Experiment on dynamic mechanical properties of steel and macro-polypropylene hybrid fibers reinforced recycled aggregate concrete under uniaxial compression
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摘要: 为探究钢-塑钢混杂纤维再生混凝土的压缩动态力学性能,设计了A、B和C 3个系列混杂纤维再生混凝土包含3种再生粗骨料取代率和5种体积分数1.5vol%的钢纤维和塑钢混掺纤维组合,采用了4种加载应变率。试验表明:随应变率增加,混杂纤维再生混凝土峰值应力、弹性模量和压缩韧性增大,峰值应变减小。相较于基准组,在相同应变率下3个系列中的含纤维组峰值应力最大增幅分别为23%、16%和16%;峰值应变最大增幅分别为19%、12%和13%;弹性模量最大增幅分别为15%、14%和35%;压缩韧性最大增幅分别为46%、32%和37%。在试验应变率范围内,再生粗骨料显著提高峰值应力、弹性模量和压缩韧性的应变率敏感性,对峰值应变的应变率敏感性并无明显影响;掺入纤维降低混凝土峰值应变和弹性模量的应变率敏感性,提高普通混凝土峰值应力和压缩韧性的应变率敏感性,降低再生混凝土峰值应力和压缩韧性的应变率敏感性。提出的动态损伤本构模型考虑了纤维增强系数、再生粗骨料取代率和应变率,计算结果与试验结果吻合较好。Abstract: The dynamic compressive behavior of hooked-end steel (HES) and macro-polypropylene (MPP) hybrid fibers reinforced recycled aggregate concrete (HyF/RAC) was studied. Three series (A, B and C) of HyF/RAC specimens were designed, which include three different recycled coarse aggregates (RCA) replacement ratios and five combinations of hybrid fibers at the total volume fraction of 1.5vol%, and four different strain rates were conducted. The results show that with the increase of strain rate, the peak stress, elastic modulus and compressive toughness increase, while the peak strain decreases. Compared with control groups in three series under the same strain rate, the largest increases of peak stress for specimens with fibers are 23%, 16% and 16%, respectively. The largest increases of peak strain are 19%, 12% and 13%, respectively. The largest increases of elastics modulus are 15%, 14% and 35%, respectively. The largest increases of compressive toughness are 46%, 32% and 37%, respectively. In the strain rate range, the strain rate sensitivity of peak stress, elastic modulus and compressive toughness increase with the RCA, and the RCA replacement ratio does not affect the strain rate sensitivity of peak strain. The strain rate sensitivity of peak strain and elastics modulus decrease with the addition of fiber. The addition of fibers enhances the strain rate sensitivity of peak stress and compressive toughness for natural aggregate concrete (NAC). While the strain rate sensitivity of peak stress and compressive toughness decrease for recycled aggregate concrete (RAC). The dynamic constitutive damage model was proposed considering the reinforcing index of fibers, RCA replacement ratio and strain rates. And all the models agree well with the experimental curves.
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超高性能混凝土(Ultra-high-performance con-crete,UHPC)是一种新型水泥基复合材料,具有超高强度、超高韧性、超高耐久性,被广泛应用于桥梁工程、建筑工程、军事防护工程等领域[1–3]。与普通混凝土相比,UHPC与钢筋具有更优异的黏结性能[4-5],从而可以解决实际工程中搭接区段钢筋搭接强度不足的问题,有效缩短钢筋的搭接长度,延缓搭接区裂缝的发展。因此,UHPC被广泛应用于桥面板连接[6-8]、钢筋搭接缺陷加固[9]和预制混凝土构件连接[10]等。
钢筋与混凝土之间的黏结是保证钢筋混凝土构件充分发挥作用的关键,中心拉拔试验、对拉搭接试验和梁式试验是目前研究钢筋与UHPC黏结性能的主要方法。通过中心拉拔试验发现[11-17],钢筋锚固长度、钢纤维掺量、保护层厚度是影响其黏结强度的主要参数。随着黏结长度的增加,钢筋应力分布越不均匀,钢筋与UHPC的平均黏结强度逐渐减小;随着混凝土保护层厚度的增加,钢筋与UHPC的黏结性能显著提高,但钢筋直径以及钢纤维掺量对其的影响尚未有统一的结论。
