domain: spi phys

text:
S Aligholi
email: aligholisaeed1@gmail.com

L Ponson

A R Torabi

Q B Zhang

A new methodology inspired from the theory of critical distances for determination of inherent tensile strength and fracture toughness of rock materials

Keywords: Critical distance, Intrinsic tensile strength, Fracture toughness, Point method, Notch mechanics

Measuring the intrinsic fracture properties of quasi-brittle materials like rocks is of great importance and at the same time a major issue for engineers. In this study, we explore the ability of the Theory of Critical Distances (TCD) to determine accurately both the tensile strength and the fracture toughness. To this end, we conduct ring tests and semi-circular bend tests on four rock types including a red sandstone, a white coarsegrained marble, a fine-grained granite and a coarse-grained granite. This selection covers sedimentary, metamorphic and igneous rock types with different grain sizes. The experimental data are analysed using a new methodology developed from the so-called Point Method (PM), a particular form of the TCD, from which we infer the intrinsic tensile strength and the fracture toughness of the studied rock materials. Our results are compared with those obtained from the methodology recommended by ISRM that is modified to take into account the finite notch root radius used in our experiments. The comparison is successful, supporting that the newly developed methodology is suitable to determine the intrinsic tensile strength and the fracture toughness of rock materials.

Introduction

Rocks are archetypes of quasi-brittle materials. Under compression, they generally show a rather extended non-linear regime owing to the spreading of micro-fractures before final failure takes place. Under traction, they fail through the propagation of a crack that grows through the coalescence of micro-cracks localized at the crack tip vicinity in the so-called process zone. If the spatial extent of the process zone is small with respect to the specimen size, this phenomenon is then appropriately described by the theory of Linear Elastic Fracture Mechanics (LEFM). [START_REF] Lawn | Fracture of Brittle Solids[END_REF] Within the LEFM framework, we introduce the fracture toughness 𝐾 Ic that quantifies the ability of the material to resist to crack growth. Alternatively, one can seek to determine the stress level at which the material fails in traction, thus defining the material tensile strength. This is of particular relevance in absence of an initial crack in the structure. [START_REF] Myer | Extensile cracking in porous rock under differential compressive stress[END_REF][START_REF] Haimson | ISRM suggested methods for rock stress estimation-Part 3: hydraulic fracturing (HF) and/or hydraulic testing of pre-existing fractures (HTPF)[END_REF] However, defining an intrinsic (specimen independent) tensile strength for quasi-brittle solids is a rather difficult challenge, as the load-bearing capacity of quasi-brittle specimens is known to strongly depend on its size, [START_REF] Weibull | A statistical distribution function of wide applicability[END_REF][START_REF] Bazant | Fracture and Size Effect in Concrete and Other Quasibrittle Materials[END_REF] and often overlooked in engineering practice. [START_REF] Perras | A review of the tensile strength of rock: concepts and testing[END_REF] Owing to their quasi-brittle nature, rock made structures can give rise to catastrophic failures. Therefore, the accurate determination of their failure properties is key to assess the structural resistance of rock masses, an important issue in many rock engineering practices such as tunnelling, rock cutting processes, hydraulic fracturing and rock slope stability. [START_REF] Justo | Application of the Theory of Critical Distances for the Fracture Assessment of a Notched Limestone Subjected to Different Temperatures and Mixed Mode with Predominant Mode I Loading Conditions[END_REF] In the following, the term structural properties is used when the geometrical features of the specimens or bodies do play a significant role on top of the intrinsic properties that depend only on the microstructural features of the rock materials as well as the surrounding environment. [START_REF] Atkinson | Subcritical crack growth in geological materials[END_REF] A suitable solution for defining the tensile strength of rocks consists in considering the characteristic stress level at which the material fails within the process zone of a stress concentrator or a running crack. According to the Cohesive Zone Model (CZM) for brittle cracks, [START_REF] Barenblatt | The mathematical theory of equilibrium cracks in brittle fracture[END_REF][START_REF] Xu | Numerical simulations of fast crack growth in brittle solids[END_REF] the so-called cohesive strength 𝜎 c of the material is then related to the material fracture toughness via the cohesive length ℓ c (or process zone size along the crack propagation direction) through the relation 𝐾 Ic ∝ 𝜎 c √ℓ c . [START_REF] Barenblatt | The mathematical theory of equilibrium cracks in brittle fracture[END_REF] Although appealing, this definition raises serious experimental issues: how to determine the stress level at the tip of stress concentrators, as the process zone is hardly larger than 1 mm in most quasi-brittle specimens.

Theoretically speaking, specimens without stress concentrator could be tested under direct tension to determine the material tensile strength while specimens with sharp cracks can be used to measure the material fracture toughness. However, in practice, such a procedure is neither reliably achievable nor practical. On the one hand, it turns out that specimens without stress concentrators cannot be used to determine the tensile strength of rock materials. The reasons behind this observation have been largely discussed by researchers e.g. [START_REF] Weibull | A statistical distribution function of wide applicability[END_REF][START_REF] Hudson | Tensile strength and the ring test[END_REF] and relates to the stochastic (defect driven) nature of tensile failure. On the other hand, preparing sharp cracks in rock specimens is a challenging task.

Considering these issues, new methodologies based on different concepts have been proposed to reliably determine the tensile strength and fracture toughness of different materials including rocks. One of them is the so-called Theory of Critical Distances (TCD) based on notch mechanics. It aims at providing simple and practical tools to engineers including rock engineers. TCD includes a group of theories used for predicting the effects of stress concentrators on material behaviour under mechanical loads. [START_REF] Taylor | The theory of critical distances[END_REF] The TCD can take different forms and has been used with success in a wide range of engineering problems to determine or predict properties of different materials including composites, [START_REF] Ibáñez-Gutiérrez | Effect of fibre content and notch radius in the fracture behaviour of short glass fibre reinforced polyamide 6: an approach from the Theory of Critical Distances[END_REF][START_REF] Ibáñez-Gutiérrez | On the influence of moisture content on the fracture behaviour of notched short glass fibre reinforced polyamide 6[END_REF] metals, [START_REF] Susmel | Fatigue design in the presence of stress concentrations[END_REF][START_REF] Susmel | On the use of the theory of critical distances to predict static failures in ductile metallic materials containing different geometrical features[END_REF][START_REF] Pereira | Low and ultra-low-cycle fatigue behavior of X52 piping steel based on theory of critical distances[END_REF] polymers, [START_REF] Cicero | Prediction of fracture loads in PMMA Unotched specimens using the equivalent material concept and the theory of critical distances combined criterion[END_REF][START_REF] Peron | A novel approach for assessing the fatigue behavior of PEEK in a physiologically relevant environment[END_REF] and rocks. [START_REF] Cicero | Analysis of notch effect on the fracture behaviour of granite and limestone: an approach from the theory of critical distances[END_REF][START_REF] Jenkins | Stress field behavior induced by hydraulic fracture in shale reservoirs: a practical view on cluster spacing[END_REF][START_REF] Justo | Notch effect on the fracture of several rocks: application of the theory of critical distances[END_REF][START_REF] Justo | Notch effect and fracture load predictions of rock beams at different temperatures using the theory of critical distances[END_REF] The TCD can circumvent the experimental difficulties encountered when it comes to determining the intrinsic tensile strength and fracture toughness of quasi-brittle materials. With such an approach, specimens without stress concentrators or perfectly sharp cracks are not required to determine these properties, as we will show in the following. Furthermore, notch mechanics can be applied to modify the effect of a round-tip notch on apparent fracture toughness of materials and provide engineers with accurate values of fracture toughness, [START_REF] Creager | Elastic field equations for blunt cracks with reference to stress corrosion cracking[END_REF][START_REF] Gomez | Failure criteria for linear elastic materials with Unotches[END_REF][START_REF] Lazzarin | A generalized stress intensity factor to be applied to roundedV-shaped notches[END_REF][START_REF] Tanné | Crack nucleation in variational phase-field models of brittle fracture[END_REF] as also detailed later.

