Statistical Damage Theory Research Articles

Evaluating Rock Brittleness Through a Novel Statistical Damage Theory-Based Index: A Case Study on Sandstones

Evaluating the brittleness of rocks or rock masses is a fundamental problem in geotechnical engineering. This study proposed a new index that expresses brittleness as the rate of damage development in rock. The brittleness index was derived from statistical damage theory. It depends on the four material parameters, i.e., the peak strain, peak strength, Poisson’s ratio and elastic modulus. The validity of the proposed brittleness index was confirmed through two case studies, including triaxial compression test results for coals subjected to varying confining pressures and for sandstones at various temperatures. Uniaxial compression experiments were then performed on rock-like materials to examine the effects of model size and joint dip angle on rock brittleness using the proposed brittleness index. Results show that the brittleness of the jointed specimens varies in a complex pattern with the model size and joint dip angle. Generally, the brittleness index initially reduces and then grows with the increasing joint dip angle, and larger specimens tend to be more brittle. Furthermore, large specimens containing horizontal or vertical joints are particularly susceptible to brittle damage. The proposed brittleness index has merits such as a clear physical meaning and simple expression, making it a valuable tool for evaluating rock brittleness.

Study on mechanical properties and damage mechanism of alkali-activated slag concrete

The cement industry has become one of the important sources of greenhouse gas emissions, and the alkali-activated slag (AAS) has the same mechanical properties and lower carbon emissions as cement in the production process. Therefore, it can be used as an alternative binder for low-carbon buildings. The effect of alkali concentration (mass ratio of Na2O to binder) on the macro-mechanical properties and meso-damage mechanism of alkali-activated slag concrete (AASC) under uniaxial compression was investigated in this work. Six kinds of concrete with alkali concentration of 2 %, 4 %, 6 %, 8 %, 10 % and 12 % were prepared. The microscopic characteristics of AASC were characterized by nuclear magnetic resonance (NMR), scanning electron microscopy (SEM), x-ray diffraction (XRD) and x-ray energy dispersive spectroscopy (EDS); the meso-damage evolution mechanism of AASC was discussed by statistical damage theory. The results showed that the peak stress and elastic modulus of AASC specimens increased, and the microstructure became denser with the increase of alkali concentration. However, the mechanical properties of the specimen deteriorated to some extent when the alkali concentration exceeded 6 %; for example, the peak stress decreased by 24.00 MPa when the alkali concentration was increased from 6 % to 12 % at the age of 28d. Based on the statistical damage theory, the effective stress concept was introduced to quantitatively analyze the mesoscopic damage evolution of AASC. Both fracture and yielding damage modes were considered. By exploring the regular changes of the mesoscopic parameters, the connection between the macroscopic mechanical properties and the mesoscopic damage mechanism was established. Based on the above mechanical properties and mesoscopic damage results, the alkali concentration of 6 % can be used as the best mixture proportions of AASC.

Study on multi-scale dynamic mechanical properties of metakaolin modified recycled aggregate concrete under coupling conditions of sulfate attack and dry-wet cycles

To explore the potential of modified recycled aggregate concrete for use in construction projects, the evolution of its properties in the combined effects of environment and loading was investigated. This paper considered various factors, including the replacement amount of metakaolin (0 %, 15 %), dry-wet cycles (0, 20, 60, and 120 cycles), and the dynamic strain rate (ε˙ = 10−5/s, 10−4/s, 10−3/s, and 10−2/s), and conducted uniaxial compression, scanning electron microscopy (SEM), and X-ray diffraction (XRD) tests. The experimental findings demonstrated that the peak stress and elastic modulus of recycled aggregate concrete (RAC) exhibited a pattern of initial increase followed by a decrease before and after 20 dry-wet cycles, a concurrent upward trend was observed in both parameters with an escalation in strain rate and the incorporation of metakaolin (MK), and no uniform trend was observed for the peak strain. The test results from SEM and XRD revealed that the persistent formation of sulfate crystals and expansive substances like gypsum and ettringite were the primary factors contributing to the deterioration of specimen damage as cycle counts escalated. A quantitative analysis was conducted to investigate the mesoscopic damage mechanism associated with the dynamic mechanical properties and durability evolution process of metakaolin-based RAC under sulfate erosion. The correlation between the macroscopic mechanical properties and the microscopic changes of RAC was explored using statistical damage theory.

