Home Technology Statistical Variations and New Correlation Models to Predict the Mechanical Behavior and Ultimate Shear Strength of Gypsum Rock
Article Open Access

Statistical Variations and New Correlation Models to Predict the Mechanical Behavior and Ultimate Shear Strength of Gypsum Rock

  • Ahmed Mohammed EMAIL logo and Wael Mahmood
Published/Copyright: August 11, 2018
Download Article (PDF)
Become an author with De Gruyter Brill
Submit Manuscript Author Information
Open Engineering
From the journal Open Engineering Volume 8 Issue 1

Abstract

In this study, over 1000 data from the several research studies was used to characterize and compare the density, strengths, modulus, flexural strength, porosity and the ultimate shear strengths of the calcium rocks. The gypsum rock data were statistically analyzed, quantified and compared with the limestone rock data. The ranges of the densities for gypsum rock (CaSO4·2H2O) and limestone rock (CaCO3) were 2.10 to 2.83 gm/cm3 and 1.70 to 2.75 gm/cm3, respectively. The compressive and tensile strengths of the gypsum and limestone rocks varied from 2 MPa to 250 MPa and 1.8 MPa to 25 MPa, respectively. Vipulanandan correlation model was effective in relating the modulus of elasticity, flexural strength, with the relevant strengths of the rocks. A new nonlinear Vipulanandan failure criterion was developed to better quantify the tensile strength, pure shear (cohesion) strength and predict the maximum shear strength limit with applied normal stress on the gypsum and limestone rocks. The prediction of the failure models for the two rock types was also compared to the Mohr-Coulomb failure model. The Vipulanandan failure model predicted the maximum shear strength limit was, as the Mohr-Coulomb failure model does not have a limit on the maximum shear strength. With the Vipulanandan failure model based on the available data, the maximum shear strengths predicted for the gypsum and limestone rocks were 64 MPa and 114 MPa, respectively.

Keywords: Calcium rocks; Strength; Flexural strength; Failure model; Statistical analysis; Modeling

1 Introduction

For constructing of various types of infrastructures on the rocks, it is important to better quantify the mechanical properties and failure criteria for the rocks. Also for the hydraulic fracturing of rocks in a cost-effective way it is important to design the drilling and fracturing processes using the rock properties including the tensile strength and failure criteria. Mohr-Coulomb criteria is used to characterize the failure of the rocks, with over predicting the tensile strength and no limit on the maximum shear strength tolerance with the applied normal stress on the rocks (Singh et al. 2015). In addition, there is very limited property correlation in the literature for the rocks. Hence, there is a need for developing improved failure criteria and property correlations for the rocks (Omar 2017).

Gypsum rock is a soft, sulfate mineral composed of calcium sulfate dehydrate, with the chemical formula CaSO4·2H2O and it covers more than 20% of the earth crust, seven million square kilometers of gypsum rock covering by highly soluble gypsum-bearing rocks (Dreybrodt et al., 2002). Gypsum is a soluble mineral that deposits from natural water because of evaporation. Gypsum can transform to anhydrite by losing water, or anhydrite can transform to gypsum by the addition of water (Salih 2013). Several studies on gypsum rock showed that it experiences the various problems due change in the physical and mechanical properties of gypsum rock due to the dissolution of gypsum, so there is a need for prediction the mechanical properties of such a problematic type of the rock, such as gypsum rock (Klimchouk, 1996; Johnson, 2005; Salih et.al. 2017).

Limestone is another Sedimentary rock, composed mainly of skeletal fragments of marine organisms such as coral, forams and mollusks. Its major materials are the minerals calcite and aragonite, which are different crystal forms of calcium carbonate (CaCO3) (Milliman et al. 2012; Abdelaali et al. 2013). The limestone was popular as a building stone, because of its regional availability, its mechanical properties, and its attractive appearance, and was used in many types of buildings. Also, limestone has been used in several types of constructions such as, concrete aggregate, asphalt pavement aggregate, asphalt surface treatments, road base, structural fill, railroad ballast, riprap and drainage and erosion control (Bednarik et al. 2014).

Uniaxial unconfined compressive strength (UCS), the most widely used parameter to evaluate rock strength, requires expensive and time-consuming testing with sample preparation (Karakus et al. 2005). Many researchers have introduced several empirical equations for determination of rock strength from simple physical properties. By means of such properties, rock strength may be determined in an easy, quick, and inexpensive manner during field investigations (Sabatakakis et al. 2008; Rajabzadeh et al. 2012). In rock engineering, most applied rock classification systems are based on mechanical parameters such as uniaxial unconfined compressive strength, tensile strength (σt) and Young’s modulus or deformability modulus (E) (Selçuk et al. 2016). Tensile strength and compression modulus are two important parameters in rock mechanics, and are used in the initiation and propagation of fractures in hydraulic fracture modeling (Meng and Pan 2007; Vipulanandan and Mohammed 2015a). The tensile strength was determined for two gypsum and limestone rocks using the standard test method for direct tensile strength from American Society of the International Association for Testing and Materials (ASTM) (Jensen 2016). The unconfined compressive strength (UCS) and angle of internal friction (φ) of gypsum and limestone rocks are key parameters needed to address a range of geomechanical problems ranging from limiting wellbore instabilities during drilling, to assessing sanding potential and quantitatively constraining stress magnitudes using observations of wellbore failure (Moos et al., 2003; Zoback et al. 2003).

1.1 Vipulanandan models

The Vipulanandan correlation model has been used to predict the behavior of cement and polymer modified soils (Usluogullari et al. 2011). Vipulanandan et al. (2007-2015) used the relationship to represent the variation of in-situ vertical stress and logarithmic undrained shear strength of the soft marine and deltaic clays; the Vipulanandan rheological model was also used to predict the rheological properties with the electrical resistivity of nanoclay modified bentonite-drilling muds. For shear thinning fluids, the shear stress-shear strain-rate relationship is nonlinear with a limit of the maximum shear stress tolerance. Vipulanandan and Mohammed (2014) used the Vipulanandan rheological model to predict the maximum shear stress limit for the bentonite drilling mud modified with the polymer. Similar trends were observed in many other engineering and environmental applications and have been modeled using the Vipulanandan correlation and rheological models (Mohammed 2016 - 2017).

1.2 Objectives

The focus of this study was to develop relationships for the tensile strength, compressive modulus, and failure model for the gypsum and limestone rocks using over 1000 data collected from the literature. The specific topics of interest are as follows:

  1. To qualify the statistical variation in the density, compressive strength, tensile, modulus of elasticity, flexural strength, and porosity of gypsum and limestone rocks.

  2. To investigate and quantify the correlation relation between the compression strength and tensile strength of gypsum and limestone rocks using the Vipulanandan correlation model.

  3. To investigate and quantify the correlation between the compressive modulus of elasticity, flexural strength, and porosity with compressive and tensile strengths respectively of gypsum and limestone rocks using the Vipulanandan correlation model.

  4. Quantify the shear failure strength of the rocks using the new Vipulanandan failure model and compare the prediction to the Mohr-Coulomb failure model.

2 Methods and materials

2.1 Data collection

This study focused on the behavior of gypsum rock compared with limestone rock collected from several research studies (Swapnil et al. 2004; Vipulanandan et al. 2009; Nam et al 2010; Salih et al 2017). The properties of interest were density, compressive and tensile strengths, flexural strength, compressive modulus, porosity and shear strength of the gypsum and limestone rocks. The density study focused on the statistical distribution and the range of variation using 97 data on gypsum and 184 data on limestone. The number of data collected based on the properties of the rock was also a clear indicator of the interest in rock behavior around the world.

2.2 Modeling

2.2.1 Vipulanandan correlation model

The nonlinear correlation between the rock properties was investigated using the Vipulanandan correlation model for the gypsum and limestone rocks. Based on the inspection of the data collected the following relationship was selected:

Y=Yo+X(A+B⋆X)(1)

where:

X = Depended variable (example: tensile strength, compressive modulus, porosity, flexural strength).

Y0, A and B = model parameters depend on the material.

