# Prediction of Stress–Strain Curves Based on Hydric Non-Destructive Tests on Sandstones

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{3}, and the real density values range between 2550 and 2670 kg/m

^{3}. These monuments’ building stones show values of ultrasonic pulse velocity of compressional waves (UPV) of about 1200 to 2200 m/s perpendicular to the bedding and between circa 1300 to 2600 m/s parallel to the bedding. The latter reaches 4000 m/s on Angkor sandstones.

^{3}(1900–2580 kg/m

^{3}). The average value of capillarity coefficient given by samples with open porosities between 17.5–22.5% is 400 g/m

^{2}s

^{1/2}, i.e., circa 24 kg/m

^{2}h

^{1/2}(0.4 kg/m

^{2}s

^{1/2}). The test of water absorption under low pressure (Karsten tube) obtained on samples with 17% open porosity gave an average value of about 30 mL in 600 s, i.e., about 35 kg/m

^{2}h

^{1/2}. The previous value was obtained by the slopes of the linear regression of the initial part of the experimental curves. The average value of compressive strength given by samples with open porosity values of 20–22.5% is around 30 MPa. The average value of drilling resistance recorded in tests with a drilling rotation speed of 600 rpm and an advancing rate of 10 mm/min downward a drill hole of 5 mm of diameter, on samples with 17–26% open porosity, is about 22.5 N, i.e., 1.15 MPa.

^{2}hod

^{0.5}) on Arkose, Hořice, and Petrín sandstones. The average compressive strength values of these sandstones obtained on cubic samples comprising a 50-mm-long edge range between circa 27 and 19 MPa. The compressive strength and the elastic modulus play a major role in the numerical simulation of the structural behavior of historical buildings and monuments. These mechanical parameters of rock reservoirs are also very important in the design and construction of oil and gas wells.

_{P}) and/or the Schmidt rebound number.

## 2. Selection of Rock Materials

^{2}h

^{1/2}(A), 2.4 kg/m

^{2}h

^{1/2}(B), 6.2 kg/m

^{2}h

^{1/2}(C), and 26 kg/m

^{2}h

^{1/2}(M). These two lithotypes are designated as lithic arkose [1,15], according to Folk [44]. Varieties A and B included in lithotype A + B have about 30–32% quartz and 34–40% carbonates, and varieties C and M of the lithotype C + M have about 20–25% carbonates and 40–51% quartz. Both lithotypes were classified as lithic arkose with carbonate cement. Well-defined lineation is exhibited macroscopically in lithotype A + B. The A variety exhibited one major orientation of mica minerals, and the variety B showed no preferred orientations once the lineation was randomly distributed on thin sections under a polarizing microscope. Lineation was not identified in variety M, and thin sections of variety M showed two major orientations of mica minerals. The two lithotypes have around 4–6% of mica minerals.

## 3. Experimental Methodology and Results

#### 3.1. Introduction

#### 3.2. Preparation of Samples

^{3}and have a height-to-length ratio of two. Lithotype A + B showed macroscopic laminations and lineations, aligned parallel to the major axis. The prismatic specimens were randomly cut as no macroscopic lineations were found in the M variety sandstone [1].

#### 3.3. Evaluation Tests on Physical Properties

^{3}were the main specifications of the pipe. The results were obtained from the water absorption under low pressure graphs (mass of fluid per area of its absorption) as a function of the square root of time, and the water absorption coefficient was given by the slope of a linear trend fitted to the first part of the test curve. The values of this parameter are usually given in the following sensitive units: kg/m

^{2}/√h.

^{2}h

^{1/2}) and M (31.8 kg/m

^{2}h

^{1/2}). The bulk density values of sandstones range between 2179 kg/m

^{3}(variety M) and 2594 kg/m

^{3}(variety A) [1]; this author presented pore size distribution of sandstone varieties B and M, obtained by mercury intrusion porosimetry (MIP). Microporosity, defined as the percentage of radii of voids smaller than 7.5 μm [47], is 80–85% in variety B and about 75% in variety M.

#### 3.4. Monotonic Compression Tests

^{−6}strain/mm.

_{c}) and strain at failure (ε

_{r}) of sandstones are shown on Table 3.

## 4. Modeling of Compression Behavior

#### 4.1. Ludovico-Marques Global Model Used on Sandstones

_{c}is the compressive strength of rocks, and f(ε/ε

_{R}) is the shape function dependent on the strain ε normalized by the strain at failure (ε

_{R}).

_{c}, and is indicated as follows:

_{R}, and the coefficient 1.47 tends to 1.5.

