## 1 Introduction  
In the 1970s, [PERSON] [1] presented the Strain Energy Density factor (S) as the product of the Strain Energy Density by a certain distance (from the point of singularity). More recently, [PERSON] and co-authors (e.g., [2; 3; 4; 5]) proposed the Average Strain Energy Density (ASED) criterion as a combination of [PERSON]'s model with the concept of elementary structural volume suggested by [PERSON] [6]. As stated in [2], the (local) ASED criterion is based on the quantification of the Strain Energy Density averaged over a control volume defined at the notch tip. The ASED approach is based on the idea that failure occurs when the mean value of the elastic strain energy (W) referred to a volume (or an area, in plane problems) is equal to a critical value (W\({}_{\text{c}}\)) [2], which is a material property. Thus, the ASED criterion is finally a relatively simple failure criterion that may be applied to linear-elastic materials and has been widely validated in a significant number of brittle and quasi-brittle structural materials (e.g., [4; 7; 8]). Its simplicity is especially evident when using the expressions of W gathered in [5].  
Meanwhile, [PERSON] proposed the Equivalent Material Concept (EMC) [9] to equate a real ductile material (thus, exhibiting elastic-plastic behaviour) with an equivalent (virtual) brittle material displaying elastic behaviour. This proposal allows elastic assessment tools, which are generally simpler, to be applied to nonlinear (ductile) materials. Some examples may be found in the literature, where the EMC is used in combination with the Theory of Critical Distances (TCD, e.g., [10]), the Maximum Tangential Stress criterion (MTS, e.g.,[11]) or the Average Stress Energy Density Criterion (e.g., [12]), among others. The results are promising, with the different resulting combined criteria (EMC-TCD, EMC-MTS and EMC-ASED) providing accurate predictions of fracture loads in different types of materials. Moreover, [PERSON] has recently proposed the Fictitious Material Concept [13], which has a similar purpose (the use of linear-elastic tools in nonlinear situations) but would seem to be more oriented to materials developing a larger nonlinear behaviour. All this suggests that there are materials which are linear-elastic enough to be analysed without any need for the EMC, there are materials for which the EMC may be used to obtain accurate analyses of fracture processes by using simple linear-elastic tools and, finally, there may be materials for which the use of the EMC is not enough to obviate the nonlinear behaviour of the real material being analysed.  
With the aim of providing a full view of the capacities and limitations of the EMC-ASED criterion for the estimation of fracture loads in structural materials containing U-notches, this criterion is applied here to various materials (and hundreds of fracture specimens) with different levels of ductility. Thus, Section 2 describes the materials being analysed and gathers a brief description of the EMC, the ASED criterion and the combined EMC-ASED criterion; Section 3 presents the results provided by the EMC-ASED criterion; Section 4 develops the corresponding discussion; and, finally, Section 5 gathers the main conclusions.