Analysis on damage mechanical characteristics of marble exposed to high temperature

Abstract

Study on the mechanical response of rock under the combined effects of high temperature and mechanical load using the method of damage mechanics is a relatively new research direction of rock mechanics. The thermal damage and mechanical damage are indicated by the uniaxial compression mechanical test of marble after exposure to high temperature. The overall damage evolution equation of rock is deduced and high-temperature damage constitutive equation is established on the basis of the macroscopic phenomenological damage mechanics and nonequilibrium statistical methods. The evolution law of the damage characteristics of rock after exposure to high temperature, taking the temperature and mechanical load as the control variable, is discussed. The results show that the effect of temperature on the mechanical properties of marble can be accurately described through the definition of thermal damage using elastic modulus. The high-temperature damage constitutive model curves are similar to the experimental curves, reflecting well the brittleness of marble. Additionally, the constitutive relations of rock after exposure to high temperature have close relationships with the elastic modulus, compressive strength, and peak strain. The overall damage of rock changes along two evolutionary pathways of high temperature and strain, reflecting the mutual combined influences of high temperature and strain on the material damage extension, can accurately reveal the damage mechanical behavior and the damage propagation law of rock material after exposure to high temperature.

Keywords

Rock mechanics thermal damage mechanical damage overall damage combined effects mechanical properties

Introduction

Rock is a material widely used in engineering (Brotóns et al., 2013), and accurate measurements of its mechanical properties are essential foundations for rock engineering (Li et al., 2014). In actual engineering, rock material is always in a certain environment, experiencing the effects of different weathering conditions. There are many factors that can cause rock weathering (Althaus et al., 1994; Basu et al., 2009; Malaga-Starzec et al., 2006; Talesnick and Shehadeh, 2007; Vasarhelyi, 2005; Veniale et al., 2008), including change of temperature, which is one of the most important factors (Althaus et al., 1994; Liang et al., 2006; Tian et al., 2012). There are many natural internal micro pores and micro cracks in the rock. In addition, a large number of new micro cracks are produced under the action of mechanical load and temperature, the expansion of these micro cracks will cause deterioration in the internal properties of the rock (Eberhardt et al., 1999; Ferrero and Marini, 2001; Homand et al., 2000; Nejati and Ghazvinian, 2014; Rigopoulos et al., 2011; Yavuz et al., 2010; Yukutake, 1989), then damage deformation in the internal structure of rock and various physical and chemical changes will occur (Bernal and Gunes, 2003; Chen et al., 2012; Luo et al., 2014). Therefore, the study on damage degradation mechanisms and the mechanical response of rock material subjected to mechanical load under the influence of high temperature has very important guiding significance for numerous rock mass engineering constructions.

So far, there has already been a great deal of research on damage of rocks subjected to mechanical load at room temperature (Eslami et al., 2012; Heap and Faulkner, 2008; Heap et al., 2009; Luo et al., 2014; Martin and Chandler, 1994). In contrast, study on the damage evolution mechanism of rock subjected to high temperature is still insufficient. In this report, the mechanical damage caused by mechanical load and thermal damage caused by temperature effect are discussed based on the results of uniaxial compression mechanical experiments on rock samples exposed to different temperatures. The high-temperature damage constitutive equation is established on the basis of the combined effects of temperature and mechanical load, and is verified by the test results. In addition, the effects of high temperature and mechanical load on the damage extension are studied. Damage mechanical properties and the damage propagation law of rock material after exposure to high temperature are also analyzed.

Thermodynamics characteristics of marble after exposure to high temperature

The rock sample used in the test is marble, extracted from a quarry in Qinling underground engineering in China, whose main mineral components are 90% dolomite, 3% calcite, 3% muscovite, 1% talc, 2% amphibole, and 1% quartz. The average density of the marble at room temperature is 2.60 g/cm³. Rock samples whose longitudinal wave velocity is relatively close to each other are selected prior to test and processed to be 50 mm × 100 mm cylinder (International Society for Rock Mechanics (ISRM), 1979). The test temperature is designed to fall into the following seven groups: room temperature, 100℃, 200℃, 400℃, 600℃, 800℃, and 1000℃, each group comprised of no less than three specimens and with the heating rate of 10℃/min. The predetermined temperature, once reached, is kept constant for 2 h. After that, specimens are left in the furnace to cool down to room temperature. During the test, the sample is properly placed on the test machine and the axial load is applied at a loading rate of 0.8 MPa/s until failure occurred.