Y = Independent variable (example: compression strength).

Based on the data collected the correlations were either linear or nonlinear for the material properties of interest. As shown in Fig. 1, the relationship proposed in Eq. 1 can be used to represent various linear and nonlinear trends based on the values of the parameters A and B. When parameters A and B are positive, the relationship was hyperbolic. The linear relationship is represented by Eq. 1 when B = zero and A will take any value. When parameters A and B are negative, the inverse hyperbolic relationship is obtained.

Figure 1 Vipulanandan mode the linear and nonlinear responses of rock
Figure 1

Vipulanandan mode the linear and nonlinear responses of rock

2.2.2 Mohr-Coulomb failure model

The Mohr-Coulomb failure criterion represents the linear relationship between the shear strength of a rock and the applied normal stress on the failure plane. This relation is as follows:

τ=τo+σntanϕ(2)

where τ is the shear strength, σn is the normal stress, τ0 the pure shear strength (cohesion), and φ is the slope of the internal friction angle.

From Eq. (3)

dτdσn=tan⁡φ(3)
d2τdσn2=0(4)

when σn → ∞; τ = ∞

Hence, Mohr-Coulomb failure model does not satisfy the upper limit condition for the shear strength tolerance of the materials.

2.2.3 Vipulanandan new failure model

Based on years of experience and reviewing the material shear strength versus applied normal stress is nonlinear, also there is limit to the maximum shear stress tolerance for all the materials, and hence following model and conditions are proposed:

τ=τo+σnC+D⋆σn(5)
dτdσn=C+Dσn−Dσn(C+Dσn)2=C(C+Dσn)2>0⇒C>0d2τdσn2=−2(C+Dσn)−3⋆CD=−2CD(C+Dσn)3<0,⇒D>0

For pure tension when τ = 0 ⇒

0=τo+σtC+D⋆σt(6)
σt=−Cτo1+Dτo(7)

Eq. (8) is similar to Eq. (1) in characterizing the tensile strength of rocks.

Also, when σn → ∞

τ∞=τ0+11D(8)

Hence, this model (Eq. 8) has a limit on the maximum shear stress the rocks will tolerate at relatively high normal stress.

2.3 Comparison of model predictions

In order to compare the model predictions statistical quantification such as coefficient of determination (R2) and the root mean square error (RMSE) were used and quantified as follows:

R2=∑i(xi−x¯)(yi−y¯)∑i(xi−x¯)2⋆∑i(yi−y¯)22(9)
RMSE=∑i=1n(yi−xi)2N(10)

where yi = experimental value; xi = predicted value from the model; ȳ = mean of the actual test values; x̄ = mean of calculated values and N is the number of data points.

3 Results and analyses

3.1 Density (γ)

Based on the 97 and 184 data for the gypsum and limestone rocks (Table 1, Table 2 and Table 3), the mean values of density were 2.27 gm/cm3 (N = 97) and 2.49 gm/cm3 (N = 184) respectively, as shown in Fig. 2 and Fig. 3. The histogram shows the relationship between the class of data (each class equal to mean value + 0.5 g/cm3) and the number of times that range of value (frequency distribution of the data) was observed. In this study, the statistical details and the histograms were developed for each type of rock density data sets to identify the distribution as shown in Fig. 2 and 3, respectively.

Table 1

Summary of gypsum rock properties

ReferenceLocationDensity (gm/cm3)Tensile Strength, σt (MPa)Compressive Strength, σc (MPa)Flexural Strength, σf(MPa)Shear Strength, τ (MPa)Elastic Modulus, E (GPa)Flexural Strength, σf (MPa)Porosity, n (%)
Bell (1981)UK-------3.4-5.3
Singh et al. (1987)--1.8-2.326-33-15.8-30-0.9-2.9
Chen (1989)USA--------
Papadopoulos et al. (1994)Greece--4 -12-0.7-3--
Bajpmar et al. (2011)Turkey--2.4-5.21.14-2.50.9-1.3--
Heidari et al. (2012)Iran-3.5-5.53.8-30----
Liang et al. (2012)---8-14-0.5-343-8.1--
Salih (2013)Iraq2.1-2.31-2.3-43.20.6-10.1-1.2-2.6-
Selçuk et al. (2015)Turkey2.3-2.4-14-19----
Meng et al. (2007)China------0.9-2.7
Yin et al. (2017)China2.4-2.83-12.8-34-10-26-6.6-8.1
Salih et al. (2017)Iraq1.8-2.78-2-20-0.8-51.4-3.4-
Remark7 countriesVaried between 2.1 to 2.83Varied between 1.8 to 5.5 MPaVaried between 2 to 43.2 MPaVaried between 0.6 to 10.1MPaVaried between 0.5 to 34 MPaVaried between 0.7 to 30 GPaVaried between 1.2 to 3.4 MPaVaried between 0.92 to 8.1 %

Table 2

Summary of limestone rock properties

ReferenceLocationDensity (gm/cm3)Tensile Strength, σt (MPa)Compressive Strength, σc (MPa)Shear Strength, τ (MPa)Compression Modulus, E (GPa)Flexural Strength, σf (MPa)Porosity, n (%)
Schmidt (1976)USA-4.6-5.78--29-34.4-
Sachpazis (1990)Greek--83-102-26-47-
Tugrul et al. (1999)Turkey2.5-2.67-84-12610-4021.6-50.4-
Katz et al. (2000)USA2.07-2.7-32-95-12-39-
Kahraman (2007)Turkey2.67-2.69-75-124--
Fjar et al. (2008)USA2.67 – 2.726 –2530-25020-70-
Zhang (2002)Sweden-2.65-10.1----
Sarno et al. (2009)USA1.9-2.2------
Aoki et al. (2008)Japan2.3-2.8-22-203----
Heap (2009)UK---22-39---
Guimin et al. (2012)China---5-12---
Richard et al. (2012)USA--6-193----
Rajabzadeh et al. (2012)Iran2.31-2.725-1433-102-7.0-17.7-0.4-0.6
Ozturk et a l. (2013)Turkey--42-175----
Okewale et al. (2013)Nigeria2.64-2.751.65-3.359.4-68.6----
Kurtulus et al. (2015)Turkey1.8-2.353.9-7.8--38-64--
Momeni et al. (2015)Malaysia2.57-2.71-26-73.4----
Parent et al. (2015)France1.71-2.46-53-98-8.3-40.4-0.2-0.7
Tuncay et al. (2016)Turkey1.81-2.2----12-4-
Akram at al. (2017)Pakistan--36.5-63---
Salih et al. (2017)Iraq2.66-2.73-28-200-13.5-41.5--
Remark13 countriesVaried between 1.71 to 2.75 gm/cm3Varied between 1.65 to 25 MPaVaried between 6 to 250 MPaVaried between 5 to 40 MPaVaried between 7 to 70 GPaVaried between 0.027 to 2 MPa.m0.5Varied between 0.2-0.7 %

Table 3

Statistical variation of rock properties

Statistical ParametersDensity (gm/cm3)Compressive Strength, σc (MPa)Tensile Strength, σt (MPa)Compression Modulus, E (GPa)Flexural Strength, σf (MPa)Porosity, n (%)
GypsumNo. of Data979785641926
Range2.10-2.832-431.8-5.50.7-301.2-3.40.92-8
Mean (μ)2.2718.33.51.923
Std. Deviation (σ)0.02101.181.20.512.34
COV (%)1.765534.261.43.9277
LimestoneNo. of Data184184961072135
Range1.71-2.756-2501.6-257.0-704-120.2-0.7
Mean (μ)2.4968.46.525.008.10.38
Std. Deviation (σ)0.2636.61.511.652.300.135
COV (%)10.453.523.146.628.335

Figure 2 Largest Extreme Value Distribution of density for gypsum rock
Figure 2

Largest Extreme Value Distribution of density for gypsum rock

Figure 3 Weibull distribution for the density for Limestone
Figure 3

Weibull distribution for the density for Limestone

  1. Gypsum rock

    Based on 97 data for gypsum, the density (γ) varied from 2.10 gm/cm3 to 2.83 gm/cm3 with the standard deviation of 0.02 gm/cm3 and coefficient of variation (COV) of 1.76 % are summarized in Table 3. The histogram in Fig. 2 shows the number of data points in each range of density that were considered. Over 60% of the total of γ was between 2.24 gm/cm3 and 2.32 gm/cm3. Different distribution tests for the densities for gypsum were performed. Based on the Anderson-Darling statistic (AD) and P-value (hypothesis testing), largest extreme value distribution for the densities for gypsum was observed, as shown in Fig. 2.