#### 4.2. Correlations Between Compressive Strength, Strain at Failure, and Coefficient of Water Absorption Under Low Pressure

^{2}) of Equation (5) is 0.964. However, on less than 30% of sandstone specimens, the absolute differences between experimental and predicted compressive strength values, obtained through Equation (5), were higher than 15%.

_{R}) with the coefficient of water absorption under low pressure on the samples of sandstone varieties are shown in Figure 6. As the coefficient of water absorption under low pressure obtained by the Karsten pipe decreases, the compressive strain at failure increases. This shows a clear trend between the variation of the compressive strain at failure and the variation of water absorption.

^{2}) of Equation (6) is 0.788.

_{R}) obtained between each experimental data of A, B, C, and M sandstone varieties and their predicted data is shown on Table 4. Only around 10% of strain at failure (ε

_{R}) data of 40 specimens had absolute differences between the predicted and experimental values higher than 15%, and 20% of them differed by more than 10%; the average of the absolute difference of all samples was around 5%.

#### 4.3. Analysis of Experimental and Predicted Results

_{c}) must be replaced by Equation (5), and the strain at compressive strength (ε

_{R}) by Equation (6). The following Equation (7) enables a prediction of the compressive strength and the definition of the relation between stress-strain curves and the water absorption under low-pressure for sandstones:

