2017 Vol. 37, No. 2
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2017, 37(2): 169-179.
doi: 10.11883/1001-1455(2017)02-0169-11
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Abstract:
The strength of a material is traditionally understood as its ability to resist flow/deformation and breakage (brittle fracture or ductile rupture) under applied load. The breakage, though looking like an abrupt occurrence, actually results from a strain-rate/time dependent process of damage evolution. The difficulty in studying dynamic damage evolution lies in that the damage evolution and the flow/deformation process are coupled and influence each other. It was found that the dynamic evolution of macro-damage could be successfully described by the thermo-activated damage evolution model. Based on this model, the damage evolution and the flow/deformation can be decoupled from the experimentally measured apparent stress-strain curve with damage evolution, and the related material parameters can be determined. Such an approach is then generalized to the objective study of the TCM pulse. The normal and pathological constitutive relations of the pulse wave system can be inversely determined by pulse wave signals, and then the degree of deviation from the normal condition (illness state) of patients can be diagnosed, and their illness state can be regarded as a kind of generalized damage. The same approach is further generalized to the study of earthquake prediction through the 'pulse-taking for the earth'. Combined with the load-unload response ratio theory, by measuring the seismic wave signals on three adjacent positions, the nonlinear constitutive load-response curve with damage evolution can be inversely determined for the tectonic plates of the earth concerned, then the degree of the damage can be finally discriminated, which is the key information for an improved earthquake prediction.
The strength of a material is traditionally understood as its ability to resist flow/deformation and breakage (brittle fracture or ductile rupture) under applied load. The breakage, though looking like an abrupt occurrence, actually results from a strain-rate/time dependent process of damage evolution. The difficulty in studying dynamic damage evolution lies in that the damage evolution and the flow/deformation process are coupled and influence each other. It was found that the dynamic evolution of macro-damage could be successfully described by the thermo-activated damage evolution model. Based on this model, the damage evolution and the flow/deformation can be decoupled from the experimentally measured apparent stress-strain curve with damage evolution, and the related material parameters can be determined. Such an approach is then generalized to the objective study of the TCM pulse. The normal and pathological constitutive relations of the pulse wave system can be inversely determined by pulse wave signals, and then the degree of deviation from the normal condition (illness state) of patients can be diagnosed, and their illness state can be regarded as a kind of generalized damage. The same approach is further generalized to the study of earthquake prediction through the 'pulse-taking for the earth'. Combined with the load-unload response ratio theory, by measuring the seismic wave signals on three adjacent positions, the nonlinear constitutive load-response curve with damage evolution can be inversely determined for the tectonic plates of the earth concerned, then the degree of the damage can be finally discriminated, which is the key information for an improved earthquake prediction.
2017, 37(2): 180-185.
doi: 10.11883/1001-1455(2017)02-0180-06
Abstract:
A type of 3D-Hopkinson bar dynamic testing system for investigating the dynamic behavior of concrete and rock under static triaxial loading was developed. The hydraulic servo control system employed in the testing system could provide independent triaxial pressure up to 100 MPa, and the split Hopkinson pressure bar employed in the impacting direction could provide dynamic loads. The dynamic responses on six faces of the cube specimen can be recorded by six bars in three dimensions, the lateral deformation states of the specimen under dynamic loading were obtained and analyzed by using this novelty Hopkinson bar. The dynamic compressive behaviors of C30 concrete under different static triaxial loading were investigated.
A type of 3D-Hopkinson bar dynamic testing system for investigating the dynamic behavior of concrete and rock under static triaxial loading was developed. The hydraulic servo control system employed in the testing system could provide independent triaxial pressure up to 100 MPa, and the split Hopkinson pressure bar employed in the impacting direction could provide dynamic loads. The dynamic responses on six faces of the cube specimen can be recorded by six bars in three dimensions, the lateral deformation states of the specimen under dynamic loading were obtained and analyzed by using this novelty Hopkinson bar. The dynamic compressive behaviors of C30 concrete under different static triaxial loading were investigated.
2017, 37(2): 186-193.
doi: 10.11883/1001-1455(2017)02-0186-08
Abstract:
A complex unsteady multiphase flow field is produced in the process of solid rocket gas jet driven liquid column. In this work, to study the temperature-reducing effect of the liquid column on the jet flow field of the solid rocket motor and reveal the flow evolution and the interaction between gas and water, the coupling flow and phase transition process of the gas and liquid column are simulated using the VOF multiphase flow model coupled with the FLUENT software. The results are compared with the calculation results of the jet flow field where no liquid column exists. The calculation results show that the pressure, temperature and velocity fluctuation in the jet flow field decreased when the liquid column is taken as an equilibrium body. The turbulence in tensity in the jet flow field is reduced. The development of the axial displacement of the jet flow field is reduced by the vaporization of the gas and the liquid column as well as the blocking action of the liquid column. The peak pressure is reduced by 0.9 MPa, the peak temperature by 503 K and the peak velocity by 291 m/s in the core area of the jet flow field, thus verifying the temperature-reducing effect of the liquid column on the gas jet flow field.
A complex unsteady multiphase flow field is produced in the process of solid rocket gas jet driven liquid column. In this work, to study the temperature-reducing effect of the liquid column on the jet flow field of the solid rocket motor and reveal the flow evolution and the interaction between gas and water, the coupling flow and phase transition process of the gas and liquid column are simulated using the VOF multiphase flow model coupled with the FLUENT software. The results are compared with the calculation results of the jet flow field where no liquid column exists. The calculation results show that the pressure, temperature and velocity fluctuation in the jet flow field decreased when the liquid column is taken as an equilibrium body. The turbulence in tensity in the jet flow field is reduced. The development of the axial displacement of the jet flow field is reduced by the vaporization of the gas and the liquid column as well as the blocking action of the liquid column. The peak pressure is reduced by 0.9 MPa, the peak temperature by 503 K and the peak velocity by 291 m/s in the core area of the jet flow field, thus verifying the temperature-reducing effect of the liquid column on the gas jet flow field.
