Research Article Analysis of Fatigue Life of PMMA at

Research ArticleAnalysis of Fatigue Life of PMMA at Different FrequenciesBased on a New Damage Mechanics Model

Aifeng Huang1 Weixing Yao2 and Fang Chen2

1 Research Institute of Unmanned Aircraft Nanjing University of Aeronautics and Astronautics Nanjing 210016 China2 State Key Laboratory of Mechanics and Control of Mechanical Structures Nanjing University of Aeronautics and AstronauticsNanjing 210016 China

Correspondence should be addressed to Weixing Yao wxyaonuaaeducn

Received 12 March 2014 Revised 28 April 2014 Accepted 4 May 2014 Published 11 June 2014

Academic Editor Jose Merodio

Copyright copy 2014 Aifeng Huang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Low-cycle fatigue tests at different frequencies and creep tests under different stress levels of Plexiglas Resist 45 were conductedCorrespondingly the creep fracture time S-N curves cyclic creep and hysteresis loop were obtainedThese results showed that thefatigue life increases with frequency at low frequency domain After analysis it was found that fatigue life is dependent on the loadrate and is affected by the creep damage In addition a new continuum damagemechanics (CDM)model was established to analyzecreep-fatigue life where the damage increment nonlinear summation rule was proposed and the frequency modification was madeon the fatigue damage evolution equation Differential evolution (DE) algorithmwas employed to determine the parameters withinthe model The proposed model described fatigue life under different frequencies and the calculated results agreed well with theexperimental results

1 Introduction

Because of its high impact resistance sufficient strength andexcellent transmittance PMMA material has wide applica-tion Since many structures must withstand cyclic loadingduring their service lives the research on fatigue of PMMA isattracting an increased interest [1ndash3]

PMMA is a typical kind of viscoelastic material The fati-gue properties of viscoelastic materials are sensitive to tem-perature and cyclic frequency which is one of the biggestdifferences from conventional metal material This paper isconcerned with the frequency sensitivity Lots of researchershave investigated the fatigue property of PMMA and otherpolymers at different frequencies Cheng et al [2] conductedPMMA fatigue crack propagation (FCP) experiments atfrequencies of 1 10 50 and 100Hz Test temperature rangedfrom minus30 to 100∘C Their results showed that FCP ratesdecrease as cyclic frequency increases until temperatureapproaches the glass transition region Jia et al [4] and Feng etal [5] performed fatigue tests on PMMA used in aircraft andordinary industrial grade they all observed that FCP ratesdecrease as cyclic frequency increases Luo et al [6] carried

out several fatigue tests of YB-3 PMMA under the frequencyof 01 1 and 10Hz the results showed that fatigue lifeincreased with increasing frequency although the increasewas not so notable in high frequency range Wang et al [7]also conducted PMMA fatigue tests at the frequency of 1 2and 5Hz and the difference was that fatigue life was observedto decrease with frequency except at high stress level

The fatigue life of PMMA varies with frequency indis-putably and that its fatigue crack propagation rate decreaseswith frequency is a consistent conclusion of most experi-ments However there is not such consistent relationshipbetween fatigue life and loading frequency Many researchanalyzed the influence mechanism of frequency on fatiguebehavior It can be summedup to three attributions hystereticheating creep and rate sensitivity Under cyclic loading thehysteresis loop is formed by out of phase stresses and strainsdue to high internal damping of polymers Hysteresis loopindicates energy dissipation or heat generation named hys-teretic heating If frequency is high hysteretic heat cannot bedissipated rapidly enough because of low thermal conductiv-ity of polymers and thus temperature will rise up Hysteretic

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 352676 8 pageshttpdxdoiorg1011552014352676

2 Mathematical Problems in Engineering

Table 1 Typical property values (at 23∘C and 50 relative humidity) of Plexiglas Resist 45

Density Transmittance Tensile strengthModulus ofelasticity

(short-term value)Poissonrsquos rate Nominal

elongation at break

119 gcm3 91 60MPa 2700MPa 041 10

heating will soften the material and both stiffness andstrength will decrease This is the reason for fatigue lifedecreasing with frequency in some experiments Creep phe-nomenon is significant and common in polymers Duringfatigue process with nonzero mean stress creep deformationand creep damage can be observed apparently even withoutload holding More creep damage accumulates in one cycleunder lower frequency for longer time per cycle This willresult in decrement of fatigue life as frequency decreasesMechanical properties of viscoelastic material are dependentupon strain rate For polymers strain rate effect is an import-ant property as creep In unidirectional tension or compres-sion and impact research field strain rate effect has beenpaid quite more attention Strain rate hardening and strainrate strengthening phenomenon of PMMA were observed inlots of experiments [8 9] It is well reasoned to deduce thatfrequency is higher strain rate is necessarily higher andthen stiffness and tensile strength will be higher and fatiguestrength will be higher too That is to say at the same loadlevel frequency is higher and fatigue life will be larger Unfor-tunately rate effect in PMMA fatigue was only mentionedoccasionally the attention paid to rate effect was far fromenough In general fatigue life variation with frequency isrelated to frequency range stress level and other parametersand is determined by the dominate mechanism For examplehysteretic heat is notable at high frequency creep damagecontributes more during lower frequency and rate effect mayplay a part in a wide frequency range

Hysteretic heating in polymers was noticed and investi-gated from an early period many works have been done [1011] For creep damage equations similar to Paris law includ-ing fatigue and creep have been proposed to describe theFCP rate [12 13] Luo et al [6] established a creep-fatigue lifemodel based on continuum damage mechanics Howeverrate effect consideration appears lacking in the fatigue lifeprediction model

