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Art. 1 – NOTCH EFFECT ON THE FRACTURE OF SEVERAL ROCKS: APPLICATION OF THE THEORY OF THE CRITICAL DISTANCES – Justo, J.; Castro, J.; Cicero, S.; Sánchez-Carro, M.A.; Husillos, R. (2017). Theoretical and Applied Fracture Mechanics 90, 251-258. DOI: 10.1016/j.tafmec.2017.05.025.

ABSTRACT: The paper analyses the fracture behaviour of several rocks, namely a sandstone, a limestone and two marbles, one of them being a Carrara marble. The experimental program comprises in total 216 fracture specimens, tested in 4-point bending conditions and including specimens with notch radii varying from 0.15 mm up to 15 mm. The notch effect is analysed through the evolution of the apparent fracture toughness and the application of the Theory of Critical Distances.

The present study aims to generalize a previous study on a granite and a limestone to a broader range of rocks. The point and line methods of the Theory of the Critical Distances successfully explain the notch effect on the fracture specimens. The value of the critical distance of these rocks is of the order of mm. Finally, the results show a correlation between the microstructural features of the rocks, specifically the grain size, and their critical distances.

Art. 2 – ENERGY-BASED APPROACH FOR FRACTURE ASSESSMENT OF SEVERAL ROCKS CONTAINING U-SHAPED NOTCHES THROUGH THE APPLICATION OF THE SED CRITERION – Justo, J.; Castro, J.; Cicero, S. (2018) International Journal of Rock Mechanics and Mining Sciences. DOI: 10.1016/j.ijrmms.2018.07.013

ABSTRACT: This work presents an energetic continuum approach for the fracture assessment of rocks containing U-shaped notches and subjected to Mode I loading conditions. Three different methodologies are proposed, all based on the premise that brittle failure will occur when the average strain energy density over a certain control area reaches a critical value that only depends on the material, as stated by the Strain Energy Density (SED) criterion.

The first method proposed (A) deals with the application of the SED criterion through an expression with a series of already tabulated parameters, which are particularised for the analysed rocks by rational extrapolation. By contrast, the second method (B) aims to obtain numerically the previously extrapolated parameters, and the third method (C) directly relates the strain energy density with the applied load, without the use of those parameters.

The research is based on the results obtained from an exhaustive experimental programme comprising 300 fracture specimens tested in four-point bending conditions. These tests combine parallelepiped samples made of six different types of rocks (two marbles, two limestones, a sandstone and a granite) and containing eight different notch radii (varying from 0.15 mm up to 15 mm).

Thus, this work aims to show the potential, capacity and limitations of the SED criterion in rock fracture analyses, by comparing the experimentally obtained fracture loads to those predicted by the three proposed methodologies.

Art. 3 – NOTCH EFFECT AND FRACTURE LOAD PREDICTIONS OF ROCK BRAMS AT DIFFERENT TEMPERATURES USING THE THEORY OF CRITICAL DISTANCES – Justo, J.; Castro, J.; Cicero, S. (2020) International Journal of Rock Mechanics and Mining Sciences. DOI: 10.1016/j.ijrmms.2019.104161

ABSTRACT: This work aims to analyse the fracture behaviour of rocks with U-shaped notches subjected to mode I loading and to different temperature conditions. To this end, the so called Theory of Critical Distances (TCD) is applied and four different types of isotropic rocks are studied: a Floresta sandstone, a Moleano limestone, a Macael marble and a Carrara marble. This study attempts to extend a previous work of the authors where the TCD was successfully applied to U-notched components subjected to mode I loading conditions at room temperature. In this case, the effect of temperature is considered as a new variable.

The research comprises, in total, more than 790 four-point bending tests and 144 tensile splitting (Brazilian) tests. The latter include 6 disc-shaped specimens for each rock and temperature (6 different temperatures), while the four-point bending tests consist of at least 6 SENB specimens for each rock, notch radius (8 different notch radii varying from 0.15 mm to 15 mm) and temperature (4 different temperatures) combination. The temperatures considered in this study vary from room temperature up to 250 °C, which is a common range in geothermal applications.

Temperature has proven to be a significant parameter when analysing the fracture behaviour of the four selected rocks. Its influence on the tensile strength and fracture toughness of the rocks is clear and reveals common patterns. However, no apparent tendencies are shown on the influence of temperature on the critical distance ( $L$ ). Likewise, the application of the TCD has led to relatively accurate fracture predictions and notch effect analyses at different temperature conditions.