Study on the dynamic constitutive model of coal at different depths in Pingdingshan mining area

The environment of deep in-situ occurrence is complex, and secondary dynamic disasters occur frequently. Exploring the dynamic constitutive models of coal at different depths is the foundation for deep resource development, which can help promote the safe and efficient mining of coal resources at different depths and reveal the mechanism of sudden dynamic disasters. Dynamic mechanical testing of coal at different depths of the Pingdingshan mine (China) was carried out using a split Hopkinson pressure bar system with confining pressure. On the basis of statistical damage theory, dynamic constitutive models of coal at different depths were explored, and the theoretical models were compared and verified using the experimental results. An expression relating to the infinitesimal element strength of coal and loading rate was derived, and the parameters σ1dp, Ed, and ε1dp were determined. Inversion verification of the model showed that the experimental results agree well with the theoretical results prior to the peak stress. These research results are expected to be applied to the exploitation of deep resources, laying a foundation for the law of deep earth science and the construction of deep engineering.

Mechanical behavior and damage constitutive relationship of basalt-brucite hybrid fiber reinforced low-heat cement concrete

In this study, the mechanical properties and microstructure of basalt-brucite hybrid fiber reinforced low heat cement concrete was explored. The results indicated that the mechanical properties of low heat cement concrete first increase and then decrease with the increase of hybrid fibers volume fraction, and the cubic compressive strength, splitting tensile strength, and axial compressive strength of the optimal strengthening group, that volume fraction is 0.20% for basalt fiber and 0.15% for brucite fiber, respectively, increased by 40.16%, 70.61%, and 56.16%, respectively. Under a reasonable volume fraction of hybrid fibers, the mechanical properties of concrete could be significantly improved. By comparing the bonding effect between fibers and concrete, fibers exhibited different distribution patterns in concrete matrix, that could improve the compactness of concrete matrix through filling effect, optimize internal structure, and delay the connection and expansion of microcracks. The causes of the bonding effect between fibers and concrete was discussed from the perspective of chemical bonding effects. Based on the above analysis, the reinforcement mechanism of hybrid fibers on concrete was explored. Based on statistical damage theory, a damage constitutive equation for hybrid fiber reinforced low heat cement concrete considering fiber volume fraction was established by modifying the model parameters, and the model curve fitted well with the experimental curve.

The Dynamic Mechanical Properties and Damage Constitutive Model of Ultra-High-Performance Steel-Fiber-Reinforced Concrete (UHPSFRC) at High Strain Rates.

A high strain rate occurs when the strain rate exceeds 100 s-1. The mechanical behavior of materials at a high strain rate is different from that at middle and low strain rates. In order to study the dynamic compressive mechanical properties of ultra-high-performance steel-fiber-reinforced concrete (UHPSFRC) at high strain rates, an electro-hydraulic servo universal testing machine and a separate Hopkinson pressure bar (SHPB) with a diameter of 120 mm were used, respectively. A quasi-static compression test (strain rate 0.001 s-1) and impact compression test with a strain rate range of 90~200 s-1 were carried out to study the failure process, failure mode, and stress-strain curve characteristics of UHPSFRC at different strain rates and quantify the strain rate strengthening effect and fiber toughening effect. Based on the statistical damage theory and energy conversion principle, a dynamic damage constitutive model considering the effects of strain rate and fiber content was constructed. The results showed that the rate correlation of UHPSFRC and the fiber toughening properties showed a certain coupling competition mechanism. When the fiber content was less than 1.5%, with an increase in the steel fiber content, the crack initiation and propagation time of the specimen was extended, and the strain rate sensitivity gradually decreased. When the fiber content was 2%, the impact compressive strength of the specimen was optimal. Compared with UHPC, the dynamic increase factor (DIF) of UHPSFRC was significantly lower. The dynamic damage constitutive model established in this paper, considering the influence of strain rate and fiber content, has a good applicability and can describe the mechanical behavior of UHPSFRC at a high strain rate.