中心拉拔试验因其操作简便、易于实现而被广泛的应用于黏结试验中,但中心拉拔试验存在“拱效应”,测得的黏结强度往往偏高,因此许多学者采用对拉搭接试验对钢筋与UHPC的黏结性能进行研究。Lagier等[18]通过对拉拔出试验研究了钢纤维掺量对搭接接头黏结强度的影响,研究表明:相比于普通混凝土,钢纤维增强UHPC可以有效缩短钢筋的搭接长度;纤维体积分数为4vol%时,直径为25 mm的钢筋在12d的搭接长度(d为钢筋直径)下即可达到屈服,且劈裂裂纹得到了有效控制。方志等[19]通过39个对拉搭接试验,研究了纵筋搭接长度、搭接钢筋间距、配箍率和活性粉末混凝土(Reactive powder concrete,RPC)强度对RPC与钢筋之间的黏结性能的影响,研究表明:搭接长度是影响其破坏模式的主要因素;随着搭接长度的增加会导致搭接强度稍有降低,而RPC强度、配箍率和钢筋净距的增加会导致搭接强度的提高。马福栋等[20]设计了21组对拉搭接试验和3组直接拔出锚固试验,研究了搭接长度、纤维掺量、配箍率对变形钢筋和UHPC黏结性能的影响,结论与文献[19]基本一致,并且建立了钢筋/UHPC锚固和搭接长度的简化算法。
中心拉拔试验和对拉搭接试验均未考虑实际梁构件中弯曲应力的影响,梁式搭接试验是最接近实际受力的试验方法,课题组前期工作[20]发现,对拉搭接试验与中心拉拔试验得出的黏结强度比值为0.66~0.69,说明中心拉拔试验会高估钢筋与UHPC的黏结强度,而对拉搭接试验未考虑实际梁构件中弯曲应力的影响。为此,有学者通过整浇UHPC梁式搭接试验来研究钢筋与UHPC的搭接黏结性能[21–24],研究表明:UHPC可以有效减小钢筋搭接长度;钢纤维掺量、钢筋搭接长度是影响搭接黏结强度的重要因素。本文通过8根局部后浇UHPC连接的搭接梁和1根局部后浇C80混凝土连接的搭接梁,研究钢筋搭接长度、钢纤维掺量和机械锚固措施对搭接梁中高强钢筋与UHPC黏结性能的影响,并与其他两种试验方法进行对比,研究弯矩作用对受拉钢筋黏结强度的影响,为后浇UHPC搭接梁应用于装配式建筑提供依据。
1. 试验概况
1.1 试件设计与制作
试验共设计了8根UHPC搭接梁和1根C80混凝土搭接梁,试验变参为钢筋搭接长度(3d、8d、12d)、钢纤维掺量(2vol%、3vol%)、机械锚固措施(90°弯钩、锚固板、单面贴焊、两面贴焊)。梁截面尺寸为200 mm×400 mm,梁纵筋采用HRB500级钢筋,直径为20 mm,箍筋及架立筋采用HPB300级钢筋,其直径均为8 mm,为保证搭接梁后浇界面新旧混凝土良好的黏结性能,设置了6 mm的粗糙面。
梁构件的尺寸和配筋图见图1和图2,锚固措施见图3,试件的设计参数及钢筋应力计算结果详见表1,根据文献[17-19, 23-24],本文取搭接长度3d为主变参,保证搭接梁发生黏结破坏。
表 1 梁式搭接试验试件参数设计及钢筋应力计算结果Table 1. Parameter design of beam lap test specimen and results of reinforcement stress calculationNumber Type L Lap form Vf/vol% Peak load/kN fs/MPa Yield or not B1
B2C80
UHPC3d
3dStraight rebar lap
Straight rebar lap2
223.5
95.2—
187Not
NotB3 UHPC 8d Straight rebar lap 2 207.8 449 Not B4 UHPC 12d Straight rebar lap 2 231.1 494 Not B5 UHPC 3d Straight rebar lap 3 145.3 302 Not B6 UHPC 3d Hook treatment 2 297.4 560 Yield B7 UHPC 3d Anchor plate 2 124.1 257 Not B8 UHPC 3d One side weld 2 124.3 257 Not B9 UHPC 3d Two side weld 2 188.0 403 Not Notes: Type—Type of post-cast concrete in lap section; UHPC—Ultra-high-performance concrete; L—Lap length (Lap form-different mechanical anchorage measures); Vf—Fibre volume fraction; fs—Calculated tensile strength of rebar. 试件的制作主要分为两步:(1)先浇筑预制梁部分,采用C40混凝土浇筑而成,用隔板与泡沫胶将搭接区和预制梁分隔开,预留出后浇部分;(2)待普通混凝土部分硬化后,取掉搭接区的隔板,并用高压水枪将泡沫胶清洗干净,将两侧键槽(图4(b))处的普通混凝土面进行凿毛处理,清洗干净后安装侧模,将制备好的UHPC在此区段进行浇筑,待收面抹平之后覆盖保鲜膜防止产生干缩裂缝。为保证其足够的黏结强度,试件搭接区养护28天之后进行加载。
1.2 试验材料性能
试验采用的UHPC主要由水泥、粉煤灰、硅灰、石英砂和水组成。水泥选用P·O 52.5级硅酸盐水泥;粉煤灰为I级粉煤灰;考虑纤维随机分布的影响,采用级配石英砂,粒径范围为0.08~3.25 mm;硅灰选用无定型超细灰白色球状粉末;减水剂采用某公司的聚羧酸高效减水剂;使用端勾型镀铜钢纤维,长度为13 mm,直径是0.22 mm,长径比为60,抗拉强度大于
2850 MPa。测得新拌UHPC的坍落度大于220 mm,扩展度约为560 mm。采用边长为100 mm的立方体试块和100 mm×100 mm×300 mm的棱柱体试块分别测量UHPC 的立方体抗压强度和棱柱体抗压强度。采用“哑铃型”试块测试UHPC的单轴抗拉强度,厚度为130 mm,试件尺寸及加载装置见图5,UHPC的力学性能见表2,钢筋的力学性能见表3。
1.3 加载制度与测点布置
1.3.1 加载制度及装置
本次试验在冠腾自动化技术有限公司制造的YAS-1000型微机控制电液伺服压力试验机上进行,加载方式为三分点静力加载。采用位移控制加载,加载速率为0.2 mm/min,当荷载降至极限荷载的85%时,停止加载。试验加载装置如图6所示。
表 2 超高性能混凝土(UHPC)材料性能Table 2. Material properties of UHPCVf/vol% fcu/MPa fc/MPa ft/MPa 2 123.3 113.2 6.22 3 135.6 122.7 7.01 Notes: fcu—Cubic compressive strength; fc—Prismatic compressive strength; ft—Tensile strength. 表 3 钢筋力学性能Table 3. Mechanical properties of reinforcementStrength grade Diameter/
mmYield strength/
MPaUltimate strength/
MPaHPB300 8 357 529 HRB500 20 560 715 1.3.2 测点布置
