6.3 L’Aquila earthquake. Bull Seismol Soc Am2012; 102(6): 2436–2451.
71.
MicheleFDMayJPeraD, et al. Spectral element numerical simulation of the 2009 L’Aquila earthquake on a detailed reconstructed domain. Geophys J Int2022; 230(1): 29–49.
72.
GaudiosiIMonacoFDMilanaG, et al. Site effects in the Aterno river valley (L’Aquila, Italy): comparison between empirical and 2D numerical modelling starting from April 6th 2009 Mw 6.3 earthquake. B Earthq Eng2014; 12(2): 697–716.
73.
MargheritiLChiaraluceLVoisinC, et al. Rapid response seismic networks in Europe: lessons learnt from the L’Aquila earthquake emergency. Ann Geophys2011; 54: 392–399.
74.
BarchiMLavecchiaGGaladiniF, et al. Sintesi delle conoscenze sulle faglie attive in italia centrale: Parametrizzazione ai fini della caratterizzazione della pericolosità sismica, 2000.
75.
CarminatiELustrinoMCuffaroM, et al. Tectonics, magmatism and geodynamics of Italy: what we know and what we imagine, in: Beltrando, M., Peccerillo, A., Mattei, M., Conticelli, S., Doglioni, C. (Eds.), The Geology of Italy: tectonics and life along plate margins. J Virtual Expl, Electronic Edition2010; 36: 1441–8142.
76.
CavinatoGPPGCellesPGD. Extensional basins in the tectonically bimodal central apennines fold-thrust belt, Italy: response to corner flow above a subducting slab in retrograde motion. Geology1999; 27(10): 955–958.
77.
CosentinoDAstiRNocentiniM, et al. New insights into the onset and evolution of the central apennine extensional intermontane basins based on the tectonically active L’Aquila basin (Central Italy). Bull Cl Sci Math Nat Sci Math2017; 129(9–10): 1314–1336.
78.
BoncioPLavecchiaGPaceB. Defining a model of 3D seismogenic sources for seismic hazard assessment applications: the case of central apennines (Italy). J Seismol2004; 8(9–10): 407–425.
79.
DevotiRPietrantonioGPisaniAR, et al. Present day kinematics of Italy. J Virtual Expl2010; 36(2): 1–14.
80.
PondrelliSSalimbeniSEkströmG, et al. The Italian CMT dataset from 1977 to the present. Phys Earth Planet Inter2006; 159(3): 286–303.
81.
GaladiniFGalliP. Active tectonics in the central Apennines (Italy) – input data for seismic hazard assessment. Nat Hazards2000; 22(3): 225–268.
82.
MalikanMEremeyevVA. On time-dependent nonlinear dynamic response of micro-elastic solids. Int J Eng Sci2023; 182: 103793.
83.
ApuzzoABarrettaRFabbrocinoF, et al. Axial and torsional free vibrations of elastic nano-beams by stress-driven two-phase elasticity. J Appl Comput Mech2019; 5(2): 402–413.
84.
BarrettaRFaghidianSALucianoR. Longitudinal vibrations of nano-rods by stress-driven integral elasticity. Mech Adv Mater Struc2019; 26(15): 1307–1315.
85.
HeiduschkeK. On tensor projections, stress or stretch vectors and their relations to Mohr’s three circles. Math Mech Complex Syst2024; 12(2): 173–216.
86.
SharmaBLEremeyevVA. Wave transmission across surface interfaces in lattice structures. Int J Eng Sci2019; 145: 103173.
87.
PlacidiLGirolamoFDFedeleR. Variational study of a Maxwell–Rayleigh-type finite length model for the preliminary design of a tensegrity chain with a tunable band gap. Mech Res Commun2024; 136: 104255.
88.
PlacidiLSherbinyMGEBaragattiP. Experimental investigation for the existence of frequency band gap in a microstructure model. Math Mech Complex Syst2022; 9(4): 413–421.
89.
MikhasevGErbaşBEremeyevVA. Anti-plane shear waves in an elastic strip rigidly attached to an elastic half-space. Int J Eng Sci2023; 184: 103809.
90.
AskesHWallaceAR. Dynamic Bergan–Wang theory for thick plates. Math Mech Complex Syst2022; 10(2): 191–204.
91.
KoKYSolyaevY. Explicit benchmark solution for topology optimization of variable-thickness plates. Math Mech Complex Syst2023; 11(3): 381–392.
92.
AndreausUdell’isolaFGiorgioI, et al. Numerical simulations of classical problems in two-dimensional (non) linear second gradient elasticity. Int J Eng Sci2016; 108: 34–50.
93.
dell’isolaFEugsterSRFedeleR, et al. Second-gradient continua: from Lagrangian to Eulerian and back. Math Mech Solids2022; 27(12): 2715–2750.
94.
SolyaevYLurieSBarchiesiE, et al. On the dependence of standard and gradient elastic material constants on a field of defects. Math Mech Solids2020; 25(1): 35–45.
95.
PlacidiLRosiGGiorgioI, et al. Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials. Math Mech Solids2014; 19(5): 555–578.
96.
RosiGGiorgioIEremeyevVA. Propagation of linear compression waves through plane interfacial layers and mass adsorption in second gradient fluids. J Appl Math Mech2013; 93(12): 914–927.
97.
YangBBacciocchiMFantuzziN, et al. Wave propagation in periodic nano structures through second strain gradient elasticity. Int J Mech Sci2023; 260: 108639.
98.
CornacchiaFFabbrocinoFFantuzziN, et al. Analytical solution of cross- and angle-ply nano plates with strain gradient theory for linear vibrations and buckling. Mech Adv Mater Struc2021; 28(12): 1201–1215.
99.
