# American Institute of Mathematical Sciences

July  2013, 12(4): 1705-1729. doi: 10.3934/cpaa.2013.12.1705

## A cohesive crack propagation model: Mathematical theory and numerical solution

 1 Applied Mathematics II, Martensstr. 3, D-91054 Erlangen, Germany, Germany 2 Chair of Applied Mechanics, Egerlandstr. 5, D-91058 Erlangen, Germany 3 Applied Mathematics II, Martensstr. 3, D-91058 Erlangen, Germany

Received  February 2011 Revised  November 2011 Published  November 2012

We investigate the propagation of cracks in 2-d elastic domains, which are subjected to quasi-static loading scenarios. As we take cohesive effects along the crack path into account and impose a non-penetration condition, inequalities appear in the constitutive equations describing the elastic behavior of a domain with crack. In contrast to existing approaches, we consider cohesive effects arising from crack opening in normal as well as in tangential direction. We establish a constrained energy minimization problem and show that the solution of this problem satisfies the set of constitutive equations. In order to solve the energy minimization problem numerically, we apply a finite element discretization using a combination of standard continuous finite elements with so-called cohesive elements. A particular strength of our method is that the crack path is a result of the minimization process. We conclude the article by numerical experiments and compare our results to results given in the literature.
Citation: G. Leugering, Marina Prechtel, Paul Steinmann, Michael Stingl. A cohesive crack propagation model: Mathematical theory and numerical solution. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1705-1729. doi: 10.3934/cpaa.2013.12.1705
##### 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. Specovius-Neugebauer, 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 crack-tip deformation and fracture for plane-strain crack growth in elastic-plastic solids,, J Mech Phys Solids, 26 (1978), 163. Google Scholar [14] M. Fleming, Y. A. Chu, B. Moran and T. Belytschko, Enriched element-free 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, Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation 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 rate-dependent cohesive model for simulating dynamic crack propagation in brittle materials,, Eng Fract Mech, 72 (2005), 1383. Google Scholar [29] G. Geissler and M. Kaliske, Time-dependent 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, "Infinite-dimensional Optimization and Optimal Design,", 2003., (). Google Scholar [36] A. W鋍hter and L. T. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale 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 metal-ceramic interfaces,, Int J Solid Struct, 39 (2002), 2827. Google Scholar [40] A. Banerjea and J. R. Smith, Origins of the universal binding-energy 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. Specovius-Neugebauer, 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 crack-tip deformation and fracture for plane-strain crack growth in elastic-plastic solids,, J Mech Phys Solids, 26 (1978), 163. Google Scholar [14] M. Fleming, Y. A. Chu, B. Moran and T. Belytschko, Enriched element-free 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, Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation 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 rate-dependent cohesive model for simulating dynamic crack propagation in brittle materials,, Eng Fract Mech, 72 (2005), 1383. Google Scholar [29] G. Geissler and M. Kaliske, Time-dependent 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, "Infinite-dimensional Optimization and Optimal Design,", 2003., (). Google Scholar [36] A. W鋍hter and L. T. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale 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 metal-ceramic interfaces,, Int J Solid Struct, 39 (2002), 2827. Google Scholar [40] A. Banerjea and J. R. Smith, Origins of the universal binding-energy relation,, Phys Rev B, 37 (1988), 6632. Google Scholar
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