Experiment and modeling on thecompressive behaviors for porous silicon nitride ceramics

Experiment and modeling on thecompressive behaviors for porous silicon nitride ceramics
Experiment and modeling on thecompressive behaviors for porous silicon nitride ceramics

Experiment and modeling on the compressive behaviors

for porous silicon nitride ceramics

Zixing Lu a,n,Qiang Liu a,Haitao Han a,Dahai Zhang b

a Institute of Solid Mechanics,Beihang University,Beijing100191,China

b National Key Laboratory of Advanced Functional Composite Materials Technology,Beijing100076,China

a r t i c l e i n f o

Article history:

Received14November2011

Received in revised form

6June2012

Accepted17August2012

Available online24August2012

Keywords:

Porous Si3N4ceramics

Compression

Finite element

Young’s modulus

Strength

a b s t r a c t

Two types of porous Si3N4ceramics with different porosity are fabricated by gel casting technique,

where the high pure Al2O3and Y2O3are selected as sintering additive.The effective Young’s modulus

and compression strength are tested by compressive experiments,respectively.The present emphasis

is placed on the mechanical characterization of porous Si3N4ceramics by employing?nite element(FE)

method.Extracting the primary features of bonded networks,the microstructure of real material is

reconstructed in the numerical model.For the obtained materials with different porosity,their

compressive behaviors are modeled by FE simulation,respectively.Afterwards,the effective Young’s

modulus and compression strength are calculated from the numerical https://www.360docs.net/doc/d25982605.html,pared with

experiment data,the calculated results provide a suf?cient accordance.Moreover,the modeled failure

mechanism in microstructure is also veri?ed by experimental observation.Utilizing the present FE

model,the in?uences of grain aspect ratio and properties of grain boundaries on the effective Young’s

modulus and compression strength are also investigated,which provides an insight into the relation-

ship between microstructure and macro-mechanical properties for porous Si3N4ceramics.

&2012Elsevier B.V.All rights reserved.

1.Introduction

Porous Si3N4ceramics with combined network of rod-like b-

Si3N4grains and grain boundaries exhibit unique properties,such

as high strength,good thermal shock resistance,damage toler-

ance[1–5].Due to their excellent mechanical properties and

thermal properties,porous Si3N4ceramics have been widely used

in the engineering applications such as?ltering materials,separa-

tion membranes,catalyst supports,and high-temperature insula-

tions.However,the mechanical properties still cannot meet the

high reliability and performance speci?cation required for the

advanced engineering applications especially when the porosity is

relatively high.It is thus that various processing techniques have

been reported to manufacture porous Si3N4ceramics with high

performance[1,2,5–10].

As one of the typical fragile brittle materials,the mechanical

properties of porous Si3N4ceramics have been investigated

extensively[1–5,7,11–13].The?exural strength has been exten-

sively tested in the open literatures and the failure mechanisms

are classi?ed into two categories by Inagaki et al.[1],which

include bridging of the crack by unbroken?brous reinforcing

grains that is partially debonded from the matrix(or grain

bridging)and frictional pull-out of?brous grains(or grain pull-

out).Similar conclusions are also reported by other authors[7,12].

Compared with the abundant works on bending experiment,the

compressive behaviors of porous Si3N4ceramics are seldom

studied.In addition,the emphasis of this study is placed on the

numerical simulation of porous Si3N4ceramics.However,the

available literatures show that the FE model based upon micro-

structure is not adequate to simulate the bending behavior of real

material unless the multi-scale analysis is employed[14,15].

Generally,so-called multi-scale analysis is not a feasible scheme

before the simulation of uniaxial properties is carried out.Those

are the reasons that the compressive behaviors of porous Si3N4

ceramics are investigated in this study.

Some empirical functions can be used to express the relations

between mechanical properties of porous ceramics and porosity

[16–19].Actually,the available literatures reveal that the

mechanical properties of porous Si3N4ceramics mostly depends

on microstructure and the composition of grain boundary phase

[11,20,21].It is therefore essential to construct the connections

between micro-parameters and the macroscopic mechanical

properties.To our knowledge,the theoretical or numerical mod-

els for porous Si3N4ceramics are scarce compared with other type

of porous materials(honeycombs,foams,etc).However,the

methods used to analyze other types of porous ceramics could

supply some references.The modulus-porosity relations were

obtained by Pabst et al.[22,23]based upon the theory of

Contents lists available at SciVerse ScienceDirect

journal homepage:https://www.360docs.net/doc/d25982605.html,/locate/msea

Materials Science&Engineering A

0921-5093/$-see front matter&2012Elsevier B.V.All rights reserved.

https://www.360docs.net/doc/d25982605.html,/10.1016/j.msea.2012.08.081

n Corresponding author.Tel.:t861082315707;fax:t861082318501.

E-mail address:luzixing@https://www.360docs.net/doc/d25982605.html,(Z.Lu).

Materials Science&Engineering A559(2013)201–209

Hashin–Shtrikman bounds,and the theoretical relations were consistent well with experimental data of several types of porous ceramics.Despite this,this analyzed model was only valid when the porosity was less than0.5.Besides,it could not be employed to predict the strength of porous ceramics.Utlizing a method of mesomechanical approach,Sadowski and Samborski[24]mod-eled the failure strength of porous Al2O3ceramics under uniaxial tension and compression.Furthermore,Sadowski and Samborski [25]proposed a method based upon the mesomechanical model and phenomenological model to demonstrate the development of damage state for two polycrystalline porous ceramics(MgO and Al2O3).The drawback was that their method was only valid when the porosity was in the range of0–0.2.Except for theoretical analysis,the FE method is another effective tool for studying the mechanical properties of porous ceramics.Roberts and Garboczi [26],respectively established three types of FE models by over-lapping solid spheres,overlapping spherical pores and overlap-ping ellipsoidal pores.In the porosity range of0–0.5,these FE models could give a good approach for the effective Young’s modulus of porous ceramics.Based upon the works of Roberts and Garboczi,Bruno et al.[27]explored the effects of porosity and pore morphology on the conversion of macro-to micro-strain and stress.Their modeled results compared well with experiment test, showing that the ratio of macro-strain to average micro-strain was affected by pore morphological.In order to characterize the microstructure of real material,Bartuli et al.[28]constructed a FE model for porous ZrO2ceramics based on the digital image and simulated the overall elastic properties.They found that the real microstructure played a fundamental role in determining the actual stress distribution within the porous structure.

The aforementioned literatures show that the precise recon-struction of microstructure of real material is very signi?cant for simulating the mechanical properties of porous Si3N4ceramics. However,the mentioned models cannot reveal the bonded structure of porous Si3N4ceramics,which is not adequate to demonstrate their mechanical properties.In addition,the above models are only valid for the porous ceramics with porosity of less than50%.It is contrast that the porosity of porous Si3N4 ceramics in engineering applications is possibly higher such as the thermal insulation material.Considering the problem,we try to construct a FE model which can simulate the real microstruc-ture of porous Si3N4ceramics.Meanwhile,the compressive behaviors of porous Si3N4ceramics are seldom tested or modeled up to now.All these are the motive of this work,studying the compressive behaviors of porous Si3N4ceramics utilizing both experiment and FE method.First,the effective Young’s modulus and strength will be measured by compressive experiment for the samples with different porosity.A FE model with combined network of b-Si3N4grains and grain boundaries is then build up based on the scanning electron microscope(SEM)observation. That is followed the analysis of numerical results and comparison with the experimental data.Finally,the microstructure para-metric study of FE model is conducted and the conclusions are summarized.

2.Experimental procedure

In this study,the gel casting technique is employed in fabrication of porous Si3N4ceramics.The commercially available Si3N4powder(a-Si3N4purity:494%,mean particle size:0.37m m) was selected.The sintering additive is consist of Al2O3(purity: 499%,mean particle size: 1.07m m)and Y2O3(purity: 499%,mean particle size:4.74m m).First,the polyacrylamide, dispersant(ammonium salt of poly)and monomers(acrylamide and N,N0-methylenebisacrylamide,AM and MBAM)were completely dissolved in deionizer water by stirring.The above Si3N4powder and sintering additive were poured into the premix solution in a plastic bottle,and then the slurry was milled with high-purity silicon nitride balls for about12h.After degassed,the milled slurry was cast into a glass mold and kept at651C for about40min to polymerize to form gelled body.Next,the gelled part was removed from the mold and dried to remove the solvent system.Through the drying process at251C of drying tempera-ture and98%of relative humidity,the sample were placed in a heating furnace(High multi-5000,Fijidempa Co.Ltd.,Osaka, Japan).Here the two-step sintering process was employed:the samples were heated from220to5501C at a heating rate of101C/h under air atmosphere and then temperature was kept at5501C for 3h;sintering was subsequently carried out at a heating rate of 101C/min and1h holding time was?nally performed at18001C in a nitrogen atmosphere.

For the gel casting technique,it is well known that the pores of sintered body mainly originate from the residual micro-space of the organic processing aids in the green body during organic binder burnout.Therefore,the sintering samples with different porosity can be fabricated by merely controlling the content of monomers. The in?uences of the monomer content on the performances of green body and sintered body of porous Si3N4ceramics have been reported in the work of Yu et al.[29].In this study,two types of porous Si3N4ceramics are obtained by adjusting the monomer content.The open-porosity of the sintered samples was measured by the Archimedes method.A scanning electric microscope(SEM, Model S-570,Hitachi)was used to characterize the microstructures of the sintered samples.The compressive behaviors of porous Si3N4 ceramics were measured in a Universal Test Machine(MTS880, MTS System Corporation,USA).The specimens used for compres-sion test were shaped into block,with the dimensions of16?8?8mm.For each type of porous material,six specimens were selected for compressive test.