Art. 4 – DISCRETE NUMERICAL ANALYSES OF GRAIN SIZE INFLUENCE ON THE FRACTURE OF NOTCHED ROCK BEAMS – Justo, J.; Konietzky, H.; Castro, J. (2020) Computers and Geotechnics. DOI: 10.1016/j.compgeo.2020.103680

ABSTRACT: This paper studies the influence of the grain size and the notch effect on the fracture of U-shaped notched rock beams through the variation of the apparent fracture toughness. The research is based on an exhaustive campaign that comprises numerical simulations of 300 four-point bending tests, 30 uniaxial compression tests and 60 tensile splitting (Brazilian) tests. Non-porous, isotropic ideal and equivalent rocks with 5 different uniform grain sizes are modelled using the distinct element method, where the rocks are modelled as a discontinuous material, defining explicitly the grains and the boundary conditions. Several notch radii are simulated and the corresponding variation in the apparent fracture toughness is observed. The notch effect is interpreted using the Theory of Critical Distances (TCD), which uses a material intrinsic property called the critical distance ( $L$ ) to evaluate the stress field around the notch tip. The paper shows the variation of the fundamental rock properties with the grain size, the applicability of the TCD to evaluate the notch effect and the correlation between the critical distance and the grain size.

Art. 5 – MECHANICAL PROPERTIES OF 4 ROCKS AT DIFFERENT TEMPERATURES AND FRACTURE ASSESSMENT USING THE STRAIN ENERGY DENSITY CRITERION – Justo, J.; Castro, J. (2021) Geomechanics for Energy and Environment. DOI: 10.1016/j.gete.2020.100212

ABSTRACT: This work aims to analyse the fracture behaviour of rocks with U-shaped notches subjected to mode I loading and to different temperature conditions. To this end, an energy-based approach is used called the Strain Energy Density (SED) criterion. This study attempts to extend a previous work of the authors where the SED criterion was successfully applied to U-notched components subjected to mode I loading conditions at room temperature. In this case, the effect of temperature is considered as a new variable.

The research analyses four different types of isotropic rocks with different lithologies, namely a Floresta sandstone, a Moleano limestone, a Macael marble and a Carrara marble. An exhaustive laboratory campaign was performed to define the main mechanical properties of the selected rocks at different temperatures. In total, 144 tensile splitting (Brazilian) tests, 120 uniaxial compression tests, 410 thermal expansion measurements and more than 790 four-point bending tests have been executed under different thermal conditions. On the other hand, the range of temperatures analysed varies from room temperature up to 250 ºC, which is a common band in geothermal applications.

Temperature has proven to be a significant parameter when analysing the fracture behaviour of the four selected rocks. Its influence on the main mechanical properties of the rocks (tensile strength, fracture toughness, compressive strength, Young’s modulus, Poisson’s ratio) has been studied and similar trends have been observed for the marbles, but different or even opposite ones for the sandstone and limestone. Overall, the application of the SED criterion has led to relatively accurate fracture predictions under different temperature conditions. This methodology assumes a linear-elastic behaviour of the rocks at the studied range of temperatures. For this reason, the failure load predictions become less accurate when non-linearities are not negligible, as in the case of the Carrara marble.

Art. 5 – 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 – Justo, J.; Castro, J.; Cicero, S. (2021) Rock Mechanics and Rock Engineering. DOI: 10.1007/s00603-021-02365-7

ABSTRACT: This work aims to assess the fracture of U-notched limestone samples subjected to mixed mode I + II loading conditions with a predominant mode I influence, both at room temperature and at 250 °C. This analysis is based on the use of the Theory of the Critical Distances, and more specifically on the use of the Line Method, considering both an analytical and a numerical approach for the definition of the stress fields. An experimental campaign of almost 400 three-point bending tests has been performed as a basis for the fracture assessment of the limestone, using Single Edge Notched Bend (SENB) specimens with notch radii varying from 0.15 mm up to 15 mm, different temperature conditions and variable loading positions. The Theory of Critical Distances has successfully been applied to study the experimental results. The analytical and numerical stress fields for pure mode I fracture assessments provide similar accurate results both at 23 °C and 250 °C. Similarly, the mixed mode (I + II) fracture assessments allow the critical distance ( $L$ ) to be characterised for different mode mixities ( $M_{e}$ ), using the stress field around the notch tip obtained from the numerical models. Comparing the values of the critical distance against the mode mixity in isolation, a slight decrease of $L$ is observed as it approaches pure mode I conditions ( $M_{e}$ = 1). However, if the results are analysed separately for each notch radius, $L$ seems to be relatively constant with $M_{e}$ . In parallel, a certain influence of the notch radius on the critical distance is appreciated, which shows an increment both at 23 °C and 250 °C.

LECTURES

Lec. 1 – NOTCH EFFECT ON THE FRACTURE OF SEVERAL ROCKS: APPLICATION OF THE THEORY OF THE CRITICAL DISTANCES – Justo, J.; Castro, J.; Cicero, S.; Sánchez-Carro, M.A.; Husillos, R. (2017). International Symposium on Notch Fracture. Santander, 29-31 March 2017, pp. 135-142. ISBN: 978-84-617-9463-8.

Lec. 2 – ROCK FRACTURE IN GEOTHERMAL ENERGY SOURCES. APPLICATION OF THE THEORY OF THE CRITICAL DISTANCES. – Justo, J.; Castro, J.; Sagaseta, C. (2017). EIDEIC. 17 May 2017. Santander.

