Rock Crack Initiation Triggered by Energy Digestion

Rock Crack Initiation Triggered by Energy Digestion | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Rock Crack Initiation Triggered by Energy Digestion Lei Yan, Jian Chang, Ebelia Manda, Helin Li, Qian Wang, Yangfan Jing This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4198169/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 01 Jul, 2024 Read the published version in Scientific Reports → Version 1 posted 12 You are reading this latest preprint version Abstract The stress intensity factors with the kinds of critical models are usually obtained from the strength test on laboratory specimens to characterize the rock strength and fracture-ability. However, crack initiation phenomenology for rock material has not been comprehensively understood. In this study, energy theory—an essential character of material property changes—is drawn to evaluate rock crack initiation and study the correlation between rock strength and energy digestion. Uniaxial compression test was conducted on different hardness coefficients rock samples under quasi-static loading conditions and the results of energy evolution, non-linear cumulative digestion, and stored ultimate energy were obtained and compared with a series of results for sandstone. A novel algorithm for the evaluation of rock crack initiation has been proposed and the concept of energy digestion index (EDI) using the ratio of ultimate energy over accumulated energy has been developed to characterize the rock strength. The result shows that crack initiation is related to the maximum growth rate of energy cumulation. Simultaneously, the increasing rate of variable elasticity modulus of rock material gets the peak. The defined EDI has a negative correlation with the rock material's strength and mechanical properties. The relationships between final strength and EDI at crack initiation were also discussed and an assessment of the remaining strength life is introduced using the predicted value by fitting the EDI, leading to a new understanding and exploration for in-situ monitoring of rock damage and failure prediction. Crack initiation Energy digestion Index Energy evolution Rock strength Rock mechanics Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 1. Introduction Disasters related to brittle fracture of rock occur frequently during the civil excavation process. Events such as rock bursts and slope landslides are often triggered by exceeding the ultimate strength of rock during tuning, sloping, mining excavation, or other civil constructions. As a typical kind of heterogeneous natural body formed by mineral debris under geological action. Some internal elements of the rock material are damaged and fatigued as a response to external force, and a large number of initial micro-cracks form and grow, eventually leading to instability and failure. Many researches have been conducted on micromechanical damage fatigue based on laboratory experiments. Griffith conducted a systematic study on stress and crack expansion when cracks occur in the internal structure of the material as early as 1920. The criterion for crack expansion under bidirectional compression was obtained by establishing a crack model for plane compression. Then, from the perspective of energy release, the energy balance theory, which is based on the ideal brittle material, is proposed, which differs from the traditional concept of material strength in the continuum[ 1 ]. CD Martin et al. [ 2 – 4 ] found that the failure of brittle rocks can be classified into five stages based on the stress-strain curve: compaction closure stage, elastic deformation stage, stable crack expansion stage, accelerated crack expansion stage, and after the peak as showed in Fig. 1 . Rock microscopic damage can now be directly observed using new technologies such as computerized tomography (CT) and nuclear magnetic resonance (NMR)[ 5 – 7 ], In-situ monitoring of rock behavior remains difficult to achieve[ 8 ]. Indirect observation methods such as acoustic emission (AE), allow the passive monitoring of the elastic ultrasonic wave that propagates in the material as a result of the crack formation’s abrupt release of energy and the energy dissipation in the process of rock damage by studying the correlation between acoustic emission signals and mechanical properties of rock[ 9 – 13 ], shown in Fig. 1 . However, it is difficult to strictly correspond to the microscopic damage and reveal the process of damage evolution with non-destructive [ 14 , 15 ]and have not yet been universally formed for the most commonly used Prediction evaluation criteria method[ 16 ]. More recently, the perspective of energy evolution has provided a new idea for the study of crack evolution. Shao et al.[ 17 ] and Salari et al.[ 18 ] proposed that the formation and development of cracks lead to dilatation, and the rock damage is characterized by volumetric strain energy. Further, the damage evolution can be characterized by elastic bulk modulus and plastic strain energy release rate[ 19 – 21 ]. A large number of laboratory experiments and theoretical analyses have confirmed the intrinsic relationship between strain energy dissipation and release in the process of rock deformation and failure[ 20 , 22 – 25 ]. Based on the above findings, the stress-strain curve of hysteresis loops obtained by uniaxial compression has been used to characterize the strain energy. However, due to the energy dissipation caused by compaction and plastic damage at an uncertainty of the stage of occurrence [ 26 ], the hysteresis curve cannot be identified; other side, the process of energy transformation has the characteristics of self-inhibition, bifurcation, and chaos, and the energy accumulation is iterative growth[ 27 ]. Consequently, the accumulated elastic energy calculated by the hysteresis curve has a large deviation. These work and theoretical research results have fully demonstrated the energy evolution characteristics of rock under load, which has laid a foundation for theoretical research in the laboratory, but it is still not practical in engineering in-situ monitoring and disaster prediction. In this study, we aimed to overcome these limitations by using the energy digestion index (EDI) a novel algorithm, based on energy theory to determine the trigger point of intact rock crack initiation to investigate the relationship between rock strength and energy digestion. Through laboratory rock mechanics tests and the energy accumulation process of loading samples, the results show that once the increased rate of energy digestion reaches its maximum, rock crack initiation occurs. By using EDI, we were able to accurately model the relationship between rock strength and energy digestion, leading to a more comprehensive understanding and exploration of In-situ monitoring of rock damage and failure prediction. 2. Energy evolution characteristics 2.1 Energy conversion analysis According to the principle of conservation of energy, the process of external work on the sample is accompanied by energy conversion; when the material is an ideal brittle rock, the external work is completely converted into elastic strain energy and stored within the material. The digested energy is to overcome the rock's intrinsic cohesion in the phase of crack evolution, which means there is no extra energy dissipation until rock strength reaches the peak[ 10 ]. Due to the heterogeneity of rock materials, not all external work is converted into elastic strain energy and stored inside the material, and it is accompanied by other forms of energy dissipation, such as elastic strain energy[ 28 ], acoustic emission[ 29 – 31 ] that releases energy in the form of elastic waves phenomenon: deformation energy[ 32 ], which causes the internal state of the material to change, thermal energy[ 33 ], and so on. This portion of the energy is primarily used to reconstruct the internal stress field distribution, as well as to generate and decimate micro-cracks, which have been observed by the acoustic emission study of rock specimens under uniaxial loading[ 23 , 26 , 29 – 32 , 34 , 35 ]. The uniaxial compression process is accompanied by a certain level of AE ringing count events before approximately 50% of compressive strength, and the occurrence of the events is random[ 10 ]. Xie Heping[ 23 ] assumed that there is no heat exchange between the rock and the outside in the process, according to the first law of thermodynamics, its form is given in the following: $$U={U^e}+{U^d}$$ 1 $${U^e}=\frac{1}{2}({\sigma _1}{\varepsilon _1}+{\sigma _2}{\varepsilon _2}+{\sigma _3}{\varepsilon _3})=\frac{1}{{2E}}[\sigma _{1}^{2}+\sigma _{2}^{2}+\sigma _{3}^{2} - 2\nu ({\sigma _1}{\text{ }}{\sigma _2}+{\sigma _1}{\sigma _3}+{\sigma _3}{\sigma _2})]$$ 2 where U is the total energy density of external workmanship, U e is the elastic strain energy density, and U d is the dissipated energy density. The one characteristic curve of uniaxial compression stress and strain obtained, shown in Fig. 2 , is used as an example to illustrate the energy evolution of the compacted and the elastic deformation stage before the crack initiation stress threshold. 2.2 Measures of elastic strain energy density Four assumptions are considered in this paper, that: (i) the rock material is perfectly elastic (no energy dissipation in elastic deformation and all elastic strain energy released after force removed) ; (ii) define the unit volume V and the unit has a closed surface S , \(\delta W\) and \(\delta U\) are the work of the external force and the increment of internal energy for unit volume, respectively; (iii) the external force is applied constantly, regardless of the effect of strain rate, to ensure that the unit body is in equilibrium at any time; (iv) the kinetic energy changes are ignored; These are guaranteed to be a quasi-static loading in uniaxial compression. Based on the above assumptions, the stress tensor is decomposed into spherical and deviatoric stress tensors; the strain tensor is decomposed into spherical and deviatoric strain tensors, that is, strain energy is decomposed into the energy of volume \({W_1}\) and the energy of shape change \({W_2}\) , the generalized form given in the following, respectively. $${W_1}=\iiint_{V} {\left( {{X_u}+{Y_v}+{Z_w}} \right)}dV$$ 3 $${W_2}=\iint_{S} {\left( {{X^*}u+{Y^*}v+{Z^*}w} \right)dS}$$ 4 Where , , defined as the external force on per unit volume element, \({X^*}\) , \({Y^*}\) , \({Z^*}\) are the internal force on the surface of the unit element, the external force work W is entirely transformed into elastic strain energy and is defined as: $$W={W_1}+{W_2}=\iiint_{V} {\left( {{X_u}+{Y_v}+{Z_w}} \right)}dV+\iint_{S} {\left( {{X^*}u+{Y^*}v+{Z^*}w} \right)dS}$$ 5 The work done by the external force acting on the unit body to produce a small displacement \(\delta W\) , is expressed as: $$\delta W=\delta {W_1}+\delta {W_2}=\iiint_{V} {{X_i}}\delta {u_i}dV+\iint_{S} {X_{i}^{*}}\delta {u_i}dS$$ 6 After substituting into the balance equation and boundary conditions, it is known from the divergence theorem: $$\delta W=\iiint_{V} {{\sigma _{ij}}}\delta {u_{i,j}}dV$$ 7 The increase of internal energy is found: $$\delta U=\iiint_{V} {{\sigma _{ij}}}\delta {\varepsilon _{i,j}}dV$$ 8 Known from Green’s formula, and let: $$~~\frac{{\partial {u_0}\left( {{\varepsilon _i}_{j}} \right)}}{{\partial {\varepsilon _i}_{j}}}={\sigma _{ij}}$$ 9 Integrate the formula can easily find: $$\int_{0}^{{{u_0}\left( {{\varepsilon _{ij}}} \right)}} {d{u_0}} ={u_0}\left( {{\varepsilon _{ij}}} \right) - {u_0}\left( 0 \right)$$ 10 where \({u_0}\left( {{\varepsilon _{ij}}} \right),{u_0}\left( 0 \right)\) respectively indicate the elastic strain energy density after loading and before loading. Let \({u_0}\left( 0 \right)=0\) , and the strain energy density is determined by its stress and strain increase, which can be defined as: $${u_0}({\varepsilon _i}_{j})=\int_{0}^{{{\varepsilon _{ij}}}} {{\sigma _{ij}}} d{\varepsilon _{ij}}$$ 11 The unit volume strain energy density of a rock sample can be expressed as: $$\frac{{dW}}{{dV}}=\int_{0}^{{{\varepsilon _{ij}}}} {{\sigma _{ij}}} d{\varepsilon _{ij}}$$ 12 2.3 Characteristics of strain energy evolution and initiation energy According to Qingbin Meng et al.[ 28 ] energy self-inhibition evolution model and Zheng Zaisheng's [ 36 ] research on the nonlinear fitting formula of energy transfer in rock deformation under uniaxial compression with a certain stress level \(\sigma\) . The rate of increment of the accumulated strain energy density was given[ 27 , 34 , 37 ] by: $$\frac{1}{{{u^i}}}\cdot \frac{{d{u^i}}}{{d\sigma }}={a_i}({u^c}^{i} - {u^i}^{0}) - {b_i}{u^i}$$ 13 The formula \({u^c}^{i}\) represents the energy density converted from external input driving energy, \({u^i}^{0}\) is the lowest input strain energy density threshold, and, \({a_i}\) , \({b_i}\) is the coefficient, which respectively reflects the degree of energy conversion and inhibition. Integrate \({u^i}\) and find: $${u^i}=\frac{{{a_i}({u^c}^{i} - {u^i}^{0}){e^{{a_i}}}{{^{{({u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{\sigma }}}{{{c_1}{b_i}[1+{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{\sigma }]}}$$ 14 with \({c_1}\) , the integral constant. Based on elasticity modulus \({E_i}\) and formula Eq. 13 , find: $${E_i}=\frac{{{\sigma ^2}}}{{{u^i}}}$$ 15 Derivative and let the first derivative be 0 as follows: $$\frac{{d{E_i}}}{{d\sigma }}=\frac{{2\sigma {u^i} - {{\left( {{u^i}} \right)}^\prime }{\sigma ^2}}}{{{{\left( {{u^i}} \right)}^2}}}=0$$ 16 Eq. 14 substituted and organized, can find: $${\sigma _i}=\frac{2}{{{a_i}\left( {{u^c}^{i} - {u^i}^{0}} \right) - {b_i}{u^i}}}=\frac{2}{{{a_i}\left( {{u^c}^{i} - {u^i}^{0}} \right) - \frac{{{a_i}({u^c}^{i} - {u^i}^{0}){e^{{a_i}}}{{^{{({u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{{\sigma _i}}}}}{{1+{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{{\sigma _i}}}}}}}=\frac{{2{a_i}\left( {{u^c}^{i} - {u^i}^{0}} \right)}}{{1+{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{{\sigma _i}}}}}$$ 17 After being simplified, when the elastic modulus reaches the maximum value, the strain energy accumulated \({u_i}\) and Stress \({\sigma _i}\) are obtained: $${u_i}=c{e^{ - \frac{1}{2}}}{\sigma _i}$$ 18 And $${\sigma _i}=\frac{{2{a_i}\left( {{u^c}^{i} - {u^i}^{0}} \right)}}{{1+{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{{\sigma _i}}}}}$$ 19 with the integral constant. 