Since each rock specimen roughly underwent through four different stages, namely compaction, elasticity, plastic, and destruction, each group of stress–strain curves of the rock specimens has a similar distribution shape. Hence, this paper lists only the typical stress–strain curves shown in Figure 1. These results reveal that from room temperature to 600℃, the temperature changes have no obvious effect on the stress–strain curves, the marble samples all show a sudden brittle failure occurring at the peak stress points, and plastic deformation is not obvious. Whereas, when the temperature reaches 800℃, the stress–strain curve is concave, indicating that the ductility of marble is significantly enhanced. After the stress reaches its peak, the strain still experiences a slow increase, showing clear plastic behavior and post-peak behavior. Therefore, there may exist a brittle to ductile transition threshold temperature in the stress–strain curves of marble between 600 and 800℃.

Uniaxial compression strengths of marble specimens after exposure to different temperatures are shown in Figure 2. In terms of overall trend, from 100℃ to 600℃, the uniaxial compressive strengths are increased by 24.74%, 1.73%, 18.53%, 7.14%, respectively, compared with room temperature. At this time, the temperature has an enhanced effect on marble. When temperature exceeds 800℃, uniaxial compression strengths show a decreasing trend with increasing temperature, from 800℃ to 1000℃, with a drop of 15.58%, 81.87%, respectively, compared with room temperature. At this time, the temperature has a weakening effect on marble. The combination among particles inside marble is not very close, and there exist many gaps as well as a large number of micro cracks (Hoxha and Homand, 2000; Wong, 1985; Zhao, 1998). Two different effects of temperature on the mechanical properties of marble have been identified. On the one hand, thermal deformation may cause the close of some primary micro cracks, with concomitant decrease in the number of micro cracks, increase in the degree of compaction, improvement of the contact state among mineral grains, strengthening of the friction characteristics, and enhancement of the bearing capacity of the specimen. On the other hand, because of the difference among the thermal expansion coefficients of the internal mineral grains of the rock, the deformation of the mineral particles under high temperature will lead to the generation of thermal stress among the mineral particles (Jansen et al., 1993), which may diminish the bearing capacity of the specimen. In addition, due to the great discreteness inside the marble, the open degrees of micro cracks of discrete rock samples are also different (Hajpal, 2002), thus the rock strengths still show certain discreteness even at the same temperature.

The relationship between peak strain and temperature is shown in Figure 3, where it is obvious that the peak strain increases with rising temperature. At 1000℃, the peak strain is increased by 200% compared to room temperature. The relationship between elastic modulus and temperature is shown in Figure 4. In this study, the elastic modulus was calculated by fitting the approximate straight line segment of the stress–strain curve prior to peak stress. The elastic modulus exhibits a discrete distribution, but, on the whole, decreases linearly with increasing temperature. At 1000℃, elastic modulus is only 1.06 GPa, indicating that this high temperature has a great influence on the deformation process of marble.

Damage characteristics of marble after exposure to high temperature

The combined effects of temperature and mechanical load can produce numerous micro cracks in rock material, and the expansion of micro cracks leads to the damage inside rock, including the thermal damage caused by temperature and mechanical damage caused by mechanical load.

Thermal damage

The thermal stress induced by temperature change will inevitably produce a large number of micro cracks. These micro cracks gradually expand with the increase of temperature, resulting in the significant decrease in elastic modulus, causing damage to the rock. Therefore, in this study, the elastic modulus is chosen to define the damage variable, which is used to describe the temperature effect on the mechanical properties of the rock material. Additionally, we propose the concept of thermal damage according to the macroscopic phenomenological damage mechanics previously described (Heap and Faulkner, 2008; Martin and Chandler, 1994)

D T = 1 - \frac{E T}{E 0}

(1)

where

D T

stands for the thermal damage of rock sample after exposure to high temperature,

E T

stands for the elastic modulus of rock sample after exposure to high temperature,

E 0

stands for the elastic modulus of rock sample at room temperature. In order to represent the damage effect of high temperature on marble, the thermal damage at room temperature is assumed to be 0 or negligible.

