Review Articles

Cohesive Zone Models: A Critical Review of Traction-Separation Relationships Across Fracture Surfaces

[+] Author and Article Information
Kyoungsoo Park

School of Civil & Environmental Engineering,
Yonsei University,
50 Yonsei-ro, Seodaemun-gu,
Seoul, Korea
e-mail: k-park@yonsei.ac.kr

Glaucio H. Paulino

Department of Civil &
Environmental Engineering,
University of Illinois at Urbana-Champaign,
205 North Mathews Avenue,
Urbana, IL 61801
e-mail: paulino@illinois.edu

1Corresponding author.

Manuscript received July 14, 2011; final manuscript received November 15, 2012; published online February 5, 2013. Editor: Harry Dankowicz.

Appl. Mech. Rev 64(6), 060802 (Feb 05, 2013) (20 pages) doi:10.1115/1.4023110 History: Received July 14, 2011; Revised November 15, 2012

One of the fundamental aspects in cohesive zone modeling is the definition of the traction-separation relationship across fracture surfaces, which approximates the nonlinear fracture process. Cohesive traction-separation relationships may be classified as either nonpotential-based models or potential-based models. Potential-based models are of special interest in the present review article. Several potential-based models display limitations, especially for mixed-mode problems, because of the boundary conditions associated with cohesive fracture. In addition, this paper shows that most effective displacement-based models can be formulated under a single framework. These models lead to positive stiffness under certain separation paths, contrary to general cohesive fracture phenomena wherein the increase of separation generally results in the decrease of failure resistance across the fracture surface (i.e., negative stiffness). To this end, the constitutive relationship of mixed-mode cohesive fracture should be selected with great caution.

Copyright © 2011 by ASME
Your Session has timed out. Please sign back in to continue.