2017, 37(2): 194-200.
doi: 10.11883/1001-1455(2017)02-0194-07
Abstract:
In this work, to obtain a water mist with a finer particle size under lower pressures, reduce the running costs of the explosion-suppressing spray system, and improve its efficiency and applicability, we designed a transparent organic glass pipeline (120 mm×120 mm×840 mm) as the experimental platform for gas explosion. Then N2 and fine water mist was pressed into the pipeline using a twin-fluid nozzle and experimental study of methane explosion suppression by nitrogen twin-fluid water mist was carried out adjusting the spray pressure and spray time and the explosion-suppression effectiveness of the twin-fluid water mist was investigated via analysis of the flame speed and the gas explosion overpressure. The results show that this twin-fluid water mist has a high explosion-suppressing efficiency, capable of reducing the damage degree of gas explosion. With the extension of the spray time, the peak value of the explosion flame speed, the peak overpressure and the average pressure rise rate decreased gradually. When the pressure of N2 was 0.4 MPa and the spray time was 3 s, the peak value of the velocity decreased by 60.39%, and the peak overpressure decreased by 37.76%.
In this work, to obtain a water mist with a finer particle size under lower pressures, reduce the running costs of the explosion-suppressing spray system, and improve its efficiency and applicability, we designed a transparent organic glass pipeline (120 mm×120 mm×840 mm) as the experimental platform for gas explosion. Then N2 and fine water mist was pressed into the pipeline using a twin-fluid nozzle and experimental study of methane explosion suppression by nitrogen twin-fluid water mist was carried out adjusting the spray pressure and spray time and the explosion-suppression effectiveness of the twin-fluid water mist was investigated via analysis of the flame speed and the gas explosion overpressure. The results show that this twin-fluid water mist has a high explosion-suppressing efficiency, capable of reducing the damage degree of gas explosion. With the extension of the spray time, the peak value of the explosion flame speed, the peak overpressure and the average pressure rise rate decreased gradually. When the pressure of N2 was 0.4 MPa and the spray time was 3 s, the peak value of the velocity decreased by 60.39%, and the peak overpressure decreased by 37.76%.
2017, 37(2): 201-207.
doi: 10.11883/1001-1455(2017)02-0201-07
Abstract:
To find out the exact effects of boundary conditions on premixed CH4+2O2 detonation characteristics, detonation experiments were conducted respectively in two cylindrical tubes with inner diameters of 63.5 mm and 50.8 mm and one rectangle tube. The detonation velocity curves were obtained using a signal detection system, and cellar patterns drawn from the smoked-foil records were obtained using a digital image processing program. Quantitative irregularities of CH4+2O2 detonation and cell size data under different initial pressures in the three tubes were analyzed and compared. It is found that the average velocities in the three tubes always closely resembled vcj, showing that the detonation velocity is mainly determined by the initial detonation pressure and mixture type rather than by the boundary conditions. In addition, judging by the detonation velocity curves, the limit pressures of the tubes with inner diameter of 50.8 and 63.5 mm are 5 and 4.05 kPa, respectively. Therefore, the boundary conditions can influence the limit pressure. Quantitative irregularities exert little significant difference under different boundary conditions of the tubes. Besides, as the detonation needs to compensate for the energy loss with the increase of the tube diameter, there is a greater number of spin heads in the tubes.
To find out the exact effects of boundary conditions on premixed CH4+2O2 detonation characteristics, detonation experiments were conducted respectively in two cylindrical tubes with inner diameters of 63.5 mm and 50.8 mm and one rectangle tube. The detonation velocity curves were obtained using a signal detection system, and cellar patterns drawn from the smoked-foil records were obtained using a digital image processing program. Quantitative irregularities of CH4+2O2 detonation and cell size data under different initial pressures in the three tubes were analyzed and compared. It is found that the average velocities in the three tubes always closely resembled vcj, showing that the detonation velocity is mainly determined by the initial detonation pressure and mixture type rather than by the boundary conditions. In addition, judging by the detonation velocity curves, the limit pressures of the tubes with inner diameter of 50.8 and 63.5 mm are 5 and 4.05 kPa, respectively. Therefore, the boundary conditions can influence the limit pressure. Quantitative irregularities exert little significant difference under different boundary conditions of the tubes. Besides, as the detonation needs to compensate for the energy loss with the increase of the tube diameter, there is a greater number of spin heads in the tubes.
2017, 37(2): 208-214.
doi: 10.11883/1001-1455(2017)02-0208-07
Abstract:
Under a blasting load, the front side adjacent to blasting in an existing tunnel surrounded by layered rocks is apt to be destroyed, and therefore it is of critical importance to keep the tunnel safe from the destructive effects of the surrounding rocks cracks and damages that occur in a blasting disturbance. In this work, based on elastic mechanics and structural mechanics, we established a simplified mechanical model for the most dangerous point of the tunnel s blasting side, and obtained the stress formula that can calculate the stress at this point. We also analyzed the regular influence of such different factors as the ratio of the tunnel radius to the thickness of layer, the inclination angle, the central angle and depth of the surrounding layered rocks on the stress. In addition, using the theory of the maximum tensile stress, we calculated the surrounding rocks‘ critical vibration velocity that ensures the stability of the tunnel and obtained, according to the attenuation patterns of seismic wave propagation, the maximum charge of detonation for the construction of a new tunnel.
Under a blasting load, the front side adjacent to blasting in an existing tunnel surrounded by layered rocks is apt to be destroyed, and therefore it is of critical importance to keep the tunnel safe from the destructive effects of the surrounding rocks cracks and damages that occur in a blasting disturbance. In this work, based on elastic mechanics and structural mechanics, we established a simplified mechanical model for the most dangerous point of the tunnel s blasting side, and obtained the stress formula that can calculate the stress at this point. We also analyzed the regular influence of such different factors as the ratio of the tunnel radius to the thickness of layer, the inclination angle, the central angle and depth of the surrounding layered rocks on the stress. In addition, using the theory of the maximum tensile stress, we calculated the surrounding rocks‘ critical vibration velocity that ensures the stability of the tunnel and obtained, according to the attenuation patterns of seismic wave propagation, the maximum charge of detonation for the construction of a new tunnel.
2017, 37(2): 215-220.
doi: 10.11883/1001-1455(2016)05-0215-06
Abstract:
In the present work, we carried out an experimental study of the influence of the coal dust categories, concentration and particle size on the coal dust-CO/H2 hybrid mixture's deflagration characteristics in a semi-open environment. The results indicate that the influence of the dust on the hybrid mixture's deflagration is mainly the outcome of the competitive relation between two opposite effects, the heat absorption for volatile releasing and the reaction of the released volatile. For the high volatile coal dust, the deflagration reaction of the released volatile dominates, which makes the deflagration severity gradually rise with the rising of the volatile content; while for the low volatile coal dust, the heat absorption for the volatile dominates, which reduces the deflagration severity. For bitumite like the Yinbei coal, with the increase of the dust concentration, the deflagration severity varies with a U-shaped tendency, while for the coke dust with a low volatile content, the change of the deflagration severity with the dust concentration is not obvious.
