Technical Papers
Aug 12, 2023

A Gradient-Enhanced Damage Model with a New Equivalent Strain Based on the Menétrey–Willam Function

Publication: Journal of Engineering Mechanics
Volume 149, Issue 10

Abstract

A new gradient-enhanced damage model is developed based on a new equivalent strain. This new equivalent strain is derived using the Menétrey–Willam yield function. The proposed model is implemented in a three-dimensional framework via 10-node tetrahedral element. Excessive spread of damage observed in the gradient-enhanced model is prevented by adopting a novel variable and anistropic nonlocal length scale. Performance of the model is investigated by comparing a wide variety of benchmark examples used in the literature. Obtained results show that proposed model is able to characterize the complex stress states, such as mixed-mode fracturing and nonplanar cracking as well as tensile-dominated fracturing.

Get full access to this article

View all available purchase options and get full access to this article.

Data Availability Statement

All data and models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

References

Ayhan, B., E. Lale, and N. Çelik. 2022. “Size effect analysis of concrete beams under bending using crack-band approach.” Politeknik Dergisi 25 (2): 605–613. https://doi.org/10.2339/politeknik.762634.
Bazant, Z. P., F. C. Caner, I. Carol, M. D. Adley, and S. A. Akers. 2000. “Microplane model m4 for concrete. I: Formulation with work-conjugate deviatoric stress.” J. Eng. Mech. 126 (9): 944–953. https://doi.org/10.1061/(ASCE)0733-9399(2000)126:9(944).
Bažant, Z. P. 1994. “Nonlocal damage theory based on micromechanics of crack interactions.” J. Eng. Mech. 120 (3): 593–617. https://doi.org/10.1061/(ASCE)0733-9399(1994)120:3(593).
Bažant, Z. P., and G. Di Luzio. 2004. “Nonlocal microplane model with strain-softening yield limits.” Int. J. Solids Struct. 41 (24): 7209–7240. https://doi.org/10.1016/j.ijsolstr.2004.05.065.
Bažant, Z. P., and B. H. Oh. 1983. “Crack band theory for fracture of concrete.” Mat. Constr. 16 (3): 155–177. https://doi.org/10.1007/BF02486267.
Benedetti, L., M. Cervera, and M. Chiumenti. 2017. “3D numerical modelling of twisting cracks under bending and torsion of skew notched beams.” Eng. Fract. Mech. 176 (May): 235–256. https://doi.org/10.1016/j.engfracmech.2017.03.025.
Bolander, J., Jr., and H. Hikosaka. 1995. “Simulation of fracture in cement-based composites.” Cem. Concr. Compos. 17 (2): 135–145. https://doi.org/10.1016/0958-9465(94)00028-W.
Bongers, G. 2011. “A stress-based gradient-enhanced damage model.” M.S. thesis, Dept. of Civil Engineering and Geoscience, Delft Univ. of Technology.
Brokenshire, D. 1996. “A study of torsional fracture tests.” Ph.D. thesis, Dept. of Civil Engineering, Cardiff Univ.
Bui, Q., and F. Lani. 2011. “Non-local implicit gradient damage models with low-order finite elements.” Int. J. Numer. Methods Biomed. Eng. 27 (6): 962–976. https://doi.org/10.1002/cnm.1347.
Caner, F. C., and Z. P. Bažant. 2013. “Microplane model M7 for plain concrete. I: Formulation.” J. Eng. Mech. 139 (12): 1714–1723. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000570.
Cedolin, L., and G. Cusatis. 2008. “Identification of concrete fracture parameters through size effect experiments.” Cem. Concr. Compos. 30 (9): 788–797. https://doi.org/10.1016/j.cemconcomp.2008.05.007.
Chen, Y., M. E. Mobasher, and H. Waisman. 2022a. “Dynamic soil consolidation model using a nonlocal continuum poroelastic damage approach.” Int. J. Numer. Anal. Methods Geomech. 46 (3): 486–528. https://doi.org/10.1002/nag.3309.
Chen, Y., M. E. Mobasher, T. You, and H. Waisman. 2022b. “Non-local continuum damage model for poro-viscoelastic porous media.” Int. J. Rock Mech. Min. Sci. 159 (Nov): 105212. https://doi.org/10.1016/j.ijrmms.2022.105212.
Cosserat, E., and F. Cosserat. 1909. Théorie des corps déformables. Paris: A. Hermann et fils.
Cusatis, G., and E. A. Schauffert. 2009. “Cohesive crack analysis of size effect.” Eng. Fract. Mech. 76 (14): 2163–2173. https://doi.org/10.1016/j.engfracmech.2009.06.008.
De Borst, R., and J. Pamin. 1996. “Gradient plasticity in numerical simulation of concrete cracking.” Eur. J. Mech. A Solids 15 (2): 295–320.
De Borst, R., J. Pamin, R. Peerlings, and L. Sluys. 1995. “On gradient-enhanced damage and plasticity models for failure in quasi-brittle and frictional materials.” Comput. Mech. 17 (1–2): 130–141. https://doi.org/10.1007/BF00356485.
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.
Di Luzio, G. 2007. “A symmetric over-nonlocal microplane model m4 for fracture in concrete.” Int. J. Solids Struct. 44 (13): 4418–4441. https://doi.org/10.1016/j.ijsolstr.2006.11.030.
