Implicit–Explicit Integration of Gradient-Enhanced Damage Models
Publication: Journal of Engineering Mechanics
Volume 145, Issue 7
Abstract
Quasi-brittle materials exhibit strain softening. Their modeling requires regularized constitutive formulations to avoid instabilities on the material level. A commonly used model is the implicit gradient-enhanced damage model. For complex geometries, it still shows structural instabilities when integrated with classical backward Euler schemes. An alternative is the implicit–explicit (IMPL-EX) integration scheme. It consists of the extrapolation of internal variables followed by an implicit calculation of the solution fields. The solution procedure for the nonlinear gradient-enhanced damage model is thus transformed into a sequence of problems that are algorithmically linear in every time step. Therefore, they require one single Newton–Raphson iteration per time step to converge. This provides both additional robustness and computational acceleration. The introduced extrapolation error is controlled by adaptive time-stepping schemes. This paper introduced and assessed two novel classes of error control schemes that provide further performance improvements. In a three-dimensional compression test for a mesoscale model of concrete, the presented scheme was about 40 times faster than an adaptive backward Euler time integration.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The research was supported by the Federal Institute for Materials Research and Testing, Berlin and by the German Research Foundation (DFG) under project Un224/7-1. Additionally, the research leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. 320815 [ERC Advanced Grant Project “Advanced tools for computational design of engineering materials” (COMP-DES-MAT)].
References
Amestoy, P. R., I. S. Duff, J. Koster, and J.-Y. L’Excellent. 2001. “A fully asynchronous multifrontal solver using distributed dynamic scheduling.” SIAM J. Matrix Anal. Appl. 23 (1): 15–41. https://doi.org/10.1137/S0895479899358194.
Amestoy, P. R., A. Guermouche, J.-Y. L’Excellent, and S. Pralet. 2006. “Hybrid scheduling for the parallel solution of linear systems.” Parallel Comput. 32 (2): 136–156. https://doi.org/10.1016/j.parco.2005.07.004.
Bazand, Z. P., and G. Pijaudier-Cabot. 1988. “Nonlocal continuum damage, localization instability and convergence.” J. Appl. Mech. 55 (2): 287–293. https://doi.org/10.1115/1.3173674.
Bažant, Z. P., and T. B. Belytschko. 1985. “Wave propagation in a strain-softening bar: Exact solution.” J. Eng. Mech. 111 (3): 381–389. https://doi.org/10.1061/(ASCE)0733-9399(1985)111:3(381).
Bažant, Z. P., T. B. Belytschko, and T.-P. Chang. 1984. “Continuum theory for strain-softening.” J. Eng. Mech. 110 (12): 1666–1692. https://doi.org/10.1061/(ASCE)0733-9399(1984)110:12(1666).
Bažant, Z. P., and M. Jirásek. 2002. “Nonlocal integral formulations of plasticity and damage: Survey of progress.” J. Eng. Mech. 128 (11): 1119–1149. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:11(1119).
Blanco, S., J. Oliver, and A. Huespe. 2007. “Contribuciones a la simulación numérica del fallo material en medios tridimensionales mediante la metodologa de discontinuidades fuertes de continuo.” Ph.D. thesis, Dept. of Strength of Materials and Structural Engineering, Polytechnic Univ. of Catalonia.
Carol, I., and Z. P. Bazand. 1997. “Damage and plasticity in microplane theory.” Int. J. Solids Struct. 34 (29): 3807–3835. https://doi.org/10.1016/S0020-7683(96)00238-7.
Carol, I., P. C. Prat, and C. M. López. 1997. “Normal/shear cracking model: Application to discrete crack analysis.” J. Eng. Mech. 123 (8): 765–773. https://doi.org/10.1061/(ASCE)0733-9399(1997)123:8(765).
Cazes, F., G. Meschke, and M.-M. Zhou. 2016. “Strong discontinuity approaches: An algorithm for robust performance and comparative assessment of accuracy.” Int. J. Solids Struct. 96: 355–379. https://doi.org/10.1016/j.ijsolstr.2016.05.016.
Desmorat, R. 2016. “Anisotropic damage modeling of concrete materials.” Int. J. Damage Mech. 25 (6): 818–852. https://doi.org/10.1177/1056789515606509.
De Vree, J., W. Brekelmans, and M. Van Gils. 1995. “Comparison of nonlocal approaches in continuum damage mechanics.” Comput. Struct. 55 (4): 581–588. https://doi.org/10.1016/0045-7949(94)00501-S.
DIN (Deutsches Institut für Normung) 2008. Concrete, reinforced and prestressed concrete structures—Part 2: Concrete—Specification, performance, production and conformity—Application rules for DIN EN 206. DIN 1045-2. Berlin: German Institute for Standardization.
Graça-e Costa, R., J. Alfaiate, D. da Costa, and L. Sluys. 2012. “A non-iterative approach for the modelling of quasi-brittle materials.” Int. J. Fract. 178 (1): 281–298. https://doi.org/10.1007/s10704-012-9768-1.
Jirásek, M. 2007. “Mathematical analysis of strain localization.” Revue européenne de génie civ. 11 (7–8): 977–991. https://doi.org/10.1080/17747120.2007.9692973.
Jirásek, M., and T. Zimmermann. 1998. “Rotating crack model with transition to scalar damage.” J. Eng. Mech. 124 (3): 277–284. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:3(277).