This paper is organized as follows. First, we present the studied rock materials as well as the experimental and analytical methods adopted for this study. A brief theoretical background on the methodology employed to analyse the ring tests and bending tests carried in this study is provided in Section 2. Section 3 presents our main results including a discussion. Finally, the conclusions of our study are drawn in Section 4.

Materials and Methods

Four different rock types including a red sandstone, a white coarse-grained marble, a fine-grained granite and a coarse-grained granite are selected for this study. This selection covers sedimentary, metamorphic and igneous rock types with different grain sizes. The PM form of the TCD is applied to measure accurately the failure properties of these rock materials including tensile strength and fracture toughness. To check the validity of the proposed PM, the fracture toughness of the tested rocks is also measured according to the ISRM Suggested Method [START_REF] Kuruppu | ISRM-suggested method for determining the Mode I static fracture toughness using semi-circular bend specimen[END_REF] modified to take into account the finite radius of the notch used in our experiments.

A modified version of the PM based on CZM

The PM is the simplest form of the TCD. [START_REF] Taylor | Predicting the fracture strength of ceramic materials using the theory of critical distances[END_REF] Its failure criterion has been defined by Taylor 30 as follows: 'Failure will occur when the stress at a distance 𝐿/2 from the notch root is equal to 𝜎 0 '. This translates as:

(1) 𝜎(𝐿 2 ⁄ ) = 𝜎 0
where L is a characteristic distance, and 𝜎 0 is the inherent tensile strength of the material. If the stress distribution ahead of a stress concentrator and the characteristic distance are known, then the inherent tensile strength can be determined. As justified in Appendix A, the material fracture toughness 𝐾 Ic can finally be estimated from the relation:

(

2) 𝐿 = 1 𝜋 ( 𝐾 Ic 𝜎 0 ) 2
Although the PM has been successfully applied to a large range of fracture problems, it remains a phenomenological method. [START_REF] Taylor | The Theory of Critical Distances: A New Perspective in Fracture Mechanics[END_REF] Interestingly, it is intimately connected to the CZM of failure, which rigorously extends LEFM to elasto-damageable solids. In its simplest version, CZM introduces a cohesive stress 𝜎 c , below which the material behaves elastically and beyond which it does not sustain any mechanical load. This approach predicts the spatial extent of the fracture process zone, also called the cohesive zone, through the Dugdale-Barenblatt (D-B) formula (see Appendix A): [START_REF] Barenblatt | The mathematical theory of equilibrium cracks in brittle fracture[END_REF][START_REF] Dugdale | Yielding of steel sheets containing slits[END_REF] (3)

ℓ c = 𝜋 8 ( 𝐾 Ic 𝜎 c ) 2
This formula is almost identical to Eq. ( 2) up to a constant 𝜋 2 /8 ≈1. 23. On top of it, considering the tensile stress distribution 𝜎(𝑟) = 𝐾 I /√2𝜋𝑟 ahead of a running crack as predicted by LEFM, one infers the relation 𝜎(4ℓ c 𝜋 2 ⁄ ) = 𝜎 c that is similar to Eq. (1). In the following, we use Eq. (3) instead of Eq. ( 2), as it derives from a well-identified assumption, namely the existence of a unique stress level that provides both the elastic limit and the failure threshold of the material, but we use the following PM based methodology to determine both 𝜎 c and ℓ c .

Specimens with different notch geometries are loaded up to failure. Following Eq. ( 1), the point of intersection of the stress distribution ahead of the stress concentrators at the onset of failure is expected to provide the material tensile strength. Following the previous interpretation of the PM based on CZM, two extreme stress concentrators, i.e. a sharp notch (very high-stress concentration) and a flat free surface (no stress concentration), are best suited. However, in practice, machining very sharp notches and initiating a crack from a flat free surface are quite difficult to achieve in rock materials.

To circumvent these difficulties, Semi-Circular Bend (SCB) specimens with a notch root radius of about 350m and ring specimens with an inner radius of around 14mm are used to produce the highest and lowest possible stress concentrations, respectively. Despite the discrepancy between these specimens and the perfect concentrators expected theoretically, our method provides accurate values of tensile strength, as we will show in Section 3.

Ring test

Rock rings are used in the following as the low-stress concentrator specimens. This test geometry has been used in the past to measure the apparent tensile strength of rocks and other brittle materials. [START_REF] Hobbs | An assessment of a technique for determining the tensile strength of rock[END_REF][START_REF] Zhang | Measurement of tensile strength of nuclear graphite based on ring compression test[END_REF] Note however the apparent tensile strength is a structure-dependent property rather than an inherent material property. [START_REF] Hudson | Tensile strength and the ring test[END_REF][START_REF] Bai | DEM investigation of the fracture mechanism of rock disc containing hole(s) and its influence on tensile strength[END_REF] The difference between the value of the apparent tensile strength and 𝜎 c results from the combination of three factors: (1) the probabilistic nature of the resistance of materials to tensile loading; (2) the complexity of the failure process involving the initiation of a crack by damage accumulation before it can propagate; and (3) the calculated stress following a linear elastic assumption may not be the 'real' stress experienced by the material. [START_REF] Hudson | Tensile strength and the ring test[END_REF] The minimum diameter of the internal hole that could be drilled into the sandstone and the marble is about 3mm, while it is about 6mm for granites (Fig. 1-c). Rings with four different inner diameters are prepared for the sandstone and the marble, whereas three different ring specimens are prepared for granites. Moreover, normal disk specimens with no hole are also prepared and tested for all rock types. At least three different specimens for any geometry are tested and the average of calculated tensile strengths for each rock type/geometry is used for further analyses. The outer diameter and thickness of the rings/disks are around 75 and 30 mm, respectively. Note that, following the analysis of Fillon, [START_REF] Filon | The stresses in a circular ring[END_REF] the ratio of the inner to the outer diameter of our ring specimens is less than or equal to 0.4 so that the tensile mode of failure dominates over the compressive one. [START_REF] Hobbs | An assessment of a technique for determining the tensile strength of rock[END_REF] The driving rate of the cross-head for all our tests is set to 0.05 mm/min.

The apparent tensile strength 𝜎 max is defined as the maximum stress level applied locally to the material at the onset of failure, assuming that it behaves elastically everywhere. It then follows:

(4)

𝜎 max = 𝑃 max 𝜋𝐵𝑅 0 [6 + 38(𝑅 𝑅 0 ⁄ ) 2 ]
that provides the tensile stress applied to the inner surface of the specimen at the applied failure load 𝑃 max . For disk specimens for which 𝑅 = 0, the maximum applied stress is located at the center of the specimen and follows:

(5)

𝜎 max = 𝑃 max 𝜋𝐵𝑅 0
Here, 𝐵 is the ring thickness while 𝑅 and 𝑅 0 are the inner and outer radii of the ring, respectively.

Following Torabi et al., [START_REF] Torabi | Size effects on brittle fracture of Brazilian disk samples containing a circular hole[END_REF] Kirsch's solution together with Hobbs' correction [START_REF] Hobbs | An assessment of a technique for determining the tensile strength of rock[END_REF] are used to determine the tensile stress distribution 𝜎 𝑥 (𝑦) along the loading axis 𝑦 (see the schematic of the ring specimen shown in Fig. 1-a for the definition of the axes 𝑥 and 𝑦):

(6) 𝜎 𝑥 (𝑦) = 𝜎 max 2 (2 -2 𝑅 2 𝑦 2 + 12 𝑅 4 𝑦 4 ) 𝐹 corr
Here, 𝐹 corr is a correction factor that should be taken into account for sufficiently large 𝑅 𝑅 0 ⁄ ratios, which follows:

(7)

𝐹 corr = 1 + 19 3 ( 𝑅 𝑅 0 ) 2 .
In the course of the ring experiments, we observe an interesting phenomenon that we would like to discuss.