  2. Limestone

    A total of 184 data for limestone was analyzed and it varied from 1.71 gm/cm3 to 2.75 gm/cm3 with the standard deviation of 0.26 gm/cm3 and coefficient of variation (COV) of 10.4% are summarized in Table 3. The histogram (Fig. 3) showed almost 77% of the total γ was between 2.3 gm/cm3 and 2.6 gm/cm3. More than 40% of the total of γ was 2.7 gm/cm3. Different distribution tests for the densities of gypsum were performed. Based on the Anderson-Darling statistic (AD) and P-value (hypothesis testing), Weibull distribution for the densities for gypsum was observed and are shown in Fig. 3.

3.2 Strength properties

3.2.1 Compressive strength (σc)

Based on the 97 and 184 data of the gypsum and limestone rocks collected from literature (Table 1, 2 and 3), the compressive strength data for gypsum varied from 2 MPa to 43 MPa with a mean of 18.3 MPa, the standard deviation of 10 MPa and coefficient of variation (COV) of 55% as summarized in Table 3. Based on the 184 compressive strength for limestone data (Table 3), it varied from 6.0 MPa to 250 MPa with a mean of 68.4 MPa, the standard deviation of 36.6 MPa and coefficient of variation (COV) of 53.5%, as summarized in Table 3.

3.2.2 Tensile strength (σt)

With 85 and 96 data of the gypsum and limestone rocks (Table 1, 2 and 3), the tensile strength data for gypsum varied from 1.8 MPa to 5.5 MPa with a mean of 3.5 MPa, the standard deviation of 1.18 MPa and coefficient of variation (COV) of 34.2% as summarized in Table 3. From the 96 tensile strength data for limestone, it varied from 1.6 MPa to 25 MPa with a mean of 6.5 MPa, standard deviation of 1.5 MPa and coefficient of variation (COV) of 23.1% as summarized in Table 3.

3.2.3 Compressive modulus (E)

Based on the 64 and 107 data of the gypsum and limestone rocks (Table 1, 2 and 3), the compressive modulus data for gypsum varied from 0.7 GPa to 30 GPa with a mean of 1.9 GPa, the standard deviation of 1.2 GPa and coefficient of variation (COV) of 61.4% as summarized in Table 3. Depending on 107 compressive modulus data for limestone, the data varied from 7.0 GPa to 70 GPa with a mean of 25.0 GPa, the standard deviation of 11.65 GPa and coefficient of variation (COV) of 46.6% as summarized in Table 3.

3.2.4 Flexural strength (σf)

From the 26 and 21 data of the gypsum and limestone rocks (Table 1, 2 and 3), the flexural strength data for gypsum varied from 1.2 MPa to 3.4 MPa with a mean of 2 1.2 MPa, the standard deviation of 0.51 MPa and coefficient of variation (COV) of 3.9% as summarized in Table 3. Based on 21 flexural strength data for limestone, the data varied from 4 MPa to 12 MPa with a mean of 8.1 MPa, the standard deviation of 2.30 MPa and coefficient of variation (COV) of 28.3% as summarized in Table 3.

3.2.5 Porosity (n %)

Based on the 26 and 35 data of the gypsum and limestone rocks (Table 1, 2 and 3), the porosity data for gypsum varied from 0.92% to 8% with a mean of 3%, the standard deviation of 2.34% and coefficient of variation (COV) of 77% as summarized in Table 3. Depending on 35 porosity data for limestone, the data varied from 0.2% to 0.7% with a mean of 0.38%, the standard deviation of 0.135% and coefficient of variation (COV) of 35% as summarized in Table 3.

3.3 Property correlations

  1. The relationship between compressive strength (σc) and density (γ)

    1. Gypsum

      Total of 86 gypsum data were used (Table 4) to investigate the correlation between the density (γ) and compressive strength (σc) using Eq. 1, as shown in Fig. 4(a). With the Vipulanandan correlation model (Eq. 1) the model parameters Y0, A, B, coefficient of determination (R2) and root mean square error (RMSE) were 411, 0.005, 0, 0.94 and 0.811 MPa respectively as summarized in Table 4.

      Table 4

      Model parameters for physical and mechanical properties relationship of rocks

      Vipulanandan correlation model (Eq.1)
      depended Variable (Y- axis)In depended Variable (X- axis)Type of rockYoABRMSER2No. of DataFig. No.
      Compressive Strength, σc (MPa)Density, γ (gm/cm3)Gypsum4110.0050.00.8110.94864 (a)
      Limestone----1864 (b)
      Tensile Strength, σt (MPa)Compressive Strength, σc (MPa)Gypsum02.880.00.110.99865 (a)
      Limestone0.02.160.110.650.80965 (b)
      Compression Modulus, E (GPa)Compressive Strength, σc (MPa)Gypsum2.300.00.0510.97526 (a)
      Limestone0.303.00.0045.500.801076 (b)
      Flexural Strength, σf (MPa)Compressive Strength, σc (MPa)Gypsum14.200.080.971927 (a)
      Limestone0.7517.30.020.940.83217 (b)
      Porosity, n (%)Compressive Strength, σc (MPa)Gypsum12.30.870.050.320.98268 (a)
      Limestone6.00.170.6110.1030.94358 (b)

      Figure 4 Variation of compressive strength and density of (a) gypsum and (b) limestone
      Figure 4

      Variation of compressive strength and density of (a) gypsum and (b) limestone

      Figure 5 Variation of tensile and compressive strength of (a) gypsum and (b) limestone
      Figure 5

      Variation of tensile and compressive strength of (a) gypsum and (b) limestone

      Figure 6 Variation of modulus of elasticity with compressive strength of (a) gypsum and (b) limestone
      Figure 6

      Variation of modulus of elasticity with compressive strength of (a) gypsum and (b) limestone

      Figure 7 Variation of compressive and flexural strengths of (a) gypsum rock and (b) Limestone
      Figure 7

      Variation of compressive and flexural strengths of (a) gypsum rock and (b) Limestone

      Figure 8 Variation of porosity with compressive strengths (a) gypsum and (b) limestone
      Figure 8

      Variation of porosity with compressive strengths (a) gypsum and (b) limestone

    2. Limestone

      Based on 86 limestone data (Table 3), no direct correlation was observed between the density (γ) and compressive strength (σc) as shown in Fig. 4(b).

  2. The relationship between tensile strength (σt) and compressive strength (σc)

    1. Gypsum

      Total of 86 gypsum data were used (Table 4) to investigate the correlation between the tensile strength (σt) and compressive strength (σc) using Eq. 1 as shown in Fig. 5(a). With the Vipulanandan correlation model (Eq.1) the model parameters Y0, A, B, the coefficient of determination (R2) and root mean square error (RMSE) were 0, 2.88, 0, 0.99 and 0.11 MPa respectively as summarized in Table 4.

    2. Limestone

      Based on 96 limestone data (Table 4) and the property correlation was investigated using Model 1. With the Vipulanandan correlation model (Eq. 1) the model parameters Y0, A, B, the coefficient of determination (R2) and root mean square error (RMSE) were 0, 2.16, 0.11, 0.80 and 0.65 MPa respectively as summarized in Table 4. With the increase in σc of the limestone rock, the σt increased from 1.8 to 8.5 when the rock compressive strength increased to 200 MPa as shown in Fig. 5(b).