2.4 Energy digestion index (EDI) From the concept of strain energy density, with isothermal conditions, the unit volume strain energy density can be expressed as: $$\frac{{dW}}{{dV}}=\int_{0}^{{{\varepsilon _{ij}}}} {{\sigma _{ij}}} d{\varepsilon _{ij}}$$ 20 The above formula shows that the strain energy density of a unit can be determined by the stress and strain of the unit. The elastic stage of the stress-strain curve obtained from the experiment is partially enlarged, as shown in Fig. 3 . When the rock sample is loaded to point B in the uniaxial compression test, the elastic strain reaches a maximum, and the rock cracks. The area AOBD can be used to express the external force input energy during this process. The sample material is generally thought to be in the elastic stage at this point[ 35 , 38 , 39 ]. When the load is unloaded at point B, the formed unloading curve essentially coincides with the loading curve, indicating that no new crack damage is generated as a result of external force and no energy is dissipated[ 40 – 42 ]. The strain energy digested by the rock sample is represented by the area AODBC. Area ACB represents the portion of the external force's work that has not been converted into strain energy. At this point, the energy generated by the external force can be divided into two parts: the strain energy \({U^e}\) converted and digested by the rock sample, and the energy loss \({U^a}\) the external force work cannot be converted into strain energy. The rock material's energy digestion index Eq. 20 is established by analyzing the proportion of the energy digested in the total input energy. This formula is used to characterize the degree of response of the rock material to external loads and reflect the properties of the material itself. The energy digestion index EDI can be expressed as: $$D=\frac{{{U^e}}}{U}=\frac{{{U^e}}}{{{U^a}+{U^e}}} \times 100\%$$ 21 The above formula shows that when a rock material is subjected to an external load, the energy generated by the external force is not completely converted into elastic strain energy and stored inside the material due to differences in the rock material with microstructure and ability to withstand the outside; in this process, the mechanical properties of rock materials are characterized by the energy conversion and digestion ability. 3 Compressive strength tests 3.1 Specimens preparation All of the rock specimens examined were obtained from a Chinese coal mine. Sandstones with varying proportions of fine sandstone and siltstone as the main constituents exist. The specimens were processed into cylinders with a diameter of 50 mm and a length-to-diameter ratio of 2 (Fig. 4 ) using the ISRM-recommended method[ 43 ]. After careful grinding with a grinder and sandpaper, the top and bottom surfaces have flatness of less than 0.02 mm and parallelism of less than 0.05 mm. 9 specimens were tested due to the discreteness of the test data. 3.2 Laboratory uniaxial compression testing The tests were carried out using the WDW-300 digital electro-hydraulic servo system's equipment (Fig. 4 ) and the specimen was placed in the test machine. The loading rate of 0.0015 mm/s, with continuous loading until the specimen breaks. The test is terminated when the residual strength equals the 60% peak strength or maximum load 240KN. In the test, 9 specimens were loaded uniaxially, and all the stress-strain data were recorded. 4 Result and discussion 4.1 The result of Compressive strength test Axial stress-strain curves for the three kinds of rock specimens under uniaxial compression are shown in Fig. 5 , which are similar in geometry, and the axial stress-axial strain behavior can be approximately divided into four typical stages, i.e., fissure closure, elastic deformation, crack growth and propagation, and strain-softening. In the process of uniaxial compression of the specimen, the stress-strain characteristics obtained are similar to those of C.D. Martin et al.[ 2 – 4 ] in the stress-strain study of brittle rocks, including: the compaction closure stage, elastic deformation stage, stable crack expansion stage, crack accelerated expansion stage, and post-peak stage (shown in Fig. 1 ); the division of each stage of the failure process depends on 3 important stress thresholds: crack initiation \({\delta _{ci}}\) ,crack evolution \({\delta _{cd}}\) ,ultimate strength \({\delta _c}\) . When the load is lower, the internal pores and cracks of the rock are compressed and sealed, and the process of the rock with better compactness is not obvious. The elastic properties of the rock body conform to Hooke's law, and the load reaches the threshold stress, which is 30–50% of the peak strength[ 4 ]. Continue to load and enter the stage of stable crack propagation. After reaching the threshold stress that is 70–80% of the peak strength, the crack enters rapid and unstable expansion, macroscopic shear failure occurs, and finally reaches the peak strength. The process of fissure compaction and closure exists in every test, but it is not significant. The reason is due to the heterogeneity of the natural rock sample itself and the existence of inevitable internal defects and fissures. The original fissures and fissures in this stage defects are further compacted, and the volume of the original crack defects decreases and tends to zero, and enters the elastic stage; in the elastic stage, some curves show an almost completely elastic state, and the stress-strain curves show linear elastic characteristics; while most of the curves are almost completely elastic. It appears as a downward concave phenomenon, that is, the slope increases as the load continues, the elastic deformation exhibits nonlinear characteristics, and the one-to-one correspondence between stress and strain is still maintained: $$\sigma =E\cdot \varepsilon =f\left( \varepsilon \right)$$ 22 At the stage of fissure closure, the uniaxial compression stress-strain curve of the sample shows the downward concave and the initial nonlinear deformation at low-stress levels, which results from the closure of some primary pores and voids in the sample with the increasing compression stress. At the stage of elastic deformation, the loading begins with the increase of axial stress after the primary fissure closure, and the elastic deformation dominates the stress-strain curve of the sample. However, the slope of the three curves, i.e., the elastic modulus, has a big difference, which results from the component of the tested rock material. The rigidity of granite samples is the largest, and that of sandstone samples is the smallest. At the stage of crack growth and propagation, the sample continuously produces the stress concentration near the tips of the initial crack, which results in the initiation and propagation of some new cracks. Therefore, the stress-strain curve departs from the elastic behavior and shows distinctly nonlinear deformation. For granite and limestone samples, this stage is short, while it is long for sandstone samples. At the stage of strain-softening, the macroscopic crack in the rock sample comes out rapidly and the post-peak behavior in the stress-strain curves shows a rapid drop. At this time, the sample can only support a lower axial stress, even though no axial stress. But the slope of the three curves, i.e., the softening modulus, in Fig. 6 , differences in the end stage of elastic deformation of each specimen, and the post-elastic deformation curves of higher strength specimen are more abrupt than low strength ones. 4.2 Modulus of elasticity The modulus of elasticity is variable. The point elastic modulus and strain increase with the stress adding; it is different from the constant elastic modulus of completely elastic material, that is, under external load, the external work is not 100% energy transfer, part of the energy is lost, and it causes the specimen such internal structure changes, resulting in strain hardening[ 36 ]. 4.3 Compressive strength and stiffness Through the analysis of the stress-strain curves of different specimens, it is found that the initial stage of the test, that is, the pressure-tight and stage curves have good consistency, and there is no obvious sudden change, indicating that the integrity of the sample is better, and the original cracks are less, which meets the requirements of the sample. After the elastic deformation stage, the uniaxial compression stress-strain curves of different specimens have obvious differences, and the material properties have changed significantly, as shown in Fig. 6 . The three groups of stress-strain curves all show variable elastic modulus in the pre-peak stage and the characteristics of failure after the sample reaches the ultimate strength as the external load continues to increase and there is a significant difference in strength. Compared with the SAN-B group, the SAN-A and SAN-C groups are more sensitive to the external load due to the smaller stress and strain under the action of the external load. The mechanical properties of different sandstone specimens are shown in Table 1 . It can be seen that the strength of different sandstone specimens under uniaxial compression is \(6.7 - 12.1MPa\) . According to the rock's Protodyakonov index F , the specimens are divided into three groups: SAN-A, SAN-B, and SAN-C, with three specimens in each group, and identified in numbers 1, 2, and 3. The average of group SAN-A \(F=0.7\) , the average of group SAN-B \(F=1.2\) , and the average of group SAN-C \(F=0.8\) . Table 1 The result of compressive strength test and dispersion Specimens Serial Number Cracking Threshold Avg. Value Ultimate Strength Avg. Ultimate strength Avg. Density Modulus of Elasticity) Protodyakonov Coefficient σ ci /MPa MPa σ c /MPa MPa kg/m 3 E c /MPaⅹ10 2 F 1 3.168 6.710 4.00 SAN-A 2 2.956 3.02 8.780 7.37 1941.64 4.94 0.70 3 2.926 6.630 4.26 1 7.139 11.30 8.03 SAN-B 2 6.579 7.03 14.39 12.11 2156.38 9.01 1.20 3 7.376 10.65 8.78 1 4.452 9.140 6.65 SAN-C 2 4.005 4.18 7.030 8.30 2011.37 5.33 0.80 3 4.085 8.740 4.95 4.4 Energy Digestion Index (EDI) According to the grouping of specimens with different Protodyakonov coefficients, the energy digestion index, sample cracking stress threshold, cracking energy threshold, and uniaxial compressive strength data of each specimen can be calculated, as shown in Table 2 . From Table 2 , in the pre-cracking stage of the specimens with different Protodyakonov coefficients, the energy digestion index is relatively concentrated and the overall level is 78%-95% higher, the ultimate strength during failure is relatively discrete, ranging from 6.71 to 14.39Mpa; Group SAN-A samples average Protodyakonov coefficient F = 0.7, average energy digestion index D = 93.57%; SAN-B group average Protodyakonov coefficient F = 1.2, average energy digestion index D = 80.90%; SAN-C group average Protodyakonov coefficient F = 0.8, average energy digestion index D = 89.04%. It can be seen that the higher the EDI of the specimens, the lower the average hardness and the lower the ultimate strength. Table 2 Samples of EDI and energy intensity value Specimens Serials Number EDI Cracking Stress Threshold Cracking Strain Energy Threshold Cracking Strain Ultimate strength Cracking Threshold / Ultimate strength D /% /MPa /MJ•m − 3 /% /MPa /% 1 92.643 3.168 1.297 0.76 6.710 47.21% SAN-A 2 92.704 2.956 0.9669 0.6 8.780 33.67% 3 95.357 2.926 1.206 0.7 6.630 44.13% 1 82.697 7.139 2.882 0.89 11.300 63.18% SAN-B 2 81.075 6.579 2.142 0.73 14.390 45.72% 3 78.924 7.376 2.685 0.84 10.650 69.26% 1 85.159 4.452 1.469 0.67 9.140 48.71% SAN-C 2 93.524 4.005 1.632 0.75 7.030 56.97% 3 88.448 4.085 1.715 0.85 8.740 46.74% To investigate the relationship between energy digestion characteristics, rock strength, and sensitivity to external forces, the EDI and uniaxial compressive strength of each sample were tested under quasi-static loading conditions[ 38 ]. The EDI and ultimate strength data are shown in a rectangular coordinate system, and the results are fitted, as illustrated in Fig. 7 . In Fig. 7 , the EDI of each rock sample and the uniaxial compression peak strength reflect the significant correlation between the strength of the rock sample and the energy digestion character. Concerning the law that the lower ultimate compressive strength with the higher the EDI in the elastic stage, the explanation for this phenomenon is that when the stress reaches the cracking stress, the energy obtained by the rock sample is all in the form of elastic strain energy accumulated in the material, and the internal damage can be ignored, the faster and more strain energy accumulates, the lower the strength of the rock, the easier it is to fail. The less of the accumulated elastic strain energy, the poorer the rock's ability to digest introduced energy, and the rock has better strength characteristics because large-scale crack formation and destruction require more energy. When the EDI reaches the ultimate value, the rock cracking sharply, and the elastic strain energy accumulated in the rock material is suddenly released in a short time, accompanied by macroscopic manifestations such as acoustic emission and debris ejection. 