Shear fracture behavior and damage constitutive model of rock joints considering the effect of pre-peak cyclic loading

The research on mechanical properties and fracture nalysed of rock joints under dynamic loading disturbance is essential for the stability analysis of rock mass engineering. In this work, a series of pre-peak cyclic shear tests considering the number of cycles and cycle amplitude are conducted for rock joints. Firstly, the shear fracture nalysed and damage evolution law of rock joints are investigated in detail. Then, based on the statistical damage theory, the novel damage constitutive model of rock joints under pre-peak cyclic shear loading is established assuming that the microscopic element damage of rock joints material obeys the Weibull distribution. Finally, the real-time response of acoustic emission and the failure mode of joint surface are nalysed. The results demonstrate that the peak and residual shear stress are positively correlated with the cyclic amplitude. The cumulative cyclic displacement grows with the increase in the number of cycles and cycle amplitude. Moreover, the developed damage model considering the effect of pre-peak cyclic loading is verified by comparing the shear test results with the theoretical values, which are in good agreement. The variation of the shear wear zone of joint surface with cyclic parameters is discussed by binarization calculations. The results of this work can provide a valuable theoretical basis for further research on the shear fracture mechanism of rock joints under dynamic loading disturbance and mitigating geological disasters.

Validation of a damage constitutive model based on logistic model dedicated to the mechanical behavior of the rock

The establishment of mechanical models of rocks can provide crucial guidance to the surrounding rocks stability analysis in the rock engineering. According to statistical damage theory and Mohr–Coulomb strength criterion, the Logistic model is used to yield the damage evolution equation and further establish statistical damage constitutive model for describing the whole failure process of rocks. Then, the rationality of the proposed damage model is proved by comparing with the different test curves, and damage evolution laws are summarized into four stages, including: slow-growth stage, rapid-growth stage, constant-speed-similar growth stage and speed-reducing growth stage. Physical meanings of model parameters r′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${r}{\\prime}$$\\end{document} and m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m$$\\end{document} are also discussed. Finally, to verify the application of this proposed model, this damage constitutive model is developed and implemented to simulate the tunnel excavation and analyze the stability of surrounding rocks by using Fortran, python and Abaqus. The displacement data on site is used to prove the superiority of proposed damage constitutive model compared with the existing ones in Abaqus. The research outcomes presented in this paper can also provide useful references for the theory and application of rock mechanics.

Study on mechanical properties and mesoscopic damage mechanism of metakaolin modified recycled aggregate concrete

This research explored the effect of water-binder ratio (w/b) and curing age on the mechanical properties, microstructure, and damage mechanism of modified recycled aggregate concrete (RAC) with metakaolin (MK). The consequence appeared that the mechanical properties of the specimens were positively correlated with curing age, but negatively correlated with w/b. In addition, the incorporation of MK increased the peak stress and peak strain of the specimens with w/b of 0.65 by an average of 17.46% and 8.42% at six ages. Through microscopic test, it was found that the mechanical properties of the specimen were related to the hydration reaction, and the continuously generated gel products made the matrix structure more compact. Based on the statistical damage theory, the influence of w/b, age and MK on the damage evolution of mesoscopic fracture and yield were quantitatively analyzed, and a correlation between macroscopic mechanical properties and the mesoscopic damage mechanism was established.

Study on the energy and damage characteristics of gas-containing coal under confining pressure unloading process

Research on energy and damage evolution patterns of gas-containing coal under constricting pressure unloading conditions is urgently needed in the process of deep underground mining and is crucial for understanding the mechanisms underlying coal and gas composite rockburst occurrences. Prior reaching peak stress, cyclic loading and unloading experiments were carried out on gas-containing coal specimens under varied confining pressures and unloading circumstances. According to the experimental findings, gas pressure significantly degrades the mechanical characteristics of coal specimens, with a higher gas pressure causing worsening of the mechanical properties. The degree of mechanical property deterioration in coal specimens caused by gas pressure steadily reduces as confining pressure rises, indicating that confining pressure has an inhibitory influence on the deterioration of mechanical characteristics brought on by gas. Additionally, the degradation of coal mechanical characteristics is a result of the unloading of confining pressures. The peak stress and elastic modulus of coal decrease as confining pressure unloading increases in magnitude. Prior to the peak stress, the capacity of coal to convert external work into elastic energy is mostly indicated by the compression storage energy coefficient, whereas the peak elastic energy predominantly denotes the upper bound of compressed storage elastic energy for coal. Based on the Weibull statistical damage theory and employing elastic energy as the distribution variable, a statistical damage model is proposed. Comparisons of coal damage evolution curves indicate that both the confining pressures unloading process and gas pressure exert a promoting effect on the damage evolution of coal.

