2. Benchmark solution#

The reference solution is an analytical solution taken from [bib3], itself inspired by a unidimensional study proposed in [bib1] and more generally based on the energetic approach to rupture proposed by G. A. Frankfurt and J. J. Marigo [bib2]. We will not go into the details of calculating this solution, we will just present the analytical value of the overall response of the structure. The imposed displacement

_images/Object_18.svg

depending on the corresponding force

_images/Object_19.svg

worth:

\(U(F)\mathrm{=}\frac{\mathit{Fl}}{2\pi {R}_{f}L\mu }+\mathit{sign}(F){\psi }^{\text{'}\mathrm{-}1}(\frac{∣F∣}{2\pi {R}_{f}L})\) eq 2-1

where

_images/Object_21.svg

denotes the Lamé coefficient (

_images/Object_22.svg

here),

_images/Object_23.svg

the cracking energy density and where \(l={R}_{f}\mathrm{ln}(R/{R}_{f})\) is a characteristic length of the structure that is decisive for the sudden or gradual evolution of decohesion.

The inverse of the derivative of the surface energy density takes the following values depending on whether we adopt the cohesive law CZM_EXP (i.e. the elements with discontinuity), the law CZM_LIN_REG (i.e. the joint elements) or the law CZM_TAC_MIX (i.e. the interface elements). (see documentation [R7.02.14] and [R7.02.11]).

CZM_EXP: \({\psi }^{\text{'}-1}(x)=-\frac{{G}_{c}}{{\sigma }_{c}}\mathrm{ln}(\frac{x}{{\sigma }_{c}})\)

CZM_LIN_REG or CZM_TAC_MIX: \({\psi }^{\text{'}-1}(x)=2\frac{{G}_{c}}{{\sigma }_{c}}(1-\frac{x}{{\sigma }_{c}})\)

2.1. Bibliographical references#

  1. CHARLOTTE M., FRANCFORT G.A., MARIGO J.J., and TRUSKINOVSKY L.: Revisiting brittle fracture as an energy minimization problem: comparison of Griffith and Barenblatt surface energy models. Proceedings of the Symposium on « Continuous Damage and Fracture » The data science library, Elsevier, edited by A. BENALLAL, Paris, pp. 7-18, (2000).

  1. FRANCFORT G.A. and MARIGO J.J.: Revisiting brittle fracture as an energy minimization problem. Mr Mech. Phys. Solids, 46 (8), pp. 1319-1342 (1998).

  2. laverne J.: Energetic formulation of rupture by cohesive force models: theoretical considerations and numerical implementations, Doctoral thesis of the University of Paris13, November 2004.