The authors of this paper performed fatigue test at 00301 and 05Hz with Plexiglas Resist 45 and recorded itsmechanical response at different frequencies to accumulatetest data towards a better understanding of the frequencyinfluence on PMMA fatigue behavior and relevant theoreticalresearch It is found that the temperature rise is insignificantat frequencies nomore than 05Hz [1] Load frequency in thispaper was low enough to take no account of hystereticheating Thus the problem could be simplified to investigatepure frequency influence To assess creep damage during thefatigue process creep tests were carried out Moreover amodifiedCDMcreep-fatiguemodelwas proposed to describefatigue life at low frequency The proposed model can also beused in the condition with load holding

15 10

30

45

220

R75

R6

(a)

250

55

15

45 R75

(b)

Figure 1 Geometry and dimensions (inmm) of specimen (a) creepspecimen and (b) fatigue specimen

2 Experimental

21Material and Specimen Plexiglas Resist 45 sheet which isdeveloped andproduced byRohmCompany is a kind of solidplastic sheet extruded from impact-modified PMMA Somematerial properties of Plexiglas Resist 45 are presented inTable 1

The creep and fatigue specimens (see Figure 1) are 3mmthick and are designed according to the standard of highpolymers mechanics test [14]

22 Creep Test In this paper creep tests were carried outunder constant load To reduce test cost creep tests werecarried out on the self-designed equipment which applyconstant load by hanging the weight A camera was utilized tomonitor and record the fracture time A gauge length with aninitial value of about 55mm was drawn on the specimenbeforehand In some interval the gauge length was measuredby microcaliper Although the error of this measuringmethod was big the approximate shape of creep curve couldbe obtained because of big creep deformation of specimenFive levels of nominal stress were used in creep tests 30MPa33MPa 36MPa 37MPa and 40MPaThree specimens weretested at each stress level

23 Fatigue Test Fatigue tests were run by applying sinu-soidal tensile loads with constant amplitude in load controlmode on a MTS809 servohydraulic testing machine Theenvironment temperature was controlled at about 27∘C

Mathematical Problems in Engineering 3

0

05

1

15

2

25

minus1

minus05

145 15 155 16 165 17 175

log(t

(h))

log(120590(MPa))

Figure 2 Creep fracture time at different stresses

throughout all tests 30 33 39 42 and 44MPa were respec-tively chosen as maximum stress Three series of tests wereconducted with cyclic load frequency values of 003 01and 05Hz To prevent specimens from buckling under com-pressive loading tension to tension load cycles were appliedwith a load ratio 001 for all tests During the load controlledfatigue tests in order to observe cyclic creep and hystere-sis loop axial strains were measured by MTS63213F-20extensometer with 10mmbase length Full stress strain cycleswere recorded at fixed time intervals

3 Results and Discussions

31 Creep Result The creep curve of Plexiglas Resist 45 wascomposed of three typical stages with rapid strain rate ininitial stage and before fracturingThe fracture time was pre-sented in Figure 2 where the log (fracture time) was plottedversus log (stress) The fracture time covered 4 hours to 200hours and decreased steeply with increasing stress As shownin Figure 2 the data points were nearly in a line

32 Fatigue Test Result

321 Fatigue Life The results of fatigue tests were reported inFigure 3 where 120590max was plotted versus log 119873 These threecurves were far from coincidence with fatigue life increasingwith increasing cyclic frequency

322 Cyclic Creep Hysteresis loop is always observed tomove gradually along the strain axis during nonzero meanstress fatigue process of polymers and this indicates thatcyclic creep strain accumulates gradually For different mate-rials load holding time or frequency has a different effect oncyclic creep rate For some high polymers frequency exhibitsan acceleration effect on cyclic creep rate [15] This paper

20

25

30

35

40

45

50

2 25 3 35 4log(N)

003Hz01Hz05Hz

120590m

ax(M

Pa)

Figure 3 Fatigue life and 119878-119873 curve at different frequencies

attempted to observe the cyclic creep of Plexiglas Resist 45 atdifferent frequencies Because the stress ratio of fatigue test inthis paper was close to zero (119877 = 001) the elastic componentincluded in minimum (min) displacement was very smallthe min displacement could be treated approximately ascyclic creep displacementThemin displacements at differentfrequencies were presented in Figure 4 (120590max = 33MPa) andFigure 5 (120590max = 44MPa) The results at other stress levelswere similar It could be seen from Figures 4(a) and 5(a) thatfrequency slightly accelerated the cyclic creep rate of the firststage For the second stage of the major part of cyclic creepcurve frequency had no notable acceleration effect Thecomparisons between Figures 4(a) and 4(b) and Figures 5(a)and 5(b) demonstrated that cyclic creep was more related totime than to cycles

323 Hysteresis Loop Because mechanical properties of vis-coelastic material such as elastic modulus and viscous dampare related to strain rate hysteresis loop at different frequen-cies exhibits some differences This paper investigated thefrequency influence on hysteresis loop The hysteresis loopof 120590max = 39MPa at half-life was presented in Figure 6 Theresults at other stress levels were similar The area withinhysteresis loop was calculated by integrationThe integrationresults showed that the area did not vary with frequency sig-nificantly The test results by Wang et al [7] on Plexiglas alsodemonstrated that frequency did not notably affect the area ofthe hysteresis loop However as seen in Figure 6 the ldquosloperdquoof hysteresis loop decreased with frequency Modulus 119864 wasdefined as 119864 = Δ120590Δ120576 and the calculated 119864 was dotted inFigure 7 where a fitted power functional curve of 119864 was plot-ted As shown in Figure 7 modulus increased with frequency


0 5000 10000 15000 200000

005

01

015

02

025

03

035

04

045

d(m

m)

003Hz01Hz05Hz

Time (s)

(a)

0

005

01

015

02

025

03

035

04

045

d(m

m)

003Hz01Hz05Hz

0 500 1000 1500 2000 2500

Cycles

(b)