为测量梁受力过程中,梁跨中混凝土应变是否满足平截面假定,在梁跨中等间距布置5个混凝土应变片,间距为80 mm,最上端应变片距顶面为40 mm;为了得到搭接梁在加载过程中的挠度变化情况,在跨中、两加载点下方和两侧支座上方处各布置1个位移传感器,共5个,具体布置如图6所示。
在搭接段每根受拉纵筋上均布置应变片,考虑到搭接段长度为3d (60 mm)时,搭接长度过小,因此将应变片布置在预制混凝土梁靠近键槽处,共布置了4个测点,应变片布置示意图如图7所示。
2. 试验结果及分析
2.1 试验现象及破坏模式
2.1.1 后浇C80混凝土(B1)
如图8(a)所示,加载初期试件处于弹性工作阶段,当加载至19 kN时,后浇界面开始出现明显的裂缝,继续加载,后浇界面裂缝变宽,其他未出现明显裂缝;当加载至24 kN时,达到峰值荷载;随着加载位移增大,后浇界面裂缝宽度逐渐增大,钢筋与后浇混凝土的黏结性能不断劣化,梁式试件承载力显著下降,试件后浇区发生黏结破坏。
2.1.2 后浇UHPC (B2~B9)
(1)不同钢筋搭接长度(B2、B3、B4)
如图8(b)所示,试件B2 (3d)加载至41 kN时,梁底后浇接缝处出现第一条裂缝;继续加载,混凝土梁段受拉区亦出现弯曲裂缝,后浇接缝处裂缝向上延伸;加载至95 kN (峰值荷载)时,试件裂缝数量不再增加;继续加载,后浇界面处裂缝变宽,钢筋与UHPC黏结强度退化并发生滑移,受拉纵筋没有屈服,跨中挠度增加,试件破坏。
如图8(c)、图8(d)所示,试件B3 (8d)和B4 (12d)的破坏过程基本相似,均经历了后浇区接缝开裂、混凝土梁段开裂、剪跨区出现斜裂缝、钢筋与UHPC黏结退化和滑移。破坏模式的差异主要体现在混凝土梁段的弯曲裂缝数量,裂缝数量与梁的受弯承载力呈正相关,受UHPC搭接区域提供的总黏结力控制。3个试件中受拉纵筋均未屈服,原因主要是搭接长度的不足导致钢筋的抗拔能力较低,钢筋还未发挥作用可能已经产生了滑移,而由于搭接钢筋间净距过小,使钢筋没有被UHPC充分包裹,导致钢筋与UHPC的有效黏结面积减小,从而未能充分发挥UHPC的高黏结特性,因此内部裂缝还未扩展至UHPC面层表面,试件就已经发生了破坏。
(2)不同钢纤维掺量(B2、B5)
如图8(e)所示,试件B5 (3vol%)加载至54 kN时,右加载点附近处出现第一条裂缝,随后,后浇接缝处才出现竖向裂缝;随着荷载继续增加,表现出与试件B2 (2vol%)相似的破坏特征;加载至145 kN (峰值荷载)时,试件破坏,受拉纵筋没有屈服,在梁端可以观察到一定数量的弯曲裂缝,但与B2试件相比并未明显增多。钢纤维掺量的增加,可以提高后浇界面与梁端普通混凝土的黏结性能,因此第一条裂缝出现在了加载点处附近,相比于钢筋搭接长度,钢纤维掺量对钢筋与UHPC黏结性能的影响有限,试件表面弯曲裂缝数量没有明显增加。
(3)不同锚固措施(B6~B9)
如图8(f)~8(i)所示,试件B6、B7、B8和B9分别采用90°弯钩、锚固板、单面贴焊和双面贴焊4种不同的机械锚固措施处理。
试件B6加载至37 kN时,梁底部后浇界面处首先出现对称竖向裂缝;继续加载,混凝土梁段不断出现新的弯曲裂缝,裂缝逐渐延伸变宽;加载至264 kN时,受拉钢筋屈服,梁段裂缝宽度明显增大;加载至297 kN (峰值荷载)时,试件裂缝不再增加,继续加载,最后受压区混凝土被压碎,试件表现出典型的适筋梁破坏特征。试件B6搭接区所提供的黏结力由平直段、弯钩段和竖直段三部分提供,有效黏结面积明显增大,总黏结力显著高于试件B7、B8和B9,因此受拉纵筋能够发生屈服,表现出明显不同于B7、B8和B9试件的破坏模式。
试件B7、B8、B9表现出相似的破坏过程,与试件B2 (直筋搭接)相比,锚具的设置主要增强了钢筋与UHPC后期的黏结强度储备,只有在试件发生较大变形的情况下才能发挥作用,因此在前期破坏过程上并没有显著的区别,梁段弯曲裂缝的数量也没有明显增加。但锚具会对周围UHPC产生一定的压力作用,在最后破坏时,可以观察到搭接段UHPC面层处有黏结裂缝产生。同时,锚具的设置增强了钢筋与UHPC的黏结强度,因此记录到的钢筋应变更大,但此时受拉纵筋仍没有屈服。试件B7、B8、B9破坏模式的主要差异体现在双面贴焊进一步减小了试件B9混凝土的保护层厚度,因此在最终呈现的破坏模式上,可以在搭接区面层上观察到更多的黏结裂缝。
综上,试件B1的破坏源于后浇区钢筋和混凝土的黏结破坏;除试件B6因受压区混凝土压碎导致破坏以外,其余试件的破坏均由UHPC浇筑区的黏结退化引起。
2.2 荷载-挠度曲线分析
搭接梁的荷载-挠度曲线如图9所示,各个搭接梁的峰值荷载及钢筋屈服情况见表1。
(1)试件B2的峰值荷载是B1的4.05倍,可认为搭接长度为3d时UHPC与钢筋的黏结强度较混凝土提高了3倍;峰值荷载后,C80混凝土搭接区混凝土保护层发生劈裂破坏后残余黏结强度约为峰值黏结强度的40%,而UHPC搭接区表现出更高的残余黏结强度比(约68%),其原因是UHPC中的钢纤维桥连作用延缓了保护层的劈裂。
(2)试件B2、B3和B4的荷载-挠度曲线见图9(b)。加载初期试件处于弹性阶段,曲线基本重合,试件开裂后刚度下降,搭接长度越大黏结强度退化越缓慢,刚度下降越缓慢。由表1可知,与试件B2相比,试件B3和B4的峰值荷载分别提高了118.3%和142.8%,但单位黏结长度提供的黏结力下降,与常规中心拉拔和对拉搭接试验结果吻合。
(3)由图9(c)可见,试件未开裂前均处于弹性阶段,B5试件开裂后刚度并无明显下降。由表1可得,当钢纤维掺量从2vol%增长到3vol%,试件的峰值荷载提高了52.6%。随着钢纤维掺量的增加,提高了UHPC的抗拉强度和抗裂性能,乱向分布的钢纤维可以抑制内部微裂纹的发展,缓冲裂缝端部的应力集中,从而提高了钢筋与UHPC的黏结性能,试件的峰值荷载更高,但明显小于钢筋搭接长度的增加对搭接梁力学性能的影响。
(4)由表1可知,与试件B2相比,采用不同机械锚固措施的试件B6~B9,峰值荷载分别提高了212.4%、97.5%、30.4%、30.6%。荷载-挠度曲线见图9(d),试件B6~B9在开裂前均处于弹性阶段,开裂后的荷载-挠度曲线表现出明显的不同。试件B6在达到峰值荷载前钢筋已经屈服,并且峰值荷载显著高于其他试件,由于焊点的失效,B8、B9试件在达到峰值荷载后,黏结强度出现了明显的退化,荷载突然下降,采用锚固板措施的B7试件,在达到峰值荷载后,仍然有着足够的黏结强度储备,并未出现黏结强度的迅速退化。
2.3 应变分析
2.3.1 跨中混凝土应变分析
图10所示为搭接梁在较小的搭接长度(3d)和不同的机械锚固措施下跨中混凝土的应变变化情况,开裂之前,应变沿截面高度分布基本成线性增长,开裂之后,略有波动,但仍呈现线性增长的趋势,因此搭接梁跨中混凝土应变情况基本符合平截面假定。
2.3.2 搭接段钢筋应变分析
图11所示为搭接梁荷载-跨中钢筋应变曲线,在这里选取了3组峰值荷载较高的试件进行分析,B2为对比试件。可以看到,峰值荷载最高的B6搭接梁,钢筋应变达到了屈服应变,符合前述搭接梁的破坏模式,而峰值荷载低于B6的其余搭接梁,钢筋均未屈服。
2.4 钢筋应力计算
2.4.1 计算假定
本文根据搭接梁达到峰值荷载时轴力与弯矩的平衡条件计算出搭接梁中钢筋的最大拉应力[23, 25],基本假定如下:
(1)搭接梁跨中UHPC正截面平均应变按照线性规律分布,即截面应变符合平截面假定;
(2)受压区UHPC取三角形应力分布,即处于弹性工作阶段;
(3) UHPC与受拉钢筋在受拉区共同承担拉应力,UHPC的应力-应变曲线采用理想弹塑性模型,UHPC初裂以后其拉应力保持不变[25]。
2.4.2 计算过程
根据截面的应力以及应变分布图12,可以得到两个平衡方程:
CUHPC+Cs=TUHPC+Ts (1) M=CUHPC(23xc)+Cs(xc−d′)+TUHPC(m2)+Ts(h0−xc) (2) m=xcεtεu⩽ (3) 式中: {T_{{\text{UHPC}}}} 为受拉区UHPC的合拉力; {T_{\text{s}}} 为受拉钢筋的合力; {C_{{\text{UHPC}}}} 为受压区UHPC的合压力; {C_{\text{s}}} 为受压钢筋的合力; M 为搭接梁所受的外力矩; {x_{\text{c}}} 为梁截面的受压区高度; {d'} 为受压钢筋合力作用点到梁外边缘的距离; {\varepsilon _{\text{t}}} 为UHPC的计算拉应变; {\varepsilon _{\text{u}}} 为极限压缩纤维处UHPC的压应变; h 为搭接梁截面高度; {h_{\text{0}}} 为搭接梁截面有效高度,即受拉钢筋合力作用点到梁外边缘的距离;m为受拉钢筋合力点到中和轴的距离,即UHPC的有效受拉区高度。
图 12 钢筋应力计算示意图Figure 12. Schematic diagram of reinforcement stress calculationh —Beam height; b—Beam width; {h_{\text{0}}} —Effective height of beam; {d'} —Distance from the point of action of the combined forces of the compression reinforcement to the outer edge of the beam; {A_{\text{s}}} —Cross-sectional area of tensile reinforcement; {A_{\text{s}}^{\prime}} —Cross-sectional area of compression reinforcement; {x_{\text{c}}} —Height of compression zone of beam section; m —The depth of the extreme UHPC tensile fiber below the neutral axis; {\varepsilon _{\text{t}}} —Calculated tensile strain of UHPC; {\varepsilon _{\text{u}}} —Compressive strain of UHPC at the extreme compression fiber; {\varepsilon _{\text{s}}} —Actual strain in tensile reinforcement; {\varepsilon _{\text{s}}^{\prime}} —Actual strain in compression reinforcement; {f_{\text{y}}} —Yield stress of tensile reinforcement; {f_{\text{y}}^{\prime}} —Yield stress of compression reinforcement; {f_{\text{t}}} —Measured uniaxial tensile strength of dumbbell specimens; {\sigma _{\text{c}}} —Compressive stress of UHPC at the extreme compression fiber; {T_{\text{s}}} —Combined force of tensile reinforcement; {T_{{\text{UHPC}}}} —Combined force of UHPC in the tension zone; {C_{\text{s}}} —Combined force of compression reinforcement; {C_{{\text{UHPC}}}} —Combined pressure of UHPC in the pressure zone受压区UHPC的合力 {C_{{\text{UHPC}}}} 可按下式进行计算:
{C_{{\text{UHPC}}}} = \frac{1}{2}{x_{\text{c}}}{\varepsilon _{\text{u}}}{E_{\text{c}}}b (4) 式中: b 为搭接梁截面宽度; {E_{\text{c}}} 为UHPC弹性模量,采用文献[26]中所拟合出来的 {f_{{\text{cu}}}} 与 {E_{\text{c}}} 之间的关系式进行计算:
E_{\text{c}}=\frac{10^5}{1.76+73.34/f_{\text{cu}}} (5) 受拉区UHPC的合力 {T_{{\text{UHPC}}}} 可按下式进行计算:
{T_{{\text{UHPC}}}} = {f_{\text{t}}}bm (6) 式中, {f_{\text{t}}} 为哑铃型试件实测出的单轴抗拉强度。
受压钢筋的合力 {C_{\text{s}}} 可用下式进行计算:
C_{\mathrm{s}}=\left\{\begin{array}{l} A_{\mathrm{s}}^{\prime} f_{\mathrm{y}}^{\prime}, f_{\mathrm{s}}^{\prime} \geqslant f_{\mathrm{y}}^{\prime} \\ A_{\mathrm{s}}^{\prime} f_{\mathrm{s}}^{\prime}=A_{\mathrm{s}}^{\prime} \varepsilon_{\mathrm{s}}^{\prime} {E_{\text{s}}^{\prime}}=A_{\mathrm{s}}^{\prime} \varepsilon_{\mathrm{u}}\left(\frac{x_{\mathrm{c}}-d^{\prime}}{x_{\mathrm{c}}}\right) E_{\mathrm{s}}^{\prime}, f_{\mathrm{s}}^{\prime} \leqslant f_{\mathrm{y}}^{\prime} \end{array}\right. (7) 式中: {A_{\text{s}}^{\prime}} 为受压钢筋的截面面积; {f_{\text{y}}^{\prime}} 为受压钢筋的屈服应力; {f_{\text{s}}^{\prime}} 为受压钢筋的实际应力; {\varepsilon _{\text{s}}^{\prime}} 为受压钢筋的实际应变; {E_{\text{s}}^{\prime}} 为受压钢筋的弹性模量,取210 GPa。
