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      黏彈性材料動(dòng)態(tài)力學(xué)性能的分?jǐn)?shù)階時(shí)溫等效模型與驗(yàn)證

      2017-10-13 06:27:07李占龍孫大剛
      關(guān)鍵詞:彈性動(dòng)態(tài)分?jǐn)?shù)

      李占龍,宋 勇,孫大剛,章 新,孫 寶

      ?

      黏彈性材料動(dòng)態(tài)力學(xué)性能的分?jǐn)?shù)階時(shí)溫等效模型與驗(yàn)證

      李占龍,宋 勇※,孫大剛,章 新,孫 寶

      (太原科技大學(xué)機(jī)械工程學(xué)院,太原 030024)

      為構(gòu)造一種更加精準(zhǔn)的黏彈性時(shí)溫等效模型,基于黏彈性阻尼材料的分?jǐn)?shù)階本構(gòu)關(guān)系和Vogel-Fulcher-Tammann黏度方程,結(jié)合WLF(Williams-Landel-Ferry)方程,提出了黏彈性阻尼材料動(dòng)態(tài)力學(xué)性能的分?jǐn)?shù)階時(shí)溫等效模型(fractional time-temperature superposition model,F(xiàn)TTSM),導(dǎo)出頻率轉(zhuǎn)化因子,并給出參數(shù)識(shí)別方法。在DMA(dynamic thermomechanical analysis)試驗(yàn)數(shù)據(jù)的基礎(chǔ)上,對比研究了FTTSM和WLF兩種模型,并應(yīng)用分?jǐn)?shù)階Kelvin-Voigt模型對二者在參考溫度5 ℃的主曲線進(jìn)行了驗(yàn)證。結(jié)果表明,F(xiàn)TTSM和WLF所表征的頻率轉(zhuǎn)化因子在溫度范圍(–80~80 ℃)內(nèi)最大相對誤差為0.984 4%,F(xiàn)TTSM主曲線和WLF主曲線相對于Kelvin-Voigt模型理論預(yù)測值的均方根誤差(RMSE)分別為1.291和1.834 MPa,驗(yàn)證了FTTSM的精確性。另外FTTSM主曲線擴(kuò)展的最低頻域低于WLF主曲線2個(gè)數(shù)量級(jí),證明FTTSM對黏彈性動(dòng)態(tài)力學(xué)特性具有更廣泛的預(yù)測能力。為黏彈性材料動(dòng)態(tài)性能預(yù)測、物理老化、蠕變損傷演化機(jī)理等研究提供理論參考。

      車輛;減振;模型;黏彈性材料;動(dòng)態(tài)力學(xué)性能;分?jǐn)?shù)階;時(shí)溫等效模型;WLF方程

      0 引 言

      黏彈性減振緩沖結(jié)構(gòu)廣泛用于農(nóng)業(yè)工程車輛的振動(dòng)噪聲控制領(lǐng)域,如履帶拖拉機(jī)的黏彈性懸架、座椅和發(fā)動(dòng)機(jī)的橡膠懸置等[1-2]。黏彈性阻尼材料動(dòng)態(tài)力學(xué)性能對環(huán)境溫度和激勵(lì)頻率具有較強(qiáng)的依賴性,這使得在黏彈性減振緩沖結(jié)構(gòu)設(shè)計(jì)過程中必須考慮二者的影響。因此,黏彈性阻尼材料動(dòng)態(tài)力學(xué)性能的溫頻影響規(guī)律是該領(lǐng)域的研究重點(diǎn)之一。

      目前對黏彈性阻尼材料動(dòng)態(tài)力學(xué)頻率影響的研究主要是基于本構(gòu)方程出發(fā),通過拉式變換將其從實(shí)數(shù)域推演到復(fù)數(shù)域,分離其實(shí)虛部,從而獲得包含頻率變量的儲(chǔ)能模量、耗能模量和損耗因子等動(dòng)態(tài)力學(xué)性能[3-5]。Nutting[6]最早發(fā)展了一種分?jǐn)?shù)指數(shù)模型來描述橡膠的應(yīng)力松弛現(xiàn)象,隨后Gemant[7]和Bosworth等[8]首次提出了黏彈性介質(zhì)的分?jǐn)?shù)階導(dǎo)數(shù)模型,之后國內(nèi)外學(xué)者對黏彈性分?jǐn)?shù)階模型開展了大量研究,并取得了諸多有益的結(jié) 論[9-10]。唐振寰等[11]提出了五參數(shù)分?jǐn)?shù)導(dǎo)數(shù)橡膠隔振器本構(gòu)模型,推演到頻域并進(jìn)行了參數(shù)識(shí)別。李占龍等[12]建立了考慮形狀參數(shù)的分?jǐn)?shù)階黏彈性振子模型,并將其運(yùn)用到履帶車輛黏彈性懸架的動(dòng)態(tài)分析。Wharmby等[13]基于分?jǐn)?shù)階導(dǎo)數(shù)建立了黏彈性材料的修正Maxwell本構(gòu)方程,通過拉式變換獲得其頻響函數(shù)。Cao等[14]提出了分?jǐn)?shù)階加權(quán)分布參數(shù)Maxwell模型,并通過拉式逆變換獲得其時(shí)域響應(yīng)。