5 Conclusions Using the uniaxial compression test, stress-strain curve analysis, and energy calculation analysis of a specific mine sandstone sample, this article comes to the following results: The test results for each sample clearly show a linear relationship between the fissure initiation threshold stress, expansion threshold stress, and peak strength; however, because each sample is unique, there are differences in the length of the strain process experienced at different stages. The strain energy absorbed by the rock sample per unit volume grows throughout the elastic deformation stage as the external load work increases continuously, exhibiting nonlinear growth features. First, there is an increase in strain energy digestion, followed by a drop. This phenomenon demonstrates how the rock sample material's capacity to absorb and store elastic strain energy is reflected in how incompletely the work produced by external forces is converted into strain energy. The article's suggested dimensionless parameter of the strain energy digest index in the elastic stage has a strong linear correlation with the rock sample's peak compressive strength. Additionally, it is evident from fitting that there is a clear linear relationship between the sample material and the strain energy that the rock sample digests during the early elastic deformation stage of uniaxial compression. The intensity increases with less strain energy being digested. Lastly, it must be clarified that full compaction and the absence of natural fissures cannot be guaranteed by the test volume and sample material. The article's conceptions and findings, which mostly highlight the fact that typical rock materials frequently have a linear elastic stage, are still preliminary. The primary cause of the observed nonlinearity in the measurement is the loss of entropy; however, quantifying the energy lost in this process is a challenging task due to its complexity. The energy difference between the beginning of the elastic phase and the end indicates the energy dissipation and loss. Second, the findings of the experimental investigation offer insightful information about the strength and deformation properties of rock materials, and they examine the impact of these properties on their uniaxial strength. To give further references for stability evaluation, the follow-up work will examine the energy digestion law of the in-situ rock mass in comparison to the energy loss properties of the rock material and the energy discrimination criterion for strain energy digestion. The study's findings show that : The material absorbs and accumulates strain energy under the influence of external forces, displaying a non-linear accumulation in a changing modulus of elasticity. The pace at which strain energy is absorbed by the rock material determines the creation of fractures, and when this rate of growth in strain energy digestion achieves its maximum, the material's growing rate of variable elasticity modulus also reaches its maximum. The mechanical qualities of the rock material are inversely correlated with the energy digestion index; that is, the lower the rock strength, the higher the energy digestion index of the rock material. The correlation between the energy digestion index and the sensitivity of the rock material to external loading is not significant. Declarations Data availability All the data generated and/or analyzed during the current study is presented in the paper. Acknowledgments This study was supported by the National Key Research and Development Program of China (No. 2020YFC1909801; 2020YFC1909101). We would like to thank anonymous reviewers for their detailed and constructive comments which helped us improve the quality of the manuscript. Author Contributions Experiments were conceived and designed by J.C. and Q.W. Samples were collected by Q.W. Data were analyzed by L.Y, E.M., and L.H.L. The manuscript was written by L.Y., and Y.F.J. reviewed the manuscript. Competing Interests: The authors declare no competing interests. Additional Information Correspondence and requests for materials should be addressed to L.Y. Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. References Griffith, A.A., VI. The phenomena of rupture and flow in solids. Philosophical transactions of the royal society of london. Series A, containing papers of a mathematical or physical character, 1921. 221(582-593): p. 163-198. Martin, C.D. and N.A. Chandler. The progressive fracture of Lac du Bonnet granite. 1994: Elsevier. Martin, C.D., Seventeenth Canadian geotechnical colloquium: the effect of cohesion loss and stress path on brittle rock strength. Canadian Geotechnical Journal, 1997. 34(5): p. 698-725. Cai, M., et al., Generalized crack initiation and crack damage stress thresholds of brittle rock masses near underground excavations. International Journal of Rock Mechanics and Mining Sciences, 2004. 41(5): p. 833-847. Hui, L., et al., Analysis of water and ice content and damage characteristics of the frozen rock during freezing based on the three-valued segmentation of CT images. Journal of Mining and Safety Engineering, 2016. 33(6): p. 1130. Li, J., et al., NMR research on deterioration characteristics of microscopic structure of sandstones in freeze–thaw cycles. Transactions of Nonferrous Metals Society of China, 2016. 26(11): p. 2997-3003. Jin, Y., et al., Simulation of NMR response of microfractures based on digital rock technology. Geoenergy Science and Engineering, 2023. 227: p. 211876. Giwelli, A., et al. In-situ nuclear magnetic resonance detection and visualisation of hydraulic fracture: A proof of concept. 2023: ARMA. Wu, G., D.Y. Wang and S.T. Zhai, Acoustic emission characteristics of sandstone after high temperature under uniaxial compression. Rock Soil Mech, 2012. 33(011): p. 3237-3242. Zhi-hui, H., et al., Acoustic emission characteristics analysis of sandstone under uniaxial compression. Nonferrous Metals Science and Engineering, 2014. 5(1): p. 91-94. Xiao, F.K., et al., Acoustic emission (AE) characteristics of fine sandstone and coarse sandstone fracture process under tension-compression-shear stress. Chinese Journal of Rock Mechanics and Engineering, 2016. 35(S2): p. 3458-3472. Niu, Y., Y. Hu and J. Wang, Cracking characteristics and damage assessment of filled rocks using acoustic emission technology. International Journal of Geomechanics, 2023. 23(4): p. 04023013. Cao, K., et al., A comprehensive model for evaluating infrared radiation and acoustic emission characteristics of sandstone fracture. Engineering Fracture Mechanics, 2023. 283: p. 109217. Melchiorre, J., et al., Acoustic Emission and Artificial Intelligence Procedure for Crack Source Localization, in Sensors. 2023. Khadivi, B., et al., Characterizing the cracking process of various rock types under Brazilian loading based on coupled acoustic emission and high-speed imaging techniques. International Journal of Rock Mechanics and Mining Sciences, 2023. 168: p. 105417. Dong, L., et al., Uncertainty investigation for the classification of rock micro-fracture types using acoustic emission parameters. International Journal of Rock Mechanics and Mining Sciences, 2023. 162: p. 105292. Shao, J., Q. Zhu and K. Su, Modeling of creep in rock materials in terms of material degradation. Computers and Geotechnics, 2003. 30(7): p. 549-555. Salari, M.R., et al., A coupled elastoplastic damage model for geomaterials. Computer methods in applied mechanics and engineering, 2004. 193(27-29): p. 2625-2643. Tsoutrelis, C.E. and G.E. Exadaktylos, Effect of rock discontinuities on certain rock strength and fracture energy parameters under uniaxial compression. Geotechnical & Geological Engineering, 1993. 11: p. 81-105. Nara, Y. and K. Kaneko, Sub-critical crack growth in anisotropic rock. International Journal of Rock Mechanics and Mining Sciences, 2006. 43(3): p. 437-453. Zhou, X.P., Triaxial compressive behavior of rock with mesoscopic heterogenous behavior: Strain energy density factor approach. Theoretical and Applied Fracture Mechanics, 2006. 45(1): p. 46-63. Xie, H., et al., Energy analysis of rock failure. Yanshilixue Yu Gongcheng Xuebao/Chin. J. Rock Mech. Eng., 2005. 24(15): p. 2603-2608. Xie, H.P., et al., Energy mechanism of deformation and failure of rock masses. Chinese Journal of Rock Mechanics and Engineering, 2008. 27(9): p. 1729-1740. Zhang, P., et al., Damage constitutive model of uniaxially compressed coal material considering energy dissipation. Journal of Materials Research and Technology, 2023. 27: p. 920-931. Wang, G., et al., Investigations on the macro‐meso mechanical properties and energy dissipation mechanism of granite shear fracture under dynamic disturbance. International Journal for Numerical and Analytical Methods in Geomechanics, 2023. 47(13): p. 2364-2384. Xiao, F., et al., Relationship between hysteresis loop and elastoplastic strain energy during cyclic loading and unloading. Chinese Journal of Rock Mechanics and Engineering, 2014. 33(9): p. 1791-1797. Zhang, Z.Z. and F. Gao, Research on nonlinear characteristics of rock energy evolution under uniaxial compression. Chinese Journal of Rock Mechanics and Engineering, 2012. 31(6): p. 1198-1207. Meng, Q., et al., Research on non-linear characteristics of rock energy evolution under uniaxial cyclic loading and unloading conditions. Environmental Earth Sciences, 2019. 78: p. 1-20. Yu, Y., et al., Energy evolution and acoustic emission characteristics of uniaxial compression failure of anchored layered sandstone. Frontiers in Earth Science, 2022. 10: p. 841598. Sun, Y., F. Yu and J. Lv, Experimental Study on Acoustic Emission Characteristics of Granite and Sandstone under Uniaxial Compression. Geofluids, 2023. 2023. Wang, Z., et al., Failure behaviour and acoustic emission characteristics of different rocks under uniaxial compression. Journal of Geophysics and Engineering, 2020. 17(1): p. 76-88. Wen, T., et al., Energy Analysis of the Deformation and Failure Process of Sandstone and Damage Constitutive Model. KSCE Journal of Civil Engineering, 2019. 23(2): p. 513-524. Li, T., et al., Nonlinear behavior and damage model for fractured rock under cyclic loading based on energy dissipation principle. Engineering Fracture Mechanics, 2019. 206: p. 330-341. Yun, C., J. Yan and C. Mian, A rock brittleness evaluation method based on energy dissipation. Chinese Journal of Theoretical and Applied Mechanics, 2015. 47(6): p. 984-993. Jin, F.N., M.R. Jiang and X.L. Gao, Defining damage variable based on energy dissipation. Chinese Journal of Rock Mechanics and Engineering, 2004. 23(12): p. 1976-1980. Zheng, Z., Energy transfer process during rock deformation. Chineses Science(Chemistry, Life Sciences & Geosciences), 1991(1): p. 104-117. Wang, C., et al., Stress–Energy Mechanism for Rock Failure Evolution Based on Damage Mechanics in Hard Rock. Rock Mechanics and Rock Engineering, 2020. 53(3): p. 1021-1037. Huang, D., R.Q. Huang and Y.X. Zhang, Experimental investigations on static loading rate effects on mechanical properties and energy mechanism of coarse crystal grain marble under uniaxial compression. Chinese Journal of Rock Mechanics and Engineering, 2012. 31(2): p. 245-255. Li, X., Y. Zuo and C. Ma, Failure criterion of strain energy density and catastrophe theory analysis of rock subjected to static-dynamic coupling loading. Yanshilixue Yu Gongcheng Xuebao/Chin. J. Rock Mech. Eng., 2005. 24(16): p. 2814-2824. Wang, J.B., et al., Analysis of energy evolution characteristics of salt rock under different loading rates. Frontiers in Earth Science, 2022. 10: p. 829185. Shekov, V. and G. Kolesnikov, Differential Energy Criterion for Brittle Fracture: Conceptualization and Application to the Analysis of Axial and Lateral Deformation in Uniaxial Compression of Rocks. Materials, 2023. 16(13): p. 4875. Wang, W., et al., Mechanical behavior and energy mechanism of salt rock with prefab fissures under uniaxial compression. Journal of Chongqing University, 2016. 39(5): p. 103-113. Fairhurst, C.E. and J.A. Hudson, Draft ISRM suggested method for the complete stress-strain curve for intact rock in uniaxial compression. International journal of rock mechanics and mining sciences (1997), 1999. 36(3): p. 279-289. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 01 Jul, 2024 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 12 May, 2024 Reviews received at journal 11 May, 2024 Reviews received at journal 28 Apr, 2024 Reviewers agreed at journal 22 Apr, 2024 Reviewers agreed at journal 17 Apr, 2024 Reviewers agreed at journal 17 Apr, 2024 Reviewers agreed at journal 17 Apr, 2024 Reviewers invited by journal 17 Apr, 2024 Editor assigned by journal 17 Apr, 2024 Editor invited by journal 17 Apr, 2024 Submission checks completed at journal 13 Apr, 2024 First submitted to journal 01 Apr, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4198169","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":290784438,"identity":"7c58dbce-3564-464c-8e4a-208cd0cc6aea","order_by":0,"name":"Lei Yan","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3UlEQVRIiWNgGAWjYHACxgMMBv/l+PmbDzAkEKsHqIXZWHLGsQRStDAwJ244kGNAnHKD42cPHPhQwAbUcubzh4c77Bj427vxW2ZwJi/h4AwDHuOZh3u3SSSeSWaQOHN2A34tQPcc5jGQkO07cHYbQ2IbM4OBRC4BLeffGBz+Y2DA2HAg5/GHxLZ6IrTcANrCYJCgOOFADoNEYtthwlokb7wxONhjcAAUyGZALcd5CPqF73yO4YMffw6AovLxx59t1XL87b34tSgcQBPgwascBOQbCCoZBaNgFIyCEQ8AAjtS7fv0TGMAAAAASUVORK5CYII=","orcid":"","institution":"Sinosteel Maanshan General Institute of Mining Research Co., Ltd.,","correspondingAuthor":true,"prefix":"","firstName":"Lei","middleName":"","lastName":"Yan","suffix":""},{"id":290784439,"identity":"468332a3-3804-431c-ae77-9764886aa4b8","order_by":1,"name":"Jian Chang","email":"","orcid":"","institution":"Sinosteel Maanshan General Institute of Mining Research Co., Ltd.,","correspondingAuthor":false,"prefix":"","firstName":"Jian","middleName":"","lastName":"Chang","suffix":""},{"id":290784441,"identity":"00a7c866-eb8c-4989-973a-1d590cc1fe68","order_by":2,"name":"Ebelia Manda","email":"","orcid":"","institution":"Copperbelt University","correspondingAuthor":false,"prefix":"","firstName":"Ebelia","middleName":"","lastName":"Manda","suffix":""},{"id":290784442,"identity":"4312d86f-2b56-444c-ac0f-42f17ca81eaa","order_by":3,"name":"Helin Li","email":"","orcid":"","institution":"Sinosteel Maanshan General Institute of Mining Research Co., Ltd.,","correspondingAuthor":false,"prefix":"","firstName":"Helin","middleName":"","lastName":"Li","suffix":""},{"id":290784445,"identity":"6db4deaf-90b5-4063-9ad8-4ef4bfb41483","order_by":4,"name":"Qian Wang","email":"","orcid":"","institution":"Sinosteel Maanshan General Institute of Mining Research Co., Ltd.,","correspondingAuthor":false,"prefix":"","firstName":"Qian","middleName":"","lastName":"Wang","suffix":""},{"id":290784446,"identity":"4095280d-1a8e-4682-9447-9c07ee63b92a","order_by":5,"name":"Yangfan Jing","email":"","orcid":"","institution":"Sinosteel Maanshan General Institute of Mining Research Co., Ltd.,","correspondingAuthor":false,"prefix":"","firstName":"Yangfan","middleName":"","lastName":"Jing","suffix":""}],"badges":[],"createdAt":"2024-04-01 04:59:16","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4198169/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4198169/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41598-024-66226-3","type":"published","date":"2024-07-02T00:29:49+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":54833603,"identity":"b843eb51-9665-452c-9d1b-cb66e8a04a9f","added_by":"auto","created_at":"2024-04-17 11:56:54","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":322818,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eStress-strain diagram of crack development stages\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-4198169/v1/0e0433c9453a4b5298110ae9.png"},{"id":54834010,"identity":"1cf1e50f-1f8f-4a04-be81-cd944bfe82c8","added_by":"auto","created_at":"2024-04-17 12:04:54","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":96313,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eThe uniaxial compressing stress-strain curve schematic diagram\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-4198169/v1/2e754d2a4477ae77591b6e0a.png"},{"id":54833601,"identity":"da53dcf7-8ae2-4ec1-9603-69cc6665f6d9","added_by":"auto","created_at":"2024-04-17 11:56:54","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":138541,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of local amplification of stress-strain curves before crack initiation\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-4198169/v1/568f90399960021f9c7d2586.png"},{"id":54833607,"identity":"1186bc86-5a76-4e10-8b30-6697cd910c65","added_by":"auto","created_at":"2024-04-17 11:56:54","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":8927611,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eCompressive strength test for specimens\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-4198169/v1/d6c42bf9c70d7d635d6b3831.png"},{"id":54834011,"identity":"92925d08-1995-475a-b20a-ee3e72c4ea75","added_by":"auto","created_at":"2024-04-17 12:04:54","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":174848,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eStress-strain curve test results of specimens\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-4198169/v1/605c4d9c5c8fdcd2bdfc7931.png"},{"id":54833605,"identity":"47ffe2d0-f036-45fb-b68f-b5507a0e6c60","added_by":"auto","created_at":"2024-04-17 11:56:54","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":281036,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eDifferences in the end stage of elastic deformation of each specimen\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-4198169/v1/916abbb1f26ee2810255febf.png"},{"id":54833606,"identity":"1b892307-454a-468f-bfac-8a36d509cfd5","added_by":"auto","created_at":"2024-04-17 11:56:54","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":164485,"visible":true,"origin":"","legend":"\u003cp\u003eFitting results of strength and energy digestion index\u003c/p\u003e","description":"","filename":"floatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-4198169/v1/4c13b6bc09a5bbe407656c18.png"},{"id":59531957,"identity":"5b9abaa8-754b-447e-8d9a-f89eca455c06","added_by":"auto","created_at":"2024-07-03 00:30:03","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":10867821,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4198169/v1/8188478c-ae28-4e1c-a1fc-0b2d17c2fd2a.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Rock Crack Initiation Triggered by Energy Digestion","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eDisasters related to brittle fracture of rock occur frequently during the civil excavation process. Events such as rock bursts and slope landslides are often triggered by exceeding the ultimate strength of rock during tuning, sloping, mining excavation, or other civil constructions.\u003c/p\u003e \u003cp\u003eAs a typical kind of heterogeneous natural body formed by mineral debris under geological action. Some internal elements of the rock material are damaged and fatigued as a response to external force, and a large number of initial micro-cracks form and grow, eventually leading to instability and failure. Many researches have been conducted on micromechanical damage fatigue based on laboratory experiments.\u003c/p\u003e \u003cp\u003eGriffith conducted a systematic study on stress and crack expansion when cracks occur in the internal structure of the material as early as 1920. The criterion for crack expansion under bidirectional compression was obtained by establishing a crack model for plane compression. Then, from the perspective of energy release, the energy balance theory, which is based on the ideal brittle material, is proposed, which differs from the traditional concept of material strength in the continuum[\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. CD Martin et al. [\u003cspan additionalcitationids=\"CR3\" citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e] found that the failure of brittle rocks can be classified into five stages based on the stress-strain curve: compaction closure stage, elastic deformation stage, stable crack expansion stage, accelerated crack expansion stage, and after the peak as showed in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eRock microscopic damage can now be directly observed using new technologies such as computerized tomography (CT) and nuclear magnetic resonance (NMR)[\u003cspan additionalcitationids=\"CR6\" citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e], In-situ monitoring of rock behavior remains difficult to achieve[\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. Indirect observation methods such as acoustic emission (AE), allow the passive monitoring of the elastic ultrasonic wave that propagates in the material as a result of the crack formation\u0026rsquo;s abrupt release of energy and the energy dissipation in the process of rock damage by studying the correlation between acoustic emission signals and mechanical properties of rock[\u003cspan additionalcitationids=\"CR10 CR11 CR12\" citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e], shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. However, it is difficult to strictly correspond to the microscopic damage and reveal the process of damage evolution with non-destructive [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]and have not yet been universally formed for the most commonly used Prediction evaluation criteria method[\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. More recently, the perspective of energy evolution has provided a new idea for the study of crack evolution. Shao et al.[\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e] and Salari et al.[\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e] proposed that the formation and development of cracks lead to dilatation, and the rock damage is characterized by volumetric strain energy. Further, the damage evolution can be characterized by elastic bulk modulus and plastic strain energy release rate[\u003cspan additionalcitationids=\"CR20\" citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]. A large number of laboratory experiments and theoretical analyses have confirmed the intrinsic relationship between strain energy dissipation and release in the process of rock deformation and failure[\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e, \u003cspan additionalcitationids=\"CR23 CR24\" citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]. Based on the above findings, the stress-strain curve of hysteresis loops obtained by uniaxial compression has been used to characterize the strain energy. However, due to the energy dissipation caused by compaction and plastic damage at an uncertainty of the stage of occurrence [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e], the hysteresis curve cannot be identified; other side, the process of energy transformation has the characteristics of self-inhibition, bifurcation, and chaos, and the energy accumulation is iterative growth[\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e]. Consequently, the accumulated elastic energy calculated by the hysteresis curve has a large deviation.\u003c/p\u003e \u003cp\u003eThese work and theoretical research results have fully demonstrated the energy evolution characteristics of rock under load, which has laid a foundation for theoretical research in the laboratory, but it is still not practical in engineering in-situ monitoring and disaster prediction.\u003c/p\u003e \u003cp\u003eIn this study, we aimed to overcome these limitations by using the energy digestion index (EDI) a novel algorithm, based on energy theory to determine the trigger point of intact rock crack initiation to investigate the relationship between rock strength and energy digestion. Through laboratory rock mechanics tests and the energy accumulation process of loading samples, the results show that once the increased rate of energy digestion reaches its maximum, rock crack initiation occurs. By using EDI, we were able to accurately model the relationship between rock strength and energy digestion, leading to a more comprehensive understanding and exploration of In-situ monitoring of rock damage and failure prediction.\u003c/p\u003e"},{"header":"2. Energy evolution characteristics","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Energy conversion analysis\u003c/h2\u003e \u003cp\u003eAccording to the principle of conservation of energy, the process of external work on the sample is accompanied by energy conversion; when the material is an ideal brittle rock, the external work is completely converted into elastic strain energy and stored within the material. The digested energy is to overcome the rock's intrinsic cohesion in the phase of crack evolution, which means there is no extra energy dissipation until rock strength reaches the peak[\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eDue to the heterogeneity of rock materials, not all external work is converted into elastic strain energy and stored inside the material, and it is accompanied by other forms of energy dissipation, such as elastic strain energy[\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e], acoustic emission[\u003cspan additionalcitationids=\"CR30\" citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e] that releases energy in the form of elastic waves phenomenon: deformation energy[\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e], which causes the internal state of the material to change, thermal energy[\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e], and so on. This portion of the energy is primarily used to reconstruct the internal stress field distribution, as well as to generate and decimate micro-cracks, which have been observed by the acoustic emission study of rock specimens under uniaxial loading[\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e, \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e, \u003cspan additionalcitationids=\"CR30 CR31\" citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e, \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e, \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eThe uniaxial compression process is accompanied by a certain level of AE ringing count events before approximately 50% of compressive strength, and the occurrence of the events is random[\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. Xie Heping[\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e] assumed that there is no heat exchange between the rock and the outside in the process, according to the first law of thermodynamics, its form is given in the following:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$U={U^e}+{U^d}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$${U^e}=\\frac{1}{2}({\\sigma _1}{\\varepsilon _1}+{\\sigma _2}{\\varepsilon _2}+{\\sigma _3}{\\varepsilon _3})=\\frac{1}{{2E}}[\\sigma _{1}^{2}+\\sigma _{2}^{2}+\\sigma _{3}^{2} - 2\\nu ({\\sigma _1}{\\text{ }}{\\sigma _2}+{\\sigma _1}{\\sigma _3}+{\\sigma _3}{\\sigma _2})]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eU\u003c/em\u003e is the total energy density of external workmanship, \u003cem\u003eU\u003c/em\u003e\u003csup\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sup\u003e is the elastic strain energy density, and \u003cem\u003eU\u003c/em\u003e\u003csup\u003e\u003cem\u003ed\u003c/em\u003e\u003c/sup\u003e is the dissipated energy density.