Abstract
We consider the quasi-static evolution of a brittle layer on a stiff substrate; adhesion between layers is assumed to be elastic. Employing a phase-field approach we obtain the quasi-static evolution as the limit of time-discrete evolutions computed by an alternate minimization scheme. We study the limit evolution, providing a qualitative discussion of its behaviour and a rigorous characterization, in terms of parametrized balanced viscosity evolutions. Further, we study the transition layer of the phase-field, in a simplified setting, and show that it governs the spacing of cracks in the first stages of the evolution. Numerical results show a good consistency with the theoretical study and the local morphology of real life craquelure patterns.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alessi, R., Freddi, F.: Failure and complex crack patterns in hybrid laminates: a phase-field approach. Compos. B Eng. 179, 107256 (2019)
Almi, S.: Irreversibility and alternate minimization in phase field fracture: a viscosity approach. Z. Angew. Math. Phys. 71(4), 21 (2020)
Almi, S., Negri, M.: Analysis of staggered evolutions for nonlinear energies in phase field fracture. Arch. Ration. Mech. Anal. 236(1), 189–252 (2020)
Almi, S., Belz, S., Negri, M.: Convergence of discrete and continuous unilateral flows for ambrosio-tortorelli energies and application to mechanics. ESAIM Math. Model. Numer. Anal. 53(2), 659–699 (2019)
Bourdin, B., Francfort, G.A., Marigo, J.-J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48(4), 797–826 (2000)
Braides, A.: Approximation of Free-Discontinuity Problems. Springer, Berlin (1998)
Braides, A., Causin, A., Solci, M.: A homogenization result for interacting elastic and brittle media. Proc. R. Soc. A 474, 20180118 (2019)
Chambolle, A.: A density result in two-dimensional linearized elasticity and applications. Arch. Ration. Mech. Anal. 167(3), 211–233 (2003)
Griffith, A.A.: The phenomena of rupture and flow in solids. Philos. Trans. Roy. Soc. Lond. 18, 163–198 (1920)
Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3–4), 251–265 (2012)
Knees, D., Negri, M.: Convergence of alternate minimization schemes for phase field fracture and damage. Math. Models Methods Appl. Sci. 27(9), 1743–1794 (2017)
Kuhn, C.: Numerical and Analytical Investigation of a Phase Field Model for Fracture. Dr.Ing. Dissertation (2013)
León-Baldelli, A.A., Bourdin, B., Marigo, J.-J., Maurini, C.: Fracture and debonding of a thin film on a stiff substrate: analytical and numerical solutions of a one-dimensional variational model. Contin. Mech. Thermodynam. 25(2), 243–268 (2013)
León-Baldelli, A.A., Babadjian, J.-F., Bourdin, B., Henao, D., Maurini, C.: A variational model for fracture and debonding of thin films under in-plane loadings. J. Mech. Phys. Solids 70, 320–348 (2014)
Mielke, A., Rossi, R., Savaré, G.: BV solutions and viscosity approximations of rate-independent systems. ESAIM Control Optim. Calc. Var. 18(1), 36–80 (2012)
Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989)
Negri, M.: A unilateral L 2-gradient flow and its quasi-static limit in phase-field fracture by alternate minimization. Adv. Calc. Var. 12(1), 1–29 (2019)
Pham, K., Marigo, J.-J.: Stability of homogeneous states with gradient damage models: size effects and shape effects in the three-dimensional setting. J. Elast. 110(1), 63–93 (2013)
Vo, T.D., Pouya, A., Hemmati, S., Tang, A.-M.: Numerical modelling of desiccation cracking of clayey soil using a cohesive fracture method. Comput. Geotech. 85, 15–27 (2017)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Negri, M. (2021). A Quasi-Static Model for Craquelure Patterns. In: Bonetti, E., Cavaterra, C., Natalini, R., Solci, M. (eds) Mathematical Modeling in Cultural Heritage. Springer INdAM Series, vol 41. Springer, Cham. https://doi.org/10.1007/978-3-030-58077-3_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-58077-3_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-58076-6
Online ISBN: 978-3-030-58077-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)