Figure 4 (a) Min displacement versus time and (b) min displace-ment versus cycles at different frequencies (120590max = 33MPa)

especially at low frequency and 119864 could be expressed as thepower function of frequency

4 CDM Model for Fatigue Life Prediction

Because the continuum damage mechanics approach cancope with the coexisting of multiform damage it is a suitableand promising method for creep-fatigue life prediction Luoet al [6] have established the creep-fatigue damage model ofYB-3 Plexiglas based on CDM additionally the pure fatiguedamage has been separated from the fatigue test includingcreep damage This paper stepped forward on their work

The main work of CDM approach to prediction fatiguelife includes establishment of damage evolution equation anddetermination of the initial and critical damage value For thesake of convenience damage initial value is mostly taken aszero119863

0= 0 and critical value as unity119863cr = 1

41 Frequency Modified Fatigue Damage Evolution EquationThe following formula is a constantly used fatigue damageevolution under simple load condition [16]

d119863d119873

=

120590

2119904+1

119886

119861(1 minus 119863)

2119904

(1)

where 119904 and 119861 are material constants 119863 is the damagevariable and 120590

119886is stress amplitude

The above analysis has concluded that because mechani-cal properties of viscoelasticmaterial are related to strain ratefatigue strengthwould increase with frequency But the quan-tity relationship of fatigue strength to frequency is difficult todetermine Experimental results in [9] showed that initialmodulus alongwith tensile strength increaseswith strain rateand they both have similar functional relationship withstrain rate It should be possible that functional relationshipbetween fatigue strength and frequency is similar to modulusand frequency According to results in [17] and this paper themodulus increases with frequency significantly when fre-quency is low and then tends to be stable at higher frequencyFigure 7 showed that modulus could be expressed as powerfunction of frequency in low frequency range Hence thispaper proposed the frequency modified fatigue damageevolution equation expressed as

d119863d119873

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863)

2119904

(2)

where 119891 is cyclic frequency and119898 is constant Because creepdamage exists inevitably in fatigue process and load fre-quency of high polymers cannot be too high it is difficult toobtain pure fatigue life directly from fatigue test The param-eters in (2) would be determined through later fitting calcu-lation of creep-fatigue life

42 Creep Damage Evolution Equation As can be seen fromcyclic creep result the shape of cyclic creep curve was similarto static creep and frequency did not accelerate the cycliccreep rate significantly It can be assumed that creep undercyclic load may occur due to the same mechanism as staticcreep Static creep damage evolution equation could beemployed to describe the pure creep damage in fatigue pro-cess Therefore Kachanov equation was utilized to describethe creep damage evolution of Plexiglas Resist 45 and thisequation has been proposed for a long time and has beenverified in many aspects [18]

d119863d119905

= 119860(

120590

1 minus 119863

)

119903

(3)

where 119860 and 119903 are material constants which can be obtainedthrough creep test


0

02

04

06

08

1

12

0 5000 10000 15000 20000

d(m

m)

Time (s)

003Hz01Hz05Hz

(a)

0

02

04

06

08

1

12

0 500 1000 1500

d(m

m)

Cycles

003Hz01Hz05Hz

(b)

Figure 5 (a) Min displacement versus time and (b) min displacement versus cycles at different frequencies (120590max = 44MPa)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 0005 001 0015 002

003Hz01Hz05Hz

P(N

)

120576

Figure 6 Hysteresis loop at different frequencies (120590max = 39MPa)

Integrating (3) with initial condition 119863 |

119905=0= 0 damage

equation can be obtained as follows

119863 = 1 minus (1 minus 119860 (119903 + 1) 120590

119903

119905)

1(119903+1)

(4)

When material is failure 119863 = 1 then fracture time 119905119888can be

written as

119905

119888=

1

119860 (119903 + 1)

120590

minus119903

(5)

21

22

23

24

25

26

27

0 02 04 06

f (Hz)

E(G

MPa

)

Figure 7 Modulus versus frequency (120590max = 39MPa)

Rewrite (5) in the following logarithm form

log (119905119888) = minus log (119860 (119903 + 1)) minus 119903 log (120590) (6)

Equation (6) showed that fracture time should be logarithmlinear with stress level 119860 = 9198119864 minus 26 119903 = 14216 could bedetermined from creep test results and the units of120590 119905 and 119905

119888

are MPa hour and hour respectively

43 Creep-Fatigue Damage Evolution Equation For high pol-ymers creep damage and fatigue damagemost often accumu-late nonlinearly because of creep-fatigue interaction duringcreep-fatigue process [15 19] To the creep-fatigue interactionquestion Lemaitre and Chaboche presented a general kinetic


equation for creep-fatigue damage problems based on theassumption that the total damage increment d119863 is the sum offatigue damage increment d119863

119891and creep damage increment

d119863119888(ie d119863 = d119863

119888+ d119863119891) [18] This model has been used

with success during the past 30 years on various classes ofmaterials and applications [20 21] Factually the damageincrement linear summation rule was not flawless for exam-ple Chaboche pointed out that this rule would overestimatecreep-fatigue interaction [22 23]

In most cases creep-fatigue interaction will be strongwhen creep damage is nearly equal to fatigue damage andwillbe weak when either creep or fatigue is dominant The inter-action intensity in a short time or in one cycle should obeythe same ruleThus increment nonlinear summation rule wasproposed here in detail fatigue damage is composed byfatigue damage itself and fatigue damage caused by creep andin the same way creep damage is the sum of creep damageitself and creep damage caused by fatigue damage Theinteraction damage is related to the relative quantity of creepdamage and fatigue damage When the sum of creep damageand fatigue damage attains critical value 1 the material willfail

d1198631015840119891

= d119863119891+ 119862 sdot d119863

119888 (7)

d1198631015840119888

= d119863119888+ 119862 sdot d119863

119891 (8)

119863

119891= 119863

119891+ d1198631015840119891

(9)