Gurtin, M. E., 1981, An Introduction to Continuum Mechanics, Academic Press, New York.
Needleman, A., 1992, “Micromechanical Modelling of Interfacial Decohesion,” Ultramicroscopy, 40(3), pp. 203–214. [CrossRef]
Park, K., Paulino, G. H., and Roesler, J. R., 2009, “A Unified Potential-Based Cohesive Model of Mixed-Mode Fracture,” J. Mech. Phys. Solids, 57(6), pp. 891–908. [CrossRef]
Needleman, A., 1987, “A Continuum Model for Void Nucleation by Inclusion Debonding,” ASME J. Appl. Mech., 54(3), pp. 525–531. [CrossRef]
Freed, Y., and Banks-Sills, L., 2008, “A New Cohesive Zone Model for Mixed Mode Interface Fracture in Bimaterials,” Eng. Fract. Mech., 75(15), pp. 4583–4593. [CrossRef]
Needleman, A., 1990, “An Analysis of Decohesion Along an Imperfect Interface,” Int. J. Fract., 42(1), pp. 21–40. [CrossRef]
Needleman, A., 1990, “An Analysis of Tensile Decohesion Along an Interface,” J. Mech. Phys. Solids, 38(3), pp. 289–324. [CrossRef]
Beltz, G. E., and Rice, J. R., 1991, “Dislocation Nucleation Versus Cleavage Decohesion at Crack Tips,” Modeling the Deformation of Crystalline Solids Presented, T. C.Lowe, A. D.Rollett, P. S.Follansbee, and G. S.Daehn, eds., The Minerals, Metals & Materials Society, Harvard University, Cambridge, MA, pp. 457–480.
Xu, X. P., and Needleman, A., 1993, “Void Nucleation by Inclusion Debonding in a Crystal Matrix,” Model. Simul. Mater. Sci. Eng., 1(2), pp. 111–132. [CrossRef]
Kanninen, M. F., and Popelar, C. H., 1985, Advanced Fracture Mechanics, Oxford University Press, New York.
Bazant, Z. P., and Cedolin, L., 1991, Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories, Oxford University Press, New York.
Anderson, T. L., 1995, Fracture Mechanics: Fundamentals and Applications, CRC Press, Boca Raton, FL.
Suresh, S., 1998, Fatigue of Materials, Cambridge University Press, New York.
Broberg, K. B., 1999, Cracks and Fracture, Academic Press, San Diego, CA.
Elliott, H. A., 1947, “An Analysis of the Conditions for Rupture Due to Griffth Cracks,” Proc. Phys. Soc., 59(2), pp. 208–223. [CrossRef]
Barenblatt, G. I., 1959, “The Formation of Equilibrium Cracks During Brittle Fracture: General Ideas and Hypotheses, Axially Symmetric Cracks,” Appl. Math. Mech., 23(3), pp. 622–636. [CrossRef]
Barenblatt, G. I., 1962, “The Mathematical Theory of Equilibrium Cracks in Brittle Fracture,” Adv. Appl. Mech., 7, pp. 55–129. [CrossRef]
Dugdale, D. S., 1960, “Yielding of Steel Sheets Containing Slits,” J. Mech. Phys. Solids, 8(2), pp. 100–104. [CrossRef]
Griffith, A. A., 1921, “The Phenomena of Rupture and Flow in Solids,” Philos. Trans. R. Soc. London, 221, pp. 163–198. [CrossRef]
Willis, J. R., 1967, “A Comparison of the Fracture Criteria of Griffith and Barenblatt,” J. Mech. Phys. Solids, 15(3), pp. 151–162. [CrossRef]
Rice, J. R., 1968, “A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks,” ASME J. Appl. Mech., 35(2), pp. 379–386. [CrossRef]
Bilby, B. A., Cottrell, A. H., and Swinden, K. H., 1963, “Spread of Plastic Yield From Notch,” R. Soc. Proc. Ser. A, 272(1350), pp. 304–314. [CrossRef]
Bilby, B. A., and Swinden, K. H., 1965, “Representation of Plasticity at Notches by Linear Dislocations Arrays,” R. Soc. Proc. Ser. A, 285(1400), pp. 22–33. [CrossRef]
Rice, J. R., 1968, “Mathematical Analysis in the Mechanics of Fracture,” Fracture: An Advanced Treatise, Vol. 2, H.Liebowitz, ed., Academic Press, New York, pp. 191–311.
Smith, E., 1974, “The Structure in the Vicinity of a Crack Tip: A General Theory Based on the Cohesive Zone Model,” Eng. Fract. Mech., 6(2), pp. 213–222. [CrossRef]
Keer, L. M., 1964, “Stress Distribution at the Edge of an Equilibrium Crack,” J. Mech. Phys. Solids, 12(3), pp. 149–163. [CrossRef]
Cribb, J. L., and Tomkins, B., 1967, “On the Nature of the Stress at the Tip of a Perfectly Brittle Crack,” J. Mech. Phys. Solids, 15(2), pp. 135–140. [CrossRef]
Smith, E., 1975, “A Generalization of Elliott's Model of a Crack Tip,” Int. J. Fract., 11(2), pp. 295–299. [CrossRef]
Hillerborg, A., Modeer, M., and Petersson, P. E., 1976, “Analysis of Crack Formation and Crack Growth in Concrete by Means of Fracture Mechanics and Finite Elements,” Cem. Concr. Res., 6(6), pp. 773–781. [CrossRef]
Boone, T. J., Wawrzynek, P. A., and Ingraffea, A. R., 1986, “Simulation of the Fracture Process in Rock With Application to Hydrofracturing,” Int. J. Rock Mech. Min. Sci., 23(3), pp. 255–265. [CrossRef]
Elices, M., Guinea, G. V., Gomez, J., and Planas, J., 2002, “The Cohesive Zone Model: Advantages, Limitations and Challenges,” Eng. Fract. Mech., 69(2), pp. 137–163. [CrossRef]
Bazant, Z. P., and Becq-Giraudon, E., 2002, “Statistical Prediction of Fracture Parameters of Concrete and Implications for Choice of Testing Standard,” Cem. Concr. Res., 32(4), pp. 529–556. [CrossRef]
Roesler, J., Paulino, G. H., Park, K., and Gaedicke, C., 2007, “Concrete Fracture Prediction Using Bilinear Softening,” Cem. Concr. Compos., 29(4), pp. 300–312. [CrossRef]