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Ludovico-Marques, M. Contribution to the Knowledge of the Effect of Crystallization of Salts in the Weathering of Sandstones. Application to the Built Heritage of Atouguia da Baleia. Ph.D. Thesis, Universidade Nova de Lisboa, Lisbon, Portugal, 2008; p. 314. (In Portuguese). [Google Scholar]
- Ludovico-Marques, M.; Chastre, C. Effect of artificial accelerated salt weathering on physical and mechanical behaviour of sandstone samples from surface reservoirs. In Handbook of Materials Failure Analysis with Case Studies from the Oil and Gas Industry; Makhlouf, H., Aliofkhazraei, M., Eds.; Butterworth Heinemann (Elsevier): Oxford, UK, 2016; pp. 215–233. [Google Scholar]
- International Charter for the Conservation and Restoration of Monuments and Sites (the Venice Charter 1964); International Council on Monuments and Sites (ICOMOS): Charenton-le-Pont, France, 1965.
- Charter, K. Magazine Trieste Contemporanea; Trieste Contemporanea: Trieste, Italy, 2000. [Google Scholar]
- Recommendations for the Analysis, Conservation and Structural Restoration of Architectural Heritage; International Council on Monuments and Sites (ICOMOS): Charenton-le-Pont, France, 2004.
- Foraboschi, P. The central role played by structural design in enabling the construction of buildings that advanced and revolutionized architecture. Constr. Build. Mater.
**2016**, 114, 956–976. [Google Scholar] [CrossRef] - Goodman, R. Introduction to Rock Mechanics, 2nd ed.; John Wiley & Sons: New York, NY, USA, 1989. [Google Scholar]
- Palchik, V. Influence of the porosity and elastic modulus on uniaxial compressive strength in soft brittle porous sandstones. Rock Mech. Rock Eng.
**1999**, 32, 303–309. [Google Scholar] [CrossRef] - Hatzor, Y.H.; Palchick, V. The influence of the grain size and porosity on the crack initiation stress and critical flaw length in dolomites. Int. J. Rock Mech. Min. Sci.
**1997**, 34, 805–816. [Google Scholar] [CrossRef] - Tugrul, A.; Zarif, I.H. Correlation of mineralogical and textural characteristics with engineering properties of selected granitic rocks from Turkey. Eng. Geol.
**1999**, 51, 303–317. [Google Scholar] [CrossRef] - Palchick, V.; Hatzor, Y.H. Crack damage stress as a composite function of porosity and elastic stiffness in dolomites and limestones. Eng. Geol.
**2002**, 63, 233–245. [Google Scholar] [CrossRef] - Palchick, V.; Hatzor, Y.H. The influence of porosity on tensile and compressive strength of porous chalks. Rock Mech. Rock Eng.
**2004**, 37, 331–341. [Google Scholar] [CrossRef] - Vasarhelyi, B.; Van, P. Influence of water content on the strength of rock. Eng. Geol.
**2006**, 84, 70–74. [Google Scholar] [CrossRef] - Yilmaz, I. Influence of water content on the strength and deformability of gypsum. Int. J. Rock Mech. Min. Sci.
**2010**, 47, 342–347. [Google Scholar] [CrossRef] - Ludovico-Marques, M.; Chastre, C.; Vasconcelos, G. Modelling the compressive mechanical behaviour of granite and sandstone historical building stones. Constr. Build. Mater.
**2012**, 28, 372–381. [Google Scholar] [CrossRef] - Heidari, M.; Torabi-Kaveh, M.; Chastre, C.; Ludovico-Marques, M.; Mohseni, H.; Akefi, H. Determination of weathering degree of the Persepolis stone under laboratory and natural conditions using fuzzy inference system. Constr. Build. Mater.
**2017**, 145, 28–41. [Google Scholar] [CrossRef] - Heidari, M.; Chastre, C.; Torabi-Kaveh, M.; Ludovico-Marques, M.; Mohseni, H. Application of fuzzy inference system for determining weathering degree of some monument stones in Iran. J. Cult. Herit.
**2017**, 25, 41–55. [Google Scholar] [CrossRef] - Chastre, C.; Ludovico-Marques, M. Nondestructive testing methodology to assess the conservation of historic stone buildings and monuments. In Handbook of Materials Failure Analysis with Case Studies from the Construction Industries; Makhlouf, H., Aliofkhazraei, M., Eds.; Butterworth Heinemann (Elsevier): Oxford, UK, 2018; pp. 255–294. [Google Scholar]
- Tan, X.; Chen, W.; Yang, J.; Cao, J. Laboratory investigations on the mechanical properties degradation of granite under freeze-thaw cycles. Cold Reg. Sci. Technol.
**2011**, 68, 130–138. [Google Scholar] [CrossRef] - Noor-E-Khuda, S.; Albermani, F.; Veidt, M. Flexural strength of weathered granites: Influence of freeze and thaw cycles. Constr. Build. Mater.
**2017**, 156, 891–901. [Google Scholar] [CrossRef] - Foraboschi, P.; Vanin, A. Experimental investigation on bricks from historical Venetian buildings subjected to moisture and salt crystallization. Eng. Fail. Anal.
**2014**, 45, 185–203. [Google Scholar] [CrossRef] - Ludovico-Marques, M.; Chastre, C. Effect of salt crystallization ageing on the compressive behavior of sandstone blocks in historical buildings. Eng. Fail. Anal.
**2012**, 26, 247–257. [Google Scholar] [CrossRef] - Malesani, P.; Vannuci, S. Decay of Pietra Serena and Pietraforte, Florentine building stones: Petrographic observations. Stud. Conserv.
**1974**, 19, 36–50. [Google Scholar] - Felix, C. Molasses et grès de Villarlod (Fribourg). Inventaire des Carrièrs Suisses de Pierre de Taille; École Polytechnique Fédérale de Lausanne-Laboratoire des Matériaux Pierreux: Lausanne, Switzerland, 1977; p. 18. [Google Scholar]
- Sorace, S. Long-term tensile and bending strength of natural building stones. Mater. Struct.
**1996**, 29, 426–435. [Google Scholar] [CrossRef] - Banchelli, A.; Fratini, F.; Germani, M.; Malesani, P.; Manganelli Del Fa, C. The sandstones of Florentine historic buildings: Individuation of the marker and determination of the supply quarries of the rocks used in some Florentine monuments. Sci. Technol. Cult. Herit.
**1997**, 6, 12–22. [Google Scholar] - Uchida, E.; Ogawa, Y.; Maeda, N.; Nakagawa, T. Deterioration of stone materials in the Angkor monuments, Cambodia. Eng. Geol.
**1999**, 55, 101–112. [Google Scholar] [CrossRef] - Tiano, P.; Valentini, E.; Exadaktylos, G.; Garrod, E.; Snethlage, R.; Wendler, E.; Singer, B.; Delgado Rodrigues, J.; Cadot-Leroux, L.; De Witte, E.; et al. Technical Annex. In Proceedings of the Workshop Drillmore—Drilling Methodologies for Monuments Restoration, Munich, Germany, 16–17 March 2000; Centro Stampa Toscana Nuova: Firenze, Italy, 2000. [Google Scholar]
- Fitzner, B.; Heinrichs, K.; La Bouchardiere, D. Weathering damage on Pharaonic sandstone monuments in Luxor—Egipt. Build. Environ.