As shown in Figs. 2-b and 2-d, the mechanical response of the ring specimen with the largest inner radius shows two peaks, the first one being larger than the second one. It turns out that full failure of the ring specimen took place in two steps. First, as the load is increased, the tensile strength of the material is reached and failure takes place at point A (see Fig. 2-c). After stress drop, the sample is still able to sustain load. As a result, the applied load increases again, starting from a lower level until it reaches a second time the tensile strength of the material at point C (see Fig. 2-c). It is interesting to notice that each half of the sample can still bear some compressive load until the tensile strength of the material is reached a second time at point C, and providing a good evidence that the sample has been split under pure tension at point A.

The first and second peaks in Fig. 2-d corresponds to the fractures labelled in Fig. 2-c and located at points A and C, respectively. From this observation, it can be concluded that ring test is suitable to measure the tensile strength. From recorded videos by high-speed cameras, we do observe that rings with smaller internal holes are always separating from point A in a tensile mode as well (Fig. 3), as expected from direct numerical simulations of failure in such specimens. [START_REF] Zhang | Measurement of tensile strength of nuclear graphite based on ring compression test[END_REF]

Semi-circular bending test

The notched semi-circular geometry is used for preparing rock specimens with high-stress concentrators.

Various methods have been used to determine the fracture toughness of rock materials. e.g. [START_REF] Kuruppu | ISRM-suggested method for determining the Mode I static fracture toughness using semi-circular bend specimen[END_REF][START_REF] Isrm | The complete ISRM suggested methods for rock characterization, testing and monitoring: 1974-2006[END_REF][START_REF] Nara | Study of subcritical crack growth in andesite using the Double Torsion test[END_REF][START_REF] Ponson | Depinning transition in the failure of inhomogeneous brittle materials[END_REF] The method suggested by ISRM [START_REF] Kuruppu | ISRM-suggested method for determining the Mode I static fracture toughness using semi-circular bend specimen[END_REF] relies on SCB specimens that is rather simple to machine and provides good repeatability. e.g. [START_REF] Chong | New specimen for fracture toughness determination for rock and other materials[END_REF][START_REF] Aliha | Application of cracked triangular specimen subjected to three-point bending for investigating fracture behavior of rock materials[END_REF][START_REF] Zhang | Effect of loading rate on fracture toughness and failure micromechanisms in marble[END_REF][START_REF] Nejati | On the directional dependency of Mode I fracture toughness in anisotropic rocks[END_REF] Herein, SCB specimens are prepared and tested according to ISRM. Multiple SCB specimens for each rock type are tested and the average generalized (or apparent) fracture toughness 𝐾 Ic U is calculated as follows:

(8) 𝐾 Ic U = 𝑌 ′ 𝑃 max √𝜋𝑎 𝐷𝐵
Here 𝑎, 𝐵, 𝐷, and 𝑃 max are the notch length, the specimen thickness, the diameter of the SCB specimen and the maximum applied load, respectively (see Fig. 4). The notch length of the tested SCB specimens is comprised between 14 to 16 mm while the notch tip radius is 350 microns. The diameter and the thickness of the SCB specimens range from 74 to 76 mm and 29 to 31 mm, respectively. Finally, 𝑌 ′ gives the non-dimensional stress intensity factor derived using the finite element method while assuming plane-strain conditions. [START_REF] Kuruppu | ISRM-suggested method for determining the Mode I static fracture toughness using semi-circular bend specimen[END_REF] where 𝑠 is the span length which is between 37 to 38 mm for all our tests while 𝛽 is equal to 2𝑎/𝐷. Creager-Paris solution [START_REF] Creager | Elastic field equations for blunt cracks with reference to stress corrosion cracking[END_REF] provides the stress distribution in SCB specimens with a blunted notch of radius 𝜌:

(10) 𝜎(𝑥, 0) = 2𝐾 U √𝜋 𝑥 + 𝜌 (2𝑥 + 𝜌) 3 2
⁄ using the coordinate system depicted in Fig. 4-b. 𝐾 U , the apparent stress intensity factor, is provided by Eq. (8) after replacing the failure load 𝑃 max by the current applied load 𝑃.

Direct fracture toughness measurement using SCB tests

To test the ability of the proposed methodology to accurately measure the fracture toughness of rock materials, we proceed to an independent measurement of 𝐾 Ic using the failure load of the semi-circular bending tests. The basic idea is to consider that at the onset of failure, the imposed stress intensity factor (determined from Eqs. ( 8) and ( 9) at the tip of the notch) reaches the fracture toughness value 𝐾 Ic . However, in our experiments, the notch tip radius is too large to be neglected. Compiling a large set of experimental data, Gomez et al. [START_REF] Gomez | Failure criteria for linear elastic materials with Unotches[END_REF] determined the ratio of the apparent fracture toughness (resulting from the finite notch root radius) over the actual material fracture toughness:

(11) 𝐾 Ic U 𝐾 Ic = √1 + 𝜋 4 𝜌 (𝐾 Ic 𝜎 c ⁄ ) 2
Here, the intrinsic tensile strength 𝜎 c is determined using the PM based methodology while 𝜌 measured from 2D slices of SCB specimens scanned by means of X-ray tomography, is found to be close to 350 microns (Fig. 4-d). 𝐾 Ic U corresponds to the apparent fracture toughness measured experimentally. As the material fracture toughness 𝐾 Ic appears on both sides of this equation, Eq. ( 11) must be solved iteratively following the procedure described in Appendix B and illustrated in Figs. 6-a and 6-b. It turns out that the ratio 𝐾 Ic /𝐾 Ic U is close to 0.95 for the four materials investigated.

Beyond the particular cases of the fracture tests carried in this study, Figs. 6-c and 6-d depicts the effect of the cohesive length in comparison to the notch root radius on the ratio 𝐾 Ic /𝐾 Ic U . In particular, it can be seen that small notch radii compared to cohesive length give rise to 𝐾 Ic ≈ 𝐾 Ic U .

Results and discussion

Size effect on tensile strength measurements using ring specimens

A natural first step in assessing the structure-independent tensile strength of the rock materials investigated is to determine the apparent (structure dependent) tensile strength 𝜎 𝑚𝑎𝑥 as a function of the ring geometry. Ring specimens with various inner radii as well as disk specimens from different rock types are tested for such a purpose. Fig. 7-a shows the value of 𝜎 𝑚𝑎𝑥 as a function of the inner hole radius as obtained after averaging over different samples. It appears that the apparent tensile strength strongly depends on the hole radius (Fig. 7-b).

This calls for a more advanced method of analysis to determine the inherent tensile strength.

Intrinsic tensile strength and material fracture toughness

The methodology described in Section 2.1 based on the SCB specimens with a notch root radius of 350 microns (high concentrator) and the ring specimens with inner radii of 13-15mm (low concentrator) is applied in Fig. 8 for the four rocks investigated. According to Eq. ( 1), the intersection point of the tensile stress distributions at the onset of failure for both ring and SCB specimens provides both the inherent tensile strength and the cohesive zone length. The fracture toughness value is then obtained from Eq. ( 3) using the D-B relationship. The results obtained for the four rocks investigated are summarized in Table 1.

The validity of the proposed methodology is now tested. First, we compare the fracture toughness value predicted by Eq. ( 3) with the fracture toughness value measured directly from the notched SCB specimen, after taking into account the effect of its finite notch root radius. For this purpose, the value of 𝜎 𝑐 determined previously is used in Eq. ( 11), providing the ratio 𝐾 Ic /𝐾 Ic U between the inherent fracture toughness and the apparent one, as explained in Sec. 2.4. The comparison shown in Table 2 is excellent. We then compare in Table 3 the cohesive zone length as measured from our method using the intersection point between both stress distributions at the onset of failure (see Fig. 8) with the one predicted from D-B Formula using the fracture toughness determined directly from the notched SCB tests and modified for the rounded notch tip effect. Here also, the agreement is very good. Last but not least, we did proceed to an independent measurement of the process zone length from statistical fractography, a technique that consists in analysing the statistics of fracture surface roughness to extract the characteristic size of the damage processes taking place at the crack tip vicinity during propagation, and found values comparable to the one determined in this study, i.e. in the range 0.7 -1 mm.