In the present work, we carried out an experimental study of the influence of the coal dust categories, concentration and particle size on the coal dust-CO/H2 hybrid mixture's deflagration characteristics in a semi-open environment. The results indicate that the influence of the dust on the hybrid mixture's deflagration is mainly the outcome of the competitive relation between two opposite effects, the heat absorption for volatile releasing and the reaction of the released volatile. For the high volatile coal dust, the deflagration reaction of the released volatile dominates, which makes the deflagration severity gradually rise with the rising of the volatile content; while for the low volatile coal dust, the heat absorption for the volatile dominates, which reduces the deflagration severity. For bitumite like the Yinbei coal, with the increase of the dust concentration, the deflagration severity varies with a U-shaped tendency, while for the coke dust with a low volatile content, the change of the deflagration severity with the dust concentration is not obvious.
2017, 37(2): 221-228.
doi: 10.11883/1001-1455(2017)02-0221-08
Abstract:
Aimed at the faults such as cartridge jamming and component dynamic fracture in the extractor system of an artillery vertical wedge breechblock, this paper was focused on transforming the extracting fault analysis to the quantitative study of the system dynamic characteristics based on the nonlinear dynamics theory. The working mechanism of the extractor system was analyzed firstly, and then a continuous contact dynamics model was established for deriving out the motion law of any components and the impact loads quantificationally during the extraction process. Moreover, the elastic wave model of the lever-type extractor was established, which can both describe the dynamic stress distribution and achieve the influences of the mechanical parameters of the extractor by the quantitative form. The numerical calculation of those theoretical models and the simulations of the virtual prototype were both conducted on the ideal extracting condition. The comparison results of the same group show that the quantitative analysis dynamic models are reasonable and effective, which provides a theoretical basis for the solution of the extracting fault and the dynamics optimization of the extractor system.
Aimed at the faults such as cartridge jamming and component dynamic fracture in the extractor system of an artillery vertical wedge breechblock, this paper was focused on transforming the extracting fault analysis to the quantitative study of the system dynamic characteristics based on the nonlinear dynamics theory. The working mechanism of the extractor system was analyzed firstly, and then a continuous contact dynamics model was established for deriving out the motion law of any components and the impact loads quantificationally during the extraction process. Moreover, the elastic wave model of the lever-type extractor was established, which can both describe the dynamic stress distribution and achieve the influences of the mechanical parameters of the extractor by the quantitative form. The numerical calculation of those theoretical models and the simulations of the virtual prototype were both conducted on the ideal extracting condition. The comparison results of the same group show that the quantitative analysis dynamic models are reasonable and effective, which provides a theoretical basis for the solution of the extracting fault and the dynamics optimization of the extractor system.
2017, 37(2): 229-236.
doi: 10.11883/1001-1455(2017)02-0229-08
Abstract:
In this work, to further study the features of the shock-wave induced flame instability, the two-dimensional Navier-Stokes (NS) equations with the single-step chemical reaction and the high resolution 9th-order weighted essentially non-oscillatory (WENO) scheme were adopted to simulate the instability of the sinusoidally premixed flame induced by incident shock waves with different Mach numbers and its reshock waves. The computational results were validated by the experimental results in the related literature. The computational results show that the evolutions of the flame are mainly influenced by both the Richtmyer-Meshkov (RM) instability and the chemical reaction. With the growth of the incident shock wave intensity, the interface instability and the chemical reaction are enhanced. To construct the parameter that can characterize the RM instability in the reactive flow, a dimensionless parameter 8338A131 that describes the interface RM instability and the chemical reactivity was proposed based on the average vorticity and the chemical reaction rate calculated in the mixing zone of the flame interface. The analysis of the parameter shows that, with the similar intensity of the incident shock wave, the logarithmic form of the parameter exhibits basically the same linear growth when an incident shock wave with a given Mach number and its reshocks successively pass through the flame interface. The linear growth rate of the logarithmic form of the parameter is also basically the same for different Mach numbers of the incident wave. Such variations of η suggest that the dimensionless parameter proposed in the present study can well characterize the intrinsic features of the flame interface development in the reactive RM instability process.
In this work, to further study the features of the shock-wave induced flame instability, the two-dimensional Navier-Stokes (NS) equations with the single-step chemical reaction and the high resolution 9th-order weighted essentially non-oscillatory (WENO) scheme were adopted to simulate the instability of the sinusoidally premixed flame induced by incident shock waves with different Mach numbers and its reshock waves. The computational results were validated by the experimental results in the related literature. The computational results show that the evolutions of the flame are mainly influenced by both the Richtmyer-Meshkov (RM) instability and the chemical reaction. With the growth of the incident shock wave intensity, the interface instability and the chemical reaction are enhanced. To construct the parameter that can characterize the RM instability in the reactive flow, a dimensionless parameter 8338A131 that describes the interface RM instability and the chemical reactivity was proposed based on the average vorticity and the chemical reaction rate calculated in the mixing zone of the flame interface. The analysis of the parameter shows that, with the similar intensity of the incident shock wave, the logarithmic form of the parameter exhibits basically the same linear growth when an incident shock wave with a given Mach number and its reshocks successively pass through the flame interface. The linear growth rate of the logarithmic form of the parameter is also basically the same for different Mach numbers of the incident wave. Such variations of η suggest that the dimensionless parameter proposed in the present study can well characterize the intrinsic features of the flame interface development in the reactive RM instability process.
2017, 37(2): 237-242.
doi: 10.11883/1001-1455(2017)02-0237-06
Abstract:
In the present work, to find out the differences in composition and the regularities in space-time distribution for coal dust residues, we carried out a coal dust explosion experiment using a horizontal pipe and analyzed the gas and solid residues of the coal dust. The results show that the content of ashes in the residues increased significantly while that of the volatile and fixed carbon decreased, with a smooth surface of the coal particle residue, accompanied by the particle cracking and bonding occurred. The study also reveals that the major components of the coal dust gas residue from explosion consist of O2, N2, CO, CO2, CH4, C2H4, C2H6, C2H2, C3H8, etc. At the source point of the explosion, the minimum concentration of oxygen was only 2.52% and that of carbon monoxide concentration was 0.38%~0.68%. The greater the distance from the explosive source point, the higher the concentration oxygen but the lower that of carbon oxide gas and hydrocarbon gas.
