Experimental and numerical study of the Kaiser effect in cyclic Brazilian tests with disk rotation

Sensitivity of the Kaiser effect to the deviations of the directions of σ₁-principal stress experienced by rock in successive loading cycles has an important impact on the application of this effect for stress measurements in rocks. The paper presents an analysis of the gradual Kaiser effect degradation with increasing deviation of the principal stress axes between loading cycles in Brazilian experiments. An experimental study was carried out to investigate the Kaiser effect in cyclic loading tests of disk specimens of a brittle limestone in diametrical compression with acoustic emission measurement. Tests were performed in which disks were loaded in two cycles without or with rotations between successive cycles. The rotation angle varied between 0° and 90°. The Kaiser effect became gradually less pronounced with increasing rotation angle, but remained detectable for angles <10°. Rotation by more than 10° resulted in complete disappearance of the effect. These experimental results were confirmed by numerical simulations using the displacement discontinuity method.     

Fracture toughness study for a brittle rock subjected to mixed mode I/II loading

A semi-disk specimen containing an angled edge crack has been used in the past for conducting fracture tests on a brittle rock named Johnstone [Fracture testing of a soft rock with semi-circular specimens under three-point bending. Part 2—mixed mode. Int J Rock Mech Min Sci Geomech Abstr 1994b;31(3):199–212]. The test specimen is appropriate for investigating brittle fracture when the rock samples are subjected to the combined effects of tension and shear along the crack line. However, the experimental results reported in Lim, Johnston, Choi, Boland [Fracture testing of a soft rock with semi-circular specimens under three-point bending. Part 2—mixed mode. Int J Rock Mech Min Sci Geomech Abstr 1994b;31(3):199–212.] are inconsistent with all of the well-known theoretical criteria available for predicting mixed mode brittle fracture. In this paper, a modified criterion is used to provide accurate predictions for the reported experimental results. The modified criterion makes use of a three-parameter model (based on K_I, K_II and T) for describing the crack tip stresses. It is shown that the non-singular stress term T has a significant role when the rock fracture tests are conducted on the semi-disk specimens.     

Generalized crack initiation and crack damage stress thresholds of brittle rock masses near underground excavations

The rock mass failure process is characterized by several distinct deformation stages which include crack initiation, crack propagation and coalescence. It is important to know the stress levels associated with these deformation stages for engineering design and practice.Extensive theoretical, experimental and numerical studies on the failure process of intact rocks exist. It is generally understood that crack initiation starts at 0.3 to 0.5 times the peak uniaxial compressive stress. In confined conditions, the constant-deviatoric stress criterion was found to describe the crack initiation stress level.Here, generalized crack initiation and crack damage thresholds of rock masses are proposed. The crack initiation threshold is defined by σ₁−σ₃=A σ_cm and the crack damage threshold is defined by σ₁−σ₃=B σ_cm for jointed rock masses, where A and B are material constants and σ_cm is the uniaxial compressive strength of the rock masses. For a massive rock mass without joints, σ_cm is equal to σ_cd, the long-term uniaxial strength of intact rock. After examining data from intact rocks and jointed rock masses, it was found that for massive to moderately jointed rock masses, the material constants A and B are in the range of 0.4 to 0.5, 0.8 to 0.9, respectively, and for moderately to highly jointed rock masses, A and B are in the range of 0.5 to 0.6, 0.9 to 1.0, respectively. The generalized crack initiation and crack damage thresholds, when combined with simple linear elastic stress analysis, assist in assessing the rock mass integrity in low confinement conditions, greatly reducing the effort needed to obtain the required material constants for engineering design of underground excavations.     

脆性岩石单轴循环加卸载试验及断裂损伤力学特性研究

 以向家坝砂岩单轴循环加卸载室内力学试验结果为基础,结合岩石内部微裂纹的细观力学分析,对脆性岩石单轴循环加卸载的应力–应变曲线特征、峰值强度及断裂损伤力学特性等进行研究。给出一种根据应力–应变曲线计算损伤变量的方法,损伤变量计算结果和声发射测试数据变化规律较为一致。试验结果表明,砂岩的循环加卸载强度要比单轴压缩强度要小很多,对于脆性岩石单轴循环加卸载的峰值强度来说,受到多种因素的影响。弹性常数计算结果表明,循环加卸载过程中泊松比逐渐增大,而弹性模量在第一次循环加卸载增大之后则缓慢减小。脆性岩石循环加卸载过程中,岩石损伤在逐渐累积,在微裂纹进入不稳定扩展阶段,岩石损伤会迅速增大,岩石宏观力学特性取决于内部微裂纹的细观力学响应。     

