武智慧^*, 牛公杰^?, 郝玉风^**, 钱建平^,*,2), 刘荣忠^** 南京理工大学机械工程学院,南京 210094
† 中国工程物理研究院总体工程研究所,四川绵阳 621999
** 辽沈工业集团有限公司产品研发中心,沈阳 110045

RESEARCH ON MODELING OF COMPRESSIVE YIELD BEHAVIOR FOR HTPB COMPOSITE BASE BLEED GRAIN¹⁾

Wu Zhihui^*, Niu Gongjie^?, Hao Yufeng^**, Qian Jianping^,*,2), Liu Rongzhong^** School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China
† Institute of Systems Engineering, China Academy of Engineering Physics, Mianyang 621999, Sichuan,China
** Product Research and Development Center of Liaoshen Industries Group Co., Ltd, Shenyang 110045, China

通讯作者: 2) 钱建平,教授,主要研究方向：弹药总体技术. E-mail：13951837475@139.com

收稿日期:2019-07-24接受日期:2019-10-21网络出版日期:2019-11-18

基金资助:

1) 国家自然科学基金资助项目.11402248

Received:2019-07-24Accepted:2019-10-21Online:2019-11-18
作者简介 About authors


摘要
目前广泛应用于底排增程技术的 HTPB 复合底排药 (composite base bleed grain,CBBG) 是一种颗粒填充含能材料,战场环境中将承受冲击、温度等载荷作用. 为研究 HTPB CBBG 冲击压缩力学性能,进行了不同温度 (233$\sim$323 K) 和应变率 (1100$\sim$7900 s$^{-1}$) 下的分离式霍普金森压杆实验. 实验结果表明,各工况下,应力应变曲线均呈现屈服-$\!$-应变硬化特征,HTPB CBBG 保持高韧性. 提高应变率和降低温度均导致相同应变下的应力幅值上升,但温度较应变率对HTPB CBBG 冲击压缩力学性能的影响更为显著. 基于所研究温度范围高于 HTPB CBBG 玻璃化转变温度,通过将水平、垂直移位因子与温度的关系表示为 WLF 方程的形式,将时温等效原理引入协同模型,并计及内应力的应变率增强效应,提出了一种新的屈服应力模型.选取参考温度,利用水平、垂直移位因子-$\!$-温度曲线和屈服应力主曲线拟合模型参数.模型预测值与实验数据对比结果表明：该模型可准确表征 233$\sim$323 K 时 HTPB CBBG 屈服应力的双线性应变率相关性,明确了较低和较高应变率时,应变率效应分别主要由内应力和驱动力贡献.
关键词： HTPB 复合底排药;冲击;压缩力学性能;时温等效原理;屈服应力

Abstract
HTPB composite base bleed grain (CBBG), which has been widely applied to the base bleed extended-range technology, is a typical particle-filled energetic material and bears both impact and temperature loads in battlefield environments. In order to investigate impact compressive mechanical properties of HTBP CBBG, split Hopkinson pressure bar experiments were conducted at various temperatures and strain rates, ranging from 233 to 323 K and from 1100 to 7900 s$^{-1}$. True stress-true strain curves shows that HTPB CBBG yields and then deforms plastically with strain hardening effect and maintains high toughness under each experimental condition. The stress value at a certain strain increases with the increase of strain rate and the decrease of temperature, but temperature has a more significant influence on impact compressive mechanical behaviors of HTPB CBBG than strain rate. On the one hand, the time-temperature superposition principle was introduced into the cooperative model by taking the correlations between horizontal/vertical shift factor and temperature as WLF function-type equations based on the fact that the temperature range discussed here was higher than the glassy transition temperature of HTPB CBBG. One the other hand, the enhancement effect of strain rate of internal stress was also taken into consideration, and then a new stress model was proposed. The smooth horizontal shift factor-temperature curve, vertical shift factor-temperature curve and master curve of yield stress were built at a reference temperature according to experimental results to obtain the parameters in the proposed model. The comparison between the model prediction and experimental data indicates that the developed model can precisely describe the bilinear dependence of yield stress on strain rate at temperatures of 233$\sim $323 K. The proposed model points out that the strain rate effect is derived from internal stress at low strain rates while it is derived from drive stress at high strain rates.
Keywords：HTPB composite base bleed grain;impact;compressive mechanical properties;the time-temperature superposition principle;yield stress


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本文引用格式
武智慧, 牛公杰, 郝玉风, 钱建平, 刘荣忠. HTPB复合底排药压缩屈服应力模型研究¹⁾. 力学学报[J], 2019, 51(6): 1810-1819 DOI:10.6052/0459-1879-19-200
Wu Zhihui, Niu Gongjie, Hao Yufeng, Qian Jianping, Liu Rongzhong. RESEARCH ON MODELING OF COMPRESSIVE YIELD BEHAVIOR FOR HTPB COMPOSITE BASE BLEED GRAIN¹⁾. Chinese Journal of Theoretical and Applied Mechanics[J], 2019, 51(6): 1810-1819 DOI:10.6052/0459-1879-19-200