These results call for a few comments. First, the intrinsic tensile strength varies in the range 8 -25 MPa for the different rock materials investigated. This is somehow larger, however comparable to the values reported in the literature for such materials. [START_REF] Perras | A review of the tensile strength of rock: concepts and testing[END_REF][START_REF] Li | The Brazilian disc test for rock Mechanics applications: review and new insights[END_REF] Note that using smaller hole radius for the low stress concentrator gives larger values of 𝜎 𝑐 , as inferred from Fig. 9 where the tensile stress distribution at the onset of failure is represented for the different specimen geometries. First, considering stronger stress concentrator is not compatible with the justification of Eq. ( 1) that requires the combination of a high and a low stress concentrator (see Section 2.1). Second, it leads to smaller values of cohesive length, of the order of a few hundred of microns, that do not match with the results inferred from the statistical analysis of the fracture surfaces.

We then would like to discuss the fracture toughness values measured for the four rocks investigated. Our methodology provides accurate fracture toughness values, in agreement with values of 𝐾 Ic determined directly from the notched SCB specimens using the ISRM suggested method. Afterwards, it turns out that the value of the apparent fracture toughness obtained with a notch root radius less than 500 microns as suggested by ISRM already provides a rather good estimate of 𝐾 Ic for the rocks investigated. Overall, precision achieved by both methods is remarkable.

Discussion

So far, the results are interpreted and it is concluded that PM is successful in order to measuring intrinsic tensile strength and material fracture toughness, especially when the D-B formula is being used to determine the material fracture toughness. This conclusion can open new doors for future researches and needs further enlightening. The main questions should be answered concerning these results are: 1) Why PM is successful?

2) Why D-B formula is giving better results?

To answer these fundamental questions, first, we need to give a brief background of PM, and both original and developed methods used to calculate the characteristic length L. As discussed in section 2, Eq. ( 1) introduced by Peterson [START_REF] Peterson | Notch-sensitivity[END_REF] is the main failure criterion of PM. This formula is considering a material dependent characteristic length inferring the estimated stress for a particular geometry at a distance L/2 from its concentrator is equal to inherent tensile strength of the material. In this argument, L is a constant characteristic length depends on intrinsic properties of a material, and is independent from geometry of specimen. Therefore, for homogeneous materials, stress distribution, as a function of distance from concentrator, of any two different geometries would intersect at a point showing material properties. The abscissa of this point is half of the material characteristic length and its ordinate is intrinsic tensile strength.

Although PM is successful in practice, from above presentation, there are two major facts lacking applicability and supportive theoretical arguments. First, materials are not homogeneous and there should be always some rooms for experimental calibrations, even though one uses the highest and lowest possible stress concentrators for determining the intersection point as it is done in this study. Second, how the L should be determined to further estimate material fracture toughness and why L/2 is corresponding to material tensile strength. The first issue concerning applicability of this model is out of scope of this study and will be addressed in a future work. From the results of this study, the second issue turns out to be very important and can increase the accuracy of PM with some modifications. Not solid, but it is reasonable to consider the stress at half of the characteristic length L would be equal to intrinsic tensile strength. It is somehow representing the average stress over L that lead to failure of material. It is notable that this argument is close to CZM assumptions for derivation of Eq. 3 (refer to Appendix A).

Barenblatt [START_REF] Barenblatt | The mathematical theory of equilibrium cracks in brittle fracture[END_REF] and Dugdale 31 separately and at the same time have developed basis for the CZM. Their models have different theoretical arguments and physics but treat the problem with similar procedures.

Barenblatt model is looking at the problem at microscopic scale and considers inter-molecular cohesive stresses at a large enough area for applying continuum fracture mechanics, and is suitable for brittle materials. Dugdale model is a macroscopic model and considers perfectly plastic material behaviour inside the process zone ahead of crack tip. In these models, the process zone (the cohesive zone in Barenblatt model or the plastic zone in Dugdale model) in direction of applied load (y) is small compared to its length in crack propagation direction (x). Moreover, in Barenblat model the length of cohesive zone is small in comparison to crack length ℓ c ≪ 𝑎, and the distribution of cohesive stress 𝜎 c in the cohesive zone for a given material is always the same and independent of the external load. [START_REF] Gross | Fracture Mechanics With an Introduction to Micromechanics[END_REF] These two models in the most simplified scenario (strip or line model) will be end up with the same closed form solution, and this is why Eq. ( 3) referred to as D-B formula (refer to Appendix A).

Overall, considering CZM and PM descriptions it makes sense to employ D-B formula instead of Eq. ( 2)

for calculating the characteristic length. On the one hand, PM asserts L is material dependent and can be determined by testing specimens from same material but different geometries. On the other hand, Barenblatt model argues distribution of 𝜎 c in the cohesive zone for a given material is always the same and depends on material properties. Finally, although these formulas considering different stress distributions over L or ℓ c , both

Eq. ( 1) and Eq. ( 3) are considering average stress at L/2 at the moment of failure, and it seems D-B model assumptions are closer to reality.

Conclusions

In this study, a TCD based methodology is examined to determine two key mechanical properties of rock materials namely intrinsic tensile strength and material fracture toughness. The first and foremost conclusion is that PM form of TCD is a suitable means to reliably determine intrinsic tensile strength and material fracture toughness of different rock types. According to our results, PM is very reliable if the cohesive length ℓ c is considered as the characteristic length L in this method.

Following the results of this study, it turns out that plane disk specimens without stress concentrators cannot be used to measure tensile strength of rock materials, and tensile strength is underestimated if plane specimens are used. However, it could provide engineers with a safe and conservative estimation despite the fact that it would often increase the costs of a project. From the observations in the course of ring experiments, it can be concluded that ring test is a suitable means to measure apparent tensile strength of rock materials. Tensile strength of rocks revealed to depend on their structural properties due to the facts discussed by Hudson. 11 However, if a specific value should be reported for a particular rock type and is needed by analytical or numerical solutions, then intrinsic tensile strength of the rock can be determined following the procedure in this study with the aid of newly developed PM.

Brittle nature of rock materials is a major issue for fabricating sharp notch in SCB specimens to successfully determine material fracture toughness. In this study, notch mechanics and practical developments in similar materials were introduced to circumvent this difficulty. From the experimental observations and comparison with different methods, it is being suggested that Gomez et al. [START_REF] Gomez | Failure criteria for linear elastic materials with Unotches[END_REF] formula can be used to successfully rectify the notch root radius effect on determining fracture toughness of rock materials. However, if the notch root radius is smaller than the cohesive length, the ISRM suggested method [START_REF] Kuruppu | ISRM-suggested method for determining the Mode I static fracture toughness using semi-circular bend specimen[END_REF] is a reliable method for determining fracture toughness of rock materials. Based on the results of this study, the cohesive length is around 1mm for rock materials. Therefore, if the notch width is less than 1mm or notch root radius is less than 500 microns, as specified in the ISRM suggested method, [START_REF] Kuruppu | ISRM-suggested method for determining the Mode I static fracture toughness using semi-circular bend specimen[END_REF] then the material fracture toughness measured by this method is reasonably close to the real value.

Although, the results are satisfying, there is a mismatch between the actual location and the considered intersection point for estimating the intrinsic tensile strength because of material heterogeneities and theoretical assumptions. This is why fracture toughness values estimated from SCB tests modified for notch root effect and developed PM are a bit different. The question remains open in this study is that how this issue can be rectified and if there is any way to measure L or ℓ c for different materials to get the best possible results. In other words, the length of the fracture process zone in the direction of crack propagation or the cohesive length ℓ c should be quantified to determine the actual stress at the tip of cohesive zone right before failure of a material in order to precisely measure the material fracture toughness. Further investigations and future researches are required to exactly quantify the length of fracture process zone and answer this question.

