zbMATH — the first resource for mathematics

Geometrically nonlinear and consistently linearized embedded strong discontinuity models for 3D problems with an application to the dissection analysis of soft biological tissues. (English) Zbl 1088.74541
The authors model the bulk material by a free-energy function suitable for describing anisotropic mechanical responses within the finite strain domain. To describe the relationship between the bulk material and the traction acting at the discontinuity, the authors propose a cohesive material model capturing transversely isotropic responses. They assume that all inelastic deformation takes place at the displacement discontinuity. Hence, the traction vector is given in terms of the gap displacement, the spatial normal vector and a scalar damage variable. In contrast to previous studies, where the parameters representing the discontinuity were treated as global variables, the authors follow the more established concept of static condensation in order to achieve a generalized displacement model. They present three types of finite element models with embedded discontinuities, which are based on so-called statically optimal symmetric, kinematically optimal symmetric and statically and kinematically optimal non-symmetric formulations, and use constant-strain tetrahedral elements.

74L15 Biomechanical solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
92C10 Biomechanics
Full Text: DOI
[1] Alfano, G.; Crisfield, M.A., Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues, Int. J. numer. methods engrg., 50, 1701-1736, (2001) · Zbl 1011.74066
[2] Armero, F.; Garikipati, K., Recent advances in the analysis and numerical simulation of strain localization in inelastic solids, (), 547-561
[3] Armero, F.; Garikipati, K., An analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids, Int. J. solids struct., 33, 2863-2885, (1996) · Zbl 0924.73084
[4] Barenblatt, G.I., The mathematical theory of equilibrium of cracks in brittle fracture, Adv. appl. mech., 7, 55-129, (1962)
[5] Bažant, Z.P.; Pijaudier-Cabot, G., Nonlocal continuum damage, localization instability and convergence, J. appl. mech., 55, 287-293, (1988) · Zbl 0663.73075
[6] Belytschko, T.; Fish, J.; Engelmann, A., A finite element with embedded localization zones, Comput. methods appl. mech. engrg., 70, 59-89, (1988) · Zbl 0653.73032
[7] G. Bolzon, A. Corigliano, An interface variables formulation for embedded-crack finite elements, in: D.R.J. Owen, E. Oñate, E. Hinton (Eds.), Computational plasticity: Fundamentals and Applications, Proceedings of the Fifth International Conference, International Center for Numerical Methods in Engineering (CIMNE), Barcelona, 1997, pp. 1617-1624
[8] Camacho, G.T.; Ortiz, M., Computational modelling of impact damage in brittle materials, Int. J. solids struct., 33, 2899-2938, (1996) · Zbl 0929.74101
[9] de Borst, R., Some recent issues in computational failure mechanics, Int. J. numer. methods engrg., 52, 63-95, (2001)
[10] de Borst, R.; Sluys, L.J.; Hüllhaus, H.B.; Pamin, J., Fundamental issues in finite element analysis of localization of deformation, Engrg. comput., 10, 99-121, (1993)
[11] Dugdale, D.S., Yielding of steel sheets containing slits, J. mech. phys. solids, 8, 100-104, (1960)
[12] Dvorkin, E.N.; Cuitiño, A.M.; Gioia, G., Finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions, Int. J. numer. methods engrg., 30, 541-564, (1990) · Zbl 0729.73209
[13] Fleisch, M.; Meier, B., Management and outcome of students in 1998: long-term outcome, Cardio. rev., 7, 215-218, (1999)
[14] Holzapfel, G.A., Nonlinear solid mechanics. A continuum approach for engineering, (2000), John Wiley & Sons Chichester · Zbl 0980.74001
[15] Holzapfel, G.A., Structural and numerical models for the (visco)elastic response of arterial walls with residual stresses, (), CISM Courses and Lectures No. 441, International Centre for Mechanical Sciences · Zbl 1151.74383
[16] Holzapfel, G.A.; Gasser, T.C.; Ogden, R.W., A new constitutive framework for arterial wall mechanics and a comparative study of material models, J. elastic., 61, 1-48, (2000) · Zbl 1023.74033
[17] G.A. Holzapfel, C.A.J. Schulze-Bauer, M. Stadler, Mechanics of angioplasty: wall, balloon and stent, in: J. Casey, G. Bao (Eds.), Mechanics in Biology, New York, 2000, The American Society of Mechanical Engineers (ASME), AMD-vol. 242/BED-vol. 46, pp. 141-156
[18] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (2000), Dover New York
[19] Jirásek, M., Comparative study on finite elements with embedded discontinuities, Comput. methods appl. mech. engrg., 188, 307-330, (2000) · Zbl 1166.74427
[20] Jirásek, M.; Zimmermann, T., Embedded crack model: part I. basic formulation, Int. J. numer. methods engrg., 50, 1269-1290, (2001) · Zbl 1013.74068
[21] Krajcinovic, D., Damage mechanics, (1996), North-Holland Amsterdam · Zbl 1111.74491
[22] Losordo, D.W.; Rosenfield, K.; Pieczek, A.; Baker, K.; Harding, M.; Isner, J.M., How does angioplasty work? serial analysis of human iliac arteries using intravascular ultrasound, Circulation, 86, 1845-1858, (1992)
[23] H.R. Lotfi, Finite element analysis of fracture in concrete and masonry structures, Ph.D. Thesis, University of Colorado, Boulder, 1992
