# 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:

show all references

##### References:
 [1] Rejeb Hadiji, Ken Shirakawa. 3D-2D asymptotic observation for minimization problems associated with degenerate energy-coefficients. Conference Publications, 2011, 2011 (Special) : 624-633. doi: 10.3934/proc.2011.2011.624 [2] Roman VodiČka, Vladislav MantiČ. An energy based formulation of a quasi-static interface damage model with a multilinear cohesive law. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1539-1561. 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) : 767-799. 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) : 631-642. 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) : 3195-3212. doi: 10.3934/dcdsb.2018281 [6] Tan Bui-Thanh, 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) : 1139-1155. doi: 10.3934/ipi.2013.7.1139 [7] Sari Lasanen. Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns. Inverse Problems & Imaging, 2012, 6 (2) : 267-287. doi: 10.3934/ipi.2012.6.267 [8] Sari Lasanen. Non-Gaussian statistical inverse problems. Part I: Posterior distributions. Inverse Problems & Imaging, 2012, 6 (2) : 215-266. doi: 10.3934/ipi.2012.6.215 [9] Tianxiao Wang. Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I. Mathematical Control & Related Fields, 2019, 9 (2) : 385-409. 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) : 903-920. 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) : 1155-1188. 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) : 29-46. 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) : 2093-2124. 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) : 1-17. doi: 10.3934/dcds.2019242 [15] 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) : 79-89. doi: 10.3934/naco.2015.5.79 [16] Luchuan Ceng, Qamrul Hasan Ansari, Jen-Chih Yao. Extragradient-projection method for solving constrained convex minimization problems. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 341-359. doi: 10.3934/naco.2011.1.341 [17] Yu-Ning 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) : 1507-1519. doi: 10.3934/jimo.2016.12.1507 [18] David Yang Gao. Solutions and optimality criteria to box constrained nonconvex minimization problems. Journal of Industrial & Management Optimization, 2007, 3 (2) : 293-304. doi: 10.3934/jimo.2007.3.293 [19] 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) : 1-8. doi: 10.3934/jimo.2019022 [20] Bernard Bonnard, Jean-Baptiste Caillau, Olivier Cots. Energy minimization in two-level dissipative quantum control: Th e integrable case. Conference Publications, 2011, 2011 (Special) : 198-208. doi: 10.3934/proc.2011.2011.198

2018 Impact Factor: 0.925