×

zbMATH — the first resource for mathematics

A variational model based on isogeometric interpolation for the analysis of cracked bodies. (English) Zbl 1423.74055
Summary: A variational model for the analysis of crack evolution is presented. The method considers strong discontinuities that evolve according to the principles of cohesive fracture mechanics. A novel isogeometric interpolation scheme is presented that, differently from previous proposals, inserts the fracture modifying the blending properties of the interpolation. A method for tracking the discontinuity is also proposed, based on a local distortion of the parametrization of the geometry obtained determining the position of the control points of the isogeometric interpolation as solution of a suitable minimization problem.

MSC:
74A45 Theories of fracture and damage
74S05 Finite element methods applied to problems in solid mechanics
74R10 Brittle fracture
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alfaiate, J.; Pires, E. B.; Martins, J., A finite element analysis of non-prescribed crack propagation in concrete, Computers & Structures, 63, 1, 17-26, (1997) · Zbl 0899.73512
[2] Alfaiate, J.; Simone, A.; Sluys, L., Non-homogeneous displacement jumps in strong embedded discontinuities, International Journal of Solids and Structures, 40, 5799-5817, (2003) · Zbl 1059.74548
[3] Ambrosio, L.; Tortorelli, V., Approximation of functionals depending on jumps by elliptic functionals via γ-convergence, Communications on Pure and Applied Mathematics, 43, 8, 999-1036, (1990) · Zbl 0722.49020
[4] Amor, H.; Marigo, J.-J.; Maurini, C., Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments, Journal of the Mechanics and Physics of Solids, 57, 8, 1209-1229, (2009) · Zbl 1426.74257
[5] Andreaus, U.; Baragatti, P., Fatigue crack growth, free vibrations, and breathing crack detection of aluminium alloy and steel beams, Journal of Strain Analysis for Engineering Design, 44, 595-608, (2009)
[6] Andreaus, U.; Dell’isola, F.; Porfiri, M., Piezoelectric passive distributed controllers for beam flexural vibrations, Journal of Vibration and Control, 10, 5, 625-659, (2004) · Zbl 1078.74026
[7] Andreaus, U.; Giorgio, I.; Lekszycki, T., A 2-D continuum model of a mixture of bone tissue and bio-resorbable material for simulating mass density redistribution under load slowly variable in time, ZAMM, (2013) · Zbl 1303.74029
[8] Auffray, N., dell’Isola, F., Eremeyev, V., Madeo, A., & Rosi, G. (2013). Analytical continuum mechanics ã la Hamilton-Piola: Least action principle for second gradient continua and capillary fluids. Mathematics and Mechanics of Solids, http://dx.doi.org/10.1177/1081286513497616. · Zbl 1327.76008
[9] Belytschko, T.; Moës, N.; Usui, S.; Parimi, C., Arbitrary discontinuities in finite elements, International Journal for Numerical Methods in Engineering, 50, 993-1013, (2001) · Zbl 0981.74062
[10] Bourdin, B.; Francfort, G. A.; Marigo, J. J., Numerical experiments in revisited brittle fracture, Journal of the Mechanics and Physics of Solids, 48, 797-826, (2000) · Zbl 0995.74057
[11] Bourdin, B.; Francfort, G. A.; Marigo, J. J.J., The variational approach to fracture, Journal of Elasticity, 91, 5-148, (2008) · Zbl 1176.74018
[12] Buffa, A.; Cho, D.; Sangalli, G., Linear independence of the T-spline blending functions associated with some particular T-meshes, Computer Methods in Applied Mechanics and Engineering, 199, 23-24, 1437-1445, (2010) · Zbl 1231.65027
[13] Chambolle, A.; Francfort, G.; Marigo, J.-J., When and how do cracks propagate ?, Journal of the Mechanics and Physics of Solids, 57, 1614-1622, (2009) · Zbl 1371.74016
[14] Ciancio, D.; Carol, I.; Cuomo, M., Crack opening conditions at corner nodes in FE analysis with cracking along mesh lines, Engineering Fracture Mechanics, 74, 1963-1982, (2007)
[15] Contrafatto, L.; Cuomo, M., A new thermodynamically consistent continuum model for hardening plasticity coupled with damage, International Journal of Solids and Structures, 39, 6241-6271, (2002) · Zbl 1032.74509