引言

底排增程弹底部的 HTPB 复合底排药 (composite base bleed grain,CBBG) 为还原剂端羟基聚丁二烯 (hydroxyl-terminated polybutadiene,HTPB) 基体内填充氧化剂高氯酸铵 (ammonium perchlorate,AP) 颗粒的热固性弹性体,其通过燃烧向弹底喷射气流,降低底阻,实现增程. 除轴向过载外,HTPB CBBG 在内弹道阶段和出炮口瞬间,还分别经历了膛压引起的压缩高过载和急速泄压引起的拉伸高过载,载荷工况相比复合推进剂经历的炮射过载更为恶劣. 高过载导致的大变形,影响了底排药柱燃烧规律,降低了增程效率,甚至出现初始损伤恶化引起的掉药的极端情况. 为进行 HTPB CBBG 结构完整性分析,有必要开展 HTPB CBBG 冲击力学性能研究. 刘志林等^[1] 研究了室温时 2900$\sim$4300 s$^{ - 1}$ 下 HTPB CBBG 压缩力学性能.由于战场环境温度的差异性 (233$\sim $323 K)^[2] 及 HTPB CBBG 具有的黏性特征,其力学性能的温度效应是另一个需要考虑的因素.

目前,众多****已对多种含能材料的力学性能进行了实验及理论研究^[3].如 PBX 炸药压缩、拉伸力学性能对比研究^[4]、室温-$\!$-高温下压缩力学性能对比研究^[5],HTPB 推进剂室温-$\!$-低温下压缩、拉伸力学性能研究^[6-10],CMDB推进剂压缩力学性能的温度及应变率效应分析^[10-11],HTPB 推进剂室温-$\!$-高温下蠕变过程的时温等效性^[12]等.

屈服点是高聚物力学性能转变的标记点,屈服应力是分析高聚物力学性能的重要指标. 聚合物DMA测试数据的研究表明,由准静态加载到冲击加载过程中,聚合物在某一应变水平上应力值随应变率增加呈现的双线性或线性-$\!$-指数式增长源于大分子热激活运动的时温相关性^[10-11,13]. 分子运动理论认为热激活链段运动的多样性导致高聚物具有松弛时间谱,不同加载条件下主导松弛时间不同,而小幅度链段运动越受限,材料宏观应力增长越显著^[13-14].

现有的屈服应力模型多限于研究温度 $T$ 低于高聚物玻璃化转变温度 $T_{\rm g}$ 的情况,即 $T\!<\!T_{\rm g}$,少有文献提出模型以描述 $T\!>\!T_{\rm g}$ 时高聚物的屈服应力特性. 最早的 Eyring 模型^[15] 基于分子跃迁理论,解释了塑性流动过程中的剪切应力特性源于分子的热激活响应.Bauwens-Crowet 等^[16] 证实该理论对屈服应力的描述同样有效,并利用包含两个 Eyring 模型的 Ree-Eyring 模型描述了聚氯乙烯屈服应力的应变率和温度相关性.该模型将屈服应力的时温相关性解释为大分子链 $\alpha$ 运动和 $\beta $ 运动的相继受限.杨龙^[10] 运用简化的 Ree-Eyring 模型描述了常温下CMDB推进剂屈服应力的双线性应变率增强效应.另一方面,Omar 等^[17]利用 Fotheringham 和 Cheery 提出的协同模型^[18] 准确描述了 3 种高聚物在冲击加载时屈服应力的应变率相关性.Richeton 模型^[19] 在协同模型基础上,通过依次引入 Arrhenius 方程和 WLF 方程,提出了一种跨越 $T_{\rm g}$ 的屈服应力模型. $ T<T_{\rm g}$ 时,该模型已被众多研究者采用,效果理想.但 $T>T_{\rm g}$ 下应用该模型时,需额外获取 $T<T_{\rm g}$ 时屈服应力值,并确定 $T_{\rm g}$步骤较为繁琐,目前尚未获得广泛应用.Gueguen 等^[20]、Gomez-del 等^[21]通过将 $T<T_{\rm g}$ 时 Richeton 模型中的热激活参量广义化,描述了 $T>T_{\rm g}$ 时聚丙烯等的屈服应力,虽然获得了较好的效果,但物理意义难以解释. 孙朝翔^[11]通过修正Richeton模型的内应力和活化能,提出了一种可以描述跨越 $T_{\rm g}$ 的 CDMB 推进剂屈服应力模型. 邓小秋等^[22]、Gomez-del等^[23] 分别针对有机玻璃和环氧树脂的屈服应力特性对比了 Ree-Eyring 模型和 Richeton 模型的准确性.其他经典模型还有 Argon 等^[24] 提出的基于分子链段旋转-$\!$-取向的分子间阻力模型等.

