3D finite fracture mechanics under mode I loading: the flat elliptical crack

Comptes Rendus Mécanique Pietro Cornetti, Vladislav Mantiˇc and Zohar Yosibash 3D finite fracture mechanics under mode I loading: the flat elliptical crack Volume 353 (2025), p. 725-745 Online since: 16 June 2025 https://doi.org/10.5802/crmeca.302 This article is licensed under the Creative Commons Attribution 4.0International License. http://creativecommons.org/licenses/by/4.0/ CENTRE MERSENNE The Comptes Rendus. Mécanique are a member of the Mersenne Center for open scientific publishing www.centre-mersenne.org — e-ISSN : 1873-7234 Comptes Rendus. Mécanique 2025, Vol. 353, p. 725-745 https://doi.org/10.5802/crmeca.302 Research article / Article de recherche 3D finite fracture mechanics under mode I loading: the flat elliptical crack Mécanique de la rupture finie en 3D sous chargement en mode I : la fissure elliptique plane Pietro Cornetti ,∗,a, Vladislav Mantiˇc ,band Zohar Yosibash ,c aDepartment of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy bGrupo de Elasticidad y Resistencia de Materiales, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos s/n, Sevilla, 41092, Spain cComputational Mechanics and Experimental Biomechanics Lab, School of Mechanical Engineering, Iby and Aladar Fleischman Faculty of Engineering, Tel Aviv University, Ramat Aviv 69978, Israel E-mail: [email protected] (P. Cornetti) Abstract. The determination of the remote stress causing crack propagation in an infinite 3D domain with an embedded flat elliptical crack is here revisited in the framework of the Coupled Criterion of Finite Fracture Mechanics. We started reviewing Linear Elastic Fracture Mechanics approaches, which differ by accounting for different infinitesimal crack growths. Then, we provide the solution based on Finite Fracture Mechanics: if the elliptical flaw is sufficiently small, the crack grows along iso-stress lines. For larger sizes, other crack growths may take place. Thus, the present investigation shows that assuming an iso-stress crack front may effectively provide the exact Finite Fracture Mechanics solution, particularly for small defects; on the other hand, it can be wrong for larger size, providing moreover un-conservative predictions. However, for the geometry at hand, it yields failure stress estimates differing from the actual one by a few percents. Thus, the iso-stress assumption, conjectured by Leguillon [D. Leguillon, “An attempt to extend the 2D coupled criterion for crack nucleation in brittle materials to the 3D case”, Theor. Appl. Fract. Mech. 74 (2014), pp. 7–17]— implying strong simplifications in the numerical implementation of the coupled criterion in 3D problems— seems to be largely justified by the present results. Moreover, regardless of the initial crack size, the finite growth predicted by the model results in a new elliptical crack shape closer to the circular one, meaning the eccentricity consistently decreases as the crack propagates. Résumé. La détermination de la contrainte à distance provoquant la propagation d’une fissure dans un domaine 3D infini contenant une fissure elliptique plane est ici revisitée dans le cadre du Critère Couplé de la Mécanique de la Rupture Finie. Nous commençons par passer en revue les approches de la Mécanique Linéaire de la Rupture, qui diffèrent selon la prise en compte de différentes croissances infinitésimales de la fissure. Ensuite, nous présentons la solution basée sur la Mécanique de la Rupture Finie : si le défaut elliptique est suffisamment petit, la fissure croît le long de lignes iso-contraintes. Pour des tailles plus grandes, d’autres modes de croissance de fissure peuvent se produire. Ainsi, cette étude montre que supposer un front de fissure iso-contraintes peut effectivement fournir la solution exacte en Mécanique de la Rupture Finie, en particulier pour les petits défauts; en revanche, cela peut être erroné pour des défauts de plus grande taille, entraînant de surcroît des prédictions non conservatrices. Toutefois, pour la géométrie considérée, ∗Corresponding