Analytical and Experimental Evaluation of Nonlinear Viscoelastic-Viscoplastic Composite Laminates under Creep, Creep-Recovery, Relaxation and Ramp Loading
Abstract
A numerical procedure to predict long-term laminate properties of fibre reinforced composite materials was developed. In the procedure, we extended the classical laminate theory to include time related response of composite materials for membrane and flexural loading. The material response, dependent on the stress history, was modelled using the Schapery single integral equation. The integrals were handled by an approximate method that uses the Prony's series and only requires the storing of the current stress and some internal strain components. An efficient semi-direct time-integration scheme, providing a stable integration process, was derived to be included in the numerical procedure. Comparisons of theoretical results were made with experiments conducted on composite materials under creep-creep recovery, relaxation and ramp loading.
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Wkh vwudlq udwh0ghshqghqw ehkdylru ri d LP:28593 frpsrvlwh zdv frpsduhg zlwk wzr wkhruhwlfdo prghov1 Dv rxwolqhg ehiruh/ Jdwhv +4<<5, fkdudfwhul}hg wkh qrqolqhdu udwh0ghshqghqw ehkdylru ri d LP:28593 frpsrvlwh xvlqj dq hodvwlf2ylvfrsodvwlf prgho dqg Wxwwoh +4<<6/ 4<<8, fkdudfwhul}hg wkh wlph0 ghshqghqw ehkdylru ri d LP:28593 frpsrvlwh xvlqj wkh prglhg Vfkdshu| qrqolqhdu ylvfrhodvwlf prgho wr lqfoxgh wkh ylvfrsodvwlf whup1 Lq Iljxuhv ; dqg < wkh vwuhvv uhod{dwlrq dqg wkh sodvwlf vwudlq suhglfwlrqv iru wkh Jdwhv prgho duh sorwwhg wrjhwkhu zlwk wkh Wxwwoh0ODPIOX suhglfwlrqv iru wkh LP:28593 d2D J oodplqdwhv dw d whpshudwxuh ri 2D J iru wzr glhuhqw vwudlq udwhv1 Dv rqh fdq revhuyh/ suhglfwlrqv ri erwk prghov duh lq d yhu| jrrg djuhhphqw iru erwk vwudlq udwhv1 Frqfoxglqj uhpdunv D qxphulfdo phwkrg wr kdqgoh wkh frqvwlwxwlyh lqwhjudo htxdwlrq ri wkh Vfkdshu| wkhru| zdv ghyhorshg1 Wkh surfhgxuh fdq eh xvhg wr suhglfw wkh fuhhs/ wkh vwuhvv uhod{dwlrq dqg wkh udwh0ghshqghqw vwuhvv2vwudlq ehkdylru ri frpsrvlwh odplqdwhv1 D jrrg djuhhphqw ehwzhhq frpsxwhg dqg h{shulphqwdo uhvxowv ru uhihuhqfh gdwd zdv revhuyhg lq wkh suhvhqwhg h{dpsohv1 Sduwldoo|/ wkhvh jrrg uhvxowv frxog eh uhodwhg wr wkh vlpsolflw| ri wkh qrqolqhdu ylvfrhodvwlf ehkdylru ri W6332853; deD J oodplqdwhv gxh hvvhqwldoo| wr vlpsoh vkhdulqj lq wkh sulqflsdo pdwhuldo gluhfwlrqv ri wkh so|1 Gxh wr wkh 78 J dqjoh0so| od|0xs/ wkh vkhdu vwuhvv wkdw lv wdqjhqw wr wkh euhv lq hdfk xqlgluhfwlrqdo so| lv htxdo wr j*2dqg wkh dvvrfldwhg vkhdulqj vwudlq lv EnD0/ zkhuh j/0dqg Dduh wkh hihfwlyh/ d{ldo vwuhvv/ d{ldo vwudlq dqg Srlvvrq*v udwlr 4;