受拉钢筋的合力 {T_{\text{s}}} 可用下式进行计算:
{T_{\text{s}}} = \left\{ \begin{gathered} {A_{\text{s}}}{f_{\text{y}}},{f_{\text{s}}} \geqslant {f_{\text{y}}} \\ {A_{\text{s}}}{f_{\text{s}}} = {A_{\text{s}}}{\varepsilon _{\text{s}}}{E_{\text{s}}} = {A_{\text{s}}}{\varepsilon _{\text{u}}}\left( {\frac{{{h_0} - {x_{\text{c}}}}}{{{x_{\text{c}}}}}} \right){E_{\text{s}}},{f_{\text{s}}} \leqslant {f_{\text{y}}} \\ \end{gathered} \right. (8) 式中: {A_{\text{s}}} 为受拉钢筋截面面积; {f_{\text{y}}} 为受拉钢筋的屈服应力; {f_{\text{s}}} 为受拉钢筋的实际应力; {\varepsilon _{\text{s}}} 为受拉钢筋的实际应变; {E_{\text{s}}} 为受拉钢筋的弹性模量,取200 GPa。
2.4.3 计算结果分析
表1为搭接梁达到峰值荷载时的钢筋应力计算结果,从表1和图9可以看出,仅有B6试件钢筋发生了屈服,表明在3d的搭接长度下,同时采用90°弯钩这一机械锚固措施,可以充分发挥出高强钢筋的作用。
3. 黏结性能分析及对比
3.1 黏结强度计算
搭接梁中钢筋与UHPC的平均黏结强度 {\tau _{\text{u}}} 可以用搭接梁达到峰值荷载时的钢筋应力计算[9, 23]:
{\tau _{\text{u}}} = \frac{{{A_{\text{b}}}{f_{\text{s}}}}}{{{\text{π}} {d_{\text{b}}}{l_{{\text{sp}}}}}} (9) 式中: {A_{\text{b}}} 为钢筋的横截面积; {d_{\text{b}}} 为受拉钢筋的直径; {f_{\text{s}}} 为峰值荷载时钢筋的最大拉应力; {l_{{\text{sp}}}} 为钢筋的搭接长度。
课题组前期进行了5组中心拉拔试验和9组对拉搭接试验研究[27],钢筋与UHPC的黏结强度采用下式进行计算:
{\tau _{\text{u}}} = \frac{F}{{{\text{π}} {d_{\text{b}}}{l_{{\text{sp}}}}}} (10) 式中: {\tau _{\text{u}}} 为对拉搭接试验的黏结强度; F 为试验的峰值荷载。
由表4可得,在混凝土保护层厚度和钢筋直径相同的情况下,对拉搭接试验得出的黏结强度与中心拉拔试验比值平均值为0.69;梁式搭接试验与对拉搭接试验得出的黏结强度比值平均值为0.84。
表 4 UHPC-高强钢筋梁式试验对比Table 4. Test comparison of UHPC-high strength rebar beam lap testNumber Type L Vf/vol% Lap form Center pull-out test Brace lap test Beam lap test τu2/τu1 τu3/τu2 τu1/MPa Failure mode τu2/MPa Failure mode τu3/MPa Failure mode B1
B2C80
UHPC3d
3d0
2Straight rebar lap
Straight rebar lap19.2
35.6SPF
SPF12.1
23.1SPF — BOF 0.63 — SPF 15.6 BOF 0.65 0.68 B3 UHPC 8d 2 Straight rebar lap 20.8 SPF 16.2 SPF 14.0 BOF 0.78 0.86 B4 UHPC 12d 2 Straight rebar lap 14.4 RF 12.3 RF 10.3 BOF 0.85 0.84 B5 UHPC 3d 3 Straight rebar lap 49.7 SPF 27.6 SPF 25.2 BOF 0.56 0.91 B6 UHPC 3d 2 Hook treatment — — 50.7 RF 46.7 BEF — 0.92 B7 UHPC 3d 2 Anchor plate — — 25.2 SPF 21.4 BOF — 0.85 B8 UHPC 3d 2 One side weld — — 27.1 SPF 21.4 BOF — 0.79 B9 UHPC 3d 2 Two side weld — — 28.4 SPF 33.6 BOF — 1.18 Notes: All the above specimens are made of HRB500 grade rebar, diameter is 20 mm, the concrete protective layer is 1.5d; τu1—Bond strength obtained by center poll-out test; τu2—Bond strength obtained by brace lap test; τu3—Bond strength obtained by beam lap test; SPF and RF represent the splitting pull-out failure and steel bar rupture failure respectively; BOF and BEF represent the bonding failure of steel bars and the bending failure of lap beams respectively. 中心拉拔试验无法准确模拟钢筋搭接时真实的受力状况,在钢筋拉拔过程中UHPC受压,受UHPC内部“拱作用”产生挤压力的影响,使钢筋拉拔过程中的机械咬合力和摩阻力增大,同时加载端UHPC还受到加载装置中钢板的端部约束,故中心拉拔试件所得到的黏结强度高于对拉搭接试验;课题组前期[27]进行的4根钢筋对拉搭接试验,消除了中心拉拔试验中拱作用和端部约束作用的影响,使UHPC在钢筋拉拔过程中受到拉应力的作用,是一种改进的试验方法,但以上两种试验方法均无法反映出实际构件中弯曲应力对钢筋与UHPC黏结强度的削弱作用,因此得出的黏结强度仍高于贴近实际受力情况的梁式搭接试验。
对于两面贴焊试件,出现了梁式搭接试验得出的黏结强度高于对拉搭接试验,可能是由于对拉搭接试验加载过程中,钢筋出现偏心的影响,从而削弱了钢筋与UHPC的黏结强度,受限于试验数据较少,无法得出具体的原因,因此取平均值时排除B9试件,后续通过进一步试验分析其原因。
3.2 不同试验方法下各因素对黏结强度的影响