      另外,時(shí)溫等效原理(time-temperature superposition, TTS)是黏彈性阻尼材料溫頻效應(yīng)研究的重要手段之一,通過該原理可將黏彈性阻尼材料動(dòng)態(tài)性能的溫度譜或頻率譜通過一定的規(guī)則獲得參考溫度和頻率下廣域譜線。目前大多關(guān)于時(shí)溫等效的研究集中在WLF(Williams- Landel-Ferry)方程參數(shù)的確定及應(yīng)用擴(kuò)展,WLF預(yù)測數(shù)據(jù)與實(shí)際結(jié)果的誤差和對新材料性能預(yù)測、動(dòng)態(tài)力學(xué)模型的驗(yàn)證等[15-17]。Paulo等[18]利用時(shí)溫等效原理研究了在恒剪切率情況下輪胎橡膠的流變力學(xué)行為,獲得了與試驗(yàn)一致的結(jié)果。Lin等[19]應(yīng)用WLF方程建立了形狀記憶線性醚型聚氨酯的溫度和可逆相的關(guān)系。Jacek等[20]討論了WLF中參數(shù)在分子層面的影響性(包括分子黏性作用能,鏈剛度和聚合物摩爾質(zhì)量)。鄭健龍等[21]利用WLF方程擴(kuò)展了試驗(yàn)數(shù)據(jù)的頻率范圍,對其提出的瀝青黏彈性損傷模型進(jìn)行了驗(yàn)證。朱凡等[22]借用WLF方程對小麥面筋系統(tǒng)流變行為進(jìn)行了預(yù)測,為面筋相關(guān)產(chǎn)品制作工藝在更大范圍內(nèi)實(shí)現(xiàn)提供了理論依據(jù)。劉博涵等[23]擬合得到了車用PVB(polyvinyl butyral)薄膜材料WLF公式,為該類材料進(jìn)一步在人員保護(hù)和車輛安全方面研究提供了基礎(chǔ)的材料參數(shù)。張針粒等[24]基于時(shí)溫等效原理提出了黏彈性材料頻率譜—溫度譜鏡像關(guān)系,并導(dǎo)出溫度譜的六參數(shù)分?jǐn)?shù)階模型,用DMA(dynamic thermomechanical analysis)試驗(yàn)驗(yàn)證了其正確性。

      綜上所述,分?jǐn)?shù)階導(dǎo)數(shù)是描述黏彈性材料“彈阻”中間性態(tài)的有效數(shù)學(xué)構(gòu)架,WLF方程是表征黏彈性溫頻效應(yīng)的有效手段,二者本質(zhì)上均為黏彈性本構(gòu)力學(xué)行為的演化形式,因此提出如下科學(xué)假設(shè):分?jǐn)?shù)階模型與WLF方程存在著內(nèi)聯(lián)因素,由此構(gòu)造的分?jǐn)?shù)階時(shí)溫等效模型可更加精確地描述黏彈性材料多溫變頻動(dòng)態(tài)力學(xué)特性。鑒于此,本文從自由體積理論出發(fā),綜合黏彈性材料分?jǐn)?shù)階導(dǎo)數(shù)關(guān)系和WLF方程,推導(dǎo)分?jǐn)?shù)階時(shí)溫等效模型(fractional time-temperature superposition model, FTTSM),并用DMA試驗(yàn)對其進(jìn)行驗(yàn)證。

      1 模型建立

      1.1 時(shí)溫等效原理

      黏彈性材料的時(shí)溫等效原理指同一材料的動(dòng)態(tài)力學(xué)性能既可用恒頻下的溫度譜表征,也可用恒溫下的頻率譜表征。根據(jù)該原理,可實(shí)現(xiàn)等溫曲線向參考溫度下平移,擴(kuò)展參考溫度下頻率譜范圍,其關(guān)系式為

      (2)

      進(jìn)一步有

      由式(3)知,頻率轉(zhuǎn)化因子α為實(shí)現(xiàn)溫度下動(dòng)態(tài)力學(xué)性能頻率譜平移到參考T下的平移量,如圖1所示,1、2下的頻率譜可通過相應(yīng)的頻率轉(zhuǎn)化因子α1、α2向參考溫度T平移(圖1a),進(jìn)而獲得該材料在參考溫度T下的寬頻性能(圖1b),即主曲線,擴(kuò)展參考溫度下對動(dòng)態(tài)性能的預(yù)測能力。

