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On the backgrounds of the theory of mHessian equations
A cohesive crack propagation model: Mathematical theory and numerical solution
1.  Applied Mathematics II, Martensstr. 3, D91054 Erlangen, Germany, Germany 
2.  Chair of Applied Mechanics, Egerlandstr. 5, D91058 Erlangen, Germany 
3.  Applied Mathematics II, Martensstr. 3, D91058 Erlangen, Germany 
References:
[1] 
A. A. Griffith, The phenomena of rupture and flow in solids, Philos Trans R Soc Lond A, 221 (1921), 16398. Google Scholar 
[2] 
G. R. Irwin, Fracture, in "Encyclopedia of Physics: Elasticity and Plasticity" (S. Fluegge ed.), SpringerVerlag Berlin, (1958), 55190. Google Scholar 
[3] 
G. I. Barenblatt, The mathematical theory of equilibrium cracks in brittle fracture, Advan. Appl. Mech., 7 (1962), 55129. Google Scholar 
[4] 
H. Stumpf and K. Ch. Le, Variational principles of nonlinear fracture mechanics, Acta Mech, 83 (1990), 2537. Google Scholar 
[5] 
G. A. Maugin and C. Trimarco, Pseudomomentum and material forces in nonlinear elasticity: variational formulations and application to brittle fracture, Acta Mech, 94 (1992), 128. Google Scholar 
[6] 
J. D. Eshelby, The continuum theory of lattice defects, in "Progress in Solids State Physics" (F. Seitz and D. Turnbull eds.), New York: Academic Press, Volume 3, 1956. Google Scholar 
[7] 
J. R. Rice, A path independent integral and the approximate analysis of strain concentraction by notches and cracks, J. Appl. Mech., 35 (1968), 37986. Google Scholar 
[8] 
M. Buliga, Energy minimizing brittle crack propagation, J Elast, 52 (1999), 201238. Google Scholar 
[9] 
N. Kikuchi and J. T. Oden, "The Variational Approach to Fracture," SIAM, 1988. Google Scholar 
[10] 
A. M. Khludnev and V. A. Kovtunenko, "Analysis of Cracks in Solids," WIT Press, 1999. Google Scholar 
[11] 
B. Bourdin, G. A. Francfort and J.J. Marigo, "Contact Problems in Elasticity," Springer, 2008. Google Scholar 
[12] 
S. A. Nazarov and M. SpecoviusNeugebauer, Use of the energy criterion of fracture to determine the shape of a slightly curved crack, Journal of Applied Mechanics and Technical Physics, 47 (2006), 714723. Google Scholar 
[13] 
J. R. Rice and E. P. Sorensen, Continuing cracktip deformation and fracture for planestrain crack growth in elasticplastic solids, J Mech Phys Solids, 26 (1978), 16386. Google Scholar 
[14] 
M. Fleming, Y. A. Chu, B. Moran and T. Belytschko, Enriched elementfree Galerkin methods for crack tip fields, Int J Numer Methods Eng, 40 (1997), 14831504. Google Scholar 
[15] 
T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing, Int J Numer Methods Eng, 45 (1999), 601620. Google Scholar 
[16] 
D. R. Curran, L. Seaman, T. Cooper and D. A. Shockey, Micromechanical model for comminution and granular flow of brittle material under high strain rate application to penetration of ceramic targets, Int J Impact Eng, 13 (1993), 5383. Google Scholar 
[17] 
A. Needleman, A continuum model for void nucleation by inclusion debonding, J. Appl. Mech., 54 (1987), 525531. Google Scholar 
[18] 
D. S. Dugdale, Yielding of steel sheets containing slits, J Mech Phys Solids, 8 (1960), 100104. Google Scholar 
[19] 
V. Tvergaard and J. W. Hutchinson, The influence of plasticity on mixed mode interface toughness, J Mech Phys Solids, 41 (1993), 11191135. Google Scholar 
[20] 
