A variational approach for a nonlinear 1-dimensional second gradient continuum damage model. (English) Zbl 1341.74016

Placidi, Luca

Contin. Mech. Thermodyn. 27, No. 4-5, 623-638 (2015).

Summary: A 1-dimensional second gradient damage continuum theory is presented within the framework of the variational approach. The action is intended to depend not only with respect to the first gradient of the displacement field and to a scalar damage field, but also to the field of the second gradient of the displacement. Constitutive prescriptions of the stiffness (the constitutive function in front of the squared first gradient term in the action functional) and of the microstructural material length (i.e., the square of the constitutive function in front of the squared second gradient term in the action functional) are prescribed in terms of the scalar damage parameter. On the one hand, as in many other works, the stiffness is prescribed to decrease as far as the damage increases. On the other hand, the microstructural material length is prescribed, in contrast to a certain part of the literature, to increase as far as the damage increases. This last assumption is due to the interpretation that a damage state induces a microstructure in the continuum and that such a microstructure is more important as far as the damage increases. At a given value of the damage parameter, the behavior is referred to second gradient linear elastic material. However, the damage evolution makes the model not only nonlinear but also inelastic. The second principle of thermodynamics can be considered by assuming that the scalar damage parameter does not decrease its value in the process of deformation, and this implies a dissipation for the elastic strain energy. It is finally remarked that damage initiation, in this second gradient continuum damage model, can be induced not only from a prescribed initial lack of stiffness, but also from an external concentrated double force or from suitable boundary conditions, and these last options have advantages that are discussed in the paper. For example, in this last case, it is possible to initiate the damage in the neighborhood of the boundaries, that is, the case in most of the empirical situations. Simple numerical simulations are also presented in order to show the exposed concepts.

MSC:

74A45	Theories of fracture and damage
74M25	Micromechanics of solids
74R99	Fracture and damage

Keywords:

damage; variation; microstructure; strain; gradient

References:

