A phase-field description of brittle fracture is employed in the reported four-point bending analyses of monolithic and laminated glass plates. Our aims are: (i) to compare different phase-field fracture formulations applied to thin glass plates, (ii) to assess the consequences of the dimensional reduction of the problem and mesh density and refinement, and (iii) to validate for quasi-static loading the time-/temperature-dependent material properties we derived recently for two commonly used polymer foils made of polyvinyl butyral or ethylene-vinyl acetate. As the nonlinear response prior to fracture, typical of the widely used Bourdin-Francfort-Marigo model, can lead to a significant overestimation of the response of thin plates under bending, the numerical study investigates two additional phase-field fracture models providing the linear elastic phase of the stress-strain diagram. The typical values of the critical fracture energy and tensile strength of glass lead to a phase-field length-scale parameter that is challenging to resolve in the numerical simulations. Therefore, we show how to determine the fracture energy concerning the applied dimensional reduction and the value of the length-scale parameter relative to the thickness of the plate. The comparison shows that the phase-field models provide very good agreement with the measured stresses and resistance of laminated glass, despite the fact that only one/two cracks are localised using the quasi-static analysis, whereas multiple cracks evolve during the experiment. It was also observed that the stiffness and resistance of the partially fractured laminated glass can be well approximated using a 2D plane-stress model with initially predefined cracks, which provides a better estimation than the one-glass-layer limit.

Department of Mechanics Faculty of Civil Engineering Czech Technical University Prague Thakurova 7 166 29 Prague 6 Dejvice Czech Republic

Zobrazit více v PubMed

Haldimann M., Luible A., Overend M. Structural Use of Glass; Structural Engineering Documents. Volume 10 IABSE; Zürich, Switzerland: 2008.

Ledbetter S.R., Walker A.R., Keiller A.P. Structural use of glass. J. Archit. Eng. 2006;12:137–149. doi: 10.1061/(ASCE)1076-0431(2006)12:3(137). DOI

Zhao C., Yang J., Wang X.E., Azim I. Experimental investigation into the post-breakage performance of pre-cracked laminated glass plates. Constr. Build. Mater. 2019;224:996–1006. doi: 10.1016/j.conbuildmat.2019.07.286. DOI

Bonati A., Pisano G., Carfagni G.R. Redundancy and robustness of brittle laminated plates. Overlooked aspects in structural glass. Compos. Struct. 2019;227:111288. doi: 10.1016/j.compstruct.2019.111288. DOI

Overend M., De Gaetano S., Haldimann M. Diagnostic interpretation of glass failure. Struct. Eng. Int. 2007;17:151–158. doi: 10.2749/101686607780680790. DOI

Calderone I., Davies P., Bennison S.J., Xiaokun H., Gang L. Glass Performance Days (Tampere, 2009) Glaston Finland/GPD; Tampere, Finland: 2009. Effective laminate thickness for the design of laminated glass; pp. 12–15.

Galuppi L., Royer-Carfagni G.F. Effective thickness of laminated glass beams: New expression via a variational approach. Eng. Struct. 2012;38:53–67. doi: 10.1016/j.engstruct.2011.12.039. DOI

Galuppi L., Royer-Carfagni G. The effective thickness of laminated glass plates. J. Mech. Mater. Struct. 2012;7:375–400. doi: 10.2140/jomms.2012.7.375. DOI

Galuppi L., Royer-Carfagni G. Enhanced effective thickness of multi-layered laminated glass. Compos. Part B. 2014;64:202–213. doi: 10.1016/j.compositesb.2014.04.018. DOI

Vallabhan C.G., Das Y., Magdi M., Asik M., Bailey J.R. Analysis of laminated glass units. J. Struct. Eng. 1993;119:1572–1585. doi: 10.1061/(ASCE)0733-9445(1993)119:5(1572). DOI

Aşık M.Z. Laminated glass plates: Revealing of nonlinear behavior. Comput. Struct. 2003;81:2659–2671. doi: 10.1016/S0045-7949(03)00325-0. DOI

Aşık M.Z., Tezcan S. A mathematical model for the behavior of laminated glass beams. Comput. Struct. 2005;83:1742–1753. doi: 10.1016/j.compstruc.2005.02.020. DOI