广州粘土真三轴试验研究

通过对广州原状粘土不同围压和不同中主应力系数下的周结不排水真三轴剪切试验,分析结构性粘土在固结不排水剪切状态下的应力与应变、强度与变形特性,并研究偏应力、孔隙水压力,应力比,有效主应力比值随不同围压和中主应力系数的变化规律。     

Effects of loading rate on rock fracture

By means of a wedge loading applied to a short-rod rock fracture specimen tested with the MTS 810 or SHPB (split Hopkinson pressure bar), the fracture toughness of Fangshan gabbro and Fangshan marble was measured over a wide range of loading rates, =10⁻²–10⁶ MPa m^1/2 s⁻¹. In order to determine the dynamic fracture toughness of the rock as exactly as possible, the dynamic Moiré method and strain–gauge method were used in determining the critical time of dynamic fracture. The testing results indicated that the critical time was generally shorter than the transmitted wave peak time, and the differences between the two times had a weak increasing tendency with loading rates. The experimental results for rock fracture showed that the static fracture toughness K_Ic of the rock was nearly a constant, but the dynamic fracture toughness K_Id of the rock ( ≥10⁴ MPa m^1/2 s⁻¹) increased with the loading rate, i.e. log(K_Id)=a log +b. Macroobservations for fractured rock specimens indicated that, in the section (which was perpendicular to the fracture surface) of a specimen loaded by a dynamic load, there was clear crack branching or bifurcation, and the higher the loading rate was, the more branching cracks occurred. Furthermore, at very high loading rates ( ≥10⁶ MPa m^1/2 s⁻¹) the rock specimen was broken into several fragments rather than only two halves. However, for a statically fractured specimen there was hardly any crack branching. Finally, some applications of this investigation in engineering practice are discussed.     

Numerical simulation of the excavation damaged zone around an opening in brittle rock

The micromechanics-based damage model proposed by Golshani et al. [A micromechanical model for brittle failure of rock and its relation to crack growth observed in triaxial compression tests of granite. Mech Mater 2006;38:287–303] is extended so that time-dependent behavior of brittle material can be taken into account, with special attention to the numerical analysis of an excavation damaged zone (EDZ) around an opening, which is a major concern in assessing the safety of underground repositories. The present model is capable of reproducing the three characteristic stages of creep behavior (i.e., primary, secondary, and tertiary creep) commonly observed in the laboratory creep tests. The sub-critical microcrack growth parameters (i.e., n and A) can be determined for Inada granite by fitting the numerical results of elapse time to failure versus the creep stress ratio curve with the experimental data under both dry and wet conditions. It is found that moisture has a significant influence on the parameter A rather than the parameter n. Use of the extended model makes it possible to analyze not only the extension of microcrack length, but also the development of EDZ around an opening as a function of time. The damaged zones mainly develop in the sidewalls of the opening in the case that the vertical stress σ₂₂ is larger than the horizontal stress σ₁₁.     