  3. The relationship between compressive strength (σc) and compressive modulus (E)

    1. Gypsum

      Using 64 gypsum data (Table 4), the variation of σc with E of the gypsum rock was represented using the Vipulanandan correlation model relationship (Eq. 1) and the model parameters Y0, A, B, coefficient of determination (R2) and root mean square error (RMSE) were 3.1 GPa, 2.30, 0, 0.97 and 0.051 GPa respectively as summarized in Table 4. With the increase in σc of rocks, the E linearly increased as shown in Fig. 6(a).

    2. Limestone

      Based on the total of 107 limestone data were collected from various research studies (Table 3) and the property correlation was investigated. With increasing of σc of the limestone rock, the E nonlinearly increased as shown in Fig. 6(b). The change in the σc with E for the rock was represented using the Vipulanandan correlation model relationship (Eq. 1) and the model parameters Y0, A, B, the coefficient of determination (R2) and root mean square error (RMSE) were 0.30 GPa, 3.0, 0.004 GPa–1, 0.80 and 5.50 GPa respectively as summarized in Table 4.

  4. The relationship between compressive strength (σc) and flexural strength (σf)

    1. Gypsum

      Total of 19 gypsum data (Table 3) was used for the property correlation study. With increasing in σc of the rock, the σf increased as shown in Fig. 7(a). The variation of n with the σf for the gypsum rock was represented using the Vipulanandan correlation model relationship (Eq. 1) and the model parameters Y0, A, B, the coefficient of determination (R2) and root mean square error (RMSE) were 0.62, 14.2, 0, 0.97 and 0.08 MPa respectively as summarized in Table 4.

    2. Limestone

      Based on the 21 limestone data (Table 3) collected from the literature, with increasing in σc of the rock, the σf also increased as shown in Fig. 7(b). The variation of n with the σf for the gypsum rock was represented using the Vipulanandan correlation model relationship (Eq. 1) and the model parameters Y0, A, B, the coefficient of determination (R2) and root mean square error (RMSE) were 0.75, 17.3, 0.02, 0.83 and 0.94 MPa respectively as summarized in Table 4.

  5. The relationship between compressive strength (σc) and porosity (n)

    1. Gypsum

      Total of 26 gypsum data (Table 3) was used for the property correlation study. With increasing in σc of the rock, the porosity nonlinearly decreased as shown in Fig. 8(a). The variation of n with the σc for the gypsum rock was represented using the Vipulanandan correlation model relationship (Eq. 1) and the model parameters Y0, A, B, the coefficient of determination (R2) and root mean square error (RMSE) were 12.3%, 0.87, 0.05, 0.98 and 0.32% respectively as summarized in Table 4.

    2. Limestone

      Based on the 35 limestone data (Table 3) collected from the literature, with increasing in σc of the rock, the porosity nonlinearly decreased as shown in Fig. 8(b). The variation of n with the σc for the gypsum rock was represented using the Vipulanandan correlation model relationship (Eq. 1) and the model parameters Y0, A, B, the coefficient of determination (R2) and root mean square error (RMSE) were 6%, 0.17, 0.611, 0.94 and 0.103 % respectively as summarized in Table 4.

3.4 Shear stress-normal stress failure models

Shear stress-normal stress relationships were predicted using the Vipulanandan failure model and compared with the Mohr–Coulomb failure model as shown in Fig. 9.

Figure 9 Vipulanandan failure mode compared to Mohr-Coulomb model
Figure 9

Vipulanandan failure mode compared to Mohr-Coulomb model

3.4.1 Mohr-Coulomb failure model

  1. Gypsum

    The failure behavior of gypsum was modeled with the Mohr-Coulomb failure criteria (Eq. (2)) using 113 data as shown in Fig. ??(a). The coefficient of determination (R2) and root mean square of error (RMSE) were 0.80 and 2.7 MPa respectively as summarized in Table 5. The cohesion (τ0), the angle of internal friction (φ) and tensile strength (σt) of the gypsum rock were 6.2 MPa, 33∘ and 4.6 MPa respectively as summarized in Table 5.

    Figure 10 Variation of shear and normal stress of (a) gypsum and (b) limestone
    Figure 10

    Variation of shear and normal stress of (a) gypsum and (b) limestone

    Table 5

    Model parameters for shear and normal stresses relationship of rocks

    Mohr-Coulomb modlel (Eq.2)Vipulanandan failure model (Eq.5)
    To (MPa)ϕ (o)σt (MPa)RMSE (MPa)R2τo (MPa)σt (MPa)CDτmax. (MPa) (Eq. 8)RMSE (MPa)R2No. of DataFig. No.
    6.2334.62.70.805.15.010.016641.890.8011310 (a)
    2.7285.03.40.972.64.81.260.0091143.20.944410 (b)

  2. Limestone

    The failure behavior of limestone was modeled with the Mohr-Coulomb failure criteria (Eq. (3)) using 44 data as shown in Fig. ??(b). The coefficient of determination (R2) and root mean square of error (RMSE) were 0.97 and 3.4 MPa respectively as summarized in Table 5. The yield stress (τ0), angle of internal friction (φ) and tensile strength (σt) of the limestone was 2.7 MPa, 28∘ and 5 MPa respectively as summarized in Table 5.

3.4.2 Vipulanandan failure model

  1. Gypsum

    The failure behavior of gypsum with 113 was predicted using the Vipulanandan failure model (Eq. (5)) as shown in Fig. 10(a). The coefficient of determination (R2) and root mean square of error (RMSE) were 0.80 and 1.89 MPa respectively as summarized in Table 5. The cohesion (τ0) and tensile strength (σt) of the limestone was 5.1 MPa and 5 MPa respectively as summarized in Table 5. The model parameters C and D for gypsum was 1 and 0.016 respectively as summarized in Table 5.

  2. Limestone

    The failure behavior of limestone using 44 was modeled using the Vipulanandan failure model (Eq.‘ (5)) as shown in Fig. 10(b). The coefficient of determinatio (R2) and root mean square of error (RMSE) were 0.94 and 3.2 MPa respectively as summarized in Table 5. The yield stress (τ0) and tensile strength (σt) of the limestone was 2.6 MPa and 4.8 MPa respectively as summarized in Table 5. The model parameter C and D for limestone was 1.26 and 0.01 respectively as summarized in Table 5.

3.5 Maximum shear strength (τmax.)

Based on Eq. 8, the Vipulanandan failure model has a limit on the maximum shear stress (τmax for the rocks at relatively high normal stress. The τmax for gypsum and limestone rocks were 64 MPa and 114 MPa respectively as summarized in Table 5.

4 Conclusions

The focus of this study was to characterize and compare the calcium rock behavior based on the density, properties, flexural strength and failure strength. Based on over 1000 data collected data from the literature studies, analytical models were developed, and the following conclusions are advanced:

  1. The statistical analyses clearly indicated the variation in the density, tensile strength, compressive strength, compressive modulus and flexural strength of the two types of rocks investigated. The variation in the properties were quantified based on the coefficient of variation.

  2. There was a positive correlation between the compressive strength with tensile strength and compressive modulus of elasticity of the rocks using the Vipulanandan correlation model.

  3. The flexural strength of the rocks showed good correlation with the compressive strength of the rocks using the Vipulanandan correlation model.

  4. The new Vipulanandan failure criterion predicted the failure of the rocks very well and the maximum shear strength for the rocks. The maximum shear stress (τmax) for gypsum and limestone rocks 64 MPa and 114 MPa respectively based on the data used in this study.

List of symbols

ASTM

International Association for Testing and Materials

τ

(cohesion) shear strength

τ0

Pure shear strength

φ

Slope of the internal friction angle.

γ

Density

COV

coefficient of variation

AD

Based on the Anderson-Darling statistic

KI

Mode-1 Fracture toughness

R2

coefficient of determination

RMSE

root mean square error

Acknowledgement

The Civil Engineering Department, University of Sulaimani, supported this study.