3.Finite element model

During sintering,the a-Si3N4was transformed into the b-Si3N4 phase and a unique pore structure forms as the growth of b-Si3N4 grains.Fig.1shows the SEM photos of porous Si3N4ceramics, which is consisted of b-Si3N4grains with random orientation. The rod-like b-Si3N4grains are bonded by the Y–Si–Al–O–N glass phase and form an interlocked microstructure.It is generally accepted that the growth of b-Si3N4grains are primarily determined by the sintering condition.As mentioned earlier,the porosity of present sintering body is controlled by merely adjusting the mono-mer content.Consequently,the morphology of b-Si3N4grains is almost consistent for the two types of porous Si3N4ceramics with different porosity.As shown in Fig.1,the grain size of samples with different porosity nearly has no difference.Different from other types of porous materials(honeycomb,foams,etc.),the microstruc-ture of porous Si3N4ceramics is random and cannot be represented by a unit model,which results in dif?culty of mechanical character-ization.In addition,the available researches indicate that mechan-ical properties of porous Si3N4ceramics are extremely determined by their complex microstructure.It is therefore essential to establish a representative volume element(RVE)model based upon the SEM photos in order to reveal mechanical behaviors of porous Si3N4 ceramics.

3.1.Timoshenko beam assumptions

The morphology of b-Si3N4grains have signi?cant in?uences on the mechanical properties of porous Si3N4ceramics,which have been reported in the literatures[30,31].Therefore,the

Z.Lu et al./Materials Science&Engineering A559(2013)201–209 202

characterization of b -Si 3N 4grains plays a pivot role in the construction of numerical model.However,several problems exist for the reconstruction of real microstructure and some simpli?cations are required.For our numerical model,crucial approaches to real microstructure are presented in the following.

First,the grains are idealized as cylinders despite that most of b -Si 3N 4grains are acicular.Second,the equivalent diameters and lengths of b -Si 3N 4grains are not uniform based on the SEM photos (Fig.1).Nevertheless,it is not feasible to obtain real distribution function of grain aspect ratio by experimental obser-vation.Instead,the mean grain aspect ratio is adopted and the value is calculated by measuring grains lie in the plane of SEM photos.It is necessary to point out that many SEM photos are selected and Fig.1is only one of them.As the sintering condition is consistent for porous Si 3N 4ceramics,the grain aspect ratio is rarely affected by the porosity in this study,which has been veri?ed by Fig.1.As a result,we assume the mean value of grain aspect ratio is constant to porosity,which is taken as 12.

Another problem is the choice of element type when numer-ical simulation is carried out.Considering the location and orientation of grains are random,the model domain size must be large enough to evaluate overall mechanical response.In other words,the amount of grains in RVE model will be extensive,

which reaches an order of 103in this study.Clearly,local strain and stress ?elds can be modeled more precisely if the Solid element is employed.However,the excessive amount of elements leads to great computational challenge.Considering the value of grain aspect ratio is up to 12,b -Si 3N 4grains can be modeled as the Beam element to engineering accuracy,which can greatly improve computational ef?ciency.Owing to the beam assumption of b -Si 3N 4grains,grain boundaries bonding rod-like grains together cannot be modeled as the Solid element.Otherwise,the mismatching of freedom degree,caused by the Beam element and Solid element,has negative in?uences on accuracy of FE simulation.Different with the stretching or bending deformation mechanism of rod-like grains,the dominant deformation of grain boundaries is shearing or torsion when macro-sample is under compression.Herein the Timoshenko beam is recommended to model the mechanical behaviors of grain boundaries,which includes deformation mechanism of bending,stretching,shearing and torsion.

3.2.Generation of numerical model

In this study,the compressive behaviors of FE model are determined in ANSYS 12.1.According to the aforementioned simpli?cation,the grains are modeled as the Beam 188with circular section,satisfying the speci?c aspect ratio.The diameter is taken as D gr ?0.65m m,con?rmed by the mean value based upon the width of grains in the SEM photos.Similarly,the grain boundaries are also equivalent to the Beam Element with the same diameter as grains in the FE model despite that the real micro-morphology is more complex.The procedure to generate the FE model in this study follows the steps listed below:i.Lines with random locations and orientations are created in a three-dimensional domain corresponding to the size of the RVE model with the volume V .The length of lines can be calculated by the speci?ed grain diameter and grain aspect ratio of b -Si 3N 4grains.

ii.Any two lines,when their distance is lower than the speci?ed value (D 0),are connected by their common perpendicular.The D 0represents the maximum distance between the bonded grains,and it is taken as D 0?3D gr .The isolated lines,which are not connected by other lines,are deleted.iii.The actual porosity of FE model is calculated by

P ?

V gr ,s tV bnd ,s

V

e1T

where the V gr,s ,V bnd,s are the volumes of grains and grain boundaries,respectively.If the error of actual porosity of the FE model relative to the prede?ned one is lower than 0.002,the procedure is continued to next step,otherwise return step i.

iv.Set the element attribute (sectional area,material attribute,

etc.)and mesh the above model.Finally,a three-dimensional FE model can be established and a representative model generated by the aforementioned steps is shown in Fig.2.

The mechanical properties of b -Si 3N 4grains are based upon the experimental measurement of commercial b -Si 3N 4whisker [32],which are E gr ?382.2GPa for the Young’s modulus and s lim,gr ?13.72GPa for the tensile strength in the elongated direction,respectively.As known from the available literatures,interfacial debonding is one of typical microstructure failure for porous Si 3N 4ceramics [33,34].Generally,interfacial debonding is determined by the mechanical properties of interface between grain and grain boundaries.Nevertheless,the present simpli?ed model consisted

of

Fig.1.The SEM microstructure of porous Si 3N 4ceramics;(a)the sample with 52.0%porosity,(b)the sample with 62.4%porosity.

Z.Lu et al./Materials Science &Engineering A 559(2013)201–209203

the Beam element cannot be competent.A compromised treatment is adopted that interfacial debonding is also considered as the fracture of grain boundaries.Accordingly,the grain boundary phase in FE model is actually an equivalent material,whose mechanical properties includes combined in?uences of grain boundaries and interfaces.The mechanical properties of Y–Si–Al–O–N glass have been investigated in some literatures [35–38],showing that the elastic modulus is about 128–165GPa and the hardness is 8.0–10.7GPa.Unlike the scheme used for grains in FE model,it is dif?cult to specify the mechanical parameters of grain boundaries.Despite that the main chemical constituent is known as Y–Si–Al–O–N glass phase,mechanical properties of grain boundaries are weakened because the impurities concentrated in the glass phase during sintering process of porous Si 3N 4ceramics [39].Due to the mentioned reason,the material parameters of grain boundaries cannot be speci?ed directly.In this study,a compromised method is adopted as follows:For one type of experimental sample,the corresponding numerical model is constructed according to the measured porosity.The FE simulation is conducted to model the mechanical response of real material in conjunction with the values of Young’s modulus and fracture strength of equivalent grain boundaries can be calculated.Finally,the numerical model corre-sponding to another type of material is established,and then is utilized to predict the measured values of experimental samples.It is important to point out that the equivalent grain boundaries include the effects of interfaces,which is not emphasized again in the followed text.

The effective Young’s modulus and compression strength of porous Si 3N 4ceramics will be modeled in this study.For the simulation of compression strength,a method so-called Element Birth and Death in ANSYS 12.1is employed.As the grains and grain boundaries exhibit the features of fragile materials,their failure criterions are both selected as the maximum principal stress rule,which can be expressed as

s max ,gr 4s lim ,gr s max ,bnd 4s lim ,bnd

e2T

where s max ,gr ,s lim ,gr are the maximum principal stress and tensile strength of b -Si 3N 4grains,respectively;s max ,bnd ,s lim ,bnd are the maximum principal stress and fracture strength of grain boundaries,respectively.The beam elements satisfying Eq.(2)are killed under compression and the killed elements remain in the model but contribute near-zero stiffness value to the overall structure.The overall stiffness of FE model decreases as the amount of killed elements increases,thus a peak stress exists in the compressive response calculated by the above

mentioned method,which is just considered as the compression strength in this study.

4.Results and discussion

4.1.Numerical simulation for experiment

In this study,two types of porous Si 3N 4ceramics with different porosity were fabricated.By the Archimedes method,the measured values of porosity are 52.0and 62.4%,respectively.Here we name the sample with the porosity of 52.0%as Sample 1and the other one as Sample 2.For each type of sample,six specimens were tested and the measured Young’s modulus and compression strength are listed in Table 1as a mean value and with the standard https://www.360docs.net/doc/d25982605.html,pared with mechanical properties of b -Si 3N 4whisker,it shows that the mechanical properties of porous Si 3N 4ceramics are weaken sharply by the random pore inside the materials.For mechanical characterization of porous Si 3N 4ceramics,the FE simulation is conducted as the following scheme.For sample 1,the corresponding numerical model is established and FE simulation is carried out.Obviously,the different parametric sets of mechanical properties of grain boundaries yield different overall response of FE model.Hereby,the effective Young’s modulus and fracture strength of grain boundaries can be con?rmed by ?tting numerical results with the experimental data of Sample 1.Afterwards,the mechanical proper-ties of Sample 2can be predicted,and then are used to compare with the measured values of Sample 2.