Shear fracture (Mode II) of brittle rock

Mode II fracture initiation and propagation plays an important role under certain loading conditions in rock fracture mechanics. Under pure tensile, pure shear, tension- and compression-shear loading, the maximum Mode I stress intensity factor, K_Imax, is always larger than the maximum Mode II stress intensity factor, K_IImax. For brittle materials, Mode I fracture toughness, K_IC, is usually smaller than Mode II fracture toughness, K_IIC. Therefore, K_Imax reaches K_IC before K_IImax reaches K_IIC, which inevitably leads to Mode I fracture. Due to inexistence of Mode II fracture under pure shear, tension- and compression-shear loading, classical mixed mode fracture criteria can only predict Mode I fracture but not Mode II fracture. A new mixed mode fracture criterion has been established for predicting Mode I or Mode II fracture of brittle materials. It is based on the examination of Mode I and Mode II stress intensity factors on the arbitrary plane θ,K_I(θ) and K_II(θ), varying with θ(−180°θ+180°), no matter what kind of loading condition is applied. Mode I fracture occurs when (K_IImax/K_Imax)<1 or 1<(K_IImax/K_Imax)<(K_IIC/K_IC) and K_Imax=K_IC at θ_IC. Mode II fracture occurs when (K_IImax/K_Imax)>(K_IIC/K_IC) and K_IImax=K_IIC at θ_IIC. The validity of the new criterion is demonstrated by experimental results of shear-box testing.Shear-box test of cubic specimen is a potential method for determining Mode II fracture toughness K_IIC of rock since it can create a favorable condition for Mode II fracture, i.e. K_IImax is always 2–3 times larger than K_Imax and reaches K_IIC before K_Imax reaches K_IC. The size effect on K_IIC for single- and double-notched specimens has been studied for different specimen thickness B, dimensionless notch length a/W (or 2a/W) and notch inclination angle α. The test results show that K_IIC decreases as B increases and becomes a constant when B is equal to or larger than W for both the single- and double-notched specimens. When a/W (or 2a/W) increases, K_IIC decreases and approaches a limit. The α has a minor effect on K_IIC when α is within 65–75°. Specimen dimensions for obtaining a reliable and reproducible value of K_IIC under shear-box testing are presented. Numerical results demonstrate that under the shear-box loading condition, tensile stress around the notch tip can be effectively restrained by the compressive loading. At peak load, the maximum normal stress is smaller than the tensile strength of rock, while the maximum shear stress is larger than the shear strength in the presence of compressive stress, which results in shear failure.     

脆性岩石破坏试验研究

 对不同加载速率控制条件下标准试样以及带中心圆孔的花岗岩岩板进行单轴压缩试验,研究岩石破坏的全过程并进行声发射特征分析。试验结果表明：岩石材料破坏过程是内部微裂纹产生和扩展过程的宏观反映;声发射信号与应力–应变曲线有良好的对应关系,根据声发射信号可以判断岩石内部裂纹扩展演化的情况;在不同的加载速率条件下对应不同的承载能力和不同的破坏形态。根据试验结果,建立弹脆性损伤本构模型,基于ABAQUS平台,采用与试验一致的控制条件对带孔岩板进行数值模拟,并与试验结果进行比较。结果表明,数值模拟真实地反映了岩石变形破坏的全过程,研究成果对研究脆性岩石的破坏以及脆性岩石的岩爆机制具有重要的指导意义。     

Non-dilatant deformation and failure mechanism in two Long Valley Caldera rocks under true triaxial compression

We conducted laboratory rock strength experiments in two ultra-fine-grained brittle rocks, hornfels and metapelite, which together are the major constituent of the Long Valley Caldera (California, USA) basement in the 2025–2996 m depth range. Both rocks are banded, and have very low porosity. Uniaxial compression tests at different orientations with respect to banding planes reveal that while the hornfels compressive strength is nearly isotropic, the metapelite possesses distinct anisotropy. Conventional triaxial tests in these rocks reveal that their respective strengths in a specific orientation increase approximately linearly with confining pressure. True triaxial compression experiments in specimens oriented at a consistent angle to banding, in which the magnitudes of the least (σ₃) and the intermediate (σ₂) principal stresses are different but kept constant during testing while the maximum principal stress is increased until failure, exhibit a behavior unlike that previously observed in other rocks under similar testing conditions. For a given magnitude of σ₃, compressive strength σ₁ does not vary significantly in both Long Valley rock types, regardless of the applied σ₂, suggesting little or no intermediate principal stress effect. Strains measured in all three principal directions during loading were used to obtain plots of σ₁ versus volumetric strain. These are consistently linear almost to the point of rock failure, suggesting no dilatancy. The phenomenon was corroborated by SEM inspection of failed specimens that showed no microcrack development prior to the emergence of one through-going shear failure plane steeply dipping in the σ₃ direction. The strong dependency of compressive strength on the intermediate principal stress in other crystalline rocks was found to be related to microcrack initiation upon dilatancy onset, which rises with increased σ₂ and retards the failure process. We infer that strength independence of σ₂ in the Long Valley rocks derives directly from their non-dilatant deformation.     