119863

119888= 119863

119888+ d1198631015840119888

(10)

119863tot = 119863119888 + 119863119891 (11)

where 119862 = 119888

1015840 exp(2radicd119863119891sdot d119863119888(d119863119891+ d119863119888)) 1198881015840 is a slight-

adjustment parameter and has a value of about 1 Let 119862 equiv 1damage increment linear summation rule will be obtainedLet 119886 = d119863

119888d119863119891 then 119862 can be written as 119862 =

119888

1015840 exp(2radic119886(1+119886)) Function119862 is symmetric to 119886 = 1 that is119862(119886) = 119862(1119886) The value of exp(2radic119886(1 + 119886)) varies from 1to 2718 and reaches maximum value at 119886 = 1 (d119863

119891= d119863119888)

In most cases the value is slightly greater than 1 and thisindicates that there is no big difference between the proposeddamage summation rule and the widely used damage incre-ment linear summation rule But the expression of 119862 in thispaper is more flexible and is consistent with the general rulethat interaction is strongest when creep damage and fatiguedamage are in the same level Variable value 119862 along withslight-adjustment coefficient 1198881015840 canmake the calculated resultagree better with experimental result

Substituting (2) and (3) into (7) and (8) respectivelycreep-fatigue damage evolution equation would be obtainedGenerally one cycle is chosen as integrate step size 119863

119888is

taken as constant value during one cycle d119863119888obtained by

integrating (3) in one cycle is substituted to (8) then

44 Parameter Determination It is difficult to determine theparameters in creep-fatigue damage model Differential evo-lution (DE) algorithm was employed to solve this problemParameters determined by DE were presented in Table 2 DE

25

30

35

40

45

0 2000 4000 6000

120590m

ax(M

Pa)

N

Figure 8 Fatigue life of test and fitted result (◻ test result red05Hz black 01 Hz blue 003Hz mdash Model 1 - - - Model 2 ndash ndashModel 3)

algorithm presented by Stron and Price is a relatively newpopulation based stochastic optimization approach [24] As abranch of evolution algorithm DE has a similar theory toother evolution algorithms including population initializa-tionmutation crossover and selection operationThe crucialdifference is that DE uses the difference of randomly sampledpairs of vectors in the population for its mutation opera-tor and other evolutionary algorithms use predeterminedprobability distribution functions DE algorithm has beenattracting increasing attention for it is simple fast and robust

45 Comparison with Conventional Model Compared toconventional creep-fatigue damage mechanical model twomodifications have been made in this paper First frequencymodification was introduced into pure fatigue evolutionequation Second nonlinear summation of damage incre-ment was proposed The comparison between the fittedresults by modified model and conventional model waspresented in Table 2 and Figure 8 Model 1 was the pro-posed model Model 2 turned nonlinear damage incrementsummation back to linear that is 119862 equiv 1 and Model 3cleared away frequency modification from fatigue evolutionequation As shown by the results the fitted result of Model 1agreed best with tested result Model 3 agreed worst Thedifference between Model 2 and Model 1 was quite small itwas because in all tests of this paper creep damage was quitesmall compared to fatigue damage and the values of 119862 inModel 1 were close to 1 The comparison also indicated thatfatigue strength rate dependence contributed largely tofatigue life frequency sensitivity of Plexiglas Resist 45

5 Conclusion

Tensile fatigue tests at three frequencies (003 01 and 05Hz)were performed in MTS809 and the results showed thatfatigue life increases with frequency Frequency had little


Table 2 Comparison of equation and 1198772 of three models

Model 1 Model 2 Model 3

Equations

Initial119873 = 0119863119891

= 0119863119888

= 0 Initial119873 = 0119863tot = 1 Initial119873 = 0119863tot(119873) = 0

End119863tot(119873) = 1 End119863tot(119873) = 1 End119863tot(119873) = 1

Δ119863

119891

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863

119891

)

Δ119863

119891

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863tot)Δ119863

119891

=

120590

2119904+1

119886

119861 (1 minus 119863tot)

Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863c)

119903

d119905 Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863tot)

119903

d119905 Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863tot)

119903

d119905

119862 = 119888

1015840 exp(radic

Δ119863

119888

+ Δ119863

119891

05(Δ119863

119888

+ Δ119863

119891

)

) 119863tot = 119863tot + Δ119863119891 + Δ119863119888 119863tot = 119863tot + Δ119863119891 + Δ119863119888

119863

119891

= 119863

119891

+ Δ119863

119891

+ 119862 sdot Δ119863

119888

119863

119888

= 119863

119888

+ Δ119863

119888

+ 119862 sdot Δ119863

119891

119863tot = 119863119888 + 119863119891

ParametersFor three models 119860 = 9198119864 minus 26 119903 = 14216 units (120590 120590

119886

MPa 119879 119905 hour)119904 = 11497 119898 = 03316 119904 = 10592 119898 = 034047 119904 = 11436

119861 = 190990790 1198881015840 = 09964 119861 = 99955327 119861 = 99767982

119877

2 098 095 066

impact on cyclic creep rate and the area within hysteresisloop but modulus increased with frequency

Based on the analysis of test data and the viscoelasticmechanical properties of Plexiglas creep damage along withcreep-fatigue interaction and fatigue strength rate-depend-ence these two factors should be concerned with the lifeprediction A creep-fatigue damage model including damageincrement nonlinear summation and frequencymodificationwas presented for life analysisThe comparison of test and cal-culation result showed the capability of the proposed modelto describe the fatigue life of Plexiglas Resist 45 at differentfrequencies

Disclosure

Aifeng Huang is the first author

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by National Natural Science Foun-dation of China (no 51275241)

References

[1] Editorial Committee of Fatigue and Fracture Atlas of PMMAFatigue and Fracture Atlas of PMMA Science Press BeijingChina 1987

