
Previous Article
Estimates of solutions for the parabolic $p$Laplacian equation with measure via parabolic nonlinear potentials
 CPAA Home
 This Issue

Next Article
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), 163. Google Scholar 
[2] 
G. R. Irwin, Fracture,, in, (1958), 551. Google Scholar 
[3] 
G. I. Barenblatt, The mathematical theory of equilibrium cracks in brittle fracture,, Advan. Appl. Mech., 7 (1962), 55. Google Scholar 
[4] 
H. Stumpf and K. Ch. Le, Variational principles of nonlinear fracture mechanics,, Acta Mech, 83 (1990), 25. 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), 1. Google Scholar 
[6] 
J. D. Eshelby, The continuum theory of lattice defects,, in, (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), 379. Google Scholar 
[8] 
M. Buliga, Energy minimizing brittle crack propagation,, J Elast, 52 (1999), 201. 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), 714. 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), 163. 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), 1483. Google Scholar 
[15] 
T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing,, Int J Numer Methods Eng, 45 (1999), 601. 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), 53. Google Scholar 
[17] 
A. Needleman, A continuum model for void nucleation by inclusion debonding,, J. Appl. Mech., 54 (1987), 525. Google Scholar 
[18] 
D. S. Dugdale, Yielding of steel sheets containing slits,, J Mech Phys Solids, 8 (1960), 100. Google Scholar 
[19] 
V. Tvergaard and J. W. Hutchinson, The influence of plasticity on mixed mode interface toughness,, J Mech Phys Solids, 41 (1993), 1119. Google Scholar 
[20] 
G. T. Camacho and M. Ortiz, Computational modelling of impact damage in brittle materials,, Int J Solids Struct, 33 (1996), 2899. Google Scholar 
[21] 
X.P. Xu and A. Needleman, Numerical simulations of fast crack growth in brittle solids,, Mech Phys Solids, 42 (1994), 1397. Google Scholar 
[22] 
M. Ortiz and A. Pandolfi, Finitedeformation irreversible cohesive elements for threedimensional crackpropagation analysis,, Int J Numer Methods Eng, 44 (1999), 1267. 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), 192. 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), 511. Google Scholar 
[25] 
V. A. Kovtunenko, Nonconvex problem for crack with nonpenetration,, ZAMM Z. Angew. Math. Mech., 85 (2005), 242. Google Scholar 
[26] 
D. Hull, An introduction to composite materials,, in, (1981), 1. 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), 43. 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), 1383. Google Scholar 
[29] 
G. Geissler and M. Kaliske, Timedependent cohesive zone modelling for discrete fracture simulation,, Eng Fract Mech, 77 (2010), 153. 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), 99. 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), 15. 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), 1209. 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), 944. 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, (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), 25. 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), 491. 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), 2827. Google Scholar 
[40] 
A. Banerjea and J. R. Smith, Origins of the universal bindingenergy relation,, Phys Rev B, 37 (1988), 6632. 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), 163. Google Scholar 
[2] 
G. R. Irwin, Fracture,, in, (1958), 551. Google Scholar 
[3] 
G. I. Barenblatt, The mathematical theory of equilibrium cracks in brittle fracture,, Advan. Appl. Mech., 7 (1962), 55. Google Scholar 
[4] 
H. Stumpf and K. Ch. Le, Variational principles of nonlinear fracture mechanics,, Acta Mech, 83 (1990), 25. 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), 1. Google Scholar 
[6] 
J. D. Eshelby, The continuum theory of lattice defects,, in, (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), 379. Google Scholar 
[8] 
M. Buliga, Energy minimizing brittle crack propagation,, J Elast, 52 (1999), 201. 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), 714. 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), 163. 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), 1483. Google Scholar 
[15] 
T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing,, Int J Numer Methods Eng, 45 (1999), 601. 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), 53. Google Scholar 
[17] 
A. Needleman, A continuum model for void nucleation by inclusion debonding,, J. Appl. Mech., 54 (1987), 525. Google Scholar 
[18] 
D. S. Dugdale, Yielding of steel sheets containing slits,, J Mech Phys Solids, 8 (1960), 100. Google Scholar 
[19] 
V. Tvergaard and J. W. Hutchinson, The influence of plasticity on mixed mode interface toughness,, J Mech Phys Solids, 41 (1993), 1119. Google Scholar 
[20] 
G. T. Camacho and M. Ortiz, Computational modelling of impact damage in brittle materials,, Int J Solids Struct, 33 (1996), 2899. Google Scholar 
[21] 
X.P. Xu and A. Needleman, Numerical simulations of fast crack growth in brittle solids,, Mech Phys Solids, 42 (1994), 1397. Google Scholar 
[22] 
M. Ortiz and A. Pandolfi, Finitedeformation irreversible cohesive elements for threedimensional crackpropagation analysis,, Int J Numer Methods Eng, 44 (1999), 1267. 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), 192. 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), 511. Google Scholar 