References

[1] Abdelaali, R., Abderrahim, B., Mohamed, B., Yves, G., Abderrahim, S., Mimoun, H., and Jamal, S. (2013). Prediction of porosity and density of calcarenite rocks from P-wave velocity measurements. International Journal of Geosciences.Search in Google Scholar

[2] Akram, M. S., Farooq, S., Naeem, M., and Ghazi, S. (2017). Prediction of mechanical behaviour from mineralogical composition of Sakesar limestone, Central Salt Range, Pakistan. Bulletin of Engineering Geology and the Environment, 2(76), 601-615.10.1007/s10064-016-1002-3Search in Google Scholar

[3] Aoki, H., and Matsukura, Y. (2008). Estimating the unconfined compressive strength of intact rocks from Equotip hardness. Bulletin of Engineering Geology and the Environment, 67(1), 23-29.10.1007/s10064-007-0116-zSearch in Google Scholar

[4] Başpınar, M. S. and E. Kahraman (2011). “Modifications in the properties of gypsum construction element via addition of expanded macroporous silica granules.” Construction and Building materials 25(8): 3327-333310.1016/j.conbuildmat.2011.03.022Search in Google Scholar

[5] Bell, F.G. (1981) ‘Geotechnical Properties of Some Evaporitic Rocks’, Bulletin 24 of the International Association of Engineering Geology, pp. 137-144.10.1007/BF02595264Search in Google Scholar

[6] Bednarik, M., Moshammer, B., Heinrich, M., Holzer, R., Laho, M., Rabeder, J., and Unterwurzacher, M. (2014). Engineering geological properties of Leitha Limestone from historical quarries in Burgenland and Styria, Austria. Engineering Geology, 176, 66-78.10.1016/j.enggeo.2014.04.005Search in Google Scholar

[7] Chen, J. F. (1989). The development of the Cracked-Chevron-Notched Brazilian Disc method for rock fracture toughness measurement and tunnelling machine performance prediction.Search in Google Scholar

[8] Dreybrodt, W., Romanov, D. and Gabrovsek, F. (2002) ‘Karstification below Dam Sites: a Model of Increasing Leakage from Reservoirs’, Environmental Geology, 42, pp. 518-524.10.1007/s00254-001-0514-7Search in Google Scholar

[9] Fjar, E., Holt, R. M., Raaen, A. M., Risnes, R., and Horsrud, P. (2008). Petroleum related rock mechanics (Vol. 53). Elsevier.Search in Google Scholar

[10] Guimin, Z., Yinping, L., Chunhe, Y., and Wenjun, J. (2012). Relationship Between Shear Stress and Shear Strain at Post-peak Curves of Rocks Subjected to Direct Shear Tests. In 46th US Rock Mechanics/Geomechanics Symposium. American Rock Mechanics Association.Search in Google Scholar

[11] Han, D. H., Nur, A., and Morgan, D. (1986). Effects of porosity and clay content on wave velocities in sandstones. Geophysics, 51(11), 2093-2107.10.1190/1.1893163Search in Google Scholar

[12] Heidari, M., Khanlari, G. R., Kaveh, M. T., & Kargarian, S. (2012). Predicting the uniaxial compressive and tensile strengths of gypsum rock by point load testing. Rock mechanics and rock engineering, 45(2), 265-273.10.1007/s00603-011-0196-8Search in Google Scholar

[13] Heap, M. J. (2009). Creep: Time dependent brittle deformation in rocks (Doctoral dissertation, UCL (University College London)), London, England.Search in Google Scholar

[14] Johnson, K. S. (2005). Salt dissolution and subsidence or collapse caused by human activities. Reviews in Engineering Geology, 16, 101-110.10.1130/2005.4016(09)Search in Google Scholar

[15] Jensen, S. S. (2016). Experimental Study of Direct Tensile Strength in Sedimentary Rocks (Master’s thesis, NTNU).Search in Google Scholar

[16] Katz, O., Reches, Z., and Roegiers, J. C. (2000). Evaluation of mechanical rock properties using a Schmidt Hammer. International Journal of Rock Mechanics and Mining Sciences, 37(4), 723-728.10.1016/S1365-1609(00)00004-6Search in Google Scholar

[17] Kahraman, S. (2007). The correlations between the saturated and dry P-wave velocity of rocks. Ultrasonics, 46(4), 341-348.10.1016/j.ultras.2007.05.003Search in Google Scholar

[18] Karakus, M., Kumral, M., and Kilic, O. (2005). Predicting elastic properties of intact rocks from index tests using multiple regression modelling. International Journal of Rock Mechanics and Mining Sciences, 42(2), 323-330.10.1016/j.ijrmms.2004.08.005Search in Google Scholar

[19] Klimchouk, A. (1996) ‘The Dissolution and Conversion of Gypsum and Anhydrite’ Int. J. Speleol., 25 (3-4), pp. 21-36.10.5038/1827-806X.25.3.2Search in Google Scholar

[20] Kurtuluş, C., Sertçelik, F., and Sertçelik, I. (2016). Correlating physico-mechanical properties of intact rocks with P-wave velocity. Acta Geodaetica et Geophysica, 51(3), 571-582.10.1007/s40328-015-0145-1Search in Google Scholar

[21] Liang, W., Yang, X., Gao, H., Zhang, C., Zhao, Y., & Dusseault, M. B. (2012). Experimental study of mechanical properties of gypsum soaked in brine. International Journal of Rock Mechanics and Mining Sciences, 53, 142-15010.1016/j.ijrmms.2012.05.015Search in Google Scholar

[22] Meng, Z., and Pan, J. (2007). Correlation between petrographic characteristics and failure duration in clastic rocks. Engineering geology, 89(3), 258-265.10.1016/j.enggeo.2006.10.010Search in Google Scholar

[23] Milliman, J., Müller, G., and Förstner, F. (2012). Recent sedimentary carbonates: Part 1 marine carbonates. Springer Science and Business Media.Search in Google Scholar

[24] Moos, D., Peska, P., Finkbeiner, T., and Zoback, M. (2003). Comprehensive wellbore stability analysis utilizing quantitative risk assessment. Journal of Petroleum Science and Engineering, 38(3), 97-109.10.1016/S0920-4105(03)00024-XSearch in Google Scholar

[25] Momeni, E., Armaghani, D. J., Hajihassani, M., and Amin, M. F. M. (2015). Prediction of uniaxial compressive strength of rock samples using hybrid particle swarm optimization-based artificial neural networks. Measurement, 60, 50-63.10.1016/j.measurement.2014.09.075Search in Google Scholar

[26] Mohammed, A. S., and Vipulanandan, C. (2014). Compressive and tensile behavior of polymer treated sulfate contaminated CL soil. Geotechnical and Geological Engineering, 32(1), 71-83.10.1007/s10706-013-9692-9Search in Google Scholar

[27] Mohammed, A., and Vipulanandan, C. (2015). Testing and modeling the short-term behavior of Lime and Fly Ash treated sulfate contaminated CL soil. Geotechnical and Geological Engineering, 33(4), 1099-1114.10.1007/s10706-015-9890-8Search in Google Scholar

[28] Mohammed, A. S. (2016). Effect of temperature on the rheological properties with shear stress limit of iron oxide nanoparticle modified bentonite-drilling muds. Egyptian Journal of Petroleum.10.1016/j.ejpe.2016.10.018Search in Google Scholar

[29] Mohammed, A. S. (2017a). Electrical resistivity and rheological properties of sensing bentonite-drilling muds modified with lightweight polymer. Egyptian Journal of Petroleum.10.1016/j.ejpe.2017.01.002Search in Google Scholar

[30] Mohammed, A. (2017 b). Vipulanandan model for the rheological properties with ultimate shear stress of oil well cement modified with nanoclay, 10.1016/j.ejpe.2017.05.007.Search in Google Scholar

[31] Mohammed, A. S. (2017 c). Property Correlations and Statistical Variations in the Geotechnical Properties of (CH) Clay Soils. Geotechnical and Geological Engineering, 1-15.10.1007/s10706-017-0418-2Search in Google Scholar

[32] Nam, M. S., and Vipulanandan, C. (2010). Relationship between Texas Cone Penetrometer Tests and Axial Resistances of Drilled Shafts Socketed in Clay Shale and Limestone. Journal of Geotechnical and Geoenviromental Engineering, 136(8), 1161-1165.10.1061/(ASCE)GT.1943-5606.0000318Search in Google Scholar