[1]	Alibert, J.-J.; Seppecher, P.; dell’Isola, F., Truss modular beams with deformation energy depending on higher displacement gradients, Math. Mech. Solids, 8, 51-73, (2003) · Zbl 1039.74028
[2]	Altenbach, H.; Eremeyev, V.A.; Lebedev, L.P.; Rendon, L.A., Acceleration waves and ellipticity in thermoelastic micropolar media, Arch. Appl. Mech., 80, 217-227, (2010) · Zbl 1271.74251
[3]	Amor, H.; Marigo, J.-J; Maurini, C., Reguralized formulation of the variational brittle fracture with unilateral contact: numerical experiment, J. Mech. Phys. Solids, 57, 1209-1229, (2009) · Zbl 1426.74257
[4]	Andreaus, U.; Baragatti, P., Cracked beam identification by numerically analysing the nonlinear behaviour of the harmonically forced response, J. Sound Vib., 330, 721-742, (2011)
[5]	Andreaus, U., Baragatti, P.: Fatigue crack growth, free vibrations, and breathing crack detection of aluminium alloy and steel beams. J. Strain Anal. Eng. Des. 44(7), 595-608 (2009) · Zbl 1039.74028
[6]	Andreaus, U.; Baragatti, P., Experimental damage detection of cracked beams by using nonlinear characteristics of forced response, Mech. Syst. Signal Process., 31, 382-404, (2012)
[7]	Andreaus, U.; Colloca, M.; Iacoviello, D., An optimal control procedure for bone adaptation under mechanical stimulus, Control Eng. Pract., 20, 575-583, (2012)
[8]	Andreaus, U.; Colloca, M.; Iacoviello, D.; Pignataro, M., Optimal-tuning PID control of adaptive materials for structural efficiency, Struct. Multidiscip. Optim., 43, 43-59, (2011)
[9]	Andreaus, U., Giorgio, I., Lekszycki, T.: A 2-D continuum model of a mixture of bone tissue and bioresorbable material for simulating mass density redistribution under load slowly variable in time. Zeitschrift fur Angewandte Mathematik und Mechanik (ZAMM), 1-23. (2014) doi:10.1002/zamm.201200182 · Zbl 1303.74029
[10]	Auffray, N.; Bouchet, R., Brechet, derivation of anisotropic matrix for bi-dimensional strain-gradient elasticity behaviour, Int. J. Solids Struct., 46, 440-454, (2009) · Zbl 1168.74319
[11]	Auffray, N.; Bouchet, R.; Brechet, Y., Strain gradient elastic homogenization of bidimensional cellular media, Int. J. Solids Struct., 47, 1698-1710, (2010) · Zbl 1194.74281
[12]	Benallal, A.; Billardon, R.; Lamaitre, J., Continuum damage mechanics and local approach to fracture: numerical procedures, Comput. Methods Appl. Mech. Eng., 92, 141-155, (1991) · Zbl 0850.73283
[13]	Bersani, A.M., Giorgio, I., Tomassetti, G.: Buckling of an elastic hemispherical shell with an obstacle. Contin. Mech. Thermodyn. 25(2-4), 443-467 (2013) · Zbl 1343.74020
[14]	Bourdin, B.; Francfort, G.A.; Marigo, J.-J., The variational approach to fracture, J. Elast., 91, 5-148, (2008) · Zbl 1176.74018
[15]	Bui, Q.V., Initiation of damage with implicit gradient-enhanced damage models, Int. J. Solids Struct., 47, 2425-2435, (2010) · Zbl 1196.74172
[16]	Buliga, M., Energy minimizing brittle crack propagation, J. Elast., 52, 201-238, (1999) · Zbl 0947.74055
[17]	Carcaterra, A.; Ciappi, E., Prediction of the compressible stage slamming force on rigid and elastic system impacting over the water surface, Nonlinear Dyn., 21, 193-220, (2000) · Zbl 0991.74027
[18]	Carcaterra, A.; Ciappi, E.; Iafrati, A.; Campana, E.F., Shock spectral analysis of elastic systems impacting on the water surface, J. Sound Vib., 229, 579-605, (2000)
[19]	Chaboche, J.L., Continuum damage mechanics: part I—general concepts, J. Appl. Mech. Trans. ASME, 55, 59-64, (1988)
[20]	Comi, C., A non-local model with tension and compression damage mechanics, Eur. J. Mech. A Solids, 20, 1-22, (2001) · Zbl 0982.74005
[21]	Contrafatto, L.; Cuomo, M., A globally convergent numerical algorithm for damaging elasto-plasticity based on the multiplier method, Int. J. Numer. Methods Eng., 63, 1089-1125, (2005) · Zbl 1155.74406
[22]	Contrafatto, L., Cuomo, M.: A numerical algorithm for the prediction of growth and propagation of interfaces. In: Õnate, E., Owen , D.R.J. (eds.) X International Conference on Computational Plasticity, COMPLAS, Barcelona, Spain, 2-4 September 2009 · Zbl 1155.74406
