A hemi-variational formulation for a damaged non-homogeneous Timoshenko beam is proposed here for the purpose of fast simulation of the properties of a dam. The dam is therefore modeled as a damaged non-homogeneous Timoshenko beam embedded in 2D space. The damage evolution and the mechanics of the beam are governed both by the hemi-variational principle and by the assumption on the form of the deformation energy functional, that comprehends the dissipative part owing to damage-aging effects. In the formulation, the Karush–Kuhn–Tucker (KKT) condition is derived and the damage, which locally decreases the stiffness (i.e., axial, bending, and shear stiffness) of the beam, becomes relevant once a measure of the elastic energy, called the normalized undamaged elastic energy, reaches a certain threshold that is constitutively prescribed ab initio. The aging effect is assumed by reducing such a threshold in the neighborhood of the bottom of the beam, where the concentration of the chemical and physical attacks is higher. As a result, we show a method for the calculation of the mechanical failure condition. In addition, we observe that, even though this threshold is assumed to decrease in a large region far from the bottom of the dam, the damage is confined at the bottom of it. Thus, we have proved that the trapezoidal shape that is used in the dam design is useful not only to decrease the state of stress in the elastic regime, but also to confine the effects of damage evolution owing to the aging effects.
DeangeliCGianiGChiaiaB, et al. Dam failures. WIT Transactions on State-of-the-art in Science and Engineering2009; 36.
2.
FantilliAPFrigoBChiaiaB.A simplified approach to the evaluation of the strength of old concrete. Proc Inst Civil Eng Construct Mater2018; 171(6): 257–266.
3.
FrigoBFantilliAPDe BiagiV, et al. A protocol to assess the seismic criticality of existing small concrete dams. Struct Infrastruct Eng2018; 14(9): 1197–1206.
4.
dell’IsolaFVolkovIAIgumnovLA, et al. Estimating fatigue related damage in alloys under block-type non-symmetrical low-cycle loading. In New Achievements in Continuum Mechanics and Thermodynamics. Berlin: Springer, 2019, pp. 81–92.
5.
Dell’IsolaFWoŹniakC.On phase transition layers in certain micro-damaged two-phase solids. Int J Fract1997; 83(2): 175–189.
6.
ZhongHLinGLiH.Numerical simulation of damage in high arch dam due to earthquake. Front Architect Civil Eng China2009; 3(3): 316–322.
7.
De BiagiV. Structural behavior of a metallic truss under progressive damage. Int J Solids Struct2016; 82: 56–64.
8.
SpagnuoloMBarczKPfaffA, et al. Qualitative pivot damage analysis in aluminum printed pantographic sheets: Numerics and experiments. Mech Res Commun2017; 83: 47–52.
9.
FairbairnEMRibeiroFLLopesLE, et al. Modelling the structural behaviour of a dam affected by alkali–silica reaction. Commun Numer Meth Eng2006; 22(1): 1–12.
10.
CarpinteriAChiaiaBCornettiP.A scale-invariant cohesive crack model for quasi-brittle materials. Eng Fract Mech2002; 69(2): 207–217.
11.
CornettiPSaporaACarpinteriA.Short cracks and V-notches: Finite fracture mechanics vs. cohesive crack model. Eng Fract Mech2016; 168: 2–12.
12.
FantilliAPValliniP.A cohesive interface model for the pullout of inclined steel fibers in cementitious matrixes. J Adv Concrete Technol2007; 5(2): 247–258.
13.
AsproneDChiaiaBBiagiVD, et al. Implementation of progressive damage in finiteelement codes for the assessment of robustness. In ECCOMAS Congress 2016.
14.
CarpinteriAChiaiaB.Embrittlement and decrease of apparent strength in large-sized concrete structures. Sadhana2002; 27(4): 425.
15.
BazantZNovákD.Nonlocal Weibull theory and size effect in failures at fracture initiation. Fract Mech Concrete Struct2001; : 659–664.
16.
