浅谈小学数学模型教学“定模——建模——固模——破模”四步走操作模式的实践与思考

本文从小学数学知识的发展体系（教材）、教师具体的教学操作活动和学生数学学习发展的角度,对＂如何实施有效的数学模型建构教学＂进行了全面深入的思考,提出并系统论述了数学模型建构教学＂定模——建模——固模——破模＂四步走教学操作的观点.其中,＂定模＂通过课例分析,论述了准确定模是实施建模教学的关键,并对教材中数学模型常见的呈现方式进行了筛分.＂建模＂则着重从教学程序设计、教材使用、数学发展、算法优化等四个方面论述了建模教学的注意问题.＂固模＂和＂破模＂论述了分层练习设计和教材习题的优化利用、实现模型思想内化并形成能力的做法.

高中数学科学建模的策略研究

文章旨在研究高中数学科学建模策略,主要采用要素取向建模和行为取向建模的方法构建解题思维模型,在解题思维模型的基础上构建破模要素,并运用构建的思维模型解决高考真题,以期提升学生的综合思维水平,提高学生的学科关键能力.

Strength and damage evolution mechanism of rock mass with holes under cyclic loading

The damage and failure law of rock mass with holes is of great significance to the stability control of roadways. This study investigates the mechanical properties and failure modes of porous rock masses under cyclic loading, elucidates the acoustic emission (AE) characteristics and their spatial evolution, and establishes the interrelation among AE, stress, strain, time, and cumulative damage. The results reveal that the rock mass with holes and the intact rock mass show softening and hardening characteristics after cyclic loading. The plastic strain of the rock mass with holes is smaller than that of the intact rock mass, and the stress −strain curve shows hysteresis characteristics. Under uniaxial compression, the pore-bearing rock mass shows the characteristics of higher ringing count, AE energy, b-value peak, and more cumulative ringing count in the failure stage, while it shows lower characteristics under cyclic action. At the initial stage of loading, compared with the intact rock mass, the pore-containing rock mass shows the characteristics of a low b-value. The AE positioning and cumulative damage percentage are larger, and the AE positioning is denser around the hole. The specimen with holes is mainly shear failure, and the complete specimen is mainly tensile shear failure.

Failure mechanisms and destruction characteristics of cemented coal gangue backfill under compression effect of non-uniform load

Backfill mining is one of the most important technical means for controlling strata movement and reducing surface subsidence and environmental damage during exploitation of underground coal resources. Ensuring the stability of the backfill bodies is the primary prerequisite for maintaining the safety of the backfilling working face, and the loading characteristics of backfill are closely related to the deformation and subsidence of the roof. Elastic thin plate model was used to explore the non-uniform subsidence law of the roof, and then the non-uniform distribution characteristics of backfill bodies’ load were revealed. Through a self-developed non-uniform loading device combined with acoustic emission (AE) and digital image correlation (DIC) monitoring technology, the synergistic dynamic evolution law of the bearing capacity, apparent crack, and internal fracture of cemented coal gangue backfills (CCGBs) under loads with different degrees of non-uniformity was deeply explored. The results showed that: 1) The uniaxial compressive strength (UCS) of CCGB increased and then decreased with an increase in the degree of non-uniformity of load (DNL). About 40% of DNL was the inflection point of DNL-UCS curve and when DNL exceeded 40%, the strength decreased in a cliff-like manner;2) A positive correlation was observed between the AE ringing count and UCS during the loading process of the specimen, which was manifested by a higher AE ringing count of the high-strength specimen. 3) Shear cracks gradually increased and failure mode of specimens gradually changed from “X” type dominated by tension cracks to inverted “Y” type dominated by shear cracks with an increase in DNL, and the crack opening displacement at the peak stress decreased and then increased. The crack opening displacement at 40% of the DNL was the smallest. This was consistent with the judgment of crack size based on the AE b-value, i. e., it showed the typical characteristics of “small b-value-large crack and large b-value-small crack”. The research results are of significance for preventing the instability and failure of backfill.