It is clear that numerical results of FE models are scattered due to randomness of microstructure.In other words,the FE models with same parametric set (porosity,grain aspect ratio,etc.)yield different Young’s modulus and compression strength.Conse-quently,it is important to evaluate the convergence of calculated data by FE simulation.Thereafter,the convergent value is employed to compare with the measured data.In this study,we obtain the convergent value of the effective Young’s modulus and compression strength by analyzing tendency of random average versus random number.Here,the random average is de?ned as

Q ?P N

i ?1Q

i

e3TIn this equation,represents the random average of N times of numerical results calculated by a series of FE models with the same parametric set.According to Monte Carlo method,the random average gradually approaches to a convergent value as the amount of numerical samples increases.When the relative error of the two random averages in sequence is lower than 3%,the corresponding average is considered as the convergent value.According to the aforementioned scheme,the mechanical properties of equivalent grain boundaries are derived by ?tting the experimental data of Sample 1.The effective Young’s modulus and fracture strength of grain boundaries are taken as E bnd ?160GPa and s bnd ?1.80GPa ,respectively.The calculated Young’s modulus is the range of represented value published in the open literatures [35–38],which veri?es the ef?ciency of the present FE model to a certain extent.For porous Si 3N 4ceramics,the calcu-lated Young’s modulus and compression strength are shown

in

Fig.2.The represebtative FE model with combined network of rod-like grains and grain boundaries for porous Si 3N 4ceramics.

Table 1

Porosity,the effective Young’s modulus and the compression strength of porous Si 3N 4ceramics fabricated in this study.Sample type

Porosity (%)

Effective Young’s modulus (GPa)Compression strength (MPa)152.012.572.855.878.7262.4

6.871.1

32.475.6

Z.Lu et al./Materials Science &Engineering A 559(2013)201–209

204

Figs.3and 4,respectively.Meanwhile,the measured data are also provided for comparison.In the two ?gures,the hollow circles represent the discrete FE results with different porosity.Due to the random microstructure,numerical results are dispersive and the scattered degree of FE results is comparable with the standard deviation of experimental data.In addition,the mean values of Young’s modulus and compression strength are determined by the method of Eq.(3).In Figs.3and 4,the ‘line-symbol’curve reveals the relations of mean Young’s modulus and compression strength versus porosity,https://www.360docs.net/doc/d25982605.html,pared with the mean value of experimental data,the present numerical results provide a good accordance.As illustrated in Fig.3,the calculated relation of the mean Young’s modulus versus porosity is nonlinear for porous Si 3N 4ceramics.By contrast,the mean compression strength changes nearly linearly versus porosity in Fig.4.It is thus indicated that the increment of porosity have a more deleterious in?uence on the compression strength than that for the effective Young’s modulus of porous Si 3N 4ceramics.

Under compression,the failure of porous Si 3N 4ceramics behaves the fragile fracture.In order to reveal the failure mechan-ism of real material,the evolution of microstructure is demon-strated by FE model in Fig.5.As illustrate in Fig.5(a),the intact

structure of porous Si 3N 4ceramics is present,where the cyan and blue solids represent b -Si 3N 4grains and grain boundaries,respec-tively.It shows that grains are bonded by grain boundaries,forming a connected network.With lower fracture strength than grains,grain boundaries are prone to become failure when the compressive load is applied on FE model.Clearly,the amount of damaged grain boundaries gradually rises as the compressive load increases.When the concentrated domain consisted of damaged grain boundaries traverse FE model as shown in Fig.5(b),the peak compressive stress is obtained and porous Si 3N 4ceramics behaves the fragile fracture.By observing the damaged solids in Fig.5(b),it shows that macro-failure of real material is contributed to the damage of grain boundaries.The FE simulation indicates that overall compression strength is governed by the interfacial debonding and the fracture of b -Si 3N 4grains play a secondary role.In Fig.6,the SEM photo of fracture surface is present,verifying the conclusions derived by FE simulation.The phenomenon is some different from what observed in the bending experiment,which exhibit the failure features of grain bridging and grain pull-out [38].That is because the initiation and propagation of crack always occur in the tensile region of beam for typical fragile materials in bending,and hence the features of ?exural fracture surface are approximately similar to the one caused by the tensile failure mechanism.

As a whole,the numerical results accord well with experi-mental measurement in both overall compressive response and failure mechanism of microstructure,which support the present FE model.Moreover,the performance of numerical convergence is also important for random model.The tendency of random average versus amount of numerical samples,corresponding to the effective Young’s modulus and compression strength of porous Si 3N 4ceramics,are,respectively illustrated in Fig.7and 8.Note that the parametric set of FE models is same as those used to simulate experiment in Section 4.1.The two ?gures show that the effective mechanical properties calculated by FE simulation approach to constant values as the increment of numerical samples,which indicates the present FE model possesses good convergent https://www.360docs.net/doc/d25982605.html,paring the convergence speed of FE models with different porosity,it shows that the FE model with higher porosity is prone to yield convergent numerical results.A possible explanation is that the FE model with lower porosity possesses more grains in quantity.Clearly,the randomness of numerical results is primarily attributed to random distribution of orientation and position of grains.More grains mean that more bonded structures with different morphology can be generated.Sequentially,the calculated results of these FE models corre-sponding to lower porosity become more scattered.Observing the convergent process of numerical results,it also shows that the mean value obtained by 16numerical samples is accurate enough to evaluate mechanical properties of porous Si 3N 4ceramics.A higher deviation could be caused if the sample amount is less,for example the maximum relative error of random average to the obtained convergent value is up to 14%in Fig.7.It is thus veri?ed that the statistical analysis of the numerical results derived by different samples is crucial to the followed investigations.

4.2.In?uences of microstructure parameters

It shows that the calculated results provide a good consistence to experiment,which supports the present FE model.For mechanical characterization of porous Si 3N 4ceramics,another objective is to explore the relations of microstructure and the macro-properties.In present work,we mainly evaluate the quantitative in?uences on overall Young’s modulus and compression strength,caused by grain aspect ratio and mechanical properties of grain

boundaries.

https://www.360docs.net/doc/d25982605.html,parison of the effective Young’s modulus between numerical simula-tion and the experimental

results.

https://www.360docs.net/doc/d25982605.html,parison of the compression strength between numerical simulation and the experimental results.

Z.Lu et al./Materials Science &Engineering A 559(2013)201–209205

As one of the most important microstructure parameters of porous Si 3N 4ceramics,the morphological features of grains have been studied in the open literatures.The grain size is affected by

experimental technology and the growth of b -Si 3N 4grains have been explored previously [40–42].Fig.9shows that the in?uences of grain aspect ratio on the effective Young’s modulus of porous Si 3N 4ceramics.Interestingly,the tendency of Young’s modulus versus grain aspect varies when the porosity is different.For porous Si 3N 4ceramics with 60%porosity,there exists an optimal grain aspect ratio which can lead to the maximum Young’s modulus.By contrast,the tendency is different for the samples with 70or 80%porosity.Herein,the effective Young’s modulus increases all along as grain aspect ratio becomes larger.Never-theless,the rising trend becomes ?at when the grain aspect ratio is over 12.It is thus inferred that the effective Young’s modulus would reach a peak at larger grain aspect ratio.Accordingly,the optimal grain aspect ratio can also be obtained for the material of with higher porosity.The existence of optimal grain aspect ratio can be understood as follows.For porous Si 3N 4ceramics,the internal structure is a connected network as a result of grains bonded by grain boundaries.In order to reveal the effects of grain aspect ratio on mechanical properties of porous Si 3N 4ceramics more clearly,their microstructure is simpli?ed as a two-di-mensional structure in Fig.10.Obviously,grains are easier to be bonded together if both the amount and length of grains are larger,and hence the overall structure is stiffer.It is well known that higher grain aspect ratio means that grains are longer,

which

Fig.5.(a)The intact structure before compression,(b)the damaged solids in microstructure after compression:the cyan and blue solids represent grain and grain boundaries,respectively.(For interpretation of the references to color in this ?gure legend,the reader is referred to the web version of this

article.)

Fig.6.The SEM fracture surface of porous Si 3N 4ceramics under compression

test.

Fig.7.The convergent process of the average Young’s modulus versus the capacity of numerical

samples.

Fig.8.The convergent process of the average compression strength versus the capacity of numerical samples.

Z.Lu et al./Materials Science &Engineering A 559(2013)201–209

206

leads to the increment of stiffness of overall structure.However,the total volume of solids is ?xed when porosity of material is speci?c.Consequently,the increment of grain length inversely results in the decrease of grain amount,which will diminish overall stiffness.Considering the two categories of reasons,the effective Young’s modulus of porous Si 3N 4ceramic ?rstly increase and then decrease as grain aspect ratio becomes larger.As a result,an optimal grain aspect ratio can be obtained for the evaluation of effective Young’s modulus.

Fig.11shows the variation of the calculated compression strength following with the increment of grain aspect ratio,and the tendency is similar to what illustrated in Fig.9.However,there exist some differences between the two ?gures.The optimal grain aspect ratio resulting in the maximum compression strength is 12,which is lower than the value for the effective Young’s modulus.Another difference is that a maximum value of compression strength can be obtained for 70%porosity when the grain aspect ratio is in the range of 4–20.As shown in Fig.10(b),the segment between different bonding points in the same grain will become longer when grain aspect ratio is much higher.When porous Si 3N 4ceramics is subjected to load,the slender grain can be considered as the beam with multi-supports.It is known that the maximum bending stress is located at the cross section close to supports.Clearly,the supports (grain boundaries)would bear higher stress and become failure more easily if the distance between different bonding points is larger.Consequently,this is

another negative in?uence of increasing grain aspect ratio on compression strength.Nevertheless,the in?uence can be neglected for the analysis of overall https://www.360docs.net/doc/d25982605.html,bined with decrease of grain amount,the increments of grain aspect ratio have more signi?cant weakening effects on the compression strength of porous Si 3N 4ceramics.That is why the optimal grain aspect ratio is lower for compression strength.Moreover,the tendency of Young’s modulus versus grain aspect ratio is thus distinct from that for compression strength at the porosity of 70%.Note that the anomalous phenom-ena only occurs in the case of grain aspect ratio is rather large.