TurcoEdell’IsolaFRizziN, et al. Fiber rupture in sheared planar pantographic sheets: Numerical and experimental evidence. Mech Res Commun2016; 76: 86–90.
17.
ChabocheJL.Continuum damage mechanics: Part I. General concepts. J Appl Mech1988; 55(1): 59–64.
18.
AmorHMarigoJJMauriniC.Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments. J Mech Phys Solids2009; 57(8): 1209–1229.
BourdinBFrancfortGMarigoJJ.The variational approach to fracture. J Elasticity2008; 91(1): 5–148.
21.
FrancfortGMarigoJJ.Griffith theory of brittle fracture revisited: Merits and drawbacks. Latin Amer J Solids Struct2005; 2(1): 57–64.
22.
FrancfortGMarigoJJ.Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids1998; 46(8): 1319–1342.
23.
LiTMarigoJJGuilbaudD, et al. Gradient damage modeling of brittle fracture in an explicit dynamics context. Int J Numer Meth Eng2016; 108(11): 1381–1405.
24.
LiTMarigoJJGuilbaudD, et al. Variational approach to dynamic brittle fracture via gradient damage models. Appl Mech Mater784: 334–341.
25.
MarigoJJMauriniCPhamK.An overview of the modelling of fracture by gradient damage models. Meccanica2016; 51: 3107–3128.
26.
PhamKAmorHMarigoJJ, et al. Gradient damage models and their use to approximate brittle fracture. Int J Damage Mech2011; 20(4): 618–652.
27.
SicsicPMarigoJJ.From gradient damage laws to Griffith’s theory of crack propagation. Journal of Elasticity2013; 113: 55–74.
28.
MisraALucaPTurcoE.Variational methods for discrete models of granular materials. Encylopedia of Continuum Mechanics. Berlin: Springer-Verlag, 2020.
29.
PlacidiLBarchiesiE.Energy approach to brittle fracture in strain-gradient modelling. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences2018; 474(2210): 20170878.
30.
PlacidiLBarchiesiEMisraA.A strain gradient variational approach to damage: A comparison with damage gradient models and numerical results. Math Mech Complex Syst2018; 6(2): 77–100.
31.
PlacidiLBarchiesiEMisraA, et al. Variational methods in continuum damage and fracture mechanics. Encylopedia of Continuum Mechanics. Berlin: Springer-Verlag, 2020.
32.
PlacidiLMisraABarchiesiE.Simulation results for damage with evolving microstructure and growing strain gradient moduli. Continuum Mech Thermodynam2019; 31(4): 1143–1163.
33.
PlacidiLMisraABarchiesiE.Two-dimensional strain gradient damage modeling: A variational approach. Z angew Math Phys2018; 69(3): 56.
34.
Del PieroGLancioniGMarchR. A variational model for fracture mechanics: Numerical experiments. J Mech Phys Solids2007; 55(12): 2513–2537.
35.
Dell’IsolaFPlacidiL.Variational principles are a powerful tool also for formulating field theories. In Variational Models and Methods in Solid and Fluid Mechanics. Berlin: Springer, 2011.
36.
FreddiFRoyer-CarfagniG.Regularized variational theories of fracture: A unified approach. J Mech Phys Solids2010; 58(8): 1154–1174.
37.
GiacominiA.Ambrosio–Tortorelli approximation of quasi-static evolution of brittle fractures. Calculus Variations Partial Differ Eqns2005; 22(2): 129–172.
38.
Dell’IsolaFGuarascioMHutterK.A variational approach for the deformation of a saturated porous solid. A second-gradient theory extending Terzaghi’s effective stress principle. Arch Appl Mech2000; 70(5): 323–337.
39.
Dell’IsolaFHutterK.Continuum mechanical modelling of the dissipative processes in the sediment–water layer below glaciers. C R Acad Sci Ser IIB Mech Phys Chem Astron1997; 325(8): 449–456.
40.
CuomoM.Continuum damage model for strain gradient materials with applications to 1D examples. Continuum Mech Thermodynam2019; 31(4): 969–987.
41.
