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Form Approved OMB No. 0704-0188
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1. REPORT DATE
3. DATES COVERED 2. REPORT TYPE
2003
00-00-2003 to 00-00-2003
4. TITLE AND SUBTITLE
5a. CONTRACT NUMBER
Multiscale Considerations in Modeling of Nonlinear Elastomers
5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER
6. AUTHOR(S)
5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
North Carolina State University,Center for Research in Scientific Computation,Raleigh,NC,27695-8205 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)
8. PERFORMING ORGANIZATION REPORT NUMBER
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12. DISTRIBUTION/AVAILABILITY STATEMENT
Approved for public release; distribution unlimited 13. SUPPLEMENTARY NOTES 14. ABSTRACT
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17
Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18
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3
Linear fit to load cell cdata for unfilled rubber sample, small weight
10
3.5
3
Data 2.5 Square Window FFT
Force (pounds)
2
2
1.5
10
Aprx
1
10
1 Data 0.5
0 0
Aprx 0
0.5
1
1.5
2 2.5 Time (seconds)
3
3.5
4
10
0
5
10
15 Frequency (hz)
20
25
30
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data computed
data computed 250 250
200
stress
stress
200
150
150 100
100 50
50 0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2
0.3
0.4
0.5
0.6 strain
0.7
0.8
0.9
1
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ï
:
ihrH u
+! h! 3 lb tip mass, nonlinear g
, gv
e
power spectrum for 3 lb tip mass
3
10
8
data computed
data computed 7
6 2
10
~
log(100|f |)
Force (pounds)
5
4
3
1
10 2
1
0
0
0
0.5
1
1.5
2 Time (sec)
2.5
3
3.5
4
10
0
5
10
15 Frequency (hz)
20
25
30
Í ca yZj>V:Q¥Ë Í a/TXT=Yg]^TQqÑj>\oalS>y]!S>YZS>qcalS>Q]mV ×NØ ×)' Ë ]% RºachQ&>Ymh]macSC]mi>i>VoY:each]^T:alYZS ]mS ÿI xT=`3QiFYRQV\:iFQ_¦T=Voj>h YmWpT=`3Q]mi3i>V:Yuealh]^T=acYZS-]mSâ -T=`>Qg]^T=]WNYZV]qlalym`ET:qcv fgqlqcQ-\=]mhi>qlQ XacT:`C!qcÿT=alih]m\:\ Ò o! Ò % Ò Ñ ÓWâ Þ â âHÕ Þéèê â ÑÓ âHÕ Þ â ó 8 âHÕ Þ â X`>QV:QmzUacSymQS>QV=]mq z^T=`3QÁ> QV:acu^]^T:acumQ\Y^W ×(' acS ² ² {]mV:Q>al\²T=Voalÿ>j3T:alYZS>]mqUalStT=`>Qú\:QS3\:Qs>Q\:_V:alÿFQQ ]mVoqlalQV º`>Q ig]mV:]mhQT=QVa+> QSUT=acf>_ ]^T:alYZS i3V:YZÿ>qcQh p]m\C\:YmqcuZQ'j>\:acS>y * e* YZi3T=achacû ]^T:alYZS V:YZjeT=alS3Q\ *vei3al_ ]^qV:Q\:j>q/T=\|WNYmVxT=`3Q&a+> QSUT=acf>_ ]^T:alYZS i>V:YZÿ3qlQh ]mVoQ&\:`3YXS acS Í alymj>V:Qt>zÎX`3QV:Q!]qlÿbT:alih]m\o\ R]m\!j>\:Q' ]^TrT:`>QÿFYmT:T:YZh YmWXT:`>Q\=]mhi>qlQ º`>Qa>QSUT=a/fg_ ]kT=alYmSi>V:Ymÿ>qlQh R]m\]mql\oYV:j>S YZS(g]^T=] YZÿ3T=]malS3Q WNV:YZh QWe iÎQV:alhQSUT:\tXa/T=` }qcÿ ]mS¥ qlÿT=aciÊh]m\:\t]M¥>Q-T=YT=`>QqlacyZ`UT=qcvCfgqcqlQ-\=]^hi>qcQ ÊQWNYmj>S ]\:alhalql]mVtyZYeY% ]myZVoQQhQSETÿFQTwúQQS T=`3Q§> ]^TQhY%> Qq$¦ }L¨ x YRQ¦uZQVzpT=`3Q a+3QSUT=a/fgQ _YLQ|R_acQSUT=\ RQV:QS3YmT|QSUT=alVoQq/v_YZS3\:al\²T=QSET%]m_VoYZ\:\xQ6e iFQV:achQSUT=\zÎ]mq/T=`>YZj3yZ`T=`>Q¦v \o`>YZj>q > Q\:_VoalÿFQT=`>Qt\=]mhQh]^T=QV:a£]^q º`3al\ºu^]mV:al]^T=acYZSal\ºi>VoYZÿg]mÿ3qcv¶_]mj>\oQb ÿUvT:`>QÆ\:j>ÿ>\²TyZQ\alSUumYZqcumQ! alST:`>QpQW3 iFQVoalhQSUT=\zE]mSr \oj>yZyZQ\oT=\T:`g]^TT:`>QºhY²> Qqe\o`>YZj>q ÿFQtVoQfgS3Q T:Y]m__YZj>SUTXWNYZVpT=`>Q\:Q#> acþÎQVoQS>_Q\ QXVoQiFQ ]^T:Q T:`>QXQ6e iFQV:achQSUT Xa/T=`]Æ`>alyZ`3qcvtfgqlqcQ! \=]mhi>qcQº`g] ueacS>yt]Æ }^tqcÿ!T:alirh]m\:\ wS!T=`>ac\ _ ]^\:Q|Ymj>V*ÿÎQ\oTRf3TÁ> Qi>al_¦T=Q' alS Í acyZj>V:Qx ]%( `g]m\R> QT:QVoalYZV:]^T=Q' º`Ej>\úúQxT:j>V:S3Q T:Y]thY%> acf>_ ]^T:alYZS YmW(YZj>VXhY²> QqÎT:YYZÿeTi>V:Yue alh]^T:alYZS¶T:Y!T:`>QWNYZV:_Qkg ]^Tal\thY²> a/fg_ ]kT=alYmSp]m\hY^T=acu^]^T:Q ÿUv-YZj>VtQ ]mVoqlalQVQþÎYZVoT:\]^\túQqcqú]^\tT:`>YZ\:QV:QiÎYmVoT=Q' alS T=`3Q qla/T=QV=]^T:j>V:QYZS \:alhalql]mV&i>VoYZÿ>qcQh\acSEumYZqcuLacS>yÊ`UvL\oT:QVoQT=ac_QþÎQ_¦T=\]^S acSET:QVoSg]mq \oT=V:]malS hY²> Qql\ WNYZV > Q¦TQq> ac\:_j3\:\:acYZS>\5alS¦ }e¨i Q S>Q6L T(_YZS3\:a+3 QV:Q't T:`>Q hY%> Qqmj>\:acS>y TwúY alSUT=QV:Sg]mqZu^]mVoa£]mÿ3qlQ\ 8 Ó 0 8 â 9 Ó 0 9 XacT:`¬> acþÎQVoQSUT > Q_]vi>]mV=]mhQT:QVo\ 0 8 â 0 9 â yZa/uZQS ÿUv ó ó Ö 9lÒ ;lÒ Ò acSnO Q o% Îw Ñ 9 Ô Ï)ÐAÚsÛ Ñ 9 Ô Ñ Ï Ð/×NØ Ñ µÙ Ï Ð 8 Ù Ï)Ð 9 Ó ÑÓ ÑÓAÑ©Õ Ñ©Õ Ñ©Õ ó ó Ý / Þ N Ù % Ò 9Ò o ~! )× ' Ñ Ñ òó 8 Þ Ô 0 8 ó 8 % Ó Ñ©Õ â ÑÓAÑVÕ
}ê
:
ihrH u
+! h! highly filled rubber sample
highly filled sample
30
30
25
25
20
20 Force (pounds)
Force (pounds)
computed data
15
15
10
10
5
5
0
0
0.1
0.2
0.3
0.4
0.5 Time (sec)
0.6
0.7
0.8
0.9
1
0
0
0.2
0.4
0.6
0.8 Time (sec)
1
1.2
1.4
1.6
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:
ihrH u
+! h!