      圖1 時(shí)溫等效示意圖

      Fig.1 Sketch map of time-temperature superposition

      因此,欲根據(jù)時(shí)溫等效原理,通過平移不同溫度下的頻率譜,獲得參考溫度下長程或超長程頻段頻率譜,頻率轉(zhuǎn)化因子α至關(guān)重要。

      1.2 WLF方程

      WLF方程為經(jīng)典的時(shí)溫轉(zhuǎn)換模型,是從大量非晶態(tài)聚合物時(shí)溫轉(zhuǎn)換實(shí)事中總結(jié)出來的,其表達(dá)式為[25]

      式中1、2為常數(shù)。

      1.3 分?jǐn)?shù)階時(shí)溫等效模型

      依據(jù)Vogel-Fulcher-Tammann黏度方程[26]

      式中為黏度,Pa×s;1、為常數(shù);0為臨界溫度,℃。

      上式體現(xiàn)了黏彈性材料自由體積隨溫度的變化與其在0時(shí)表現(xiàn)的不連續(xù)性,當(dāng)溫度大于0時(shí),自由體積可在黏彈性材料內(nèi)重新分布而沒有能量變化。

      同理,在參考溫度T時(shí),方程(5)為

      由于黏彈性材料的力學(xué)行為介于理想胡克彈簧(應(yīng)力正比于應(yīng)變的零階導(dǎo)數(shù),)和理想牛頓流體(應(yīng)力正比于應(yīng)變的一階導(dǎo)數(shù),),故可推測具有“彈阻”共性的黏彈性材料的應(yīng)力正比于應(yīng)變的分?jǐn)?shù)階導(dǎo)數(shù),即

      (7)

      式中為應(yīng)力,N/m2;為應(yīng)變;為材料參數(shù)。

      對式(7)進(jìn)行拉普拉斯變換得

      則復(fù)模量為

      其模為

      (10)

      進(jìn)而黏度為

      (12)

      式中ω為參考圓頻率,rad/s。

      對式(12)、(13)等式兩端取自然對數(shù)得

      (15)

      式(14)、(15)相減得

      (17)

      上式為黏彈性材料分?jǐn)?shù)階時(shí)溫互轉(zhuǎn)模型的初級(jí)形式,包含互轉(zhuǎn)量(,)和(ω,T)以及常系數(shù),故應(yīng)確定(-0)(T-0)的值。

      根據(jù)WLF方程可知

      由于式(18)中α、1和2均為常數(shù),故(-T)也為常數(shù)。另外,根據(jù)α的定義

      (19)

      故lg(ω/)也為常數(shù),那么式(17)的右端也為常數(shù),又因(-T)和都為常數(shù),可知(-0)(T-0)為常數(shù),即

      因此,式(17)簡化為FTTSM的最簡形式

      (21)

      根據(jù)頻率轉(zhuǎn)化因子定義(式(3)),利用式(21)獲得FTTSM的頻率轉(zhuǎn)化因子

      2 模型分析

      2.1 參數(shù)識(shí)別

      由式(22)可知,F(xiàn)TTSM模型頻率轉(zhuǎn)化因子α包含常數(shù)和¢,其中為材料參數(shù),可由材料本構(gòu)力學(xué)行為確定,¢為環(huán)境參數(shù),可由材料溫度譜數(shù)據(jù)擬合獲得,其流程為:

      通過靜態(tài)拉伸試驗(yàn)獲得材料在小應(yīng)變下的應(yīng)力應(yīng)變曲線,利用式(7)對其進(jìn)行擬合獲得材料參數(shù)。

      通過DMA試驗(yàn)獲得材料動(dòng)態(tài)性能組溫度下的頻率譜,選取同一動(dòng)態(tài)性能所對應(yīng)的溫頻參數(shù),利用式(21)對其進(jìn)行擬合獲得環(huán)境參數(shù)¢。

      將所得材料參數(shù)和環(huán)境參數(shù)¢帶入式(22)即可求得頻率轉(zhuǎn)換因子α。

      2.2 參數(shù)分析

      分別討論FTTSM中材料參數(shù)和環(huán)境參數(shù)¢對頻率轉(zhuǎn)化因子α的影響規(guī)律,見圖2。圖2a為α在¢=0.2時(shí)隨0.2、0.4、0.6、0.8、1.0的變化趨勢,圖2b為α在0.7時(shí)隨的¢=0、0.2、0.4、0.6、0.8、1.0變化趨勢。

      注:B¢為環(huán)境參數(shù);α為材料參數(shù)