In the present work, to find out the differences in composition and the regularities in space-time distribution for coal dust residues, we carried out a coal dust explosion experiment using a horizontal pipe and analyzed the gas and solid residues of the coal dust. The results show that the content of ashes in the residues increased significantly while that of the volatile and fixed carbon decreased, with a smooth surface of the coal particle residue, accompanied by the particle cracking and bonding occurred. The study also reveals that the major components of the coal dust gas residue from explosion consist of O2, N2, CO, CO2, CH4, C2H4, C2H6, C2H2, C3H8, etc. At the source point of the explosion, the minimum concentration of oxygen was only 2.52% and that of carbon monoxide concentration was 0.38%~0.68%. The greater the distance from the explosive source point, the higher the concentration oxygen but the lower that of carbon oxide gas and hydrocarbon gas.
2017, 37(2): 243-248.
doi: 10.11883/1001-1455(2017)02-0243-06
Abstract:
In the present work, we developed a novel method combining the forward Lagrange method and the transfer function method to process the data of ramp wave compression experiments. Compared with traditional methods, this method is more suitable to process the data containing materials' complex behaviors and produces more accurate results. Meanwhile, this method has lower precision requirement regarding the initial gauss of testing material parameters. The feasibility and robustness of the transfer function method was analyzed, and the application of the forward Lagrange method and the transfer function method were examined in forward data processing of strength measurement experiments under ramp wave compression.
In the present work, we developed a novel method combining the forward Lagrange method and the transfer function method to process the data of ramp wave compression experiments. Compared with traditional methods, this method is more suitable to process the data containing materials' complex behaviors and produces more accurate results. Meanwhile, this method has lower precision requirement regarding the initial gauss of testing material parameters. The feasibility and robustness of the transfer function method was analyzed, and the application of the forward Lagrange method and the transfer function method were examined in forward data processing of strength measurement experiments under ramp wave compression.
2017, 37(2): 249-254.
doi: 10.11883/1001-1455(2017)02-0249-06
Abstract:
The characteristic dimension of the explosion-generated cavity in the soil is the most important influence factor for the feature of the far field seismic wave. In this paper we established a quasi-static model to predict the characteristic dimension of the cavity. When a spherical charge was detonated in the homogeneous, incompressible, infinite and elastic mediums, the analytical expressions of the fine crushing zone and the radial fissures zone were shown in this model. In addition, we also used this model to calculate the dimension of the divisions in different conditions. Finally, we conducted a comparison among the quasi-static model, the dynamic model and the experiments. The errors range of the quasi static model was 5.4%-16.0%. The results show that our model can be used to predict exactly the dimension of the cavity.
The characteristic dimension of the explosion-generated cavity in the soil is the most important influence factor for the feature of the far field seismic wave. In this paper we established a quasi-static model to predict the characteristic dimension of the cavity. When a spherical charge was detonated in the homogeneous, incompressible, infinite and elastic mediums, the analytical expressions of the fine crushing zone and the radial fissures zone were shown in this model. In addition, we also used this model to calculate the dimension of the divisions in different conditions. Finally, we conducted a comparison among the quasi-static model, the dynamic model and the experiments. The errors range of the quasi static model was 5.4%-16.0%. The results show that our model can be used to predict exactly the dimension of the cavity.
2017, 37(2): 255-261.
doi: 10.11883/1001-1455(2017)02-0255-07
Abstract:
To accurately characterize such important characteristics as the peak velocity and the energy distribution in different frequency ranges of the blast vibration signal, this signal's trend after the time integral has to be removed. In this paper, the trapezoidal numerical integration of the measured blast vibration acceleration signal was carried out, the blast vibration velocity signal after the time integral as a method for the wavelet basis was proposed, the trend of the blast vibration velocity signal after the time integral was removed using this method, and the characteristics of energy distribution in different frequency ranges of the signal after the trend removal was analyzed. The results show that the trend of the blast vibration velocity signal after the time integral was successfully removed using the pattern adapted wavelet method. Compared with the frequency spectra analysis based on the conventional Fourier transform, the energy analysis based on the wavelet transform had a higher frequency resolution and was more suitable for the analysis that satisfied a higher requirement of frequency resolution for the signal. The more widely divided the different frequency ranges, the higher the degree of the energy distribution correlation in different frequency ranges between the measured blast acceleration signal and the blast vibration velocity signal and, on the other hand, the less widely divided, the lower the degree of the correlation.
To accurately characterize such important characteristics as the peak velocity and the energy distribution in different frequency ranges of the blast vibration signal, this signal's trend after the time integral has to be removed. In this paper, the trapezoidal numerical integration of the measured blast vibration acceleration signal was carried out, the blast vibration velocity signal after the time integral as a method for the wavelet basis was proposed, the trend of the blast vibration velocity signal after the time integral was removed using this method, and the characteristics of energy distribution in different frequency ranges of the signal after the trend removal was analyzed. The results show that the trend of the blast vibration velocity signal after the time integral was successfully removed using the pattern adapted wavelet method. Compared with the frequency spectra analysis based on the conventional Fourier transform, the energy analysis based on the wavelet transform had a higher frequency resolution and was more suitable for the analysis that satisfied a higher requirement of frequency resolution for the signal. The more widely divided the different frequency ranges, the higher the degree of the energy distribution correlation in different frequency ranges between the measured blast acceleration signal and the blast vibration velocity signal and, on the other hand, the less widely divided, the lower the degree of the correlation.
2017, 37(2): 262-268.
doi: 10.11883/1001-1455(2017)02-0262-07
Abstract:
The digital-laser dynamic caustics system in combination with a static-dynamic loading device was utilized in the blasting fracturing test, in which the PMMA specimens underwent four kinds of vertical static stresses (0, 3, 6 and 9 MPa, respectively) with the strictly same total charge. By using the fracture mechanics theory, the mechanism of the fracture and propagation behaviors of the cracks along the static principle stresses was analyzed. The result indicates that the stress concentration is first created under the pre-applied vertical stress field around the borehole, where the maximum tensile stress is located on the borehole wall corresponding to the far-field maximum principle stress direction. Then, when disturbed by the dynamic loads induced by blasting, the crack is precociously initiated from the maximum tensile stress point and extends along the maximum principle stress direction. Furthermore, the crack velocity increases accordingly with higher vertical pre-static stresses; the greater the crack velocity increases, the higher the stress intensity factor of the crack tip.