3.2.1 钢筋搭接长度(B2、B3、B4)
由图13可见,随着黏结长度增大,3种试验方法得到的黏结强度差异越小,原因主要是黏结长度越大,中心拉拔试件受“压力拱”效应影响越小,荷载偏心和界面弯曲应力的不利影响也变小;3种试验方法得到的黏结强度均随黏结长度的增大而减小,由于随着搭接长度的增加,黏结应力的分布越不均匀,采用直筋搭接的方式,越靠近搭接钢筋末端,黏结应力越小,搭接长度的增加导致黏结应力较小的区域增加,从而钢筋的平均黏结强度相应降低。
3.2.2 钢纤维掺量(B2、B5)
由图14可以看出,随着钢纤维掺量的增大,3种试验方法下得到的黏结强度均随之增大,这是由于随着钢纤维掺量的增加,提高了UHPC的抗拉强度和抗裂性能,乱向分布的钢纤维可以抑制内部微裂纹的发展,缓冲裂缝端部的应力集中;但随着钢纤维掺量的增大,中心拉拔试验与其他种试验方法所得到的黏结强度差异较大,这是因为单根钢筋的拔出试验中,钢筋被UHPC充分包裹,所以这种作用效应更能充分发挥,而搭接钢筋则由于钢筋间净距较小,提升并不明显。
3.2.3 机械锚固措施(B2、B6、B7、B8、B9)
设置了机械锚固措施以后,钢筋与UHPC的黏结强度主要由两部分组成,即直锚段和锚头段两部分,因此在相同的搭接长度下,采取机械锚固措施处理后的的试件,钢筋与UHPC具有更高的黏结强度。由表4和图15可以看出,对拉搭接试验和梁式搭接试验具有相同的规律性,以梁式搭接试验为例,采用90°弯钩处理后的试件B6黏结强度提升最为明显,提升了199.4%,原因是采用弯钩处理后的试件,弯钩处的UHPC受到局部挤压作用,得益于UHPC较高的抗压强度,钢筋与UHPC的黏结强度显著提高。
采用锚固板、单面贴焊、两面贴焊处理后的试件B7、B8和B9,黏结强度提高幅度均小于90°弯钩处理后的试件,分别提高了37.2%、115.4%。从对拉搭接试验中试件的破坏模式来看,在3d的搭接长度下,上述3个试件均发生劈裂破坏,劈裂裂缝的产生削弱了钢筋与UHPC的黏结强度,发生劈裂破坏的原因一方面是由于锚板和短筋的存在会减小混凝土保护层厚度,另一方面是由于钢筋搭接长度过小,锚板和短钢筋与周围UHPC的局部挤压作用在受力前期就十分明显,虽然试件B6也会出现局部挤压作用,但弯钩处钩形面面积较大,并不会在前期就产生显著的应力集中现象,导致前期黏结刚度不足, 而发生劈裂破坏。在梁式搭接试验中,B6试件并未在搭接区段观察到劈裂裂缝,但B7、B8和B9试件均在搭接区段观察到了少量的劈裂裂缝。
3.3 高强钢筋与UHPC临界搭接长度计算
结合式(9)和对拉搭接试验拟合得出的钢筋搭接黏结强度计算公式(11)[27],并考虑弯矩对黏结强度的影响,可以得到梁式搭接试验中高强钢筋与UHPC的临界搭接长度计算公式(13),计算结果见表5。
表 5 UHPC-高强钢筋搭接长度计算Table 5. Lap length calculation of UHPC-high strength rebarLap
lengthVf/vol% Mechanical anchoring measures 3 Straight
rebar lapHook treatment Anchor plate One side weld lsy 9.8d 11.6d 6.7d 9.4d 8.6d lsu 13.7d 16.0d 10.9d 13.6d 12.6d Notes: lsy—Minimum lap length of steel bar yield; lsu—Minimum lap length of steel bar rupture. \tau_{\mathrm{ul}}=\left\{\begin{array}{l} \left(0.39+1.69 \dfrac{d}{l_{\mathrm{s}}}\right)\left(3.22+0.72 \dfrac{c}{d}\right) f_{\mathrm{t}} \\ \phi\left(0.39+1.69 \dfrac{d}{l_{\mathrm{s}}}\right)\left(3.22+0.72 \dfrac{c}{d}\right) f_{\mathrm{t}}+\psi \dfrac{f_{\mathrm{t}} d}{{\text{π}} l_{\mathrm{s}}} \end{array}\right. (11) {\tau _{\text{u}}} = k{\tau _{{\text{ul}}}} (12) 式中: {\tau _{\text{u}}} 为梁式搭接试验的黏结强度;ls为钢筋搭接长度; {\tau _{{\text{ul}}}} 为对拉搭接试验的黏结强度;c为UHPC保护层厚度; k=\tau_{\text{u3}}/\tau_{\text{u2}} ,取平均值0.84; \phi 为带端部锚固措施钢筋的黏结应力折减系数,带弯钩和锚固板的 \phi 为1.05、贴焊短筋的 \phi 取为1.10;锚固板系数 \psi 取8,弯钩系数 \psi 取23、贴焊短筋 \psi 取9。
{\dfrac{l_{\text{s}}}{d}=\left\{\begin{gathered}\dfrac{f}{4kf_{\text{t}}\left(1.26+0.28c/d\right)}-4.33 \\ \dfrac{f}{4kf_{\text{t}}\phi\left(1.26+0.28c/d\right)}-\dfrac{\psi}{\text{π}\phi\left(1.26+0.28c/d\right)}-4.33 \\ \end{gathered}\right.} (13) 式中: {l_{\text{s}}}/d 为相对搭接长度, f 取钢筋的屈服强度 {f_{\text{y}}} ,可以得到临界屈服搭接长度 {l_{{\text{sy}}}} ; f 取钢筋的极限强度 {f_{\text{u}}} ,可以得到临界极限搭接长度 {l_{{\text{su}}}} 。
由式(13)可得,本文采用HRB500级钢筋所计算出直筋的临界和极限搭接长度分别为11.6d和16d,详细计算结果见表5。
4. 结 论
通过9根梁式搭接试验,研究了钢筋搭接长度、钢纤维掺量、机械锚固措施对搭接梁中高强钢筋与超高性能混凝土(UHPC)黏结性能的影响,所得到的结论如下:
(1)与搭接区后浇C80混凝土相比,高强钢筋与UHPC具有更优异的黏结性能,可以更好发挥出高强钢筋的性能,提升搭接梁的受力性能;
(2)随着钢筋搭接长度的增加,搭接梁的峰值荷载逐渐提高,但搭接长度越长,钢筋与UHPC的黏结应力分布越不均匀,平均黏结强度逐渐降低;
(3)钢纤维掺量从2vol%增加到3vol%,UHPC的增韧、阻裂效果增强,搭接梁的黏结强度以及峰值荷载相应提高;
(4)采用机械锚固措施处理后的搭接梁具有更高的黏结强度和峰值荷载,其中采用90°弯钩处理的搭接梁,峰值荷载和黏结强度最高,最后破坏时钢筋屈服,因此在3d的搭接长度下(d为钢筋直径),同时采用钢筋弯钩处理可以充分发挥高强钢筋的作用,以用于实际工程中;