\u003c/p\u003e \u003cp\u003eThe one characteristic curve of uniaxial compression stress and strain obtained, shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, is used as an example to illustrate the energy evolution of the compacted and the elastic deformation stage before the crack initiation stress threshold.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Measures of elastic strain energy density\u003c/h2\u003e \u003cp\u003eFour assumptions are considered in this paper, that: \u003cem\u003e(i)\u003c/em\u003e the rock material is perfectly elastic (no energy dissipation in elastic deformation and all elastic strain energy released after force removed) ; \u003cem\u003e(ii)\u003c/em\u003e define the unit volume \u003cem\u003eV\u003c/em\u003e and the unit has a closed surface \u003cem\u003eS\u003c/em\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\delta W\\)\u003c/span\u003e\u003c/span\u003eand \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\delta U\\)\u003c/span\u003e\u003c/span\u003eare the work of the external force and the increment of internal energy for unit volume, respectively; \u003cem\u003e(iii)\u003c/em\u003e the external force is applied constantly, regardless of the effect of strain rate, to ensure that the unit body is in equilibrium at any time;\u003cem\u003e(iv)\u003c/em\u003e the kinetic energy changes are ignored; These are guaranteed to be a quasi-static loading in uniaxial compression.\u003c/p\u003e \u003cp\u003eBased on the above assumptions, the stress tensor is decomposed into spherical and deviatoric stress tensors; the strain tensor is decomposed into spherical and deviatoric strain tensors, that is, strain energy is decomposed into the energy of volume \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({W_1}\\)\u003c/span\u003e\u003c/span\u003eand the energy of shape change\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({W_2}\\)\u003c/span\u003e\u003c/span\u003e, the generalized form given in the following, respectively.\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$${W_1}=\\iiint_{V} {\\left( {{X_u}+{Y_v}+{Z_w}} \\right)}dV$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$${W_2}=\\iint_{S} {\\left( {{X^*}u+{Y^*}v+{Z^*}w} \\right)dS}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere\u003cspan class=\"InlineEquation\"\u003e\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003c/span\u003edefined as the external force on per unit volume element, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X^*}\\)\u003c/span\u003e\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Y^*}\\)\u003c/span\u003e\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Z^*}\\)\u003c/span\u003e\u003c/span\u003e are the internal force on the surface of the unit element, the external force work \u003cem\u003eW\u003c/em\u003e is entirely transformed into elastic strain energy and is defined as:\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$W={W_1}+{W_2}=\\iiint_{V} {\\left( {{X_u}+{Y_v}+{Z_w}} \\right)}dV+\\iint_{S} {\\left( {{X^*}u+{Y^*}v+{Z^*}w} \\right)dS}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe work done by the external force acting on the unit body to produce a small displacement\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\delta W\\)\u003c/span\u003e\u003c/span\u003e, is expressed as:\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\delta W=\\delta {W_1}+\\delta {W_2}=\\iiint_{V} {{X_i}}\\delta {u_i}dV+\\iint_{S} {X_{i}^{*}}\\delta {u_i}dS$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAfter substituting into the balance equation and boundary conditions, it is known from the divergence theorem:\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\delta W=\\iiint_{V} {{\\sigma _{ij}}}\\delta {u_{i,j}}dV$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe increase of internal energy is found:\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$\\delta U=\\iiint_{V} {{\\sigma _{ij}}}\\delta {\\varepsilon _{i,j}}dV$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eKnown from Green\u0026rsquo;s formula, and let:\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$$~~\\frac{{\\partial {u_0}\\left( {{\\varepsilon _i}_{j}} \\right)}}{{\\partial {\\varepsilon _i}_{j}}}={\\sigma _{ij}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIntegrate the formula can easily find:\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$$\\int_{0}^{{{u_0}\\left( {{\\varepsilon _{ij}}} \\right)}} {d{u_0}} ={u_0}\\left( {{\\varepsilon _{ij}}} \\right) - {u_0}\\left( 0 \\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({u_0}\\left( {{\\varepsilon _{ij}}} \\right),{u_0}\\left( 0 \\right)\\)\u003c/span\u003e\u003c/span\u003e respectively indicate the elastic strain energy density after loading and before loading.\u003c/p\u003e \u003cp\u003eLet\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({u_0}\\left( 0 \\right)=0\\)\u003c/span\u003e\u003c/span\u003e, and the strain energy density is determined by its stress and strain increase, which can be defined as:\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$${u_0}({\\varepsilon _i}_{j})=\\int_{0}^{{{\\varepsilon _{ij}}}} {{\\sigma _{ij}}} d{\\varepsilon _{ij}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe unit volume strain energy density of a rock sample can be expressed as:\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e\n$$\\frac{{dW}}{{dV}}=\\int_{0}^{{{\\varepsilon _{ij}}}} {{\\sigma _{ij}}} d{\\varepsilon _{ij}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Characteristics of strain energy evolution and initiation energy\u003c/h2\u003e \u003cp\u003eAccording to Qingbin Meng et al.[\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e] energy self-inhibition evolution model and Zheng Zaisheng's [\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e] research on the nonlinear fitting formula of energy transfer in rock deformation under uniaxial compression with a certain stress level\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\sigma\\)\u003c/span\u003e\u003c/span\u003e. The rate of increment of the accumulated strain energy density was given[\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e, \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e, \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e] by:\u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e\n$$\\frac{1}{{{u^i}}}\\cdot \\frac{{d{u^i}}}{{d\\sigma }}={a_i}({u^c}^{i} - {u^i}^{0}) - {b_i}{u^i}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe formula \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({u^c}^{i}\\)\u003c/span\u003e\u003c/span\u003erepresents the energy density converted from external input driving energy, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({u^i}^{0}\\)\u003c/span\u003e\u003c/span\u003e is the lowest input strain energy density threshold, and, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({a_i}\\)\u003c/span\u003e\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({b_i}\\)\u003c/span\u003e\u003c/span\u003e is the coefficient, which respectively reflects the degree of energy conversion and inhibition.\u003c/p\u003e \u003cp\u003eIntegrate \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({u^i}\\)\u003c/span\u003e\u003c/span\u003e and find:\u003cdiv id=\"Equ14\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ14\" name=\"EquationSource\"\u003e\n$${u^i}=\\frac{{{a_i}({u^c}^{i} - {u^i}^{0}){e^{{a_i}}}{{^{{({u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{\\sigma }}}{{{c_1}{b_i}[1+{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{\\sigma }]}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e14\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewith\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({c_1}\\)\u003c/span\u003e\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003c/span\u003ethe integral constant.\u003c/p\u003e \u003cp\u003eBased on elasticity modulus\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E_i}\\)\u003c/span\u003e\u003c/span\u003e and formula Eq.\u0026nbsp;\u003cspan refid=\"Equ13\" class=\"InternalRef\"\u003e13\u003c/span\u003e, find:\u003cdiv id=\"Equ15\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ15\" name=\"EquationSource\"\u003e\n$${E_i}=\\frac{{{\\sigma ^2}}}{{{u^i}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e15\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eDerivative and let the first derivative be 0 as follows:\u003cdiv id=\"Equ16\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ16\" name=\"EquationSource\"\u003e\n$$\\frac{{d{E_i}}}{{d\\sigma }}=\\frac{{2\\sigma {u^i} - {{\\left( {{u^i}} \\right)}^\\prime }{\\sigma ^2}}}{{{{\\left( {{u^i}} \\right)}^2}}}=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e16\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eEq.\u0026nbsp;\u003cspan refid=\"Equ14\" class=\"InternalRef\"\u003e14\u003c/span\u003e substituted and organized, can find:\u003cdiv id=\"Equ17\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ17\" name=\"EquationSource\"\u003e\n$${\\sigma _i}=\\frac{2}{{{a_i}\\left( {{u^c}^{i} - {u^i}^{0}} \\right) - {b_i}{u^i}}}=\\frac{2}{{{a_i}\\left( {{u^c}^{i} - {u^i}^{0}} \\right) - \\frac{{{a_i}({u^c}^{i} - {u^i}^{0}){e^{{a_i}}}{{^{{({u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{{\\sigma _i}}}}}{{1+{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{{\\sigma _i}}}}}}}=\\frac{{2{a_i}\\left( {{u^c}^{i} - {u^i}^{0}} \\right)}}{{1+{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{{\\sigma _i}}}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e17\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAfter being simplified, when the elastic modulus reaches the maximum value, the strain energy accumulated\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({u_i}\\)\u003c/span\u003e\u003c/span\u003eand Stress\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma _i}\\)\u003c/span\u003e\u003c/span\u003eare obtained:\u003cdiv id=\"Equ18\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ18\" name=\"EquationSource\"\u003e\n$${u_i}=c{e^{ - \\frac{1}{2}}}{\\sigma _i}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e18\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAnd\u003cdiv id=\"Equ19\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ19\" name=\"EquationSource\"\u003e\n$${\\sigma _i}=\\frac{{2{a_i}\\left( {{u^c}^{i} - {u^i}^{0}} \\right)}}{{1+{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{{\\sigma _i}}}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e19\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewith \u003cspan class=\"InlineEquation\"\u003e\u003c/span\u003ethe integral constant.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e2.4 Energy digestion index (EDI)\u003c/h2\u003e \u003cp\u003eFrom the concept of strain energy density, with isothermal conditions, the unit volume strain energy density can be expressed as:\u003cdiv id=\"Equ20\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ20\" name=\"EquationSource\"\u003e\n$$\\frac{{dW}}{{dV}}=\\int_{0}^{{{\\varepsilon _{ij}}}} {{\\sigma _{ij}}} d{\\varepsilon _{ij}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e20\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe above formula shows that the strain energy density of a unit can be determined by the stress and strain of the unit. The elastic stage of the stress-strain curve obtained from the experiment is partially enlarged, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eWhen the rock sample is loaded to point B in the uniaxial compression test, the elastic strain reaches a maximum, and the rock cracks. The area AOBD can be used to express the external force input energy during this process. The sample material is generally thought to be in the elastic stage at this point[\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e, \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e, \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e]. When the load is unloaded at point B, the formed unloading curve essentially coincides with the loading curve, indicating that no new crack damage is generated as a result of external force and no energy is dissipated[\u003cspan additionalcitationids=\"CR41\" citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eThe strain energy digested by the rock sample is represented by the area AODBC. Area ACB represents the portion of the external force's work that has not been converted into strain energy. At this point, the energy \u003cspan class=\"InlineEquation\"\u003e\u003c/span\u003egenerated by the external force can be divided into two parts: the strain energy \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({U^e}\\)\u003c/span\u003e\u003c/span\u003econverted and digested by the rock sample, and the energy loss \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({U^a}\\)\u003c/span\u003e\u003c/span\u003e the external force work cannot be converted into strain energy. The rock material's energy digestion index Eq.