**2003**, 38, 1089–1103. [Google Scholar] [CrossRef] - Zoghlami, K.; Gomez-Graz, D.; Alvarez, A. Petrophysical characterization and durability of Miocenic sandstones used in Roman aqueduct of Zaghouan—Cartaghe building. In Lectures and Proceedings of 6th International Symposium on the Conservation of Monuments in the Mediterranean Basin; Aires-Barros, L., Zezza, F., Eds.; IST: Lisboa, Portugal, 2004; pp. 385–389. [Google Scholar]
- Heinrichs, K. Diagnose der Verwitterungsschaden an den Felsmonumenten der Antiken Stadt Petra/Jordanien. Ph.D. Thesis, Geologisches Institut/RWTH Aachen, Aachen, Germany, 2005. [Google Scholar]
- Ehrenberg, S.; Nadeau, P. Sandstone versus carbonate petroleum reservoirs: A global perspective on porosity-depth and porosity-permeability relationships. AAPG Bull.
**2005**, 89, 435–445. [Google Scholar] [CrossRef] - Jizba, D. Mechanical and Acoustical Properties of Sandstones and Shales. Ph.D. Thesis, Stanford University, Stanford, CA, USA, 1991. [Google Scholar]
- Folk, R. Petrology of Sedimentary Rocks; Hemphill Publishing: Austin, TX, USA, 1974; p. 184. [Google Scholar]
- Vernik, L.; Bruno, M.; Bovberg, C. Empirical relations between compressive strength and porosity of siliciclastic rocks. Int. J. Rock Mech. Min. Sci. Geomech. Abstr.
**1993**, 30, 677–681. [Google Scholar] [CrossRef] - Moos, D.; Zoback, M.; Bailey, L. Feasibility study of the stability of open hole multilaterals, Cook Inlet, Alaska. In SPE Mid-Continent Operations Symposium 1999; Society of Petroleum Engineers: Oklahoma City, OK, USA, 2001. [Google Scholar]
- Cnudde, V. Exploring the Potencial of X ray Tomography as a Non-Destructive Research Tool in Conservation Studies of Natural Building Stones. Ph.D. Thesis, University of Gent, Gent, Belgium, 2005. [Google Scholar]
- Hasnikova, H.; Zima, P. Comparative testing of natural stones used as a building material. In Experimental Methods and Numerical Simulation in Engineering Sciences, Proceedings of XIIIth Bilateral Czech/German Symposium, Telč, Czech Republic, 5–8 June 2012; Jirousek, O., Kyty, D., Eds.; University Centre Telč: Telč, Czech Republic, 2012. [Google Scholar]
- Ludovico-Marques, M. A fast and less expensive test to determine permeability-related parameters on well’s drilled cuttings. In Advances in Petroleum Engineering and Petroleum Geochemistry, Advances in Science, Technology & Innovation; Springer Nature Switzerland AG: Cham, Switzerland, 2019; pp. 13–16. [Google Scholar]
- Fener, M.; Kahraman, K.; Bilgil, A.; Gunaydin, O. A comparative evaluation of indirect methods to estimate the compressive strength of rocks. Rock Mech. Rock Eng.
**2005**, 38, 329–343. [Google Scholar] [CrossRef] - Chang, C. Empirical rock strength logging in boreholes penetrating sedimentary formations. Geol. Earth Environ. Sci.
**2004**, 7, 174–183. [Google Scholar] - Vasconcelos, G.; Lourenço, P.B.; Alves, C.A.; Pamplona, J. Ultrasonic evaluation of the physical and mechanical properties of granites. Ultrason
**2008**, 48, 453–466. [Google Scholar] [CrossRef] [Green Version] - Selcuk, L.; Nar, A. Prediction of uniaxial compressive strength of intact rocks using ultrasonic pulse velocity and rebound-hammer number. Q. J. Eng. Geol. Hydrogeol.
**2015**, 49, 67–75. [Google Scholar] [CrossRef] - Yasar, E.; Erdogan, Y. Estimation of rock physicomechanical properties using hardness methods. Eng. Geol.
**2004**, 71, 281–288. [Google Scholar] [CrossRef] - RILEM. Recommended tests to measure the deterioration of stone and to assess the effectiveness of treatment methods. Mat. Const. Bourdais Dunoud
**1980**, 13, 175–253. [Google Scholar] - EN 1936. Natural Stone Test Method-Determination of Real Density and Apparent Density, and of Total and Open Porosity; European Committee for Standardization: Brussels, Belgium, 1999. [Google Scholar]
- Pellerin, F. La porosimetrie au mercure apliquee a l’ estude geotechnique des sols et des roches. Bull. Lias. Ponts Chaussés
**1980**, 106, 105–116. [Google Scholar] - Fairhurst, C.; Hudson, J. Draft ISRM suggested method for the complete stress–strain curve for intact rock in uniaxial compression. International Society for Rock Mechanics Commission on Testing Methods. Int. J. Rock Mech. Min. Sci.
**1999**, 36, 279–289. [Google Scholar] - ASTM D7012. Standard Test Method for Compressive Strength and Elastic Moduli of Intact Rock Core Specimens under Varying States of Stress and Temperatures; ASTM Standards: West Conshohocken, PA, USA, 2010. [Google Scholar]
- Eberhardt, E.; Stead, D.; Stimpson, B. Quantifying progressive pre-peak brittle fracture damage in rock during uniaxial compression. Int. J. Rock Mech. Min. Sci.
**1999**, 36, 361–380. [Google Scholar] [CrossRef] - Rocha, M. Mecânica das Rochas; Laboratório Nacional de Engenharia Civil: Lisboa, Portugal, 1981. (In Portuguese) [Google Scholar]
- Zoback, M.; Byerlee, J. The effect of Microcrack Dilatancy on the Permeability of Westerly Granite. J. Geophys. Res.
**1975**, 80, 752–755. [Google Scholar] [CrossRef] - Keaney, G.; Meredith, P.; Murrell, S. Laboratory study of permeability evolution in a ‘tight’ sandstone under non-hydrostatic stress conditions. Rock Mech. Pet. Eng.
**1998**, 1, 329–335. [Google Scholar] - Heiland, J.; Raab, S. Experimental investigation of the influence of differential stress on permeability of a lower permian (Rotliegend) sandstone deformed in the brittle deformation field. Phys. Chem. Earth Part A
**2001**, 26, 33–38. [Google Scholar] [CrossRef] - Popp, T.; Kern, H.; Schulze, O. Evolution of dilatancy and permeability in rock salt during hydrostatic compaction and triaxial deformation. J. Geophys. Res.
**2001**, 106, 4061–4078. [Google Scholar] [CrossRef] - Heiland, J. Laboratory testing of coupled hydro-mechanical processes during rock deformation. Hydrogeol. J.
**2003**, 11, 122–141. [Google Scholar] [CrossRef]