}}
h YZS>YmT:YZS>al_acTwv YmWºT:`>Q¶S>YmS>qlacS>Q ]mVÆWNj>S>_¦T=acYZS>\ ×4Ø ×(' ×('ED wS3\oT=Q]M{zT:`>Qv]mV:Q]m\:\oj>hQ'T=Y\:]^T=ac\oWÍvÊ] qlYL_ ]^q ali>\o_acT:ûi>V:YZiFQV²Twv ]mS YmT:`>QV!]m\o\:j>hi3â T:alYZS>â \!\oalhø alq£]^VtT:Y T:`>YZ\oQCj3\:Q ÿÎQ¦WNYZV:Q QVoQh]mV: T=`>]^TR\oalhalql]mV(T:Q_S>aEj>Q\RúQVoQ|\oj>__Q\:\²WNj>qlq/vQhi>qlYvZQ'T:YQ\²T_Q¦Øj>S>a+Ej>QS>Q\o\ºYmW{úQ ]m \:Ymqlj3T:alYZS>\ WNYZVRqcalS>Q]mV QuZYmqlj3T:alYZSQ Ejg]^T:alYZS>\RYmW7\:Q_YZS YZ V >QVRalS alS ¦ }kM ¨Ñ]m¥S WNYZVR\oQhalqcalS>Q]mVú\oQ_YZS YZ V >QVQuZYmqlj3T:alYZS Q Ejg]^T:alYZS>\X`>QVoQbT=`3QbS>YZS3qlalS3Q ]mV!T=QV:h \=]^T:alÓ\²fgQ\]-yZqlYmÿg]mq ali>\o_acT:û _YZS >a/T=alYmS alS ¦ }m8 ¨ wÜ S ¦ }m8 ¨úúQj>\oQ ý\oalhalq£]^V|T=Q_S>+a Ej>Q\&T=Y \oT=¥j 3vÊ]bS>YZS>qcalS>Q]mVÆÿÎQ]mh Q Ejg]^T:alYZS-X`>QVoQT=`3Q S>YZS3qlalS3Q ]mVoacTwv\=]^T:al\of>Q\XYZS>q/v]!qlYL_ ]mq aci>\:_a/T=acYZS Å
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âHÕ Þéèê â ÑÓ Hâ Õ Þ â ó
âHÕ Þ ø
Load cell
extra mass
hammer hit
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8 Þ ÑÓ X`>QV:Q l V:Qi>V:Q\:QSETrT:`>Q¶_YZqlqcQ_T:Q >]^TES>YXS _YeuQ R_alQSUT=\ º`>QWQ E iFQV:achQSUTR]m\riFQVoWNYmV:hQ ÊXa/T=`qcalyZ`UT:qcvfgqcql'Q ]mS `>alyZ`3qcvfgqcqlQ V:j>ÿ>ÿFQV\=]mhi>qcQ\Xa/T=` 3acþÎQV:QSUT ]mhYZj>SUT=\YmWX6Q LT=V:]h]m\o\ wS yZQS>QV:]mq zúQ¶WNYZj>¥S T=`g]^TT=`>Q >]^TYWQ LT=V=]h]m\o\ YZS¶T:`>Q\=]mhi>qcQ%_YZj3+q ¶ÿFQÆ]mi>i>VouY 3ach]kT=Q ¶úQqcqÎXa/T=`¶T=`3QÆ]mÿÎYumQh%Y >QqÎQumQSX`>QST=`3Q%ueac\:_YeQq£]m\²T=al_ V:Q\:iFYZS>\oQ ×)' ac\º]m\:\oj>hQ T=Y!ÿÎQÆqcalS>Q]mV º`3al\ºal\ºS3YmTX\:j>Voi>V:ac\:acS>y>z>\oalS>_QÆT:`>Qh4] ealh&j>h \oT:V=]macSqlQumQqc\ alST:`>Q\oQTwvLiÎQtYmW(6Q eiFQV:achQSUT=\XRQV:QtÿÎQqlY ~ %j>VXÿÎQ\oT f>TXWNYZVºT=`>QÆ`>acyZ`>q/vfgqcqlQ \:]mhi3qlQ%XacT:` S>Y6Q LT:V=]h]m\o\al&\ >Qi>al_¦T='Q ÊacS Í alyZj>VoQ!~ YúQumQVz(X`>QSÊWQ eT:V=]h]^\:\p]^\tacSUuZYZq/uZQ T=YC]m_Q¦WNYZV:h]^T=acYZS>\]mS ý\²T=V=]^alSqcQuZQql\zT=`>QWQ 3iFQVoalhQSU±T >a S>YmT&i>V:YuL+a >Q\:j>a/Tqc`Q g]kT_Q M] 3acT=acYZSg]mq ²T=acqcT:alS>y ® h%Y >Q\ÿFQ_]mhQCWQ e_acT:Q T=`g]kT_YZj>q S3YmTÿÎQC]m__YZj>SUT:Q WNYmVrÿEvýT:`>QC]mÿFYumQ YZS>Q/ >achQS>\:acYZSg]mqÎh%Y >Qq A225 (b) 1 2
computed data 1
force
0
−1
−2
−3
−4
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
time
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Acceleration data and the approximated values
3
2 computed data
computed data
1
acceleration (g)
acceleration (g)
2
0 −1 −2
1
0
−1
−3 −4
0
0.01
0.02
0.03
0.04
0.05 time (sec)
0.06
0.07
0.08
0.09
−2
0.1
0
0.01
0.02
Comparison of load cell data and the model predicted values
0.05 time (sec)
0.06
0.07
0.08
0.09
0.1
15 computed data
computed data
10 load cell (lb)
10 load cell (lb)
0.04
Comparison of load cell data and the model predicted values
20
0 −10 −20 −30
0.03
5 0 −5 −10
0
0.01
0.02
0.03
0.04
0.05 time (sec)
0.06
0.07
0.08
0.09
−15
0.1
0
0.01
0.02
0.03
0.04
0.05 time (sec)
0.06
0.07
0.08
0.09
0.1
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