A novel meso-damage constitutive model of rock under true triaxial stress with three-dimensional cracking strength, threshold and closure effect

Deep underground engineering is in a true three-dimensional stress state, and the adjustment of the three-dimensional stress state caused by engineering excavation will induce the fracture or even instability of the surrounding rock. However, three-dimensional mechanical model research suitable for the stability analysis of deep surrounding rock is very scarce. Therefore, a series of tests under different true triaxial stresses on two rocks (rhyodacite and marble) were conducted, and the characteristic strength (crack stable propagation initiation stress, crack unstable propagation initiation stress and peak strength) and deformation characteristics were further analyzed. After that, using the Lemaitre strain equivalence hypothesis and rock statistical damage theory, a new statistical damage constitutive model at true triaxial stress states was proposed, which introduced the three-dimensional strength criterion Modified Wiebols Cook to characterize the three-dimensional strength of the rock microelement. Therefore, the intermediate principal stress can be reasonably considered. The damage threshold, initial compaction effect and residual strength of the rock microelement at different true triaxial stress conditions were also considered. Then the relationships between the proposed model parameters and σ2 and σ3 were analyzed. Furthermore, sensitivity analysis of the influence of parameters m and F0 in proposed model on the shape of rock stress–strain curve and peak strength was also investigated. The comparison between the results predicted by proposed model and the experimental data shows that the new model established in this study can well simulate the prepeak and postpeak deformation characteristics of rock and the intermediate principal stress effect under true triaxial stress conditions.

Numerical Simulation Study on the Constitutive Model of Fully-Graded Concrete Based on Statistical Damage Theory

A statistical damage model (SDM) of fully-graded concrete was created using statistical damage theory, based on the mechanical properties of axial tension and axial compression of the material. The SDM considers two damage modes, fracture and yield, and explains the intrinsic connection between the mesoscopic damage evolution mechanism and the macroscopic nonlinear mechanical behavior of fully-graded concrete. The artificial bee colony (ABC) algorithm was used to obtain the optimal parameter combination through an intelligent search of parameters εa, εh, εb and H in the constitutive model by taking the test data as the target value, and the sum of the squares of the differences between the target value and the predicted value as the objective function. The SDM numerical simulation model of fully-graded concrete is proposed by compiling subroutines in FORTRAN by constructing two modules of data model and damage analysis. The numerical results under uniaxial and biaxial forces are in agreement with the experimental results, which verifies the accuracy of the program. The model also analyzes the characteristics of mesoscopic damage evolution and predicts the mechanical properties under triaxial forces. The results show that the proposed numerical simulation model can reflect the salient features for fully-graded concrete under uniaxial, biaxial and triaxial loading conditions, and the evolution law of mesoscopic parameters. Therefore, the proposed model serves as a basis for the refined finite element analysis of hydraulic fully-graded concrete structures and reveals the mesoscopic damage mechanism of concrete under different load environments.

Stability analysis of jointed rock mass in large underground caverns based on multi-scale analysis method

Random defects at different scales are the crucial cause of damage evolution and cascade transfer in rocks. For a large-scale underground cavern, the rock mesoscopic heterogeneity and macroscopic random joints are inevitable factors of rock stability, which needs to be further studied. In this study, a multi-scale analysis method for the stability of jointed rock mass in large underground caverns is proposed. The statistical damage theory is used to describe the mesoscopic heterogeneous rock damage. Monte Carlo simulation and measured joint geometric information are combined to reconstruct the macroscopic random joint network of rock mass. The integrated analysis of mesoscopic heterogeneity of rock and the macroscopic response of random jointed rock mass is performed by Rock Failure Process Analysis (RFPA) system. The research shows that the existence of joint affects the failure mode of rock mass and strengthens the heterogeneity. The representative elementary volume (REV) of jointed rock mass and its equivalent mechanical parameters are determined and used in the stability analysis of a large-scale underground cavern project. Results show a good agreement with the in-situ monitoring deformation of surrounding rock mass, which verifies the effectiveness of the rock mass multi-scale analysis method.

