# 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 and 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), 163-98. [2] G. R. Irwin, Fracture, in "Encyclopedia of Physics: Elasticity and Plasticity" (S. Fluegge ed.), Springer-Verlag Berlin, (1958), 551-90. [3] G. I. Barenblatt, The mathematical theory of equilibrium cracks in brittle fracture, Advan. Appl. Mech., 7 (1962), 55-129. [4] H. Stumpf and K. Ch. Le, Variational principles of nonlinear fracture mechanics, Acta Mech, 83 (1990), 25-37. [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), 1-28. [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. [7] J. R. Rice, A path independent integral and the approximate analysis of strain concentraction by notches and cracks, J. Appl. Mech., 35 (1968), 379-86. [8] M. Buliga, Energy minimizing brittle crack propagation, J Elast, 52 (1999), 201-238. [9] N. Kikuchi and J. T. Oden, "The Variational Approach to Fracture," SIAM, 1988. [10] A. M. Khludnev and V. A. Kovtunenko, "Analysis of Cracks in Solids," WIT Press, 1999. [11] B. Bourdin, G. A. Francfort and J.-J. Marigo, "Contact Problems in Elasticity," Springer, 2008. [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), 714-723. [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), 163-86. [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), 1483-1504. [15] T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing, Int J Numer Methods Eng, 45 (1999), 601-620. [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), 53-83. [17] A. Needleman, A continuum model for void nucleation by inclusion debonding, J. Appl. Mech., 54 (1987), 525-531. [18] D. S. Dugdale, Yielding of steel sheets containing slits, J Mech Phys Solids, 8 (1960), 100-104. [19] V. Tvergaard and J. W. Hutchinson, The influence of plasticity on mixed mode interface toughness, J Mech Phys Solids, 41 (1993), 1119-1135. [20] G. T. Camacho and M. Ortiz, Computational modelling of impact damage in brittle materials, Int J Solids Struct, 33 (1996), 2899-2938. [21] X.-P. Xu and A. Needleman, Numerical simulations of fast crack growth in brittle solids, Mech Phys Solids, 42 (1994), 1397-1434. [22] M. Ortiz and A. Pandolfi, Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis, Int J Numer Methods Eng, 44 (1999), 1267-1282. [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), 192-198. [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), 511-519. [25] V. A. Kovtunenko, Nonconvex problem for crack with nonpenetration, ZAMM Z. Angew. Math. Mech., 85 (2005), 242-251. [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, 1-246. [27] J. C. J. Schellekens and R. de Borst, On the numerical integration of interface elements, Int J Numer Methods Engng, 36 (1993), 43-66. [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), 1383-1410. [29] G. Geissler and M. Kaliske, Time-dependent cohesive zone modelling for discrete fracture simulation, Eng Fract Mech, 77 (2010), 153-169. [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), 99-121. [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), 15-29. [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), 1209-1229. [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), 944-960. [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. [35] M. Burger, "Infinite-dimensional Optimization and Optimal Design," 2003. [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), 25-57. [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), 491-516. [38] V. A. Kovtunenko, A hemivariational inequality in crack problems, Optimization, doi: 10.1080/02331934.2010.534477. [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), 2827-2855. [40] A. Banerjea and J. R. Smith, Origins of the universal binding-energy relation, Phys Rev B, 37 (1988), 6632-6645.

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##### References:
 [1] A. A. Griffith, The phenomena of rupture and flow in solids, Philos Trans R Soc Lond A, 221 (1921), 163-98. [2] G. R. Irwin, Fracture, in "Encyclopedia of Physics: Elasticity and Plasticity" (S. Fluegge ed.), Springer-Verlag Berlin, (1958), 551-90. [3] G. I. Barenblatt, The mathematical theory of equilibrium cracks in brittle fracture, Advan. Appl. Mech., 7 (1962), 55-129. [4] H. Stumpf and K. Ch. Le, Variational principles of nonlinear fracture mechanics, Acta Mech, 83 (1990), 25-37. [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), 1-28. [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. [7] J. R. Rice, A path independent integral and the approximate analysis of strain concentraction by notches and cracks, J. Appl. Mech., 35 (1968), 379-86. [8] M. Buliga, Energy minimizing brittle crack propagation, J Elast, 52 (1999), 201-238. [9] N. Kikuchi and J. T. Oden, "The Variational Approach to Fracture," SIAM, 1988. [10] A. M. Khludnev and V. A. Kovtunenko, "Analysis of Cracks in Solids," WIT Press, 1999. [11] B. Bourdin, G. A. Francfort and J.-J. Marigo, "Contact Problems in Elasticity," Springer, 2008. [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), 714-723. [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), 163-86. [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), 1483-1504. [15] T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing, Int J Numer Methods Eng, 45 (1999), 601-620. [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), 53-83. [17] A. Needleman, A continuum model for void nucleation by inclusion debonding, J. Appl. Mech., 54 (1987), 525-531. [18] D. S. Dugdale, Yielding of steel sheets containing slits, J Mech Phys Solids, 8 (1960), 100-104. [19] V. Tvergaard and J. W. Hutchinson, The influence of plasticity on mixed mode interface toughness, J Mech Phys Solids, 41 (1993), 1119-1135. [20] G. T. Camacho and M. Ortiz, Computational modelling of impact damage in brittle materials, Int J Solids Struct, 33 (1996), 2899-2938. [21] X.-P. Xu and A. Needleman, Numerical simulations of fast crack growth in brittle solids, Mech Phys Solids, 42 (1994), 1397-1434. [22] M. Ortiz and A. Pandolfi, Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis, Int J Numer Methods Eng, 44 (1999), 1267-1282. [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), 192-198. [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), 511-519. [25] V. A. Kovtunenko, Nonconvex problem for crack with nonpenetration, ZAMM Z. Angew. Math. Mech., 85 (2005), 242-251. [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, 1-246. [27] J. C. J. Schellekens and R. de Borst, On the numerical integration of interface elements, Int J Numer Methods Engng, 36 (1993), 43-66. [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), 1383-1410. [29] G. Geissler and M. Kaliske, Time-dependent cohesive zone modelling for discrete fracture simulation, Eng Fract Mech, 77 (2010), 153-169. [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), 99-121. [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), 15-29. [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), 1209-1229. [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), 944-960. [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. [35] M. Burger, "Infinite-dimensional Optimization and Optimal Design," 2003. [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), 25-57. [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), 491-516. [38] V. A. Kovtunenko, A hemivariational inequality in crack problems, Optimization, doi: 10.1080/02331934.2010.534477. [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), 2827-2855. [40] A. Banerjea and J. R. Smith, Origins of the universal binding-energy relation, Phys Rev B, 37 (1988), 6632-6645.
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