[23]	Contrafatto, L.; Cuomo, M.; Fazio, F., An enriched finite element for crack opening and rebar slip in reinforced concrete members, Int. J. Fract., 178, 33-50, (2012)
[24]	Contro, R.; Poggi, C.; Cazzani, A., Numerical analysis of fire effects on beam structures, Eng. Comput. (Swansea, Walles), 5, 53-58, (1988)
[25]	Cuomo, M., Nicolosi, A.: A poroplastic model for hygro-chemo-mechanical damage of concrete. In: Computational Modelling of Concrete Structures—Proceedings of EURO-C 2006, pp. 533-542 (2006)
[26]	Vree, J.H.P.; Brekelmans, W.A.M.; Gils, M.A.J., Comparison of nonlocal approaches in continuum damage mechanics, Comput. Struct., 55, 581-588, (1995) · Zbl 0919.73187
[27]	del Piero, G.: A variational approach to fracture and other inelastic phenomena. J. Elast. 112, 3-77 (2013) · Zbl 1394.74163
[28]	dell’Isola, F.; Guarascio, M.; Hutter, K., A variational approach for the deformation of a saturated porous solid. A second gradient theory extending terzaghi’s effective stress principle, Arch. Appl. Mech., 70, 323-337, (2000) · Zbl 0981.74016
[29]	dell’Isola, F.; Kosiński, W., Deduction of thermodynamic balance laws for bidimensional nonmaterial directed continua modelling interphase layers, Arch. Mech., 45, 333-359, (1993) · Zbl 0815.73007
[30]	dell’Isola, F.; Madeo, A.; Placidi, L., Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua, Z. Angew. Math. Mech., 92, 52-71, (2012) · Zbl 1247.74031
[31]	dell’Isola, F., Madeo, A., Seppecher, P.: Boundary conditions at fluid-permeable interfaces in porous media: a variational approach. Int. J. Solids Struct. 46(17), 3150-3164 (2009) · Zbl 1167.74393
[32]	dell’Isola, F.; Placidi, L.; dell’Isola, F. (ed.); Gavrilyuk, S. (ed.), Variational principles are a powerful tool also for formulating field theories, 52-71, (2011), NewYork
[33]	dell’Isola, F.; Romano, A., On a general balance law for continua with an interface, Ric. Math., 35, 325-337, (1986) · Zbl 0629.76004
[34]	dell’Isola, F.; Romano, A., On the derivation of thermomechanical balance equations for continuous systems with a nonmaterial interface, Int. J. Eng. Sci., 25, 1459-1468, (1987) · Zbl 0624.73001
[35]	dell’Isola, F.; Seppecher, P., Edge contact forces and quasi-balanced power, Meccanica, 32, 33-52, (1997) · Zbl 0877.73055
[36]	dell’Isola, F.; Seppecher, P., The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power, Comptes Rendus de l’Academie de Sciences Serie IIb: Mecanique, Physique, Chimie, Astronomie, 321, 303-308, (1995) · Zbl 0844.73006
[37]	dell’Isola, F.; Seppecher, P.; Madeo, A., How contact interactions may depend on the shape of Cauchy cuts in nth gradient continua: approach a la D’alambert, Zeitschrift fur Angewandte Mathematik und Physik, 63, 1119-1141, (2012) · Zbl 1330.76016
[38]	dell’Isola, F.; Vidoli, S., Continuum modelling of piezoelectromechanical truss beams: an application to vibration damping, Arch. Appl. Mech., 68, 1-19, (1998) · Zbl 0908.73067
[39]	Eremeyev, V.A., Acceleration waves in micropolar elastic media, Dokl. Phys., 50, 204-206, (2005)
[40]	Eremeyev, V.A.; Freidin, A.B.; Sharipova, L.L., Nonuniqueness and stability in problems of equilibrium of elastic two-phase bodies, Dokl. Phys., 48, 359-363, (2003)
[41]	Eremeyev, V.A.; Freidin, A.B.; Sharipova, L.L., The stability of the equilibrium of two-phase elastic solids, PMM J. Appl. Math. Mech., 71, 61-84, (2007)
[42]	Eremeyev, V.A.; Pietraszkiewicz, W., Phase transitions in thermoelastic and thermoviscoelastic shells, Arch. Mech., 61, 41-67, (2009) · Zbl 1273.74371
[43]	Eremeyev, V.A.; Pietraszkiewicz, W., The nonlinear theory of elastic shells with phase transitions, J. Elast., 74, 67-86, (2004) · Zbl 1058.74058
[44]	Eremeyev, V.A.; Pietraszkiewicz, W., Thermomechanics of shells undergoing phase transition, J. Mech. Phys. Solids, 59, 1395-1412, (2011) · Zbl 1270.74125
[45]	Ferretti, M., Madeo, A., dell’Isola, F., Boisse, P.: Modeling the onset of shear boundary layers, in fibrous composite reinforcements by second-gradient theory. Zeitschrift fur Angewandte Mathematik und Physik, 1-26 (2013) · Zbl 1176.74128