本文进行了 323 K、301 K(室温)、253 K 和 233 K 下 HTPB CBBG 霍普金森压杆实验,应变率范围 1100$\sim $7900 s$^{ - 1}$.结合其力学性能曲线,分析了不同应变率和温度下 HTPB CBBG 冲击压缩力学性能的异同. 着重研究了应变率和温度对屈服应力值的影响规律,并基于时温等效原理修正了协同模型,同时考虑了内应力的应变率增强效应,建立了一种新的适用于 $T>T_{\rm g}$ 时的屈服应力模型,以期为进一步建立 HTPB CBBG 本构模型提供理论基础.

1 不同温度的分离式霍普金森压杆实验

1.1 试件材料

本文研究的 HTPB CBBG 密度为 1.54 g/cm$^{3}$,HTPB 基体和 AP 颗粒质量比例为 20 $:$ 73,其余为固化剂、增塑剂等. 为提高 HTPB CBBG 的燃烧效率,HTPB 基体内包含有粗 (210 $\sim$ 250 $\mu$m)、细 (110 $\sim$ 150 $\mu$m) 和超细 (10 $\sim$ 15 $\mu$m) 3 种粒径的 AP 颗粒. 试件由辽阳庆阳特种化工集团制备.

1.2 实验方案

本文采用图 1 所示的分离式霍普金森压杆 (SHPB) 系统进行 HTPB CBBG 冲击加载实验. 试件变形过程若满足常应变率加载和动态应力平衡的实验前提,试件工程应力 $\sigma_{\rm e}$、工程应变 $\varepsilon_{\rm e}$ 及加载应变率 $\dot{\varepsilon }$ 可通过二波法计算^[25]

(1)$ \sigma _{\rm e} (t) = \dfrac{A_0 }{A_{\rm s} }E_0 \varepsilon _{\rm T} (t)$
(2)$\varepsilon _{\rm e} (t) = - \dfrac{2C_0 }{L_{\rm s} }\int_0^t \varepsilon _{\rm R} (t) d t$
(3)$\dot {\varepsilon }(t) = - \dfrac{2C_0 }{L_{\rm s} }\varepsilon _{\rm R} (t)$
式中, $\varepsilon_{\rm I}(t)$, $\varepsilon_{\rm R}(t)$, $\varepsilon _{\rm T}(t)$ 分别为入射波、反射波和透射波历史信号;$A_{0}$,$E_{0}$,$C_{0}$ 分别为杆件横截面积、杨氏模量、波速;$A_{\rm s}$,$L_{\rm s}$ 分别为试件初始横截面积和长度.假设材料不可压缩,且压应变、压应力均取为正,根据 $\sigma =\sigma_{\rm e} (1-\varepsilon_{\rm e})$ 和 $\varepsilon=-\ln(1-\varepsilon_{\rm e})$ 可求真应力 $\sigma$、真应变 $\varepsilon $.

图1

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图1SHPB 实验装置

Fig.1Schematic diagrams of SHPB system



HTPB CBBG 较软,属于典型的低波阻、低波速材料,为增强透射信号强度,透射杆材料为低模量高强度的超硬铝 7A04-T6,同时,采用高灵敏度系数的半导体应变片记录入射波、反射波和透射波波形.撞击杆、入射杆、透射杆长度分别为 300,1500, 1500 mm,密度为 2810 kg/m$^3$,直径为 14.5 mm,杨氏模量为 71 GPa. 为使实验前提条件成立,需利用整形器将峰值震荡的矩形入射波整形成具有一定下降时 沿和加载平台的光滑梯形波^[25-26].本文所用整形器为直径 10 mm、厚 1 mm 的纸板. 采用直径为 10 mm,厚度为3 mm 的薄试件以利于试件尽快达到应力平衡状态.试件与压杆的接触面采用凡士林 润滑.

为对置于入射杆和透射杆之间的试件进行加热和降温,本文设计了小型发热陶瓷高温箱和液氮低温箱.温控可靠性试验结果表明,箱内试件所在位置温度波动范围在 2 K 以内,且加热或制冷过程中在杆件试件端形成的温度梯度未对一维应力波的传播产生影响. 实验温度包括 323 K、301 K (室温)、253 K 和 233 K. 试件于温度箱内保温半 小时后进行冲击加载实验.各工况下均完成3次有效实验,并取实验结果的平均值作为 HTPB CBBG 的力学性能曲线.

2 实验结果与分析

2.1 损伤扩展

图 2 为加载前后 HTPB CBBG 试件宏、细观结构对比情况,其中,细观结构照片采用扫描电子显微镜 (SEM) 获取,SEM 型号为 JOEL JSM-6380LV,拍摄区域为试件中心部分. 结合图 2 可知,加载前,试件按压时富有弹性,基体致密,颗粒包裹完好,细观上可见个别孔洞、颗粒脱湿等初始损伤;加载后,试件出现宏观裂纹,基体松散,细观上表现为基体大量断裂破坏,进而引起颗粒脱湿,同时伴随颗粒穿晶断裂失效. 由此可见,冲击加载下,试件应变足够大时,损伤扩展明显,强度降低,而损伤本质上是不可逆能量耗散的热力学过程,并与塑性形变紧密相关^[27-28],这是研究 HTPB CBBG 大变形时的力学性能需要考虑的因素.