author ISSN (électronique) : 1873-7234 https://comptes-rendus.academie-sciences.fr/mecanique/ 726 Pietro Cornetti et al. cela donne des estimations de contrainte de rupture ne différant que de quelques pourcents de la valeur réelle. Ainsi, l’hypothèse d’iso-contraintes, avancée par Leguillon [D. Leguillon, “An attempt to extend the 2D coupled criterion for crack nucleation in brittle materials to the 3D case”, Theor. Appl. Fract. Mech. 74 (2014), pp. 7-17], impliquant des simplifications importantes dans l’implémentation numérique du critère couplé dans des problèmes 3D, semble largement justifiée par les résultats présents. En outre, quelle que soit la taille initiale de la fissure, la croissance finie prédite par le modèle aboutit à une nouvelle forme elliptique de la fissure, plus proche d’un cercle, ce qui signifie que l’excentricité diminue systématiquement au fur et à mesure de la propagation de la fissure. Keywords. Coupled criterion, Finite fracture mechanics, 3D linear elastic fracture mechanics, Elliptical cracks, Quasi-brittle materials. Mots-clés. Critère couplé, Mécanique de la rupture finie, Mécanique de la fracture élastique linéaire en 3D, Fissures elliptiques, Matériaux quasi-fragile. Funding. NEWFRAC Project (Marie Sklodowska-Curie grant agreement No. 861061), Pazy Research Foundation, Spanish Ministry of Science and Innovation, European Regional Development Fund (PID2021123325OB-I00). Manuscript received 6 February 2025, revised 17 May 2025, accepted 19 May 2025. 1. Introduction The Coupled Criterion of Finite Fracture Mechanics (CCFFM), introduced for the first time by Leguillon [1] in 2002, has proven to be an effective, yet simple, fracture criterion for obtaining the failure load in a variety of structural problems, spanning from size effect (e.g. [2]) to stress concentration/intensification (e.g. [3–5]) in homogeneous materials, from composite materials (e.g. [6–8]) to bonded joints (e.g. [9]), from static loadings to dynamic (e.g. [10,11]) and fatigue (e.g. [12,13]) loadings. With respect to Linear Elastic Fracture Mechanics (LEFM), a major advantage is its applicability to any geometry, cracked or plain (i.e. not only cracked). With respect to more sophisticated models like the Cohesive Crack Model (CCM) or the Phase Field (PF) model for fracture, the numerical implementation of the CCFFM is usually much easier, often allowing for an analytical or semi-analytical solution for the problem at hand. Moreover, the CCFFM has proven to be often in agreement with CCM and PF models: for a comparison between CCFFM and CCM the reader is referred to [14–19]; and for a comparison between CCFFM and PF to [20–24]. Most of the CCFFM applications address two-dimensional problems, where cracks/V-notch tips are straight lines through-the-thickness. However, recently, attention has also been focussed to 3D problems, starting from Leguillon’s pioneering work [25]. The application of the CCFFM to 3D problems is challenging because the finite crack advance, differently from the 2D case, can be of any shape [26]. To overcome this difficulty, researchers often assumed a finite crack growth occurring along an iso-stress line, see e.g. [27–29]. One of our main purposes is investigating this assumption for a model problem allowing an analytical derivation in the framework of the CCFFM: a flat elliptical flaw in an infinite linear elastic medium subjected to a remote tensile stress orthogonal to the crack plane. The linear elastic stress-strain solution for an elliptical flaw under remote tensile stress dates back to Green and Sneddon [30], following (and including) the one for a penny-shaped crack provided by Sneddon [31]. Later, Irwin [32] provided the Stress Intensity Factor (SIF) values along the elliptical crack front. More recent contributions related to the stress field in the vicinity of the crack front can be found in [33,34]. For what concerns crack propagation, Lazarus [35], among different flaw shapes, considered the elliptical one and analysed the crack growth for a brittle fracture (assuming propagation where the SIF reaches the fracture toughness and regularising the crack front) and fatigue (using Paris’ equation). More numerical/practical contributions along with experimental data validation (under cycling loading) about crack propagation from Pietro Cornetti et al. 727 flat elliptical cracks can be found, e.g., in [36–38]. Finally, we refer to a recent numerical investigation where the CCFFM criterion has been exploited to deal with free edge delamination in angle-ply laminates, assuming a semi-elliptical crack shape, under static [39], fatigue [40] and thermal [41] loadings. These papers provide details on the numerical implementation of the Coupled Criterion for geometries similar to the one addressed herein. The paper is organised as follows. In Section 2 we focus on LEFM, deriving the general expression of the mode I failure stress in the presence of a flaw of any shape under the assumption of an infinitesimal iso-stress crack growth. Then, we specify the failure stress for the flat elliptical crack, providing its closed form estimate by means of SIFs. Thereafter, we show the same result can be achieved by evaluating the Strain Energy Release Rate (SERR). Following this latter procedure, we also consider the infinitesimal elliptical crack growth along the minor axis alone, showing it provides failure stress estimates lower than the iso-stress one. In Section 3 we derive the failure stress provided by the CCFFM for the flat elliptical crack. We assume the finite crack propagation to be characterised by an elliptical crack front of any shape/size. Thus, the new crack front is characterised by two parameters (the increments of the semi-axes); CCFFM implies solving a minimisation problem upon the variation of these two parameters. It will be shown that, based on the crack shape and size, two different scenarios (i.e. iso-stress and minor-axis crack propagations) can occur and the corresponding fracture stress is finally provided. The results are commented and in Section 4 some conclusions are drawn. 2. LEFM approach We first provide the failure stress according to LEFM for a flat crack with an arbitrary shape subject to a mode I loading, then specify it for the elliptical crack. Two procedures, based on SIF or SERR, are outlined and exploited. 2.1. Planar crack of arbitrary shape: iso-stress crack growth Let us consider a planar crack of arbitrary shape in an infinite body, made by a homogeneous isotropic linear elastic brittle material. Let the plane of the crack be (x,y). The remote loading is a uniform tensile stress σin the zdirection as in Figure 1. Under this assumption the crack is in pure mode I condition (or—the same—in opening mode, since, because of isotropy, the displacement field is symmetrical to the crack plane). Except for the penny-shaped crack, the SIF varies along the contour Cof the crack, i.e. KI=KI(s), sbeing the curvilinear abscissa along the contour C(Figure 1b). Denoting by KI,min and the by KI,max the minimum and maximum value of the SIF respectively, we can write: KI,min ≤KI(s)≤KI,max. (1) Assume KI=KI(s) is available in an analytic or numerical form, and the fracture toughness KIc of the material is known. One is interested in σf, the remote stress causing crack growth, i.e. failure, according to LEFM. Because of mode I, the crack expands in its plane (x,y). However, if one wishes to consider the Griffith’s SERR, there are an infinite possible shapes of infinitesimal crack growth, unlike in 2D domains where only a collinear crack growth is possible along an infinitesimal length da(abeing the crack length). A reasonable starting point is assuming (yet, an assumption) an infinitesimal crack growth defined by an iso-stress line. Since the asymptotic stress field in the direction normal to the crack contour (ris the coordinate along the normal ˆ nstarting from the crack contour C, see Figure 1b) is: σz∼ =KI p2πr(2) 728 Pietro Cornetti et al. Figure 1. A planar crack of arbitrary shape in an infinite 3D domain under uniform tensile stress normal to the