\u0026nbsp;\u003cspan refid=\"Equ20\" class=\"InternalRef\"\u003e20\u003c/span\u003e is established by analyzing the proportion of the energy digested in the total input energy. This formula is used to characterize the degree of response of the rock material to external loads and reflect the properties of the material itself. The energy digestion index EDI can be expressed as:\u003cdiv id=\"Equ21\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ21\" name=\"EquationSource\"\u003e\n$$D=\\frac{{{U^e}}}{U}=\\frac{{{U^e}}}{{{U^a}+{U^e}}} \\times 100\\%$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e21\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe above formula shows that when a rock material is subjected to an external load, the energy generated by the external force is not completely converted into elastic strain energy and stored inside the material due to differences in the rock material with microstructure and ability to withstand the outside; in this process, the mechanical properties of rock materials are characterized by the energy conversion and digestion ability.\u003c/p\u003e \u003c/div\u003e"},{"header":"3 Compressive strength tests","content":"\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Specimens preparation\u003c/h2\u003e \u003cp\u003eAll of the rock specimens examined were obtained from a Chinese coal mine. Sandstones with varying proportions of fine sandstone and siltstone as the main constituents exist. The specimens were processed into cylinders with a diameter of 50 mm and a length-to-diameter ratio of 2 (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e) using the ISRM-recommended method[\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e]. After careful grinding with a grinder and sandpaper, the top and bottom surfaces have flatness of less than 0.02 mm and parallelism of less than 0.05 mm. 9 specimens were tested due to the discreteness of the test data.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Laboratory uniaxial compression testing\u003c/h2\u003e \u003cp\u003eThe tests were carried out using the WDW-300 digital electro-hydraulic servo system's equipment (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e) and the specimen was placed in the test machine. The loading rate of 0.0015 mm/s, with continuous loading until the specimen breaks. The test is terminated when the residual strength equals the 60% peak strength or maximum load 240KN. In the test, 9 specimens were loaded uniaxially, and all the stress-strain data were recorded.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"4 Result and discussion","content":"\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e4.1 The result of Compressive strength test\u003c/h2\u003e \u003cp\u003eAxial stress-strain curves for the three kinds of rock specimens under uniaxial compression are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, which are similar in geometry, and the axial stress-axial strain behavior can be approximately divided into four typical stages, i.e., fissure closure, elastic deformation, crack growth and propagation, and strain-softening. In the process of uniaxial compression of the specimen, the stress-strain characteristics obtained are similar to those of C.D. Martin et al.[\u003cspan additionalcitationids=\"CR3\" citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e] in the stress-strain study of brittle rocks, including: the compaction closure stage, elastic deformation stage, stable crack expansion stage, crack accelerated expansion stage, and post-peak stage (shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e); the division of each stage of the failure process depends on 3 important stress thresholds: crack initiation\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\delta _{ci}}\\)\u003c/span\u003e\u003c/span\u003e,crack evolution\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\delta _{cd}}\\)\u003c/span\u003e\u003c/span\u003e,ultimate strength\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\delta _c}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eWhen the load is lower, the internal pores and cracks of the rock are compressed and sealed, and the process of the rock with better compactness is not obvious. The elastic properties of the rock body conform to Hooke's law, and the load reaches the threshold stress, which is 30\u0026ndash;50% of the peak strength[\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eContinue to load and enter the stage of stable crack propagation. After reaching the threshold stress that is 70\u0026ndash;80% of the peak strength, the crack enters rapid and unstable expansion, macroscopic shear failure occurs, and finally reaches the peak strength. The process of fissure compaction and closure exists in every test, but it is not significant. The reason is due to the heterogeneity of the natural rock sample itself and the existence of inevitable internal defects and fissures. The original fissures and fissures in this stage defects are further compacted, and the volume of the original crack defects decreases and tends to zero, and enters the elastic stage; in the elastic stage, some curves show an almost completely elastic state, and the stress-strain curves show linear elastic characteristics; while most of the curves are almost completely elastic. It appears as a downward concave phenomenon, that is, the slope increases as the load continues, the elastic deformation exhibits nonlinear characteristics, and the one-to-one correspondence between stress and strain is still maintained:\u003cdiv id=\"Equ22\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ22\" name=\"EquationSource\"\u003e\n$$\\sigma =E\\cdot \\varepsilon =f\\left( \\varepsilon \\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e22\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAt the stage of fissure closure, the uniaxial compression stress-strain curve of the sample shows the downward concave and the initial nonlinear deformation at low-stress levels, which results from the closure of some primary pores and voids in the sample with the increasing compression stress. At the stage of elastic deformation, the loading begins with the increase of axial stress after the primary fissure closure, and the elastic deformation dominates the stress-strain curve of the sample. However, the slope of the three curves, i.e., the elastic modulus, has a big difference, which results from the component of the tested rock material. The rigidity of granite samples is the largest, and that of sandstone samples is the smallest. At the stage of crack growth and propagation, the sample continuously produces the stress concentration near the tips of the initial crack, which results in the initiation and propagation of some new cracks. Therefore, the stress-strain curve departs from the elastic behavior and shows distinctly nonlinear deformation. For granite and limestone samples, this stage is short, while it is long for sandstone samples. At the stage of strain-softening, the macroscopic crack in the rock sample comes out rapidly and the post-peak behavior in the stress-strain curves shows a rapid drop. At this time, the sample can only support a lower axial stress, even though no axial stress. But the slope of the three curves, i.e., the softening modulus, in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e, differences in the end stage of elastic deformation of each specimen, and the post-elastic deformation curves of higher strength specimen are more abrupt than low strength ones.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Modulus of elasticity\u003c/h2\u003e \u003cp\u003eThe modulus of elasticity is variable. The point elastic modulus and strain increase with the stress adding; it is different from the constant elastic modulus of completely elastic material, that is, under external load, the external work is not 100% energy transfer, part of the energy is lost, and it causes the specimen such internal structure changes, resulting in strain hardening[\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e].\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Compressive strength and stiffness\u003c/h2\u003e \u003cp\u003eThrough the analysis of the stress-strain curves of different specimens, it is found that the initial stage of the test, that is, the pressure-tight and stage curves have good consistency, and there is no obvious sudden change, indicating that the integrity of the sample is better, and the original cracks are less, which meets the requirements of the sample. After the elastic deformation stage, the uniaxial compression stress-strain curves of different specimens have obvious differences, and the material properties have changed significantly, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe three groups of stress-strain curves all show variable elastic modulus in the pre-peak stage and the characteristics of failure after the sample reaches the ultimate strength as the external load continues to increase and there is a significant difference in strength. Compared with the SAN-B group, the SAN-A and SAN-C groups are more sensitive to the external load due to the smaller stress and strain under the action of the external load.\u003c/p\u003e \u003cp\u003eThe mechanical properties of different sandstone specimens are shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. It can be seen that the strength of different sandstone specimens under uniaxial compression is\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(6.7 - 12.1MPa\\)\u003c/span\u003e\u003c/span\u003e. According to the rock's Protodyakonov index \u003cem\u003eF\u003c/em\u003e, the specimens are divided into three groups: SAN-A, SAN-B, and SAN-C, with three specimens in each group, and identified in numbers 1, 2, and 3. The average of group SAN-A \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(F=0.7\\)\u003c/span\u003e\u003c/span\u003e, the average of group SAN-B\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(F=1.2\\)\u003c/span\u003e\u003c/span\u003e, and the average of group SAN-C \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(F=0.8\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eThe result of compressive strength test and dispersion\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"9\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eSpecimens\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eSerial Number\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eCracking Threshold\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eAvg. Value\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eUltimate Strength\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eAvg. Ultimate strength\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eAvg. Density\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003eModulus of Elasticity)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c9\"\u003e \u003cp\u003eProtodyakonov Coefficient\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eσ\u003csub\u003eci\u003c/sub\u003e /MPa\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMPa\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eσ\u003csub\u003ec\u003c/sub\u003e /MPa\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eMPa\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003ekg/m\u003csup\u003e3\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003eE\u003csub\u003ec\u003c/sub\u003e /MPaⅹ10\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c9\"\u003e \u003cp\u003e\u003cem\u003eF\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.168\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e6.710\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e4.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSAN-A\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.956\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e8.780\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.37\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1941.64\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e4.94\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e0.70\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.926\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e6.630\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e4.26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e7.139\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e11.30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e8.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSAN-B\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e6.579\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e7.