图2

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图2试件加载前后宏、细观结构对比

Fig.2Comparison of macro and micro structures of specimens before and after impact loading

2.2 数据可靠性检验

图3(a)为 301 K 下 SHPB 实验的典型三波图,显然,反射波呈现平台,则由式 (3) 可知,应变率几乎恒定.入射杆与试 件接触端面受力为 $F_{1}(t) =( \varepsilon _{\rm I}(t)+\varepsilon_{\rm R}(t))E_{0}A_{0}$,透射杆与试件接触端面受力为 $F_{2}(t)=\varepsilon _{\rm T}(t)E_{0}A_{0}$,图3(b)中 $F_{1}(t)$ 和 $F_{2}(t)$ 的对比结果显示,二者基本保持一致,表明试件达到了动态应力平衡. 由此可见,入射波整形有效,测得的 HTPB CBBG 力学性能曲线可靠.

为进一步分析试件的常应变率加载情况,图 4 给出了图 3(a) 对应的 $\dot{\varepsilon }$-$\varepsilon $ 曲线,平均应变率为 5890 s$^{-1}$. 应变率波动范围在 $\pm$5% 内时,可认为试件达到常应变率加载,则真应变为 0.092$\sim$0.76 时(记为 AB 段),对应 的 $\sigma $-$\varepsilon$ 曲线足够准确,可反映 HTPB CBBG 的力学性能. 而真应变为 0$\sim$0.092 时(记为 OA 段),为实现常应变率加载的过程,对应的 $\sigma$-$ \varepsilon$ 曲线可能出现剧烈抖动的情况,这时,该段 $\sigma$-$\varepsilon $ 曲线可通过将AB段对应的 $\sigma$-$\varepsilon $ 曲线拟合、外推至原点获得 ^[11].

图3

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图3SHPB 实验原始数据

Fig.3Original data in SHPB experiments

图4

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图4$\dot {\varepsilon}$-$\varepsilon $ 曲线

Fig.4$\dot {\varepsilon }$-$\varepsilon $ curve

2.3 冲击压缩力学性能

图 5 对比了不同温度下,HTPB CBBG 冲击压缩$\sigma$-$\varepsilon $ 曲线. 当试件变形至横截面超过压杆横截面时,真应变满足 $d_{\rm s }\exp(\varepsilon/2)> d_{0}$ ($d_{\rm s}$ 为试件初始直径,$d_{0}$ 为压杆直径)^[29],则 $\varepsilon >0.743$ 时的 $\sigma -\varepsilon$曲线应截去,且该截断应变与常应变率加载下的最大应变较为接近. 需要指出,应变率较低时,在加载波时长内,试件尚未变形至屈服应变便发生卸载, $\sigma -\varepsilon$ 曲线无法展现屈服阶段. 由图 5 可知,HTPB CBBG 压缩力学性能显著依赖于应变率和温度. 相同应变下的应力幅值随着应变率的升高和温度的降低而增大,符合时温等效规律：应变率增加或温度降低,大分子链及链段相对运动受限,抵抗变形能力增加,刚性 增强.

图5

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图5不同温度下,HTPB CBBG 冲击压缩 $\sigma $-$\varepsilon $ 曲线

Fig.5$\sigma$-$\varepsilon $ curves at various temperatures under impact compressive loading for HTPB CBBG



屈服点通常取为$\sigma$-$\varepsilon$ 曲线拐点或 $\sigma$-$\varepsilon$ 曲线开始趋于平坦的点^[14],如图 6 所示. 以屈服点为界, $\sigma$-$\varepsilon$ 曲线分为初始弹性阶段和后屈服应变硬化阶段,两阶段曲线的斜率分别为初始模量 $E$ 和切线模量 $E_{\rm t}$. 温度相同时,随着应变率的增大,初始模量明显提高.各工况下,屈服应变范围为 0.102 5$\sim$0.118 2,为方便起见,后文将冲击加载实验下的屈服应力$\sigma_{\rm y}$取为平均屈服应变 0.11 时 (已在图 5 中用虚线标出)的应力值;后屈服阶段的曲线形态保持不变,切线模量为 11.74$\sim$15.39 MPa,说明在所研究的应变率和温度范围内,HTPBCBBG 力学性能未发生显著变化. 屈服点后至试件破坏前,尚未高度取向的大分子链及链段的松弛行为占主导^[30-31],结合损伤的削弱作用,最终导致切线模量低于初始模量.温度为 233 K、应变率为 7600 s$^{-1}$时, $\varepsilon=0.73$,表明此时 HTPB CBBG 仍具有高韧性.