Figure 5 shows the relationship between the thermal damage of marble and temperature. The thermal damage evolution equation can be calculated as follows

D T = - 0.0045 + 2.32 (\frac{T}{1000}) - 4.46 (\frac{T}{1000}) 2 + 3.11 (\frac{T}{1000}) 3 (R 2 = 0.962)

(2)

At 200℃, the thermal damage reaches a peak due to the conditions created when the heating temperature does not exceed 200℃, under such conditions the moisture inside the rock seeks to escape outwardly along the pores during the evaporation process, leading to the extension of pores. From 200℃ to 400℃, the thermal damage decreases slightly because, by this time in the process, the escape of moisture has been completed and the rock particles continue to swell until the pores are filled, which leads to the rock samples being able to continue to withstand a greater pressure at this stage. Above 600℃, the thermal stress among the mineral particles will lessen the bearing capacity of the specimen and cause the thermal damage to increase sharply.

The relationship between uniaxial compressive strength and thermal damage is presented in Figure 6, where it is clearly seen that, with increasing thermal damage, the compressive strength increases first and then decreases, and it reaches its peak in the vicinity of the thermal damage of 0.22. Because the main factor affecting thermal damage is the temperature, the variation of compressive strength with thermal damage, to a certain extent, reflects the influence of temperature on the rock strength.

Damage strain energy release rate

Assuming that the material properties of rock are isotropic, the damage strain energy release rate is obtained using the following formula (Lemaitre and Desmorat, 2005)

θ = \frac{σ rq 2}{2 E T (1 - D T) 2} [\frac{2}{3} (1 + μ) + 3 (1 - 2 μ) (\frac{σ m}{σ rq}) 2]

(3)

where θ stands for the damage strain energy release rate; μ stands for Poisson's ratio of rock material;

σ m

stands for the average stress,

σ rq

stands for the equivalent stress. Under uniaxial stress state, i.e.,

σ 1 = - σ

σ 2 = σ 3 = 0

σ m = - \frac{1}{3} σ

σ rq = σ

, the damage strain energy release rate can be expressed as

θ = \frac{σ 2}{2 E T (1 - D T) 2}

(4)

By uniaxial compression test data, according to equation (4), the changing law of damage strain energy release rate along with the change of temperature can be calculated as shown in Figure 7. The damage strain energy release rate, which stands for the generalized force threshold value, increases with increasing temperature. For instance, below 800℃, the damage strain energy release rate changes little, whereas above 800℃, it increases rapidly. Thus, 800℃ may be the threshold temperature of the damage strain energy release rate.

Mechanical damage

By taking into account the heterogeneity of the rock material and the uneven distribution in infinitesimal strength of the rock material, and assuming that the rock strength obeys Weibull distribution (Weibull, 1951), the corresponding probability density function of strength is obtained

p (ɛ) = \frac{m}{ɛ 0} (\frac{ɛ}{ɛ 0}) m - 1 e [- (\frac{ɛ}{ɛ 0}) m]

(5)

where

p (ɛ)

stands for the infinitesimal strength distribution function; m and

ɛ 0

stand for the parameters related to rock material, which reflect the different response characteristics of the rock material when subjected to the external load; ɛ stands for the infinitesimal strain unit.

The damage is caused by uneven destruction of local infinitesimal unit. Assuming that the number of infinitesimal unit, which has been destructed is n, the total number of infinitesimal unit is N, then the damage variable can be defined as $D ɛ = \frac{n}{N}$ , and the number of infinitesimal unit that has been destructed in interval of $[ɛ, ɛ + d ɛ]$ is $NP (ɛ) d ɛ$ . The damage variable is expressed as

D ɛ = \frac{n (ɛ)}{N} = \frac{\int_{0}^{ɛ} NP (ɛ) d ɛ}{N} = 1 - e [- (\frac{ɛ}{ɛ 0}) m]

(6)

where

D ɛ

stands for the mechanical damage of rock.

According to strain equivalence principle proposed by Lemaitre (1986), the rock damage constitutive equation at room temperature can be obtained as follows

σ = E 0 ɛ (1 - D ɛ) = E 0 ɛ e [- (\frac{ɛ}{ɛ 0}) m]

(7)

Because the slope at the peak stress points of stress–strain curve is zero, we can get

\frac{d σ}{d ɛ} | ɛ = ɛ c = E 0 e [- (\frac{ɛ c}{ɛ 0}) m] [1 - m (\frac{ɛ c}{ɛ 0}) m] = 0

(8)

The relationship between m and $ɛ 0$ can be determined according to equation (8)

ɛ 0 = ɛ c (\frac{1}{m}) - \frac{1}{m}

(9)

where

σ c

and

ɛ c

stand for the peak stress and peak strain of the rock, respectively.

Substituting equation (9) into equation (6), the damage evolution equation of rock can be obtained, i.e.