[16] Cottrell, J.; Hughes, T.; Bazilevs, Y., Isogeometric analysis: toward integration of CAD and FEA, (2009), Wiley · Zbl 1378.65009
[17] Dal Maso, G.; Francfort, G. A.; Toader, R., Quasistatic crack growth in nonlinear elasticity, Archive for Rational Mechanics and Analysis, 176, 165-225, (2005) · Zbl 1064.74150
[18] dell’Isola, F.; Gouin, H.; Seppecher, P., Radius and surface tension of microscopic bubbles by second gradient theory, Comptes rendus de l’Académie des Sciences Série IIb, 320, 211-216, (1995) · Zbl 0833.76004
[19] dell’Isola, F.; Guarascio, M.; Hutter, K., Variational approach for the deformation of a saturated porous solid. A second-gradient theory extending terzaghi’s effective stress principle, Archive of Applied Mechanics, 75, 323-337, (2000) · Zbl 0981.74016
[20] dell’Isola, F.; Madeo, A.; Placidi, L., Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua, Zeitschrift fur Angewandte Mathematik und Mechanik (ZAMM), 92, 1, 52-71, (2012) · Zbl 1247.74031
[21] dell’Isola, F.; Romano, A., A phenomenological approach to phase transition in classical field theory, International Journal of Engineering Science, 25, 11-12, 1469-1475, (1987) · Zbl 0629.73006
[22] dell’Isola, F.; Seppecher, P., The relationship between edge contact forces, double force and interstitial working allowed by the principle of virtual power, Comptes rendus de l’Académie des Sciences Serie IIb, 321, 303-308, (1995) · Zbl 0844.73006
[23] dell’Isola, F.; Seppecher, P.; Madeo, A., How contact interactions may depend on the shape of Cauchy cuts in N-th gradient continua: approach à la D’alembert, Zeitschrift fur Angewandte Mathematik und Physik (ZAMP), 63, 1119-1141, (2012) · Zbl 1330.76016
[24] Dias-da Costa, D.; Alfaiate, J.; Sluys, L.; Júlio, E., Towards a generalization of a discrete strong discontinuity approach, Computer Methods in Applied Mechanics and Engineering, 198, 3670-3681, (2009) · Zbl 1230.74177
[25] Dias-da Costa, D.; Alfaiate, J.; Sluys, L.; Júlio, E., A comparative study on the modeling of discontinuous fracture by means of enriched nodal and element techniques and interface elements, International Journal of Fracture, 161, 97-119, (2010) · Zbl 1273.74531
[26] Eremeyev, V.; Freidin, A.; Sharipova, L., Nonuniqueness and stability in problems of equilibrium of elastic two-phase bodies, Doklady Physics, 48, 7, 359-363, (2003)
[27] Eremeyev, V.; Pietraszkiewicz, W., The nonlinear theory of elastic shells with phase transitions, Journal of Elasticity, 74, 1, 67-86, (2004) · Zbl 1058.74058
[28] Eremeyev, V.; Pietraszkiewicz, W., Phase transitions in thermoelastic and thermoviscoelastic shells, Archive of Mechanics, 61, 1, 41-67, (2009) · Zbl 1273.74371
[29] Eremeyev, V.; Pietraszkiewicz, W., Thermomechanics of shells undergoing phase transition, Journal of Mechanics and Physics, 59, 7, 1395-1412, (2011) · Zbl 1270.74125
[30] Forest, S., Micromorphic approach for gradient elasticity, viscoplasticity, and damage, Journal of Engineering Mechanics, 135, 3, 117-131, (2009)
[31] Forest, S.; Sievert, R., International Journal of Solids and Structures, 43, 7224-7245, (2006)
[32] Francfort, G.; Larsen, C., Existence and convergence for quasi-static evolution in brittle fracture, Communications on Pure and Applied Mathematics, 56, 1465-1500, (2003) · Zbl 1068.74056
[33] Francfort, G. A.; Marigo, J. J.J., Revisiting brittle fracture as an energy minimization problem, Journal of the Mechanics and Physics of Solids, 46, 8, 1319-1342, (1998) · Zbl 0966.74060
[34] Griffith, A., The phenomena of rupture and flow in solids, Philosophical Transactions of the Royal Society of London, CCXXI-A, 163-198, (1920)
[35] Hakim, V.; Karma, A., Laws of crack motion and phase-field models of fracture, Journal of the Mechanics and Physics of Solids, 57, 342-368, (2009) · Zbl 1421.74089
[36] Hughes, T.; Cottrell, J.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 194, 4135-4195, (2005) · Zbl 1151.74419
[37] Kiendl, J.; Bletzinger, K.-U.; Linhard, J.; Wuchner, R., Isogeometric shell analysis with kirchoff-love elements, Computer Methods in Applied Mechanics and Engineering, 189, 3902-3914, (2009) · Zbl 1231.74422