图6

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图6屈服点的确定及 $\sigma -\varepsilon $ 曲线区域划分

Fig. 6Definitions of yield point and two regions of $\sigma -\varepsilon $ curve



在一定应变率范围内,应变率的提高有利于抑制损伤的发展,但当应变率足够高、应变足够大时,试件损伤愈发恶劣,如图 2 所示. 这种现象可解释为,由于裂纹产生比裂纹演化消耗更多能量,随着应变率的提高,传递至试件的能量增加,瞬时加载过程中积聚的大量能量倾向于通过新生裂纹的方式消耗,表现为应变率越高,裂纹越多,损伤越严重^[10,32].

图 7 为屈服应力的应变率和温度相关性,其中准静态实验数据利用万能试验机获取,试验机型号为 SANS CMT6503,试件 直径为 10 mm,厚度为 4 mm. 试件载荷历程 $f(t)$ 和位移历程 $u(t)$ 分别由力传感器和位移传感器记录,并通过 $\sigma(t)=f(t)/[A_{0}(1-u(t)/L_{s})]$ 和 $\varepsilon(t)=-\ln(1-u(t)/L_{s})$ 计算得到 $\sigma $-$\varepsilon$曲线.图 7 表明,各温度下,屈服应力随着对数应变率的增大呈现显著的双线性增长,据此可知,由于高聚物大分子运动层级繁多,即使$T>T_{\rm g}$,只要应变率(或温度)跨度足够大,仍可在该区域内观察到类似 $\alpha$ 运动受限和 $\alpha $, $\beta$ 运动共同受限而导致的力学参量急速增长现象. 将高应变率下的屈服应力值分别拟合为对数应变率和温度的线性函数 $y=Ax+B$,表 1 为各拟合曲线斜率 $A$ 和决定系数 $R^{2}$.对比表 1 中各因素变化时 $A$ 的变化趋势可知,降低温度直接导致屈服应力值的应变率敏感性显著增强,反之,增大应变率未对屈服应力值的温度敏感性产生显著影响. 可见,在所研究工况下,相对于应变率,温度(大分子链及链 段运动活性)对 HTPB CBBG 压缩力学性能的影响更为显著.

图7

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图7屈服应力与应变率、温度的相关性,内图为准静态压缩学性能曲线

Fig.7Yield stress as a function of strain rate at various temperatures and the insert presents results of quasi-static compressive experiments



Table 1
表1
表1$A$ 和 $R^{2}$
Table 1$A$ and $R^{2}$

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3 屈服应力模型

3.1 模型建立

Fotheringham 和 Cheery 提出的协同模型$^{^[18,33]}$ 为

(4)$\sigma _{\rm y} (T,\dot {\varepsilon }) = \sigma ^\ast (T) + \sigma ^{\rm d}(T,\dot{\varepsilon })$
(5)$\sigma ^{\rm d} (T,\dot {\varepsilon }) = \dfrac{2kT}{V}\sinh ^{ - 1}\Big(\dfrac{\dot{\varepsilon }}{\dot {\varepsilon }^\ast }\Big)^{\tfrac{1}{n}}$
式中, $\sigma^{\ast}$ 为内应力,是 $T$ 的函数,表征变形历史引起的大分子链及链段的剪切作用对材料当前应力的贡献,为弹性回复力,$\sigma^{\rm d}$ 为驱动应力,是实际导致屈服后塑性变形的应力部分,又称为有效应力,表征大分子热激活运动的时温敏感性,$k$ 为玻尔兹曼常数,$V$ 为活化体积,$\dot {\varepsilon }^\ast$ 为特征应变率,$n$为协同指数.

高聚物具有的时温等效性表明,通过沿应变率轴平移移位因子 $\lg(\alpha_{T})$,可实现力学参量在某一温度 $T$ 下的曲线与参考温度 $T_{0}$ 下的曲线叠合,从而获得该力学参量主曲线.WLF 方程将 $\lg(\alpha_{T})$ 表示为^[34-35]

(6)$\lg (\alpha _T ) = - \dfrac{C_1 (T - T_0 )}{C_2 + (T - T_0 )}$
式中,$C_{1}$,$C_{2}$ 为水平移位参数. 该式适用范围为 $T_{\rm g}\sim T_{\rm g}+100$ K. 根据文献^[36] 可知,对于HTPB推进剂 (HTPB 基体和 AP 颗粒质量比为 17:83),$T_{\rm g}$约为 215$\sim$229 K,据此推断,在所研究温度范围内,可应用 WLF 方程分析 HTPB CBBG 的时温等效性.