[33] Omar, M. (2017). Empirical correlations for predicting strength properties of rocks–United Arab Emirates. International Journal of Geotechnical Engineering, 11(3), 248-261.10.1080/19386362.2016.1214339Search in Google Scholar

[34] Okewale, I. A., and Olaleye, B. M. Correlation of Strength Properties of Limestone Deposit in Ogun State, Nigeria with Penetration Rate Using Linear Regression Analysis for Engineering Applications.Search in Google Scholar

[35] Ozturk, C. A., and Nasuf, E. (2013). Strength classification of rock material based on textural properties. Tunnelling and Underground Space Technology, 37, 45-54.10.1016/j.tust.2013.03.005Search in Google Scholar

[36] Papadopoulos, Z., Kolaiti, E., & Mourtzas, N. (1994). The effect of crystal size on geotechnical properties of Neogene gypsum in Crete. Quarterly Journal of Engineering Geology and Hydrogeology, 27(3), 267-273.10.1144/GSL.QJEGH.1994.027.P3.07Search in Google Scholar

[37] Parent, T., Domede, N., Sellier, A., and Mouatt, L. (2015). Mechanical characterization of limestone from sound velocity measurement. International Journal of Rock Mechanics and Mining Sciences, 79, 149-156.10.1016/j.ijrmms.2015.08.009Search in Google Scholar

[38] Rajabzadeh, M. A., Moosavinasab, Z., and Rakhshandehroo, G. (2012). Effects of rock classes and porosity on the relation between uniaxial compressive strength and some rock properties for carbonate rocks. Rock mechanics and rock engineering, 45(1), 113-122.10.1007/s00603-011-0169-ySearch in Google Scholar

[39] Richard, T., Dagrain, F., Poyol, E., and Detournay, E. (2012). Rock strength determination from scratch tests. Engineering Geology, 147, 91-100.10.1016/j.enggeo.2012.07.011Search in Google Scholar

[40] Sachpazis, C. I. (1990). Correlating Schmidt hardness with compressive strength and Young’s modulus of carbonate rocks. Bulletin of Engineering Geology and the Environment, 42(1), 75-83.Search in Google Scholar

[41] Sabatakakis, N., Koukis, G., Tsiambaos, G., and Papanakli, S. (2008). Index properties and strength variation controlled by microstructure for sedimentary rocks. Engineering Geology, 97(1), 80-90.10.1016/j.enggeo.2007.12.004Search in Google Scholar

[42] Sarno, A., Farah, R., Hudyma, N., and Hiltunen, D. R. (2009). Relationships between index and physical properties of weathered Ocala limestone. In 43rd US Rock Mechanics Symposium and 4th US-Canada Rock Mechanics Symposium. American Rock Mechanics Association.Search in Google Scholar

[43] Salih, N. B. (2013). Stability of dams constructed on problematic substrates (Doctoral dissertation, Brunel University School of Engineering and Design PhD Theses).Search in Google Scholar

[44] Salih, N., and Mohammed, A. (2017). Characterization and modeling of long-term stress-strain behavior of water confined pre-saturated gypsum rock in Kurdistan Region, Iraq. Rock mechanics and geotechnical engineering, 9 (5), 10.1016/j.jrmge.2017.03.009.10.1016/j.jrmge.2017.03.009Search in Google Scholar

[45] Schmidt, R. A. (1976). Fracture-toughness testing of limestone. Experimental Mechanics, 16(5), 161-167.10.1007/BF02327993Search in Google Scholar

[46] Selçuk, L., & Kayabali, K. (2015). Evaluation of the unconfined compressive strength of rocks using nail guns. Engineering Geology, 195, 164-171.10.1016/j.enggeo.2015.06.014Search in Google Scholar

[47] Selçuk, L., and Nar, A. (2016). Prediction of uniaxial compressive strength of intact rocks using ultrasonic pulse velocity and rebound-hammer number. Quarterly Journal of Engineering Geology and Hydrogeology, 49(1), 67-75,10.1144/qjegh2014-094Search in Google Scholar

[48] Singh, M. V., Chhabra, R., & Abrol, I. P. (1987). Interactions between applications of gypsum and zinc sulphate on the yield and chemical composition of rice grown on an alkali soil. The Journal of Agricultural Science, 108(2), 275-279.10.1017/S0021859600079260Search in Google Scholar

[49] Singh, M., Samadhiya, N. K., Kumar, A., Kumar, V., and Singh, B. (2015). A nonlinear criterion for triaxial strength of inherently anisotropic rocks. Rock Mechanics and Rock Engineering, 48(4), 1387-1405.10.1007/s00603-015-0708-zSearch in Google Scholar

[50] Swapnil, K., Kim, Min-Gu and Vipulanandan, C. (2004) “Nondestructive Properties of Clay shale and Limestone in Dallas, Texas”, Proceeding (CD), ARA/NARMS 04-559, American Rock Mechanics Association, Houston, Texas, June 6, 2004.Search in Google Scholar

[51] Tiwari, R. P., and Rao, K. S. (2007). Response of an anisotropic rock mass under polyaxial stress state. Journal of materials in civil engineering, 19(5), 393-403.10.1061/(ASCE)0899-1561(2007)19:5(393)Search in Google Scholar

[52] Tugrul, A., and Zarif, I. H. (2000). Engineering aspects of limestone weathering in Istanbul, Turkey. Bulletin of Engineering Geology and the Environment, 58(3), 191-206.10.1007/s100640050075Search in Google Scholar

[53] Tuncay, E. B., Kilinçarslan, Ş., & Yağmurlu, F. (2016). Investigation of Usability as Aggregate of Different Originated Rocks. In IOP Conference Series: Earth and Environmental Science (Vol. 44, No. 2, p. 022002). IOP Publishing.Search in Google Scholar

[54] Usluogullari, O. F., and Vipulanandan, C. (2011). Stress-strain behavior and California bearing ratio of artificially cemented sand. Journal of Testing and Evaluation, 39(4), 1-9.Search in Google Scholar

[55] Vipulanandan, C., Ahossin Guezo, Y. J., and Bilgin, Ö. (2007). Geotechnical properties of marine and deltaic soft clays. In Advances in Measurement and Modeling of Soil Behavior, ASCE, GSP 173, pp. 1-13.10.1061/9780784409176Search in Google Scholar

[56] Vipulanandan, C. and Nam, E. (2009), “Drilled Shaft Socketed in Uncemented Clay Shale,” Proceedings, Foundation Congress 2009, Contemporary Topics in Deep Foundations, ASCE, GSP 185, pp. 151-158.10.1061/41021(335)19Search in Google Scholar

[57] Vipulanandan, C., and Mohammed, A. S. (2014). Hyperbolic rheological model with shear stress limit for acrylamide polymer modified bentonite-drilling muds. Journal of Petroleum Science and Engineering, Vol. 122, 38-47.10.1016/j.petrol.2014.08.004Search in Google Scholar

[58] Vipulanandan, C., and Mohammed, A. S. (2015 a). Characterizing the Hydraulic Fracturing Fluid Modified with Nano Silica Proppant. AADE-15-NTCE-38, CD Proceeding, San Antonio, Texas, April 2015.Search in Google Scholar

[59] Vipulanandan, C., and Mohammed, A. (2015 b) Effect of nanoclay on the electrical resistivity and rheological properties of smart and sensing bentonite drilling muds. Journal of Petroleum Science and Engineering, 130, 86-95.10.1016/j.petrol.2015.03.020Search in Google Scholar

[60] Vipulanandan, C., and Mohammed, A. (2015 c) XRD and TGA, Swelling and Compacted Properties of Polymer Treated Sulfate Contaminated CL Soil. Journal of Testing and Evaluation, 44(6), 2270-2284.10.1520/JTE20140280Search in Google Scholar

[61] Vipulanandan, C., and Mohammed, A. (2015 d) Smart cement modified with iron oxide nanoparticles to enhance the piezoresistive behavior and compressive strength for oil well applications. Smart Materials and Structures, 24(12), 125020.10.1088/0964-1726/24/12/125020Search in Google Scholar