(5)根据搭接梁在峰值荷载下轴力与弯矩的平衡条件,计算出受拉钢筋的最大应力,进一步得出钢筋与UHPC的平均黏结强度,并与中心拉拔和对拉搭接试验结果进行对比,为梁式搭接试件中钢筋与UHPC的黏结强度计算提供准确的理论依据。
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表 1 端钩钢纤维(HES)和塑钢纤维(MPP)属性
Table 1 Properties of hooked-end steel fiber (HES) and macro-polypropylene fiber (MPP)
Fiber type Equivalent
diameter/mmLength/mm Aspect
ratioDensity/(kg·m−3) Tensile strength/MPa Young’s modulus/GPa HES 0.75 35 47 7800 1120 200.0 MPP 0.94 28 30 910 580 5.5 表 2 钢-塑钢混杂纤维再生混凝土(HyF/RAC)配合比设计
Table 2 Designed mix proportions of hybrid fiber reinforced recycled aggregate concrete (HyF/RAC)
kg/m3 Mix Notation Water Cement Sand RCA NCA HES MPP AW A0 NAC 251 467 639 0 1044 0 0.00 0 A1 1.5%HES/NAC 251 467 639 0 1044 117 0.00 0 A2 1.25%HES-0.25%MPP/NAC 251 467 639 0 1044 98 2.28 0 A3 1.0%HES-0.5%MPP/NAC 251 467 639 0 1044 78 4.55 0 A4 0.75%HES-0.75%MPP/NAC 251 467 639 0 1044 59 6.83 0 A5 0.5%HES-1.0%MPP/NAC 251 467 639 0 1044 39 9.10 0 B0 RAC(50%) 251 467 639 522 522 0 0.00 20 B1 1.5%HES/RAC(50%) 251 467 639 522 522 117 0.00 20 B2 1.25%HES-0.25%MPP/RAC(50%) 251 467 639 522 522 98 2.28 20 B3 1.0%HES-0.5%MPP/RAC(50%) 251 467 639 522 522 78 4.55 20 B4 0.75%HES-0.75%MPP/RAC(50%) 251 467 639 522 522 59 6.83 20 B5 0.5%HES-1.0%MPP/RAC(50%) 251 467 639 522 522 39 9.10 20 C0 RAC(100%) 251 467 639 1044 0 0 0.00 40 C1 1.5%HES/RAC(100%) 251 467 639 1044 0 117 0.00 40 C2 1.25%HES-0.25%MPP/RAC(100%) 251 467 639 1044 0 98 2.28 40 C3 1.0%HES-0.5%MPP/RAC(100%) 251 467 639 1044 0 78 4.55 40 C4 0.75%HES-0.75%MPP/RAC(100%) 251 467 639 1044 0 59 6.83 40 C5 0.5%HES-1.0%MPP/RAC(100%) 251 467 639 1044 0 39 9.10 40 Notes: NAC—Natural aggregate concrete; RAC—Recycled aggregate concrete; RCA—Recycled coarse aggregate; NCA—Natural coarse aggregate; AW—Absorbed water; In iHES-jMPP/RAC(k), i—Volume fraction of HES, j—Volume fraction of MPP, k—Replacement ratio by mass of RCA. 表 3 不同应变率下HyF/RAC峰值应力
{\sigma ^{\text{p}}} Table 3 Results of peak stress
{\sigma ^{\text{p}}} of HyF/RAC at different strain ratesMix 10−5 s−1 10−4 s−1 10−3 s−1 5×10−3 s−1 {\sigma ^{\text{p}}} /MPa CV/% {\sigma ^{\text{p}}} /MPa CV/% {\sigma ^{\text{p}}} /MPa CV/% {\sigma ^{\text{p}}} /MPa CV/% A0 31.3 1.58 32.4 2.84 33.7 3.15 34.7 0.00 A1 32.4 2.53 34.0 2.72 38.8 4.01 39.3 1.03 A2 32.9 4.31 36.3 0.98 39.1 1.27 40.1 5.14 A3 34.9 0.17 39.0 2.90 41.3 0.34 41.9 0.84 A4 31.5 3.59 32.8 5.40 35.2 4.22 36.9 4.41 A5 32.7 5.19 34.4 3.29 36.6 2.61 37.3 5.13 B0 28.3 1.00 29.1 3.87 32.4 1.46 34.0 2.29 B1 29.6 2.15 30.8 1.15 34.6 4.71 35.7 4.56 B2 31.9 1.30 33.6 5.27 34.2 3.93 36.1 3.73 B3 31.2 0.23 32.4 1.97 35.7 1.98 37.2 6.84 B4 30.4 1.51 32.3 1.99 35.3 3.13 36.1 0.78 B5 31.6 1.59 33.7 1.70 37.3 0.57 38.3 2.22 C0 24.6 8.57 26.9 2.74 30.6 4.48 32.5 0.22 C1 26.2 4.04 27.8 3.89 31.6 1.14 32.4 3.45 C2 26.8 2.24 28.8 0.72 32.2 2.64 32.8 1.72 C3 28.4 2.69 31.1 0.68 33.0 2.69 34.4 3.63 C4 27.8 0.95 30.0 6.21 32.7 2.60 33.1 5.32 C5 26.3 0.79 27.3 2.23 30.6 5.46 32.7 0.43 Note: CV—Coefficients of variation. 表 4 不同应变率下HyF/RAC峰值应变
{\varepsilon ^{\text{p}}} Table 4 Results of peak strain