[25] 
V. A. Kovtunenko, Nonconvex problem for crack with nonpenetration,, ZAMM Z. Angew. Math. Mech., 85 (2005), 242. Google Scholar 
[26] 
D. Hull, An introduction to composite materials,, in, (1981), 1. 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), 43. 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), 1383. Google Scholar 
[29] 
G. Geissler and M. Kaliske, Timedependent cohesive zone modelling for discrete fracture simulation,, Eng Fract Mech, 77 (2010), 153. 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), 99. 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), 15. 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), 1209. 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), 944. 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, (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), 25. 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), 491. 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), 2827. Google Scholar 
[40] 
A. Banerjea and J. R. Smith, Origins of the universal bindingenergy relation,, Phys Rev B, 37 (1988), 6632. Google Scholar 
[1] 
Rejeb Hadiji, Ken Shirakawa. 3D2D asymptotic observation for minimization problems associated with degenerate energycoefficients. Conference Publications, 2011, 2011 (Special) : 624633. doi: 10.3934/proc.2011.2011.624 
[2] 
Roman VodiČka, Vladislav MantiČ. An energy based formulation of a quasistatic interface damage model with a multilinear cohesive law. Discrete & Continuous Dynamical Systems  S, 2017, 10 (6) : 15391561. doi: 10.3934/dcdss.2017079 
[3] 
Xavier Dubois de La Sablonière, Benjamin Mauroy, Yannick Privat. Shape minimization of the dissipated energy in dyadic trees. Discrete & Continuous Dynamical Systems  B, 2011, 16 (3) : 767799. doi: 10.3934/dcdsb.2011.16.767 
[4] 
Tianliang Hou, Yanping Chen. Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements. Journal of Industrial & Management Optimization, 2013, 9 (3) : 631642. doi: 10.3934/jimo.2013.9.631 
[5] 
Jiayu Han. Nonconforming elements of class $L^2$ for Helmholtz transmission eigenvalue problems. Discrete & Continuous Dynamical Systems  B, 2018, 23 (8) : 31953212. doi: 10.3934/dcdsb.2018281 
[6] 
Tan BuiThanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems & Imaging, 2013, 7 (4) : 11391155. doi: 10.3934/ipi.2013.7.1139 
[7] 
Sari Lasanen. NonGaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns. Inverse Problems & Imaging, 2012, 6 (2) : 267287. doi: 10.3934/ipi.2012.6.267 
[8] 
Sari Lasanen. NonGaussian statistical inverse problems. Part I: Posterior distributions. Inverse Problems & Imaging, 2012, 6 (2) : 215266. doi: 10.3934/ipi.2012.6.215 
[9] 
Tianxiao Wang. Characterizations of equilibrium controls in time inconsistent meanfield stochastic linear quadratic problems. I. Mathematical Control & Related Fields, 2019, 9 (2) : 385409. doi: 10.3934/mcrf.2019018 
[10] 
Anass Belcaid, Mohammed Douimi, Abdelkader Fassi Fihri. Recursive reconstruction of piecewise constant signals by minimization of an energy function. Inverse Problems & Imaging, 2018, 12 (4) : 903920. doi: 10.3934/ipi.2018038 
[11] 
Riccardo Adami, Diego Noja, Nicola Visciglia. Constrained energy minimization and ground states for NLS with point defects. Discrete & Continuous Dynamical Systems  B, 2013, 18 (5) : 11551188. doi: 10.3934/dcdsb.2013.18.1155 
[12] 
M. Montaz Ali. A recursive topographical differential evolution algorithm for potential energy minimization. Journal of Industrial & Management Optimization, 2010, 6 (1) : 2946. doi: 10.3934/jimo.2010.6.29 
[13] 
Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete & Continuous Dynamical Systems  B, 2019, 24 (5) : 20932124. doi: 10.3934/dcdsb.2019086 
[14] 
Haïm Brezis. Remarks on some minimization problems associated with BV norms. Discrete & Continuous Dynamical Systems  A, 2019, 0 (0) : 117. doi: 10.3934/dcds.2019242 
[15] 
Yuan Shen, Xin Liu. An alternating minimization method for matrix completion problems. Discrete & Continuous Dynamical Systems  S, 2018, 0 (0) : 00. doi: 10.3934/dcdss.2020103 
[16] 
Sanming Liu, Zhijie Wang, Chongyang Liu. Proximal iterative Gaussian smoothing algorithm for a class of nonsmooth convex minimization problems. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 7989. doi: 10.3934/naco.2015.5.79 
[17] 
Luchuan Ceng, Qamrul Hasan Ansari, JenChih Yao. Extragradientprojection method for solving constrained convex minimization problems. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 341359. doi: 10.3934/naco.2011.1.341 
[18] 
YuNing Yang, Su Zhang. On linear convergence of projected gradient method for a class of affine rank minimization problems. Journal of Industrial & Management Optimization, 2016, 12 (4) : 15071519. doi: 10.3934/jimo.2016.12.1507 
[19] 
David Yang Gao. Solutions and optimality criteria to box constrained nonconvex minimization problems. Journal of Industrial & Management Optimization, 2007, 3 (2) : 293304. doi: 10.3934/jimo.2007.3.293 
[20] 
Jie Sun, Honglei Xu, Min Zhang. A new interpretation of the progressive hedging algorithm for multistage stochastic minimization problems. Journal of Industrial & Management Optimization, 2017, 13 (5) : 18. doi: 10.3934/jimo.2019022 
2018 Impact Factor: 0.925
Tools
Metrics
Other articles
by authors
[Back to Top]