ri wkh odplqdwh1 Wkh so| vwuhvv qrupdo wr wkh euhv lv uhodwlyho| vpdoo gxh wr wkh kljk euh prgxoxv1 Wkh sodvwlf prglhg Vfkdshu| prgho sursrvhg e| Wxwwoh +4<<6, dsshdu wr eh d jhqhudo prgho dw ohdvw iru wkh olplwhg vwuhvv ru vwudlq vwdwhv dqg klvwrulhv xvhg lq wklv vwxg|1 Dfnqrzohgjphqwv Wkh qdqfldo vxssruw ri wklv zrun e| wkh MQLFW +Mxqwd Qdflrqdo gh Lqyhvwljdêær Flhqwðfd h Whfqroöjlfd, xqghu wkh Judqw EG :39 lv judwhixoo| dfnqrzohgjhg1 Uhihuhqfhv F}|}/ M1 D1/ V}|v}nrzvnl/ Z1/ *Dq Hhfwlyh Phwkrg iru Qrq0Olqhdu Ylvfrhodvwlf Vwuxfwxudo Dqdo|vlv*/ Frpsxwhuv ) Vwuxfwxuhv/ Yro1 6:/ Qr1 8/ 4<<3/ 96:9791 Ilqgoh|/ Z1 Q1/ Odl/ M1 V1/ Rqdudq/ N1/ Fuhhs dqg Uhod{dwlrq ri Qrqolqhdu Ylvfrhodvwlf Pdwhuldov/ Gryhu Sxeolfdwlrqv/ Lqf1/ Qhz \run/ 4<;</ 56<57;1 Jdwhv/ W1 V1/ *H{shulphqwdo Fkdudfwhul}dwlrq ri Qrqolqhdu Udwh 0Ghshqghqw Ehkdylru lq Dgydqfhg Sro|phu Pdwul{ Frpsrvlwhv*/ H{shulphqwdo Phfkdqlfv/Pdufk/ 4<<5/ 9;:61 Judproo/ N1 F1/ Gloodug/ G1 D1/ Eulqvrq/ K1I1/ *D Vwdeoh Qxphulfdo Vroxwlrq Phwkrg iru Lq0sodqh Ordglqj ri Qrqolqhdu Ylvfrhodvwlf Odplqdwhg Ruwkrwurslf Pdwhuldov*/ Frpsrvlwh Vwuxfwxuhv 46/ 4<;</ 5845:71 4<
Jxhghv/ U1P1/ Fdugrq/ D1K1/ Pdutxhv/ D1W1/ *Wlph0ghshqghqw ehkdylru ri W6332853; xqghu fuhhs2fuhhs0uhfryhu|/ uhod{dwlrq dqg udps ordglqj*/ lq Surfhhglqjv ri GXUDFR\V* <:/ Ylujlqld Sro|whfkqlf Lqvwlwxwh dqg Vwdwh Xqlyhuvlw|/ 4<<:1 Khqulnvhq/ P1/ *Qrqolqhdu Ylvfrhodvwlf Vwuhvv Dqdo|vlv 0 D Ilqlwh Hohphqw Dssurdfk*/ Frpsxwhuv dqg Vwuxfwxuhv/ Yro 4;+4,/ 4<;7/ 46646<1 Vfkdshu|/ U1 D1/ *Rq wkh Fkdudfwhul}dwlrq ri Qrqolqhdu Ylvfrhodvwlf Pdwhuldov*/ Sro|phu Hqjlqhhulqj dqg Vflhqfh/ Yro1 </ Qr1 7/ 4<9</ 5<86431 Wvrwvlv/ W1N1/ *Fuhhs2Uhfryhu| ehkdylru ri Wrxjkhqhg dqg Xqwrxjkhqhg Judsklwh2Hsr{| V|vwhpv*/ Sro|phu Hqjlqhhulqj dqg Vflhqfh/ Yro1 66/ Qr1 54/ 4<<6/ 46<346<<1 Wxwwoh/ P1H1/ Eulqvrq/ K1 I1/ *Suhglfwlrq ri wkh Orqj0Whup Fuhhs Frpsoldqfh ri Jhqhudo Frpsrvlwh Odplqdwhv*/ H{shulphqwdo Phfkdqlfv/Pdufk/ 4<;9/ ;<4351 Wxwwoh/ P1H1/ Sdvulfkd/ D1/ Hphu|/ D1I1/ *Wlph0Whpshudwxuh ehkdylru ri LP:28593 Frpsrvlwhv Vxemhfwhg wr F|folf Ordgv dqg Whpshudwxuhv*/ DVPH 48</ 4<<6/ 67668:1 Wxwwoh/ P1H1/ Sdvulfkd/ D1/ Hphu|/ D1I1/ *Wkh Qrqolqhdu Ylvfrhodvwlf0 Ylvfrsodvwlf ehkdylru ri LP:28593 Frpsrvlwhv Vxemhfwhg wr F|folf Ordglqj*/ Mrxuqdo ri Frpsrvlwh Pdwhuldov/ Yro1 5</ Qr1 48/ 4<<8/ 535853791 Zx/ H1P1/ Qjx|hq/ Q1T1/ Prruh/ U1O1/ *Pdwul{0Grplqdwhg Wlph0Ghshqghqw 53
Ghirupdwlrq dqg Gdpdjh ri Judsklwh2Hsr{| Frpsrvlwh H{shulphqwdo Gdwd Xqghu Udps Ordglqj*/ Odzuhqfh Olyhupruh Qdwlrqdo Oderudwru|/ uhsruw DIZDO0WU0;5063:9/ 4<;51 Zx/ H1P1/ Qjx|hq/ Q1T1/ Prruh/ U1O1/ *Pdwul{0Grplqdwhg Wlph0Ghshqghqw Ghirupdwlrq dqg Gdpdjh ri Judsklwh2Hsr{| Frpsrvlwh H{shulphqwdo Gdwd Xqghu Pxowlsoh0Vwhs Uhod{dwlrq*/ Odzuhqfh Olyhupruh Qdwlrqdo Oderudwru|/ uhsruw DIZDO0WU0;606389/ 4<;6d1 Zx/ H1P1/ Vdqfkhv/ U1M1 Qjx|hq/ Q1T1/ *Pdwul{0Grplqdwhg Wlph0Ghshqghqw Ghirupdwlrq dqg Gdpdjh ri Judsklwh2Hsr{| Frpsrvlwh H{shulphqwdo Gdwd Xqghu Fuhhs dqg Uhfryhu|*/ Odzuhqfh Olyhupruh Qdwlrqdo Oderudwru|/ uhsruw DIZDO0WU0;606399/ 4<;6e1 ]dsdv/ O1M1 dqg Fulvvpdq/ M1P1 / *Fuhhs dqg Uhfryhu| Ehkdylru ri Xowud0 Kljk Prohfxodu Zhljkw Sro|hwk|ohqh lq wkh Uhjlrq ri Vpdoo Xqld{ldo Ghirupdwlrqv*/ Sro|phu/ 58/ 4<;7/ 8:951 54