D ɛ = 1 - e [- \frac{1}{m} (\frac{ɛ}{ɛ c}) m]

(10)

Construction and validation of the high-temperature damage constitutive equation

According to the strain equivalence principle (Lemaitre, 1986), we can get the damage constitutive relation of rock material

σ = E 0 ɛ (1 - D) = E 0 (1 - D T) (1 - D ɛ) ɛ

(11)

where D stands for the overall damage of rock.

The overall damage of the rock material changes along two evolutionary pathways of high temperature and strain, reflecting the mutual coupled influences of high temperature and strain on the material damage extension, and can truly reveal the damage mechanical behavior and the damage propagation law of the rock material after exposure to high temperature. Combining equations (1), (10), and (11), the overall damage evolution equation can also be obtained as

D = D T + D ɛ - D T D ɛ = 1 - \frac{E T}{E o} e [- \frac{1}{m} (\frac{ɛ}{ɛ c}) m]

(12)

Substituting equation (12) into equation (11), the high-temperature damage constitutive equation can be obtained

σ = E T e [- \frac{1}{m} (\frac{ɛ}{ɛ c}) m] ɛ

(13)

The high-temperature damage constitutive model curve and the experimental curve, according to equation (13), are shown in Figure 8. We can see that the general shape of the fitting curve is similar to that of the experimental curve, reflecting well the brittleness of marble. Before peak stress, the rock sample presents elastic deformation, and after peak stress, the strength drops rapidly, until it reaches zero. However, there are also certain deviations between the fitting curve and the experimental curve, mainly in two aspects. On the one hand, before the peak, the compaction stage does not exist in the fitting curve under the same strain, and the stress value of the fitting curve is greater than that of experimental curve. On the other hand, after the peak, the difference between the fitting curve and the experimental curve is obvious, this is mainly because in addition to the strong interaction among adjacent particles, there also exist strong interaction among distant particles. Due to the fact that the constitutive model is a continuous change function, it cannot reflect the damage localization process in rocks. The constitutive model itself and the test process, both, may cause certain deviations. Therefore, in order to establish a more suitable model that better reflects the actual rock deformation, the constitutive model, as well as the reliability of the experiment, need to be improved to a certain extent.

We also calculate the m value at the different temperatures by fitting the stress–strain curve of marble according to equation (13). The variation of m with temperature is shown in Figure 9, which indicate that below 600℃, m exhibits the volatility change, and at 600℃, it is increased by 1.4%, compared with room temperature. Above 600℃, m decreases approximately linear with temperature increasing, and at 1000℃, it is only 1.68. The results also indicate that the stress at arbitrary point is closely related to the elastic modulus, the compression strength, the peak strain, and the strain at that point. The rock damage shape parameter m is closely related to the material properties, and plays a key role in the nonlinear behavior of rock deformation, namely, the smaller m is, the more the rock tends to undergo plastic failure; the greater m is, the more the rock tends to undergo brittle failure.

Overall damage

Equation (12) suggests that the combination of high temperature and mechanical load exacerbates the rock damage, leading to significant nonlinear characteristics on rock deformation. However, although high temperature results in local internal rock damage, the fault slip and dislocation of rock grains under mechanical load may limit the expansion to some extent. Thus, the overall damage may be reduced under the combined effects of high temperature and mechanical load. Accordingly, to study the change in overall damage, the key is to find the overall damage evolution law of the rock. To this end, the overall damage evolution curve was determined according to uniaxial compression test mechanical parameters of the rock as depicted in Figure 10. The results on this figure indicate that, on the whole, the overall damage of marble increases rapidly with increasing temperature, especially under conditions of high temperature, when the damage is significantly different to that observed at room temperature. At 1000℃, the damage variable is close to 1, when the strain is very low, indicating that the marble is close to collapse due to the high temperature. Additionally, under the same temperature, the damage increases with the increase of strain. In the initial stage, when the rock is subjected to mechanical load, the micro pores and micro cracks are gradually closed and the density is increased. Then, the rock goes into the linear stage and once the deformation reaches a certain degree, the damage begins to evolve, stably expanding, and subsequently accelerating until the damage variable reaches 1. Along with the process of damage evolution, micro cracks sprout, expand, and converge until macroscopic cracks appear.