(A1) 𝜎(𝑟) = 𝜎√ 𝑎 2𝑟

if 𝑟 ≪ 𝑎 i.e. for the comparatively close points to the crack tip for an applied tensile stress 𝜎.

According to LEFM for the same conditions mode I stress intensity factor 𝐾 I can be calculated: where 𝐾 I is given in Eq. (A2) and 𝐾 I c can be solved using Eq. A6. Now, the equilibrium can be rewritten as follows:

(A8) 𝜎√𝜋𝑐 = 2𝜎 c √ 𝑐 𝜋 cos -1 𝑎 𝑐 ; → √𝜋𝑐 (𝜎 - 2𝜎 c 𝜋 cos -1 𝑎 𝑐 ) = 0; ∴ 𝑎 𝑎 + ℓ c = cos ( 𝜋𝜎 2𝜎 c )
Finally, by two reasonable assumptions including ℓ c ≪ 𝑎 and 𝜎 ≪ 𝜎 c this equilibrium can be solved for ℓ c :

(A9) 1 - ℓ c 𝑎 = 1 - 𝜋 2 𝜎 2 8𝜎 c 2 ; (B1) 𝐾 𝐼𝑐 𝑈 𝐾 𝐼𝑐 = √1 + 𝜋 4 𝜌 𝑙 ch
where 𝑙 𝑐ℎ is a characteristic length given in Eq. B2:

(B2) 𝑙 ch = (𝐾 Ic 𝜎 c ⁄ ) 2 .
Then, an iterative process for estimating 𝐾 Ic can be presented in four steps as follows: The larger the 𝜌 or the smaller the 𝑙 ch , the greater the number of required iterations for convergence (see Fig.

4-a).

This method has a limitation that is connected to the ratio of 𝜌 𝑙 ch ⁄ . Based on some numerical examples, it turns out that this iterative process works well if 𝜌 is smaller or slightly larger than 𝑙 ch . It is notable that if 𝜌 = 𝑙 ch , then 𝐾 Ic 𝐾 c U ⁄ ≈ 0.5 (after convergence).

List of Figures

Failure

  of SCB specimens is recorded by means of a high-speed camera (Fig 5-a). It can be clearly seen that the fracture initiates from the notch tip and propagates parallel to the axis of application of the forces, as expected. Typical load-extension curves obtained for different rock types are shown in Fig 5-b.

  Fig.A2shows this simplified situation for a crack of length 2(𝑎 + ℓ c ) = 2𝑐 in an infinite body under uniaxial tensile stress 𝜎. Then, using superposition of the problem, and right before crack propagation, the following equilibrium could be reached:

  a) estimating the 𝑙 ch using Eq. (B2) by assuming 𝐾 Ic is equal to the measured generalized fracture toughness from experiment; b) estimating the material fracture toughness by replacing the measured generalized fracture toughness from experiment, notch tip radius 𝜌, and the calculated 𝑙 ch from the first step into Eq. (B1); c) updating the 𝑙 ch by replacing the estimated material fracture toughness from the second step into Eq. (B2); and d) repeating this loop several times until old and new 𝑙 ch values and accordingly material fracture toughness values converge.

Fig. 1 .

 1 Fig. 1. Ring experiment: (a) Schematic of the ring specimen; (b) A marble ring with an inner radius of 15mm under compression; and (c) Specimens with the minimum inner radii before testing.

Fig. 3 .Fig. 4 .

 34 Fig.3. Sequence of high-speed images taken from a sandstone ring with an inner radius of 3 mm showing symmetric fracture propagation from point A, as depicted in Fig.2-c.

Fig. 5 .

 5 Fig. 5. (a) Sequence of high-speed images taken from a fine-grained granite SCB specimen showing crack initiation and growth from the initial notch tip; (b) Mechanical response of the notched SCB specimens for the four rocks investigated.

Fig. 6 .

 6 Fig. 6. Graphical guides for modifying the notch root radius effect on material fracture toughness: a) A 3D guide showing the iterative process; and b) A guide for estimating 𝐾 Ic as a function of notch tip radius, intrinsic tensile strength and generalized fracture toughness. c) 𝐾 Ic as a function of notch tip radius, the cohesive length and generalized fracture toughness; and d) a single master curve for evaluating 𝐾 Ic as a function of ℓ c 𝜌 ⁄ ratio and generalized fracture toughness.

Fig. 7 .

 7 Fig. 7. Failure load 𝑃 max (a) and apparent tensile strength 𝜎 max (b) of the different rocks investigated as obtained from the different fracture tests.

Fig. 8 .

 8 Fig. 8. Application of the PM for the determination of the tensile strength: Stress distribution against distance for the two geometries displaying the highest and the lowest stress concentrations for (a) sandstone, (b) marble, (c) fine-grained granite and (d) coarse-grained granite. The point of intersection of both curves provide the intrinsic tensile strength as well as the cohesive length, as illustrated for marble in the panel (b).

Fig. 9 .Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .

 91234 Fig. 9. Tensile Stress distribution against distance at the onset of failure for different fracture test geometries and different materials: a) sandstone; b) marble; c) fine-grained granite; and d) coarse-grained granite.

Fig. 5 .

 5 Fig. 5. (a) Sequence of high-speed images taken from a fine-grained granite SCB specimen showing crack initiation and growth from the initial notch tip; (b) Mechanical response of the notched SCB specimens for the four rocks investigated.

Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .c

 6789 Fig. 6. Graphical guides for modifying the notch root radius effect on material fracture toughness: a) A 3D guide showing the iterative process; and b) A guide for estimating 𝐾 Ic as a function of notch tip radius, intrinsic tensile strength and generalized fracture toughness. c) 𝐾 Ic as a function of notch tip radius, the cohesive length and generalized fracture toughness; and d) A single master curve for evaluating 𝐾 Ic as a function of ℓ c 𝜌 ⁄ ratio and generalized fracture toughness.

  y

Table 1 .

 1 x The intrinsic tensile strength, the cohesive half-length and the material fracture toughness, determined by the developed PM.

	Rock type	𝜎 c [MPa]	ℓ c /2 [mm]	𝐾 Ic [MPa.m 0.5 ]
	Sandstone	8.4	0.53	0.44
	Marble	15.4	0.51	0.78
	Fine grained granite	24.0	0.39	1.07
	Coarse grained granite	19.8	0.48	0.98

Table 2 .

 2 Comparison of measured generalized fracture toughness 𝐾 Ic U [MPa.m 0.5 ] and modified fracture toughness 𝐾 Ic [MPa.m 0.5 ] values with those obtained using the common and developed PMs.𝐾 Ic (Gomez et al.[START_REF] Gomez | Failure criteria for linear elastic materials with Unotches[END_REF] ) 𝐾 Ic (common PM) 𝐾 Ic (developed PM)

	Rock type	𝐾 Ic U (ISRM 28 )
	Sandstone	0.45	0.43	0.49	0.44
	Marble	0.80	0.76	0.86	0.78
	Fine grained granite	1.08	1.00	1.19	1.07
	Coarse grained granite 1.02	0.97	1.11	0.98

Table 3 .

 3 The cohesive length ℓ c [mm] as per D-B formula determined both from SCB tests modified for the rounded notch tip effect and the developed PM.