Except for grain aspect ratio,grain boundaries also have sig-ni?cant effects on the mechanical properties of porous Si 3N 4ceramics.In this study,the mechanical properties of grain bound-aries include the effective Young’s modulus and fracture strength,which are con?rmed by ?tting experimental as mentioned earlier.Considering the relative stiffness (E bnd /E gr )is generally lower than 1for the realistic material,we only exploring the in?uence of stiffness of grain boundaries in the range of 0.2–1.0of the relative stiffness.The effective Young’s modulus of porous Si 3N 4ceramics increases when the grain boundaries become stiffer as shown in Fig.12.However,the enhanced effects are not evident when the relative stiffness is larger than 0.8for the three types of numerical samples with different porosity.Clearly,grain boundaries is stiffer,the overall stiffness is higher.It is understood that the macro-deformation of porous Si 3N 4ceramics is consisted of the shearing or torsion deformation mechanism of grain boundaries and bending deformation mechanism of slender https://www.360docs.net/doc/d25982605.html,pared with the bending deformation of grains,the shearing or torsion deformation mechanism of grain boundaries have less in?uences on macro-deformation.When the stiffness of grain boundaries is high enough,the macro-deformation of porous Si 3N 4ceramics is almost fully governed by the bending deformation mechanism of grains,and hence the overall stiffness becomes insensitive to stiffening of grain boundaries.In particular,the effective Young’s modulus of porous Si 3N 4ceramics with lower porosity increases more rapidly,which is determined by more volume fraction of grain boundaries at lower porosity.

Fig.13shows the in?uences of stiffness of grain boundaries on compression strength of porous Si 3N 4ceramics.When the frac-ture strength of grain boundaries is relatively low (weak grain boundaries),it is interesting that the tendency is distinctively different from what illustrated in https://www.360docs.net/doc/d25982605.html,pared with the in?uences of relative stiffness on the effective Young’s modulus,the compression strength inversely decreases as the

relative

Fig.9.The in?uence of the grain aspect ratio on the effective Young’s modulus of porous Si 3N 4

ceramics.Fig.10.The schematic diagram of the microstructures of porous Si 3N 4ceramics (a)the sample with lower grain ratio;(b)the sample with higher aspect

ratio.

Fig.11.The in?uence of the grain aspect ration on the compression strength of porous Si 3N 4ceramics.

Z.Lu et al./Materials Science &Engineering A 559(2013)201–209207

stiffness increases in the range of 0.2–1.0.It is indicated that improving the stiffness of grain boundaries has a deleterious effects on compression strength,which can be explained in the following.As demonstrated in Figs.5and 6,grain boundaries are easier to become failure than grains because the fracture strength of grain boundaries is much lower than tensile strength of grains.It is well known that the bearing stress of grain boundaries increases as the relative stiffness increases.Accordingly,the increasing Young’s modulus of grain boundaries inversely diminishes overall compression strength.In deed,the maximum principal stress of b -Si 3N 4grains is higher than that of grain boundaries as b -Si 3N 4grains is much stiffer than grain bound-aries.Consequently,the compressive failure of porous Si 3N 4ceramics will be governed by fracture of b -Si 3N 4grains if fracture strength of grain boundaries was high enough (strong grain boundaries).Right now,stiffer grain boundaries lead to the unloading effects of grains when overall structure is under compression.It is thereby inferred that increasing relative stiff-ness of grain boundaries has enhanced effects on the overall compression strength.Here we assume that grain boundaries possessed the same fracture strength as b -Si 3N 4grains,

whereafter numerical results are illustrated in Fig.14.It shows that the increment of stiffness of grain boundaries actually has a positive effect on overall compression strength as our prediction.However,the enhancement effect is not evident.

Fig.15reveals the in?uences of fracture strength of grain boundaries on the compression strength of porous Si 3N 4ceramics.The increasing fracture strength of grain boundaries results in higher compression strength,whereas the enhanced effects become less signi?cant when the relative strength (s bnd /s gr )is beyond 0.4.As porous Si 3N 4ceramics is consisted of random grains bonded by grain boundaries,the compression strength depends on the weakest solids of microstructure.When the relative strength is low,the macroscopic compression strength is restricted to the fracture strength of grain boundaries.Clearly,the overall compression strength rise rapidly as the relative strength increases.As mentioned earlier,the compression strength will be governed by the failure of both grains and grain boundaries as the relative strength becomes larger.Furthermore,it is entirely determined by the tensile strength of grains when the relative strength is high enough.In that case,the overall compression strength becomes insensitive to the increment

of

Fig.12.The in?uence of the stiffness of grain boundaries on the effective Young’s modulus of porous Si 3N 4

ceramics.

Fig.13.The in?uence of the striffness of grain boundaries on the compression strength of porous Si 3N 4ceramics wtith weak grain

boundaries.

Fig.14.The in?uence of the stiffness of grain boundaries on the compression strength of porous Si 3N 4ceramics with strong grain

boundaries.

Fig.15.The in?uence of the strength of grain boundaries on the compression strength of porous Si 3N 4ceramics.

Z.Lu et al./Materials Science &Engineering A 559(2013)201–209

208

relative strength.As hinted earlier,the proportion of bonded points(grain boundaries)in microstructure is larger in the case of lower porosity.Similarly,the increasing relative strength have more signi?cant enhanced effects on compression strength for the porous Si3N4ceramics with lower porosity as shown in Fig.15. It is noteworthy that the value of relative strength utilized to simulate the experimental data is equal to0.131.When the relative strength of grain boundaries is larger than0.131, the overall compression strength will rise rapidly.In that case,the fracture of grain boundaries is the dominant failure mechanism, which is consistent with the experimental phenomena in the aforementioned section.

5.Conclusions

Two types of porous Si3N4ceramics with52.4and62.0%porosity are fabricated,respectively.The compressive behaviors are?rstly tested by experiment,and the measured Young’s modulus and compression strength are expressed as the mean value and with the standard deviation.Based upon the SEM photos,the FE model consisted of grains with random orientations and grain boundaries is https://www.360docs.net/doc/d25982605.html,pared with experimental measurement,the numerical results provide a good accordance.In the last section, the in?uences of micro-parameters on macro-mechanical properties are studied.Finally,the conclusions are summarized as follows: 1)The mechanical properties of porous Si3N4ceramics are weakened

by the increment of porosity.The dominant failure mechanism in microstructure is interfacial debonding,in contrast that the fracture of b-Si3N4grains only plays a secondary role.

2)The present FE model reveals the random structure of rod-like

grains bonded by grain boundaries.By observing the conver-gent process of numerical results,it indicates that FE model exhibit a good performance of numerical https://www.360docs.net/doc/d25982605.html,-pared with experimental data,the calculated results provide a good accordance.Meanwhile,the scattered degree of numer-ical results is also comparable with the standard deviation of experimental data.The FE simulation shows that the inter-facial debonding is the dominant failure mechanism in micro-structure,which is veri?ed by the SEM observation.

3)The numerical results show that there exist optimal grain aspect

ratio to obtain the maximum Young’s modulus and compression strength for porous Si3N4ceramics.However,the value of optimal grain aspect ratio corresponding to the effective Young’s modulus is different from that for compression strength.That is mainly because the in?uences of bonded structure on overall stiffness and strength are somewhat different.Moreover,the in?uences of grain aspect ratio on the overall mechanical proper-ties vary from porosity of material.

4)The mechanical properties of grain boundaries have signi?cant

effects on the macro-properties of porous Si3N4ceramics.

When grain boundaries are weak,the increasing stiffness of grain boundaries results in the increment of the effective Young’s modulus,whereas it has deleterious in?uences on compression strength.By contrast,the stiffer grain boundaries also have positive in?uences on compression strength if the fracture strength of grain boundaries is high enough.As the fracture strength of grain boundaries increases,FE simulation shows that the dominant failure mechanism in microstructure

changes from the interfacial debonding to the fracture of grains.The overall compression strength becomes insensitive to grain boundaries when the relative strength is larger than0.4.

Acknowledgments

The authors thank the support from the National Natural Science Foundation of China(11272030,10932001,11072015,and 11002011)and the Scienti?c Research Key Program of Beijing Municipal Commission of Education(No.KZ201010005003)and the Fundamental Research Funds for the Central Universities(No. YWF-10-02-002).

References

[1]Y.Inagaki,N.Kondo,T.Ohji,J.Eur.Ceram.Soc.22(2002)2489–2494.

[2]F.S.Li,W.C.Zhou,H.J.Hu,F.Luo,D.M.Zhu,Ceram.Int.35(2009)3169–3173.

[3]F.Chen,Q.Shen,J.M.Schoenung,L.Zhang,Mater.Sci.Eng.A497(2008)

495–500.

[4]D.S.Park,T.W Roh,B.D Han,H.D Kim,C.Park,J.Eur.Ceram.Soc.22(2002)

535–543.

[5]S.Y.Shan,Q.M.Jia,L.H.Jiang,Y.M.Wang,J.F.Yang,Ceram.Int.35(2009)

3371–3374.