G. T. Camacho and M. Ortiz, Computational modelling of impact damage in brittle materials, Int J Solids Struct, 33 (1996), 28992938. Google Scholar 
[21] 
X.P. Xu and A. Needleman, Numerical simulations of fast crack growth in brittle solids, Mech Phys Solids, 42 (1994), 13971434. Google Scholar 
[22] 
M. Ortiz and A. Pandolfi, Finitedeformation irreversible cohesive elements for threedimensional crackpropagation analysis, Int J Numer Methods Eng, 44 (1999), 12671282. Google Scholar 
[23] 
M. E. Walter, G. Ravichandran and M. Ortiz, Computational modeling of damage evolution in unidirectional fiber reinforced ceramic matrix composites, Computational Mechanics, 20 (1997), 192198. Google Scholar 
[24] 
J. Mergheim, E. Kuhl and P. Steinmann, A hybrid discontinuous Galerkin/interface method for the computational modelling of failure, Commun Numer Methods Eng, 20 (2004), 511519. Google Scholar 
[25] 
V. A. Kovtunenko, Nonconvex problem for crack with nonpenetration, ZAMM Z. Angew. Math. Mech., 85 (2005), 242251. Google Scholar 
[26] 
D. Hull, An introduction to composite materials, in "Cambridge Solid State Science Series" (R. W. Cahn, M. W. Thompson and I. M. Ward eds.), Cambridge University Press, 1981, 1246. Google Scholar 
[27] 
J. C. J. Schellekens and R. de Borst, On the numerical integration of interface elements, Int J Numer Methods Engng, 36 (1993), 4366. Google Scholar 
[28] 
F. Zhou, J.F. Molinari and T. Shioya, A ratedependent cohesive model for simulating dynamic crack propagation in brittle materials, Eng Fract Mech, 72 (2005), 13831410. Google Scholar 
[29] 
G. Geissler and M. Kaliske, Timedependent cohesive zone modelling for discrete fracture simulation, Eng Fract Mech, 77 (2010), 153169. Google Scholar 
[30] 
R. De Borst, L. J. Sluys, H.B. Mühlhaus and J. Pamin, Fundamental issues in finite element analyses of localization of deformation, Engineering Computations, 10 (1993), 99121. Google Scholar 
[31] 
M. Prechtel, P. Leiva Ronda, R. Janisch, A. Hartmaier, G. Leugering, P. Steinmann and M. Stingl, Simulation of fracture in heterogeneous elastic materials with cohesive zone models, Int J Fract, 168 (2011), 1529. Google Scholar 
[32] 
H. Amor, J.J. Marigo and C. Maurini, Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments, J. Mech. Phys. Solids, 57 (2009), 12091229. Google Scholar 
[33] 
M. Prechtel, G. Leugering, P. Steinmann and M. Stingl, Towards optimization of crack resistance of composite materials by adjustment of fiber shapes Reference, Eng Fract Mech, 78 (2011), 944960. Google Scholar 
[34] 
N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods," SIAM Studies in Applied Mathematics, SIAM Philadelphia, 1988. Google Scholar 
[35] 
M. Burger, "Infinitedimensional Optimization and Optimal Design,", 2003., (). Google Scholar 
[36] 
A. W鋍hter and L. T. Biegler, On the implementation of a primaldual interior point filter line search algorithm for largescale nonlinear programming, Mathematical Programming, 106 (2006), 2557. Google Scholar 
[37] 
M. Hintermüller, V. A. Kovtunenko and K. Kunisch, Obstacle problems with cohesion: a hemivariational inequality approach and its efficient numerical solution, SIAM J Optim, 21 (2011), 491516. Google Scholar 
[38] 
V. A. Kovtunenko, A hemivariational inequality in crack problems,, Optimization, (). Google Scholar 
[39] 
N. Chandra, H. Li, C. Shet and H. Ghonem, Some issues in the application of cohesive zone models for metalceramic interfaces, Int J Solid Struct, 39 (2002), 28272855. Google Scholar 