Duddu, R., and H. Waisman. 2013. “A nonlocal continuum damage mechanics approach to simulation of creep fracture in ice sheets.” Comput. Mech. 51 (6): 961–974. https://doi.org/10.1007/s00466-012-0778-7.
Eringen, A., and D. Edelen. 1972. “On nonlocal elasticity.” Int. J. Eng. Sci. 10 (3): 233–248. https://doi.org/10.1016/0020-7225(72)90039-0.
Eringen, A. C. 1965. Linear theory of micropolar elasticity. Lafayette, IN: School of Aeronautics, Astronautics and Engineering Sciences, Defense Technical Information Center Document, Purdue Univ.
Geers, M., R. De Borst, W. Brekelmans, and R. Peerlings. 1998. “Strain-based transient-gradient damage model for failure analyses.” Comput. Methods Appl. Mech. Eng. 160 (1–2): 133–153. https://doi.org/10.1016/S0045-7825(98)80011-X.
Geers, M., R. Peerlings, W. Brekelmans, and R. De Borst. 2000. “Phenomenological nonlocal approaches based on implicit gradient-enhanced damage.” Acta Mech. 144 (1–2): 1–15. https://doi.org/10.1007/BF01181824.
Giry, C., F. Dufour, and J. Mazars. 2011. “Stress-based nonlocal damage model.” Int. J. Solids Struct. 48 (25): 3431–3443. https://doi.org/10.1016/j.ijsolstr.2011.08.012.
Grassl, P., and M. Jirásek. 2006. “Damage-plastic model for concrete failure.” Int. J. Solids Struct. 43 (22–23): 7166–7196. https://doi.org/10.1016/j.ijsolstr.2006.06.032.
Grassl, P., D. Xenos, M. Jirásek, and M. Horák. 2014. “Evaluation of nonlocal approaches for modelling fracture near nonconvex boundaries.” Int. J. Solids Struct. 51 (18): 3239–3251. https://doi.org/10.1016/j.ijsolstr.2014.05.023.
Grassl, P., D. Xenos, U. Nyström, R. Rempling, and K. Gylltoft. 2013. “Cdpm2: A damage-plasticity approach to modelling the failure of concrete.” Int. J. Solids Struct. 50 (24): 3805–3816. https://doi.org/10.1016/j.ijsolstr.2013.07.008.
Havlásek, P., P. Grassl, and M. Jirásek. 2016. “Analysis of size effect on strength of quasi-brittle materials using integral-type nonlocal models.” Eng. Fract. Mech. 157 (May): 72–85. https://doi.org/10.1016/j.engfracmech.2016.02.029.
Hoover, C. G., and Z. P. Bažant. 2014. “Cohesive crack, size effect, crack band and work-of-fracture models compared to comprehensive concrete fracture tests.” Int. J. Fract. 187 (1): 133–143. https://doi.org/10.1007/s10704-013-9926-0.
Hoover, C. G., Z. P. Bažant, J. Vorel, R. Wendner, and M. H. Hubler. 2013. “Comprehensive concrete fracture tests: Description and results.” Eng. Fract. Mech. 114 (Dec): 92–103. https://doi.org/10.1016/j.engfracmech.2013.08.007.
Jefferson, A. D., B. Barr, T. Bennett, and S. Hee. 2004. “Three dimensional finite element simulations of fracture tests using the craft concrete model.” Comput. Concr. Int. J. 1 (3): 261–284. https://doi.org/10.12989/cac.2004.1.3.261.
Jirasek, M., S. Rolshoven, and P. Grassl. 2004. “Size effect on fracture energy induced by non-locality.” Int. J. Numer. Anal. Methods Geomech. 28 (7–8): 653–670. https://doi.org/10.1002/nag.364.
Jirásek, M., and M. Bauer. 2012. “Numerical aspects of the crack band approach.” Comput. Struct. 110 (Nov): 60–78. https://doi.org/10.1016/j.compstruc.2012.06.006.
Krayani, A., G. Pijaudier-Cabot, and F. Dufour. 2009. “Boundary effect on weight function in nonlocal damage model.” Eng. Fract. Mech. 76 (14): 2217–2231. https://doi.org/10.1016/j.engfracmech.2009.07.007.
Kupfer, H., H. K. Hilsdorf, and H. Rusch. 1969. “Behavior of concrete under biaxial stresses.” J. Proc. 66: 656–666.
Lale, E., and G. Cusatis. 2021. “Symmetric high order microplane model for damage localization and size effect in quasi-brittle materials.” Int. J. Numer. Anal. Methods Geomech. 45 (10): 1458–1476. https://doi.org/10.1002/nag.3209.
Lale, E., X. Zhou, and G. Cusatis. 2017. “Isogeometric implementation of high-order microplane model for the simulation of high-order elasticity, softening, and localization.” J. Appl. Mech. 84 (1): 011005. https://doi.org/10.1115/1.4034784.
Lee, J., and G. L. Fenves. 1998. “Plastic-damage model for cyclic loading of concrete structures.” J. Eng. Mech. 124 (8): 892–900. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:8(892).
Liu, P., Z. Gu, Y. Yang, and X. Peng. 2016. “A nonlocal finite element model for progressive failure analysis of composite laminates.” Composites, Part B 86 (Feb): 178–196. https://doi.org/10.1016/j.compositesb.2015.09.061.
Mazars, J. 1984. “Application de la mécanique de l’endommagement au comportement non linéaire et à la rupture du béton de structure.” Ph.D. thesis, Laboratoire de Mecanique et Technologie, Université Pierre et Marie Curie-Paris 6, France.
Menétrey, P., and K. Willam. 1995. “Triaxial failure criterion for concrete and its generalization.” Struct. J. 92 (3): 311–318. https://doi.org/10.14359/1132.