Ivanov I.V. Analysis, modelling, and optimization of laminated glasses as plane beam. Int. J. Solids Struct. 2006;43:6887–6907. doi: 10.1016/j.ijsolstr.2006.02.014. DOI

Foraboschi P. Behavior and failure strength of laminated glass beams. J. Eng. Mech. 2007;133:1290–1301. doi: 10.1061/(ASCE)0733-9399(2007)133:12(1290). DOI

Koutsawa Y., Daya E.M. Static and free vibration analysis of laminated glass beam on viscoelastic supports. Int. J. Solids Struct. 2007;44:8735–8750. doi: 10.1016/j.ijsolstr.2007.07.009. DOI

Foraboschi P. Analytical model for laminated-glass plate. Compos. Part B. 2012;43:2094–2106. doi: 10.1016/j.compositesb.2012.03.010. DOI

Schulze S.H., Pander M., Naumenko K., Altenbach H. Analysis of laminated glass beams for photovoltaic applications. Int. J. Solids Struct. 2012;49:2027–2036. doi: 10.1016/j.ijsolstr.2012.03.028. DOI

Teotia M., Soni R. Applications of finite element modelling in failure analysis of laminated glass composites: A review. Eng. Fail. Anal. 2018;94:412–437. doi: 10.1016/j.engfailanal.2018.08.016. DOI

Chen S., Zang M., Wang D., Yoshimura S., Yamada T. Numerical analysis of impact failure of automotive laminated glass: A review. Compos. Part B. 2017;122:47–60. doi: 10.1016/j.compositesb.2017.04.007. DOI

Lancioni G., Royer-Carfagni G. The variational approach to fracture mechanics. A practical application to the French Panthéon in Paris. J. Elast. 2009;95:1–30. doi: 10.1007/s10659-009-9189-1. DOI

Martínez-Pañeda E., Golahmar A., Niordson C.F. A phase-field formulation for hydrogen assisted cracking. Comput. Methods Appl. Mech. Eng. 2018;342:742–761. doi: 10.1016/j.cma.2018.07.021. DOI

Natarajan S., Annabattula R.K. Modeling crack propagation in variable stiffness composite laminates using the phase-field method. Compos. Struct. 2019;209:424–433.

Alessi R., Freddi F. Phase-field modelling of failure in hybrid laminates. Compos. Struct. 2017;181:9–25. doi: 10.1016/j.compstruct.2017.08.073. DOI

Alessi R., Freddi F. Failure and complex crack patterns in hybrid laminates: A phase-field approach. Compos. Part B. 2019;179:107256. doi: 10.1016/j.compositesb.2019.107256. DOI

Freddi F., Mingazzi L. Phase Field Simulation of Laminated Glass Beam. Materials. 2020;13:3218. doi: 10.3390/ma13143218. PubMed DOI PMC

Naumenko K., Eremeyev V.A. A layer-wise theory for laminated glass and photovoltaic panels. Compos. Struct. 2014;112:283–291. doi: 10.1016/j.compstruct.2014.02.009. DOI

Eisenträger J., Naumenko K., Altenbach H., Meenen J. A user-defined finite element for laminated glass panels and photovoltaic modules based on a layer-wise theory. Compos. Struct. 2015;133:265–277. doi: 10.1016/j.compstruct.2015.07.049. DOI

Hána T., Janda T., Schmidt J., Zemanová A., Šejnoha M., Eliášová M., Vokáč M. Experimental and numerical study of viscoelastic properties of polymeric interlayers used for laminated glass: Determination of material parameters. Materials. 2019;12:2241. doi: 10.3390/ma12142241. PubMed DOI PMC

Griffith A.A. The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. A. 1921;221:163–198.

Bourdin B., Francfort G.A., Marigo J.J. Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids. 2000;48:797–826. doi: 10.1016/S0022-5096(99)00028-9. DOI

Francfort G.A., Marigo J.J. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids. 1998;46:1319–1342. doi: 10.1016/S0022-5096(98)00034-9. DOI

Amor H., Marigo J.J., Maurini C. Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments. J. Mech. Phys. Solids. 2009;57:1209–1229. doi: 10.1016/j.jmps.2009.04.011. DOI

Wu J.Y., Nguyen V.P., Nguyen C.T., Sutula D., Bordas S., Sinaie S. Phase field modeling of fracture. Adv. Appl. Mechanics-Multi-Scale Theory Comput. 2020;53:1–183.