Petersson, P. E., 1981, “Crack Growth and Development of Fracture Zones in Plain Concrete and Similar Materials,” Tech. Report No. LUTVDG/TVBM-1006, Lund Institute of Technology, Sweden.
Wittmann, F. H., Rokugo, K., Bruehwiler, E., Mihashi, H., and Simonin, P., 1988, “Fracture Energy and Strain Softening of Concrete as Determined by Means of Compact Tension Specimens,” Mater. Struct., 21(1), pp. 21–32. [CrossRef]
Guinea, G. V., Planas, J., and Elices, M., 1994, “A General Bilinear Fit for the Softening Curve of Concrete,” Materiaux Constr., 27(2), pp. 99–105. [CrossRef]
Park, K., Paulino, G. H., and Roesler, J. R., 2008, “Determination of the Kink Point in the Bilinear Softening Model for Concrete,” Eng. Fract. Mech., 75(13), pp. 3806–3818. [CrossRef]
Jenq, Y. S., and Shah, S. P., 1985, “Two Parameter Fracture Model for Concrete,” JSCE J. –Eng. Mech., 111(10), pp. 1227–1241. [CrossRef]
Bazant, Z. P., and Kazemi, M. T., 1990, “Determination of Fracture Energy, Process Zone Length and Brittleness Number From Size Effect, With Application to Rock and Concrete,” Int. J. Fract., 44(2), pp. 111–131. [CrossRef]
Shah, S. P., Swartz, S. E., and Ouyang, C., 1995, Fracture Mechanics of Concrete: Applications of Fracture Mechanics to Concrete, Rock and Other Quasi-Brittle Materials, Wiley-Interscience, New York.
van Mier, J. G. M., 1996, Fracture Processes of Concrete: Assessment of Material Parameters for Fracture Models, CRC Press, Boca Raton, FL.
Bazant, Z. P., and Planas, J., 1998, Fracture and Size Effect in Concrete and other Quasibrittle Materials, CRC Press, Boca Raton, FL.
Li, V. C., Stang, H., and Krenchel, H., 1993, “Micromechanics of Crack Bridging in Fibre-Reinforced Concrete,” Mater. Struct., 26(162), pp. 486–494. [CrossRef]
Park, K., Paulino, G. H., and Roesler, J., 2010, “Cohesive Fracture Model for Functionally Graded Fiber Reinforced Concrete,” Cem. Concr. Res., 40(6), pp. 956–965. [CrossRef]
Hui, C. Y., Ruina, A., Long, R., and Jagota, A., 2011, “Cohesive Zone Models and Fracture,” J. Adhes., 87(1), pp. 1–52. [CrossRef]
Kramer, E. J., 1983, “Microscopic and Molecular Fundamentals of Crazing,” Crazing in Polymers (Advances in Polymer Science), Vol. 52–53, H.Kausch, ed., Springer-Verlag, Berlin, Germany, pp. 1–56.
Tijssens, M. G. A., van der Giessen, E., and Sluys, L. J., 2000, “Modeling of Crazing Using a Cohesive Surface Methodology,” Mech. Mater., 32(1), pp. 19–35. [CrossRef]
Estevez, R., Tijssens, M. G. A., and der Giessen, E. V., 2000, “Modeling of the Competition Between Shear Yielding and Crazing in Glassy Polymers,” J. Mech. Phys. Solids, 48(12), pp. 2585–2617. [CrossRef]
Allen, D. H., and Searcy, C. R., 2001, “Micromechanical Model for a Viscoelastic Cohesive Zone,” Int. J. Fract., 107(2), pp. 159–176. [CrossRef]
Hui, C. Y., Ruina, A., Creton, C., and Kramer, E. J., 1992, “Micromechanics of Crack Growth Into a Craze in a Polymer Glass,” Macromolecules, 25(15), pp. 3948–3955. [CrossRef]
Hui, C. Y., and Kramer, E. J., 1995, “Molecular Weight Dependence of the Fracture Toughness of Glassy Polymers Arising From Crack Propagation Through a Craze,” Polym. Eng. Sci., 35(5), pp. 419–425. [CrossRef]
Hong, S., Chew, H. B., and Kim, K.-S., 2009, “Cohesive-Zone Laws for Void Growth – I. Experimental Field Projection of Crack-Tip Crazing in Glassy Polymers,” J. Mech. Phys. Solids, 57(8), pp. 1357–1373. [CrossRef]
Bolander, J. E., and Sukumar, N., 2005, “Irregular Lattice Model for Quasistatic Crack Propagation,” Phys. Rev. B, 71(9), p. 094106. [CrossRef]
Li, S., and Ghosh, S., 2006, “Extended Voronoi Cell Finite Element Model for Multiple Cohesive Crack Propagation in Brittle Materials,” Int. J. Numer. Methods Eng., 65(7), pp. 1028–1067. [CrossRef]
Bishop, J. E., 2009, “Simulating the Pervasive Fracture of Materials and Structures Using Randomly Close Packed Voronoi Tessellations,” Comput. Mech., 44(4), pp. 455–471. [CrossRef]
Walter, M. E., Ravichandran, G., and Ortiz, M., 1997, “Computational Modeling of Damage Evolution in Unidirectional Fiber Reinforced Ceramic Matrix Composites,” Comput. Mech., 20(1–2), pp. 192–198. [CrossRef]
Carpinteri, A., Paggi, M., and Zavarise, G., 2005, “Snap-Back Instability in Micro-Structured Composites and its Connection With Superplasticity,” Strength, Fract. Complexity, 3(2–4), pp. 61–72.
Needleman, A., Borders, T. L., Brinson, L., Flores, V. M., and Schadler, L. S., 2010, “Effect of an Interphase Region on Debonding of a CNT Reinforced Polymer Composite,” Compos. Sci. Technol., 70(15), pp. 2207–2215. [CrossRef]
Ngo, D., Park, K., Paulino, G. H., and Huang, Y., 2010, “On the Constitutive Relation of Materials With Microstructure Using a Potential-Based Cohesive Model for Interface Interaction,” Eng. Fract. Mech., 77(7), pp. 1153–1174. [CrossRef]
Paulino, G. H., Jin, Z. H., and Dodds, R. H., 2003, “Failure of Functionally Graded Materials,” Comprehensive Structural Integrity, Vol. 2, B.Karihaloo and W. G.Knauss, eds., Elsevier, The Netherlands, pp. 607–644.
Erdogan, F., and Sih, G. C., 1963, “On the Crack Extension in Plates Under Plane Loading and Transverse Shear,” ASME J. Basic Eng., 85(4), pp. 519–525. [CrossRef]