The digital-laser dynamic caustics system in combination with a static-dynamic loading device was utilized in the blasting fracturing test, in which the PMMA specimens underwent four kinds of vertical static stresses (0, 3, 6 and 9 MPa, respectively) with the strictly same total charge. By using the fracture mechanics theory, the mechanism of the fracture and propagation behaviors of the cracks along the static principle stresses was analyzed. The result indicates that the stress concentration is first created under the pre-applied vertical stress field around the borehole, where the maximum tensile stress is located on the borehole wall corresponding to the far-field maximum principle stress direction. Then, when disturbed by the dynamic loads induced by blasting, the crack is precociously initiated from the maximum tensile stress point and extends along the maximum principle stress direction. Furthermore, the crack velocity increases accordingly with higher vertical pre-static stresses; the greater the crack velocity increases, the higher the stress intensity factor of the crack tip.
2017, 37(2): 269-273.
doi: 10.11883/1001-1455(2017)02-0269-05
Abstract:
A two-stage model is proposed herein to predict the through-thickness cone cracking of a reinforced concrete slab struck transversely by a flat-ended projectile at relatively low velocities. The first stage is penetration, in which the resistive pressure from the concrete medium during the penetration process is composed of two parts: quasi-static resistive pressure and dynamic resistive pressure arising from velocity effect. The second stage is cone cracking, in which the dynamic resistive load is obtained through the punch shear strength for the concrete slab loaded quasi-statically by introducing a dynamic enhancement factor. A semi-analytical equation is derived for predicting the critical impact energy that causes the through-thickness cone cracking of the reinforced concrete slab subjected to impact by the flat-nosed projectile. It transpires that the present model predictions are in good agreement with available test data.
A two-stage model is proposed herein to predict the through-thickness cone cracking of a reinforced concrete slab struck transversely by a flat-ended projectile at relatively low velocities. The first stage is penetration, in which the resistive pressure from the concrete medium during the penetration process is composed of two parts: quasi-static resistive pressure and dynamic resistive pressure arising from velocity effect. The second stage is cone cracking, in which the dynamic resistive load is obtained through the punch shear strength for the concrete slab loaded quasi-statically by introducing a dynamic enhancement factor. A semi-analytical equation is derived for predicting the critical impact energy that causes the through-thickness cone cracking of the reinforced concrete slab subjected to impact by the flat-nosed projectile. It transpires that the present model predictions are in good agreement with available test data.
2017, 37(2): 274-282.
doi: 10.11883/1001-1455(2017)02-0274-09
Abstract:
In this paper, the auto-ignition mechanism in the gaseous detonation propagation of the stoichiometric H2-air detonable mixture in a straight tube was numerically studied using an overall one-step chemical reaction model and a detailed chemical reaction model based on the two-dimensional Euler equations. Meanwhile, the ignition delay times predicted by different models under different pressures and at different temperatures were compared and the propagation process of triple-shock points and the cell sizes were investigated. The results demonstrated that the cell sizes are proportional to the ignition delay times, and the ignition delay time in the induction zone is consistent with the average movement period of the triple-shock points. The leading shock compresses the detonable gas and then both the temperature and the pressure of the gas rise. The gas with high temperature and pressure soon finishes the process of auto-ignition, and a lot of heat is released during the ignition to maintain the detonation propagation, which means the auto-ignition mechanism ensures the self-sustained detonation propagation. The ignition delay time is considered as a chemical time scale characterizing the chemical reaction. The period of the movement of the triple-shock points is a characteristic time scale of shock dynamics. The coupling of these two time scales is a principal mechanism in gaseous detonation propagation.
In this paper, the auto-ignition mechanism in the gaseous detonation propagation of the stoichiometric H2-air detonable mixture in a straight tube was numerically studied using an overall one-step chemical reaction model and a detailed chemical reaction model based on the two-dimensional Euler equations. Meanwhile, the ignition delay times predicted by different models under different pressures and at different temperatures were compared and the propagation process of triple-shock points and the cell sizes were investigated. The results demonstrated that the cell sizes are proportional to the ignition delay times, and the ignition delay time in the induction zone is consistent with the average movement period of the triple-shock points. The leading shock compresses the detonable gas and then both the temperature and the pressure of the gas rise. The gas with high temperature and pressure soon finishes the process of auto-ignition, and a lot of heat is released during the ignition to maintain the detonation propagation, which means the auto-ignition mechanism ensures the self-sustained detonation propagation. The ignition delay time is considered as a chemical time scale characterizing the chemical reaction. The period of the movement of the triple-shock points is a characteristic time scale of shock dynamics. The coupling of these two time scales is a principal mechanism in gaseous detonation propagation.
2017, 37(2): 283-290.
doi: 10.11883/1001-1455(2017)02-0283-08
Abstract:
To improve the design of the underwater multi-layered protective bulkhead structure, we carried out several simulations to investigate the characteristics of the damaging load on the void cabin with a multi-layered protective bulkhead subjected to underwater contact explosion. We adopted a typical three-tank structure model for our examination of the characteristics, conducted their analysis using the Dytran software, obtained a simplified model of the load, and derived by fitting the calculation formula of the quasi-static pressure of the load in the smooth pulse stage in the void cabin. The results from the calculation show that the damaging load in the void cabin can be characterized as two stages on a time scale, i.e. the air expansion diffusion stage and the smooth pulse stage, and as two areas in spatial distribution, i.e. the normal reflection area and the Mach reflection area. The loads on the normal reflection area are the initial shock-wave load followed by the quasi-static gas pressure and those on the Mach reflection area are mainly quasi-static gas pressure.
To improve the design of the underwater multi-layered protective bulkhead structure, we carried out several simulations to investigate the characteristics of the damaging load on the void cabin with a multi-layered protective bulkhead subjected to underwater contact explosion. We adopted a typical three-tank structure model for our examination of the characteristics, conducted their analysis using the Dytran software, obtained a simplified model of the load, and derived by fitting the calculation formula of the quasi-static pressure of the load in the smooth pulse stage in the void cabin. The results from the calculation show that the damaging load in the void cabin can be characterized as two stages on a time scale, i.e. the air expansion diffusion stage and the smooth pulse stage, and as two areas in spatial distribution, i.e. the normal reflection area and the Mach reflection area. The loads on the normal reflection area are the initial shock-wave load followed by the quasi-static gas pressure and those on the Mach reflection area are mainly quasi-static gas pressure.