**Figure 1.**Sandstone monuments in the world. (

**a**) Luxor temple in Tebas (Upper Egypt); (

**b**) El Deir (“The Monastery”) in Petra (Jordan); (

**c**) Carthage aqueduct in Tunisia; (

**d**) Angkor Wat in Cambodia; (

**e**) St. Nicolas Cathedral in Switzerland; (

**f**) Birkenfeld Monastery in Germany; (

**g**) Strozzi Place in Florence (Italy).

**Figure 2.**St. Leonard Church: View of the main façade (West) of the sandstone national monument in Atouguia da Baleia with its Gothic vault.

**Figure 3.**Hand specimen view of the four varieties of sandstone A, B, C, and M collected close to the monuments, identified on images. These varieties have average values of water absorption coefficient of, respectively: 0.8 kg/m

^{2}h

^{1/2}, 2.4 kg/m

^{2}h

^{1/2}, 6.2 kg/m

^{2}h

^{1/2}, and 26 kg/m

^{2}h

^{1/2}.

**Figure 4.**Testing equipment for the absorption of water under low pressure: Karsten glass pipe on sandstone samples inside a glass tray.

**Figure 5.**Relationship between compressive strength (σ

_{c}) and coefficient of water absorption under low pressure (k) obtained from sandstone samples.

**Figure 6.**Variation between the strain at failure (ε

_{R}) and the water absorption (k) obtained on sandstone specimens.

**Figure 7.**Analytical modeling of results of curves of variety A sandstone specimens. The dashed black curves are the stress-strain diagrams of experimental results, and the thicker black curves are the analytical stress–strain diagrams: (

**a**) Specimen APN; (

**b**) Specimen AP9; (

**c**) Specimen AP5.

**Figure 8.**Analytical modeling of an experimental curve of variety B sandstone sample (BP3). The dashed black curve is the stress-strain diagram of experimental results and the thicker black curve is the analytical stress–strain-diagram.

**Figure 9.**Analytical modeling of data from variety M sandstone specimens diagrams. The dashed black curves are the stress-strain diagrams of experimental results and the thicker black curves are the analytical stress–strain diagrams: (

**a**) Specimen MP2; (

**b**) Specimen MP3; (

**c**) Specimen MP5; (

**d**) Specimen MP1.

Countries | Tunisia | Egypt | Jordan | Switzerland | Germany | Italy | Cambodja | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Origin | Aqueduct (Zoghlami et al., 2004) | Gebel Silsila (Fitzner et al., 2003) | Petra tombs (Heinrichs, 2005) | Fribourg (“Suisse plateau”) (Félix, 1977) | Birkenfeld Monastery (Tiano et al., 2000) | Strozzi Palace (Malesani and Vannucci, 1974) (Banchelli et al., 1997) | Angkor (Uchida et al., 1999) | ||||||

Lithology | Medium to fine grained sandstones | Fine grained sandstones (white to yellow brown) | Medium to fine grained sandstones (several colors, including white) | Blue molasse | Yellow molasse | Villarlod sandstone | Sandstones with rock fragments (green to gray) | Lithic arenites with carbonate cement (Pietraforte) | Grey yellow sandstone | Red sandstone | Green grauwacke | ||