[62] Vipulanandan, C., and Mohammed, A. (2015 e) Smart cement rheological and piezoresistive behavior for oil well applications. Journal of Petroleum Science and Engineering, 135, 50-58.10.1016/j.petrol.2015.08.015Search in Google Scholar

[63] Vipulanandan, C., and Mohammed, A. (2017a). Rheological Properties of Piezoresistive Smart Cement Slurry Modified With Iron-Oxide Nanoparticles for Oil-Well Applications. Journal of Testing and Evaluation, 45(6).10.1520/JTE20150443Search in Google Scholar

[64] Vipulanandan, C., Mohammed, A., and Samuel, R. G. (2017). Smart Bentonite Drilling Muds Modified with Iron Oxide Nanoparticles and Characterized Based on the Electrical Resistivity and Rheological Properties with Varying Magnetic Field Strengths and Temperatures. OTC-MS-270626.10.4043/27626-MSSearch in Google Scholar

[65] Yin, S., Zhou, W., Shan, Y., Ding, W., Xie, R., & Guo, C. (2017). Assessment of the geostress field of deep-thick gypsum cap rocks: A case study of Paleogene Formation in the southwestern Tarim Basin, NW China. Journal of Petroleum Science and Engineering, 154, 76-90.10.1016/j.petrol.2017.04.021Search in Google Scholar

[66] You, M. (2015). Strength criterion for rocks under compressive-tensile stresses and its application. Journal of Rock Mechanics and Geotechnical Engineering, 7(4), 434-43910.1016/j.jrmge.2015.05.002Search in Google Scholar

[67] Zhang, Z. X. (2002). An empirical relation between mode I fracture toughness and the tensile strength of rock. International Journal of Rock Mechanics and Mining Sciences, 39(3), 401-406.10.1016/S1365-1609(02)00032-1Search in Google Scholar

[68] Zoback, M. D., Barton, C. A., Brudy, M., Castillo, D. A., Finkbeiner, T., Grollimund, B. R., and Wiprut, D. J. (2003). Determination of stress orientation and magnitude in deep wells. International Journal of Rock Mechanics and Mining Sciences, 40(7), 1049-1076.10.1016/j.ijrmms.2003.07.001Search in Google Scholar

Received: 2018-03-17
Accepted: 2018-04-26
Published Online: 2018-08-11

© 2018 Ahmed Mohammed and Wael Mahmood, published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

  1. Regular Article
  2. Real-scale comparison between simple and composite raw sewage sampling
  3. 10.1515/eng-2018-0017
  4. The risks associated with falling parts of glazed facades in case of fire
  5. Implementation of high speed machining in thin-walled aircraft integral elements
  6. Evaluating structural crashworthiness and progressive failure of double hull tanker under accidental grounding: bottom raking case
  7. Influence of Silica (SiO2) Loading on the Thermal and Swelling Properties of Hydrogenated-Nitrile-Butadiene-Rubber/Silica (HNBR/Silica) Composites
  8. Statistical Variations and New Correlation Models to Predict the Mechanical Behavior and Ultimate Shear Strength of Gypsum Rock
  9. Analytic approximate solutions to the chemically reactive solute transfer problem with partial slip in the flow of a viscous fluid over an exponentially stretching sheet with suction/blowing
  10. Thermo-mechanical behavior simulation coupled with the hydrostatic-pressure-dependent grain-scale fission gas swelling calculation for a monolithic UMo fuel plate under heterogeneous neutron irradiation
  11. Optimal Auxiliary Functions Method for viscous flow due to a stretching surface with partial slip
  12. Vibrations Analysis of Rectangular Plates with Clamped Corners
  13. Evaluating Lean Performance of Indian Small and Medium Sized Enterprises in Automotive Sector
  14. FPGA–implementation of PID-controller by differential evolution optimization
  15. Thermal properties and morphology of polypropylene based on exfoliated graphite nanoplatelets/nanomagnesium oxide
  16. A computer-based renewable resource management system for a construction company
  17. Hygrothermal Aging of Amine Epoxy: Reversible Static and Fatigue Properties
  18. The selected roof covering technologies in the aspect of their life cycle costs
  19. Influence of insulated glass units thickness and weight reduction on their functional properties
  20. Structural analysis of conditions determining the selection of construction technology for structures in the centres of urban agglomerations
  21. Selection of the optimal solution of acoustic screens in a graphical interpretation of biplot and radar charts method
  22. Subsidy Risk Related to Construction Projects: Seeking Causes
  23. Multidimensional sensitivity study of the fuzzy risk assessment module in the life cycle of building objects
  24. Planning repetitive construction projects considering technological constraints
  25. Identification of risk investment using the risk matrix on railway facilities
  26. Comparison of energy parameters of a centrifugal pump with a multi-piped impeller in cooperation either with an annular channel and a spiral channel
  27. Influence of the contractor’s payment method on the economic effectiveness of the construction project from the contractor’s point of view
  28. Special Issue Automation in Finland
  29. Diagnostics and Identification of Injection Duration of Common Rail Diesel Injectors
  30. An advanced teaching scheme for integrating problem-based learning in control education
  31. A survey of telerobotic surface finishing
  32. Wireless Light-Weight IEC 61850 Based Loss of Mains Protection for Smart Grid
  33. Smart Adaptive Big Data Analysis with Advanced Deep Learning
  34. Topical Issue Desktop Grids for High Performance Computing
  35. A Bitslice Implementation of Anderson’s Attack on A5/1
  36. Efficient Redundancy Techniques in Cloud and Desktop Grid Systems using MAP/G/c-type Queues
  37. Templet Web: the use of volunteer computing approach in PaaS-style cloud
  38. Using virtualization to protect the proprietary material science applications in volunteer computing
  39. Parallel Processing of Images in Mobile Devices using BOINC
  40. “XANSONS for COD”: a new small BOINC project in crystallography
  41. Special Issue on Sustainable Energy, Engineering, Materials and Environment
  42. An experimental study on premixed CNG/H2/CO2 mixture flames
  43. Tidal current energy potential of Nalón river estuary assessment using a high precision flow model
  44. Special Spring Issue 2017
  45. Context Analysis of Customer Requests using a Hybrid Adaptive Neuro Fuzzy Inference System and Hidden Markov Models in the Natural Language Call Routing Problem
  46. Special Issue on Non-ferrous metals and minerals
  47. Study of strength properties of semi-finished products from economically alloyed high-strength aluminium-scandium alloys for application in automobile transport and shipbuilding
  48. Use of Humic Sorbent from Sapropel for Extraction of Palladium Ions from Chloride Solutions
  49. Topical Issue on Mathematical Modelling in Applied Sciences, II
  50. Numerical simulation of two-phase filtration in the near well bore zone
  51. Calculation of 3D Coordinates of a Point on the Basis of a Stereoscopic System
  52. The model of encryption algorithm based on non-positional polynomial notations and constructed on an SP-network
  53. A computational algorithm and the method of determining the temperature field along the length of the rod of variable cross section
  54. ICEUBI2017 - International Congress on Engineering-A Vision for the Future
  55. Use of condensed water from air conditioning systems
  56. Development of a 4 stroke spark ignition opposed piston engine
  57. Development of a Coreless Permanent Magnet Synchronous Motor for a Battery Electric Shell Eco Marathon Prototype Vehicle
  58. Removal of Cr, Cu and Zn from liquid effluents using the fine component of granitic residual soils
  59. A fuzzy reasoning approach to assess innovation risk in ecosystems
  60. Special Issue SEALCONF 2018
  61. Brush seal with thermo-regulating bimetal elements
  62. The CFD simulation of the flow structure in the sewage pump
  63. The investigation of the cavitation processes in the radial labyrinth pump
  64. Testing of the gaskets at liquid nitrogen and ambient temperature
  65. Probabilistic Approach to Determination of Dynamic Characteristics of Automatic Balancing Device
  66. The design method of rubber-metallic expansion joint
  67. The Specific Features of High-Velocity Magnetic Fluid Sealing Complexes
  68. Effect of contact pressure and sliding speed on the friction of polyurethane elastomer (EPUR) during sliding on steel under water wetting conditions
  69. Special Issue on Advance Material
  70. Effect of thermo-mechanical parameters on the mechanical properties of Eurofer97 steel for nuclear applications
  71. Failure prediction of axi-symmetric cup in deep drawing and expansion processes
  72. Characterization of cement composites based on recycled cellulosic waste paper fibres
  73. Innovative Soft Magnetic Composite Materials: Evaluation of magnetic and mechanical properties
  74. Statistical modelling of recrystallization and grain growth phenomena in stainless steels: effect of initial grain size distribution
  75. Annealing effect on microstructure and mechanical properties of Cu-Al alloy subjected to Cryo-ECAP
  76. Influence of heat treatment on corrosion resistance of Mg-Al-Zn alloy processed by severe plastic deformation
  77. The mechanical properties of OFHC copper and CuCrZr alloys after asymmetric rolling at ambient and cryogenic temperatures
https://doi.org/10.1515/eng-2018-0026
Keywords for this article
Calcium rocks; Strength; Flexural strength; Failure model; Statistical analysis; Modeling
Creative Commons
BY-NC-ND 4.0