crack plane (a). Crack geometry (b). Figure 2. Asymptotic stress field ahead the crack front (a) and iso-stress crack growth (b). The same stress level (e.g. σ0) is achieved at different distances from the contour C, larger where KIis larger, smaller where KIis smaller (see Figure 2a). Where KIis maximum, the stress σ0is achieved at a distance (∆a)max: (∆a)max ∼ = K2 I,max 2πσ2 0 (3) while in a generic point we have: (∆a)∼ =K2 I 2πσ2 0 (4) Dividing Equation (3) by (4), one obtains: ∆a (∆a)max ∼ =µKI KI,max ¶2 =G Gmax (5) Pietro Cornetti et al. 729 where the last expression is a consequence of Irwin’s relationship G=K2 I/E′,Gbeing the SERR and E′the Young modulus of the material under plane strain conditions. As ∆atends to zero, Equation (5) defines the shape of the infinitesimal iso-stress crack growth (see Figure 2b). Expressing the Griffith infinitesimal energy balance according to LEFM (Gcbeing the material fracture energy) by an integral along the curve C, one obtains: IC G(s)∆ads=IC Gc∆ads(6) Dividing both sides of Equation (6) by (∆a)max and substituting Equation (5) into Equation (6), one obtains the condition for crack growth: Giso =IC G2(s)ds IC G(s)ds=Gc(7) The ratio between the integrals is somehow an equivalent-2D SERR, since crack growth occurs whenever this value reaches the material fracture energy Gc, like in 2D problems. We named it Giso since it is the equivalent-2D SERR under the assumption of iso-stress crack growth. Note that, from a mathematical point of view, Giso is the contra-harmonic mean (sum of the squared values divided by sum of values, see Appendix A) of the SERR values evaluated along the crack contour. Among different averages (i.e. harmonic, geometric, arithmetic, quadratic, etc.) the contra-harmonic mean is the highest one, thus affected by large values and close to the maximum value of the variable. By Irwin’s relationship we can get also the equivalent-2D SIF KI,iso, which provides the failure stress when it equals the material fracture toughness KIc: KI,iso =v u u u u u t IC K4 I(s)ds IC K2 I(s)ds=KIc (8) 2.2. Elliptical crack: iso-stress crack growth by SIF values As a particular case, we consider the elliptical flat crack shown in Figure 3a. The failure stress according to LEFM assuming an iso-stress crack growth is here obtained by the results in Section 2.1 (Equation (8)). The ellipse in Figure 3b is defined by semi-axes aand b(a≥b) defined as: x2 a2+y2 b2=1 (9) or in parametric form (0 ≤ϕ<2π): P(ϕ)=(x=acosϕ y=bsinϕ(10) The aspect of the ellipse is univocally defined either by the aspect ratio γ=b/a(0 <γ<1) or by the eccentricity k=p1−(b/a)2(0 <k<1). The SIF along the crack front is [32]: KI=σpπb E(k2) 4 q1−k2cos2ϕ(11) where E(k2) is the complete elliptic integral of the second kind (see Appendix B). From Equation (11) the maximum value of the SIF KI,max is at point B(i.e. on the minor axis, ϕ=π/2) while the minimum value of the SIF KI,min is at point A(i.e. on the major axis, ϕ=0): KI,max =(KI)B=σpπb E(k2)(12) 730 Pietro Cornetti et al. Figure 3. An elliptical (planar) crack in an infinite medium under uniform tensile stress normal to the crack plane (a). Crack geometry (b). KI,min =(KI)A=σb E(k2)rπ a=KI,maxsb a(13) For what concerns KI,max, Equation (12) encompasses the limit cases of a penny-shaped crack (k=0 or γ=1): KI=2 πσpπb(14) constant all around the crack front, and of a through-the-thickness crack of length 2b(corresponding to k=1 or γ=0): KI=σpπb(15) To obtain the failure stress, we have to compute the 2D-equivalent SIF by means of Equation (8). Using the parametric expression of the ellipse (Equation (10)), the equivalent SIF (Equation (8)) reads: KI,iso =v u u u u tRπ/2 0K4 I(ϕ)° ° ° dP dϕ° ° °dϕ Rπ/2 0K2 I(ϕ)° ° ° dP dϕ° ° °dϕ=σpπb E(k2)v u u tRπ/2 0¡1−k2cos2ϕ¢3/2 dϕ Rπ/2 0¡1−k2cos2ϕ¢dϕ(16) where the double symmetry of the ellipse has been exploited to limit the integration interval to [0,π/2]. By analytical manipulations, the integrals in Equation (16) can be cast in terms of complete elliptic integrals of first (K(k2)) and second (E(k2)) kind (see Appendix B) as: KI,iso =2σpb E(k2)s2(2−k2)E(k2)−(1−k2)K(k2) 3(2−k2)(17) Thus, according to LEFM and iso-stress crack growth, failure is achieved whenever the above quantity reaches the material fracture toughness. Hence, the corresponding failure stress (σf)LEFM-iso is: (σf)LEFM-iso =p3 2 KIc pbE(k2)s2−k2 2(2−k2)E(k2)−(1−k2)K(k2)(18) Pietro Cornetti et al. 731 Note that, as expected for any LEFM approach, the failure stress tends to infinity as the crack vanishes (i.e. σf→∞as b→0). 