Lec. 3 – FINITE ELEMENT ANALYSES FOR FRACTURE ASSESSMENT ON ROCKS CONTAINING I-SHAPED NOTCHES: AN ENERGETIC CONTINUUM APPROACH. – Justo, J.; Castro, J.; Sagaseta, C. (2017). European PLAXIS Users’ Meeting. 18-19 May 2017. Schrobenhausen, Alemania.

ABSTRACT: This work presents finite element analyses in plane strain conditions for fracture assessment on rocks containing U-shaped notches. The notch fracture assessment is performed using the elastic stresses from conventional finite element analyses and an energetic approach based on the Strain Energy Density (SED) criterion. The method is applied to several rocks subjected to Mode I loading conditions and containing U-shaped notches whose radii vary from 0.15 mm up to 15 mm. The volume-based SED criterion statesthat brittle failure will occur when the average value of the strain energy density over a certain control volume (which becomes an area in plane strain conditions) reaches a critical value dependent on the material. Under this premise, the experimentally obtained fracture loads are compared to those predicted by the SED criterion, checking its capacity and limitations.

The study requires a hybrid model that implements the numerical analysis of the stress field at the notch tip performed by a Finite Element Model in PLAXIS 2D under plane strain conditions and assuming a linear elastic behaviour of the rock, with a mathematical software like Matlab that allows assessing the notch fracture through the SED criterion.

This work aims to show the capabilities of PLAXIS as a traditional (non-geotechnical) finite element code and the potential and simplicity of the SED criterion in rock fracture analysis among others, which may lead to future lines of work within the field of fracture mechanics of rocks using continuum finite element analyses.

Lec. 4 – APPLICATION OF THE THEORY OF THE CRITICAL DISTANCES AND THE STRAIN ENERGY DENSITY CRITERION FOR ROCK FRACTURE PREDICTION AT DIFFERENT TEMPERATURES. – Justo, J.; Castro, J.; Sagaseta, C. (2018). Winter School on Geomechanics for Energy and the Environment. 22-24 January 2018. Villars-sur-Ollon, Switzerland.

ABSTRACT: This research deals with the study and assessment of rock fracture in geothermal energy sources. Based on the development that this type of energy is undertaking, it is intended to advance in the knowledge of the brittle failure of rocks at the usual temperatures in geothermal applications (approximately between 20 and 250 ºC). Two different methodologies have been chosen with this purpose: the Theory of the Critical Distances (TCD), a local failure criterion, and the Strain Energy Density (SED) criterion, an energetic continuum approach. Both constitute two of the most recent developments in fracture mechanics, and its applicability has been widely studied in several materials like steels and polymers. However, despite the potential of the two proposed methodologies in the analysis of fracture processes, their suitability in the field of heterogeneous and brittle materials like rocks requires further research.

The study is based on a systematic and exhaustive experimental program which includes, among others: simple compression tests, Brazilian tests, triaxial tests and 3-point and 4-point bending tests with notch radii varying from 0.15 mm to 15 mm (to consider the notch effect), all of them both at room temperature and at higher temperatures up to 250 ºC. Additionally, a detailed microstructural analysis of each of the selected rocks is contemplated to facilitate the interpretation of the obtained results.

With all this, it is intended to increase the “know-how” associated to the characterization of rock mass fracture phenomena, which might lead to an improvement in the reliability and to a reduction in the financial costs of the drilling operations corresponding to geothermal energy projects.

Lec. 5 – ROCK FRACTURE IN GEOTHERMAL ENERGY SOURCES. – Justo, J.; Castro, J.; Sagaseta, C. (2018). EIDEIC. 17 May 2018. Santander.

Lec. 6 – COMPARISON BETWEEN AN ENERGETIC CONTINUUM APPROACH AND A LOCAL FAILURE CRITERION FOR ROCK FRACTURE PREDICTION. – Justo, J.; Castro, J.; Cicero, S.; Sánchez-Carro, M.A.; Sagaseta, C. (2018). Proc. of the 2018 European Rock Mechanics Symposium. EUROCK2018: Geomechanics and Geodynamics of Rock Masses (Litvinenko, V. ED.). Saint Petersburg, Russia 22-26 May 2018. CRC Press, London. pp. 1203-1208. ISBN 978-1-138-61645-5.

ABSTRACT: This paper aims to compare two different approaches for fracture assessment on rocks containing U-shaped notches and subjected to Mode I loading conditions.First, the Strain Energy Density (SED) criterion has beenconsidered, which is based on an energetic continuum approach for rock fracture predictions. Likewise, the Theory of the Critical Distances (TCD) has also been applied, which can be considered as a local failure criterion based on the stress field near the notch tip. Both methods properly describe the rock fracture behaviour and provide relatively accurate and reliable estimations. Thus, the rock fracture predictions provided by these two methods have been analysed and compared in this paper, trying to focus on the particularities of each one.The research is based on the results obtained from an exhaustive experimental program comprising 78 fracture specimens of a Moleano Limestone. Thisprogramincludes 12 Brasilian tests, 12 simple compression tests and 54 parallelepiped specimens tested in 4-point bending conditions with 8 different notch radii varying from 0.15 mm up to 15 mm in order to consider the possible notch effects.

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