Constitutive Damage Model of Foamed Lightweight Concrete Using Statistical Damage Theory.

Foamed lightweight concrete has been applied in different fields of civil engineering because of its superior properties, but the related research considering internal pore damage is limited. Based on statistical damage theory and considering the uneven distribution of fracture damage and strength between the pores of light concrete, a damage constitutive model of foamed lightweight concrete was established based on the Weibull function. The parameters of the damage model were determined through a triaxial compression test, and the rationality was verified by combining the existing test data. Comparative tests show that the theoretical calculation results of the proposed statistical damage model of foamed light soil are consistent with the general trend of the experimental results, reflecting the value of the peak stress and strain and describing the overall development law of the stress and strain. The best fit was obtained when the confining pressure was 0.3 MPa and the density was 700 kg·m-3. The suggested damage constitutive method is highly applicable, which is of great significance to the microscopic mechanical analysis of foamed light concrete and the structural design in civil engineering.

Softening/Hardening Damage Model and Numerical Implementation of Seabed Silt-Steel Interface in Yellow River Underwater Delta

The interaction between soil and structure is a research hotspot in ocean engineering, and the shear performance of interfaces is an essential factor affecting the bearing capacity of offshore structures. Taking the Yellow River Underwater Delta as the research area, the Softening/Hardening damage model of the silt–steel interface and the determination method of model parameters are proposed based on the statistical damage theory. Through the interface monotonic shear test under the conditions of different normal stress, roughness and water content, the shear mechanical properties and volumetric deformation laws on the silt–steel interface are analyzed, and the damage model parameters are obtained. Finally, a FRIC subroutine for the damage model was developed based on ABAQUS. The research results indicate the following: (1) The interface between silt and steel exhibits two characteristics, softening/hardening and shear shrinkage/expansion, under different conditions. Roughness significantly impacts interfacial cohesion, while water content mainly affects the internal friction angle. (2) The softening model based on the classic rock damage model can better simulate the stress–strain relationship of the silt–steel interface under high normal stress and low water content. In contrast, the hardening model based on the classic hyperbola model can better simulate the stress–strain relationship under low normal stress and high water content. The calculated results of the softening/hardening model agree with the experimental results, and the model has 7 parameters. (3) The developed FRIC subroutine can effectively simulate the nonlinear mechanical behavior of the interface between silt and steel. The research results provide a reference for exploring the stability analysis of offshore structures considering interface weakening effects.

A Damage Constitutive Model for a Jointed Rock Mass under Triaxial Compression

Due to the different structural characteristics and in situ environment, jointed rock masses encounter different failure mechanisms. Because of the joints, jointed rock masses show peculiar characteristics such as anisotropy and weakening. To describe the deformation and failure mechanism in jointed rock masses, a novel damage constitutive model of rock mass is proposed here considering the geometric parameters and mechanical properties of joints. A total damage variable is derived on the basis of the strain equivalence hypothesis, which combines the Weibull statistical damage theory for the strength of the rock elements and the fracture mechanics model for joints. This new damage variable reflects the coupled damage inflicted by two damage states, one is initial damage induced by prefabricated joints and the other is joint damage and rock mesoscopic damage under loading. The damage evolution path revealed by the new damage variable corresponds to the evolution of mechanical properties and behaviors induced by changes in the rock mass structure. Afterward, the computational formulation scheme for the model parameters is deduced using the extremum method. The model parameters, m and F0, have a clear physical meaning. The parameter m describes the ductility–brittleness characteristics of jointed rock masses, and F0 reflects the level of strength. This model describes the effect of joints and strain on the evolution of rock mass damage, model parameters, deformation, and failure characteristics. The mechanical behavior of rock mass described by the damage model and the change law of model parameters are consistent with the published experimental results.