[2] W M Cheng G A Miller J A Manson R W Hertzberg andL H Sperling ldquoMechanical behaviour of poly (methyl meth-acrylate)mdashpart 2 the temperature and frequency effects on

the fatigue crack propagation behaviourrdquo Journal of MaterialsScience vol 25 no 4 pp 1924ndash1930 1990

[3] W Liu and X J Yang ldquoDamage evolution with growing cycliccreep and life prediction of MDYB-3 PMMArdquo Fatigue ampFracture of EngineeringMaterials amp Structures vol 36 no 6 pp483ndash491 2012

[4] J-H Jia Y-Z Li and J Xiao ldquoStudy on fatigue crack propaga-tion behavior of YB-MD-3 PMMA platesrdquo Journal of Aeronau-tical Materials vol 26 no 5 pp 109ndash112 2006

[5] L Feng Z Yin S Gan et al ldquoFatigue crack propagation ofPMMArdquo Polymer Materials Science and Engineering vol 16 no6 pp 121ndash123 2000

[6] C-L Luo W-X Yao and Z-L Zhang ldquoSeparation model offatigue-damage and creep-damage of polymethyl methacrylatefor fatigue life predictionrdquo Journal of Nanjing University ofAeronautics amp Astronautics vol 37 no 6 pp 741ndash744 2005

[7] H Wang Y Wang X Yi and J Zhuang ldquoFrequency responseregularity of cyclic toughness and S-N curve in polymethrymethacrylaterdquoMechanical Science and Technology for AerospaceEngineering vol 30 no 5 pp 699ndash702 2011

[8] W M Cheng G A Miller J A Manson R W Hertzberg andL H Sperling ldquoMechanical behaviour of poly(methyl meth-acrylate)mdashpart 1 tensile strength and fracture toughnessrdquoJournal of Materials Science vol 25 no 4 pp 1917ndash1923 1990

[9] H Wu G Ma and Y Xia ldquoExperimental study on mechanicalproperties of PMMA under unidirectional tensile at low andintermediate strain ratesrdquo Journal of Experimental Mechanicsvol 20 no 2 pp 193ndash198 2005

[10] M D Skibo R W Hertzberg and J A Manson ldquoThe effect oftemperature on the frequency sensitivity of fatigue crack prop-agation in polymersrdquo Fracture vol 3 pp 1127ndash1133 1977

[11] A Bernasconi and R M Kulin ldquoEffect of frequency uponfatigue strength of a short glass fiber reinforced polyamide 6a superposition method based on cyclic creep parametersrdquoPolymer Composites vol 30 no 2 pp 154ndash161 2009


[12] Y Hu J Summers A Hiltner and E Baer ldquoCorrelation offatigue and creep crack growth in poly(vinyl chloride)rdquo Journalof Materials Science vol 38 no 4 pp 633ndash642 2003

[13] M Parsons E V Stepanov A Hiltner and E Baer ldquoEffect ofstrain rate on stepwise fatigue and creep slow crack growth inhigh density polyethylenerdquo Journal of Materials Science vol 35no 8 pp 1857ndash1866 2000

[14] F Xu Polymer Material Mechanics Test Science Press BeijingChina 1988

[15] AMVinogradov and S Schumacher ldquoCyclic creep of polymersandpolymer-matrix compositesrdquoMechanics of CompositeMate-rials vol 37 no 1 pp 29ndash34 2001

[16] H Wu Damage Mechanics National Defence Industry PressBeijing China 1990

[17] J Shi ldquoMeasurement of elastic modulus of plexiglassrdquo Journalof Suzhou College of Education no 3 pp 83ndash84 1996

[18] J Lemaitre and J L Chaboche Solid Mechanics of Materialstranslated by T Yu Y Wu National Defence Industry PressBeijing China 1997

[19] Z Song W Huang and M Liu ldquoStudy of the mutual damageof polystyrene both in fatigue and creeprdquo Polymer MaterialsScience and Engineering vol 19 no 1 pp 188ndash190 2003

[20] X J Yang C L Chow and K J Lau ldquoA unified viscoplasticfatigue damage model for 63Sn-37Pb solder alloy under cyclicstress controlrdquo International Journal of Damage Mechanics vol12 no 3 pp 225ndash243 2003

[21] H Xiao X Li Y Yan N Liu and Y Shi ldquoDamage behaviorof SnAgCu solder under thermal cyclingrdquo Rare Metal Materialsand Engineering vol 42 no 2 pp 221ndash226 2013

[22] LM Kachanov Introduction to ContinuumDamageMechanicsMartinus Nijhoff Publishers DordrechtTheNetherlands 1986

[23] J-L Chaboche and F Gallerneau ldquoAn overview of the damageapproach of durability modelling at elevated temperaturerdquoFatigue and Fracture of Engineering Materials amp Structures vol24 no 6 pp 405ndash418 2001

[24] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient Heuristic for global optimization over continuous spacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997

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Table 1 Typical property values (at 23∘C and 50 relative humidity) of Plexiglas Resist 45

Density Transmittance Tensile strengthModulus ofelasticity

(short-term value)Poissonrsquos rate Nominal

elongation at break

119 gcm3 91 60MPa 2700MPa 041 10

heating will soften the material and both stiffness andstrength will decrease This is the reason for fatigue lifedecreasing with frequency in some experiments Creep phe-nomenon is significant and common in polymers Duringfatigue process with nonzero mean stress creep deformationand creep damage can be observed apparently even withoutload holding More creep damage accumulates in one cycleunder lower frequency for longer time per cycle This willresult in decrement of fatigue life as frequency decreasesMechanical properties of viscoelastic material are dependentupon strain rate For polymers strain rate effect is an import-ant property as creep In unidirectional tension or compres-sion and impact research field strain rate effect has beenpaid quite more attention Strain rate hardening and strainrate strengthening phenomenon of PMMA were observed inlots of experiments [8 9] It is well reasoned to deduce thatfrequency is higher strain rate is necessarily higher andthen stiffness and tensile strength will be higher and fatiguestrength will be higher too That is to say at the same loadlevel frequency is higher and fatigue life will be larger Unfor-tunately rate effect in PMMA fatigue was only mentionedoccasionally the attention paid to rate effect was far fromenough In general fatigue life variation with frequency isrelated to frequency range stress level and other parametersand is determined by the dominate mechanism For examplehysteretic heat is notable at high frequency creep damagecontributes more during lower frequency and rate effect mayplay a part in a wide frequency range