[46]	Forest, S., Micromorphic approach for gradient elasticity, viscoplasticity, and damage, J. Eng. Mech., 135, 117-131, (2009)
[47]	Fremond, M.; Nedjar, B., Damage, gradient of damage and principle of virtual power, Int. J. Solids Struct., 33, 1083-1103, (1996) · Zbl 0910.73051
[48]	Giorgio, I.; Culla, A.; Del Vescovo, D., Multimode vibration control using several piezoelectric transducers shunted with a multiterminal network, Arch. Appl. Mech., 79, 859-879, (2009) · Zbl 1176.74128
[49]	Iafrati, A.; Carcaterra, A.; Ciappi, A.; Campana, E.F., Hydroelastic analysis of a simple oscillator impacting the free surface, J. Ship Res., 44, 278-289, (2000)
[50]	Kachanov, L.M.: Rupture time under creep conditions. translated in Int. J. Fract. 97, xi-xviii 1999 (1958) · Zbl 1273.74371
[51]	Krajcinovic, D., Continuous damage mechanics revisited: basic concepts and definitions, J. Appl. Mech., 52, 829-834, (1985)
[52]	Krajcinovic, D.; Rinaldi, A., Statistical damage mechanics 1. theory, J. Appl. Mech., 72, 76-85, (2005) · Zbl 1111.74491
[53]	Lorentz, E.; Andrieux, S., A variational formulation for nonlocal damage models, Int. J. Plast., 15, 119-138, (1999) · Zbl 1024.74005
[54]	Luongo, A., Mode localization in dynamics and buckling of linear imperfect continuous structures, Nonlinear Dyn., 25, 133-156, (2001) · Zbl 1021.74017
[55]	Luongo, A.; Paolone, A., Multiple scale analysis for divergence-Hopf bifurcation of imperfect symmetric systems, J. Sound Vib., 218, 527-539, (1998) · Zbl 1235.34129
[56]	Madeo, A.; Djeran-Maigre, I.; Rosi, G.; Silvani, C., The effect of fluid streams in porous media on acoustic compression wave propagation, transmission, and reflection, Contin. Mech. Thermodyn., 25, 173-196, (2013) · Zbl 1343.76068
[57]	Madeo, A., dell’Isola, F., Darve, F.: A continuum model for deformable, second gradient porous media partially saturated with compressible fluids. J. Mech. Phys. Solids 61(11), 2196-2211 (2013) · Zbl 0844.73006
[58]	Madeo, A.; Gavrilyuk, S., Propagation of acoustic waves in porous media and their reflection and transmission at a pure-fluid/porous-medium permeable interface, Eur. J. Mech. A Solids, 29, 897-910, (2010)
[59]	Madeo, A.; George, D.; Lekszycki, T.; Nierenberger, M.; Remond, Y., A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodeling, Comptes Rendus Mecanique, 340, 575-589, (2012)
[60]	Madeo, A.; Lekszycki, T.; dell’Isola, F., A continuum model for the bio-mechanical interactions between living tissues and bio-resorbable graft after bone reconstructive surgery, Comptes Rendus Mecanique, 339, 625-640, (2011)
[61]	Maurini, C.; dell’Isola, F.; Pouget, J., On models of layered piezoelectric beams for passive vibration control, Journal de Physique IV France, 115, 307-316, (2004)
[62]	Maurini, C.; Pouget, J.; dell’Isola, F., Extension of the Euler-Bernoulli model of piezoelectric laminates to include 3D effects via a mixed approach, Comput. Struct., 84, 1438-1458, (2006)
[63]	Maurini, C.; Pouget, J.; dell’Isola, F., On a model of layered piezoelectric beams including transverse stress effect, Int. J. Solids Struct., 41, 4473-4502, (2004) · Zbl 1079.74569
[64]	Misra, A., Effect of asperity damage on friction behavior of single fracture, Eng. Fract. Mech., 69, 1997-2014, (2002)
[65]	Misra, A.; Yang, Y., Micromechanical model for cohesive materials based upon pseudo-granular structure, Int. J. Solids Struct., 47, 2970-2981, (2010) · Zbl 1196.74161
[66]	Nedoushan, R.J.; Farzin, M.; Mashayekhi, M., A micro-structural model for prediction of void initiation in superplastic forming, Int. J. Damage Mech., 5, 403, (1996)
[67]	Paas, M.H.J.W.; Oomens, C.W.J.; Schreurs, P.J.G.; Janssen, J.D., The mechanical behaviour of continuous media with stochastic damage, Eng. Fract. Mech., 36, 255-266, (1990)
[68]	Pasic, H., A unified approach of fracture and damage mechanics to fatigue damage problems, Int. J. Solids Struct., 29, 1957-1968, (1992) · Zbl 0775.90147