Discussions

Rock aggregated with particles or crystal cemented together is a nonhomogeneous material that contains numerous micro cracks and micro holes within it (Bhasin and Kaynia, 2004; Liu and Xu, 2013). The changes in the macroscopic mechanical characteristics result from changes in the microstructure (Indraratna et al., 2010; Liu and Xu, 2014; Wu et al., 2013). Rocks in practical engineering are generally subjected to joint action of two or more factors. From the perspective of material mechanics, high-temperature effect has weakened the connection among rock particles, namely the degradation in material performance over time. From the perspective of structural engineering, the degradation in material performance will influence the rock strength, particularly the deformation characteristics and the structural safety. The high temperature or mechanical load both can increase the overall damage, while the combined effects of high temperature and mechanical load will lessen the overall damage.

The high-temperature damage constitutive model of rock was constructed in this study to investigate the combined effects of high temperature and mechanical load and adopted to characterize the complex relationships among the high temperature, mechanical load and damage. The damage mechanical properties of rock predicted by the damage constitutive model appear to conform to the actual situation, and thus also provide a new method to study the damage mechanism of rock under the combined effects of high temperature and mechanical load.

Conclusions

In this study, the method of damage mechanics is applied to investigate the thermodynamic of rock material, and to establish the high-temperature damage constitutive model of rock. The damage mechanical behavior and the damage propagation law of rock material at high temperatures are analyzed. The following conclusions could be drawn:

The temperature changes, from room temperature to 600℃, have no obvious effect on the stress–strain curves. But when the temperature reaches 800℃, the stress–strain curve is concave.

By the definition of thermal damage using an elastic modulus, the thermal damage evolution equation is established. Eight hundred degree Celsius may be the threshold temperature of the damage strain energy release rate.

Considering the combined effects of high temperature and mechanical load, the high-temperature damage constitutive equation of rock is established. The high-temperature damage constitutive equation of rock after exposure to high temperature has close relationships with the elastic modulus, the compression strength, and the peak strain.

Although the high temperature results in local internal rock damage, the fault slip and dislocation of rock grains under mechanical load may limit the expansion to a certain extent. Thus, the overall damage may be reduced under the combined effects of high temperature and mechanical load.

The overall damage of marble changes along two evolutionary pathways of high temperature and strain. At or above 1000℃, the damage variable is close to 1, when the strain is very low, indicating that the marble is close to collapse due to the high temperature. Moreover, under the same temperature, the damage increases with the increase of the strain.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work has been supported by The National Natural Science Foundation of China (No. 51378497).

References

Althaus

Friz-Ttipfer

Lempp

(1994) Effects of water on strength and failure mode of coarse-grained granites at 300℃. Rock Mechanics and Rock Engineering 27(1): 1–21.

Basu

Celestino

Bortolucci

(2009) Evaluation of rock mechanical behaviors under uniaxial compression with reference to assessed weathering grades. Rock Mechanics and Rock Engineering 42: 73–93.

Bernal

Gunes

(2003) Flexibility based approach for damage characterization: Benchmark application. Journal of Engineering Mechanics ASCE 130(1): 61–70.

Bhasin

Kaynia

(2004) Static and dynamic simulation of a 700-m high rock slope in western Norway. Engineering Geology 71: 213–226.

Brotóns

Tomás

Ivorra

(2013) Temperature influence on the physical and mechanical properties of a porous rock: San Julian's calcarenite. Engineering Geology 167: 117–127.

Chen

Shao

(2012) Coalescence of fractures under uni-axial compression and fatigue loading. Rock Mechanics and Rock Engineering 45: 241–249.

Eberhardt

Stead

Stimpson

(1999) Quantifying pre-peak progressive brittle fracture damage in rock during uniaxial compression. International Journal of Rock Mechanics and Mining Sciences 36: 361–380.

Eslami

Hoxha

Grgic

(2012) Estimation of the damage of a porous limestone using continuous wave velocity measurements during uniaxial creep tests. Mechanics of Materials 49: 51–65.

Ferrero

Marini

(2001) Experimental studies on the mechanical behaviour of two thermal cracked marbles. Rock Mechanics and Rock Engineering 34(1): 157–166.

10.

Hajpal

(2002) Changes in sandstones of historical monuments exposed to fire or high temperature. Fire Technology 38: 373–382.

11.

Heap

Faulkner

(2008) Quantifying the evolution of static elastic properties as crystalline rock approaches failure. International Journal of Rock Mechanics and Mining Sciences 45(4): 564–573.

12.

Heap

Vinciguerra

Meredith

(2009) The evolution of elastic moduli with increasing crack damage during cyclic stressing of a basalt from Mt. Etna volcano. Tectonophysics 471(1–2): 153–160.