	Rock type	Experimental	Developed PM
	Sandstone	1.03	1.07
	Marble	0.96	1.01
	Fine grained granite	0.68	0.78
	Coarse grained granite	0.94	0.96

SCB test: High stress concentration Ring test: Low stress concentration Applying Point Method Performing experiments Modifying PM Validating the results

	Graphical Abstract PDF
	P							Stress distribution vs.	𝜎 Τ 𝓁 c 2 = 𝜎 c
									distance from concentrator
	a	B	R 0
	s	P		Creager-Paris solution 𝜋 2𝑑 + 𝜌 Τ 3 2 𝜎 𝑑, 0 = 2𝐾 𝐼𝑐 𝑈 𝑑 + 𝜌 SCB test:	𝜎 c	fracture toughness Estimating
	y							0	𝓁 𝑐 /2	d
		x	2R 2R 0	Ring test:
				𝜎 𝑑, 0 =	𝜎 𝑚𝑎𝑥 2	2 -2	𝑅 2 𝑑 2 + 12	𝑅 4 𝑑 4 1 +	3 19	𝑅 0 𝑅	2
				Kirsch's solution and Hobbs' correction

𝐾 𝐼𝑐 = 𝑓(𝓁 𝑐 , 𝜎 𝑐 ) 𝐾 𝐼𝑐 (Experimental) ≈ 𝐾 𝐼𝑐 (Developed PM)

Acknowledgements S. A. wishes to acknowledge the support from Australian Government Research Training Program (RTP) Scholarship and the Monash International Tuition Scholarship (MITS). This research was financially supported by the Australian Research Council (DE200101293), Australian Synchrotron, the MASSIVE HPC facility (www.massive.org.au), and the Monash Centre for Electron Microscopy (MCEM).

Declaration of competing interest

The authors would like to declare that there is no conflict of interests regarding publication of this article.

Appendix A

Following Taylor, 30 derivation of Eq. (2) starts by Westergaard's equation [START_REF] Westergaard | Bearing Pressures and Cracks[END_REF] that provides estimation of tensile stress 𝜎(𝑟) in the direction of crack propagation as a function of distance r from the crack tip, for a through-thickness crack of a half-length a in an infinite body. The equation can be read as:

Appendix B

The iterative method for estimating material fracture toughness 𝐾 Ic from Gomez et al. [START_REF] Gomez | Failure criteria for linear elastic materials with Unotches[END_REF] practical formula (Eq. 11) can be presented as follows. First of all, Eq. 11 can be divided in the two following formulas: Table 1. The intrinsic tensile strength, the cohesive half-length and the material fracture toughness, determined by the developed PM.

List of Tables


Edouard Pauwels

On the nature of Bregman functions

Let C ⊂ R n be convex, compact, with nonempty interior and h be Legendre with domain C, continuous on C. We prove that h is Bregman if and only if it is strictly convex on C and C is a polytope. This provides insights on sequential convergence of many Bregman divergence based algorithm: abstract compatibility conditions between Bregman and Euclidean topology may equivalently be replaced by explicit conditions on h and C. This also emphasizes that a general convergence theory for these methods (beyond polyhedral domains) would require more refinements than Bregman's conditions.

Introduction

Let C ⊂ R n be convex compact with nonempty interior, h : C → R be convex. We assume throughout the article that h is Legendre:

• h is essentially smooth: continuously differentiable on intC, such that ∥∇h∥ goes to +∞ when approaching the boundary of C.

• h is strictly convex on intC.

We will also assume throughout the text that h is continuous on C. The Bregman divergence associated to h is then for all y ∈ C and x ∈ intC, D h (y, x) = h(y) -h(x) -⟨∇h(x), y -x⟩ .

Fejérian sequences:

Let f : C → R be convex proper lower semi-continuous. Consider the problem

min x∈C f (x) (1)
and denote by S ⊂ C, the solution set of [START_REF] Alvarez | Hessian Riemannian gradient flows in convex programming[END_REF]. The following definition is adapted from the Euclidean setting [START_REF] Combettes | Fejér-monotonicity in convex optimization[END_REF].

Definition 1. Let (x k ) k∈N be a sequence in intC, it is called Fejérian, or Fejér, for problem (1) if

• All accumulation points are in S.

• For all y ∈ S, D h (y, x k ) has a finite limit as k → ∞.

Actually, Definition 1 may be given for a general closed convex set S ⊂ C (see e.g. [START_REF] Combettes | Fejér-monotonicity in convex optimization[END_REF] which may be directly adapted). We stick to the optimization problem [START_REF] Alvarez | Hessian Riemannian gradient flows in convex programming[END_REF] for simplicity since it already cover a wide range of algorithms.

Algorithmic examples

We list below some of the most common algorithms generating Fejérian sequences in convex optimization.

Mirror descent: the algorithm is due to Nemirovsky [START_REF] Nemirovsky | Problem Complexity and Method Efficiency in Optimization[END_REF]. Assume that f is Lipschitz. Initialize x 0 ∈ intC and, given sequence of positive step sizes (α k ) k∈N , iterate

x k+1 = ∇h * (∇h(x k ) -α k v k ), (2)
v k ∈ ∂f (x k ). (3)
Assuming +∞ k=0 α k = +∞ and +∞ k=0 α 2 k < +∞ the resulting algorithmic sequence is Fejér. In this case equation (4.22) in [START_REF] Beck | Mirror descent and nonlinear projected subgradient methods for convex optimization[END_REF] ensures that accumulation points are in S and equation (4.21) combined with [Lemma 2, Section 2.2] [START_REF] Polyak | Introduction to optimization[END_REF] ensures that D(y, x k ) has a limit for all y ∈ S.

Bregman gradient and NoLips algorithm: Assume that f is C 1 on an open set containing C and h -αf is convex for some α > 0. Initialize x 0 ∈ intC and iterate

x k+1 = ∇h * (∇h(x k ) -α∇f (x k )). (4)
In this case, for all y ∈ S, D h (y, x k ) is non increasing as k grows [4, Lemma 5]. This extends to the NoLips algorithm proposed for composite objectives in [START_REF] Bauschke | A descent lemma beyond Lipschitz gradient continuity: first-order methods revisited and applications[END_REF]. Similarly, the work presented in [START_REF] Alvarez | Hessian Riemannian gradient flows in convex programming[END_REF] presents a continuous time variant of Fejér property in the context of Hessian-Riemannian gradient flows.

Proximal minimization with D-functions: Assume that f is proper lower-semicontinuous. Initialize x 0 ∈ intC and, given sequence of positive step sizes (α k ) k∈N , iterate

x k+1 = arg min x∈C α k f (x) + D h (x, x k ). (5)
In this case, for all y ∈ S, D h (y, x k ) is non increasing as k grows [10, Lemma 3.3] and if +∞ k=0 α k = +∞, then the resulting sequence has all its accumulation point in S [10, Theorem 3.4].

Alternating projection: Beyond optimization problem (1), in his foundational paper [START_REF] Bregman | The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming[END_REF], Bregman considers an alternating projection algorithm generating a Fejér sequence to find an element in the intersection of convex sets.

Sequential convergence analysis

The Fejérian property can be used to prove convergence of the sequence similarly as in the Euclidean case, in relation to Opial's Lemma (see [START_REF] Combettes | Fejér-monotonicity in convex optimization[END_REF] for an overview). This argument is valid provided that the topology encoded by D h is equivalent to the Euclidean topology. This is always true in the interior of C by strict convexity, but further assumptions need to be made to ensure that this also holds at the boundary. The first condition is

(x k ) k∈N ⊂ intC, y ∈ C D h (y, x k ) → k→∞ 0 ⇒ x k → k→∞ y (A)
We know from continuity of h on its domain that strict convexity of h on the whole domain C (not only the interior) is sufficient for (A) [START_REF] Kiwiel | Free-steering relaxation methods for problems with strictly convex costs and linear constraints[END_REF]Lemma 2.16]. If we assume the opposite implication

(x k ) k∈N ⊂ intC, y ∈ C, x k → k→∞ y ⇒ D h (y, x k ) → k→∞ 0 (B)
then we have the following Féjer argument: assume that (A) and (B) hold true, then any Fejérian sequence (x k ) k∈N (Definition 1) converges. Indeed, an accumulation point x exists by compacity, it must be in S by Definition 1, from property (A), up to a subsequence, D h (x, x k ) → 0, but Definition 1 ensures that D h (x, x k ) converges so its limit must be 0 and property (B) ensures that x k → x.