[69]	Pham, K.; Marigo, J.-J., Approche variationnelle de l’endommagement: I. LES concepts fondamentaux, C. R. Mécanique, 338, 191-198, (2010) · Zbl 1300.74046
[70]	Pham, K.; Marigo, J.-J., Approche variationnelle de l’endommagement : II. LES modèles à gradient, C. R. Mécanique, 338, 199-206, (2010) · Zbl 1300.74047
[71]	Pham, K.; Marigo, J.-J.; Maurini, C., The issue of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models, J. Mech. Phys. Solids, 59, 1163-1190, (2011) · Zbl 1270.74015
[72]	Pietraszkiewicz, W.; Eremeyev, V.A.; Konopińska, V., Extended non-linear relations of elastic shells undergoing phase transitions, ZAMM, 87, 150-159, (2007) · Zbl 1146.74032
[73]	Placidi, L.; dell’Isola, F.; Ianiro, N.; Sciarra, G., Variational formulation of pre-stressed solid fluid mixture theory, with an application to wave phenomena, Eur. J. Mech. A Solids, 27, 582-606, (2008) · Zbl 1146.74012
[74]	Placidi, L.; Rosi, G.; Giorgio, I.; Madeo, A., Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second gradient materials, Math. Mech. Solids, 92, 1-24, (2013)
[75]	Rinaldi, A., A rational model for 2D disordered lattices under uniaxial loading, Int. J. Damage Mech., 18, 233-257, (2009)
[76]	Rinaldi, A., Bottom-up modeling of damage in heterogeneous quasi-brittle solids, Contin. Mech. Thermodyn., 25, 359-373, (2013) · Zbl 1343.74045
[77]	Rinaldi, A., Statistical model with two order parameters for ductile and soft fiber bundles in nanoscience and biomaterials, Pys. Rev. E Stat. Nonlinear Soft Matter Phys., 83, 046126, (2011)
[78]	Rinaldi, A.; Krajcinovic, K.; Peralta, P.; Lai, Y.-C., Modeling polycrystalline microstructures with lattice models: a quantitative approach, Mech. Mater., 40, 17-36, (2008)
[79]	Rinaldi, A.; Lai, Y.-C., Statistical damage theory of 2d lattices: energetics and physical foundations of damage parameter, Int. J. Plast., 23, 1796-1825, (2007) · Zbl 1155.74402
[80]	Rinaldi, A.; Mastilovic, S.; Krajcinovic, D., Statistical damage mechanics—2 constitutive relations, J. Theor. Appl. Mech., 44, 585-602, (2006)
[81]	Rinaldi, A., Placidi, L.: A microscale second gradient approximation of the damage parameter of quasi-brittle heterogeneous lattices. Zeitschrift fuer Angewandte Mathematik und Mechanik, 1-16. (2013) doi:10.1002/zamm.201300028 · Zbl 1301.74042
[82]	Rosi, G.; Madeo, A.; Guyader, J.-L., Switch between fast and slow Biot compression waves induced by second gradient microstructure at material discontinuity surfaces in porous media, Int. J. Solids Struct., 50, 1721-1746, (2013)
[83]	Rosi, G.; Paccapeli, R.; Ollivier, F.; Pouget, J., Optimization of piezoelectric patch positioning for passive sound radiation control of plates, J. Vib. Control, 19, 658-673, (2013)
[84]	Roveri, N.; Carcaterra, A., Damage detection in structures under traveling loads by Hilbert-huang transform, Mech. Syst. Signal Process., 28, 128-144, (2012)
[85]	Silling, S.A.; Lehoucq, R.B., Convergence of peridynamics to classical elasticity theory, J. Elast., 93, 13-37, (2008) · Zbl 1159.74316
[86]	Silling, S.A.; Epton, M.; Weckner, O.; Xu, J.; Askari, E., Peridynamic states and constitutive modelling, J. Elast., 88, 151-184, (2007) · Zbl 1120.74003
[87]	Sun, X.; Wimmer, S.W.; Karrt, D.G., Shear band initiation of brittle damage materials, Int. J. Damage Mech., 5, 403, (1996)
[88]	Triantafyllidis, N.; Aifantis, E.C., A gradient approach to localization of deformation. I. hyperelastic materials, J. Elast., 16, 225-237, (1986) · Zbl 0594.73044
[89]	Yang, Y.; Misra, A., Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity, Int. J. Solids Struct., 49, 2500-2514, (2012)
[90]	Yang, Y.; Misra, A., Higher-order stress-strain theory for damage modeling implemented in an element-free Galerkin formulation, Comput. Model. Eng. Sci., 64, 1-36, (2010) · Zbl 1231.74023
[91]	Yang, Y.; Ching, W.Y.; Misra, A., Higher-order continuum theory applied to fracture simulation of nano-scale intergranular glassy film, J. Nanomech. Micromech., 1, 60-71, (2011)