Verhoosel C.V., de Borst R. A phase-field model for cohesive fracture. Int. J. Numer. Methods Eng. 2013;96:43–62. doi: 10.1002/nme.4553. DOI

Freddi F., Iurlano F. Numerical insight of a variational smeared approach to cohesive fracture. J. Mech. Phys. Solids. 2017;98:156–171. doi: 10.1016/j.jmps.2016.09.003. DOI

Ambati M., Gerasimov T., De Lorenzis L. Phase-field modeling of ductile fracture. Comput. Mech. 2015;55:1017–1040. doi: 10.1007/s00466-015-1151-4. DOI

Borden M.J., Hughes T.J.R., Landis C.M., Anvari A., Lee I.J. A phase-field formulation for fracture in ductile materials: Finite deformation balance law derivation, plastic degradation, and stress triaxiality effects. Comput. Methods Appl. Mech. Eng. 2016;312:130–166. doi: 10.1016/j.cma.2016.09.005. DOI

Bourdin B., Larsen C.J., Richardson C.L. A time-discrete model for dynamic fracture based on crack regularization. Int. J. Fract. 2011;168:133–143. doi: 10.1007/s10704-010-9562-x. DOI

Borden M.J., Verhoosel C.V., Scott M.A., Hughes T.J.R., Landis C.M. A phase-field description of dynamic brittle fracture. Comput. Methods Appl. Mech. Eng. 2012;217–220:77–95. doi: 10.1016/j.cma.2012.01.008. DOI

Kiendl J., Ambati M., De Lorenzis L., Gomez H., Reali A. Phase-field description of brittle fracture in plates and shells. Comput. Methods Appl. Mech. Eng. 2016;312:374–394. doi: 10.1016/j.cma.2016.09.011. DOI

Miehe C., Hofacker M., Schänzel L.M., Aldakheel F. Phase field modeling of fracture in multi-physics problems. Part II. Coupled brittle-to-ductile failure criteria and crack propagation in thermo-elastic-plastic solids. Comput. Methods Appl. Mech. Eng. 2015;294:486–522. doi: 10.1016/j.cma.2014.11.017. DOI

Bourdin B., Francfort G.A., Marigo J.J. The variational approach to fracture. J. Elast. 2008;91:5–148. doi: 10.1007/s10659-007-9107-3. DOI

Miehe C., Welschinger F., Hofacker M. Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations. Int. J. Numer. Methods Eng. 2010;83:1273–1311. doi: 10.1002/nme.2861. DOI

Miehe C., Schänzel L.M., Ulmer H. Phase field modeling of fracture in multi-physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids. Comput. Methods Appl. Mech. Eng. 2015;294:449–485. doi: 10.1016/j.cma.2014.11.016. DOI

Pham K., Amor H., Marigo J.J., Maurini C. Gradient damage models and their use to approximate brittle fracture. Int. J. Damage Mech. 2011;20:618–652. doi: 10.1177/1056789510386852. DOI

Mandal T.K., Nguyen V.P., Wu J.Y. Length scale and mesh bias sensitivity of phase-field models for brittle and cohesive fracture. Eng. Fract. Mech. 2019;217:106532. doi: 10.1016/j.engfracmech.2019.106532. DOI

Kuhn C., Schlüter A., Müller R. On degradation functions in phase-field fracture models. Comput. Mater. Sci. 2015;108:374–384. doi: 10.1016/j.commatsci.2015.05.034. DOI

Zhang X., Vignes C., Sloan S.W., Sheng D. Numerical evaluation of the phase-field model for brittle fracture with emphasis on the length-scale. Comput. Mech. 2017;59:737–752. doi: 10.1007/s00466-017-1373-8. DOI

Farrell P., Maurini C. Linear and nonlinear solvers for variational phase-field models of brittle fracture. Int. J. Numer. Methods Eng. 2017;109:648–667. doi: 10.1002/nme.5300. DOI

Ambati M., Gerasimov T., De Lorenzis L. A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput. Mech. 2015;55:383–405. doi: 10.1007/s00466-014-1109-y. DOI

CEN . EN572-1: 2004 Glass in Building-Basic Soda Lime Silicate Glass Products—Part 1. Standard, Deutsches Institut für Bautechnik; Berlin, Germany: 2004.