[6]D.X.Yao,Y.P Zeng,K.H.Zuo,D.L Jiang.,Int.J.Appl.Ceram.Technol.(2011),

https://www.360docs.net/doc/d25982605.html,/10.1016/j.ceramint.2011.05.035.

[7]T.Ohji,Mater.Sci.Eng.A498(2008)5–11.

[8]G.P.Jiang,J.F.Yang,J.Q.Gao,K.Niihara,Mater.Charact.60(2009)456–460.

[9]Y.F Shao,D.C Jia,B.Y.Liu,J.Eur.Ceram.Soc.29(2009)1529–1534.

[10]A.Diaz,S.Hampshire,J.Eur.Ceram.Soc.24(2004)413–419.

[11]H.H.Lu,J.L.Huang.,Ceram.Int.27(2001)621–628.

[12]J.F.Yang,Z.Y.Deng,T.Ohji,J.Eur.Ceram.Soc.23(2003)371–378.

[13]Y.Inagaki,Ando.M Shigegaki,T.Ohji,J.Eur.Ceram.Soc.24(2004)197–200.

[14]R.de Borst,Comput.Mater.Sci.43(2008)1–15.

[15]T.Sadowski,L.Marsavina,Comput.Mater.Sci.50(2011)1336–1346.

[16]K.K.Phani,S.K.Niyogi,J.Mater.Sci.Lett.6(1987)511–515.

[17]R.L.Coble,W.D.Kingery,J.Am.Ceram.Soc.39(1956)377–385.

[18]R.M.Spriggs,J.Am.Ceram.Soc.44(1961)628–629.

[19]J.F.Yang,S.Y.Shan,R.Janssen,G.Schneider,T.Ohji,C.Kanzaki,Acta Mater.53

(2005)2981–2990.

[20]H.Park,H.E.Kim,K.Niihara,J.Am.Ceram.Soc.80(1997)750–756.

[21]M.Belmonte,A.Pablos,M.I.Osendi,P.Miranzo,Mater.Sci.Eng.A475(2008)

185–189.

[22]W.Pabst,E.Gregorova,G.Ticha,J.Eur.Ceram.Soc.26(2006)1085–1097.

[23]Z.Zivcova,M.Cerny,W.Pabst,E.Gregorowa,J.Eur.Ceram.Soc.29(2009)

2765–2771.

[24]T.Sadowski,S.Samborski,Comput.Mater.Sci.28(2003)512–517.

[25]T.Sadowski,S.Samborski,Comput.Mater.Sci.43(2008)75–81.

[26]A.P.Roberts,E.J.Garboczi,J.Am.Ceram.Soc.83(2000)3041–3048.

[27]G.Bruno,A.M.Efremov,A.N.Levandovskyi,B.Clausen,J.Mater.Sci.46(2011)

161–173.

[28]C.Bartuli, E.Bemporadb,J.M.Tulliani,J.Tirrilo,J.Pulci, C.Bartuli,

M.Sebastiani,J.Eur.Ceram.Soc.29(2009)2979–2989.

[29]J.Yu,H.Wang,J.Zhang,D.Zhang,Y.Yan,J.Sol–Gel Sci.Technol.53(2010)

515–523.

[30]https://www.360docs.net/doc/d25982605.html,nge,J.Am.Ceram.Soc.56(1973)518–522.

[31]N.Hirosaki,Y.Akimune,M.Mitomo,J.Am.Ceram.Soc.76(1993)1892–1894.

[32]W.Li,Z.Jin,Z.Zhang,Prog.Chem.15(2003)264–274.

[33]P.F.Becher,et al.,Acta Mater.44(1996)3881–3893.

[34]E.Y.Sun,et al.,J.Am.Ceram.Soc.81(1998)2831–2840.

[35]T.Rouxel,M.Huger,J.L.Besson,J.Mater.Sci.27(1992)279–284.

[36]R.E.Loehman,J.Am.Ceram.Soc.62(1979)491–494.

[37]D.Yuquan,D.Zishang,J.Zhonghua,J.Non-Cryst.Solids112(1989)408–412.

[38]A.Bhatnagar,M.J.Hoffman,R.H.Daushardt,J.Am.Ceram.Soc.83(2000)

585–596.

[39]J.E.Weston,P.L.Pratt,B.Steele,J.Mater.Sci.13(1978)2137–2146.

[40]R.Raj,M.J.Hoffmann,J.Am.Ceram.Soc.80(1997)3250–3252.

[41]M.Kitayama,K.Hirao,M.Toriyama,S.Kanzaki,Acta Mater.46(1998)

6551–6557.

[42]D.S.Perera,D.Mitchell,S.Leung,J.Eur.Ceram.Soc.20(2000)789–794.

Z.Lu et al./Materials Science&Engineering A559(2013)201–209209

漏斗实验(Funnel Experiment)

漏斗实验(Funnel Experiment) 在戴明博士四日谈中,以漏斗实验来解释管理与干预问题。管理人员常因缺乏对系统变异的统计思考方式而对系统进行干预,造成问题越变越离谱。譬如,厂内的管理阶层在品质会议中要 求不良率最高的单位提出改善计划或业务会议中要求营业额退步的营业员提出对策。以前国中的 导师每周对学生评分排名,对退步的学生给与严厉的指责警告(现在应该还是一样)。但是以长期 来看不良率依然有高有低;营业额每月仍是有好有坏;学生的排名每周还是有进有退,这些数据 的变异很多是系统的正常变异,也就是所谓共同原因的变异。但是,管理人员对这些变异进行干 预,采取矫正措施,使得系统越变越复杂。例如,制程管理人员隐藏不良品使不良率好看;营业 人员虚报营业额使得帐面上好看;学生到补习班先练习考试题目使得排名进步。以上这些现象在 我们所处的工作或生活环境中屡见不鲜,我们应该先了解系统的变异是来自特殊原因或是共同原 因,再采取适当的行动。 所谓,就是假想我们有一漏斗,装在桌上约半公尺高的架上,桌上有个靶。假设我 们把一颗弹珠放入漏斗,不论我们放下的方式如何,弹珠就会以随机的方式滚下漏斗,然后由漏

斗底部掉下到靶上,再用铅笔在落点做个记号。我们利用一些简单的规则来使漏斗瞄准目标,这 些规则相当于我们在使用设备、流程或系统中作的一些决策规则。 [编辑] 漏斗实验的四种规则: 规则一:每次都不调整漏斗位置(结果:弹珠落点随机分布在目标值两侧) 规则二:根据上一次落点,调整漏斗位置(结果:弹珠落点范围较规则一大了约41%) 规则三:调整前先归回目标值(结果:弹珠落点由两侧大幅散开) 规则四:瞄准上一次的落点(结果:弹珠落点呈随机漫步到天边) 将四个规则仿真的结果绘在同张图上,可以一目了然地比较四种规则的结果。 :一个漏斗、一粒可以很容易通过漏斗的弹珠、一张桌子,最好铺上桌布。 第一次实验:规则为漏斗位置不变。首先在桌布上标出一点作为目标,开始实验。将漏斗口 瞄准目标点。保持这种状态,将弹珠由漏斗口落下50次,在弹珠每次落下的静止位置作标记。

死亡实验(the experiment)观后感

观后感 ——关于社会角色、侵犯行为、群体压力理论的思考这部由斯坦福监狱实验改编的影片,忘了剧中这个实验的目的是什么,但是它很好地诠释了社会心理学中有关于社会角色、侵犯行为、群体压力等理论。 首先关于社会角色。对于角色,社会心理学是这样定义的——一个社会身份所要求的一般行为方式及其内在的态度和价值观基础。显然,影片中将实验参与者分为看守和囚犯两类,就是重新定义了他们的社会角色。按照角色的分类,看守属于支配角色,而囚犯则属于受支配角色。既然赋予了他们一定的社会角色,那他们就必然会按照其特定的地位和所处的情境来行动,就像小孩子在玩过家家,扮演父亲的孩子会出去上班赚钱,扮演母亲的孩子会在家做家务一样,这些是他们从生活中学习模仿而来,这种行为被称作角色扮演。在剧中,角色扮演则明显体现在扮演看守的人员身上。当出现“囚犯”不服从命令的情况时,自己是看守,有义务维持监狱的秩序的思想会适时浮现于脑中,于是他们便会尽自己的职责,也许模仿某些警匪片中的行为,对囚犯加以控制,而在这个过程中,权利和义务也就诞生了。这个实验很好地向我们诠释了什么是社会角色,以及与社会角色相关联的角色扮演,甚至权利与义务是社会化过程的产物等相关的概念。 关于侵犯行为。侵犯行为也可称之为攻击行为,是指个体违反了社会主流规范的、有动机的、伤害他人的行为。在剧中,我们看到了很多暴力行为,比如把囚犯关进黑屋子,绑在椅子上等。那这些

