5. Link between global and local approaches

Lorentz [LOR 08] and Wadier [WAD 07b] position the \({G}_{p}\) energy approach in relation to global and local approaches to disruption. More recent than the global approach in fracture mechanics, the local approach aims to predict the ruin of a structure by relying on the microscopic mechanism of cleavage. It accounts for the dispersion of results through intrinsically probabilistic modeling. The Beremin model, for example, defines the probability of failure and the Weibull stress \({\mathrm{\sigma }}_{w}\) at time \(t\) by:

(5.1)\[ {P} _ {R} (t) =1-\ mathrm {exp}\ left [- {\ left (\ frac {\ underset {\ mathrm {\ tau}\ le t} {\ text {max}} {\ text {max}}} {\ mathrm {max}}} {\ mathrm {max}}} {\ max}}} {\ max}}} {\ max}}} {\ mathrm {max}}} {\ max}}} {\ mathrm {max}}} {\ mathrm {max}}} {\ max}}} {\ mathrm {max}}} {\ max}}} {\ mathrm {max}}} {\ max}}} {\ mathrm {max}}} {\ max}}}}\ right)} ^ {m}\ right]\ text {;} {\ mathrm {\ sigma}} _ {w} (\ mathrm {\ tau}) = {\ left [\ frac {1} {{1} {{V}} {V} _ {0}}}\ underset {{\ mathrm {\ sigma}}} _ {\ tau}) = {\ left [\ frac {1}} {{V}} {V} _ {0}}}\ underset {\ mathrm {\ tau})} {\ int} {\ mathrm {\ sigma}} _ {I} {I} {(\ mathrm {\ tau})} ^ {m} d\ mathrm {\ omega}\ right]}} ^ {\ frac {1} {m}}\]

with \({\mathrm{\sigma }}_{I}(t)\) and \({\mathrm{\Omega }}_{p}(t)\) the maximum principal stress and the domain being plasticized at time respectively. \({V}_{0}\) is a reference volume, \({\mathrm{\sigma }}_{c}\) is the critical constraint, and \(m\) is a dimensionless exponent. Thus, the expression of the probability of rupture is based in particular on the hypothesis of the weak link, i.e. the ruin is associated with the initiation of the most penalizing microscopic defect. For the sake of comparison between the two models, we can define a probability of breaking the \({G}_{p}\) approach by the formula:

(5.2)\[ {P} _ {R} (t) =1-\ mathrm {exp}\ left [-a {\ left (\ frac {{G} _ {p}} {{G} _ {p0}} _ {p0}}}\ right)}\ right)} {p0}}}\ right]\ text {}\]

where \(a\) and \(m\) are constants identified from the experimental results, and \({G}_{p0}\) a constant such that \({P}_{r}\) is equal to 5% when \({G}_{p}\) is equal to the value identified for a probability of rupture of 5%.

A comparison between the two approaches reveals a number of links. The quantities involved in the initiation criterion (elastic energy or stress) are similar. These quantities are averaged over a zone located in the vicinity of the notch bottom. Finally, it may be necessary to take into account the hydrostatic stress [WAD 07b].

The Beremin model allows a natural consideration of the traction/compression distinction thanks to the introduction of the maximum principal stress rather than that of elastic energy. On the other hand, the \({G}_{p}\) approach offers a simpler transition with previous approaches in the industrial field. It makes it possible to make the link with the classical global approach because it is based, like it, on energetic principles: \({G}_{p}=G\) notch in elasticity [LOR 14].