      由圖2可以發(fā)現(xiàn)α為溫度差的減函數(shù),在℃(即環(huán)境溫度小于參考溫度T)時(shí),說明在此溫度范圍內(nèi)動(dòng)態(tài)力學(xué)曲線要向參考溫度右端的高頻段移動(dòng),在℃(即環(huán)境溫度大于參考溫度)時(shí),說明在此溫度范圍內(nèi)動(dòng)態(tài)力學(xué)曲線要向參考溫度左端的低頻段移動(dòng),該現(xiàn)象與時(shí)溫等效原理中“低溫對高頻,高溫對低頻”對應(yīng)。

      3 試驗(yàn)研究

      以某履帶拖拉機(jī)黏彈性懸架的緩沖材料為對象,依據(jù)2.1節(jié)參數(shù)識(shí)別方法進(jìn)行靜態(tài)拉伸和DMA試驗(yàn),分別獲得FTTSM模型的材料參數(shù)和環(huán)境參數(shù),并對其寬域頻率范圍的動(dòng)力學(xué)性能進(jìn)行描述,獲得主曲線,并與WLF結(jié)果進(jìn)行對比,最后用黏彈性材料理論模型進(jìn)行驗(yàn)證。

      3.1 試驗(yàn)

      3.1.1 靜態(tài)拉伸試驗(yàn)

      按照GB/T9865.1[27]規(guī)定制備1型啞鈴狀試樣,依據(jù)GB/T528-1998[28],在Testometric M350-10 kN型精密拉伸儀(Testometric, 英國)上進(jìn)行靜態(tài)拉伸測試(見圖3),拉伸速率為10 mm/min,進(jìn)行2次試驗(yàn)取均值,然后對其擬合獲得FTTSM材料參數(shù)。

      圖3 拉伸儀及其夾具

      3.1.2 DMA試驗(yàn)

      DMA試驗(yàn)設(shè)備為耐馳DMA242C(Netzsch, 德國)動(dòng)態(tài)機(jī)械分析儀(圖4),其測試范圍:溫度(-150~ 650 ℃),振動(dòng)頻率(0.01~200 Hz),載荷(<10 N)。試樣尺寸:50 mm10 mm2 mm(長寬高),依照ISO 6721-1[29],采用三點(diǎn)彎曲測試模式,設(shè)定激勵(lì)頻率為:0.5、1.0、2.0、3.3、5.0、10.0 Hz,掃描溫度范圍:-120~120 ℃,升溫間隔10 ℃。通過頻率掃描和溫度掃描獲得該材料動(dòng)態(tài)力學(xué)性能的頻率譜,見圖5。

      1. 測試系統(tǒng) 2. 控制器 3. 水浴系統(tǒng) 4. 分析系統(tǒng)

      注:箭頭方向頻率依次為10.0、5.0、3.3、2.0、1.0、0.5 Hz。

      為保證參數(shù)擬合的精度,應(yīng)選擇具有明顯差別的溫頻數(shù)據(jù)。因此,在溫度譜數(shù)據(jù)中選取儲(chǔ)能模量為94 MPa時(shí)不同頻率所對應(yīng)溫度,見表1,以第一列數(shù)據(jù)為參考,利用式(21)對其擬合,獲得環(huán)境參數(shù)。

      表1 儲(chǔ)能模量為94 MPa 時(shí)的溫頻值

      3.2 FTTSM模型時(shí)溫性驗(yàn)證

      選擇(–75~75 ℃)范圍的頻率譜作為驗(yàn)算對象,溫度間隔為10 ℃,參考溫度T=5 ℃。在動(dòng)態(tài)機(jī)械分析軟件Proteus Analysis中提取WLF方程參數(shù)1=90.5,2= 515.4 ℃,繪制FTTSM和WLF的頻率轉(zhuǎn)化因子關(guān)于溫度變化的變化曲線,見圖6。由圖6可知,F(xiàn)TTSM和WLF有很好一致性,最大相對誤差為0.984 4%,說明FTTSM符合時(shí)溫等效原理。

      圖6 FTTSM和WLF在Tr=5 ℃時(shí)的αT對比

      3.3 FTTSM模型精確性驗(yàn)證

      根據(jù)時(shí)溫等效原理,即可利用頻率轉(zhuǎn)化因子將多組溫度下的試驗(yàn)頻率譜平移,獲得參考溫度下的主曲線,從而實(shí)現(xiàn)對試驗(yàn)設(shè)備無法涉及的高/低頻段的動(dòng)態(tài)力學(xué)性能進(jìn)行預(yù)測。分?jǐn)?shù)階Kelvin-Voigt黏彈性振子(fractional Kelvin-Voigt viscoelastic oscillator,KFVEO)本構(gòu)方程在較寬頻域?qū)︷椥詣?dòng)態(tài)特性試驗(yàn)數(shù)據(jù)有很好的擬合 性[30],因此本文以儲(chǔ)能模量¢為考察對象,利用KFVEO模型對試驗(yàn)頻段內(nèi)數(shù)據(jù)進(jìn)行擬合,獲得本構(gòu)方程參數(shù),然后在FTTSM主曲線的頻段內(nèi)對儲(chǔ)能模量¢進(jìn)行理論預(yù)測,實(shí)現(xiàn)對FTTSM模型的驗(yàn)證。