Wiederhorn S.M. Fracture surface energy of glass. J. Am. Ceram. Soc. 1969;52:99–105. doi: 10.1111/j.1151-2916.1969.tb13350.x. DOI

Wang X.E., Yang J., Liu Q.F., Zhang Y.M., Zhao C. A comparative study of numerical modelling techniques for the fracture of brittle materials with specific reference to glass. Eng. Struct. 2017;152:493–505. doi: 10.1016/j.engstruct.2017.08.050. DOI

Pham K., Ravi-Chandar K., Landis C. Experimental validation of a phase-field model for fracture. Int. J. Fract. 2017;205:83–101. doi: 10.1007/s10704-017-0185-3. DOI

Wu J.Y., Nguyen V.P. A length-scale insensitive phase-field damage model for brittle fracture. J. Mech. Phys. Solids. 2018;119:20–42. doi: 10.1016/j.jmps.2018.06.006. DOI

Logg A., Mardal K.A., Wells G. Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book. Volume 84 Springer; Berlin/Heidelberg, Germany: 2012.

Schmidt J. Laminated_Glass_Fracture_QS Supplementary Code for Phase-Field Fracture Modelling of Thin Monolithic or Laminated Glass Plates under Quasi-Static Bending. [(accessed on 6 November 2020)];2020 Available online: https://gitlab.com/JaraSit/laminated_glass_fracture_qs. PubMed PMC

Bleyer J., Roux-Langlois C., Molinari J.F. Dynamic crack propagation with a variational phase-field model: Limiting speed, crack branching and velocity-toughening mechanisms. Int. J. Fract. 2017;204:79–100. doi: 10.1007/s10704-016-0163-1. DOI

Raj H., Natarajan S., Annabattula R.K., Martínez-Pañeda E. Phase field modelling of crack propagation in functionally graded materials. Compos. Part Eng. 2019;169:239–248.

Hunt G.W., Baker G. Principles of localization in the fracture of quasi-brittle structures. J. Mech. Phys. Solids. 1995;43:1127–1150. doi: 10.1016/0022-5096(95)00028-H. DOI

Brocca M. Ph.D. Thesis. University of Tokyo; Tokyo, Japan: 1997. Analysis of Cracking Localization and Crack Growth Based on Thermomechanical Theory of Localization.

Audy M. Master’s Thesis. Czech Technical University in Prague, Faculty of Civil Engineering; Prague, Czech Republic: 2003. Localization of Inelastic Deformation in Problems Freeof Initial Stress Concentrators.

DIN Standards Committee . DIN 18008-1:2010-12 Glass in Building—Design and Construction Rule—Part 1: Terms and General Bases. Standard, German Institute for Standardisation; Berlin, Germany: 2010.

Duser A.V., Jagota A., Bennison S.J. Analysis of glass/polyvinyl butyral laminates subjected to uniform pressure. J. Eng. Mech. 1999;125:435–442. doi: 10.1061/(ASCE)0733-9399(1999)125:4(435). DOI

Christensen R. Theory of Viscoelasticity: An Introduction. 2nd ed. Academic Press; London, UK: 2012.

Hána T., Eliášová M., Sokol Z. For point bending tests of double laminated glass panels; Proceedings of the 24th International Conference Engineering Mechanics 2018; Svratka, Czech Republic. 14–17 May 2018; pp. 285–288. DOI

Sargado J.M., Keilegavlen E., Berre I., Nordbotten J.M. High-accuracy phase-field models for brittle fracture based on a new family of degradation functions. J. Mech. Phys. Solids. 2018;111:458–489. doi: 10.1016/j.jmps.2017.10.015. DOI

Phase-Field Fracture Modelling of Thin Monolithic and Laminated Glass Plates under Quasi-Static Bending