Stratigraphy | Miocene Fortuna Formation. | Cretaceous Qoseir Formation. | Cambrian Umm Ishrin, Formation | Ordovician Disi Formation. | Extra Alpin sea molasse. | Triassic, middle Keuper (Schilfsandstein) | Upper Cretaceous (External Ligurides) | - | - | - | |||

Composition (%) | Quartz | 69–84 | 65–90 | Matrix-rich sandstone | quartz. sandstone | 60–70 | 89 | Major | + | ++ | + | ||

Feldspars | 0–1 | 0–8 | 10–15 | 9 | Major k-feldspar and plagioclases | + | + | ||||||

Micas | 0–0.5 | + | 1 | + | - | + | |||||||

Clays | 0–10 | 0–10; *25 | - | ++ | Illite and chlorite-vermiculite | - | - | - | |||||

Calcite | 0–3 | 20–30 | - | Major calcite and dolomite | - | - | - | ||||||

others | 1–5 | 3–5 (glauconite) | 1 (chlorite) | chlorite | + (goeth.) | + (hemat., goeth) | - | ||||||

Grain size analysis | Average size (mm) | 0.15–0.42 | 0.1–0.2 | 0.17 | 0.31 | 0.4–0.5 | 0.3–0.4 | 0.1–0.2 0.2–0.3 | 0.3 (0.1–0.5) | 0.08 (max. 0.25) | 0.2–0.3 | 0.1–-0.2 | - |

Distribution | Slight to poor graded | Slight to poor graded | - | - | graded | Slight graded | Slight graded | Slight graded | Poor graded | graded |

Origin | Aqueduct (Zoghlami et al., 2004) | Gebel Silsila (Fitzner et al., 2003) | Petra Tombs (Heinrichs, 2005) | Fribourg (“Suisse Plateau”) (Félix, 1977) | Birkenfeld Monastery (Tiano et al., 2000) | Strozzi Palace (Malesani and Vannucci, 1974) (Sorace, 1996) | Angkor (Uchida et al., 1999) | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Median pore radius (µm) | 20–50 (Mode) | 75–110 10 clayey sandstone | 13 | 115 | - | - | - | 100–200 (average) | - | - | - | - |

Unimodal/multimodal | Unimodal | Unimodal | - | - | - | - | - | - | - | - | - | - |

Porosity (%) | 17.5–26; 23 (open) | 25–35 | 17.4 | 21.3 | 15 | 18.2 | 14.9 | 19.2 ± 0.7 (open) | 1.8–5.5 (open) | 13–19 (open) | 11–15 (open) | 2 (open) |

Densities: bulk ^{1}, real ^{2} (kg/m^{3}) | 1950–2150 ^{1}2550–2600 ^{2} | 1800–2000 ^{1}2600–2750 ^{2} | - | - | 2260–2280 ^{1}2670 ^{2} | 2180–2190 ^{1}2660–2670 ^{2} | 2240–2270 ^{1}2640–2660 ^{2} | 2160 ± 20 ^{1}2670 ± 10 ^{2} | 2580–2610 ^{2} | 2100–2400 ^{1} | 2100–2400 ^{1} | 2600–2700 ^{1} |

Maximum water absorption (%) | 11 ± 1.7 | - | - | - | - | - | - | 1–2.5 | - | - | - | |

Water absorption coefficient/ /capillarity (kg/m^{2}h^{1/2})or permeability (mD) | 3.1 ± 1.6 (0.51 ± 0.27 kg/m ^{2}s^{1/2}) | - | - | - | 3.2–3.4 (┴) 3.3–3.5 (║) | 5.3–5.4 (┴) 6.2–6.5 (║) | 2.2–2.4 (┴) 3.6–3.7 (║) | - | < 1 mD | - | - | - |

Expansion (mm/m) | None | - | - | - | 1.63–1.74 (┴) 1.58–1.68 (║) | 2.5–2.76 (┴) 1.71–2.10 (║) | 2.27–2.28 (┴) 1.35–1.44 (║) | - | - | - | - | - |

Compressive strength (MPa) | 15.6 ± 7.9 | - | - | - | 46.3–55.4 (┴) 46.7–51.7 (║) | 30.6–40.4 (┴) | 62.4–71.0 (┴) 51.6–57.3 (║) | 52.3 ± 10.9 | 121.2-140 | 32–44 estimated | 43 estimated | 80 estimated |

Dynamic elastic modulus (GPa) | - | - | - | - | 7.2–8.4 (║) | 17.5–18.1 (║) | 10.3–12.1 (║) | - | 38.3 | - | - | - |

Bending strength (MPa) | - | - | - | - | 2.55–2.70 (┴) | 2.49–3.65 (┴) | 9.60–9.90 (┴) | - | 9.4 | - | - | - |