Mindlin, R., and N. Eshel. 1968. “On first strain-gradient theories in linear elasticity.” Int. J. Solids Struct. 4 (1): 109–124. https://doi.org/10.1016/0020-7683(68)90036-X.
Mindlin, R., and H. Tiersten. 1962. “Effects of couple-stresses in linear elasticity.” Arch. Ration. Mech. Anal. 11 (1): 415–448. https://doi.org/10.1007/BF00253946.
Mobasher, M. E., L. Berger-Vergiat, and H. Waisman. 2017. “Non-local formulation for transport and damage in porous media.” Comput. Methods Appl. Mech. Eng. 324 (Sep): 654–688. https://doi.org/10.1016/j.cma.2017.06.016.
Negi, A., S. Kumar, and L. H. Poh. 2020. “A localizing gradient damage enhancement with micromorphic stress-based anisotropic nonlocal interactions.” Int. J. Numer. Methods Eng. 121 (18): 4003–4027. https://doi.org/10.1002/nme.6397.
Nelson, R. B. 1976. “Simplified calculation of eigenvector derivatives.” AIAA J. 14 (9): 1201–1205. https://doi.org/10.2514/3.7211.
Nguyen, G. D. 2011. “A damage model with evolving nonlocal interactions.” Int. J. Solids Struct. 48 (10): 1544–1559. https://doi.org/10.1016/j.ijsolstr.2011.02.002.
Nguyen, T. H., T. Q. Bui, and S. Hirose. 2018. “Smoothing gradient damage model with evolving anisotropic nonlocal interactions tailored to low-order finite elements.” Comput. Methods Appl. Mech. Eng. 328 (Jan): 498–541. https://doi.org/10.1016/j.cma.2017.09.019.
Nooru-Mohamed, M. B. 1993. “Mixed-mode fracture of concrete: An experimental approach.” Ph.D. thesis, Dept. of Civil Engineering and Geoscience, Delft Univ. of Technology.
Peerlings, R. H., R. de Borst, W. M. Brekelmans, and J. 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.
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.
Sarkar, S., I. Singh, B. Mishra, A. Shedbale, and L. Poh. 2019. “A comparative study and abaqus implementation of conventional and localizing gradient enhanced damage models.” Finite Elem. Anal. Des. 160 (Aug): 1–31. https://doi.org/10.1016/j.finel.2019.04.001.
Seidenfuss, M., M. Samal, and E. Roos. 2011. “On critical assessment of the use of local and nonlocal damage models for prediction of ductile crack growth and crack path in various loading and boundary conditions.” Int. J. Solids Struct. 48 (24): 3365–3381. https://doi.org/10.1016/j.ijsolstr.2011.08.006.
Shedbale, A. S., G. Sun, and L. H. Poh. 2021. “A localizing gradient enhanced isotropic damage model with ottosen equivalent strain for the mixed-mode fracture of concrete.” Int. J. Mech. Sci. 199 (Jun): 106410. https://doi.org/10.1016/j.ijmecsci.2021.106410.
Simone, A., H. Askes, and L. J. Sluys. 2004. “Incorrect initiation and propagation of failure in non-local and gradient-enhanced media.” Int. J. Solids Struct. 41 (2): 351–363. https://doi.org/10.1016/j.ijsolstr.2003.09.020.
Thai, T. Q., T. Rabczuk, Y. Bazilevs, and G. Meschke. 2016. “A higher-order stress-based gradient-enhanced damage model based on isogeometric analysis.” Comput. Methods Appl. Mech. Eng. 304 (Jun): 584–604. https://doi.org/10.1016/j.cma.2016.02.031.
Vandoren, B., and A. Simone. 2018. “Modeling and simulation of quasi-brittle failure with continuous anisotropic stress-based gradient-enhanced damage models.” Comput. Methods Appl. Mech. Eng. 332 (Apr): 644–685. https://doi.org/10.1016/j.cma.2017.12.027.
Willam, K. J. 1974. “Constitutive model for the triaxial behaviour of concrete.” In Proc., Int. Association of Bridge and Structural Engineers, Seminar on Concrete Structure Subjected to Triaxial Stresses, 1–30. Zurich, Switzerland: International Association For Bridge And Structural Engineering.
Wolff, C., N. Richart, and J.-F. Molinari. 2015. “A non-local continuum damage approach to model dynamic crack branching.” Int. J. Numer. Methods Eng. 101 (12): 933–949. https://doi.org/10.1002/nme.4837.
Wu, J. Y., J. Li, and R. Faria. 2006. “An energy release rate-based plastic-damage model for concrete.” Int. J. Solids Struct. 43 (3–4): 583–612. https://doi.org/10.1016/j.ijsolstr.2005.05.038.
Wu, J.-Y. 2017. “A unified phase-field theory for the mechanics of damage and quasi-brittle failure.” J. Mech. Phys. Solids 103 (Jun): 72–99. https://doi.org/10.1016/j.jmps.2017.03.015.
Wu, L., L. Noels, L. Adam, and I. Doghri. 2013. “An implicit-gradient-enhanced incremental-secant mean-field homogenization scheme for elasto-plastic composites with damage.” Int. J. Solids Struct. 50 (24): 3843–3860. https://doi.org/10.1016/j.ijsolstr.2013.07.022.
Zhang, H., M. E. Mobasher, Z. Shen, and H. Waisman. 2023. “A unified non-local damage model for hydraulic fracture in porous media.” Acta Geotech. 1–39. https://doi.org/10.1007/s11440-023-01873-w.