Hysteretic heating in polymers was noticed and investi-gated from an early period many works have been done [1011] For creep damage equations similar to Paris law includ-ing fatigue and creep have been proposed to describe theFCP rate [12 13] Luo et al [6] established a creep-fatigue lifemodel based on continuum damage mechanics Howeverrate effect consideration appears lacking in the fatigue lifeprediction model

The authors of this paper performed fatigue test at 00301 and 05Hz with Plexiglas Resist 45 and recorded itsmechanical response at different frequencies to accumulatetest data towards a better understanding of the frequencyinfluence on PMMA fatigue behavior and relevant theoreticalresearch It is found that the temperature rise is insignificantat frequencies nomore than 05Hz [1] Load frequency in thispaper was low enough to take no account of hystereticheating Thus the problem could be simplified to investigatepure frequency influence To assess creep damage during thefatigue process creep tests were carried out Moreover amodifiedCDMcreep-fatiguemodelwas proposed to describefatigue life at low frequency The proposed model can also beused in the condition with load holding

15 10

30

45

220

R75

R6

(a)

250

55

15

45 R75

(b)

Figure 1 Geometry and dimensions (inmm) of specimen (a) creepspecimen and (b) fatigue specimen

2 Experimental

21Material and Specimen Plexiglas Resist 45 sheet which isdeveloped andproduced byRohmCompany is a kind of solidplastic sheet extruded from impact-modified PMMA Somematerial properties of Plexiglas Resist 45 are presented inTable 1

The creep and fatigue specimens (see Figure 1) are 3mmthick and are designed according to the standard of highpolymers mechanics test [14]

22 Creep Test In this paper creep tests were carried outunder constant load To reduce test cost creep tests werecarried out on the self-designed equipment which applyconstant load by hanging the weight A camera was utilized tomonitor and record the fracture time A gauge length with aninitial value of about 55mm was drawn on the specimenbeforehand In some interval the gauge length was measuredby microcaliper Although the error of this measuringmethod was big the approximate shape of creep curve couldbe obtained because of big creep deformation of specimenFive levels of nominal stress were used in creep tests 30MPa33MPa 36MPa 37MPa and 40MPaThree specimens weretested at each stress level

23 Fatigue Test Fatigue tests were run by applying sinu-soidal tensile loads with constant amplitude in load controlmode on a MTS809 servohydraulic testing machine Theenvironment temperature was controlled at about 27∘C


0

05

1

15

2

25

minus1

minus05

145 15 155 16 165 17 175

log(t

(h))

log(120590(MPa))








20

25

30

35

40

45

50

2 25 3 35 4log(N)

003Hz01Hz05Hz

120590m

ax(M

Pa)





0 5000 10000 15000 200000

005

01

015

02

025

03

035

04

045

d(m

m)

003Hz01Hz05Hz

Time (s)

(a)

0

005

01

015

02

025

03

035

04

045

d(m

m)

003Hz01Hz05Hz

0 500 1000 1500 2000 2500

Cycles

(b)








d119863d119873

=

120590

2119904+1

119886

119861(1 minus 119863)

2119904

(1)




d119863d119873

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863)

2119904

(2)



d119863d119905

= 119860(

120590

1 minus 119863

)

119903

(3)



0

02

04

06

08

1

12

0 5000 10000 15000 20000

d(m

m)

Time (s)

003Hz01Hz05Hz

(a)

0

02

04

06

08

1

12

0 500 1000 1500

d(m

m)

Cycles

003Hz01Hz05Hz

(b)


0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 0005 001 0015 002

003Hz01Hz05Hz

P(N

)

120576



119905=0= 0 damage


119863 = 1 minus (1 minus 119860 (119903 + 1) 120590

119903

119905)

1(119903+1)

(4)


written as

119905

119888=

1

119860 (119903 + 1)

120590

minus119903

(5)

21

22

23

24

25

26

27

0 02 04 06

f (Hz)

E(G

MPa

)





119888






d119863119888(ie d119863 = d119863




d1198631015840119891

= d119863119891+ 119862 sdot d119863

119888 (7)

d1198631015840119888

= d119863119888+ 119862 sdot d119863

119891 (8)

119863

119891= 119863

119891+ d1198631015840119891

(9)

119863

119888= 119863

119888+ d1198631015840119888

(10)

119863tot = 119863119888 + 119863119891 (11)

where 119862 = 119888




119888


119891= d119863119888)



119888is




25

30

35

40

45

0 2000 4000 6000

120590m

ax(M

Pa)

N




5 Conclusion





Equations

Initial119873 = 0119863119891

= 0119863119888



Δ119863

119891

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863

119891

)

Δ119863

119891

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863tot)Δ119863

119891

=

120590

2119904+1

119886

119861 (1 minus 119863tot)

Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863c)

119903

d119905 Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863tot)

119903

d119905 Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863tot)

119903

d119905

119862 = 119888

1015840 exp(radic

Δ119863

119888

+ Δ119863

119891

05(Δ119863

119888

+ Δ119863

119891

)


119863

119891

= 119863

119891

+ Δ119863

119891

+ 119862 sdot Δ119863

119888

119863

119888

= 119863

119888

+ Δ119863

119888

+ 119862 sdot Δ119863

119891

119863tot = 119863119888 + 119863119891


119886

MPa 119879 119905 hour)119904 = 11497 119898 = 03316 119904 = 10592 119898 = 034047 119904 = 11436