[40] 
A. Banerjea and J. R. Smith, Origins of the universal bindingenergy relation, Phys Rev B, 37 (1988), 66326645. Google Scholar 
show all references
References:
[1] 
A. A. Griffith, The phenomena of rupture and flow in solids, Philos Trans R Soc Lond A, 221 (1921), 16398. Google Scholar 
[2] 
G. R. Irwin, Fracture, in "Encyclopedia of Physics: Elasticity and Plasticity" (S. Fluegge ed.), SpringerVerlag Berlin, (1958), 55190. Google Scholar 
[3] 
G. I. Barenblatt, The mathematical theory of equilibrium cracks in brittle fracture, Advan. Appl. Mech., 7 (1962), 55129. Google Scholar 
[4] 
H. Stumpf and K. Ch. Le, Variational principles of nonlinear fracture mechanics, Acta Mech, 83 (1990), 2537. Google Scholar 
[5] 
G. A. Maugin and C. Trimarco, Pseudomomentum and material forces in nonlinear elasticity: variational formulations and application to brittle fracture, Acta Mech, 94 (1992), 128. Google Scholar 
[6] 
J. D. Eshelby, The continuum theory of lattice defects, in "Progress in Solids State Physics" (F. Seitz and D. Turnbull eds.), New York: Academic Press, Volume 3, 1956. Google Scholar 
[7] 
J. R. Rice, A path independent integral and the approximate analysis of strain concentraction by notches and cracks, J. Appl. Mech., 35 (1968), 37986. Google Scholar 
[8] 
M. Buliga, Energy minimizing brittle crack propagation, J Elast, 52 (1999), 201238. Google Scholar 
[9] 
N. Kikuchi and J. T. Oden, "The Variational Approach to Fracture," SIAM, 1988. Google Scholar 
[10] 
A. M. Khludnev and V. A. Kovtunenko, "Analysis of Cracks in Solids," WIT Press, 1999. Google Scholar 
[11] 
B. Bourdin, G. A. Francfort and J.J. Marigo, "Contact Problems in Elasticity," Springer, 2008. Google Scholar 
[12] 
S. A. Nazarov and M. SpecoviusNeugebauer, Use of the energy criterion of fracture to determine the shape of a slightly curved crack, Journal of Applied Mechanics and Technical Physics, 47 (2006), 714723. Google Scholar 
[13] 
J. R. Rice and E. P. Sorensen, Continuing cracktip deformation and fracture for planestrain crack growth in elasticplastic solids, J Mech Phys Solids, 26 (1978), 16386. Google Scholar 
[14] 
M. Fleming, Y. A. Chu, B. Moran and T. Belytschko, Enriched elementfree Galerkin methods for crack tip fields, Int J Numer Methods Eng, 40 (1997), 14831504. Google Scholar 
[15] 
T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing, Int J Numer Methods Eng, 45 (1999), 601620. Google Scholar 
[16] 
D. R. Curran, L. Seaman, T. Cooper and D. A. Shockey, Micromechanical model for comminution and granular flow of brittle material under high strain rate application to penetration of ceramic targets, Int J Impact Eng, 13 (1993), 5383. Google Scholar 
[17] 
A. Needleman, A continuum model for void nucleation by inclusion debonding, J. Appl. Mech., 54 (1987), 525531. Google Scholar 
[18] 
D. S. Dugdale, Yielding of steel sheets containing slits, J Mech Phys Solids, 8 (1960), 100104. Google Scholar 
[19] 
V. Tvergaard and J. W. Hutchinson, The influence of plasticity on mixed mode interface toughness, J Mech Phys Solids, 41 (1993), 11191135. Google Scholar 
[20] 
G. T. Camacho and M. Ortiz, Computational modelling of impact damage in brittle materials, Int J Solids Struct, 33 (1996), 28992938. Google Scholar 
[21] 
X.P. Xu and A. Needleman, Numerical simulations of fast crack growth in brittle solids, Mech Phys Solids, 42 (1994), 13971434. Google Scholar 