13.

Homand

Hoxha

Belem

(2000) Geometric analysis of damaged microcracking in granites. Mechanics of Materials 32: 361–376.

14.

Hoxha

Homand

(2000) Microstructural approach in damage modeling. Mechanics of Materials 32(6): 377–387.

15.

Indraratna

Oliveira

DAF

Brown

(2010) A shear-displacement criteria for sand-infilled rock discontinuities. Geotechnique 60: 623–633.

16.

International Society for Rock Mechanics (ISRM) (1979) SM for determining the uniaxial compressive strength and deformability of rock materials. ISRM Suggested Methods 2: 137–140.

17.

Jansen

Carlson

Young

(1993) Ultrasonic-imaging and acoustic-emission monitoring of thermally-induced microcracks in lac-du-bonnet-granite. Journal of Geophysical Research: Solid Earth 98: 22231–22243.

18.

Lemaitre

(1986) Local approach of fracture. Engineering Fracture Mechanics 25(5–6): 523–537.

19.

Lemaitre

Desmorat

(2005) Engineering damage mechanics: Ductile, creep, fatigue and brittle failures, Berlin: Springer.

20.

Huang

(2014) Strain rate dependency of coarse crystal marble under uniaxial compression: Strength, deformation and strain energy. Rock Mechanics and Rock Engineering 47: 1153–1164.

21.

Liang

Zhao

(2006) Experimental study of temperature effects on physical and mechanical characteristics of salt rock. Rock Mechanics and Rock Engineering 39(5): 469–482.

22.

Liu

(2013) Study on dynamic characteristics of marble under impact loading and high temperature. International Journal of Rock Mechanics & Mining Sciences 62: 51–58.

23.

Liu

(2014) Mechanical properties of Qinling biotite granite after high temperature treatment. International Journal of Rock Mechanics & Mining Sciences 71: 188–193.

24.

Luo

Lytton

(2014) Energy-based mechanistic approach for damage characterization of pre-flawed visco-elasto-plastic materials. Mechanics of Materials 70: 18–32.

25.

Malaga-Starzec

Akensson

Lindqvist

(2006) Microscopic and macroscopic characterization of the porosity of marble as a function of temperature and impregnation. Construction and Building Materials 20(10): 939–947.

26.

Martin

Chandler

(1994) The progressive fracture of Lac du Bonnet granite. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts 31(6): 643–659.

27.

Nejati

Ghazvinian

(2014) Brittleness effect on rock fatigue damage evolution. Rock Mechanics and Rock Engineering 47: 1839–1848.

28.

Rigopoulos

Tsikouras

Pomonis

(2011) Microcracks in ultrabasic rocks under uniaxial compressive stress. Engineering Geology 117: 104–113.

29.

Talesnick

Shehadeh

(2007) The effect of water content on the mechanical response of a high-porosity chalk. International Journal of Rock Mechanics and Mining Sciences 44: 584–600.

30.

Tian

Kempka

(2012) Physical properties of sandstones after high temperature treatment. Rock Mechanics and Rock Engineering 45: 1113–1117.

31.

Vasarhelyi

(2005) Statistical analysis of the influence of water content on the strength of the Miocene limestone. Rock Mechanics and Rock Engineering 38: 69–76.

32.

Veniale

Setti

Lodola

(2008) Diagnosing stone decay in built heritage. Facts and perspectives. Materiales de Construccion 58(289–290): 11–32.

33.

Weibull

(1951) A statistical distribution function of wide applicability. Journal of Applied Mechanics 18: 293–297.

34.

Wong

(1985) Geometric probability approach to the characterization and analysis of microcracking in rocks. Mechanics of Materials 4(3–4): 261–276.

35.

Zhu

Zhao

(2013) A further study on seismic response of a set of parallel rock fractures filled with viscoelastic materials. Geophysical Journal International 192: 671–675.

36.

Yavuz

Demirdag

Caran

(2010) Thermal effect on the physical properties of carbonate rocks. International Journal of Rock Mechanics & Mining Sciences 47: 94–103.

37.

Yukutake

(1989) Fracturing process of granite inferred from measurements of spatial and temporal variations in velocity during triaxial deformation. Journal of Geophysical Research 94: 15639–15651.

38.

Zhao

(1998) Crack pattern evolution and a fractal damage constitutive model for rock. International Journal of Rock Mechanics and Mining Sciences 35(3): 349–366.