暴力行为是否属于侵犯行为呢?在判断某行为是否是侵犯行为时,有三个标准:一是个体外在的行为,二是看该行为是否违反社会主流规范,三是个体的内在动机或意图如何。显而易见地,看守的行为动机就是对罪犯进行报复,是故意的,而且也有外在的暴力动作,但是否能将这种暴力行为归于侵犯,我仍有一丝疑虑,因为在实验之初就已经说过,囚犯会丧失部分人权,所以这样的行为有没有违反社会主流规范仍旧存在疑虑,因为“看守看管囚犯,即使他运用了怎样的手段,也是可以理解的”这种思想没有经过调查,不好判断它是否属于社会主流思想。侵犯行为是否与生俱来也是社会心理学界争论的问题。霍尔提出过“挫折——侵犯理论”,他认为挫折是指当一个人为实现某种目标而努力时遭受干扰或破坏,致使需求不能得到满足时的情绪状态,人的侵犯行为是因为个体遭受挫折而引起的。从这个实验来看,如果证实了“看守”的行为的确属于侵犯行为的话,那么可见侵犯行为确实不是与生俱来的。因为在实验之前,他们都是正常人,而是在实验过程中,由于17号的挑战对于监狱管理产生困难与挫折,看守才最终选择了暴力侵犯行为。 关于群体压力。群体借助规范的力量形成的对其成员心理上的强迫力量,就是群体压力,群体借助这种力量达到对群体成员行为的约束作用。看守将17号关入黑屋子的惩罚就是为了造成对囚犯的群体压力。然而群体压力也有积极的意义:增强群体团结,有助于群体任务的完成,对多数成员内心安全感的形成起很大作用。这点明见于最后的囚犯群体大逃亡。但群体心理对于群体内固执己见的少数人

第18章-随机实验与自然实验

1 ? 陈强,《高级计量经济学及 Stata 应用》课件,第二版,2014 年,高等教育出版社。 第 18 章 随机实验与自然实验 18.1 实 验 数 据 假设研究x 1 是否导致 y 。假定{x 1, x 2 , , x K }包含所有影响 y 的因素。 不同学科采用不同的实验方法,大致分为以下几类。 (1) 控制实验(controlled experiment):在理想的物理实验中,控制 {x 2 , , x K }全部不变,单独让x 1 变化,观察 y 的变化。 (2) 随机(控制)实验(randomized controlled experiment):

【例】医学上对新药x 疗效的实验。由于参加实验者的体质与生活 1 方式不同,不可能完全控制所有其他因素{x2 , , x K }。 随机实验将实验人群(或个体)随机地分为两组,其中“实验组”或“处理组”(treatment group)服用真药,而“控制组”(control group,也称“对照组”)服用“安慰药”(placebo)。 被试者不知道自己分在哪一组,避免心理干扰。有时科研人员也不知道被试者在哪一组,称为“双盲法”(double blind)。 【例】农学中将地块随机地分成三组(很难找到土壤条件完全一样的地块),分别给予不同的施肥量,然后考察施肥的效果。 2

(3)自然实验或准实验(natural experiment or quasi experiment): 由于某些并非为了实验目的而发生的外部突发事件,使得当事人仿佛被随机地分在了实验组或控制组。 【例】一个州通过某法律,但相邻州未通过此法律。两州民众事先不知道哪个州会通过此法律,故无法自我选择住在哪个州。从考察法律的效果而言,可近似认为民众随机选择住在哪个州,或被随机分为实验组(通过法律)与控制组(没通过法律)。 (4)思想实验(thought experiment): Milton Friedman 曾设想在小岛上通过空投货币,考察该岛的宏观经济的变化。 3

完全随机设计试验资料的方差分析-东北农业大学植物科学与技术实验

东北农业大学本科课程教学大纲 课程名称:田间试验与统计方法 英文名称:Field Experiment and Statistic-method 课程编号:01600008j 适用专业:草业科学、植物生产类 总学时数:40 总学分:2.5 大纲主撰人:李文霞 内容简介 《试验设计与统计分析》是一门收集整理数据、分析数据, 并根据数据进行推断的科学。本课程为高等农业院校农学类专业的专业基础课,主要讲授有关田间试验的基本知识和统计分析的基本方法和技能,为学习专业课程奠定基础,使学生具备承担科学试验,正确分析和评价科学试验结果及其可靠性的能力。 教学大纲 一、课堂讲授部分 (一)分章节列出标题、各章节要点及授课时数(务必将要点写清楚) 第1章绪论 一、基本内容 1.1 农业科学试验的任务和要求1学时 1.1.1 农业科学试验和田间试验 1.1.2 农业科学试验的任务和来源 1.1.3 农业科学试验的基本要求 1.2 试验误差及其控制2学时 1.2.1 试验误差 1.2.2 试验误差的来源 1.2.3试验误差的控制 1.3 生物统计学与农业科学试验1学时 1.3.1 部分生物统计学基本概念 1.3.2 生物统计学的形成与发展 1.3.3 生物统计学在农业科学试验中的作用和注意问题 二、教学目的与要求 要求学生掌握农业科学试验的基本要求、试验误差的概念、来源和控制、部分生物统计学的概念,了解农业科学试验的任务和来源、生物统计学在农业科学试验中的作用和注意问题。 三、重点与难点 重点:农业科学试验的基本要求、试验误差的概念、来源和控制、部分生物统计学的概念

难点:试验误差的概念和生物统计学的基本概念的理解 第2章试验的设计和实施 一、基本内容 2.1 试验方案1学时 2.1.1 试验方案的概念和类别 2.1.2 处理效应 2.1.3 试验方案的设计要点 2.2 试验设计原则 1.5学时 2.2.1 重复 2.2.2 随机排列 2.2.3 局部控制 2.3 小区技术0.5学时 2.3.1 小区 2.3.2 区组和小区的排列 2.3.3 保护行 2.4 常用的试验设计1学时 2.4.1 对比法设计 2.4.2 间比法设计 2.4.3 完全随机设计 2.4.4 随机区组设计 2.4.5 拉丁方设计 2.4.6 裂区设计 2.5 试验的实施(学生自学) 2.6 田间抽样(学生自学) 二、教学目的与要求 要求学生掌握试验方案、试验设计原则、小区技术和常用的试验设计,自学试验的实施和田间抽样。 三、重点与难点 重点:试验方案、试验设计原则、小区技术和常用的试验设计。 难点:试验设计原则、小区技术、试验方案的设计要点的理解。 第3章描述性统计 一、基本内容 3.1 次数分布 1.5学时 3.1.1次数分布表 定量资料、定性资料 3.1.2次数分布图 柱形图、多边形图 3.1.3其它常用统计图 结合Excel的作图向导讲解,重点柱形图和折线图 3.2 平均数1.5学时

热传导实验(Heat Conduction Experiment)

熱傳導實驗(Heat Conduction Experiment) 目的:測定各種金屬之『熱傳導係數』,並探討物質具有不同大小之熱傳導係數要如何應用。實驗設備:自己填寫 實驗方式:分別以沿『軸向』及『徑向』之熱傳導試件進行實驗,以試件內之溫度達到穩定狀態時為準,來計算金屬之熱傳導係數。 操作步驟: (1)將金屬試件(不鏽鋼或黃銅、 不鏽鋼或鋁)安裝到要進行實 驗的座位台上 (2)打開電源,選擇溫度顯示相近 的RTD測溫棒插入試件的測 溫孔,並確定測溫棒與測溫孔 緊密接觸 (3)選擇『軸向』或『徑向』之加 熱源,並調整熱率輸入視窗之 數值為20W (4)每隔5分鐘讀取每支測溫棒 之溫度,每個試件至少記錄六次共30分鐘,歸納結果時要將各個測溫點的『溫度-時間圖』畫出,並以溫度達到穩定狀態時為準,來計算金屬之熱傳導係數。 (5)更換試件重複步驟(1)~(4) 實驗數據記錄: 試件名稱:軸向熱傳導 T1(℃) T2(℃) T3(℃) T4(℃) T5(℃)測溫點 時間 5(min) 10(min) 15(min) 20(min) 25(min) 30(min) 試件名稱:軸向熱傳導 T1(℃) T2(℃) T3(℃) T4(℃) T5(℃)測溫點 時間 5(min) 10(min) 15(min) 20(min) 25(min) 30(min)

試件名稱: 徑向熱傳導 測溫點 時間 T1(℃) T2(℃) T3(℃) T4(℃) T5(℃) 5(min ) 10(min ) 15(min ) 20(min ) 25(min ) 30(min ) 試件名稱: 徑向熱傳導 測溫點 時間 T1(℃) T2(℃) T3(℃) T4(℃) T5(℃) 5(min ) 10(min ) 15(min ) 20(min ) 25(min ) 30(min ) 實驗數據圖示: (1) 用Excell 畫出各個測溫點的『溫度-時間』圖 (2) 依據(1)之圖,估計各個測溫點達到穩定狀態時的溫度,依此溫度,畫出各試件在各個 測溫點達到穩定狀態的『溫度-位置』圖(在『軸向』實驗中應有兩試件之線;在『徑向』實驗中也應有兩試件之線) 實驗數據計算: (1) 依據穩定狀態的『溫度-位置』圖,將各點連成擬合直線(不是折線),依據此直線之斜 率(『軸向』為 X T ΔΔ)(『徑向』為)ln(i o o i r r T T ?)來 計算『熱傳導係數K 』。 (2) 『軸向』公式為X T KA Q ΔΔ=;Q :輸入熱率(A :試件截面積(m 2);△T :直線上兩點之溫度差(℃);△X :直線上兩點之位置差(m ) (3) 『徑向』公式為)ln(2i o o i r r T T KL Q ?=π;L=試件厚度(m ),T i =靠近圓心處之溫度(℃),r i =靠近圓心處之半徑(m ),T o =靠外側處之溫度(℃),r o =靠外側處之半徑(m ) 結果與討論: (1) 書本上不鏽鋼的『熱傳導係數K 』約為20W/m ℃;黃銅約為100W/m ℃;鋁約為200W/m ℃,為何實驗計算出的值比書本提供的值為大?

