1. Introduction#

In this chapter we start by introducing the limits of the classical approach in fracture mechanics (§ 1.1), then we recall the objective and the interest of the \({G}_{p}\) approach (§ 1.2). Finally, we discuss the foundations of this method in (§ 1.4).

1.1. Limits of the classical approach#

In fracture mechanics, the classical parameter for defining a criterion for the initiation of an already existing crack [1] _ is the parameter \(G\), elastic energy recovery rate. Parameter \(G\) is valid for linear elasticity as well as for non-linear elasticity. That is, the physical framework and hypotheses are well defined, and the calculation formula \(G\mathrm{\theta }\) is consistent with this framework. On the other hand, the relevance of breakthrough criterion \(G=\mathit{Gc}\) with experimental results is another question, not addressed in this document. In elastoplasticity, the parameter \(G\) is generally not valid. However, if the load remains radial and monotonic, then \(G\) is valid. This hypothesis implies that the plasticity equations are equivalent to the Hencky equations (which are those for non-linear elasticity [:ref:` R4.20.01 < R4.20.01 >`]). However, cases of strictly proportional loads are quite rare (or even do not exist when it comes to structures with geometric defects such as cracks).

1.2. Objective and interest of Gp#

The objective of method \({G}_{p}\) is to define a valid priming parameter in incremental plasticity. This theory assumes a decoupling of plasticity and cracking (case of breakage by cleavage). Thus the use of \({G}_{p}\) to predict cleavage will reduce the conservatisms of the classical approach. Its definition uses free energy as well as some geometric parameters necessary to define the crack, which is modelled by a notch. This parameter \({G}_{p}\) has the unit of a surface energy density and is calculated in the area adjacent to the notch edge.

The advantage of the method is threefold: \({G}_{p}\) is a deterministic priming criterion, in other words it makes it possible to assess whether or not a crack has started. This parameter is consistent with elastic \(G\) [WAD 13, LOR 14], and it is valid in an elastoplastic framework in non-proportional loading and in discharge.

Note:

  • \({G}_{p}\) is not a criterion for stability or instability of the defect.

  • \({G}_{p}\) does not predict propagation length.

1.3. History of the development of the Gp approach#

The development of this approach, which originated in the early 2000s, came from the work of Lorentz, Wadier and Debr[ LOR00]. The crack is then modelled by a cut in the plane but the idea of introducing a notch is already mentioned in perspective and will be quickly adopted. The modeling of the crack by a notch was presented in 2003 (CR) in [WAD 03d] and in 2004 (articles) in [LOR 04] and [WAD 04].

The approach has been presented numerous times at international conferences by first presenting its interest in the tank in charge/discharge situations [WAD 00], [WAD 01a], [WAD 01b], then by showing the interpretations of experimental results illustrating the « small defect » effect [WAD 03a] [WAD 03c] then by showing the interpretations of experimental results illustrating the « small defect » effect [] [] then the effect of hot preloading [ WAD 03b] [WAD 05] [WAD 09].

A summary recalling the foundations of the Gp approach as well as the main validation results is carried out in [WAD 13]. To date, this is the only article published in an international journal, dedicated to the Gp approach. As a result, it is a reference for the Gp approach.

It should be noted that attempts to predict crack arrest using a Gp (G-delta) approach have been made in the past [WAD 07a], but this approach is no longer being pursued at present.

Since the departure into inactivity of the main player in the method (Y. Wadier), a new generation of EDF R&D engineers has taken up the torch and continues to promote the Gp approach to the scientific community [GEN 16] [JUL 17]. Recently, the Gp approach has been extended to configurations where the area around the defect tip is in compression [HAB 17].

1.4. Foundations of the Gp approach#

The \({G}_{p}\) method is placed in the context of a global energetic formulation of fragile rupture. The approach consists in determining whether defect propagation (maintaining the geometry of the defect background and with constant mechanical fields) makes it possible to reduce the total energy of the structure (principle of minimization) .To achieve this, the crack path is assumed to be known a priori. Under this hypothesis, the energy framework proposed by Frankfurt and Marigo [FRA 93, FRA 98] can be put into practice to predict the sudden onset of cracks. However, this framework has some characteristics that limit its scope [LOR 08], including undesirable scale effects and the incompatibility of the theory with loadings such as imposed forces. This is why two types of modifications have been made to solve these difficulties, while maintaining the shape of the energies introduced. On the one hand, the crack is replaced by a notch whose radius becomes a parameter of the model. On the other hand, it is examined whether the absence of propagation is a (global) minimum with fixed mechanical fields, so an overall minimization is carried out but in only one direction. The energetic nature of the proposed formulation allows the introduction of plastic mechanisms.