Static and dynamic tensile failure characteristics of rock based on splitting test of circular ring

Static and dynamic splitting tests were conducted on ring marble specimens with different internal diameters to study the tensile strength and failure modes with the change of the ratio of internal radius to external radius （ρ） under different loading rates. The results show that the dynamic tensile strength of disc rock specimen is approximately five times its static tensile strength. The failure modes of ring specimens are related to the dimension of the internal hole and loading rate. Under static loading tests, when the ratio of internal radius to external radius of the rock ring is small enough （ρ〈0.3）, specimens mostly split along the diametral loading line. With the increase of the ratio, the secondary cracks are formed in the direction perpendicular to the loading line. Under dynamic loading tests, specimens usually break up into four pieces. When the ratio ρreaches 0.5, the secondary cracks are formed near the input bar. The tensile strength calculated by Hobbs’ formula is greater than the Brazilian splitting strength. The peak load and the radius ratio show a negative exponential relationship under static test. Using ring specimen to determine tensile strength of rock material is more like a test indicator rather than the material properties.

Mechanical model for failure modes of rock and soil under compression

The failure modes of rock and soil under compression are complex phenomena that have not been explained in a mechanical perspective. However, large amounts of studies indicate that the failure modes of rock and soil samples can be categorized into eight types. In this work, the inner tensile stress and the dissipation and conversion of energy of rock and soil under compression are analyzed, then the effective conversion coefficient of energy is deduced, thus the tensile failure criterion of rock and soil under compression is established. Combined with the shear strength criterion of Mohr–Coulomb, a tensile joint shear strength criterion for rock and soil under compression is built. Therefore, a mechanical criterion model concerning the failure modes of rock and soil under compression is established and verified by tests. This model easily explains the test results in the existing literature and many natural phenomena, such as collapse.

Failure characteristics and its influencing factors of talus-derived rock mass during open-pit mining

The failure characteristic of talus-derived rock mass continues to challenge quantitative hazard assessments in open-pit mining. Physical model test was used to assess the failure modes and mechanisms on talus-derived rock mass. The different types of failure modes of the talus-derived rock mass were introduced and a possible failure mechanism relation between the failure zone and the structure of the talus-derived rock mass was also shown. The physical model test results indicate that the rainfall has significant influence on the stability and failure modes of talus-derived rock mass during open-pit mining. The development of the seepage area caused by rainfall initiates the localized failure in that particular area, and the initiation of localized instability is mainly induced by stress changes concentrated in the seepage area.

Comparative study on three dynamic modulus of elasticity and static modulus of elasticity for Lodgepole pine lumber

The dynamic and static modulus of elasticity （MOE） between bluestained and non-bluestained lumber of Lodgepole pine were tested and analyzed by using three methods of Non-destructive testing （NDT）, Portable Ultrasonic Non-destructive Digital Indicating Testing （Pundit）, Metriguard and Fast Fourier Transform （FFT） and the normal bending method. Results showed that the dynamic and static MOE of bluestained wood were higher than those of non-bluestained wood. The significant differences in dynamic MOE and static MOE were found between bulestained and non-bluestained wood, of which, the difference in each of three dynamic MOE （Ep. the ultrasonic wave modulus of elasticity, Ems, the stress wave modulus of elasticity and El, the longitudinal wave modulus of elasticity） between bulestained and non-bluestained wood arrived at the 0.01 significance level, whereas that in the static MOE at the 0.05 significance level. The differences in MOE between bulestained and non-bluestained wood were induced by the variation between sapwood and heartwood and the different densities of bulestained and non-bluestained wood. The correlation between dynamic MOE and static MOE was statistically significant at the 0.01 significance level. Although the dynamic MOE values of Ep, Em, Er were significantly different, there exists a close relationship between them （arriving at the 0.01 correlation level）. Comparative analysis among the three techniques indicated that the accurateness of FFT was higher than that of Pundit and Metriguard. Effect of tree knots on MOE was also investigated. Result showed that the dynamic and static MOE gradually decreased with the increase of knot number, indicating that knot number had significant effect on MOE value.