[24] Lotfi, H.R.; Shing, P.B., Embedded representation of fracture in concrete with mixed finite elements, Int. J. numer. methods engrg., 38, 1307-1325, (1995) · Zbl 0824.73070
[25] de Borst, R., Fracture in quasi-brittle materials: a review of continuum damage-based models, Engrg. fract. mech., 69, 95-112, (2002)
[26] Melenk, J.M.; Babuška, I., The partition of unity finite element method: basic theory and applications, Comput. methods appl. mech. engrg., 139, 289-314, (1996) · Zbl 0881.65099
[27] Miehe, C., Aspects of the formulation and finite element implementation of large strain isotropic elasticity, Int. J. numer. methods engrg., 37, 1981-2004, (1994) · Zbl 0804.73067
[28] Moës, N.; Dolbow, J.; Belytschko, T., A finite element method for crack growth without remeshing, Int. J. numer. methods engrg., 46, 131-150, (1999) · Zbl 0955.74066
[29] Needleman, A., A continuum model for void nucleation by inclusion debonding, J. appl. mech., 54, 525-531, (1987) · Zbl 0626.73010
[30] Needleman, A., An analysis of decohesion along an imperfect interface, Int. J. fract., 42, 21-40, (1990)
[31] Oliver, J., Continuum modelling of strong discontinuities in solid mechanics, (), 455-479
[32] Oliver, J., Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. part 1: fundamentals, Int. J. numer. methods engrg., 39, 3575-3600, (1996) · Zbl 0888.73018
[33] Oliver, J., Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. part 2: numerical simulation, Int. J. numer. methods engrg., 39, 3601-3624, (1996)
[34] Oliver, J., On the discrete constitutive models induced by strong discontinuity kinematics and continuum constitutive equations, Int. J. solids struct., 37, 7207-7229, (2000) · Zbl 0994.74004
[35] Oliver, J.; Cervera, M.; Manzoli, O., Strong discontinuites and continuum plasticity models: the strong discontinuity approach, Int. J. plastic., 15, 319-351, (1999) · Zbl 1057.74512
[36] Oliver, J.; Huespe, A.E.; Pulido, M.D.G.; Chaves, E., From continuum mechanics to fracture mechanics: the strong discontinuity approach, Engrg. fract. mech., 69, 113-136, (2002)
[37] Ortiz, M.; Leroy, Y.; Needleman, A., A finite element method for localized failure analysis, Comput. methods appl. mech. engrg., 61, 189-214, (1987) · Zbl 0597.73105
[38] Ortiz, M.; Pandolfi, A., Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis, Int. J. numer. methods engrg., 44, 1267-1282, (1999) · Zbl 0932.74067
[39] Rice, J.R., Mathematical analysis in the mechanics of fracture, (), 191-311 · Zbl 0214.51802
[40] J.G. Rots, Computational modeling of concrete fracture, Ph.D. Thesis, Delft University of Technology, Netherlands, 1988
[41] Simo, J.C.; Armero, F., Geometrically non – linear enhanced strain mixed methods and the method of incompatible modes, Int. J. numer. methods engrg., 33, 1413-1449, (1992) · Zbl 0768.73082
[42] Simo, J.C.; Armero, F.; Taylor, R.L., Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems, Comput. methods appl. mech. engrg., 110, 359-386, (1993) · Zbl 0846.73068
[43] J.C. Simo, J. Oliver, A new approach to the analysis and simulation of strain softening in solids, in: Z.P. Bažant, Z. Bittnar, M. Jirásek, J. Mazars (Eds.), In Fracture and Damage in Quasibrittle Structures: Proceedings of the US-Europe Workshop on Fracture and Damage in Quasibrittle Structures, E & FN Spon, London, 1994, pp. 25-39
[44] Simo, J.C.; Oliver, J.; Amero, F., An analysis of strong discontinuities induced by strain softening in rate-independent inelastic solids, Int. J. numer. methods engrg., 12, 277-296, (1993) · Zbl 0783.73024
[45] Simo, J.C.; Rifai, M.S., A class of mixed assumed strain methods and the method of incompatible modes, Int. J. numer. methods engrg., 29, 1595-1638, (1990) · Zbl 0724.73222
[46] Sluys, L.J.; Berends, A.H., Discontinuous failure analysis for mode-I and mode-II localization problems, Int. J. solids struct., 35, 4257-4274, (1998) · Zbl 0933.74060
[47] Steinmann, P.; Larsson, R.; Runesson, K., On the localization properties of multiplicative hyperelasto-plastic continua with strong discontinuities, Int. J. solids struct., 34, 969-990, (1997) · Zbl 0947.74508
[48] Sukumar, N.; Srolovitz, D.J.; Baker, T.J.; Prévost, J.-H., Brittle fracture in polycrystalline microstructures with the extended finite element method, Int. J. numer. methods engrg., 56, 2015-2037, (2003) · Zbl 1038.74652
[49] R.L. Taylor, FEAP–A finite element analysis program–version 7.3, University of California at Berkeley, 2000
[50] Tvergaard, V.; Hutchinson, J.W., The relation between crack growth resistance and fracture process parameters in elastic – plastic solids, J. mech. phys. solids, 40, 1377-1397, (1992) · Zbl 0775.73218
[51] Washizu, K., Variational methods in elasticity and plasticity, (1982), Pergamon Press Oxford · Zbl 0164.26001
[52] G.N. Wells, Discontinuous modelling of strain localization and failure, Ph.D. Thesis, Delft University of Technology, Netherlands, 2001
[53] Wells, G.N.; de Borst, R.; Sluys, L.J., A consistent geometrically non-linear approach for delamination, Int. J. numer. methods engrg., 54, 1333-1355, (2002) · Zbl 1086.74043
[54] Wells, G.N.; Sluys, L.J., Three-dimensional embedded discontinuity model for brittle fracture, Int. J. solids struct., 38, 897-913, (2001) · Zbl 1004.74065
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.