2.3. Elliptical crack: iso-stress crack growth by SERR evaluation We show in this sub-section that same results in Section 2.2 are obtained also by proper evaluation of the SERR. This procedure will be exploited in the following to remove the assumption of iso-stress crack growth. Computation of the SERR Gcan be performed without the need of SIF (i.e. without exploiting Irwin’s relationship). It is the way followed originally by Griffith in his 1921 seminal work. Accordingly, under load control, one has: G=lim ∆A→0 ∆Φ ∆A(19) To compute Giso (i.e. the SERR assuming an iso-stress crack growth) by Equation (19) one need to know (i) the iso-stress contour lines and (ii) the change in strain energy ∆Φ due to the (finite) variation ∆Aof the crack surface. These ingredients can be derived by Green and Sneddon [30] solution. Accordingly, because of the remote stress the planar elliptical crack takes the shape of an ellipsoid whose semi-axes are a,band wmax (the maximum crack opening displacement), the last one given by: wmax =2σb E(k2)E′(20) By looking at the load as a uniform remote tensile stress field plus a uniform compressive stress applied on the crack faces, and by means of Clapeyron’s theorem, the strain energy Φincrement due to the presence of the crack can be computed as: Φ=σV 2(21) where Vis the volume of the (deformed) crack (i.e. the ellipsoid), whose value is: V=4 3πa b wmax (22) Hence, by combining Equations (20) to (22): Φ=4π 3E(k2) σ2ab2 E′(23) The iso-stress lines can also be derived from Green and Sneddon [30]. From their solution, the σzstress field on the crack plane (outside the crack faces) is amenable of the following analytical expression: σz σ=1+1 E(k2)"a pξsb2+ξ a2+ξ−EÃarcsin a pa2+ξ|k2!# (24) where E(ϕ|k2) is the incomplete elliptic integral of the second kind (see Appendix B) and ξis an ellipsoidal coordinate. On the crack plane (z=0), ξ=constant (ξ≥0) corresponds to a family of ellipses with equation: x2 a2+ξ+y2 b2+ξ=1 (25) Hence, we get a relevant information: the iso-stress curves are a particular family of ellipses. More precisely, Equations (24) and (25) show that, as ξincreases from 0 to ∞, the stress value decreases from ∞to σand the corresponding isostress lines are ellipses of increasing size and decreasing eccentricity. As an example, some of them are plotted in Figure 4a for an elliptical crack with aspect ratio γ=0.5. Naming by ∆aand ∆bthe increment of the semi-axes of the generic iso-stress 732 Pietro Cornetti et al. Figure 4. Possible crack growths: (elliptical) iso-stress crack growth (a); elliptical, with increment along minor axis alone (b), named minor-axis crack growth; non-elliptical crack growth, not considered in the current investigation (c). line with respect to the ones of the original elliptical flaw, from Equation (25) the relationship between them may be obtained (see details in Appendix C): ∆a=qa2+2b∆b+(∆b)2−a(26) Let us assume the crack grows by a finite amount up to a given iso-stress line, which in turns is defined by a given ξvalue. From Equation (23), the energy variation ∆Φ is: ∆Φ =4πσ2 3E′(¡b2+ξ¢pa2+ξ E£(a2−b2)/(a2+ξ)¤−ab2 E(1−b2/a2))(27) while the newly created crack surface is (difference between elliptical areas): ∆A=πhpa2+ξpb2+ξ−abi(28) The following step is inserting Equations (27) and (28) into Equation (19). Then, the limit for ∆A→0 (i.e. ξ→0) has to be evaluated. The limit takes the undetermined form 0/0, but the application of De l’Hôpital rule along with property (B4) allows the computation of the limit. Finally equating the SERR Giso to the fracture energy