2017, 37(2): 291-298.
doi: 10.11883/1001-1455(2017)02-0291-08
Abstract:
In the present work, to find out how the inert core's physical properties influence the detonation driving of the axial scalar hollow charge, we fabricated three warheads separately stuffed with air, nylon 1011 and aluminum 12, carried out a static explosion test using the pulse X-ray imaging testing technology, and obtained the average velocity of the fragment field and the characteristics of the velocity distribution. Also, using the LS-dyna software and the ALE algorithm, we carried out numerical simulation to analyze the influences of the physical properties of three kinds of inert core's stuffings on the fragment field's shape, shockwave pressure, and initial velocity. The results show that the detonation driving is related with the shock resistance and dynamic stiffness of the stuffing: if the shock resistance of the stuffing is greater than the main charge, the impact on the shock wave pressure is more obvious; the greater the initial pressure on the shell surface, the greater the power of the main charge to drive the fragment, and the faster the fragment's velocity; and the influence of the inert core's physical properties become greater as the inert core or cavity radius increases.
In the present work, to find out how the inert core's physical properties influence the detonation driving of the axial scalar hollow charge, we fabricated three warheads separately stuffed with air, nylon 1011 and aluminum 12, carried out a static explosion test using the pulse X-ray imaging testing technology, and obtained the average velocity of the fragment field and the characteristics of the velocity distribution. Also, using the LS-dyna software and the ALE algorithm, we carried out numerical simulation to analyze the influences of the physical properties of three kinds of inert core's stuffings on the fragment field's shape, shockwave pressure, and initial velocity. The results show that the detonation driving is related with the shock resistance and dynamic stiffness of the stuffing: if the shock resistance of the stuffing is greater than the main charge, the impact on the shock wave pressure is more obvious; the greater the initial pressure on the shell surface, the greater the power of the main charge to drive the fragment, and the faster the fragment's velocity; and the influence of the inert core's physical properties become greater as the inert core or cavity radius increases.
2017, 37(2): 299-306.
doi: 10.11883/1001-1455(2017)02-0299-08
Abstract:
As a thin-walled structure with a very high content of reinforcement, a viaduct box girder is hard to break using the drilling-style blasting. In this paper, based on the demolition project of Zhuan Yang Viaduct in Wuhan, a blasting plan using the water-pressure blasting technique was proposed to demolish multi-chamber reinforced concrete box girders. The breaking process of the multi-chamber box girder was simulated using the dynamic finite element method, the role played by the shockwave and the explosive gas in the breaking process were analyzed and, on the basis of the analysis and the actual blasting results, the charge arrangement, the blasting parameters and the blasting sequence of the water pressure blasting plan were discussed.
As a thin-walled structure with a very high content of reinforcement, a viaduct box girder is hard to break using the drilling-style blasting. In this paper, based on the demolition project of Zhuan Yang Viaduct in Wuhan, a blasting plan using the water-pressure blasting technique was proposed to demolish multi-chamber reinforced concrete box girders. The breaking process of the multi-chamber box girder was simulated using the dynamic finite element method, the role played by the shockwave and the explosive gas in the breaking process were analyzed and, on the basis of the analysis and the actual blasting results, the charge arrangement, the blasting parameters and the blasting sequence of the water pressure blasting plan were discussed.
2017, 37(2): 307-314.
doi: 10.11883/1001-1455(2017)02-0307-08
Abstract:
Based on the structural features of the railgun and the theories of impact thermodynamics, a 3D gouging model containing a micro particle was established using the material point method to simulate the formation process of the rail gouging, and the gouging mechanism and some influencing factors were also analyzed. The results show that the local impact between the armature and the rails at a high velocity produces transient energy exchanges, thus simultaneously forming metal flows with high energy and high pressure that penetrate obliquely into the rail and cause the formation of gouging, for which there is a threshold velocity to produce. As the armature velocity increases, the gouging damage gets more serious. On the other hand, the galling damage occurs when the armature velocity is below the threshold velocity. Both controlling the particle size within a certain range and increasing the head angle of the armature will help to suppress the formation of gouging.
Based on the structural features of the railgun and the theories of impact thermodynamics, a 3D gouging model containing a micro particle was established using the material point method to simulate the formation process of the rail gouging, and the gouging mechanism and some influencing factors were also analyzed. The results show that the local impact between the armature and the rails at a high velocity produces transient energy exchanges, thus simultaneously forming metal flows with high energy and high pressure that penetrate obliquely into the rail and cause the formation of gouging, for which there is a threshold velocity to produce. As the armature velocity increases, the gouging damage gets more serious. On the other hand, the galling damage occurs when the armature velocity is below the threshold velocity. Both controlling the particle size within a certain range and increasing the head angle of the armature will help to suppress the formation of gouging.
2017, 37(2): 315-322.
doi: 10.11883/1001-1455(2017)02-0315-08
Abstract:
In order to investigate the dynamic fracture mechanics behavior and damage morphology of C/SiC ceramic matrix composites, dynamic splitting tensile tests on the C/SiC composites with three different volume fractions of short carbon fiber (16.0%, 21.0%, 24.8%) were carried out by the split Hopkinson pressure bar, the destructive interface part of C/SiC composites was scanned by using scanning electron microscopy and the failure characteristics and toughening mechanism of C/SiC composites were analyzed. The experimental results show that the dynamic tensile strengths of the C/SiC composite specimens with the same short carbon fiber volume fraction increase with the increasing of the impact pressure in the dynamic splitting tensile failure process, the failure of the specimens is significantly correlated with impact pressure and short carbon fiber volume fraction. When the short carbon fiber volume fraction is 16.0%, the tensile strength of the C/SiC composite specimens is the lowest. After the impact, the overall destruction of the C/SiC composite specimens is related to impact pressure and short carbon fiber volume fraction.