Tensile strength (MPa) | - | - | - | - | 1.17–1.28 (║) | 0.6–0.7 (║) | 2.24–2.74 (║) | - | 3.9 | - | - | - |

UPV of P waves (km/s) | - | 1.2–2.2 (┴) 1.3–2.6 (║) | - | - | 1.7 (┴) 1.9 (║) | 1.2–1.4 (┴) 1.7–1.8 (║) | 1.7–2.1 (┴) 2.3–2.5 (║) | - | - | 1.9–3.2 (║) | 3.9–4.0 (║) | 4.4 (║) |

Rebound number | - | - | - | - | 45 (┴) | 31 (┴) & 29 (║) | 47 (┴) & 42 (║) | - | - | 45–54 | 53 | 64 |

Drilling strength | - | 0.5–1.2 (┴) | 4.5 | 2.0 | - | - | - | 15 N | - | - | - | - |

Variety | Specimens | σ_{c} (MPa) | ε_{r} (×10^{−3}) | k (kg/m^{2}/√h) |
---|---|---|---|---|

A | AP38 | 126.4 | 5.7900 | 0.7 |

AP39 | 131.8 | 5.7931 | 0.7 | |

AP53 | 148.2 | 5.6968 | 0.8 | |

AP96 | 104.9 | 5.8542 | 0.7 | |

AP1 | 102.3 | 5.0000 | 1.3 | |

AP5 | 105.2 | 5.1000 | 1.1 | |

AP6 | 104.0 | 5.3000 | 1.0 | |

AP9 | 120.3 | 5.2000 | 0.9 | |

AP11 (N) | 136.2 | 6.2500 | 0.8 | |

AP13 (X) | 135.7 | 6.6300 | 0.9 | |

B | BP6 | 99.7 | 6.4725 | 2.4 |

BP27 | 82.6 | 6.4932 | 2.4 | |

BP32 | 83.1 | 6.8965 | 3.2 | |

BP45 | 97.3 | 6.7059 | 2.5 | |

BP72 | 98.2 | 6.5136 | 2.0 | |

BP3 | 95.0 | 7.2000 | 2.4 | |

BP13 | 105.3 | 7.8000 | 2.4 | |

BP | 94.5 | 6.5136 | 2.0 | |

C | CP18 | 47.8 | 7.4265 | 7.6 |

CP24 | 45.7 | 7.4139 | 7.4 | |

CP50 | 52.8 | 7.3366 | 5.3 | |

CP40 | 55.1 | 7.3497 | 5.3 | |

CP87 | 55.3 | 7.3627 | 5.3 | |

M | MP12 | 20.0 | 8.0615 | 23.2 |

MP13 | 20.4 | 8.0708 | 23.2 | |

MP9 | 22.0 | 8.0048 | 26.0 | |

MP10 | 22.7 | 7.9953 | 25.4 | |

MP11 | 20.7 | 8.0333 | 26.4 | |

MP12M | 22.3 | 8.0048 | 26.0 | |

MP92 | 21.3 | 8.0144 | 26.7 | |

MP109 | 19.6 | 8.0800 | 31.8 | |

MP110 | 22.4 | 8.0048 | 26.0 | |

MP111 | 21.6 | 8.0239 | 27.3 | |

MP112 | 21.3 | 8.0144 | 26.7 | |

MP113 | 22.2 | 8.0048 | 26.0 | |

MP1 | 18.7 | 7.9000 | 26.4 | |

MP2 | 20.0 | 6.7300 | 23.7 | |

MP3 | 24.5 | 7.9800 | 22.8 | |

MP5 | 17.9 | 8.8300 | 23.2 | |

MP6 | 17.6 | 7.9800 | 31.8 |

Variety | Specimens | σ_{c} (MPa) | ε_{r} (×10^{−3}) | Predicted ε_{r} (×10^{−3}) | Predicted and Experimental Absolute Difference ε_{r} (%) | Predicted σ_{c} (MPa) | Predicted and Experimental Absolute Difference σ_{c} (%) |
---|---|---|---|---|---|---|---|