带裂纹花岗岩试件三点弯变形演化及破裂机制

为建立岩石工程灾害精准预测预报体系,开展含预制裂纹花岗岩试件的三点弯试验,研究岩石试样变形演化及破裂机制。以声发射技术和数字散斑相关方法作为试验观测手段,通过矩张量反演方法研究岩石破裂类型。通过对岩石预制裂纹扩展长度及声发射振铃计数、能量和体积参数进行分析,研究岩石试件变形演化特征和破裂机制。研究结果表明:(1)三点弯岩石试样裂纹扩展具有明显的阶段性和不连续性,其过程为某次裂纹扩展 微裂纹静默积累-裂纹再次扩展;(2)预制裂纹扩展与不同破裂类型裂纹的声发射指标具有相关性。预制裂纹扩展时,张拉裂纹体积参数、能量和振铃计数随之出现;预制裂纹扩展前后,3种类型裂纹体积参数、能量和振铃计数随机产生;在预制裂纹长度保持稳定不变时,没有声发射信号产生;(3)裂纹扩展后到下一次裂纹扩展前,先是在贯通裂纹尖端附近随机出现各种破裂类型的裂纹,随着微裂纹的积累,贯通裂纹所主导的裂纹类型占比开始增加,直至产生新的贯通裂纹。     

The flattened Brazilian disc specimen used for testing elastic modulus, tensile strength and fracture toughness of brittle rocks: analytical and numerical results

The flattened Brazilian disc specimen is proposed for determination of the elastic modulus E, tensile strength σ_t and opening mode fracture toughness K_IC for brittle rocks in just one test. This paper is concerned with the theoretical analysis as well as analytical and numerical results for the formulas. According to the results of stress analysis and Griffith's strength criteria, in order to guarantee crack initiation at the centre of the specimen, which is considered to be crucial for the test validity, the loading angle corresponding to the flat end width must be greater than a critical value (2α20°). The analysis shows that, based on the recorded complete load–displacement curve of the specimen (the curve should include the ‘fluctuation’ section after the maximum load), E can be determined by the slope of the section before the maximum load, σ_t by the maximum load, and K_IC by the local minimum load immediately subsequent to the maximum load. The relevant formulas for the calculation of E, σ_t, K_IC are obtained, and the key coefficients in these formulas are calibrated by finite-element analysis. In addition, some approximate closed-form formulas based on elasticity are provided, and their accuracy is shown to be adequate by comparison with the finite-element results.     

Coupled evolutions of fracture toughness and elastic wave velocities at high crack density in thermally treated Westerly granite

A series of 20 chevron cracked notched Brazilian disc (CCNBD) samples of Westerly granite were failed in a standard Mode I tensile test at room temperature in order to evaluate the effect of thermal damage on fracture toughness. The heat treatment involved slowly heating four sets of four samples to 250, 450, 650 and 850 °C. The fifth set of samples was not thermally treated. Thermal cracking not only induced a substantial decrease of the mechanical strength, but also of the dynamic elastic properties of Westerly granite. In particular, normalized P-wave compressional velocities matched remarkably well the decreasing trend of normalized fracture toughness (K_IC). Above 450 °C, grain boundary opening and cracking, intragranular cracking and mineral grain dissection linked to the quartz α–β phase transition induced a significant increase in the total crack density. Fracture path interaction with various mineral–mineral contact types showed that fracture branching and total fracture length increased with the amount of temperature of heat treatment.Using non-interactive crack theories, dimensionless crack densities were obtained from wave velocity inversion, up to unusually high values of 10 at 850 °C. This geophysical analysis showed to be in close agreement with crack parameters determined optically, such as optical crack density determination, crack aspect ratio evolutions, and the measured sample porosity with temperature. Our results also show that only the non-interactive crack theory can predict K_IC relatively well at high crack density, by simply using dimensionless crack densities inverted from velocities. A decrease of 50% for crack densities larger than 1, 80% for crack densities larger than 5 is predicted, in close agreement with our observed experimental variation of K_IC. At the microscale, this can be interpreted by the fact that the main fracture is strongly interacting with the pre-existing microcrack fabric. These combined experimental and modeling results illustrate the importance of understanding the details of how the rock microstructure is changing in response to an external stimulus, in order to predict the simultaneous evolution of physical and mechanical properties of rock.     