Mechanical damage model and brittleness index of frozen rocks based on statistical damage theory

Mechanical damage model and brittleness index of frozen rocks based on statistical damage theory

Study on macroscopic mechanical properties and mesoscopic damage mechanism of recycled concrete with metakaolin under sodium sulfate erosion environment

The sodium sulfate resistance of recycled concrete (RC) under dry-wet (DW) cycle was studied, and the mesoscopic damage mechanism of RC under uniaxial compression was revealed by using statistical damage theory. The substitution rate of metakaolin (MK) and the number of DW cycles were considered in the test variables. The microstructure of RC was characterized by acoustic emission (AE), scanning electron microscope (SEM), X-ray diffraction (XRD) and X-ray fluorescence spectrometer (XRF). The results showed that with the increase of the number of DW cycles, the peak stress and elastic modulus of the specimen increased at first and then decreased, and the peak strain and mass change rate tended to increase. When the cycle times extended from 0 to 120, the peak stress of MK0 increased from −37.2 MPa to −50.7 MPa and then decreased to −29.1 MPa. The deterioration degree and mechanical properties of RC were improved by adding MK. The peak stress of MK15 increased by 26.4% and 16.4% respectively when it was cycled for 0 and 120 times compared with that of MK0. Based on the statistical damage theory, it was considered that there were 2 mesoscopic damage modes of fracture and yield, and the effective stress dominated the damage evolution of specimen during the failure process. The peak state and critical state were distinguished, and the mean values of σcr/σp and εcr/εp were 0.884 and 1.392 respectively. This paper provided experimental support for improving the sulfate resistance of RC, and provided theoretical basis for the use of in erosive environments.

Study on mechanical properties and damage mechanism of recycled concrete containing silica fume in freeze–thaw environment

In order to study the uniaxial compressive mechanical properties and damage evolution of recycled aggregate concrete (RAC) containing silica fume (SF) under freeze–thaw cycles, the RAC with SF replaced partial cement was prepared. The replacement rates were 0% and 9%. The RAC were subjected to 0, 50, 75, 100, 125 freeze–thaw cycles and uniaxial compression tests. The microstructure, micro-crack propagation process and reaction products were observed by acoustic emission (AE) and scanning electron microscope (SEM). The statistical damage theory was used to explore the mesoscopic damage evolution mechanism of RAC. The results showed that the peak stress and modulus of elasticity of the specimen tended to decrease, the peak strain and mass loss rate gradually increased with the increase of the number of freeze–thaw cycles(N). The compressive strength of RAC with S = 0% decreases from −41.52MPa to −11.23MPa when N increases from 0 to 125. The addition of SF produced additional volcanic ash reactions, resulting in an increase in ettringite (AFt), which reduces the porosity of RAC and is able to increase the compressive strength at different N significantly. Compressive strength increases by 21.84% and 12.06% at N = 0 and 125, respectively. Considering the existence of two mesoscopic damage modes of fracture and yield, the deformation failure mechanism of RAC was analyzed from the perspective of effective stress. The all five characteristic parameters characterizing the mesoscopic damage show significant regular change with increase of the number of cycles. The relationship between the macroscopic mechanical properties of RAC and the mesoscopic damage mechanism was established by analyzing the evolution law for the characteristic parameters.

Experimental investigation on damage and seepage of red sandstone subjected to cyclic thermal and cold treatment

For geothermal mining projects in cold regions, the problems of surrounding rock and water damage caused by thermal-cold shocks have become the main factors affecting engineering stability and construction safety. In addition, the permeability and damage characteristics of the rock mass after the thermal-cold cyclic shock action are not yet known. Therefore, in this study, seepage tests of red sandstone during complete stress-strain after thermal-cold cyclic shock were carried out to analysis the change in permeability of red sandstone during progressive fracturing. Meanwhile, the permeability model considering damage was further constructed by using the damage constitutive relationship based on statistical damage theory. And found that peak stress and elastic modulus were found to decrease with temperature and cycles, and both initial and maximum permeability increase with temperature and cycles during its gradual fracture. Moreover, the theoretical curve calculated by the permeability model is not much different from the experimental curve, which can effectively characterize the relationship between permeability and damage. The research results can provide reference value for the damage assessment, repair and reinforcement of the wellbore surrounding rock structure after thermal-cold shock cycle in the geothermal engineering of cold regions.