[38] Le, K. C., Stumpf, H., & Weichert, D. (1989). Variational priciples of fracture mechanics. Technical Report 64, Institut fur mechanic - Ruhr Universitat Bochum. · Zbl 0693.73066
[39] Lekszycki, T.; dell’Isola, F., A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable material, Zeitschrift fur Angewandte Mathematik und Mechanik ZAMM, 92, 6, 426-444, (2012) · Zbl 1241.92010
[40] Linder, C.; Armero, F., Finite elements with embedded strong discontinuities for the modeling of failure in solids, International Journal for Numerical Methods in Engineering, 72, 1391-1433, (2007) · Zbl 1194.74431
[41] Luycker, E. D.; Benson, D. J.; Belytschko, T.; Bazilevs, Y.; Hsu, M. C., X-FEM in isogeometric analysis for linear fracture mechanics, International Journal for Numerical Methods in Engineering, 87, 541-565, (2011) · Zbl 1242.74105
[42] Madeo, A.; George, D.; Lekszycki, T.; Nierenberger, M.; Rémond, Y., A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodelling, Comptes Rendus Mécanique, 340, 8, 575-589, (2012)
[43] Madeo, A.; Lekszycki, T.; dell’Isola, F., A continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery, Comptes Rendus - Mecanique, 339, 10, 625-640, (2011)
[44] Maurini, C.; Pouget, J.; dell’Isola, F., On a model of layered piezoelectric beams including transverse stress effect, International Journal of Solids and Structures, 41, 16-17, 4473-4502, (2004) · Zbl 1079.74569
[45] Melenk, J.; Babus̆ka, I., The partition of unity finite element method: basic theory and applications, Computer Methods in Applied Mechanics and Engineering, 139, 289-314, (1996) · Zbl 0881.65099
[46] Miehe, C.; Hofacker, M.; Welschinger, F., A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits, Computer Methods in Applied Mechanics and Engineering, 199, 2765-2778, (2010) · Zbl 1231.74022
[47] Miehe, C.; Welschinger, F.; Hofacker, M., Thermodynamically consistent phase-field models of fracture: variational principles and multifield FE implementations, International Journal for Numerical Methods in Engineering, 83, 1273-1311, (2010) · Zbl 1202.74014
[48] Moës, N.; Dolbow, I.; Belytschko, T., A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering, 46, 131-150, (1999) · Zbl 0955.74066
[49] Moës, N.; Stolz, C.; Bernard, P.-E.; Chevaugeon, N., A level set based model for damage growth : the thick level set approach, International Journal For Numerical Methods in Engineering, 86, 358-380, (2011) · Zbl 1235.74302
[50] Neff, P., Finite-strain elastic-plastic Cosserat theory for polycrystals with grain rotations, International Journal of Engineering Science, 44, 8-9, 574-594, (2006) · Zbl 1213.74032
[51] Neff, P., A geometrically exact planar Cosserat shell-model with microstructure: existence of minimizers for zero Cosserat couple modulus, Mathematical Models and Methods in Applied Sciences, 17, 3, 363-392, (2007) · Zbl 1119.74029
[52] Negri, M.; Ortner, C., Quasi-static crack propagation by griffith’s criterion, Mathematical Models and Methods in Applied Sciences, 18, 1895-1925, (2008) · Zbl 1155.74035
[53] Noels, L.; Radovitzky, R., A new discontinuous Galerkin method for Kirchhoff-love shells, Computer Methods in Applied Mechanics and Engineering, 197, 2901-2929, (2008) · Zbl 1194.74456
[54] Oliver, J., Modelling of strong discontinuities in solid mechanics via strain softening constitutive equations. part 1: fundamentals. part 2: numerical simulation, International Journal for Numerical Methods in Engineering, 39, 21, 3575-3623, (1996) · Zbl 0888.73018
[55] Oliver, J.; Cervera, M.; Manzoli, O., Strong discontinuities and continuum plasticity models: the strong discontinuity approach, International Journal of Plasticity, 15, 319-351, (1999) · Zbl 1057.74512
[56] Oliver, J.; Huespe, A., Continuum approach to material failure in strong discontinuity, Computer Methods in Applied Mechanics and Engineering, 193, (2003) · Zbl 1060.74507
[57] Oliver, J.; Huespe, A., Theoretical and computational issues in modelling material failure in strong discontinuity scenarios, Computer Methods in Applied Mechanics and Engineering, 193, (2004) · Zbl 1067.74505