Bauwens-Crowet 等^[16]研究聚氯乙烯的屈服应力时发现,为获得理想的屈服应力主曲线,需对实验数据同时进行水平和垂直移位. Povolo 等^[18]提出协同模型的水平和垂直移位因子满足

(7)$\Delta (\lg \dot {\varepsilon } ) = \lg (\dot {\varepsilon }^\ast (T_0 )) - \lg(\dot {\varepsilon }^\ast (T))$
(8)$\Delta \Big(\dfrac{\sigma _y }{T} \Big ) = \dfrac{\sigma ^\ast (T_0 )}{T_0 } -\dfrac{\sigma ^\ast (T)}{T}$
$T<T_{g}$ 时,Bauwes-Crowet 等^[16]、Richeton 等^[19] 假设 $\Delta(\lg \dot {\varepsilon } )$ 和 $\Delta (\dfrac{\sigma _{\rm y}}{T})$ 的温度相关性均具有 Arrhenius 方程形式. $T>T_{\rm g}$ 时,本文采用类似方法,假设水平移位因子满足 WLF 方程,有

(9)$\Delta (\lg \dot {\varepsilon } ) = - \dfrac{C_1 (T - T_0 )}{C_2 + (T - T_0)} $
同时,垂直移位因子与温度的相关性采用 WLF 方程的形式描述,即

(10)$\Delta \Big(\dfrac{\sigma _y }{T}\Big) = \dfrac{C_3 (T - T_0 )}{C_4 + (T - T_0 )}$
式中,$C_{3}$,$C_{4}$ 为垂直移位参数. 将式 (9)、式 (10) 分别代入式 (7)、式 (8),并进一步整理,得到

(11)$\dot {\varepsilon }^\ast (T) = \dot {\varepsilon }^\ast _0 \exp \Big [\dfrac{C_1 (T -T_0 )}{C_2 + (T - T_0 )}\ln 10 \Big ]$
(12)$\sigma ^\ast (T) = \dfrac{\sigma _0 }{T_0 }T - \dfrac{C_3 (T - T_0 )}{C_4 + (T - T_0)}T$
式中,$\dot {\varepsilon }^\ast _0 = \dot {\varepsilon }^\ast (T_0 )$,$\sigma_{0}=\sigma^{\ast } (T_{0})$.

此外,Boyce 等^[30]、Srivastava 等^[31]认为,$\sigma^{\ast }$ 同时具有应变率效应.对于 HTPB CBBG 而言,可修正式 (12) 为

(13)$\sigma ^\ast (T,\dot {\varepsilon }) = a\sinh ^{ - 1}\Big (\dfrac{\dot {\varepsilon }}{\dot {\varepsilon }_0}\Big )^{\tfrac{1}{m}} + \dfrac{\sigma _0 }{T_0 }T - \dfrac{C_3 (T - T_0 )}{C_4 + (T - T_0 )}T $
式中,引入常量 $\dot {\varepsilon }_0 =1$ s$^{ -1}$,$a=1$ MPa 以保持量纲一致性,$m$为材料参数.将式 (5)、式 (11)、式 (13) 代入式 (4),得到

(14)$\sigma _{\rm y} (T,\dot {\varepsilon }) = a\sinh ^{ - 1}\Big (\dfrac{\dot {\varepsilon }}{\dot{\varepsilon }_0 }\Big)^{\tfrac{1}{m}} + \dfrac{\sigma _0 }{T_0 }T - \dfrac{C_3 (T - T_0 )}{C_4 + (T - T_0)}T + qquad \dfrac{2kT}{V}\sinh ^{ - 1}\Bigg \{\dfrac{\dot {\varepsilon }}{\dot {\varepsilon }^\ast_0 \exp \Big[\dfrac{C_1 (T - T_0 )}{C_2 + (T - T_0 )}\ln 10\Big]} \Bigg\}^{\tfrac{1}{n}}$
式 (14) 即为本文提出的屈服应力模型.

3.2 结果与讨论

运用遗传算法对式 (14) 中的 9 个模型参数进行优化拟合时,给定各参数合理的取值范围,且不计屈服点前小应变范围内损伤对 HTPB CBBG 力学性能的影响^[37].

参考温度 $T_{0}$ 取 301 K,将图 7 中其他温度下的屈服应力同时进行水平和垂直移位,直至得到如图 8 所示的屈服应力主曲线,图 8 内图为该过程中各温度对应的移位因子 $\Delta (\lg \dot {\varepsilon } )$, $\Delta \Big (\dfrac{\sigma _{\rm y}}{T}\Big)$,此时,可根据目标函数式 (15),分别利用式 (9) 和式 (10) 拟合 $C_{1}$,$C_{2}$ 和 $C_{3}$, $C_{4}$

(15)$\min \sum_{i = 1}^4 {\vert (y_i^{\rm model} - y_i^{\exp } ) /y_i^{\exp } \vert } $
式中,$y_{i}^{\rm model}$ 为某温度下移位因子的模型预测值,$y_{i}^{\exp}$ 为该温度下移位因子实际值.