Articles in the same Issue

  1. Regular Article
  2. Real-scale comparison between simple and composite raw sewage sampling
  3. 10.1515/eng-2018-0017
  4. The risks associated with falling parts of glazed facades in case of fire
  5. Implementation of high speed machining in thin-walled aircraft integral elements
  6. Evaluating structural crashworthiness and progressive failure of double hull tanker under accidental grounding: bottom raking case
  7. Influence of Silica (SiO2) Loading on the Thermal and Swelling Properties of Hydrogenated-Nitrile-Butadiene-Rubber/Silica (HNBR/Silica) Composites
  8. Statistical Variations and New Correlation Models to Predict the Mechanical Behavior and Ultimate Shear Strength of Gypsum Rock
  9. Analytic approximate solutions to the chemically reactive solute transfer problem with partial slip in the flow of a viscous fluid over an exponentially stretching sheet with suction/blowing
  10. Thermo-mechanical behavior simulation coupled with the hydrostatic-pressure-dependent grain-scale fission gas swelling calculation for a monolithic UMo fuel plate under heterogeneous neutron irradiation
  11. Optimal Auxiliary Functions Method for viscous flow due to a stretching surface with partial slip
  12. Vibrations Analysis of Rectangular Plates with Clamped Corners
  13. Evaluating Lean Performance of Indian Small and Medium Sized Enterprises in Automotive Sector
  14. FPGA–implementation of PID-controller by differential evolution optimization
  15. Thermal properties and morphology of polypropylene based on exfoliated graphite nanoplatelets/nanomagnesium oxide
  16. A computer-based renewable resource management system for a construction company
  17. Hygrothermal Aging of Amine Epoxy: Reversible Static and Fatigue Properties
  18. The selected roof covering technologies in the aspect of their life cycle costs
  19. Influence of insulated glass units thickness and weight reduction on their functional properties
  20. Structural analysis of conditions determining the selection of construction technology for structures in the centres of urban agglomerations
  21. Selection of the optimal solution of acoustic screens in a graphical interpretation of biplot and radar charts method
  22. Subsidy Risk Related to Construction Projects: Seeking Causes
  23. Multidimensional sensitivity study of the fuzzy risk assessment module in the life cycle of building objects
  24. Planning repetitive construction projects considering technological constraints
  25. Identification of risk investment using the risk matrix on railway facilities
  26. Comparison of energy parameters of a centrifugal pump with a multi-piped impeller in cooperation either with an annular channel and a spiral channel
  27. Influence of the contractor’s payment method on the economic effectiveness of the construction project from the contractor’s point of view
  28. Special Issue Automation in Finland
  29. Diagnostics and Identification of Injection Duration of Common Rail Diesel Injectors
  30. An advanced teaching scheme for integrating problem-based learning in control education
  31. A survey of telerobotic surface finishing
  32. Wireless Light-Weight IEC 61850 Based Loss of Mains Protection for Smart Grid
  33. Smart Adaptive Big Data Analysis with Advanced Deep Learning
  34. Topical Issue Desktop Grids for High Performance Computing
  35. A Bitslice Implementation of Anderson’s Attack on A5/1
  36. Efficient Redundancy Techniques in Cloud and Desktop Grid Systems using MAP/G/c-type Queues
  37. Templet Web: the use of volunteer computing approach in PaaS-style cloud
  38. Using virtualization to protect the proprietary material science applications in volunteer computing
  39. Parallel Processing of Images in Mobile Devices using BOINC
  40. “XANSONS for COD”: a new small BOINC project in crystallography
  41. Special Issue on Sustainable Energy, Engineering, Materials and Environment
  42. An experimental study on premixed CNG/H2/CO2 mixture flames
  43. Tidal current energy potential of Nalón river estuary assessment using a high precision flow model
  44. Special Spring Issue 2017
  45. Context Analysis of Customer Requests using a Hybrid Adaptive Neuro Fuzzy Inference System and Hidden Markov Models in the Natural Language Call Routing Problem
  46. Special Issue on Non-ferrous metals and minerals
  47. Study of strength properties of semi-finished products from economically alloyed high-strength aluminium-scandium alloys for application in automobile transport and shipbuilding
  48. Use of Humic Sorbent from Sapropel for Extraction of Palladium Ions from Chloride Solutions
  49. Topical Issue on Mathematical Modelling in Applied Sciences, II
  50. Numerical simulation of two-phase filtration in the near well bore zone
  51. Calculation of 3D Coordinates of a Point on the Basis of a Stereoscopic System
  52. The model of encryption algorithm based on non-positional polynomial notations and constructed on an SP-network
  53. A computational algorithm and the method of determining the temperature field along the length of the rod of variable cross section
  54. ICEUBI2017 - International Congress on Engineering-A Vision for the Future
  55. Use of condensed water from air conditioning systems
  56. Development of a 4 stroke spark ignition opposed piston engine
  57. Development of a Coreless Permanent Magnet Synchronous Motor for a Battery Electric Shell Eco Marathon Prototype Vehicle
  58. Removal of Cr, Cu and Zn from liquid effluents using the fine component of granitic residual soils
  59. A fuzzy reasoning approach to assess innovation risk in ecosystems
  60. Special Issue SEALCONF 2018
  61. Brush seal with thermo-regulating bimetal elements
  62. The CFD simulation of the flow structure in the sewage pump
  63. The investigation of the cavitation processes in the radial labyrinth pump
  64. Testing of the gaskets at liquid nitrogen and ambient temperature
  65. Probabilistic Approach to Determination of Dynamic Characteristics of Automatic Balancing Device
  66. The design method of rubber-metallic expansion joint
  67. The Specific Features of High-Velocity Magnetic Fluid Sealing Complexes
  68. Effect of contact pressure and sliding speed on the friction of polyurethane elastomer (EPUR) during sliding on steel under water wetting conditions
  69. Special Issue on Advance Material
  70. Effect of thermo-mechanical parameters on the mechanical properties of Eurofer97 steel for nuclear applications
  71. Failure prediction of axi-symmetric cup in deep drawing and expansion processes
  72. Characterization of cement composites based on recycled cellulosic waste paper fibres
  73. Innovative Soft Magnetic Composite Materials: Evaluation of magnetic and mechanical properties
  74. Statistical modelling of recrystallization and grain growth phenomena in stainless steels: effect of initial grain size distribution
  75. Annealing effect on microstructure and mechanical properties of Cu-Al alloy subjected to Cryo-ECAP
  76. Influence of heat treatment on corrosion resistance of Mg-Al-Zn alloy processed by severe plastic deformation
  77. The mechanical properties of OFHC copper and CuCrZr alloys after asymmetric rolling at ambient and cryogenic temperatures
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2026 De Gruyter Brill
Downloaded on 3.3.2026 from https://www.degruyterbrill.com/document/doi/10.1515/eng-2018-0026/html
Scroll to top button