119861 = 190990790 1198881015840 = 09964 119861 = 99955327 119861 = 99767982

119877

2 098 095 066



Disclosure




Acknowledgment


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

05

1

15

2

25

minus1

minus05

145 15 155 16 165 17 175

log(t

(h))

log(120590(MPa))








20

25

30

35

40

45

50

2 25 3 35 4log(N)

003Hz01Hz05Hz

120590m

ax(M

Pa)





0 5000 10000 15000 200000

005

01

015

02

025

03

035

04

045

d(m

m)

003Hz01Hz05Hz

Time (s)

(a)

0

005

01

015

02

025

03

035

04

045

d(m

m)

003Hz01Hz05Hz

0 500 1000 1500 2000 2500

Cycles

(b)








d119863d119873

=

120590

2119904+1

119886

119861(1 minus 119863)

2119904

(1)




d119863d119873

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863)

2119904

(2)



d119863d119905

= 119860(

120590

1 minus 119863

)

119903

(3)



0

02

04

06

08

1

12

0 5000 10000 15000 20000

d(m

m)

Time (s)

003Hz01Hz05Hz

(a)

0

02

04

06

08

1

12

0 500 1000 1500

d(m

m)

Cycles

003Hz01Hz05Hz

(b)


0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 0005 001 0015 002

003Hz01Hz05Hz

P(N

)

120576



119905=0= 0 damage


119863 = 1 minus (1 minus 119860 (119903 + 1) 120590

119903

119905)

1(119903+1)

(4)


written as

119905

119888=

1

119860 (119903 + 1)

120590

minus119903

(5)

21

22

23

24

25

26

27

0 02 04 06

f (Hz)

E(G

MPa

)





119888






d119863119888(ie d119863 = d119863




d1198631015840119891

= d119863119891+ 119862 sdot d119863

119888 (7)

d1198631015840119888

= d119863119888+ 119862 sdot d119863

119891 (8)

119863

119891= 119863

119891+ d1198631015840119891

(9)

119863

119888= 119863

119888+ d1198631015840119888

(10)

119863tot = 119863119888 + 119863119891 (11)

where 119862 = 119888




119888


119891= d119863119888)



119888is




25

30

35

40

45

0 2000 4000 6000

120590m

ax(M

Pa)

N




5 Conclusion





Equations

Initial119873 = 0119863119891

= 0119863119888



Δ119863

119891

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863

119891

)

Δ119863

119891

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863tot)Δ119863

119891

=

120590

2119904+1

119886

119861 (1 minus 119863tot)

Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863c)

119903

d119905 Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863tot)

119903

d119905 Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863tot)

119903

d119905

119862 = 119888

1015840 exp(radic

Δ119863

119888

+ Δ119863

119891

05(Δ119863

119888

+ Δ119863

119891

)


119863

119891

= 119863

119891

+ Δ119863

119891

+ 119862 sdot Δ119863

119888

119863

119888

= 119863

119888

+ Δ119863

119888

+ 119862 sdot Δ119863

119891

119863tot = 119863119888 + 119863119891


119886

MPa 119879 119905 hour)119904 = 11497 119898 = 03316 119904 = 10592 119898 = 034047 119904 = 11436

119861 = 190990790 1198881015840 = 09964 119861 = 99955327 119861 = 99767982

119877

2 098 095 066



Disclosure




Acknowledgment


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5000 10000 15000 200000

005

01

015

02

025

03

035

04

045

d(m

m)

003Hz01Hz05Hz

Time (s)

(a)

0

005

01

015

02

025

03

035

04

045

d(m

m)

003Hz01Hz05Hz

0 500 1000 1500 2000 2500

Cycles

(b)








d119863d119873

=

120590

2119904+1

119886

119861(1 minus 119863)

2119904

(1)




d119863d119873

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863)

2119904

(2)



d119863d119905

= 119860(

120590

1 minus 119863

)

119903

(3)



0

02

04

06

08

1

12

0 5000 10000 15000 20000

d(m

m)

Time (s)

003Hz01Hz05Hz

(a)

0

02

04

06

08

1

12

0 500 1000 1500

d(m

m)

Cycles

003Hz01Hz05Hz

(b)


0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 0005 001 0015 002

003Hz01Hz05Hz

P(N

)

120576



119905=0= 0 damage


119863 = 1 minus (1 minus 119860 (119903 + 1) 120590

119903

119905)

1(119903+1)

(4)


written as

119905

119888=

1

119860 (119903 + 1)

120590

minus119903

(5)

21

22

23

24

25

26

27

0 02 04 06

f (Hz)

E(G

MPa

)





119888






d119863119888(ie d119863 = d119863




d1198631015840119891

= d119863119891+ 119862 sdot d119863

119888 (7)

d1198631015840119888

= d119863119888+ 119862 sdot d119863

119891 (8)

119863

119891= 119863

119891+ d1198631015840119891

(9)

119863

119888= 119863

119888+ d1198631015840119888

(10)

119863tot = 119863119888 + 119863119891 (11)

where 119862 = 119888




119888


119891= d119863119888)



119888is




25

30

35

40

45

0 2000 4000 6000

120590m

ax(M

Pa)

N




5 Conclusion





Equations

Initial119873 = 0119863119891

= 0119863119888



Δ119863

119891

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863

119891

)

Δ119863

119891

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863tot)Δ119863

119891

=

120590

2119904+1

119886

119861 (1 minus 119863tot)

Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863c)

119903

d119905 Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863tot)

119903

d119905 Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863tot)

119903

d119905

119862 = 119888

1015840 exp(radic

Δ119863

119888

+ Δ119863

119891

05(Δ119863

119888

+ Δ119863

119891

)