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e14.39\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e12.11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2156.38\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e9.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1.20\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e7.376\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e10.65\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e8.78\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.452\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e9.140\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e6.65\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSAN-C\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.005\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e7.030\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e8.30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2011.37\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e5.33\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e0.80\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.085\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e8.740\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e4.95\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e4.4 Energy Digestion Index (EDI)\u003c/h2\u003e \u003cp\u003eAccording to the grouping of specimens with different Protodyakonov coefficients, the energy digestion index, sample cracking stress threshold, cracking energy threshold, and uniaxial compressive strength data of each specimen can be calculated, as shown in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eFrom Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, in the pre-cracking stage of the specimens with different Protodyakonov coefficients, the energy digestion index is relatively concentrated and the overall level is 78%-95% higher, the ultimate strength during failure is relatively discrete, ranging from 6.71 to 14.39Mpa; Group SAN-A samples average Protodyakonov coefficient \u003cem\u003eF\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.7, average energy digestion index D\u0026thinsp;=\u0026thinsp;93.57%; SAN-B group average Protodyakonov coefficient \u003cem\u003eF\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1.2, average energy digestion index D\u0026thinsp;=\u0026thinsp;80.90%; SAN-C group average Protodyakonov coefficient \u003cem\u003eF\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.8, average energy digestion index D\u0026thinsp;=\u0026thinsp;89.04%. It can be seen that the higher the EDI of the specimens, the lower the average hardness and the lower the ultimate strength.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSamples of EDI and energy intensity value\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"8\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eSpecimens\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eSerials Number\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEDI\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eCracking Stress Threshold\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eCracking Strain Energy Threshold\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eCracking Strain\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eUltimate strength\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003eCracking Threshold\u003c/p\u003e \u003cp\u003e/ Ultimate strength\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eD /%\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e/MPa\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e/MJ\u0026bull;m\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e/%\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e/MPa\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003e/%\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e92.643\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.168\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.297\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.76\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e6.710\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e47.21%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSAN-A\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e92.704\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.956\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.9669\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e8.780\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e33.67%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e95.357\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.926\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.206\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e6.630\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e44.13%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e82.697\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e7.139\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2.882\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.89\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e11.300\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e63.18%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSAN-B\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e81.075\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e6.579\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2.142\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.73\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e14.390\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e45.72%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e78.924\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e7.376\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2.685\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.84\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e10.650\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e69.26%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e85.159\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.452\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.469\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.67\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e9.140\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e48.71%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSAN-C\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e93.524\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.005\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.632\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e7.030\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e56.97%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e88.448\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.085\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.715\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.85\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e8.740\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e46.74%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eTo investigate the relationship between energy digestion characteristics, rock strength, and sensitivity to external forces, the EDI and uniaxial compressive strength of each sample were tested under quasi-static loading conditions[\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e]. The EDI and ultimate strength data are shown in a rectangular coordinate system, and the results are fitted, as illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e, the EDI of each rock sample and the uniaxial compression peak strength reflect the significant correlation between the strength of the rock sample and the energy digestion character. Concerning the law that the lower ultimate compressive strength with the higher the EDI in the elastic stage, the explanation for this phenomenon is that when the stress reaches the cracking stress, the energy obtained by the rock sample is all in the form of elastic strain energy accumulated in the material, and the internal damage can be ignored, the faster and more strain energy accumulates, the lower the strength of the rock, the easier it is to fail. The less of the accumulated elastic strain energy, the poorer the rock's ability to digest introduced energy, and the rock has better strength characteristics because large-scale crack formation and destruction require more energy.\u003c/p\u003e \u003cp\u003eWhen the EDI reaches the ultimate value, the rock cracking sharply, and the elastic strain energy accumulated in the rock material is suddenly released in a short time, accompanied by macroscopic manifestations such as acoustic emission and debris ejection.\u003c/p\u003e \u003c/div\u003e"},{"header":"5 Conclusions","content":"\u003cp\u003eUsing the uniaxial compression test, stress-strain curve analysis, and energy calculation analysis of a specific mine sandstone sample, this article comes to the following results:\u003c/p\u003e \u003cp\u003eThe test results for each sample clearly show a linear relationship between the fissure initiation threshold stress, expansion threshold stress, and peak strength; however, because each sample is unique, there are differences in the length of the strain process experienced at different stages.\u003c/p\u003e \u003cp\u003eThe strain energy absorbed by the rock sample per unit volume grows throughout the elastic deformation stage as the external load work increases continuously, exhibiting nonlinear growth features. First, there is an increase in strain energy digestion, followed by a drop. This phenomenon demonstrates how the rock sample material's capacity to absorb and store elastic strain energy is reflected in how incompletely the work produced by external forces is converted into strain energy.\u003c/p\u003e \u003cp\u003eThe article's suggested dimensionless parameter of the strain energy digest index in the elastic stage has a strong linear correlation with the rock sample's peak compressive strength. Additionally, it is evident from fitting that there is a clear linear relationship between the sample material and the strain energy that the rock sample digests during the early elastic deformation stage of uniaxial compression. The intensity increases with less strain energy being digested.\u003c/p\u003e \u003cp\u003eLastly, it must be clarified that full compaction and the absence of natural fissures cannot be guaranteed by the test volume and sample material. The article's conceptions and findings, which mostly highlight the fact that typical rock materials frequently have a linear elastic stage, are still preliminary. The primary cause of the observed nonlinearity in the measurement is the loss of entropy; however, quantifying the energy lost in this process is a challenging task due to its complexity. The energy difference between the beginning of the elastic phase and the end indicates the energy dissipation and loss. Second, the findings of the experimental investigation offer insightful information about the strength and deformation properties of rock materials, and they examine the impact of these properties on their uniaxial strength.\u003c/p\u003e \u003cp\u003eTo give further references for stability evaluation, the follow-up work will examine the energy digestion law of the in-situ rock mass in comparison to the energy loss properties of the rock material and the energy discrimination criterion for strain energy digestion.\u003c/p\u003e \u003cp\u003e \u003cb\u003eThe study's findings show that\u003c/b\u003e:\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eThe material absorbs and accumulates strain energy under the influence of external forces, displaying a non-linear accumulation in a changing modulus of elasticity.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eThe pace at which strain energy is absorbed by the rock material determines the creation of fractures, and when this rate of growth in strain energy digestion achieves its maximum, the material's growing rate of variable elasticity modulus also reaches its maximum.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eThe mechanical qualities of the rock material are inversely correlated with the energy digestion index; that is, the lower the rock strength, the higher the energy digestion index of the rock material.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eThe correlation between the energy digestion index and the sensitivity of the rock material to external loading is not significant.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eData availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAll the data generated and/or analyzed during the current study is presented in the paper.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgments\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis study was supported by the National Key Research and Development Program of China (No. 2020YFC1909801; 2020YFC1909101). We would like to thank anonymous reviewers for their detailed and constructive comments which helped us improve the quality of the manuscript.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eExperiments were conceived and designed by J.C. and Q.W. Samples were collected by Q.W. Data were analyzed by L.Y, E.M., and L.H.L. The manuscript was written by L.Y., and Y.F.J. reviewed the manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting Interests:\u003c/strong\u003e The authors declare no competing interests.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAdditional Information\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eCorrespondence and requests for materials should be addressed to L.Y.