[58] Oliver, J.; Huespe, A.; Dias, I., Strain localization strong discontinuities and material fracture: matches and mismatches, Computer Methods in Applied Mechanics and Engineering, 241-244, 323-336, (2012) · Zbl 1353.74009
[59] Ortiz, M.; Pandolfi, A., A class of cohesive elements for the simulation of three-dimensional crack propagation, International Journal for Numerical Methods in Engineering, 44, 9, 1267-1282, (1999) · Zbl 0932.74067
[60] Patzák, B.; Jirásek, M., Adaptive resolution of localized damage in quasi-brittle materials, Journal of Engineering Mechanics, 130, 6, 720-732, (2004)
[61] Del Piero, G., A variational approach to fracture and other inelastic phenomena, Journal of Elasticity, 112, 1, 3-77, (2013) · Zbl 1394.74163
[62] Pietraszkiewicz, W.; Eremeyev, V.; Konopinska, V. Z., Extended nonlinear relations of elastic shells undergoing phase transitions, Zeitschrift fur Angewandte Mathematik und Mechanik ZAMM, 87, 2, 150-159, (2007) · Zbl 1146.74032
[63] 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, Mathematics and Mechanics of Solids, (2013) · Zbl 1305.74047
[64] Reali, A., An isogeometric analysis approach for the study of structural vibrations, Journal of Earthquake Engineering, 10, 1-30, (2006)
[65] Rinaldi, A., A rational model for 2D disordered lattices under uniaxial loading, International Journal of Damage Mechanics, 18, 233-257, (2009)
[66] Rinaldi, A., Statistical model with two order parameters for ductile and soft fiber bundles in nanoscience and biomaterials, Physical Review E, 83, 4, 046126, (2011)
[67] Rinaldi, A., Bottom-up modeling of damage in heterogeneous quasibrittle solids, Continuum Mechanics and Thermodynamics, 25, 2-4, 359-373, (2013) · Zbl 1343.74045
[68] Rinaldi, A.; Krajcinovic, D.; Mastilovic, S., Statistical damage mechanics and extreme value theory, International Journal of Damage Mechanics, 16, 1, 57-76, (2007)
[69] Rinaldi, A.; Krajcinovic, D.; Peralta, P.; Lai, Y.-C., Lattice models of polycrystalline microstructures: A quantitative approach, Mechanics of Materials, 40, 17-36, (2008)
[70] Rinaldi, A.; Lai, Y., Statistical damage theory of 2D lattices: energetics and physical foundations of damage parameter, International Journal of Plasticity, 23, 1796-1825, (2007) · Zbl 1155.74402
[71] Rosi, G.; Giorgio, I.; Eremeyev, V., Propagation of linear compression waves through plane interfacial layers and mass adsorption in second gradient fluids pubblicazione, Zeitschrift fur Angewandte Mathematik und Mechanik (ZAMM), (2013)
[72] Sciarra, G.; dell’Isola, F.; Coussy, O., Second gradient poromechanics, International Journal of Solids and Structures, 44, 20, 6607-6629, (2007) · Zbl 1166.74341
[73] Stumpf, H.; Le, K. C., Acta Mechanica, 83, 1-2, 25-37, (1990)
[74] Ventura, G.; Xu, J.; Belytschko, T., A vector level set method and new discontinuity approximations for crack growth by EFG, International Journal for Numerical Methods in Engineering, 54, 923-944, (2002) · Zbl 1034.74053
[75] Verhoosel, C. V.; Scott, M. A.; de Borst, R.; Hughes, T. J.R., An isogeometric approach to cohesive zone modeling, International Journal for Numerical Methods in Engineering, 87, 336-360, (2011) · Zbl 1242.74169
[76] Vidoli, S.; Dell’isola, F., Vibration control in plates by uniformly distributed PZT actuators interconnected via electric networks, European Journal of Mechanics A/Solids, 20, 3, 435-456, (2001) · Zbl 0988.74047
[77] Yang, Y.; Ching, W.; Misra, A., Higher-order continuum theory applied to fracture simulation of nano-scale intergranular glassy film, Journal of Nanomechanics and Micromechanics, 1, 60-71, (2011)
[78] Yang, Y.; Misra, A., Higher-order stress-strain theory for damage modeling implemented in an element-free Galerkin formulation, Computer Modeling in Engineering and Sciences, 64, 1-36, (2010) · Zbl 1231.74023
[79] Zi, G.; Belytschko, T., New crack tip elements for XFEM and application to cohesive crack, International Journal for Numerical Methods in Engineering, 57, 2221-2240, (2003) · Zbl 1062.74633
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.