[22] 
M. Ortiz and A. Pandolfi, Finitedeformation irreversible cohesive elements for threedimensional crackpropagation analysis, Int J Numer Methods Eng, 44 (1999), 12671282. Google Scholar 
[23] 
M. E. Walter, G. Ravichandran and M. Ortiz, Computational modeling of damage evolution in unidirectional fiber reinforced ceramic matrix composites, Computational Mechanics, 20 (1997), 192198. Google Scholar 
[24] 
J. Mergheim, E. Kuhl and P. Steinmann, A hybrid discontinuous Galerkin/interface method for the computational modelling of failure, Commun Numer Methods Eng, 20 (2004), 511519. Google Scholar 
[25] 
V. A. Kovtunenko, Nonconvex problem for crack with nonpenetration, ZAMM Z. Angew. Math. Mech., 85 (2005), 242251. Google Scholar 
[26] 
D. Hull, An introduction to composite materials, in "Cambridge Solid State Science Series" (R. W. Cahn, M. W. Thompson and I. M. Ward eds.), Cambridge University Press, 1981, 1246. Google Scholar 
[27] 
J. C. J. Schellekens and R. de Borst, On the numerical integration of interface elements, Int J Numer Methods Engng, 36 (1993), 4366. Google Scholar 
[28] 
F. Zhou, J.F. Molinari and T. Shioya, A ratedependent cohesive model for simulating dynamic crack propagation in brittle materials, Eng Fract Mech, 72 (2005), 13831410. Google Scholar 
[29] 
G. Geissler and M. Kaliske, Timedependent cohesive zone modelling for discrete fracture simulation, Eng Fract Mech, 77 (2010), 153169. Google Scholar 
[30] 
R. De Borst, L. J. Sluys, H.B. Mühlhaus and J. Pamin, Fundamental issues in finite element analyses of localization of deformation, Engineering Computations, 10 (1993), 99121. Google Scholar 
[31] 
M. Prechtel, P. Leiva Ronda, R. Janisch, A. Hartmaier, G. Leugering, P. Steinmann and M. Stingl, Simulation of fracture in heterogeneous elastic materials with cohesive zone models, Int J Fract, 168 (2011), 1529. Google Scholar 
[32] 
H. Amor, J.J. Marigo and C. Maurini, Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments, J. Mech. Phys. Solids, 57 (2009), 12091229. Google Scholar 
[33] 
M. Prechtel, G. Leugering, P. Steinmann and M. Stingl, Towards optimization of crack resistance of composite materials by adjustment of fiber shapes Reference, Eng Fract Mech, 78 (2011), 944960. Google Scholar 
[34] 
N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods," SIAM Studies in Applied Mathematics, SIAM Philadelphia, 1988. Google Scholar 
[35] 
M. Burger, "Infinitedimensional Optimization and Optimal Design,", 2003., (). Google Scholar 
[36] 
A. W鋍hter and L. T. Biegler, On the implementation of a primaldual interior point filter line search algorithm for largescale nonlinear programming, Mathematical Programming, 106 (2006), 2557. Google Scholar 
[37] 
M. Hintermüller, V. A. Kovtunenko and K. Kunisch, Obstacle problems with cohesion: a hemivariational inequality approach and its efficient numerical solution, SIAM J Optim, 21 (2011), 491516. Google Scholar 
[38] 
V. A. Kovtunenko, A hemivariational inequality in crack problems,, Optimization, (). Google Scholar 
[39] 
N. Chandra, H. Li, C. Shet and H. Ghonem, Some issues in the application of cohesive zone models for metalceramic interfaces, Int J Solid Struct, 39 (2002), 28272855. Google Scholar 
[40] 
A. Banerjea and J. R. Smith, Origins of the universal bindingenergy relation, Phys Rev B, 37 (1988), 66326645. Google Scholar 
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