A | AP38 | 126.4 | 5.7900 | 5.6971 | 1.6 | 146.0 | 15.5 |

AP39 | 131.8 | 5.7931 | 5.6971 | 1.7 | 146.1 | 10.9 | |

AP53 | 148.2 | 5.6968 | 5.7723 | 1.3 | 132.1 | 10.9 | |

AP96 | 104.9 | 5.8542 | 5.6971 | 2.7 | 147.6 | 40.7 | |

AP1 | 102.3 | 5.0000 | 6.0538 | 21.1 | 85.5 | 16.5 | |

AP5 | 105.2 | 5.1000 | 5.9554 | 16.8 | 96.8 | 8.0 | |

AP6 | 104.0 | 5.3000 | 5.9000 | 11.3 | 106.8 | 2.7 | |

AP9 | 120.3 | 5.2000 | 5.8393 | 12.3 | 112.0 | 6.9 | |

AP11 (N) | 136.2 | 6.2500 | 5.7723 | 7.6 | 144.9 | 6.4 | |

AP13 (X) | 135.7 | 6.6300 | 5.8393 | 11.9 | 142.8 | 5.2 | |

B | BP6 | 99.7 | 6.4725 | 6.4291 | 0.7 | 75.2 | 24.5 |

BP27 | 82.6 | 6.4932 | 6.4291 | 1.0 | 75.5 | 8.6 | |

BP32 | 83.1 | 6.8965 | 6.6131 | 4.1 | 66.9 | 19.5 | |

BP45 | 97.3 | 6.7059 | 6.4549 | 3.7 | 76.0 | 21.9 | |

BP72 | 98.2 | 6.5136 | 6.3151 | 3.0 | 84.9 | 13.5 | |

BP3 | 95.0 | 7.2000 | 6.4291 | 10.7 | 83.7 | 11.9 | |

BP13 | 105.3 | 7.8000 | 6.4291 | 17.6 | 90.7 | 13.9 | |

BP | 94.5 | 6.5136 | 6.3151 | 3.0 | 84.9 | 10.1 | |

C | CP18 | 47.8 | 7.4265 | 7.1988 | 3.1 | 41.8 | 12.5 |

CP24 | 45.7 | 7.4139 | 7.1800 | 3.2 | 42.5 | 7.1 | |

CP50 | 52.8 | 7.3366 | 6.9487 | 5.3 | 51.8 | 1.8 | |

CP40 | 55.1 | 7.3497 | 6.9487 | 5.5 | 51.9 | 5.8 | |

CP87 | 55.3 | 7.3627 | 6.9487 | 5.6 | 52.0 | 5.9 | |

M | MP12 | 20.0 | 8.0615 | 8.0317 | 0.4 | 22.5 | 12.6 |

MP13 | 20.4 | 8.0708 | 8.0317 | 0.5 | 22.5 | 10.5 | |

MP9 | 22.0 | 8.0048 | 8.1220 | 1.5 | 20.8 | 5.4 | |

MP10 | 22.7 | 7.9953 | 8.1034 | 1.4 | 21.1 | 7.1 | |

MP11 | 20.7 | 8.0333 | 8.1341 | 1.3 | 20.7 | 0.1 | |

MP12M | 22.3 | 8.0048 | 8.1220 | 1.5 | 20.8 | 6.7 | |

MP92 | 21.3 | 8.0144 | 8.1432 | 1.6 | 20.5 | 3.8 | |

MP109 | 19.6 | 8.0800 | 8.2840 | 2.5 | 18.5 | 5.6 | |

MP110 | 22.4 | 8.0048 | 8.1220 | 1.5 | 20.8 | 7.1 | |

MP111 | 21.6 | 8.0239 | 8.1609 | 1.7 | 20.2 | 6.3 | |

MP112 | 21.3 | 8.0144 | 8.1432 | 1.6 | 20.5 | 3.8 | |

MP113 | 22.2 | 8.0048 | 8.1220 | 1.5 | 20.8 | 6.3 | |

MP1 | 18.7 | 7.9000 | 8.1341 | 3.0 | 20.3 | 8.8 | |

MP2 | 20.0 | 6.7300 | 8.0485 | 19.6 | 18.5 | 7.3 | |

MP3 | 24.5 | 7.9800 | 8.0180 | 0.5 | 22.5 | 8.0 | |

MP5 | 17.9 | 8.8300 | 8.0317 | 9.0 | 24.7 | 37.8 | |

MP6 | 17.6 | 7.9800 | 8.2840 | 3.8 | 18.3 | 3.9 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ludovico-Marques, M.; Chastre, C.
Prediction of Stress–Strain Curves Based on Hydric Non-Destructive Tests on Sandstones. *Materials* **2019**, *12*, 3366.
https://doi.org/10.3390/ma12203366

**AMA Style**

Ludovico-Marques M, Chastre C.
Prediction of Stress–Strain Curves Based on Hydric Non-Destructive Tests on Sandstones. *Materials*. 2019; 12(20):3366.
https://doi.org/10.3390/ma12203366

**Chicago/Turabian Style**

Ludovico-Marques, Marco, and Carlos Chastre.
2019. "Prediction of Stress–Strain Curves Based on Hydric Non-Destructive Tests on Sandstones" *Materials* 12, no. 20: 3366.
https://doi.org/10.3390/ma12203366