锦屏深埋大理岩破裂特征与损伤演化规律

 锦屏二级引水隧洞最大主应力超过70 MPa,在如此高应力条件下,必然存在岩体强度和地应力之间的尖锐矛盾。深入认识大理岩的破裂特征,把握围岩的损伤演化规律,对于支护优化设计和评价围岩稳定性具有重要意义。在已有关于脆性岩石破裂问题研究成果的基础上,借助于裂纹体积应变拐点和大理岩体积应变拐点配合声发射测试,确定不同围压条件下大理岩的特征强度,并将特征强度在主应力空间中进行表达,形成现场可用的强度包线。利用起裂强度曲线分析损伤区的应力状态,并根据计算成果确定不同损伤区分区的范围,描述监测断面随掌子面推进过程中损伤区的演化过程。为进一步分析损伤区的演化规律,在监测断面布置声波和声发射测试,声波检测获得的松弛圈深度主要与损伤区中的破坏区相对应,而声发射测试可以获得完整的损伤区分布特征,更有利于了解围岩的损伤特征,可为支护优化设计和支护时机的选择提供更加科学的依据。     

Influence of intermediate principal stress on rock fracturing and strength near excavation boundaries—Insight from numerical modeling

The influence of the intermediate principal stress on rock fracturing and strength near excavation boundaries is studied using a FEM/DEM combined numerical tool. A loading condition of σ₃=0 and σ₁≠0, and σ₂≠0 exists at the tunnel boundary, where σ₁, σ₂, and σ₃, are the maximum, intermediate, and minimum principal stress components, respectively. The numerical study is based on sample loading testing that follows this type of boundary stress condition. It is seen from the simulation results that the generation of tunnel surface parallel fractures and microcracks is attributed to material heterogeneity and the existence of relatively high intermediate principal stress (σ₂), as well as zero to low minimum principal stress (σ₃) confinement. A high intermediate principal stress confines the rock in such a way that microcracks and fractures can only be developed in the direction parallel to σ₁ and σ₂. Stress-induced fracturing and microcracking in this fashion can lead to onion-skin fractures, spalling, and slabbing in shallow ground near the opening and surface parallel microcracks further away from the opening, leading to anisotropic behavior of the rock. Hence, consideration of the effect of the intermediate principal stress on rock behavior should focus on the stress-induced anisotropic strength and deformation behavior of the rocks. It is also found that the intermediate principal stress has limited influence on the peak strength of the rock near the excavation boundary.     

A modified Hoek–Brown failure criterion for anisotropic intact rock

The Hoek–Brown criterion parameters (σ_ci, m_i and s) are significantly influenced by the strength anisotropy of intact rock. In the present study, the criterion was modified by incorporating a new parameter (k_β) to account for the effect of strength anisotropy, thus being able to determine the strength of intact anisotropic rock under loading in different orientations of the plane of anisotropy. The range of the parameter (k_β) for the rocks tested has been analytically investigated by carrying out triaxial tests, in different orientations of the foliation plane. The proposed modification was studied for metamorphic rocks (gneiss, schist, marble), but could also be applied to other rock types exhibiting “inherent” anisotropy, e.g. sedimentary as well as igneous rocks. The proposed modified criterion is intended for use for prediction of strength of intact rock, but can also be extended to rock masses.     

Application of Weibull's theory to estimating in situ maximum stress σH by hydrofracturing

Hydrofracturing is a widely used and established method for rock stress measurement and is especially valuable at great depths. In conventional hydrofracturing (Haimson, Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 15 (1978) 167), dealing with an axi-parallel fracture, the horizontal minimum stress σ_h is obtained as the shut-in pressure and the maximum stress σ_H is calculated from the breakdown pressure or reopening pressure. It has been pointed out, however, that σ_H is not as reliable as σ_h. This paper therefore presents a new approach for estimating σ_H. In this approach the probabilistic aspects of tensile failure are considered as new sources of information, because the probability density of fracture direction may provide valuable information concerning the stress difference σ_H−σ_h. As the basic theory to describe the tensile failure of rock, we adopted the Weibull’s weakest link theory. The applicability of the theory is first verified via tensile tests on rock specimens of different shape and size, then the probabilistic approach is applied to hydrofracturing to give the probability function of breakdown and the probability density function for the fracture direction. The applicability of the proposed method is presented through numerical calculations and an example in which σ_H−σ_h is estimated from the probabilistic variability of the fracture direction.     