Note that by continuity of h on its domain, condition (B) is equivalent to ⟨∇h(x k ), y -x k ⟩ → 0 as x k → y. Conditions (A) and (B) date back to Bregman [START_REF] Bregman | The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming[END_REF] and have been extensively considered in the litterature [START_REF] Censor | An iterative row-action method for interval convex programming[END_REF][START_REF] Censor | Proximal minimization algorithm withdfunctions[END_REF][START_REF] Chen | Convergence analysis of a proximal-like minimization algorithm using Bregman functions[END_REF][START_REF] Bauschke | Legendre functions and the method of random Bregman projections[END_REF][START_REF] Kiwiel | Free-steering relaxation methods for problems with strictly convex costs and linear constraints[END_REF][START_REF] Alvarez | Hessian Riemannian gradient flows in convex programming[END_REF][START_REF] Auslender | Interior gradient and proximal methods for convex and conic optimization[END_REF][START_REF] Bauschke | A descent lemma beyond Lipschitz gradient continuity: first-order methods revisited and applications[END_REF] in the same abstract form or with adaptation to broader settings that considered here. Given h, a Legendre function, continuous on its compact domain C, if conditions (A) and (B) are satisfied, then h is called a Bregman function.

Main results

We are interested in the following question: How restrictive are conditions (A) and (B)?

In other words, how much does it take for a continuous Legendre function to be Bregman?

We provide the following answer.

Theorem 1. Let C ⊂ R n be convex, compact, with nonempty interior and h be Legendre with domain C, continuous on C. Then:

• (A) holds if and only if h is strictly convex on C.

• (B) holds if and only if C is a polytope.

Theorem 1, provides an explicit sufficient condition on h ensuring convergence of all algorithms described in Section 1.2. Indeed if C is a polytope and h is strictly convex on C, then any Fejérian sequence converges. Furthermore, Theorem 1 illustrates the fact that existing Fejér arguments for sequential convergence are only valid for polytopic domains, and convergence analysis for more general domains will require different arguments.

Theorem 1 can be obtained by combining Proposition 1, Proposition 2, Lemma 5 and Lemma 6. We make crucial use of [15, Lemma 1]. Let us mention that Theorem 1 and all our presentation is limited to compact C, but most proof arguments relate to possibly unbounded closed C so that the intermediate results convey information of independent interest for this more general case. The link between strict convexity and (A) is natural and one implication is due to Kiwiel [START_REF] Kiwiel | Free-steering relaxation methods for problems with strictly convex costs and linear constraints[END_REF]. The connection between (B) and the polytopic nature of C is the most interesting part of our results and we will start with it.

Condition (B)

Let us first illustrate failure of (B) and its relation with curvature with a simple example, which was described independently in [3, Remark 2].

Intuition: incompatibility with curvature

Set h : x → -1 -∥x∥ 2 on C ⊂ R 2 , the unit Euclidean ball in the plane. We have ∇h :

x → x 1 √ 1-∥x∥ 2 .
Considering polar coordinates in the plane, x = (r cos(θ), r sin(θ)), with e 1 the first basis vector (1, 0), we have

h(e 1 ) -h(x) -⟨∇h(x), e 1 -x⟩ = - √ 1 -r 2 + r(r -cos(θ)) √ 1 -r 2
Choosing θ(r) = arccos r -√ 1 -r 2 for r ≥ 0, as r → 1, x(r) = (r cos(θ(r)), r sin(θ(r))) goes to e 1 and D h (e 1 , x) → 1.

Actually, the curve r → x(r) converges to e 1 with a vertical tangent as shown in Figure 1. This illustrates the main mechanism of failure of condition B. Because of curvature, chords all meet the interior of C so that the directional derivative of h along chords explodes at their endpoints. Here the curve remains on a chord segment in a region of negative directional derivative along the chord.

From this illustration, one intuition is that the boundary of the compact domain C of a Bregman function should not have too much curvature. More precisely it should not have curvature accumulating anywhere. This intuition is actually correct, the key mechanism is that around extreme points, there is a lot of curvature which allows to generate behavior similar to the circle example above, see Figure 1. This cannot happen too densely otherwise this contradicts (B), and as a result, the extreme points of C have to be isolated, this is expressed in Proposition 1. On the other hand, polyhedra have a very strong structure which will enforce (B) this is Proposition 2.

Figure 1: Left: Illustration of the tangential phenomenon, closedness to the boundary and to the opposite extremity of the chord. Right: in general extreme points constitute a region of high curvature and it is possible to find neighboring points in a similar configuration. This cannot appen too much under condition (B) and in particular extreme points should not accumulate anywhere.

First implication

In this section we prove the following Proposition 1. Let C ⊂ R n be convex closed with non-empty interior and h be Legendre on C, continuous on C and satisfy (B). Then the extreme points of C are locally finite. In particular if C is bounded it is a polytope.

We start with two lemmas Lemma 1. Under the assumptions of Proposition 1, let x ∈ bdC and y ∈ C be such that x+y 2 ∈ intC. Then D h (y, (1 -λ)x + λy) → +∞ as λ → 0.

Proof : Note that in this case the open segment y) is contained in the interior of C. For any λ ∈ (0, 1), setting z λ = (1 -λ)x + λy, we have

D h (y, (1 -λ)x + λy) = h(y) -h(z λ ) -(1 -λ) ⟨∇h(z λ ), y -x⟩ .
Letting λ → 0, consider any accumulation point, d ∈ R n of ∇h(z λ ) ∥∇h(z λ )∥ . We have d ∈ N C (x) by [1, Lemma 4.2] and y -x ∈ intT C (x) (since x+y 2 ∈ intC), so that ⟨d, y -x⟩ < 0. Therefore, up to a subsequence as λ → 0, Proof of the claim. We may assume that ϵ 1 < ∥y -a∥, so that, since a ̸ = y, the points y and b define a unique line for any b ∈ B ϵ 1 (a). We also impose that ϵ 1 < ϵ. Assume toward a contradiction that for all such

D h (y, (1 -λ)x + λy) = h(y) -h(z λ ) -(1 -λ)∥∇h(z λ )∥ ∇h(z λ ) ∥∇h(z λ )∥ , y -x , → +∞ since h(z λ ) → h(x), ∥∇h(z λ )∥ → ∞
ϵ 1 > 0, there is b ∈ intC ∩ B ϵ 1 (a)

□

Proof of Proposition 1: Assume that the set of extreme points of C it is not locally finite. This means that we can find a bounded sequence (z k ) k∈N of pairwise distinct extreme points. This sequence has a converging subsequence, let y be its limit. Since the sequence has pairwise distinct elements, there is at most one k ∈ N such that y = z k and we may remove it from the sequence. In other words z k → y ∈ C as k → ∞ and z k ̸ = y for all k ∈ N. We will show that in this case, condition B is violated.

Fix k ∈ N, using Lemma 2 with a = z k , we may find which is equivalent to lim inf k→∞ ⟨∇h(x k ), y -x k ⟩ ≥ 0.

x k ∈ intC such that ∥x k -z k ∥ ≤ ∥z k -y∥ and D h (y, x k ) ≥ k. Now as k → ∞,
⟨a i , 2x -y⟩ = ⟨a i , y + 2(x -y)⟩ < b i ∀i ̸ ∈ I, ⟨a i , 2x -y⟩ = 2 ⟨a i , x⟩ -⟨a i , y⟩ ≤ 2b i -⟨a i , y⟩ = 2b i -b i = b i ∀i ∈ I,
We have shown that 0 ≤ lim inf k→∞ ⟨∇h(x k ), y -x k ⟩ ≤ lim sup k→∞ ⟨∇h(x k ), x k -y⟩ ≤ 0, so that the limit is 0. This concludes the proof. □

3 Condition (A)

One implication was given by [START_REF] Kiwiel | Free-steering relaxation methods for problems with strictly convex costs and linear constraints[END_REF]Lemma 2.16].