图8

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图8301 K 下屈服应力主曲线,内图为水平、垂直移位因子的温度相关性

Fig.8Master curve of yield stress built at 301 K and the insert presents correlations between the horizontal/vertical shift factor and temperature



$T=T_{0}$ 时,式 (14) 简化为

(16)$\sigma _{\rm y} (T_0 ,\dot {\varepsilon }) = a\sinh ^{ - 1}\Big (\dfrac{\dot {\varepsilon }}{\dot{\varepsilon }_0 }\Big)^{\tfrac{1}{m}} + \sigma _0 +\\\ \qquad \dfrac{2kT_0 }{V}\sinh ^{-1}\Big(\dfrac{\dot {\varepsilon }}{\dot {\varepsilon }^\ast _0 }\Big)^{\tfrac{1}{n}} $
此式即为 $T_{0}$ 时屈服应力主曲线的表达式. 根据式 (16) 和图 8 拟合剩余模型参数时,目标函数为

(17) $\min \sum_{j = 1}^{17} {\vert (z_j^{\rm model } - z_j^{\exp } ) / z_j^{\exp }\vert }$
式中,$z_{j}^{\rm model}$ 为某工况下屈服应力的模型预测值,$z_{j}^{\exp}$ 为该工况下屈服应力实际值. 为保证低应变率下模型仍具物理意义并提高优化算法的收敛效率,约束

(18)$ \sigma ^\ast (323) > 0$
式中,$\sigma^{\ast}$(323) 为温度 323 K 时利用式 (12) 计算的屈服应力值.

表 2 为各模型参数的拟合结果,图 8 中阴影区域为移位因子和屈服应力主曲线的拟合结果分布区间. 结合表 2 和图 8 可知,在合适的约束条件和取值范围下,各参数未出现大幅波动,且拟合效果良好,表明所建模型具有较低的参数敏感性和较高的稳定性的同时,准确性较高.

Table 2
表2
表2屈服应力模型参数
Table 2Parameters for proposed yield stress model

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将表 2 所列的一组典型拟合结果代入式 (14),得到各工况下屈服应力预测值,如图 9 所示,可见,与未修正内应力的模型预测值相比,其 有效补充了宽泛应变率下屈服应力的应变率效应. 现利用卡方检验法检验拟合优度,由 $\sum_{j =1}^{17}$ (预测值--实验值) $^2$ /预测值,得检验统计量 ${\chi}^{2} = 0.948 7$,而自由度为 $17 - 1 = 16$、置信度 $\alpha =0.95$ 时查表得 $\chi^{2}_{1 - \alpha} =26.296>\chi^{2}$,即实验值与预测值吻合良好,表明所建模型可以

图9

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图9屈服应力实验值与预测值对比

Fig.9Comparison of experimental and predicted yield stress



准确描述 233$\sim $323 K 时 HTPB CBBG 屈服应力的双线性应变率相关性：应变率较低时,应变率效应主要由内应力 $\sigma^{\ast}$ 贡献,屈服应力随应变率的增大而缓慢增大,应变率较高时,应变率效应主要由驱动应力 $\sigma^{\rm d}$贡献, 屈服应力随应变率的增大而陡升,大分子链及链段来不及通过热激活运动形成充分扩展的新构象,受限严重. 根据图 9 可以推断,在更低应变率和(或)更高温度下,屈服应力将达到渐近线 $\sigma_{\rm y}=0$,此时 HTPB CBBG 屈服现象不明显,大分子链及链段运动更加充分,``软''的力学特性突显.

4 结论

(1) 233$\sim $323 K 时,HTPB CBBG 在不同应变率下均呈现屈服-$\!$-应变硬化的力学特征,屈服应力具有双线性应变率相关性,切线模量约为 11.74$\sim$15.39 MPa.

(2)相同应变下的应力幅值随应变率的增大或温度的降低而增大,但温度对 HTPB CBBG 冲击压缩力学性的影响较应变率更加显著.

(3)所建屈服应力模型将协同模型和时温等效原理相结合,并计及内应力的应变率增强效应,可准确描述 $T>T_{\rm g}$ 时 HTPB CBBG 屈服应力的应变率和温度相关性,明确了较低和较高应变率时,应变率效应分别主要由内应力和驱动力贡献.同时,对内应力的修正提高了模型表征宽泛应变率下屈服应力应变率相关性的能力.

(4)将所建屈服应力模型进行适当外推后可推测,当温度达到 323K、应变率低至 1$\times$10$^{ - 5}$ s$^{-1}$ 时,屈服应力接近 0,屈服点将难以辨别,细观上大分子链及链段运动将更加充分.

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The tensile yield stress of plasticized starch filled with montmorillonite has been studied as a function of the temperature and the strain rate and has been compared to the yield behavior of the original matrix. Aggregated/intercalated and exfoliated nano-biocomposites, obtained from different nanofillers, have been produced and tested under uniaxial tension (tensile test). To model the nanocomposite tensile yield stress behavior, a preexisting micro-mechanically based cooperative model, which describes properly the yield of semi-crystalline polymers has been modified. According to our development, the yield behavior of nano-biocomposites is strongly dependant on the clay concentration and exfoliation ratio. Based on the thermodynamics properties, an effective activation volume and effective activation energy are computed through the Takayanagi homogenization model. The predicted results for the yield stress at low strain rates and at different temperatures are in agreement with our experimental results.

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Yielding behaviour under compressive loading of two materials based on polypropylene, an isotactic homopolymer and an ethylene&ndash;propylene block copolymer, is studied at different strain rates and temperatures. Quasi-static tests, performed in electromechanical machines, and dynamic tests, carried out in a Hopkinson bar, were compared and simultaneously analyzed to generate a master curve representative of the material yielding, assuming the strain rate-temperature superposition principle. Experimental data were fitted to equations based on the cooperative model for semi-crystalline polymers.