119863

119891

= 119863

119891

+ Δ119863

119891

+ 119862 sdot Δ119863

119888

119863

119888

= 119863

119888

+ Δ119863

119888

+ 119862 sdot Δ119863

119891

119863tot = 119863119888 + 119863119891


119886

MPa 119879 119905 hour)119904 = 11497 119898 = 03316 119904 = 10592 119898 = 034047 119904 = 11436

119861 = 190990790 1198881015840 = 09964 119861 = 99955327 119861 = 99767982

119877

2 098 095 066



Disclosure




Acknowledgment


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

02

04

06

08

1

12

0 5000 10000 15000 20000

d(m

m)

Time (s)

003Hz01Hz05Hz

(a)

0

02

04

06

08

1

12

0 500 1000 1500

d(m

m)

Cycles

003Hz01Hz05Hz

(b)


0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 0005 001 0015 002

003Hz01Hz05Hz

P(N

)

120576



119905=0= 0 damage


119863 = 1 minus (1 minus 119860 (119903 + 1) 120590

119903

119905)

1(119903+1)

(4)


written as

119905

119888=

1

119860 (119903 + 1)

120590

minus119903

(5)

21

22

23

24

25

26

27

0 02 04 06

f (Hz)

E(G

MPa

)





119888






d119863119888(ie d119863 = d119863




d1198631015840119891

= d119863119891+ 119862 sdot d119863

119888 (7)

d1198631015840119888

= d119863119888+ 119862 sdot d119863

119891 (8)

119863

119891= 119863

119891+ d1198631015840119891

(9)

119863

119888= 119863

119888+ d1198631015840119888

(10)

119863tot = 119863119888 + 119863119891 (11)

where 119862 = 119888




119888


119891= d119863119888)



119888is




25

30

35

40

45

0 2000 4000 6000

120590m

ax(M

Pa)

N




5 Conclusion





Equations

Initial119873 = 0119863119891

= 0119863119888



Δ119863

119891

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863

119891

)

Δ119863

119891

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863tot)Δ119863

119891

=

120590

2119904+1

119886

119861 (1 minus 119863tot)

Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863c)

119903

d119905 Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863tot)

119903

d119905 Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863tot)

119903

d119905

119862 = 119888

1015840 exp(radic

Δ119863

119888

+ Δ119863

119891

05(Δ119863

119888

+ Δ119863

119891

)


119863

119891

= 119863

119891

+ Δ119863

119891

+ 119862 sdot Δ119863

119888

119863

119888

= 119863

119888

+ Δ119863

119888

+ 119862 sdot Δ119863

119891

119863tot = 119863119888 + 119863119891


119886

MPa 119879 119905 hour)119904 = 11497 119898 = 03316 119904 = 10592 119898 = 034047 119904 = 11436

119861 = 190990790 1198881015840 = 09964 119861 = 99955327 119861 = 99767982

119877

2 098 095 066



Disclosure




Acknowledgment


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












d119863119888(ie d119863 = d119863




d1198631015840119891

= d119863119891+ 119862 sdot d119863

119888 (7)

d1198631015840119888

= d119863119888+ 119862 sdot d119863

119891 (8)

119863

119891= 119863

119891+ d1198631015840119891

(9)

119863

119888= 119863

119888+ d1198631015840119888

(10)

119863tot = 119863119888 + 119863119891 (11)

where 119862 = 119888




119888


119891= d119863119888)



119888is




25

30

35

40

45

0 2000 4000 6000

120590m

ax(M

Pa)

N




5 Conclusion





Equations

Initial119873 = 0119863119891

= 0119863119888



Δ119863

119891

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863

119891

)

Δ119863

119891

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863tot)Δ119863

119891

=

120590

2119904+1

119886

119861 (1 minus 119863tot)

Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863c)

119903

d119905 Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863tot)

119903

d119905 Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863tot)

119903

d119905

119862 = 119888

1015840 exp(radic

Δ119863

119888

+ Δ119863

119891

05(Δ119863

119888

+ Δ119863

119891

)


119863

119891

= 119863

119891

+ Δ119863

119891

+ 119862 sdot Δ119863

119888

119863

119888

= 119863

119888

+ Δ119863

119888

+ 119862 sdot Δ119863

119891

119863tot = 119863119888 + 119863119891


119886

MPa 119879 119905 hour)119904 = 11497 119898 = 03316 119904 = 10592 119898 = 034047 119904 = 11436

119861 = 190990790 1198881015840 = 09964 119861 = 99955327 119861 = 99767982

119877

2 098 095 066



Disclosure




Acknowledgment


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Equations

Initial119873 = 0119863119891

= 0119863119888



Δ119863

119891

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863

119891

)

Δ119863

119891

=

120590

2119904+1

119886

119861119891

119898

(1 minus 119863tot)Δ119863

119891

=

120590

2119904+1

119886

119861 (1 minus 119863tot)

Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863c)

119903

d119905 Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863tot)

119903

d119905 Δ119863

119888

= int

119873119879

(119873minus1)119879

119860(

120590

1 minus 119863tot)

119903

d119905

119862 = 119888

1015840 exp(radic

Δ119863

119888

+ Δ119863

119891

05(Δ119863

119888

+ Δ119863

119891

)


119863

119891

= 119863

119891

+ Δ119863

119891

+ 119862 sdot Δ119863

119888

119863

119888

= 119863

119888

+ Δ119863

119888

+ 119862 sdot Δ119863

119891

119863tot = 119863119888 + 119863119891


119886

MPa 119879 119905 hour)119904 = 11497 119898 = 03316 119904 = 10592 119898 = 034047 119904 = 11436

119861 = 190990790 1198881015840 = 09964 119861 = 99955327 119861 = 99767982

119877

2 098 095 066



Disclosure




Acknowledgment


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Analysis of Fatigue Life of PMMA at