Lemma 5 (Kiwiel). Let C ⊂ R n be convex closed with non-empty interior, let h be Legendre on C, continuous on C and strictly convex on C, then (A) holds.

The reverse implication follows from Lemma 3 again.

Lemma 6. Let C ⊂ R n be convex closed with non-empty interior, let h be Legendre on C, continuous on C such that (A) holds, then h is strictly convex on C.

Proof : Toward a contradiction, assume that h is not strictly convex. This means that there exists x, y ∈ C such that

h x + y 2 = h(x) + h(y) 2 , (6)
which implies that h is affine along the segment [x, y], that is

h x + y 2 + t(x -y) = h x t + 1 2 + y 1 2 -t = h(x) + h(y) 2 + t(h(x) -h(y))
for all t ∈ [-1/2, 1/2]. In particular, setting z = x+y 2 , we have

h ′ (z, x -z) = -h ′ (z, y -z) = h(x) -h(y) 2 . ( 7
)
Now consider z 0 ∈ intC and the sequence (z k ) k∈N in intC, such that for all k ∈ N, k ≥ 1, z k = 1 k z 0 + k-1 k z ∈ C. We set for all k ∈ N, k ≥ 1, y k = 2 k+1 z 0 + k-1 k+1 y ∈ C. We have for all k ≥ 1,

z k -x = 1 k z 0 + k -1 2k y + k -1 2k - 2k 2k x = 2 2k z 0 + k -1 2k y - k + 1 2k x y k -x = 2 k + 1 z 0 + k -1 k + 1 y - k + 1 k + 1 x = (z k -x) 2k k + 1 y k -z k = y k -x + x -z k = (z k -x) 2k k + 1 -1 = k -1 k + 1 (z k -x).
Using Lemma 3, we have lim sup k→∞ ⟨∇h(z k ), y k -z k ⟩ = lim sup k→∞ k -1 k + 1 ⟨∇h(z k ), z k -x⟩ = -lim inf k→∞ ⟨∇h(z k ), x -z k ⟩ ≤ h ′ (z, y -z) = -h ′ (z, x -z), so that lim sup k→∞ ⟨∇h(z k ), x -z k ⟩ ≤ h ′ (z, x -z) ≤ lim inf k→∞ ⟨∇h(z k ), x -z k ⟩ and ⟨∇h(z k ), x -z k ⟩ → h ′ (z, x -z) as k → ∞. We deduce by continuity of h using ( 6) and ( 7)

lim k→∞ h(x) -h(z k ) -⟨∇h(z k ), x -z k ⟩ = h(x) - h(x) + h(y) 2 - h(x) -h(y) 2 = 0
So we have that z k → z ̸ = x but D h (x, z k ) → 0 which shows that condition (A) does not hold. This proves the result by contraposition. □

Lemma 2 .Claim 1 .

 21 and ⟨d, y -x⟩ < 0. The result follows because d was arbitrary. This concludes the proof.□ Under the hypotheses of Proposition 1, let a ∈ bdC be an extreme points and y ∈ C, different from a. Then for any ϵ, K > 0, there exists c ∈ intC such that ∥c -a∥ ≤ ϵ D h (y, c) ≥ K.Proof : First if [a, y]∩intC ̸ = ∅, then we are in the conditions of Lemma 1, which provides the desired result (with x = a). We may therefore assume that [a, y] ⊂ bdC. There exists ϵ 1 > 0 such that for any b ∈ intC ∩ B ϵ 1 (a), the line from y to b crosses the boundary of C at x such that ∥x -a∥ ≤ ϵ.

  , such that the line from y to b either does not cross the boundary of C or it crosses it at a point at distance greater than ϵ. In both cases, since b ∈ B ϵ (a) (as we assumed ϵ 1 < ϵ), the line from y to b exits B ϵ (a) at a point x ∈ C. In this case, using a vanishing sequence of values for ϵ 1 , we can produce a sequence (b k ) k∈N converging to a such that each segment [y, b k ] can be extended up to x k ∈ C where b k ∈ [x k , y] and ∥x k -a∥ = ϵ. Passing to the limit, up to subsequences b k → a and x k → x ∈ C, we have a ∈ [x, y] where a ̸ = y and a ̸ = x which contradicts the fact that a is an extreme point of C. This proves the claim. We may choose ϵ 1 ≤ ϵ, now consider any point b ∈ intC ∩ B ϵ 1 (a), b belongs to a segment of the form [x, y] for some x ∈ bdC ∩ B ϵ (a). Note that [b, x] ⊂ B ϵ (a) so that we can apply Lemma 1 and obtain c ∈ [b, x] with D h (y, c) arbitrarily large. This proves the desired result.

  which shows that 2x -y ∈ C and concludes the proof. □ Proof of Proposition 2: By continuity of h on C, it suffices to show that lim x→y,x∈intC ⟨∇h(x), y -x⟩ = 0 for all y ∈ C.Fix y ∈ C and consider (x k ) k∈N , converging to y. C can be locally represented by a polyhedra and therefore Lemma 4 can be applied to y and C. Let ϵ > 0 be given by Lemma 4 and assume without loss of generality that ∥y -x k ∥ ≤ ϵ for all k ∈ N. We have for all k ∈ N, x k + (y -x k ) = y ∈ C so that applying Lemma 4 withd k = y -x k , which converges to 0 lim sup k→∞ ⟨∇h(x k ), y -x k ⟩ ≤ h ′ (y, 0) = 0.Furthermore, by Lemma 4, for all k ∈ N, we have 2x k -y = x k + (x k -y) ∈ C. Therefore, one may apply Lemma 4 with d k = x k -y, which also converges to 0 to obtain lim sup k→∞ ⟨∇h(x k ), x k -y⟩ ≤ h ′ (y, 0) = 0.

  we have x k → y and D h (y, x k ) → +∞ which contradicts condition B. This concludes the proof. The result follows from the following ([15, Lemma 1]). Lemma 3 (Tseng and Bertsekas). Let h : R p → R ∪ +∞, be lower semicontinuous and continuous on its domain C. Then • For any y ∈ C, there exists a nondegenrate closed ball centered at y such that C ∩ B is closed. • For any y ∈ C, and d such that y + d ∈ C and sequences x k → y and d k → d such that x k + d k ∈ C for all k, we have lim sup k→∞ h ′ (x k , d k ) ≤ h ′ (y, d). Lemma 4. Let C be a polyhedra and y ∈ C. Then there exists ϵ > 0 such that for any x ∈ C ∩ B ϵ (y), 2x -y ∈ C. Proof : Let a 1 , . . . , a m ∈ R n and b 1 , . . . , b m ∈ R such that C = {x| ⟨a i , x⟩ ≤ b i , i = 1, . . . , m}. Fix y ∈ C, and I ⊂ {1, . . . m} the set of active indices, such that ⟨a i , y⟩ = b i if and only if i ∈ I. By continuity of linear functions, there exists ϵ > 0 such that for all d with ∥d∥ ≤ 2ϵ, and all i ̸ ∈ I, ⟨a i , y + d⟩ < b i . Now for any x ∈ C such that ∥x -y∥ ≤ ϵ, we have

	□
	2.3 Reverse implication

Proposition 2. Let C ⊂ R n be convex closed with non-empty interior and locally polyhedral (represented locally by finitely many affine inequalities, for example a polytope), let h be Legendre on C, continuous on C, then (B) holds.

Acknowledgements

The author would like to thank Jérôme Bolte for his continuous support and fruitful interactions. This work supported by the AI Interdisciplinary Institute ANITI, ANR-19-PI3A-0004, Air Force Office of Scientific Research, Air Force Material Command, USAF, FA9550-19-1-7026 FA8655-22-1-7012, and ANR MaSDOL 19-CE23-0017-01.