Construction of universal quantitative models for the determination of cefoperazone sodium/sulbactam sodium for injection from different manufacturers using near-infrared reflectance spectroscopy

To develop near-infrared （NIR） reflectance spectroscopic methods for the quantitative analysis of cefoperazone sodium/ sulbactam sodium from different manufacturers for injection powder medicaments. Various powders of cefoperazone sodium/ sulbactam sodium were directly analyzed by non-destructive NIR reflectance spectroscopy using the spectrometer EQUINOX55. Two quantitative methods via integrating sphere （IS） and fiberoptic probe （FOP） models were explored from 6 batches of commercial samples and 42 batches of laboratory samples at a content ranging from 30% to 70% for cefoperazone and 60% to 20% for sulbactam. The root mean square errors of cross validation （RMSECV） and the root mean square errors of prediction （RMSEP） of IS were 1.79% and 2.85%, respectively, for cefoperazone sodium, and were 1.86% and 3.08%, respectively, for sulbactam sodium; and those of FOP were 2.93% and 2.92%, respectively, for cefoperazone sodium, and were 2.23% and 3.01%, respectively, for sulbactam sodium. Based on the ICH guidelines and Ref. 12, the quantitative models were then evaluated in terms of specificity, linearity, accuracy, precision, robustness and model transferability. The non-destructive quantitative NIR methods used in this study are applicable for rapid analysis of injectable powdered drugs from different manufacturers.

Uniform asymptotics for finite-time ruin probability in some dependent compound risk models with constant interest rate

Consider two dependent renewal risk models with constant interest rate. By using some methods in the risk theory, uniform asymptotics for finite-time ruin probability is derived in a non-compound risk model, where claim sizes are upper tail asymptotically independent random variables with dominatedly varying tails, claim inter-arrival times follow the widely lower orthant dependent structure, and the total amount of premiums is a nonnegative stochastic process. Based on the obtained result, using the method of analysis for the tail probability of random sums, a similar result in a more complex and reasonable compound risk model is also obtained, where individual claim sizes are specialized to be extended negatively dependent and accident inter-arrival times are still widely lower orthant dependent, and both the claim sizes and the claim number have dominatedly varying tails.

Contour blasting parameters by using a tunnel blast design mode

The quality of contour blasting depends on many initial blasting parameters.The parameters including blasthole diameter,rock Protodyakonov coefficient,tunnel area and distance between cracks on the tunnel face are more important.In this study,an algorithm linking between Delphi programming language and AutoCAD was created to develop a tunnel blasting model.Using this model,tunnel contour blasting passport in AutoCAD can be obtained automatically.The effects of rock Protodyakonov coefficient and cracks’distance on blastholes number and specific charge with the variation of blasthole diameter and the semi-circular tunnel face area were investigated to yield a set of equations with the highest correlations.The results show that specific charge increases as rock Protodyakonov coefficient,cracks’distance and drillhole diameter increase,but decreases when tunnel face area increases.In addition,the number of drillholes increases linearly as tunnel face area increases but decreases when drillhole diameter increases.

Trend analysis and comparison of basic parameters for tunnel blast design models

One of the most important factors influencing on a tunnel blast efficiency is the proper design of blasting pattern. Among blasting parameters, blasthole diameter and tunnel face area are more significant so that any change in these parameters could finally affect on specific charge and specific drilling. There are mainly two groups of methods for tunnel blast design categorized based on the parallel cuts and angular cuts. In this research, a software for tunnel blast design was developed to analyze the effect and sensitiveness of blasthole diameter and the tunnel face area on blasting results in different blast design models. Using the software, it is quickly possible to determine specific charge, specific drilling and number of blastholes for each blast design model. The relations between both of blasthole diameters and the tunnel face area with the above parameters in different blast design models were then investigated to yield a set of equations with the highest correlations to compare the methods. The results showed that angular method requires more blasthole numbers than parallel method in similar condition(blasthole diameter and tunnel face area). Moreover, the specific charge values yielded by the two methods are approximately the same and very close together.