In order to investigate the dynamic fracture mechanics behavior and damage morphology of C/SiC ceramic matrix composites, dynamic splitting tensile tests on the C/SiC composites with three different volume fractions of short carbon fiber (16.0%, 21.0%, 24.8%) were carried out by the split Hopkinson pressure bar, the destructive interface part of C/SiC composites was scanned by using scanning electron microscopy and the failure characteristics and toughening mechanism of C/SiC composites were analyzed. The experimental results show that the dynamic tensile strengths of the C/SiC composite specimens with the same short carbon fiber volume fraction increase with the increasing of the impact pressure in the dynamic splitting tensile failure process, the failure of the specimens is significantly correlated with impact pressure and short carbon fiber volume fraction. When the short carbon fiber volume fraction is 16.0%, the tensile strength of the C/SiC composite specimens is the lowest. After the impact, the overall destruction of the C/SiC composite specimens is related to impact pressure and short carbon fiber volume fraction.
2017, 37(2): 323-331.
doi: 10.11883/1001-1455(2017)02-0323-09
Abstract:
To investigate the effects of a T-shaped branch structure on the overpressures during the explosions of the gasoline-air mixture in a pipe, we conducted multiple parameter contrast experiments under the conditions of different initial gasoline vaper volume fractions and different initial ignition energy, and carried out visualized investigation on the flame propagation of the mixture. Our research reveal the following results: the T-shaped branch pipe can enhance the explosive overpressure of the gasoline-air mixture in a pipe, and the degree of this enhancement is closely related with the increase of the initial gasoline vaper volume fraction approximately in a parabolic trend, and with the maximum value obtained around the stoichiometric initial concentration. The maximum explosive overpressures before and after the T-shaped branch pipe increases linearly with the growth of the initial ignition energy. The effects of the T-shaped branch pipe on raising the degree of the explosion mainly originate from four factors: the wave diffraction and reflection, the turbulence enhancement, the sudden expansion of the pipe cross-section area and the reinforcement of the disturbance by the obstacles. It can be seen from the visual analysis that, when the flame propagates through the T-shaped branch pipe, the flame front is distorted seriously into creases, the flame surface area is enlarged, and the returning flames are observed, which exerts a destructive effect on the wall of the T-shaped branch pipe.
To investigate the effects of a T-shaped branch structure on the overpressures during the explosions of the gasoline-air mixture in a pipe, we conducted multiple parameter contrast experiments under the conditions of different initial gasoline vaper volume fractions and different initial ignition energy, and carried out visualized investigation on the flame propagation of the mixture. Our research reveal the following results: the T-shaped branch pipe can enhance the explosive overpressure of the gasoline-air mixture in a pipe, and the degree of this enhancement is closely related with the increase of the initial gasoline vaper volume fraction approximately in a parabolic trend, and with the maximum value obtained around the stoichiometric initial concentration. The maximum explosive overpressures before and after the T-shaped branch pipe increases linearly with the growth of the initial ignition energy. The effects of the T-shaped branch pipe on raising the degree of the explosion mainly originate from four factors: the wave diffraction and reflection, the turbulence enhancement, the sudden expansion of the pipe cross-section area and the reinforcement of the disturbance by the obstacles. It can be seen from the visual analysis that, when the flame propagates through the T-shaped branch pipe, the flame front is distorted seriously into creases, the flame surface area is enlarged, and the returning flames are observed, which exerts a destructive effect on the wall of the T-shaped branch pipe.
2017, 37(2): 332-338.
doi: 10.11883/1001-1455(2017)02-0332-07
Abstract:
Based on the finite element method, the heat conduction equation was made discrete and written as the heat transfer computation code which was then embedded into the existing impact dynamics program. The new program was applied to the numerical analysis of the long-rod projectile penetrating into AD95 ceramic targets in the range of 900-1 800 m/s, and the influence of heat transfer on penetration capability was examined. Calculations show that the calculated penetration depth is less than that by the adiabatic model when the heat transfer is taken into account in the range of 900-1 350 m/s. However, it is opposite when the velocity of the projectile comes in the range of 1 450-1 800 m/s. The results by the heat transfer model and the adiabatic model are close to each other in the range of 1 350-1 450 m/s.
Based on the finite element method, the heat conduction equation was made discrete and written as the heat transfer computation code which was then embedded into the existing impact dynamics program. The new program was applied to the numerical analysis of the long-rod projectile penetrating into AD95 ceramic targets in the range of 900-1 800 m/s, and the influence of heat transfer on penetration capability was examined. Calculations show that the calculated penetration depth is less than that by the adiabatic model when the heat transfer is taken into account in the range of 900-1 350 m/s. However, it is opposite when the velocity of the projectile comes in the range of 1 450-1 800 m/s. The results by the heat transfer model and the adiabatic model are close to each other in the range of 1 350-1 450 m/s.
2017, 37(2): 339-346.
doi: 10.11883/1001-1455(2017)02-0339-08
Abstract:
The mechanical and chemical response of explosives under low velocity impact is the basis for the evaluation of their sensitivity. The drop-weight impact apparatus equipped with the optical photography was used to achieve a frequency of 150 thousand seconds of real time observation. It was capable of distinguishing the samples of "Go" or "no Go" and observing such characteristics of deformation, crushing and breakage, jetting, ignition and combustion evolution, of RDX and HMX particles under low velocity impact. The selected photographic frames show that ignition occurred in the partially melted RDX phase. But for the HMX particles, ignition mainly occurred in the solid phase. A violent jetting phenomenon often occurs before the reaction of combustion. The occurrence of jetting primarily results from the energy released by gaseous products, which push the pulverized or melted explosives splash. The response characteristics of single and multiple granular explosives were compared. Because of the interaction of the hot spots broken, the combustion reaction of the particles is more intense than that of the single particles. The size ratio of the image to the actual length can be used to estimate combustion wave propagation velocity in each case, which is very suitable for characterizing the intensity of the macro-combustion reaction.
The mechanical and chemical response of explosives under low velocity impact is the basis for the evaluation of their sensitivity. The drop-weight impact apparatus equipped with the optical photography was used to achieve a frequency of 150 thousand seconds of real time observation. It was capable of distinguishing the samples of "Go" or "no Go" and observing such characteristics of deformation, crushing and breakage, jetting, ignition and combustion evolution, of RDX and HMX particles under low velocity impact. The selected photographic frames show that ignition occurred in the partially melted RDX phase. But for the HMX particles, ignition mainly occurred in the solid phase. A violent jetting phenomenon often occurs before the reaction of combustion. The occurrence of jetting primarily results from the energy released by gaseous products, which push the pulverized or melted explosives splash. The response characteristics of single and multiple granular explosives were compared. Because of the interaction of the hot spots broken, the combustion reaction of the particles is more intense than that of the single particles. The size ratio of the image to the actual length can be used to estimate combustion wave propagation velocity in each case, which is very suitable for characterizing the intensity of the macro-combustion reaction.