15章随机实验与自然实验-ShandongUniversity

教学用PPT ,《高级计量经济学及Stata 应用》,陈强编著,高等教育出版社,2010年 第15章 随机实验与自然实验 15.1实验数据 不同学科可能依条件的不同而采用不同的实验方法。 (1)控制实验(controlled experiment ):在理想的物理实验中,对除1x 以外的因素{}2,,K x x "全部控制不变,单独让

x变化,然后观察y变化的情况。 1 (2)随机(控制)实验(randomized controlled experiment):通常将实验人群随机地分为两组,其中“实验组”(treatment group)服用真药,而“控制组”(control group,也称“对照组”)服用“安慰药”(placebo)。 (3)自然实验或准实验(natural experiment or quasi experiment):由于某些并非为了实验目的而发生的外部突 发事件,使得当事人仿佛 ..被随机分在了实验组或控制组。

15.2 理想的随机实验 在理想的随机实验(ideal randomized experiment)中,实验组与控制组的成员决定完全随机,比如,通过抛硬币或电脑随机数来决定。故个体究竟被分在哪一组或得到多大的实验“剂量水平”(treatment level),与个体的特征或其他可能影响实验结果的因素是完全独立的。这就避免了遗漏变量偏差(omitted variable bias)。

考虑以下回归模型, i i i y x αβε=++ (15.1) 其中,i x 是完全随机地决定的。由于i x 与i ε相互独立,故 Cov(,)0i i x ε=,因此OLS 是一致的。由于i x 与i ε相互独立,故1E(|,,)0i n x x ε=",故OLS 也是无偏的。 在理想的随机实验中,X 对y 的因果效应表现在条件期望的差别,即E(|)E(|0)y X x y X =?=,也称为“实验效应”

实验安全风险分析EXPERIMENT RISK ANALYSIS

E UROPEAN S YNCHROTRON R ADIATION F ACILITY INSTALLATION EUROPEENNE DE RAYONNEMENT SYNCHROTRON EXPERIMENT RISK ANALYSIS Experimental Number:Beamline: Main Proposer: Title of the Experiment: 1EXPERIMENT (only if changes since the proposal) Classification of the sample: Radioactive Contaminant Corrosive Oxidising Explosive Biological Other: Sample Description: Crystal Powder Polycrystalline Multilayer Liquid Gas Nanoparticles Other: Container: Capillaries Flat plate Pressure cell – Type: Other: ESRF equipment to be used: Furnace Magnet Cryostat Cryogenic gas stream Refrigerator Laser High pressure Fixed temperature Other: The Safety Group must immediately be informed of all modifications made and which differ from the original proposal and this at least two weeks before your arrival on site. Your equipment has been tested by your home institute. No changes can be made before your arrival at the ESRF and until your experiment has started. Page 1 / 12- Analyse des risques de l’expérience

experiment1

实验一 电位、电压的测量和叠加定理的研究 一、实验目的 1.熟悉实验台的整体布置及交、直流电源和交、直流仪表的使用。 2.学会测量电路中各点的电位和电压的方法。 3.掌握线性电路的叠加定理。 二、实验设备 实验箱(EEL-51)(EEL-53)、恒压源、直流电压表、直流电流表 三、实验内容 1.熟悉实验台的整体布局、记录实验台的主要设备和仪表的参数。要求记录:设备的名称、规格、量程及精度。 2.熟悉直流恒压源、恒流源和直流电压表、电流表的使用。 (a)自行设计一个电路,以某点为参考点,测量电路各点的电位和两点之间的电压。具体要求: ①用三个电阻和一个电源(电压不超过8V )组成一个简单电路; ②由附录中实验箱选择电阻元件的阻值,并画出电路; ③选择参考点计算各点的电位和两点之间的电压,自行设计一个表格,将所计算的数据填入表格中。然后实际连接电路,测量电位和电压。 (b)叠加定理的研究 使用EEL-53实验箱,按叠加原理图1-1进行实验,测量每个电源(V U S 121 =,V U S 62 =) 单独作用时和共同作用时各支路电流值,填入表1-1中。 表1-1 图1-1 4 5

四、实验注意事项 1.测量直流电压应并联在被测元件上,注意正负极性。测量直流电流时应串联在被测支路中,要注意电流的方向。 2.选择测量仪表的量程,根据估算选择稍大的量程,如电流偏小,再降低量程,以保证测量的精度。注意测量仪表报警铃响时,应关闭仪表的电源,检查原因,改正后重新合上仪表的电源。 3.正确使用可调直流恒压源和恒流源,正确读数(读数以电压表测量为准,而不以电源表盘指示值为准)。 4.使用电流插头测量时应注意仪表的极性的正确连接,以及读数时"",""-+号的记录。 5.叠加定理实验中,每个电源单独作用时,去掉另一个电源,是由开关S 1 ,S 2 操作完成,而不能将直流电源短路。 五、预习思考题 1.叠加定理实验中1 S U ,2 S U 分别单独作用时实验中应如何操作? 2.如将叠加定理中电阻R 3改为二极管D 时,叠加定理是否成立? 3.电路中各点电位与选择的参考点有什么关系?任意两点之间的电压与参考点的选择有关系吗? 六、实验报告要求 1.预习报告内容的要求:实验目的、实验设备(写出具体实验箱的型号和测量仪表的型号)、实验内容及步骤、根据实验内容,具体画出线路及实验参数,计算结果,以及设计出测量用的记录表格。 2.总结报告内容的要求:除预习报告内容之外,再增加数据的误差分析或曲线比较,理论分析,故障分析和心得体会。

Reynolds_experiment雷诺实验(英文版)

Reynolds experiment Aim of the experiment 1.Observe the laminar and turbulent flow, and the process of transition from one state to the other. 2.Measure the critical Reynolds number and develop the skills on how to distinguish the pipe flow state. 3.Study the dimensional analysis method to analyze the experiment, confirming the criterion number of flow state for a non-circular pipe. Experimental apparatus 1. The figure of the apparatus Figure 1 shows the experimental apparatus and the name of each part. Figure 1. 1: Self-circulating water supply, 2: Hydraulic bench, 3: Speed controller, 4: Constant head water tank, 5: Coloured water pipe, 6: Perforated plate, 7: Overflow, 8: Experiment pipe, 9: Flow rate control valve

DOE(Design of Experiment,试验设计)

DOE 出自 MBA智库百科(https://www.360docs.net/doc/d25982605.html,/) DOE(Design of Experiment,试验设计) 目录 [隐藏] ? 1 什么是DOE ? 2 为什么需要DOE ? 3 DOE的基本原理 ? 4 DOE实验的基本策略 ? 5 DOE的步骤 ? 6 DOE的作用 ?7 DOE的方法 [编辑] 什么是DOE DOE(Design of Experiment)试验设计,一种安排实验和分析实验数据的数理统计方法;试验设计主要对试验进行合理安排,以较小的试验规模(试验次数)、较短的试验周期和较低的试验成本,获得理想的试验结果以及得出科学的结论。 试验设计源于1920年代研究育种的科学家Dr.Fisher的研究, Dr. Fisher 是大家一致公认的此方法策略的创始者, 但后续努力集其大成, 而使DOE在工业界得以普及且发扬光大者, 则非Dr. Taguchi (田口玄一博士) 莫属。 [编辑] 为什么需要DOE ?要为原料选择最合理的配方时(原料及其含 量); ?要对生产过程选择最合理的工艺参数时; ?要解决那些久经未决的“顽固”品质问题 时;

?要缩短新产品之开发周期时; ?要提高现有产品的产量和质量时; ?要为新或现有生产设备或检测设备选择最 合理的参数时等。 另一方面,过程通过数据表现出来的变异,实际上来源于二部分:一部分来源于过程本身的变异,一部分来源于测量过程中产生的变差,如何知道过程表现出来的变异有多接近过程本身真实的变异呢?这就需要进行MSA测量系统分析。 [编辑] DOE的基本原理 试验设计的三个基本原理是重复,随机化,以及区组化。 所谓重复,意思是基本试验的重复进行。重复有两条重要的性质。第一,允许试验者得到试验误差的一个估计量。这个误差的估计量成为确定数据的观察差是否是统计上的试验差的基本度量单位。第二,如果样本均值用作为试验中一个因素的效应的估计量,则重复允许试验者求得这一效应的更为精确的估计量。如 s2是数据的方差,而有n次重复,则样本均值的方差是。这一点的实际含义是,如果n=1,如果2个处理的y1 = 145,和y2 = 147,这时我们可能不能作出2个 处理之间有没有差异的推断,也就是说,观察差147-145=2可能是试验误差的结果。但如果n合理的大,试验误差足够小,则当我们观察得y1随机化是试验设计使用统计方法的基石。 所谓随机化,是指试验材料的分配和试验的各个试验进行的次序,都是随机地确定的。统计方法要求观察值(或误差)是独立分布的随机变量。随机化通常能使这一假定有效。把试验进行适当的随机化亦有助于“均匀”可能出现的外来因素的效应。 区组化是用来提高试验的精确度的一种方法。一个区组就是试验材料的一个部分,相比于试验材料全体它们本身的性质应该更为类似。区组化牵涉到在每个区组内部对感兴趣的试验条件进行比较。 [编辑] DOE实验的基本策略 策略一:筛选主要因子(X型问题化成A型问题)

experiment

Project 1 Technology Enhanced Presentations Choice 1. To begin a presentation: a.choose the New menu, then choose New. b.choose the File menu, then choose New Presentation. c.choose the File menu, then choose New. d.click the New Slide button on the toolbar. 2. To insert footer into the slide presentation: a.click the Header/Footer button on the toolbar. b.choose the View menu, choose Master, then choose Slide Master. c.choose the Edit menu, then choose Header/Footer. d.choose the View menu, then choose Header/Footer. 3. The type of animation that is caused by the change from one slide to another is a(n): a.action setting. b.custom animation. c.slide transition. d.preset animation. 4. To set a transition: a.choose the Slide Show menu, then choose Transition. b.choose the Slide Show menu, then choose Slide Transition. c.click the Transition button on the toolbar. d.choose the Format menu, then choose Transition. 5. What is the first step in setting a custom animation? a.Clicking the Animation button on the toolbar. b.Choosing the Slide Show menu. c.Clicking the mouse on a part of the text to be animate d. d.Clicking the Add Effect button on the toolbar. Essay Questions What is the purpose of having a clear beginning and ending to a presentation?