Pindera, M.-J., and Paulino, G. H., 2008, “Honoring Professor Erdogan's Seminal Contributions to Mixed Boundary-Value Problems of Inhomogeneous and Functionally Graded Materials,” ASME J. Appl. Mech., 75(5), p. 050301. [CrossRef]
Erdogan, F., and Ozturk, M., 2008, “On the Singularities in Fracture and Contact Mechanics,” ASME J. Appl. Mech., 75(5), p. 051111. [CrossRef]
Dag, S., and Ilhan, K. A., 2008, “Mixed-Mode Fracture Analysis of Orthotropic Functionally Graded Material Coatings Using Analytical and Computational Methods,” ASME J. Appl. Mech., 75(5), p. 051104. [CrossRef]
Tvergaard, V., 2002, “Theoretical Investigation of the Effect of Plasticity on Crack Growth Along a Functionally Graded Region Between Dissimilar Elastic-Plastic Solids,” Eng. Fract. Mech., 69(14–16), pp. 1635–1645. [CrossRef]
Wang, Z., and Nakamura, T., 2004, “Simulations of Crack Propagation in Elastic-Plastic Graded Materials,” Mech. Mater., 36(7), pp. 601–622. [CrossRef]
Jin, Z.-H., Paulino, G. H., and Dodds, R. H., Jr., 2003, “Cohesive Fracture Modeling of Elastic-Plastic Crack Growth in Functionally Graded Materials,” Eng. Fract. Mech., 70(14), pp. 1885–1912. [CrossRef]
Rangaraj, S., and Kokini, K., 2004, “A Study of Thermal Fracture in Functionally Graded Thermal Barrier Coatings Using a Cohesive Zone Model,” ASME J. Eng. Mater. Technol., 126(1), pp. 103–115. [CrossRef]
Zhang, Z., and Paulino, G. H., 2005, “Cohesive Zone Modeling of Dynamic Failure in Homogeneous and Functionally Graded Materials,” Int. J. Plast., 21(6), pp. 1195–1254. [CrossRef]
Kandula, S. S. V., Abanto-Bueno, J., Geubelle, P. H., and Lambros, J., 2005, “Cohesive Modeling of Dynamic Fracture in Functionally Graded Materials,” Int. J. Fract., 132(3), pp. 275–296. [CrossRef]
Jin, Z.-H., Paulino, G. H., and Dodds, R. H., 2002, “Finite Element Investigation of Quasi-Static Crack Growth in Functionally Graded Materials Using a Novel Cohesive Zone Fracture Model,” ASME J. Appl. Mech., 69(3), pp. 370–379. [CrossRef]
Shim, D.-J., Paulino, G. H., and Dodds, R. H., Jr., 2006, “J Resistance Behavior in Functionally Graded Materials Using Cohesive Zone and Modified Boundary Layer Models,” Int. J. Fract., 139(1), pp. 91–117. [CrossRef]
Brocks, W., and Cornec, A., 2003, “Guest Editorial: Cohesive Models,” Eng. Fract. Mech., 70(14), pp. 1741–1742. [CrossRef]
de Andres, A., Perez, J. L., and Ortiz, M., 1999, “Elastoplastic Finite Element Analysis of Three Dimensional Fatigue Crack Growth in Aluminum Shafts Subjected to Axial Loading,” Int. J. Solids Struct., 36(15), pp. 2231–2258. [CrossRef]
Deshpande, V. S., Needleman, A., and Van der Giessen, E., 2001, “A Discrete Dislocation Analysis of Near-Threshold Fatigue Crack Growth,” Acta Mater., 49(16), pp. 3189–3203. [CrossRef]
Roe, K. L., and Siegmund, T., 2003, “An Irreversible Cohesive Zone Model for Interface Fatigue Crack Growth Simulation,” Eng. Fract. Mech., 70(2), pp. 209–232. [CrossRef]
Maiti, S., and Geubelle, P. H., 2005, “A Cohesive Model for Fatigue Failure of Polymers,” Eng. Fract. Mech., 72(5), pp. 691–708. [CrossRef]
Ural, A., Krishnan, V. R., and Papoulia, K. D., 2009, “A Cohesive Zone Model for Fatigue Crack Growth Allowing for Crack Retardation,” Int. J. Solids Struct., 46(11–12), pp. 2453–2462. [CrossRef]
Ingraffea, A. R., Gerstle, W. H., Gergely, P., and Saouma, V., 1984, “Fracture Mechanics of Bond in Reinforced Concrete,” J. Struct. Eng., 110(4), pp. 871–890. [CrossRef]
Prasad, M. V. K. V., and Krishnamoorthy, C. S., 2002, “Computational Model for Discrete Crack Growth in Plain and Reinforced Concrete,” Comput. Methods Appl. Mech. Eng., 191(25–26), pp. 2699–2725. [CrossRef]
Koeberl, B., and Willam, K., 2008, “Question of Tension Softening Versus Tension Stiffening in Plain and Reinforced Concrete,” ASCE J. Eng. Mech., 134(9), pp. 804–808. [CrossRef]
Yang, Q. D., and Thouless, M. D., 2001, “Mixed-Mode Fracture Analyses of Plastically-Deforming Adhesive Joints,” Int. J. Fract., 110(2), pp. 175–187. [CrossRef]
Xu, C., Siegmund, T., and Ramani, K., 2003, “Rate-Dependent Crack Growth in Adhesives: I. Modeling Approach,” Int. J. Adhes. Adhes., 23(1), pp. 9–13. [CrossRef]
Alfano, M., Furgiuele, F., Leonardi, A., Maletta, C., and Paulino, G. H., 2009, “Mode I Fracture of Adhesive Joints Using Tailored Cohesive Zone Models,” Int. J. Fract., 157(1–2), pp. 193–204. [CrossRef]
Khoramishad, H., Crocombe, A. D., Katnam, K. B., and Ashcroft, I. A., 2010, “Predicting Fatigue Damage in Adhesively Bonded Joints Using a Cohesive Zone Model,” Int. J. Fatigue, 32(7), pp. 1146–1158. [CrossRef]
Miller, O., Freund, L. B., and Needleman, A., 1999, “Energy Dissipation in Dynamic Fracture of Brittle Materials,” Model. Simul. Mater. Sci. Eng., 7(4), pp. 573–586. [CrossRef]
Zhang, Z., Paulino, G. H., and Celes, W., 2007, “Extrinsic Cohesive Modelling of Dynamic Fracture and Microbranching Instability in Brittle Materials,” Int. J. Numer. Methods Eng., 72(8), pp. 893–923. [CrossRef]