Gcalong with Irwin relationship yields failure stress (σf)LEFM−iso, which coincides with Equation (18). Thus, despite the different lines of thought (local vs. global), we checked that the SIFand SERR-based procedures yield the same outcome, Equation (18). Pietro Cornetti et al. 739 Figure 8. Flowchart to determine the failure stress σffor a given flaw shape (k) and size (β). their solution is simpler and already available in the literature [16,17]. Of course, the comparison being made at constant b, the failure stress decreases as the eccentricity increases. In Figure 10b the same comparison is provided at a constant flaw area (the through-the-thickness crack case is somewhat meaningless, since Aconstant and ainfinite yield bnull). Again, as in the LEFM analysis, it is apparent that, for relatively small eccentricity, the parameter governing failure stress is the flaw area. In other words, for flaw aspect ratio b/abetween 0.5 and 1, the failure stress due to the presence of an elliptical crack is (almost) equal to the one due to a penny-shaped crack of the same area. Note that, whatever is the scenario, the elliptical crack always grows toward an elliptical shape closer to that of a circle (with respect to the original elliptical shape), i.e. the eccentricity diminishes. This is a common finding in the literature, even for original flaws of shape other than the elliptical one [35,38,43,44]. Regarding the difference between iso-stress and non-iso-stress finite crack growth, for small sizes fracture propagates actually by iso-stress lines. For larger size (i.e. β>βth) the iso-stress failure stress prediction becomes larger than the minor axis one. Let us consider for instance the case considered in Figure 7b, where k=0.8 and β=10: although it is clear that the minimum load is achieved for minor-axis growth (εa=1), it is also apparent that the failure stress corresponding to iso-stress crack growth is just slightly larger. Actually the difference between the two predictions increases with size, i.e. β. For β→ ∞, FFM reverts to LEFM and the difference between minor-axis and iso-stress predictions can be directly determined from Figure 5a. For instance, for k=0.8, the relative difference (i.e. the error made using the iso-stress assumption) is about 3%, which is almost negligible from an engineering point of view. Figure 5a also shows that the largest error takes place when β→∞and k=1, i.e. a large through-the-thickness crack: in this extreme case the difference is [p(3/8) −p(1/π)/p(1/π)] ∼ =8.5%. Thus, we can conclude that, for the geometry at hand and for ellipse aspect ratios not really close to zero (i.e. except for case b≪a), the error made by using the iso-stress crack growth assumption is reasonably small. Of course, this does not mean that this is always the case, but the present case corroborates the 740 Pietro Cornetti et al. Figure 9. Failure stress vs. dimensionless flaw size β=b/lch for different flaw shape (i.e. ellipse eccentricity kor aspect ratio γ=b/a) according to LEFM (minor axis crack growth) and to FFM: γ=0.8, k=0.6 (a); γ=0.5, k∼ =0.85 (b); γ=0.2, k∼ =0.98 (c). Threshold value of βdividing iso-stress (left) and minor axis (right) finite crack growth are also highlighted. conjecture made by Leguillon [25], i.e. iso-stress crack growth can be a reasonable and effective simplifying assumption. Finally, note that extending our analysis to a more complex stress state, such as the one occurring to the present geometry when the remote tensile stress is not normal to the crack plane, would broaden the applicability of the paper. However, given the mode mixity (I, II, III) Pietro Cornetti et al. 741 Figure 10. FFM estimates of the failure stress for different flaw shape (k∼ =0, 0.6, 0.85, 0.98, 1; i.e. γ=1, 0.8, 0.5, 0.2, 0) vs. dimensionless flaw size β=b/lch (a) and vs. dimensionless flaw area (A/l2 ch) (b). and the expected non-planar (unknown) crack growth, this is a major, challenging task going beyond the scope of the current manuscript. In this sense, it would be reasonable to start with