KotronisPCollinFBésuelleP, et al. Local second gradient models and damage mechanics: 1D post-localization studies in concrete specimens. In Bifurcations, Instabilities, Degradation in Geomechanics. Berlin: Springer, 2007, pp. 127–142.
42.
MarigoJJPhamK.Construction and analysis of localized responses for gradient damage models in a 1D setting. Vietnam J Mech2009; 31(3–4): 5651.
43.
MeschkeGLacknerRMangHA.An anisotropic elastoplastic-damage model for plain concrete. Int J Numer Meth Eng1998; 42(4): 703–727.
44.
PlacidiL.A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model. Continuum Mech Thermodynam2016; 28(1–2): 119–137.
45.
PlacidiL.A variational approach for a nonlinear 1-dimensional second gradient continuum damage model. Continuum Mech Thermodynam2015; 27(4–5): 623.
46.
Di PaolaMBilelloC. An integral equation for damage identification of euler-bernoulli beams under static loads. J Eng Mech2004; 130(2): 225–234.
47.
ArosioAPanizziSPaoliMG.Inhomogeneous Timoshenko beam equations. Math Meth Appl Sci1992; 15(9): 621–630.
48.
PiccardoGRanziGLuongoA.A complete dynamic approach to the generalized beam theory cross-section analysis including extension and shear modes. Math Mech Solids2014; 19(8): 900–924.
49.
De AngeloMPlacidiLNejadsadeghiN, et al. Non-standard Timoshenko beam model for chiral metamaterial: Identification of stiffness parameters. Mech Res Commun2020; 103: 103462.
50.
CennamoCCusanoCAngelilloM, et al. A study on form and seismic vulnerability of the dome of San Francesco di Paola in Naples. Ingegneria Sismica2018; 35(1): 88–108.
51.
CennamoCDi FioreM. Best practice of structural retrofit: the SS. Rosario church in Gesualdo, Italy. Int J Disaster Resilience Built Environment2013; 4(2): 0049.
52.
BattistaADella CorteAdell’IsolaF, et al. Large deformations of 1D microstructured systems modeled as generalized Timoshenko beams. Z angew Math Phys2018; 69(3): 52.
53.
TurcoEBarchiesiEGiorgioI, et al. A Lagrangian Hencky-type non-linear model suitable for metamaterials design of shearable and extensible slender deformable bodies alternative to Timoshenko theory. Int J Non-Lin Mech2020; 123: 103481.
54.
GiorgioIScerratoD.Multi-scale concrete model with rate-dependent internal friction. Eur J Environ Civil Eng2017; 21(7–8): 821–839.
55.
ScerratoDGiorgioIMadeoA, et al. A simple non-linear model for internal friction in modified concrete. Int J Eng Sci2014; 80: 136–152.
56.
ScerratoDGiorgioIDella CorteA, et al. A micro-structural model for dissipation phenomena in the concrete. Int J Numer Anal Meth Geomech2015; 39(18): 2037–2052.
57.
MisraA.Effect of asperity damage on shear behavior of single fracture. Eng Fract Mech2002; 69(17): 1997–2014.
58.
MisraASinghV.Micromechanical model for viscoelastic materials undergoing damage. Continuum Mech Thermodynam2013; 25: 343–358.
59.
TurcoEdell’IsolaFMisraA.A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations. Int J Numer Anal Meth Geomech2019; 43(5): 1051–1079.
60.
MisraAPoorsolhjouyP.Granular micromechanics model for damage and plasticity of cementitious materials based upon thermomechanics. Math Mech Solids2020; 25(10): 1778–1803.
61.
PoorsolhjouyPMisraA.Effect of intermediate principal stress and loading-path on failure of cementitious materials using granular micromechanics. Int J Solids Struct2017; 108: 139–152.
62.
VolkovIAIgumnovLAdell’IsolaF, et al. A continual model of a damaged medium used for analyzing fatigue life of polycrystalline structural alloys under thermal–mechanical loading. Continuum Mech Thermodynam2020; 32(1): 229–245.