Rabczuk, T., Song, J.-H., and Belytschko, T., 2009, “Simulations of Instability in Dynamic Fracture by the Cracking Particles Method,” Eng. Fract. Mech., 76(6), pp. 730–741. [CrossRef]
Pandolfi, A., Krysl, P., and Ortiz, M., 1999, “Finite Element Simulation of Ring Expansion and Fragmentation: The Capturing of Length and Time Scales Through Cohesive Models of Fracture,” Int. J. Fract., 95(1–4), pp. 279–297. [CrossRef]
Zhou, F., Molinari, J.-F., and Ramesh, K. T., 2005, “A Cohesive Model Based Fragmentation Analysis: Effects of Strain Rate and Initial Defects Distribution,” Int. J. Solids Struct., 42(18–19), pp. 5181–5207. [CrossRef]
Molinari, J. F., Gazonas, G., Raghupathy, R., Rusinek, A., and Zhou, F., 2007, “The Cohesive Element Approach to Dynamic Fragmentation: The Question of Energy Convergence,” Int. J. Numer. Methods Eng., 69(3), pp. 484–503. [CrossRef]
Reeder, J. R., and Crews, J. H., Jr., 1990, “Mixed-Mode Bending Method for Delamination Testing,” AIAA J., 28(7), pp. 1270–1276. [CrossRef]
Benzeggagh, M. L., and Kenane, M., 1996, “Measurement of Mixed-Mode Delamination Fracture Toughness of Unidirectional Glass/Epoxy Composites With Mixed-Mode Bending Apparatus,” Compos. Sci. Technol., 56(4), pp. 439–449. [CrossRef]
Banks-Sills, L., Travitzky, N., Ashkenazi, D., and Eliasi, R., 1999, “A Methodology for Measuring Interface Fracture Properties of Composite Materials,” Int. J. Fract., 99(3), pp. 143–160. [CrossRef]
Zhu, Y., Liechti, K. M., and Ravi-Chandar, K., 2009, “Direct Extraction of Rate-Dependent Traction Separation Laws for Polyurea/Steel Interfaces,” Int. J. Solids Struct., 46(1), pp. 31–51. [CrossRef]
Xu, X. P., and Needleman, A., 1994, “Numerical Simulations of Fast Crack Growth in Brittle Solids,” J. Mech. Phys. Solids, 42(9), pp. 1397–1434. [CrossRef]
Camacho, G. T., and Ortiz, M., 1996, “Computational Modelling of Impact Damage in Brittle Materials,” Int. J. Solids Struct., 33(20–22), pp. 2899–2938. [CrossRef]
Ortiz, M., and Pandolfi, A., 1999, “Finite-Deformation Irreversible Cohesive Elements for Three Dimensional Crack-Propagation Analysis,” Int. J. Numer. Methods Eng., 44(9), pp. 1267–1282. [CrossRef]
Celes, W., Paulino, G. H., and Espinha, R., 2005, “A Compact Adjacency-Based Topological Data Structure for Finite Element Mesh Representation,” Int. J. Numer. Methods Eng., 64(11), pp. 1529–1556. [CrossRef]
Celes, W., Paulino, G. H., and Espinha, R., 2005, “Efficient Handling of Implicit Entities in Reduced Mesh Representations,” J. Comput. Info. Sci. Eng., 5(4), pp. 348–359. [CrossRef]
Mota, A., Knap, J., and Ortiz, M., 2008, “Fracture and Fragmentation of Simplicial Finite Element Meshes Using Graphs,” Int. J. Numer. Methods Eng., 73(11), pp. 1547–1570. [CrossRef]
Espinha, R., Celes, W., Rodriguez, N., and Paulino, G. H., 2009, “ParTopS: Compact Topological Framework for Parallel Fragmentation Simulations,” Eng. Comput., 25(4), pp. 345–365. [CrossRef]
Wells, G. N., and Sluys, L. J., 2001, “A New Method for Modelling Cohesive Cracks Using Finite Elements,” Int. J. Numer. Methods Eng., 50(12), pp. 2667–2682. [CrossRef]
Moes, N., and Belytschko, T., 2002, “Extended Finite Element Method for Cohesive Crack Growth,” Eng. Fract. Mech., 69(7), pp. 813–833. [CrossRef]
Remmers, J. J. C., de Borst, R., and Needleman, A., 2008, “The Simulation of Dynamic Crack Propagation Using the Cohesive Segments Method,” J. Mech. Phys. Solids, 56(1), pp. 70–92. [CrossRef]
Song, J.-H., and Belytschko, T., 2009, “Cracking Node Method for Dynamic Fracture With Finite Elements,” Int. J. Numer. Methods Eng., 77(3), pp. 360–385. [CrossRef]
Paulino, G. H., Park, K., Celes, W., and Espinha, R., 2010, “Adaptive Dynamic Cohesive Fracture Simulation Using Edge-Swap and Nodal Perturbation Operators,” Int. J. Numer. Methods Eng., 84(11), pp. 1303–1343. [CrossRef]
Simo, J. C., Oliver, J., and Armero, F., 1993, “An Analysis of Strong Discontinuities Induced by Strain Softening in Rate-Independent Inelastic Solids,” Comput. Mech., 12(5), pp. 277–296. [CrossRef]
Oliver, J., Huespe, A. E., Pulido, M. D. G., and Chaves, E., 2002, “From Continuum Mechanics to Fracture Mechanics: The Strong Discontinuity Approach,” Eng. Fract. Mech., 69(2), pp. 113–136. [CrossRef]
Linder, C., and Armero, F., 2007, “Finite Elements With Embedded Strong Discontinuities for the Modeling of Failure in Solids,” Int. J. Numer. Methods Eng., 72(12), pp. 1391–1433. [CrossRef]
Carol, I., Prat, P. C., and Lopez, C. M., 1997, “Normal/Shear Cracking Model: Application to Discrete Crack Analysis,” ASCE J. Eng. Mech., 123(8), pp. 765–773. [CrossRef]
Willam, K., Rhee, I., and Shing, B., 2004, “Interface Damage Model for Thermomechanical Degradation of Heterogeneous Materials,” Comput. Methods Appl. Mech. Eng., 193(30–32), pp. 3327–3350. [CrossRef]
Caballero, A., Willam, K. J., and Carol, I., 2008, “Consistent Tangent Formulation for 3D Interface Modeling of Cracking/Fracture in Quasi-Brittle Materials,” Comput. Methods Appl. Mech. Eng., 197(33–40), pp. 2804–2822. [CrossRef]