2017, 37(2): 347-352.
doi: 10.11883/1001-1455(2017)02-0347-06
Abstract:
By considering the under-expanded jet flow from nozzle exit, the integral form Euler equations for unsteady compressible flow in the Lagrange coordinates of a moving control volume was developed. By using three-order essentially non-oscillatory (ENO) interpolations at cell interfaces, a three-order ENO finite volume method for the integral form Euler equations was presented. The Sod shock tube case and nozzle outlet under-expanded jet shock wave structures were used to test the proposed scheme. The numerical results demonstrate that the method is accurate and non-oscillatory, and it can capture the wave structures of jet flow fields including shock cell structure, slip lines, jet boundary and the triple point well. Meanwhile, the simulated Mach disk locations in wave structures coincide with the experimentally measured ones, especially the error of the first Mach disk locations in wave structures between the numerical results and the experimental results was less than 1.1%.
By considering the under-expanded jet flow from nozzle exit, the integral form Euler equations for unsteady compressible flow in the Lagrange coordinates of a moving control volume was developed. By using three-order essentially non-oscillatory (ENO) interpolations at cell interfaces, a three-order ENO finite volume method for the integral form Euler equations was presented. The Sod shock tube case and nozzle outlet under-expanded jet shock wave structures were used to test the proposed scheme. The numerical results demonstrate that the method is accurate and non-oscillatory, and it can capture the wave structures of jet flow fields including shock cell structure, slip lines, jet boundary and the triple point well. Meanwhile, the simulated Mach disk locations in wave structures coincide with the experimentally measured ones, especially the error of the first Mach disk locations in wave structures between the numerical results and the experimental results was less than 1.1%.
2017, 37(2): 353-364.
doi: 10.11883/1001-1455(2017)02-0353-12
Abstract:
An experimental system and numerical model were set up to investigate ethylene-air premix deflagration flame propagation and quenching by crimped-ribbon flame arresters in a horizontal pipe, closed at both ends. The deflagration suppression experiment showed that, when the concentration of the flammable gas was close to the stoichiometric ratio (6.6% ethylene by volume), the evolution processes of explosion pressure for the premixed gas of ethylene-air in the pipe (D=32, 80, 400mm) could be divided into four stages: isobaric combustion, slow rise, quick rise and pressure oscillation. During the explosion, due to the interaction between the reflected pressure wave and the flame, the overpressure value fluctuated several times, and the pressure oscillation lasted normally tens of milliseconds. The ethylene-air deflagration flame velocity gradually increased with the increase of the pipe diameter and the decrease of the crimp height. Furthermore, the performance of the flame arrester gradually increased with the increase of the element length. The simulation result showed that the flame front was formed in a semi-sphere shape and spread around in the form of laminar diffusion after ignition at the closed end on the left side. When the flame reached the wall, its shape enlarged under the restriction of the pipe. Then the flame velocity at the near wall gradually exceeded that at the pipe axis, and finally a "tulip" flame was formed. A big amount of heat was lost as the flame front contacted the arrester element, under the influence from the rarefaction waves formed in the reaction area, the chemical reaction rate decreased rapidly, and the flame temperature decreased gradually, which resulted in quenching. During the whole explosion process, the pressure wave and the flame velocity were accompanied by drastic fluctuations through the simulation calculation. The influence mechanism of the porosity and the element length on the flame propagation was analyzed numerically. Finally, the relationship between the deflagration flame velocity and the explosion pressure was derived based on the classic theory of the heat transfer and the experimental data. This study will serve as accurate reference for the design and selection of the crimped-ribbon flame arrester.
An experimental system and numerical model were set up to investigate ethylene-air premix deflagration flame propagation and quenching by crimped-ribbon flame arresters in a horizontal pipe, closed at both ends. The deflagration suppression experiment showed that, when the concentration of the flammable gas was close to the stoichiometric ratio (6.6% ethylene by volume), the evolution processes of explosion pressure for the premixed gas of ethylene-air in the pipe (D=32, 80, 400mm) could be divided into four stages: isobaric combustion, slow rise, quick rise and pressure oscillation. During the explosion, due to the interaction between the reflected pressure wave and the flame, the overpressure value fluctuated several times, and the pressure oscillation lasted normally tens of milliseconds. The ethylene-air deflagration flame velocity gradually increased with the increase of the pipe diameter and the decrease of the crimp height. Furthermore, the performance of the flame arrester gradually increased with the increase of the element length. The simulation result showed that the flame front was formed in a semi-sphere shape and spread around in the form of laminar diffusion after ignition at the closed end on the left side. When the flame reached the wall, its shape enlarged under the restriction of the pipe. Then the flame velocity at the near wall gradually exceeded that at the pipe axis, and finally a "tulip" flame was formed. A big amount of heat was lost as the flame front contacted the arrester element, under the influence from the rarefaction waves formed in the reaction area, the chemical reaction rate decreased rapidly, and the flame temperature decreased gradually, which resulted in quenching. During the whole explosion process, the pressure wave and the flame velocity were accompanied by drastic fluctuations through the simulation calculation. The influence mechanism of the porosity and the element length on the flame propagation was analyzed numerically. Finally, the relationship between the deflagration flame velocity and the explosion pressure was derived based on the classic theory of the heat transfer and the experimental data. This study will serve as accurate reference for the design and selection of the crimped-ribbon flame arrester.
2017, 37(2): 365-376.
doi: 10.11883/1001-1455(2017)02-0365-12
Abstract:
Based on several key issues of the elastic-plastic constitutive model, the equation of state and strength criterion, and the calculation of the strength of concrete under blast and impact loading were reviewed. Concrete being viewed as a binary material composed of matrix and pores, a model including the pore compaction and dynamic damage evolution in elastic-plastic deformation and failure were introduced. On the basis of a mathematical-experimental method, some conclusions and proposals for further reseach were made at the end of the paper.
Based on several key issues of the elastic-plastic constitutive model, the equation of state and strength criterion, and the calculation of the strength of concrete under blast and impact loading were reviewed. Concrete being viewed as a binary material composed of matrix and pores, a model including the pore compaction and dynamic damage evolution in elastic-plastic deformation and failure were introduced. On the basis of a mathematical-experimental method, some conclusions and proposals for further reseach were made at the end of the paper.