Mechanical properties and rockburst proneness of phyllite under uniaxial compression

To investigate the influence of the bedding angle,β,on the mechanical properties and rockburst proneness,uniaxial compression tests were conducted using cylindrical phyllite specimens with different bedding angles.According to the results,the peak stress,peak strain,cumulative acoustic emission counts,and potential energy of the elastic strain exhibited a U-shaped change trend.With an increase in β from 0°to 90°,the failure mode transformed from tensile splitting failure along the bedding plane to shear slip failure along the weak bedding plane.Finally,the failure mode evolved into a tensile splitting failure across the bedding plane.When β=15°,30°,and 45°,the phyllite specimens exhibited strong,slight,and moderate rockburst proneness,with strong,slight,and moderate shear slip rockbursts,respectively.When β=0°,60°,75°,and 90°,the phyllite specimens had extremely strong rockburst proneness,and an extremely strong strain rockburst occurred.

Mechanical and energy dissipation characteristics of granite under cyclic impact loading

This study investigated the effect of repeated blasting on the stability of surrounding rock during the construction of a tunnel or city underground engineering.The split Hopkinson pressure bar(SHPB)was used to carry out cyclic impact tests on granite samples,each having a circular hole,under different axial pressures,and the cumulative specific energy was proposed to characterize the damage characteristics of the rock during the cyclic impact.The mechanical properties and the energy absorbed by the granite samples under cyclic impact loads were analyzed.The results showed that under different axial pressures,the reflected waveform from the samples was characterized by“double-peak”phenomenon,which gradually changed to“single-peak”wi th the increase in damage value.The dynamic peak stress of the sample first increased and then decreased with an increase in impact times.The damage value criterion established based on the energy dissipation could well characterize the relationship between the damage and the number of impacts,which showed a slow increase,steady increase,and high-speed increase,and the damage value depended mainly on the last impact.Under the action of different axial pressures,all the failure modes of the samples were axial splitting failures.As the strain rate increased,with an increase in the dimension of the block,the sizes of the rock fragments decreased,and the fragmentation became more severe.

Thermo-mechanical coupled particle model for rock

A thermo-mechanical coupled particle model for simulation of thermally-induced rock damage based on the particle simulation method was proposed.The simulation results of three verification examples,for which the analytical solutions are available,demonstrate the correctness and usefulness of the thermo-mechanical coupled particle model.This model is applied to simulating an application example with two cases：one is temperature-independent elastic modulus and strength,while the other is temperature-dependent elastic modulus and strength.The related simulation results demonstrate that microscopic crack initiation and propagation process with consideration of temperature-independent and temperature-dependent elastic modulus and strength are different and therefore,the corresponding macroscopic failure patterns of rock are also different.On the contrary,considering the temperature-dependent elastic modulus and strength has no or little effect on the heating conduction behavior.Numerical results,which are obtained by using the proposed model with temperature-dependent elastic modulus and strength,agree well with the experimental results.This also reveals that the rock subjected to heating experiences much more cracking than the rock subjected to cooling.

Capacities and Failure Modes of Suction Bucket Foundation with Internal Bulkheads

Suction bucket foundations can be divided into four compartments by cruciform internal bulkheads,thereby yielding better capacity in certain conditions than those without internal bulkheads.As yet,no systematic study has been conducted regarding the effects of cruciform internal bulkheads on the capacities of suction bucket foundations.In this study,we established a large number of finite element models of suction bucket foundations with and without cruciform internal bulkheads and of solid embedded circular foundations.We found the uniaxial capacities and failure modes of suction bucket foundations with various depth ratios to remain basically unaffected by internal bulkheads in uniform clays.However,in inhomogeneous clay with high strength heterogeneity,we observed the uniaxial moment and horizontal capacities and corresponding failure modes of suction bucket foundations with a low depth ratio to be obviously affected by internal bulkheads.In this case,the uniaxial moment capacities,in particular,as well as the horizontal capacities of suction bucket foundations with cruciform internal bulkheads become obviously greater than those without internal bulkheads.Under combined loading,we found the failure envelopes of suction bucket foundations with and without cruciform internal bulkheads and of solid circular foundation to also be basically consistent in uniform clays.However,in inhomogeneous clay with high strength heterogeneity,cruciform internal bulkheads can obviously change the shapes of the failure envelopes of bucket foundations with a small depth ratio.We conclude that when the acting vertical load or foundation depth is relatively small,suction bucket foundations with cruciform internal bulkheads can be subjected to larger moment and horizontal loads in soft clays with high strength heterogeneity.