Segura, J. M., and Carol, I., 2010, “Numerical Modelling of Pressurized Fracture Evolution in Concrete Using Zero-Thickness Interface Elements,” Eng. Fract. Mech., 77(9), pp. 1386–1399. [CrossRef]
Gao, H., and Klein, P., 1998, “Numerical Simulation of Crack Growth in an Isotropic Solid With Randomized Internal Cohesive Bonds,” J. Meh. Phys. Solids, 46(2), pp. 187–218. [CrossRef]
Gao, H., and Ji, B., 2003, “Modeling Fracture in Nanomaterials via a Virtual Internal Bond Method,” Eng. Fract. Mech., 70(14), pp. 1777–1791. [CrossRef]
Park, K., Paulino, G. H., and Roesler, J. R., 2008, “Virtual Internal Pair-Bond (VIPB) Model for Quasi-Brittle Materials,” ASCE J. Eng. Mech., 134(10), pp. 856–866. [CrossRef]
Bazant, Z. P., 1984, “Microplane Model for Strain-Controlled Inelastic Behavior,” Mechanics of Engineering Materials, C. S.Desai and R. H.Gallagher, eds., Prentice-Hall, Englewood Cliffs, NJ, pp. 45–59.
Bazant, Z. P., and Oh, B. H., 1985, “Microplane Model for Progressive Fracture of Concrete and Rock,” ASCE J. Eng. Mech., 111(4), pp. 559–582. [CrossRef]
Bazant, Z. P., and Caner, F. C., 2005, “Microplane Model M5 With Kinematic and Static Constraints for Concrete Fracture and Anelasticity I: Theory,” ASCE J. Eng. Mech., 131(1), pp. 31–40. [CrossRef]
Silling, S. A., 2000, “Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces,” J. Mech. Phys. Solids, 48(1), pp. 175–209. [CrossRef]
Macek, R. W., and Silling, S. A., 2007, “Peridynamics via Finite Element Analysis,” Finite Elem. Anal. Des., 43(15), pp. 1169–1178. [CrossRef]
Kilic, B., Agwai, A., and Madenci, E., 2009, “Peridynamic Theory for Progressive Damage Prediction in Center-Cracked Composite Laminates,” Compos. Struct., 90(2), pp. 141–151. [CrossRef]
Sorensen, B. F., and Jacobsen, T. K., 2003, “Determination of Cohesive Laws by the J Integral Approach,” Eng. Fract. Mech., 70(14), pp. 1841–1858. [CrossRef]
Slowik, V., Villmann, B., Bretschneider, N., and Villmann, T., 2006, “Computational Aspects of Inverse Analyses for Determining Softening Curves of Concrete,” Comput. Methods Appl. Mech. Eng., 195(52), pp. 7223–7236. [CrossRef]
de Oliveira e Sousa, J. L. A., and Gettu, R., 2006, “Determining the Tensile Stress-Crack Opening Curve of Concrete by Inverse Analysis,” ASCE J. Eng. Mech., 132(2), pp. 141–148. [CrossRef]
Kwon, S. H., Zhao, Z., and Shah, S. P., 2008, “Effect of Specimen Size on Fracture Energy and Softening Curve of Concrete: Part II. Inverse Analysis and Softening Curve,” Cem. Concr. Res., 38(8–9), pp. 1061–1069. [CrossRef]
Abanto-Bueno, J., and Lambros, J., 2005, “Experimental Determination of Cohesive Failure Properties of a Photodegradable Copolymer,” Exp. Mech., 45(2), pp. 144–152. [CrossRef]
Tan, H., Liu, C., Huang, Y., and Geubelle, P. H., 2005, “The Cohesive Law for the Particle/Matrix Interfaces in High Explosives,” J. Mech. Phys. Solids, 53(8), pp. 1892–1917. [CrossRef]
Shen, B., and Paulino, G. H., 2011, “Direct Extraction of Cohesive Fracture Properties From Digital Image Correlation: A Hybrid Inverse Technique,” Exp. Mech., 51(2), pp. 143–163. [CrossRef]
Kulkarni, M. G., Geubelle, P. H., and Matou, K., 2009, “Multi-Scale Modeling of Heterogeneous Adhesives: Effect of Particle Decohesion,” Mech. Mater., 41(5), pp. 573–583. [CrossRef]
Scheider, I., 2009, “Derivation of Separation Laws for Cohesive Models in the Course of Ductile Fracture,” Eng. Fract. Mech., 76(10), pp. 1450–1459. [CrossRef]
Kulkarni, M. G., Matous, K., and Geubelle, P. H., 2010, “Coupled Multi-Scale Cohesive Modeling of Failure in Heterogeneous Adhesives,” Int. J. Numer. Methods Eng., 84(8), pp. 916–946. [CrossRef]
Zeng, X., and Li, S., 2010, “A Multiscale Cohesive Zone Model and Simulations of Fractures,” Comput. Methods Appl. Mech., 199(9–12), pp. 547–556. [CrossRef]
Tvergaard, V., 1990, “Effect of Fibre Debonding in a Whisker-Reinforced Metal,” Mater. Sci. Eng., A125(2), pp. 203–213. [CrossRef]
Tvergaard, V., and Hutchinson, J. W., 1993, “The Influence of Plasticity on Mixed Mode Interface Toughness,” J. Mech. Phys. Solids, 41(6), pp. 1119–1135. [CrossRef]
Scheider, I., and Brocks, W., 2003, “Simulation of Cup-Cone Fracture Using the Cohesive Model,” Eng. Fract. Mech., 70(14), pp. 1943–1961. [CrossRef]
Geubelle, P. H., and Baylor, J. S., 1998, “Impact-Induced Delamination of Composites: A 2D Simulation,” Compos. Part B: Eng., 29(5), pp. 589–602. [CrossRef]
Espinosa, H. D., and Zavattieri, P. D., 2003, “A Grain Level Model for the Study of Failure Initiation and Evolution in Polycrystalline Brittle Materials. Part I: Theory and Numerical Implementation,” Mech. Mater., 35(3–6), pp. 333–364. [CrossRef]
Tvergaard, V., and Hutchinson, J. W., 1992, “The Relation Between Crack Growth Resistance and Fracture Process Parameters in Elastic-Plastic Solids,” J. Mech. Phys. Solids, 40(6), pp. 1377–1397. [CrossRef]
Wei, Y., and Hutchinson, J. W., 1997, “Steady-State Crack Growth and Work of Fracture for Solids Characterized by Strain Gradient Plasticity,” J. Mech. Phys. Solids, 45(8), pp. 1253–1273. [CrossRef]