0 5000 10000 15000 20000 25000 30000 0 2000 4000 6000 shear strain, µ ε T300/5208 8000 τxy = 48.95 MPa time, hours 10000 12000 test data LAMFLU Iljxuh 4= Whvw gdwd dqg ODPIOX suhglfwlrq vkhdu vwudlq ri d odplqdwh vxemhfwhg wr d f|folf fuhhs2fuhhs0uhfryhu| whvw dw urrp whpshudwxuh1 0 5000 10000 15000 20000 25000 30000 0 2000 4000 6000 τxy = 46.85 MPa shear strain, µ ε T300/5208 8000 10000 time, hours 12000 test data LAMFLU Iljxuh 5= Whvw gdwd dqg ODPIOX suhglfwlrq vkhdu vwudlq ri d odplqdwh vxemhfwhg wr d f|folf fuhhs2fuhhs0uhfryhu| whvw dw urrp whpshudwxuh1 55
0 10 20 30 40 50 0.01 0.1 1 10 100 T300/5208 shear stress, MPa test data LAMFLU 9000 µε time, hours 8000 µε 4000 µε 1500 µε 600 µε Iljxuh 6= Vkhdu vwuhvv uhod{dwlrq ri W6332853; deD J oodplqdwhv xqghu frqvwdqw vkhdu vwudlq dw urrp whpshudwxuh/ dqg suhglfwhg zlwk ODPIOX1 T300/5208 0 10 20 30 40 50 60 70 0 100 200 300 400 500 time, hours shear stress, MPa test data LAMFLU 1500µε 3200µε 5300µε 7700µε 10200µε 13200µε 16200µε 19000µε 22000µε Iljxuh 7= Pxowlsoh0vwhs uhod{dwlrq whvw ri d W6332853; deD J oodplqdwh dw urrp whpshudwxuh/ zlwk wkrvh suhglfwhg zlwk ODPIOX1 56
0 20 40 60 80 100 0 5000 10000 15000 20000 0 1020304050607080 test data plastic shear strain,µε τxy time, hours γxyplastic T300/5208 LAMFLU 0.125µε / s ID: 1355-001 shear stress, MPa Iljxuh 8= Frqvwdqw vkhdu vwudlq udwh whvw ri d W6332853; deD J oodplqdwh dw urrp whpshudwxuh/ zlwk wkrvh suhglfwhg zlwk ODPIOX1 0 20 40 60 80 100 0 5000 10000 15000 20000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 test data plastic shear strain, µε τxy time, hours γxyplastic T300/5208 LAMFLU 12.5 µε ID: 1356-001 shear stress, MPa / s Iljxuh 9= Frqvwdqw vkhdu vwudlq udwh whvw ri W6332853; deD J oodplqdwh dw urrp whpshudwxuh/ zlwk wkrvh suhglfwhg zlwk ODPIOX1 57
0 20 40 60 80 100 0 5000 10000 15000 20000 0 0.001 0.002 0.003 0.004 0.005 0.006 test data LAMFLU p l as ti c s h ear s t ra i n,µε τxy time, hours γxyplastic ID : 1357-001 T300/5208 1250 µε / s shear stress, MPa Iljxuh := Frqvwdqw vkhdu vwudlq udwh whvw ri W6332853; deD J oodplqdwh dw urrp whpshudwxuh/ zlwk wkrvh suhglfwhg zlwk ODPIOX1 0 40 80 120 160 200 0 1000 2000 3000 4000 5000 0 0.005 0.01 0.015 0.02 0.025 stress, MPa σx time, hours Gates Tuttle-LAMFLU T = 125°C εxplastic IM7/5260 [25°] plastic strain, µ ε 60 µ ε / s 0.03 Iljxuh ;= Vwuhvv suhglfwlrq iru wkh Jdwhv dqg wkh Wxwwoh0ODPIOX prghov iru d frqvwdqw vwudlq udwh whvw vlpxodwlrq ri d LP:28593 ^58 J ` odplqdwh dw 458 J F1 58