      KFVEO本構(gòu)方程為

      式中為剛度系數(shù),N/m;為阻尼系數(shù),N/(m/s);為階數(shù);為分?jǐn)?shù)階微分算子,其Riemann-Liouville定義

      (24)

      對式(23)進(jìn)行拉式變換得

      由式(25)得復(fù)模量

      (26)

      以式(27)對¢(T=5 ℃)的試驗(yàn)頻率譜進(jìn)行擬合,獲得KFVEO參數(shù)為:0.295 1,=3.56 N/m,=0.112 7 N/(m/s)。

      分別用FTTSM和WLF對-75~75 ℃(溫度間隔10 ℃)的頻率譜相對于參考溫度T=5 ℃進(jìn)行平移(縮減頻率見表2),獲得2種方法的儲(chǔ)能模量主曲線,并用KFVEO在相同頻段進(jìn)行理論預(yù)測,見圖7。由圖7可知,較WLF主曲線,F(xiàn)TTSM主曲線更接近KFVEO理論預(yù)測值;此外,WLF主曲線和FTTSM主曲線與理論預(yù)測值的均方根誤差(RMSE)分別為1.834和1.291 MPa,驗(yàn)證了FTTSM模型的精確性。

      由表2平移量及式(2)計(jì)算可知,二者主曲線擴(kuò)展的頻段分別為:6.655 1′10-10~1.249 6′1011Hz(FTTFM),2.088 4′10-8~1.485 1′1011Hz(WLF),可見FTTSM主曲線和WLF主曲線的最高頻率在同一數(shù)量級(jí),最低頻率則相差2個(gè)數(shù)量級(jí),說明FTTSM對黏彈性動(dòng)態(tài)力學(xué)性能具有更廣泛的預(yù)測能力。另外,可以看出低溫頻率譜平移到了高頻段,高溫頻率譜平移到了低頻段,這是由于在低溫時(shí)黏彈性材料內(nèi)部運(yùn)動(dòng)鏈段被“凍結(jié)”,形變主要由高分子鏈中原子間化學(xué)鍵的鍵長、鍵角改變所產(chǎn)生,形變阻力增大,從而儲(chǔ)能模量變大;而在受高頻激勵(lì)時(shí),運(yùn)動(dòng)鏈段松弛時(shí)間大于激勵(lì)間隔,鏈段來不及運(yùn)動(dòng)而呈現(xiàn)較大儲(chǔ)能模量,符合時(shí)溫等效原理中“低溫對高頻,高溫對低頻”原則。

      表2 FTTSM和WLF的偏移量

      4 結(jié) 論

      1)提出了一種分?jǐn)?shù)階時(shí)溫等效模型(FTTSM),給出了其參數(shù)識(shí)別方法。參數(shù)分析表明:參數(shù)具有明確的物理含義,在溫度差℃時(shí),頻率轉(zhuǎn)化因子與材料參數(shù)負(fù)相關(guān),與環(huán)境參數(shù)¢正相關(guān);在溫度差℃時(shí),頻率轉(zhuǎn)化因子與材料參數(shù)正相關(guān),與環(huán)境參數(shù)¢負(fù)相關(guān)。FTTSM和WLF所表征的頻率轉(zhuǎn)化因子在溫度變化(–80~80 ℃)內(nèi)最大相對誤差為0.984 4%,說明新模型符合時(shí)溫等效原理。

      2)利用KFVEO對等寬頻段的動(dòng)態(tài)力學(xué)性能進(jìn)行理論預(yù)測,對通過FTTSM和WLF平移多組溫度下頻率譜獲得的儲(chǔ)能模量主曲線進(jìn)行驗(yàn)證。結(jié)果顯示,WLF方程和FTTSM對儲(chǔ)能模量的預(yù)測與理論值之間的均方根誤差(RMSE)分別為1.834和1.291 MPa,說明FTTSM主曲線和KFVEO理論預(yù)測值具有更好的一致性,驗(yàn)證了新模型的精確性;FTTSM擴(kuò)展的低頻段比WLF擴(kuò)展的小2個(gè)數(shù)量級(jí),而在高頻段二者在同一數(shù)量級(jí),證明新模型具有更廣泛的預(yù)測能力。

      下一步將對FTTSM在黏彈性材料多溫變頻下的動(dòng)態(tài)力學(xué)性能預(yù)測(如諾模圖)、變階阻尼機(jī)理、蠕變損傷演化規(guī)律和長期力學(xué)性能加速表征等方面應(yīng)用開展深入研究。

      [1] Geethamma V G, Asaletha R, Kalarikkal N, et al. Vibration and sound damping in polymers[J]. Resonance, 2014, 19(9): 821-833.