Tvergaard, V., and Hutchinson, J. W., 2008, “Mode III Effects on Interface Delamination,” J. Mech. Phys. Solids, 56(1), pp. 215–229. [CrossRef]
Rose, J. H., Ferrante, J., and Smith, J. R., 1981, “Universal Binding Energy Curves for Metals and Bimetallic Interfaces,” Phys. Rev. Lett., 47(9), pp. 675–678. [CrossRef]
Song, S. H., Paulino, G. H., and Buttlar, W. G., 2006, “Simulation of Crack Propagation in Asphalt Concrete Using an Intrinsic Cohesive Zone Model,” ASCE J. Eng. Mech., 132(11), pp. 1215–1223. [CrossRef]
Song, S. H., Paulino, G. H., and Buttlar, W. G., 2006, “A Bilinear Cohesive Zone Model Tailored for Fracture of Asphalt Concrete Considering Viscoelastic Bulk Material,” Eng. Fract. Mech., 73(18), pp. 2829–2848. [CrossRef]
Aragao, F. T. S., Kim, Y.-R., Lee, J., and Allen, D. H., 2011, “Micromechanical Model for Heterogeneous Asphalt Concrete Mixtures Subjected to Fracture Failure,” ASCE J. Mater. Civil Eng., 23(1), pp. 30–38. [CrossRef]
Foulk, J. W., Allen, D. H., and Helms, K. L. E., 2000, “Formulation of a Three-Dimensional Cohesive Zone Model for Application to a Finite Element Algorithm,” Comput. Methods Appl. Mech. Eng., 183(1), pp. 51–66. [CrossRef]
Nutt, S. R., and Needleman, A., 1987, “Void Nucleation at Fiber Ends in Al-SiC Composites,” Scr. Metal., 21(5), pp. 705–710. [CrossRef]
McHugh, P. E., Varias, A. G., Asaro, R. J., and Shih, C. F., 1989, “Computational Modeling of Microstructures,” FGCS, Future Gener. Comput. Syst., 5(2–3), pp. 295–318. [CrossRef]
Shabrov, M. N., and Needleman, A., 2002, “An Analysis of Inclusion Morphology Effects on Void Nucleation,” Model. Simul. Mater. Sci. Eng., 10(2), pp. 163–183. [CrossRef]
Rice, J. R., and Wang, J. S., 1989, “Embrittlement of Interfaces by Solute Segregation,” Mater. Sci. Eng., A107(1–2), pp. 23–40. [CrossRef]
Rice, J. R., 1992, “Dislocation Nucleation From a Crack Tip: An Analysis Based on the Peierls Concept,” J. Mech. Phys. Solids, 40(2), pp. 239–271. [CrossRef]
Beltz, G. E., and Rice, J. R., 1992, “Dislocation Nucleation at Metal-Ceramic Interfaces,” Acta Metall. Mater., 40(Suppl. 1), pp. S321–S331. [CrossRef]
Peierls, R., 1940, “The Size of a Dislocation,” Proc. Phys. Soc., 52(1), pp. 34–37. [CrossRef]
Sun, Y., Beltz, G. E., and Rice, J. R., 1993, “Estimates From Atomic Models of Tension-Shear Coupling in Dislocation Nucleation From a Crack Tip,” Mater. Sci. Eng., A170(1–2), pp. 67–85. [CrossRef]
Falk, M. L., Needleman, A., and Rice, J. R., 2001, “A Critical Evaluation of Cohesive Zone Models of Dynamic Fracture,” J. Phys. IV France, 11, pp. 43–50. [CrossRef]
Finot, M., Shen, Y. L., Needleman, A., and Suresh, S., 1994, “Micromechanical Modeling of Reinforcement Fracture in Particle-Reinforced Metal-Matrix Composites,” Metall. Mater. Trans., A, 25A(11), pp. 2403–2420. [CrossRef]
Needleman, A., 1997, “Numerical Modeling of Crack Growth Under Dynamic Loading Conditions,” Comput. Mech., 19(6), pp. 463–469. [CrossRef]
Needleman, A., and Rosakis, A. J., 1999, “Effect of Bond Strength and Loading Rate on the Conditions Governing the Attainment of Intersonic Crack Growth Along Interfaces,” J. Mech. Phys. Solids, 47(12), pp. 2411–2449. [CrossRef]
Zhai, J., and Zhou, M., 2000, “Finite Element Analysis of Micromechanical Failure Modes in a Heterogeneous Ceramic Material System,” Int. J. Fract., 101(1), pp. 161–180. [CrossRef]
van den Bosch, M. J., Schreurs, P. J. G., and Geers, M. G. D., 2006, “An Improved Description of the Exponential Xu and Needleman Cohesive Zone Law for Mixed-Mode Decohesion,” Eng. Fract. Mech., 73(9), pp. 1220–1234. [CrossRef]
Klein, P. A., Foulk, J. W., Chen, E. P., Wimmer, S. A., and Gao, H. J., 2001, “Physics-Based Modeling of Brittle Fracture: Cohesive Formulations and the Application of Meshfree Methods,” Theor. Appl. Fract. Mech., 37(1–3), pp. 99–166. [CrossRef]
Volokh, K. Y., 2004, “Comparison Between Cohesive Zone Models,” Commun. Numer. Methods Eng., 20(11), pp. 845–856. [CrossRef]
Alfano, G., 2006, “On the Influence of the Shape of the Interface Law on the Application of Cohesive-Zone Models,” Compos. Sci. Technol., 66(6), pp. 723–730. [CrossRef]
Song, S. H., Paulino, G. H., and Buttlar, W. G., 2006, “Influence of the Cohesive Zone Model Shape Parameter on Asphalt Concrete Fracture Behavior,” Multiscale and Functionally Graded Material 2006 (M&FGM 2006), G. H.Paulino, M.-J.Pindera, R. H.Dodds, Jr., F. A.Rochinha, E.Dave, and L.Chen, eds., AIP Conference Proceedings, Maryland, pp. 730–735. [CrossRef]
Park, K., Paulino, G. H., Celes, W., and Espinha, R., 2012, “Adaptive Mesh Refinement and Coarsening for Cohesive Zone Modeling of Dynamic Fracture,” Int. J. Numer. Methods Eng., 92(1), pp. 1–35. [CrossRef]
Park, K., and Paulino, G. H., 2012, “Computational Implementation of the PPR Potential-Based Cohesive Model in ABAQUS: Educational Perspective,” Eng. Fract. Mech., 93, pp. 239–262. [CrossRef]
Park, K., 2009, “Potential-Based Fracture Mechanics Using Cohesive Zone and Virtual Internal Bond Modeling,” Ph.D. thesis, University of Illinois at Urbana-Champaign, Urbana, IL.