Mazars, J., and G. Pijaudier-Cabot. 1989. “Continuum damage theory—Application to concrete.” J. Eng. Mech. 115 (2): 345–365. https://doi.org/10.1061/(ASCE)0733-9399(1989)115:2(345).
Oliver, J. 1989. “A consistent characteristic length for smeared cracking models.” Int. J. Numer. Methods Eng. 28 (2): 461–474. https://doi.org/10.1002/nme.1620280214.
Oliver, J., M. Cervera, S. Oller, and J. Lubliner. 1990. “Isotropic damage models and smeared crack analysis of concrete.” In Vol. 2 of Proc., 2nd Int. Conf. on Computer Aided Analysis and Design of Concrete Structures, 945–958. Swansea, UK: Pineridge Press.
Oliver, J., A. Huespe, S. Blanco, and D. Linero. 2006. “Stability and robustness issues in numerical modeling of material failure with the strong discontinuity approach.” Comput. Methods Appl. Mech. Eng. 195 (52): 7093–7114. https://doi.org/10.1016/j.cma.2005.04.018.
Oliver, J., A. Huespe, and J. Cante. 2008. “An implicit/explicit integration scheme to increase computability of non-linear material and contact/friction problems.” Comput. Methods Appl. Mech. Eng. 197 (21): 1865–1889. https://doi.org/10.1016/j.cma.2007.11.027.
Oliver, J., A. Huespe, M. Pulido, and E. Chaves. 2002. “From continuum mechanics to fracture mechanics: The strong discontinuity approach.” Eng. Fract. Mech. 69 (2): 113–136. https://doi.org/10.1016/S0013-7944(01)00060-1.
Peerlings, R., R. De Borst, W. Brekelmans, and M. Geers. 1998. “Gradient-enhanced damage modelling of concrete fracture.” Mech. Cohesive-Frict. Mater. 3 (4): 323–342. https://doi.org/10.1002/(SICI)1099-1484(1998100)3:4%3C323::AID-CFM51%3E3.0.CO;2-Z.
Peerlings, R., M. Geers, R. De Borst, and W. Brekelmans. 2001. “A critical comparison of nonlocal and gradient-enhanced softening continua.” Int. J. Solids Struct. 38 (44): 7723–7746. https://doi.org/10.1016/S0020-7683(01)00087-7.
Peerlings, R., T. Massart, and M. Geers. 2004. “A thermodynamically motivated implicit gradient damage framework and its application to brick masonry cracking.” Comput. Methods Appl. Mech. Eng. 193 (30–32): 3403–3417. https://doi.org/10.1016/j.cma.2003.10.021.
Peerlings, R. H. J., R. De Borst, W. A. M. Brekelmans, and J. H. P. De Vree. 1996. “Gradient enhanced damage for quasi-brittle materials.” Int. J. Numer. Methods Eng. 39 (19): 3391–3403. https://doi.org/10.1002/(SICI)1097-0207(19961015)39:19%3C3391::AID-NME7%3E3.0.CO;2-D.
Pham, K., H. Amor, J.-J. Marigo, and C. Maurini. 2011. “Gradient damage models and their use to approximate brittle fracture.” Int. J. Damage Mech. 20 (4): 618–652. https://doi.org/10.1177/1056789510386852.
Pijaudier-Cabot, G., and Z. P. Bažant. 1987. “Nonlocal damage theory.” J. Eng. Mech. 113 (10): 1512–1533. https://doi.org/10.1061/(ASCE)0733-9399(1987)113:10(1512).
Poh, L. H., and G. Sun. 2017. “Localizing gradient damage model with decreasing interactions.” Int. J. Numer. Methods Eng. 110 (6): 503–522. https://doi.org/10.1002/nme.5364.
Rots, J., P. Nauta, G. Kuster, and J. Blaauwendraad. 1985. Smeared crack approach and fracture localization in concrete. Delft, Netherlands: Delft Univ. of Technology.
Rots, J. G., B. Belletti, and S. Invernizzi. 2008. “Robust modeling of RC structures with an ‘event-by-event’ strategy.” Eng. Fract. Mech. 75 (3): 590–614. https://doi.org/10.1016/j.engfracmech.2007.03.027.
Simone, A., H. Askes, R. Peerlings, and L. Sluys. 2003. “Interpolation requirements for implicit gradient-enhanced continuum damage models.” Int. J. Numer. Methods Biomed. Eng. 19 (7): 563–572. https://doi.org/10.1002/cnm.597.
Titscher, T., and J. F. Unger. 2015. “Application of molecular dynamics simulations for the generation of dense concrete mesoscale geometries.” Comput. Struct. 158: 274–284. https://doi.org/10.1016/j.compstruc.2015.06.008.
Triantafyllidis, N., and E. C. Aifantis. 1986. “A gradient approach to localization of deformation. I. hyperelastic materials.” Journal of Elast. 16 (3): 225–237. https://doi.org/10.1007/BF00040814.
Unger, J. F., and S. Eckardt. 2011. “Multiscale modeling of concrete.” Arch. Comput. Methods Eng. 18 (3): 341–393. https://doi.org/10.1007/s11831-011-9063-8.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 145 • Issue 7 • July 2019
Copyright
©2019 American Society of Civil Engineers.
History
Received: May 10, 2018
Accepted: Oct 30, 2018
Published online: Apr 24, 2019
Published in print: Jul 1, 2019
Discussion open until: Sep 24, 2019
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.