Information & Authors

Information

Published In

Go to Journal of Engineering Mechanics
Journal of Engineering Mechanics
Volume 149Issue 10October 2023

History

Received: Mar 7, 2023
Accepted: Jun 25, 2023
Published online: Aug 12, 2023
Published in print: Oct 1, 2023
Discussion open until: Jan 12, 2024

Permissions

Request permissions for this article.

ASCE Technical Topics:

Authors

Affiliations

Dept. of Civil Engineering, Istanbul Technical Univ., Istanbul 34469, Turkey (corresponding author). ORCID: https://orcid.org/0000-0003-4895-5239. Email: [email protected]
Bahar Ayhan, Ph.D. [email protected]
Dept. of Civil Engineering, Istanbul Technical Univ., Istanbul 34469, Turkey. Email: [email protected]
Nilay Celik, Ph.D. [email protected]
Dept. of Civil Engineering, Istanbul Technical Univ., Istanbul 34469, Turkey. Email: [email protected]

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.

View Options

Get Access

Access content

Please select your options to get access

Log in/Register Log in via your institution (Shibboleth)
ASCE Members: Please log in to see member pricing

Purchase

Save for later Information on ASCE Library Cards
ASCE Library Cards let you download journal articles, proceedings papers, and available book chapters across the entire ASCE Library platform. ASCE Library Cards remain active for 24 months or until all downloads are used. Note: This content will be debited as one download at time of checkout.

Terms of Use: ASCE Library Cards are for individual, personal use only. Reselling, republishing, or forwarding the materials to libraries or reading rooms is prohibited.
ASCE Library Card (5 downloads)
$105.00
Add to cart
ASCE Library Card (20 downloads)
$280.00
Add to cart
Buy Single Article
$35.00
Add to cart

Get Access

Access content

Please select your options to get access

Log in/Register Log in via your institution (Shibboleth)
ASCE Members: Please log in to see member pricing

Purchase

Save for later Information on ASCE Library Cards
ASCE Library Cards let you download journal articles, proceedings papers, and available book chapters across the entire ASCE Library platform. ASCE Library Cards remain active for 24 months or until all downloads are used. Note: This content will be debited as one download at time of checkout.

Terms of Use: ASCE Library Cards are for individual, personal use only. Reselling, republishing, or forwarding the materials to libraries or reading rooms is prohibited.
ASCE Library Card (5 downloads)
$105.00
Add to cart
ASCE Library Card (20 downloads)
$280.00
Add to cart
Buy Single Article
$35.00
Add to cart

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share