单轴压缩条件下锚杆影响脆性岩体破裂的细观机制

锚杆支护系统是控制深部脆性围岩动力灾害的重要措施,但锚固理论研究仍滞后,锚杆支护下的脆性岩体破坏问题困扰着深部岩体工程实践。根据实际工程中锚杆支护下脆性围岩的浅表局部破坏特点,通过室内相似模型试验研究单轴压缩条件下锚杆杆径影响完整脆性岩体的破坏特性,试验表明,锚杆杆径对脆性岩体弹性模量和强度的提升存在最优匹配的特点,一味强调增大锚杆直径并不能达到理想的围岩控制效果;锚杆改变了脆性岩体单轴压缩破坏模式,宏观上由劈裂破坏转为剪切破坏,杆径对试样剪切破坏的程度有所影响。从细观角度,建立了含两条固有主裂纹的裂纹扩展分析模型,加锚试样单轴压缩破裂模式的改变,可以归结为锚杆锚固止裂效应对试样内部裂纹扩展的抑制作用,使翼裂纹与主裂纹长度比η变小。根据最易开裂角度ζ的计算结果,翼裂纹较长时,翼裂纹朝外载作用方向扩展,产生劈裂破坏,翼裂纹较短时,翼裂纹偏离外载作用方向扩展,产生剪切破坏。从细观上很好地解释了锚杆改变脆性岩体破裂模式的作用机制。     

三轴多级荷载下盐岩声波声发射特征与损伤演化规律研究

 采用岩石声波、声发射一体化监测装置,系统地研究三轴多级循环荷载作用下盐岩超声波波速与声发射变化特征。结果表明：(1) 岩石的超声波波速和声发射活动与应力状态呈现出良好的一致性。加载阶段,超声波波速上升,声发射活跃,卸载阶段,超声波波速下降,声发射平静,应力级数越高,这一特征越显著。(2) 盐岩的声波、声发射特征与试验围压应力密切相关。围压水平越低,应力循环试验中岩石波速变化率越大,声发射事件数量越多;围压水平越高,岩石超声波波速变化率越小,声发射事件数量越少。五级应力荷载试验中,围压条件为5,10,15,20 MPa时盐岩的声发射事件数量分别为1 026,703,361和206个,显示了“围压致密效应”。(3) 分别应用卸载模量、裂隙密度和Felicity比表征盐岩的损伤演化。结论认为：盐岩的裂隙密度和Felicity比变化与岩体承载破坏特征较为一致,可以较好地反映盐岩的损伤破裂过程,而利用卸载模量表征盐岩损伤误差较大,这是由于盐岩特殊的黏塑性变形特征造成的。     

Effects of loading rate on rock fracture: fracture characteristics and energy partitioning

By means of the Scanning Electron Microscope (SEM), an examination was performed of the fracture surfaces (including their vertical sections) of both Fangshan gabbro and Fangshan marble specimens fractured at the loading rates MPa m^1/2 s⁻¹. The results showed that one or more branching cracks near the fracture surfaces of dynamic rock specimens were clear and the cracks increased with increasing loading rates. However, such branching cracks were rarely seen near the static fracture surfaces. In addition, with the aid of the Split Hopkinson Pressure Bar (SHPB) testing system and a high-speed framing camera, the energy partitioning in the dynamic fracture process of a short rod (SR) rock specimen was analysed quantitatively. The total energy W_L absorbed by an SR specimen in the dynamic fracture process mainly consisted of the fracture and damage energy W_FD and the kinetic energy W_K of flying fragments. The energies W_L and W_K could be quantitatively calculated through stress wave measurement and high-speed photography in the SHPB testing system. Thus, the fracture and damage energy W_FD could be obtained. The results showed that: (1) the energy W_K increased with an increase in the impact speed of the striker bar or the loading rate; (2) the energy W_FD for dynamic rock fracture was markedly greater than that for static rock fracture, and the W_FD increased with an increase in the impact speed of the striker bar or the loading rate; and (3) the value W_L/W_B (W_B is the energy input into the loading system) in the case of dynamic fracture is much lower than that in the case of static fracture. In addition, the ratio decreases with an increase in the loading rate or the impact speed of the striker bar. This means that the energy utilisation decreases when the loading rate or the impact speed of the striker bar rises. Finally, some application problems are discussed in the paper.