an inclined penny shaped crack, which is a configuration investigated in the past with simpler fracture criteria, see e.g. [46]; see also [47] for recent interesting experimental data. 4. Conclusions The failure remote stress causing (unstable) crack propagation in an infinite linear elastic 3D domain containing a flat elliptical crack has been obtained in an analytical form by means of the CCFFM, under the assumption that the finite crack growth can be of any elliptical shape. It is found that finite crack growth always leads to elliptical crack geometries with lower eccentricity, i.e. the crack shape tends to that of a penny-shaped crack. Differently from other investigations available in the literature, in this 3D application of FFM we removed the assumption of isostress crack growth. Particularly for large flaws we found failure stress values lower than the ones obtained by the iso-stress assumption, which, thus, must be seen as potentially dangerous since providing non-conservative predictions. Nevertheless, the difference appears to be of a few percents and, thus, the iso-stress assumption when applying CCFFM is regarded as more 742 Pietro Cornetti et al. than reasonable in engineering practice. It is noteworthy that for small elliptical defects, the failure remote stress predicted by the present FFM procedure can be significantly smaller than that obtained by LEFM assuming infinitesimal crack growth. Notably, the failure remote stress predicted by LEFM depends on the assumed shape of the infinitesimal crack increment. Declaration of interests The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations. Acknowledgements The present manuscript is the final version of what presented in the Sevilla workshop within NEWFRAC Project (Marie Sklodowska-Curie grant agreement No. 861061), which is gratefully acknowledged by all the Authors, and later, at the International Conference of Fracture held in Atlanta, June 2023 (ICF15). In Atlanta the Authors had—as usual—fruitful and stimulating discussions about the topic investigated in the present manuscript with Prof. Dominique Leguillon: although he passed away, we wish to thank him and dedicate the present work to his memory. ZY is grateful for the partial support of this research by the Pazy Research Foundation. The research of VM was partially supported by the Spanish Ministry of Science and Innovation and the European Regional Development Fund (PID2021-123325OB-I00). Finally, PC wishes to thank also Prof. Veronique Lazarus for pleasant and useful discussions had at Sorbonne Université, where he was upon Prof. Leguillon’s invitation. Appendix A. The Contra-harmonic mean (C) was introduced by Eudoxus from Cnidus (408–355 B.C.) as the ratio between the sum of the squares of the values and the sum of the values themselves. The name is due to the fact that, if we consider two values aand b, the distance between the Arithmetic mean (A) and the Harmonic mean (H) is equal to the one between the Contraharmonic mean (C) and the Arithmetic mean (A): A(a,b)=a+b 2(A1) H(a,b)=2 1/a+1/b(A2) C(a,b)=a2+b2 a+b(A3) C(a,b)−A(a,b)=A(a,b)−H(a,b) (A4) For instance, if a=9 and b=1, A=5, H=1.8, C=8.2 (C−A=3.2 =A−H). Appendix B. The complete elliptic integral of the first kind reads: K(m)=Zπ/2 0 1 q1−msin2ϕ dϕ(B1) Pietro Cornetti et al. 743 The complete elliptic integral of the second kind reads: E(m)=Zπ/2 0q1−msin2ϕdϕ(B2) The derivative of the elliptic integral of the second kind with respect to mis: dE(m) dm=E(m)−K(m) 2m(B3) The incomplete elliptic integral of the second kind is: E¡ϕ|m¢=Zϕ 0p1−msin2ϑdϑ(B4) The relationship between the incomplete elliptic integral and its complete counterpart is: E³π 2|m´=E(m)(B5) Appendix C. Here we derive the relationship between the semi-axes increments in case of iso-stress crack growth, in dimensional—Equation (26)—and dimensionless—Equation (45)—form. From Equation (36) we have: ∆a=pa2+ξ−a(C1) while, from Equation (38): ξ=(∆b+b)2−b2(C2) Replacing (C2) into (C1), we get Equation (26). 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