CuomoMContrafattoLGrecoL. A cohesive interface model with degrading friction coefficient. Math Mech Complex Syst2024; 12(2): 113–133.
100.
PonomarevD. A generalised time-evolution model for contact problems with wear and its analysis. Math Mech Complex Syst2022; 10(3): 279–319.
101.
GrilloAStefanoSD. An a posteriori approach to the mechanics of volumetric growth. Math Mech Complex Syst2023; 11(1): 57–86.
102.
GrilloAStefanoSD. Comparison between different viewpoints on bulk growth mechanics. Math Mech Complex Syst2023; 11(2): 287–311.
103.
TepedinoM. The mechanical role of the periodontal ligament for developing mathematical models in orthodontics. Math Mech Complex Syst2023; 11(4): 525–539.
104.
OliveMAuffrayN. Symmetry classes in piezoelectricity from second-order symmetries. Math Mech Complex Syst2021; 9: 77–105.
105.
MisraAPlacidiLDell’isolaF, et al. Identification of a geometrically nonlinear micromorphic continuum via granular micromechanics. Z fur Angew Math Physik2021; 72(4): 157.
106.
SciarraGdell’IsolaFCoussyO. Second gradient poromechanics. Int J Solids Struct2007; 44(20): 6607–6629.
107.
TimofeevDBarchiesiEMisraA, et al. Hemivariational continuum approach for granular solids with damage-induced anisotropy evolution. Math Mech Solids2021; 26(5): 738–770.
108.
TurcoEdell’IsolaFMisraA. A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations. Int J Numer Anal Met2019; 43(5): 1051–1079.
109.
PlacidiLBarchiesiEDell’isolaF, et al. A granular-based elasto-plastic–damage energy formulation for strain gradient solids. In: Book of Abstracts, 2021.
110.
BarchiesiEMisraAPlacidiL, et al. Identification, based on granular micromechanics, of elastic isotropic strain gradient stiffness matrices for geometrically nonlinear deformations, 2021. Available at: https://iris.uninettunouniversity.net/handle/20.500.14086/2320
111.
NocentiniMAstiRCosentinoD, et al. Plio-quaternary geology of L’Aquila–Scoppito Basin (Central Italy). J Maps2017; 13.2: 563–574.
112.
AntonielliBSetaMDEspositoC, et al. Quaternary rock avalanches in the Apennines: new data and interpretation of the huge clastic deposit of the L’Aquila basin (Central Italy). Geomorphology2020; 361: 107194.
113.
BlumettiAM. Neotectonic investigations and evidence of paleoseismicity in the epicentral area of the January–February 1703, Central Italy, earthquakes. Perspect Paleoseismol1995; 6: 83–100.
114.
BruyelleP. Jean Demangeot—Géomorphologie des Abruzzes adriatiques. Thèse pour le doctorat ès-lettres. Centre de Recherches et documentation cartographique et géographiques, n° hors-série-1965. Hommes terres nord1966; 1(1): 120–121.
115.
GiorgioI. A variational formulation for one-dimensional linear thermoviscoelasticity. Math Mech Complex Syst2022; 4(9): 397–412.
116.
BersaniAMCaressaPCiallellaA. Numerical evidence for the approximation of dissipative systems by gyroscopically coupled oscillator chains. Math Mech Complex Syst2022; 10(3): 265–278.
117.
EremeyevVA. Surface finite viscoelasticity and surface anti-plane waves. Int J Eng Sci2024; 196: 104029.
118.
BarchiesiEMisraAPlacidiL, et al. Granular micromechanics-based identification of isotropic strain gradient parameters for elastic geometrically nonlinear deformations. Z Angew Math Mech2021; 101(11): 1–21.
119.
SoizeC. An overview on uncertainty quantification and probabilistic learning on manifolds in multiscale mechanics of materials. Math Mech Complex Syst2023; 11(1): 87–174.
120.
SalnikovV. On numerical approaches to the analysis of topology of the phase space for dynamical integrability. Chaos Solitons Fractals2013; 57: 155–161.
121.
SalnikovVHamdouniA. From modelling of systems with constraints to generalized geometry and back to numerics. J Appl Math Mech2019; 99(6): e201800218.
122.
BarchiesiEYangHTranC, et al. Computation of brittle fracture propagation in strain gradient materials by the fenics library. Math Mech Solids2021; : 325–340.
123.
FabbrocinoFFunariMFGrecoF, et al. Dynamic crack growth based on moving mesh method. Compos B Eng2019; 174: 107053.
124.
PlacidiLBarchiesiE. Energy approach to brittle fracture in strain-gradient modelling. Proc R Soc A Math Phys Eng Sci2018; 474: 20170878.
125.
PlacidiLMisraABarchiesiE. Simulation results for damage with evolving microstructure and growing strain gradient moduli. Continuum Mech Therm2019; 31(4): 1143–1163.
126.
TurcoEBarchiesiECiallellaA, et al. A new block-based approach for the analysis of damage in masonries undergoing large deformations. Continuum Mech Therm2023; 35: 1625–1654.
127.
MasiAManfrediVVonaM, et al. Prestazioni delle tamponature e tramezzature negli edifici in c.a.: Implicazioni progettuali e costruttive alla luce dell’esperienza del terremoto dell’abruzzo 2009. Progettazione Sismica2009; 3: 51–66.
128.
SaittaFClementePBuffariniG, et al. Base isolation of buildings with curved surface sliders: basic design criteria and critical issues. Adv Civil Eng2018; 2018: 1569683.
129.
SalvatoriA. Seismic base isolation: retrofitting application in structures damaged by earthquake. In: Proceedings of 16th European conference on earthquake engineering ECEE, Thessaloniki, Greece, 18-21 June, 2018.