      [2] 郭浩亮,穆希輝,楊小勇,等. 四橡膠履帶輪式車輛轉(zhuǎn)向力學(xué)性能分析與試驗(yàn)[J]. 農(nóng)業(yè)工程學(xué)報(bào),2016,32(21):79-86. Guo Haoliang, Mu Xihui, Yang Xiaoyong, et al. Mechanics properties analysis and test of four rubber tracked assembly vehicle steering system[J]. Transactions of the Chinese Society of Agricultural Engineering (Transactions of the CSAE), 2016, 32(21): 79-86. (in Chinese with English abstract)

      [3] 張瀧,劉耀儒,薛利軍,等. 基于內(nèi)變量熱力學(xué)的黏彈性本構(gòu)方程及其基本性質(zhì)研究[J]. 中國科學(xué):物理學(xué) 力學(xué) 天文學(xué), 2015,45(1):014601.Zhang Long, Liu Yaoru, Xue Lijun, et al. A viscoelastic constitutive equation and its fundamental properties based on thermodynamics with internal state variable[J]. Scientia Sinica Physics, Mechanics & Astronomy, 2015, 45(1): 014601. (in Chinese with English abstract)

      [4] Aditya Kumar, Oscar Lopez-Pamies. On the two-potential constitutive modeling of rubber viscoelastic materials[J]. Computers Rendus Mécanique, 2016, 344(2): 102-112.

      [5] 燕碧娟,孫大剛,張文軍,等. 農(nóng)業(yè)機(jī)械管狀過渡阻尼結(jié)構(gòu)參數(shù)分析及優(yōu)化[J]. 農(nóng)業(yè)工程學(xué)報(bào),2015,31(22):56-62. Yan Bijuan, Sun Dagang, Zhang Wenjun, et al. Parameter analysis and optimization of tubular transitional layer damping structure for agricultural machinery[J]. Transactions of the Chinese Society of Agricultural Engineering (Transactions of the CSAE), 2015, 31(22): 56-62. (in Chinese with English abstract)

      [6] Nutting P G. A new general law deformation[J]. Journal of the Franklin Institute, 1921, 191: 678-685.

      [7] Gemant A. On fractional differentials[J]. Philos Mag Ser, 1938, 25(168): 540-549.

      [8] Bosworth R C. A definition of plasticity[J]. Nature, 1946, 157(4): 447.

      [9] Bagley R L, Torvik P J. On the Fractional Calculus Model of Viscoelastic Behavior[J]. Journal of Rheology, 1986, 30(1): 133-155.

      [10] 孫洪廣,常愛蓮,陳文,等. 反常擴(kuò)散:分?jǐn)?shù)階導(dǎo)數(shù)建模及其在環(huán)境流動(dòng)中的應(yīng)用[J]. 中國科學(xué):物理學(xué)力學(xué)天文學(xué),2015,45(10):104702.Sun Hongguang, Chang Ailian, Chen Wen, et al. Anomalous diffusion: fractional derivative equation models and applications in environmental flows[J]. Scientia Sinica Physics, Mechanics & Astronomy, 2015, 45(10): 104702. (in Chinese with English abstract)

      [11] 唐振寰,羅貴火,陳偉,等. 含橡膠隔振器振動(dòng)系統(tǒng)動(dòng)態(tài)特性研究[J]. 南京航天航空大學(xué)學(xué)報(bào),2014,46(2):285-291. Tang Zhenhuan, Luo Guihuo, Chen Wei, et al. Dynamic characteristics of vibration system including rubber isolator[J]. Journal of Nanjing University of Aeronautics & Astronautics, 2014, 46(2): 285-291. (in Chinese with English abstract)

      [12] 李占龍,孫大剛,燕碧娟,等. 履帶式車輛黏彈性懸架分?jǐn)?shù)階模型及其減振效果分析[J]. 農(nóng)業(yè)工程學(xué)報(bào),2015,31(7):72-79. Li Zhanlong, Sun Dagang, Yan Bijuan, et al. Fractional order model of viscoelastic suspension for crawler vehicle and its vibration suppression analysis[J]. Transactions of the Chinese Society of Agricultural Engineering (Transactions of the CSAE), 2015, 31(7): 72-79. (in Chinese with English abstract)

      [13] Wharmby A W, Bagley R L. Modifying Maxwell's equations for dielectric materials based on techniques from viscoelasticityand concepts from fractional calculus[J]. International Journal of Engineering Science, 2014, 79(2): 59-80.