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003ePublisher\u0026rsquo;s note:\u003c/strong\u003e Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eGriffith, A.A., VI. The phenomena of rupture and flow in solids. Philosophical transactions of the royal society of london. Series A, containing papers of a mathematical or physical character, 1921. 221(582-593): p. 163-198.\u003c/li\u003e\n\u003cli\u003eMartin, C.D. and N.A. Chandler. The progressive fracture of Lac du Bonnet granite. 1994: Elsevier.\u003c/li\u003e\n\u003cli\u003eMartin, C.D., Seventeenth Canadian geotechnical colloquium: the effect of cohesion loss and stress path on brittle rock strength. Canadian Geotechnical Journal, 1997. 34(5): p. 698-725.\u003c/li\u003e\n\u003cli\u003eCai, M., et al., Generalized crack initiation and crack damage stress thresholds of brittle rock masses near underground excavations. International Journal of Rock Mechanics and Mining Sciences, 2004. 41(5): p. 833-847.\u003c/li\u003e\n\u003cli\u003eHui, L., et al., Analysis of water and ice content and damage characteristics of the frozen rock during freezing based on the three-valued segmentation of CT images. Journal of Mining and Safety Engineering, 2016. 33(6): p. 1130.\u003c/li\u003e\n\u003cli\u003eLi, J., et al., NMR research on deterioration characteristics of microscopic structure of sandstones in freeze\u0026ndash;thaw cycles. Transactions of Nonferrous Metals Society of China, 2016. 26(11): p. 2997-3003.\u003c/li\u003e\n\u003cli\u003eJin, Y., et al., Simulation of NMR response of microfractures based on digital rock technology. Geoenergy Science and Engineering, 2023. 227: p. 211876.\u003c/li\u003e\n\u003cli\u003eGiwelli, A., et al. In-situ nuclear magnetic resonance detection and visualisation of hydraulic fracture: A proof of concept. 2023: ARMA.\u003c/li\u003e\n\u003cli\u003eWu, G., D.Y. Wang and S.T. Zhai, Acoustic emission characteristics of sandstone after high temperature under uniaxial compression. Rock Soil Mech, 2012. 33(011): p. 3237-3242.\u003c/li\u003e\n\u003cli\u003eZhi-hui, H., et al., Acoustic emission characteristics analysis of sandstone under uniaxial compression. Nonferrous Metals Science and Engineering, 2014. 5(1): p. 91-94.\u003c/li\u003e\n\u003cli\u003eXiao, F.K., et al., Acoustic emission (AE) characteristics of fine sandstone and coarse sandstone fracture process under tension-compression-shear stress. Chinese Journal of Rock Mechanics and Engineering, 2016. 35(S2): p. 3458-3472.\u003c/li\u003e\n\u003cli\u003eNiu, Y., Y. Hu and J. Wang, Cracking characteristics and damage assessment of filled rocks using acoustic emission technology. International Journal of Geomechanics, 2023. 23(4): p. 04023013.\u003c/li\u003e\n\u003cli\u003eCao, K., et al., A comprehensive model for evaluating infrared radiation and acoustic emission characteristics of sandstone fracture. Engineering Fracture Mechanics, 2023. 283: p. 109217.\u003c/li\u003e\n\u003cli\u003eMelchiorre, J., et al., Acoustic Emission and Artificial Intelligence Procedure for Crack Source Localization, in Sensors. 2023.\u003c/li\u003e\n\u003cli\u003eKhadivi, B., et al., Characterizing the cracking process of various rock types under Brazilian loading based on coupled acoustic emission and high-speed imaging techniques. International Journal of Rock Mechanics and Mining Sciences, 2023. 168: p. 105417.\u003c/li\u003e\n\u003cli\u003eDong, L., et al., Uncertainty investigation for the classification of rock micro-fracture types using acoustic emission parameters. International Journal of Rock Mechanics and Mining Sciences, 2023. 162: p. 105292.\u003c/li\u003e\n\u003cli\u003eShao, J., Q. Zhu and K. Su, Modeling of creep in rock materials in terms of material degradation. Computers and Geotechnics, 2003. 30(7): p. 549-555.\u003c/li\u003e\n\u003cli\u003eSalari, M.R., et al., A coupled elastoplastic damage model for geomaterials. Computer methods in applied mechanics and engineering, 2004. 193(27-29): p. 2625-2643.\u003c/li\u003e\n\u003cli\u003eTsoutrelis, C.E. and G.E. Exadaktylos, Effect of rock discontinuities on certain rock strength and fracture energy parameters under uniaxial compression. Geotechnical \u0026amp; Geological Engineering, 1993. 11: p. 81-105.\u003c/li\u003e\n\u003cli\u003eNara, Y. and K. Kaneko, Sub-critical crack growth in anisotropic rock. International Journal of Rock Mechanics and Mining Sciences, 2006. 43(3): p. 437-453.\u003c/li\u003e\n\u003cli\u003eZhou, X.P., Triaxial compressive behavior of rock with mesoscopic heterogenous behavior: Strain energy density factor approach. Theoretical and Applied Fracture Mechanics, 2006. 45(1): p. 46-63.\u003c/li\u003e\n\u003cli\u003eXie, H., et al., Energy analysis of rock failure. Yanshilixue Yu Gongcheng Xuebao/Chin. J. Rock Mech. Eng., 2005. 24(15): p. 2603-2608.\u003c/li\u003e\n\u003cli\u003eXie, H.P., et al., Energy mechanism of deformation and failure of rock masses. Chinese Journal of Rock Mechanics and Engineering, 2008. 27(9): p. 1729-1740.\u003c/li\u003e\n\u003cli\u003eZhang, P., et al., Damage constitutive model of uniaxially compressed coal material considering energy dissipation. Journal of Materials Research and Technology, 2023. 27: p. 920-931.\u003c/li\u003e\n\u003cli\u003eWang, G., et al., Investigations on the macro‐meso mechanical properties and energy dissipation mechanism of granite shear fracture under dynamic disturbance. International Journal for Numerical and Analytical Methods in Geomechanics, 2023. 47(13): p. 2364-2384.\u003c/li\u003e\n\u003cli\u003eXiao, F., et al., Relationship between hysteresis loop and elastoplastic strain energy during cyclic loading and unloading. Chinese Journal of Rock Mechanics and Engineering, 2014. 33(9): p. 1791-1797.\u003c/li\u003e\n\u003cli\u003eZhang, Z.Z. and F. Gao, Research on nonlinear characteristics of rock energy evolution under uniaxial compression. Chinese Journal of Rock Mechanics and Engineering, 2012. 31(6): p. 1198-1207.\u003c/li\u003e\n\u003cli\u003eMeng, Q., et al., Research on non-linear characteristics of rock energy evolution under uniaxial cyclic loading and unloading conditions. Environmental Earth Sciences, 2019. 78: p. 1-20.\u003c/li\u003e\n\u003cli\u003eYu, Y., et al., Energy evolution and acoustic emission characteristics of uniaxial compression failure of anchored layered sandstone. Frontiers in Earth Science, 2022. 10: p. 841598.\u003c/li\u003e\n\u003cli\u003eSun, Y., F. Yu and J. Lv, Experimental Study on Acoustic Emission Characteristics of Granite and Sandstone under Uniaxial Compression. Geofluids, 2023. 2023.\u003c/li\u003e\n\u003cli\u003eWang, Z., et al., Failure behaviour and acoustic emission characteristics of different rocks under uniaxial compression. Journal of Geophysics and Engineering, 2020. 17(1): p. 76-88.\u003c/li\u003e\n\u003cli\u003eWen, T., et al., Energy Analysis of the Deformation and Failure Process of Sandstone and Damage Constitutive Model. KSCE Journal of Civil Engineering, 2019. 23(2): p. 513-524.\u003c/li\u003e\n\u003cli\u003eLi, T., et al., Nonlinear behavior and damage model for fractured rock under cyclic loading based on energy dissipation principle. Engineering Fracture Mechanics, 2019. 206: p. 330-341.\u003c/li\u003e\n\u003cli\u003eYun, C., J. Yan and C. Mian, A rock brittleness evaluation method based on energy dissipation. Chinese Journal of Theoretical and Applied Mechanics, 2015. 47(6): p. 984-993.\u003c/li\u003e\n\u003cli\u003eJin, F.N., M.R. Jiang and X.L. Gao, Defining damage variable based on energy dissipation. Chinese Journal of Rock Mechanics and Engineering, 2004. 23(12): p. 1976-1980.\u003c/li\u003e\n\u003cli\u003eZheng, Z., Energy transfer process during rock deformation. Chineses Science(Chemistry, Life Sciences \u0026amp; Geosciences), 1991(1): p. 104-117.\u003c/li\u003e\n\u003cli\u003eWang, C., et al., Stress\u0026ndash;Energy Mechanism for Rock Failure Evolution Based on Damage Mechanics in Hard Rock. Rock Mechanics and Rock Engineering, 2020. 53(3): p. 1021-1037.\u003c/li\u003e\n\u003cli\u003eHuang, D., R.Q. Huang and Y.X. Zhang, Experimental investigations on static loading rate effects on mechanical properties and energy mechanism of coarse crystal grain marble under uniaxial compression. Chinese Journal of Rock Mechanics and Engineering, 2012. 31(2): p. 245-255.\u003c/li\u003e\n\u003cli\u003eLi, X., Y. Zuo and C. Ma, Failure criterion of strain energy density and catastrophe theory analysis of rock subjected to static-dynamic coupling loading. Yanshilixue Yu Gongcheng Xuebao/Chin. J. Rock Mech. Eng., 2005. 24(16): p. 2814-2824.\u003c/li\u003e\n\u003cli\u003eWang, J.B., et al., Analysis of energy evolution characteristics of salt rock under different loading rates. Frontiers in Earth Science, 2022. 10: p. 829185.\u003c/li\u003e\n\u003cli\u003eShekov, V. and G. Kolesnikov, Differential Energy Criterion for Brittle Fracture: Conceptualization and Application to the Analysis of Axial and Lateral Deformation in Uniaxial Compression of Rocks. Materials, 2023. 16(13): p. 4875.\u003c/li\u003e\n\u003cli\u003eWang, W., et al., Mechanical behavior and energy mechanism of salt rock with prefab fissures under uniaxial compression. Journal of Chongqing University, 2016. 39(5): p. 103-113.\u003c/li\u003e\n\u003cli\u003eFairhurst, C.E. and J.A. Hudson, Draft ISRM suggested method for the complete stress-strain curve for intact rock in uniaxial compression. International journal of rock mechanics and mining sciences (1997), 1999. 36(3): p. 279-289.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Crack initiation, Energy digestion Index, Energy evolution, Rock strength, Rock mechanics","lastPublishedDoi":"10.21203/rs.3.rs-4198169/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4198169/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe stress intensity factors with the kinds of critical models are usually obtained from the strength test on laboratory specimens to characterize the rock strength and fracture-ability. However, crack initiation phenomenology for rock material has not been comprehensively understood. In this study, energy theory\u0026mdash;an essential character of material property changes\u0026mdash;is drawn to evaluate rock crack initiation and study the correlation between rock strength and energy digestion. Uniaxial compression test was conducted on different hardness coefficients rock samples under quasi-static loading conditions and the results of energy evolution, non-linear cumulative digestion, and stored ultimate energy were obtained and compared with a series of results for sandstone. A novel algorithm for the evaluation of rock crack initiation has been proposed and the concept of energy digestion index (EDI) using the ratio of ultimate energy over accumulated energy has been developed to characterize the rock strength. The result shows that crack initiation is related to the maximum growth rate of energy cumulation. Simultaneously, the increasing rate of variable elasticity modulus of rock material gets the peak. The defined EDI has a negative correlation with the rock material's strength and mechanical properties. The relationships between final strength and EDI at crack initiation were also discussed and an assessment of the remaining strength life is introduced using the predicted value by fitting the EDI, leading to a new understanding and exploration for in-situ monitoring of rock damage and failure prediction.\u003c/p\u003e","manuscriptTitle":"Rock Crack Initiation Triggered by Energy Digestion","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-04-17 11:56:49","doi":"10.21203/rs.3.rs-4198169/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2024-05-13T03:31:32+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-05-11T19:56:47+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-04-28T07:58:12+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"1793e0a1-c566-4560-973c-a5a6af8cf658","date":"2024-04-22T19:16:55+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"ca5838b4-f359-4cb3-b1fc-55c9457e7c70","date":"2024-04-17T16:40:10+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"02744841-8c34-4404-87eb-c2b9f25f3428","date":"2024-04-17T14:09:52+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"e98e375a-d04b-424a-b5bd-51aa46798020","date":"2024-04-17T08:48:51+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-04-17T08:43:11+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-04-17T08:36:49+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2024-04-17T07:17:12+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-04-13T10:09:24+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2024-04-01T04:53:59+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"0a2170d0-ba95-4a5b-a5c9-c8cb99fd5cf2","owner":[],"postedDate":"April 17th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2024-07-03T00:29:49+00:00","versionOfRecord":{"articleIdentity":"rs-4198169","link":"https://doi.org/10.1038/s41598-024-66226-3","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2024-07-02 00:29:49","publishedOnDateReadable":"July 2nd, 2024"},"versionCreatedAt":"2024-04-17 11:56:49","video":"","vorDoi":"10.1038/s41598-024-66226-3","vorDoiUrl":"https://doi.org/10.1038/s41598-024-66226-3","workflowStages":[]},"version":"v1","identity":"rs-4198169","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4198169","identity":"rs-4198169","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}