University of Cambridge > > Engineering Department - Mechanics Colloquia Research Seminars > Leapfrog in Fracture and Damage Mechanics inspired by Gap Test and Curvature-Resisting Sprain Energy

Leapfrog in Fracture and Damage Mechanics inspired by Gap Test and Curvature-Resisting Sprain Energy

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Abstract: For sixty-one years after Ray Clough’s epoch-making finite element analysis of cracks in Norfolk Dam1, there has been no completely satisfactory computational model for fracture and continuum damage. This is evidenced by recent model comparisons with many distinctive2 fracture tests, which are those that cannot be fitted closely by very different models and include the size effect, shear fracture and the new gap test6,7. The distinctive comparisons demonstrated severe limitations of the phase-field models3,4, dismal performance of peridynamics and severe innate inadequacies of the nonlocal models of integral and gradient types, while the crack band model5 (CBM) with microplane M7 constitutive damage law performed well, though less than perfectly. Compared to all models, including the nonlocal and gradient ones, the CBM is the only one that has non-problematic boundary conditions. As it transpired, the CBM performance can be enhanced by introducing an energy, named the sprain energy Φ, which augments the strain energy Ψ and characterizes material resistance to the second gradient tensor of curvature of the displacement vector field, named the sprain tensor. This tensor differs from the strain gradient tensor and, importantly, includes the gradient of material rotation tensor. In FE discretization, the derivatives of Φ yield self-equilibrated sets of nodal or body sprain forces opposing excessive localization of softening damage. Subdividing the material characteristic length into a number of finite elements allows resolving the homogenized (or smooth) strain distribution across the width of the FE crack band. This leads to the smooth Crack Band Model8 (sCBM) which, along with M7, is found to capture the big effect of crack parallel stresses on both the fracture energy and the crack front width, as evidenced by the gap test6,7. Examples of FE fits of distinctive2 data are given.

References: • 1Ray W. Clough (1962). The stress distribution in Norfolk Dam. Structures and Materials Series 100, IER Issue 19, Dept. of Civil Eng., Univ. of California Berkeley (134 pp.) (contract DA-03-050-Civeng-62-511, U.S. Army Engineer District, Little Rock). • 2Bažant, Z.P., and Nguyen, Hoang T. (2023), ``Proposal of a model index, MI, for experimental comparison of fracture and damage models.” J. of Engrg. Mechanics ASCE ; in press. • 3Bažant, Z.P., Nguyen, H.T. and Abdullah Dönmez, A., 2022, ``Critical Comparison of Phase-Field, Peridynamics, and Crack Band Model M7 in Light of Gap Test and Classical Fracture Tests.” J. of Appl. Mech. 89: 061008, 1-26. • 4Bažant, Z.P., Luo, Wen, Chau, Viet T., and Bessa, M.A. (2016). ``Wave dispersion and basic concepts of peridynamics compared to classical nonlocal models.” J. of Applied Mechanics ASME 83 (Nov.) 111004, 1-16. • 5Bažant, Z.P., Le, J.L. and Salviato, M., 2021, ``Quasibrittle Fracture Mechanics and Size Effect: A First Course” Oxford UP. • 6Nguyen, Hoang T., Pathirage, M., Cusatis, G., and Bažant, Z.P. (2020). ``Gap test of crack-parallel stress effect on quasibrittle frcture and its consequences.” ASME J . of Applied Mechanics 87 (July), 071012-1—11. • 7Nguyen, H.T., Dönmez, A. A., Bažant, Z.P., 2021. ``Structural strength scaling law for fracture of plastic-hardening metals and testing of fracture properties.” Extreme Mechanics Letters 43, 101141, 1-12. • 8Zhang, Y., and Bažant Z.P., 2023, “Smooth Crack Band Model (sCBM)—a Computational Paragon Based on Unorthodox Continuum Homogenization.” J. Appl. Mech.041007.

Collaborators: Yupeng Zhang, Houlin Xu, A. Abdullah Dönmez and Anh Nguyen Northwestern University, Evanston, IL, USA Seminar Biosketch of Zdenˇek P. Baˇzant April 5, 2023 Born and educated in Prague (Ph.D. 1963), Baˇzant joined Northwestern in 1969, where he has been W.P. Murphy Professor since 1990 and simultaneously McCormick Institute Professor since 2002, and Director of Center for Concrete and Geomaterials (1981-87). He was inducted to NAS , NAE, Am. Acad. of Arts & Sci., Royal Soc. London, the national academies of Austria, Japan, Italy, Spain, Canada, Czech Rep., Greece, India, Lombardy and Turin, Academia Europaea and Eur. Acad. Sci. & Arts. Honorary Member of: ASCE , ASME, ACI , RILEM. Received Austrian Cross of Honor for Science and Art I. Class from Pres. of Austria; 7 honorary doctorates (Prague, Karlsruhe, Colorado-Boulder, Milan, Lyon, Vienna, Ohio State); U. Minnesota); ASME Medal, ASME Timoshenko, Nadai and Warner Medals; ASCE von K´arm´an, Freudenthal, Newmark, Biot, Mindlin, TY Lin and Croes Medals, SES Prager Medal; Guggenheim Fellow; Outstanding Res. Award from Am. Soc. for Composites; RILEM L ’Hermite Medal; Exner Medal (Austria); Torroja Medal (Madrid); etc. He authored nine books, on Scaling of Struct. Strength, Creep in Concrete Str., Inelastic Analysis, Fracture and Size Effect, Stability of Structures, Concrete at High Temp., Creep & Hygrothermal Effects, Probab. Mech. of Quasibrittle Str., and Quasibrittle Fracture Mechanics. H-index: 146, 90,000 cit. (Google). In 2019 Stanford U. weighted and filtered citation survey∗ (see PLoS), he was ranked worldwide no.1 in CE and no.2 in Engrg. In 2015, ASCE established ZP Baˇzant Medal for Failure and Damage Prevention.

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