      [14] Cao Lili, Li Yan, Tian Guohui, et al. Time domain analysis of the fractional order weighted distributed parameter Maxwell model[J]. Computers and Mathematics with Applications, 2013, 66(5): 813-823.

      [15] Micha Peleg. On the use of the WLF model in polymer and foods[J]. Critical Reviews in Food Science and Nurtrition, 1992, 32(1): 59-66.

      [17] Put T A C M van der. Theoretical derivation of the WLF- and annealing equations[J]. Journal of Non-Crystalline Solids, 2010, 356(5/6): 394-399.

      [18] Paulo Lima, Sara P. Magalhaes da Silva, Jose Oliveira. Rheological properties of ground tyre rubber based thermoplasitc elastomeric blends[J]. Polymer Testing, 2015, 45(5): 56-67.

      [19] Lin L J, Chen L W. The mechanical-viscoelastic model and WLF relationship in shape memorized linear ether-type polyurethanes[J]. Journal of Polymer Research, 1999, 6(1): 35-44.

      [20] Jacek Dudowicz, Jack F Douglas, Kari F Freed. The meaning of the “universal” WLF parameters of glass- forming polymer liquids[J]. The Journal of Chemical Physics, 2015, 142(1): 1-7.

      [21] 鄭健龍,呂松濤,田小革. 基于蠕變試驗(yàn)的瀝青黏彈性損傷特性[J]. 工程力學(xué),2008,25(2):193-196. Zheng Jianlong, Lü Songtao, Tian Xiaoge. Viscoelastic damage characteristics of asphalt based on creep test[J]. Engineering Mechanics, 2008, 25(2): 193-196. (in Chinese with English abstract)

      [22] 朱凡,徐廣文,丁文平. 基于管式模型的小麥面筋系統(tǒng)流變行為的解析,農(nóng)業(yè)工程學(xué)報(bào)[J]. 2007,23(7):24-29. Zhu Fan, Xu Guangwen, Ding Wenping. Tube theory based analysis on the rheological behavior of wheat gluten dough[J]. Transactions of the Chinese Society of Agricultural Engineering (Transactions of the CSAE), 2007, 23(7): 24-29. (in Chinese with English abstract)

      [23] 劉博涵,周嘉,孫岳霆,等. 車用PVB薄膜材料動(dòng)態(tài)黏彈性的實(shí)驗(yàn)研究[J]. 汽車工程,2012,34(10):898-904. Liu Bohan, Zhou Jia, Sun Yueting, et al. An experimental study on the dynamic viscoelasticity of PVB film material for vehicle[J]. Automotive Engineering, 2012, 34(10): 898-904. (in Chinese with English abstract)

      [24] 張針粒,李世其,朱文革. 黏彈性阻尼材料動(dòng)態(tài)力學(xué)性能溫度譜模型[J]. 機(jī)械工程學(xué)報(bào),2011,47(20):135-140. Zhang Zhenli, Li Shiji, Zhu Wenge. Temperature spectrum model of dynamic mechanical properties for viscoelastic damping materials[J]. Journal of Mechanical Engineering, 2011, 47 (20): 135-140. (in Chinese with English abstract)

      [25] Bharani K. Ashokan, Jozef L. Kokini. Determination of the WLF constants of cooked soy flour and their dependence on extent of cooking[J]. Rheologica Acta, 2005, 45(6): 192-201.

      [26] Marc Bletry, Minh Thanh Thai, Yannick Champion, et al. Consistency of the free-volume approach to the homogeneous deformation of metallic glasses[J]. Comptes Rendus Mecanique, 2014, 342(5): 311-314.

      [27] GB/T 9865.1,硫化橡膠或熱塑性橡膠樣品和試樣的制備第一部分: 物理試驗(yàn)[S].

      [28] GB/T 528-1998,硫化橡膠或熱塑性橡膠—拉伸應(yīng)力應(yīng)變性能的測定[S].

      [29] ISO 6721-1, Plastics-Determination of dynamic mechanical properties-Part1: General principles[S].

      [30] Li Z L, Sun D G, Sun B, et al. Fractional model of viscoelastic oscillator and application to a crawler tractor[J]. Noise Control Engineering Journal, 2016, 64 (3): 388-402.