Developing a mathematical assessment model for blasting patterns management: Sungun copper mine

Blasting is one of the most important operations in the mining projects that has effective role in the whole operation physically and economically. Unsuitable blasting pattern may lead to unwanted events such as poor fragmentation, back break and fly rock. Multi attribute decision making(MADM) can be useful method for selecting the most appropriate blasting pattern among previously performed patterns. In this work, initially, from various already performed patterns, efficient and inefficient patterns are determined using data envelopment analysis(DEA). In the second step, after weighting impressive attributes using experts' opinion, elimination Et choice translating reality(ELECTRE) was used for ranking the efficient patterns and recognizing the most appropriate pattern in the Sungun Copper Mine, Iran. According to the obtained results, blasting pattern with the hole diameter of 15.24 cm, burden of 3 m, spacing of 4 m and stemming of 3.2 m has selected as the best pattern and has selected for future operation.

Upper bound analytical solution for surrounding rock pressure of shallow unsymmetrical loading tunnels

By combining the results of laboratory model tests with relevant flow rules, the failure mode of shallow unsymmetrical loading tunnels and the corresponding velocity field were established. According to the principle of virtual power, the upper bound solution for surrounding rock pressure of shallow unsymmetrical loading tunnel was derived and verified by an example. The results indicate that the calculated results of the derived upper bound method for surrounding rock pressure of shallow unsymmetrical loading tunnels are relatively close to those of the existing "code method" and test results, which means that the proposed method is feasible. The current code method underestimates the unsymmetrical loading feature of surrounding rock pressure of shallow unsymmetrical loading tunnels, so it is unsafe; when the burial depth is less or greater than two times of the tunnel span and the unsymmetrical loading angle is less than 45°, the upper bound method or the average value of the results calculated by the upper bound method and code method respectively, is comparatively reasonable. When the burial depth is greater than two times of the tunnel span and the unsymmetrical loading angle is greater than 45°, the code method is more suitable.

Three-dimensional upper bound limit analysis of underground cavities using nonlinear Baker failure criterion

A generalized nonlinear Baker failure criterion is employed with the upper bound limit analysis to study the surrounding rock stability of underground cavities. A three-dimensional(3D) failure mode is established by extending the two-dimensional(2D) failure mode, which offers an upper bound expression of the surrounding rock pressure. This method is validated with a series of examples before the influence of four parameters of scale parameter, curvature parameter, shift parameter and lateral pressure coefficient, on the surrounding rock pressure is analyzed. According to these results, failure ranges of the underground cavities are determined. The following conclusions are reached:(1) the proposed approach is more accurate to predict surrounding rock pressure than the Mohr-Coulomb failure criterion;(2) the surrounding rock with large scale parameter, curvature parameter, shift parameter, and lateral pressure coefficient can lead to a more stable underground cavity;(3) the failure range in 3D mode can be predicted according to the upper bound solutions.

Mechanical properties of deep sandstones under loading rate effect

The advance speed of the working face in coal mines can significantly affect the fluctuation frequency of abutment pressure in front of the coal body.Moreover,it has a certain correlation with the change of axial loading rate in coal and rock mechanics test.Therefore,uniaxial compression tests under various loading rates of 0.05,0.1,0.15,0.25,0.5 MPa/s were conducted using 2000 kN triaxial testing machine and PCI-2 acoustic emission test system to study the loading rate effect on the mechanical properties of deep sandstones.The results show that 1)the peak strength and elastic modulus of the deep sandstone increase with the loading rate increasing;2)with the loading rate increasing,the deep sandstone transforms from plastic-elastic-plastic to plastic-elastic and moreover,the failure mode gradually transfers from type I to type III;3)With the loading rate increasing,the total input strain energy,elastic strain energy,and dissipated strain energy generally increase;4)the damage variable presents the evolution characteristics of inverted“S”shape with time,and with the loading rate increasing,the damage degree of the deep sandstone is aggravated.The conclusion obtained can provide the theoretical basis for the stability control of the surrounding rock in deep engineering.