操作系统experiment3

实验2 存储管理 1. 实验目的 存储管理的主要功能之一是合理地分配空间,请求页式管理式一种常用的虚拟存储管理技术。本实验的目的是通过请求页式管理中页面置换算法的模拟设计,了解虚拟存储技术的特点,掌握请求页式存储管理的页面置换算法。 2. 实验内容 (1) 通过随机数产生一个指令序列,共320条指令,指令地址按下述原则生成 ①50%的指令是顺序执行; ②25%的指令是均匀分布在前地址部分; ③25%的指令是均匀分布在后地址部分; 具体的实施方法是 ①在[0, 319] 的指令地址之间随机选取一起点m ; ②顺序执行一条指令,即执行地址为m+1 的指令; ③在前地址[0, m+1] 中随机选取一条指令并执行,该指令的地址是m’; ④顺序执行一条指令,其地址为m’+1 ; ⑤在后地址[m’+2, 319] 中随机选取一条指令并执行; ⑥重复上述步骤①~⑤,直到执行320次指令。 (2) 将指令序列变为页地址流 设:①页面大小为1K ; ②用户页面内存容量为4页到32页; ③用户虚存容量为32K 。 在用户虚存中,按每K 存放10条指令排列虚存地址,即320条指令在虚存中的存放方式为: 第0条~第9条指令为第0页(对应虚存地址为[0, 9]); 第10条~第19条指令为第1页(对应虚存地址为[10, 19]); …… …… 第310条~第319条指令为第31页(对应虚存地址为[310, 319]); 按以上方式,用户指令可组成32页。 (3) 计算并输出下述各种算法在不同内存容量下的命中率 ①First-In-First-out (FIFO) Page Replacement ②Least-Recently-Used (LRU) Page Replacement ③Optimal Page Replacement 其中③和④为选择内容

Dr. Heidegger's Experiment读后感

Ambiguity and Uncertainty in Dr. Heidegger's Experiment Nathaniel Hawthorne's short story, Dr. Heidegger's Experiment, reveals that people have a futile and self-destructive desire to relive their pasts instead of moving onward and accepting their fates. In Dr. Heidegger's Experiment, Hawthorne uses a point of view that allows ambiguity to enter the narration. The unnamed narrator opens many aspects of the story to more than one interpretation and enhances the revelation of the theme through uncertainty as he tells of the reactions of four old people to the water of the Fountain of Youth. The narrator himself seems unsure whether the events he is relating have even occurred. "Was it delusion?" he asks, "Even while the draught was passing down their throats, it seemed to have wrought a change on their whole systems." Commenting on the many tales that have sprung up around the mysterious Dr. Heidegger, the narrator even admits his own unreliability, stating, "Some of these fables, to my shame be it spoken, might possibly be traced back to mine own veracious self; and if any passages of the present tale should startle the reader's faith, I must be content to bear the stigma of a fiction-monger." These uncertainties divorce the story's happenings from reality, enhancing the allegorical meaning of the tale. The narrator is uncertain whether Dr. Heidegger's four old subjects have attained a second youth. As they drink the water, their actions become those of youths, but have their bodies changed too? "...the three gentlemen behaved in such a manner, as proved that the water of the Fountain of Youth possessed some intoxicating qualities; unless, indeed, their exhilaration of spirits were merely a lightsome dizziness, caused by the sudden removal of the weight of years." When they lose their newfound "youth," the same doubts are shown: "A strange chillness, whether of body or spirit they could not tell, was creeping gradually over them all. They ... fancied that each fleeting moment snatched away a charm, and left a deepening furrow where none had been before. Was it an illusion? Had the changes of a life-time been crowded into so brief a space... In truth, they had [grown old]. The Water of Youth possessed merely a virtue more transient than that of wine." The Elixir of Youth is likened to an alcoholic drink, yet the effects of an actual loss of age, and later a loss of their new-found youth, are felt in the four subjects. The narrator, and thus the reader, does not know the true extent of Dr. Heidegger's "experiment." This does not obscure the truths that the subjects reactions reveal; whatever interpretation the reader chooses, the theme remains. This uncertainty also highlights the multiple meanings of certain lines. When the four old

JAVA Experiment

以下所有实验完成的环境: OS: Windows XP IDE: Eclipse Database: MySQL或SQL Server 实验一 实验名称:JAV A中循环结构 实验目的:熟悉循环结构,熟悉JA V A类的定义以及参数的传递。 实验时间:(2学时) 实验内容: 1.金字塔:Pyramid.java 在屏幕上显示一个由星型符号“*”组成的金字塔图案,示例如下: * *** ***** 实验二 实验名称:封装,继承与多态 实验目的:熟悉JA V A面向对象的三大特性。 实验时间:(4学时) 实验内容: 1.定义一个形状类(Shape)方法:计算周长,计算面积 子类: 矩形类(Rect):额外的方法:cha()计算长宽差 圆形类(Circle) 正方形类(Square)矩形的子类 生成几个不同的形状对象,放在一个Shape类型的数组里,分别求每个形状的周长和面积。如果形状对象是一个矩形,且不是正方形,则计算长宽差。 实验三 实验名称:集合 实验目的:熟悉JA V A的集合框架,熟练掌握以下接口和类的使用,Collection, Map, List,Set, SortedSet, ArrayList, LinkedList, Vector, HashMap, Hashtable等。 实验时间:(2学时) 实验内容: 1. 数组拷贝CopyArray.java 定义数组int[] a = { 1,2,3,4,5,6,7,8,9,10 }和b。 (1)将数组a中的所以元素拷贝到数组b中,打印b中元素。(用循环实现) 结果参考: 1,2,3,4,5,6,7,8,9, (2)将数组a中从第3个元素起连续5个元素拷贝到数组b中,打印b中的元素(用api中提供的数组拷贝方法实现)

第18章-随机实验与自然实验(优选.)

1 ? 陈 强,《高级计量经济学及S t a t a 应用》课件,第二版,2014年,高等教育出版社。 第1 8章 随机 实验与自然实验 18.1 实 验 数 据 假设研 究1x 是否导致y 。假定{}12,,,K x x x 包含所有影响y 的因素。 不同学科采用不同的实验方法,大致分为以下几类。 (1) 控制实验(c o n t r o l l e d e x p e r i m e n t ):在理想的物理实验中,控制{}2,,K x x 全部不变,单独让1x 变化,观察y 的变化。 (2) 随机(控制)实验(r a n d o m i z e d c o n t r o l l e d e x p e r i m e n t ):

2 【例】 医学 上对 新药 1x 疗效的 实验。 由于 参加实验者的体质 与生 活 方式不同 ,不 可能 完全 控制 所有 其他因 素 {} 2, , K x x 。 随机实 验将实验人群(或个体)随机地分为两组,其中“实验组”或“处理组” (t r e a t m e n t g r o u p )服用真药,而“控制组”(c o n t r o l g r o u p ,也称“对照组”)服用“安慰药”(p l a c e b o )。 被试 者不知道自己分在哪一组,避免心理干扰。有时科研人员也不知道被试者 在哪一组,称为“双盲法”(d o u b l e b l i n d )。 【例 】农学中将地块随机地分成三组(很难找到土壤条件完全一样的地块 ),分 别给予不同的施肥量,然后考察施肥的效果。

3 (3) 自然实验或 准实验 (n a t u r a l e x p e r i m e n t o r q u a s i e x p e r i m e n t ): 由于某些 并非 为 了实 验 目的而发生的外部突发事件,使得当事 人 仿佛被随机 地分在 了实验 组或控制组。 【例】一个州通过某法律,但相邻州未通过此法律。两州民众事先不知道哪个州会通过此法律,故无法自我选择住在哪个州。从考察法律的效果而言,可近似认为民众随机选择住在哪个州,或被随机分为实验组(通过法律)与控制组(没通过法律)。 (4) 思想实验(t h o u g h t e x p e r i m e n t ): M i l t o n F r i e d m a n 曾设想在小岛上通过空投货币,考察该岛的宏观经济的变化。

Experiment_Report

Experiment Report (实验报告) Experiment Name(实验名称) The first Experiment(第 1 次实验) Experiment Date(日期) 2014-06-17 Teacher(老师) Student ID(学号) Name(姓名) Class(班级) Score(成绩) 一.Aim and Requirement(目的和要求) Define a Class containing an overloaded method area(), which calculates area of a Square, Rectangle, and a Circle depending on the type and number of arguments passed to it. 二. Experiment Content(实验内容) 1、Write an overloaded method area(), which calculates area of a Square, Rectangle, and a Circle. 2、Write the main function to test the method. 三.Experimental Procedures(实验步骤) 1、Write an overloaded method area() to calculate area of a Circle. 2、Overload the method area() again to calculate area of a Square or a Rectangle. 3、Write the main function to test the method. 4、Run the project file, the result shows that: 四.Experimental Summary(实验小结)

相关文档
最新文档