Grahic Jump Location
Fig. 1

Schematics of the cohesive zone model

Grahic Jump Location
Fig. 2

Effective traction-separation relationships: (a) cubic polynomial, (b) trapezoidal, (c) smoothed trapezoidal, (d) exponential, (e) linear softening, and (f) bilinear softening

Grahic Jump Location
Fig. 3

Cohesive fracture separations along the local coordinate system (a) two-dimensions (Δ1, Δ2) and (b) three-dimensions (Δ1, Δ2, Δ3)

Grahic Jump Location
Fig. 4

Effective displacement-based model with a linear softening: (a) normal cohesive traction, and (b) its derivative with respect to the normal separation (Δn) for T¯=σmax(1-Δ¯) where φn=100 N/m and σmax=10 MPa

Grahic Jump Location
Fig. 5

Needleman [4] potential (Ψ) and its gradients (Tn, Tt) with respect to displacement separations (Δn, Δt); φn=100 N/m, σmax=30 MPa, and αs = 10. The gradients refer to a cubic-linear model.

Grahic Jump Location
Fig. 6

Freed and Banks-Sills [5] potential (Ψ) and its gradients (Tn, Tt) with respect to displacement separations (Δn, Δt); φn=100 N/m, and σmax=30 MPa. The gradients refer to a revisited cubic-linear model.

Grahic Jump Location
Fig. 7

Needleman [7] exponential-periodic potential and its gradients; φn=100 N/m, and σmax=30 MPa

Grahic Jump Location
Fig. 8

Beltz and Rice [8] generalized exponential-periodic potential and its gradients; φn=2γs=100 N/m, φt=γus=200 N/m, σmax=30 MPa, τmax=40 MPa, and r=0

Grahic Jump Location
Fig. 9

Xu and Needleman [9] exponential-exponential potential and its gradients; φn=100 N/m, φt=200 N/m, σmax=30 MPa, τmax=40 MPa, and r=0

Grahic Jump Location
Fig. 10

Xu and Needleman [9] exponential-exponential potential and its gradients; φn=200 N/m, φt=100 N/m, σmax=30 MPa, τmax=40 MPa, and r=0

Grahic Jump Location
Fig. 11

Xu and Needleman [9] exponential-exponential potential and its gradients; φn=100 N/m, φt=100 N/m, σmax=30 MPa, and τmax=40 MPa

Grahic Jump Location
Fig. 12

Unified mixed-mode potential (PPR) [3] and its gradients for the intrinsic cohesive zone model with φn=100 N/m, φt=200 N/m, σmax=40 MPa, τmax=30 MPa, α=5, β=1.3, λn=0.1, and λt=0.2

Grahic Jump Location
Fig. 13

Description of each cohesive interaction (Tn, Tt) region defined by the final crack opening widths (δn, δt) and the conjugate final crack opening widths (δ¯n, δ¯t); (a) Tn versus (δn,δ¯t) space; (b) Tt versus (δ¯n,δt) space

Grahic Jump Location
Fig. 14

The PPR potential [3] and its gradients for the extrinsic cohesive zone model with φn=100 N/m, φt=200 N/m, σmax=40 MPa, τmax=30 MPa, α=5, and β=1.3

Grahic Jump Location
Fig. 15

Two arbitrary separation paths for the material debonding process; (a) nonproportional path 1; (b) nonproportional path 2

Grahic Jump Location
Fig. 16

Variation of the work-of-separation considering the PPR potential [3] (φn = 100 N/m, φt = 200 N/m); (a) nonproportional path 1; (b) nonproportional path 2



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In