      Fractional time-temperature superposition model and its validation for dynamic mechanical properties of viscoelastic material

      Li Zhanlong, Song Yong※, Sun Dagang, Zhang Xin, Sun Bao

      (030024,)

      The viscoelastic damping structure is widely used in the vibration and noise control of the agricultural engineering vehicle because of its high vibration dissipation capability, simple structure and lower maintenance cost. The mechanical behavior of the viscoelastic materials displays anelastic feature and temperature-frequency dependence, so the precise dynamic modeling is the key step in the viscoelastic structure design and its vibration damping analysis process. For anelasticity, the fractional derivative defined based on global definition can precisely represent the history dependence of the system function, and be extensively applied to the viscoelastic models with fewer parameters and better data fitting ability. For temperature-frequency dependence, the time-temperature superposition (TTS) principle was adopted, according to which the frequency spectrums at various temperatures can be collapsed into a master curve at the reference temperature by multiplying the conversion factors. The master curve covers a wide reduced frequency range up to many orders of magnitudes. In this research, the fractional time-temperature superposition model (FTTSM) for dynamic mechanical properties of viscoelastic materials was proposed based on the fractional order relationship, the Vogel-Fulcher-Tammann equation and the WLF (Williams-Landel-Ferry) equation. The frequency conversion factor from FTTSM was derived and its parameter identification process was developed based on tensile test and DMA (dynamic thermo mechanical analysis) test. In order to understand the parameter influence on the conversion factor, the variation of the conversion factor was studied under the material parameter of 0.2, 0.4, 0.6, 0.8 and 1 and the environment parameter of 0, 0.4, 0.6, 0.8 and 1. For the application and validation, the tensile test was conducted following GB/T 528-1998 on the M350-10 kN type precise elongation apparatus (Testometric, Britain), after preparing the I type dumbbell-shaped sample following GB/T 9865.1. In addition, the DMA test was also carried out according to ISO 6721-1 using the DMA 242C (Netzsch, Germany), in a 3-piont bending mode with a 40 mm span between the supports, in which the sample was supported on 2 supporting edges, while the probe edge applied load to the sample. On the base of the test data, the master curves at reference temperature of 5 ℃ from FTTSM and WLF equation, constructed through horizontally superposing the isothermals at various temperatures onto the isothermal at reference temperature, were comparatively studied. Furthermore, the theoretical prediction over the same frequency span was made through fractional Kelvin-Voigt constitutive model (KFVEO) to testify the master curves. The results indicated that the frequency conversion factors from FTTSM and WLF equation showed a good consistence with the maximum error of 0.984 4% within temperature scope (-80-80 ℃), and the master curves constructed by FTTSM and WLF equation greatly extended the frequency range up to 10 decades. The RMSE (root mean square error) between the master curves from FTTSM and WLF and the KFVEO prediction value was1.291 and 1.834 respectively, which manifested the FTTSM was more precise. Regarding the extended frequency, the minimum extended frequency by FTTSM was 2 orders of magnitudes less than that by WLF equation, while the maximum extended frequency stayed at the same level for these 2 models. This indicated a higher frequency extended capacity of FTTSM. This research can provide the theoretical reference for the investigation of viscoelastic material on dynamic behavior prediction, physical aging and mechanism of creep damage evolution, and so on.

      vehicles; vibration control; models; viscoelastic material; dynamic behavior; fractional order; time-temperature superposition; WLF equation

      10.11975/j.issn.1002-6819.2017.08.012

      O328

      A

      1002-6819(2017)-08-0090-07

      2016-07-19

      2017-02-07

      國家青年科學(xué)基金資助項(xiàng)目(51305288,51405323);太原科技大學(xué)博士啟動(dòng)基金(20122050,20162005)

      李占龍,男,山西朔州人,講師,博士,主要從事農(nóng)業(yè)工程車輛黏彈性減振技術(shù)研究。太原 太原科技大學(xué)機(jī)械工程學(xué)院,030024。 Email:lizlbox@163.com

      宋 勇,男,安徽歙縣人,講師,博士,從事振動(dòng)與噪聲控制研究。太原 太原科技大學(xué)機(jī)械工程學(xué)院,030024。Email:52921460@qq.com

      李占龍,宋 勇,孫大剛,章 新,孫 寶.黏彈性材料動(dòng)態(tài)力學(xué)性能的分?jǐn)?shù)階時(shí)溫等效模型與驗(yàn)證[J]. 農(nóng)業(yè)工程學(xué)報(bào),2017,33(8):90-96. doi:10.11975/j.issn.1002-6819.2017.08.012 http://www.tcsae.org

      Li Zhanlong, Song Yong, Sun Dagang, Zhang Xin, Sun Bao.Fractional time-temperature superposition model and its validation for dynamic mechanical properties of viscoelastic material[J]. Transactions of the Chinese Society of Agricultural Engineering (Transactions of the CSAE), 2017, 33(8): 90-96. (in Chinese with English abstract) doi:10.11975/j.issn.1002-6819.2017.08.012 http://www.tcsae.org

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