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( Deng Xiaoqiu, Li Zhiqiang, Zhou Zhiwei, et al. One-dimensional yield behavior of MDYB-3 polymethyl methacrylate at different strain rate
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This paper investigates molecular-scale polymer mechanical deformation during large-strain squeeze flow of polystyrene (PS) films, where the squeeze flow gap is close to the polymer radius of gyration (R(g)). Stress-strain and creep relations were measured during flat punch indentation from an initial film thickness of 170 nm to a residual film thickness of 10 nm in the PS films, varying molecular weight (M(w)) and deformation stress rate by over 2 orders of magnitude while temperatures ranged from 20 to 125 degrees C. In stress-strain curves exhibiting an elastic-to-plastic yield-like knee, the response was independent of M(w), as expected from bulk theory for glassy polymers. At high temperatures and long times sufficient to extinguish the yield-knee, the mechanical response M(w) degeneracy was broken, but no molecular confinement effects were observed during thinning. Creep measurements in films of 44K M(w) were well-approximated by bulk Newtonian no-slip flow predictions. For extrusions down to a film thickness of 10 nm, the mechanical relaxation in these polymer films scaled with temperature similar to Williams-Landel-Ferry scaling in bulk polymer. Films of 9000K M(w), extruded from an initial film thickness of 2R(g) to a residual film thickness of 0.5R(g), while showing stress-strain viscoelastic response similar to that of films of 900K M(w), suggestive of shear-thinning behavior, could not be matched to a constitutive flow model. In general, loading rate and magnitude influenced subsequent creep extrusion depth of high-M(w) films, with deeper final extrusions for high loading rates than for low loading rates. The measurements suggest that, for high-resolution nanoimprint lithography, mold flash or final residual film thickness can be reduced for high strain and strain rate loading of high-M(w) thin films.

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( Wang Baozhen, Hu Shisheng. Research on dynamic mechanical response and constitutive model of porcine liver
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( Wang Zenghui, Li Xikui. Meso-mechanically informed macroscopic characterization of damage-healing-plasticity for granular materials
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Ultrastructural features, surface morphology and immunologic surface markers were examined on the cells of three human thymomas. The vast majority of the lymphocytes from the thymomas formed spontaneous rosettes with unsensitized sheep erythrocytes in both cell suspension and frozen tissue section and were, therefore, T cells. In addition to the lymphocytes, epithelial cells and macrophages were observed within the thymomas by transmission electron microscopy. When examined by scanning electron microscopy, most lymphocytes had virtually smooth surfaces, whereas cells believed to be epithelial in origin had surface projections.

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Abstract

Amorphous thermoplastic polymers are important engineering materials; however, their non-linear, strongly temperature- and rate-dependent elastic-viscoplastic behavior is still not very well understood, and is modeled by existing constitutive theories with varying degrees of success. There is no generally agreed upon theory to model the large-deformation, thermo-mechanically-coupled, elastic-viscoplastic response of these materials in a temperature range which spans their glass transition temperature. Such a theory is crucial for the development of a numerical capability for the simulation and design of important polymer processing operations, and also for predicting the relationship between processing methods and the subsequent mechanical properties of polymeric products. In this paper we extend our recently published theory [Anand, L., Ames, N. M., Srivastava, V., Chester, S. A., 2009. A thermo-mechanically-coupled theory for large deformations of amorphous polymers. Part I: formulation. International Journal Plasticity 25, 1474–1494; Ames, N. M., Srivastava, V., Chester, S. A., Anand, L., 2009. A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part II: applications. International Journal of Plasticity 25, 1495–1539] to fill this need.
We have conducted large strain compression experiments on three representative amorphous polymeric materials – a cyclo-olefin polymer (Zeonex-690R), polycarbonate (PC), and poly(methyl methacrylate) (PMMA) – in a temperature range from room temperature to approximately 50 °C above the glass transition temperature, ?_g, of each material, in a strain-rate range of ≈10^-4 to , and compressive true strains exceeding 100%. We have specialized our constitutive theory to capture the major features of the thermo-mechanical response of the three materials studied experimentally.
We have numerically implemented our thermo-mechanically-coupled constitutive theory by writing a user material subroutine for a widely used finite element program. In order to validate the predictive capabilities of our theory and its numerical implementation, we have performed the following validation experiments: (i) a plane-strain forging of PC at a temperature below ?_g, and another at a temperature above ?_g; (ii) blow-forming of thin-walled semi-spherical shapes of PC above ?_g; and (iii) microscale hot-embossing of channels in Zeonex and PMMA above ?_g. By comparing the results from this suite of validation experiments of some key features, such as the experimentally-measured deformed shapes and the load-displacement curves, against corresponding results from numerical simulations, we show that our theory is capable of reasonably accurately reproducing the experimental results obtained in the validation experiments.

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