{\varepsilon ^{\text{p}}} of HyF/RAC at different strain ratesMix 10−5 s−1 10−4 s−1 10−3 s−1 5×10−3 s−1 {\varepsilon ^{\text{p}}} /10−3 CV/% {\varepsilon ^{\text{p}}} /10−3 CV/% {\varepsilon ^{\text{p}}} /10−3 CV/% {\varepsilon ^{\text{p}}} /10−3 CV/% A0 2.376 4.73 2.330 8.37 2.224 1.67 2.101 9.02 A1 2.410 2.87 2.340 7.71 2.256 10.77 2.199 11.43 A2 2.413 1.55 2.390 0.74 2.378 11.52 2.321 6.02 A3 2.599 8.68 2.555 3.18 2.528 3.98 2.500 3.42 A4 2.419 2.70 2.425 6.95 2.413 10.11 2.312 11.72 A5 2.443 6.27 2.432 8.62 2.330 6.22 2.254 1.68 B0 2.593 3.67 2.469 0.79 2.370 3.18 2.325 6.08 B1 2.651 9.27 2.631 6.65 2.620 6.75 2.510 6.20 B2 2.673 3.61 2.645 9.62 2.592 5.28 2.481 9.76 B3 2.760 5.68 2.708 7.48 2.631 7.32 2.610 5.55 B4 2.583 0.39 2.544 2.43 2.544 7.28 2.533 8.45 B5 2.666 2.41 2.658 6.79 2.630 4.84 2.591 2.11 C0 2.814 4.46 2.726 3.44 2.621 2.51 2.524 2.73 C1 2.933 1.81 2.815 11.43 2.708 5.28 2.694 0.33 C2 2.950 1.47 2.813 1.33 2.769 2.75 2.733 2.20 C3 2.941 1.62 2.853 0.99 2.845 6.83 2.648 3.47 C4 3.041 3.67 2.996 4.01 2.891 0.79 2.861 9.33 C5 2.981 3.50 2.859 1.67 2.754 7.66 2.701 0.46 表 5 不同应变率下HyF/RAC弹性模量E
Table 5 Results of elastic modulus E of HyF/RAC at different strain rates
Mix 10−5 s−1 10−4 s−1 10−3 s−1 5×10−3 s−1 E/GPa CV/% E/GPa CV/% E/GPa CV/% E/GPa CV/% A0 29.5 3.86 33.2 9.20 37.6 9.87 41.2 2.37 A1 31.8 8.13 35.3 1.34 38.6 3.95 42.0 9.60 A2 32.1 7.05 34.3 1.09 38.3 4.35 41.4 4.92 A3 34.1 7.30 34.1 2.88 34.7 0.61 38.1 10.65 A4 32.5 7.19 33.7 5.58 35.7 4.14 37.4 11.49 A5 33.9 4.75 34.0 5.06 35.0 3.23 38.2 2.85 B0 24.8 7.61 26.2 8.55 28.6 3.24 30.8 2.69 B1 25.6 9.08 26.2 0.91 28.6 5.58 30.8 9.23 B2 27.1 0.22 28.1 11.70 29.4 8.73 33.9 8.52 B3 27.3 0.70 28.7 9.85 31.0 6.90 31.7 1.92 B4 28.2 1.65 29.9 3.60 33.2 0.66 33.6 6.38 B5 27.2 4.70 27.8 11.33 28.9 9.67 32.9 11.19 C0 18.7 3.06 21.9 11.63 24.9 6.18 26.7 9.24 C1 20.7 8.05 22.8 11.44 26.3 11.54 27.8 7.55 C2 23.2 7.56 24.3 2.31 26.1 1.54 29.5 10.99 C3 25.1 6.06 26.3 4.08 28.0 11.09 28.1 2.47 C4 22.9 0.56 23.3 2.43 25.2 4.69 28.3 9.75 C5 22.7 0.56 24.4 9.26 27.9 3.17 29.2 9.11 表 6 不同应变率下HyF/RAC压缩韧性T
Table 6 Results of compressive toughness T of HyF/RAC at different strain rates
Mix 10−5 s−1 10−4 s−1 10−3 s−1 5×10−3 s−1 T/MPa T/MPa T/MPa T/MPa A0 0.042 0.044 0.046 0.043 A1 0.047 0.049 0.051 0.050 A2 0.047 0.051 0.053 0.047 A3 0.055 0.058 0.060 0.062 A4 0.047 0.048 0.052 0.053 A5 0.047 0.049 0.047 0.051 B0 0.041 0.043 0.043 0.045 B1 0.046 0.048 0.050 0.052 B2 0.049 0.051 0.053 0.056 B3 0.051 0.051 0.055 0.057 B4 0.048 0.050 0.054 0.057 B5 0.050 0.051 0.057 0.058 C0 0.038 0.040 0.043 0.045 C1 0.043 0.046 0.050 0.053 C2 0.048 0.049 0.051 0.057 C3 0.052 0.053 0.055 0.052 C4 0.051 0.054 0.054 0.057 C5 0.047 0.047 0.047 0.052 表 7 现有压缩动态本构模型
Table 7 Review of dynamic constitutive models under compression
Reference Model Parameter Application range Ibrahim et al[21] \begin{gathered} y = \frac{{Ax + \left( {B - 1} \right){x^2}}}{{1 + \left( {A - 2} \right)x + B{x^2}}} \\ A\left( {A - 2} \right) - B + 1 \geqslant 0 \\ A + B > 1 \\ \end{gathered} \begin{gathered} A = 3.6\exp \left( {9.0 \times {{10}^{ - 8}}\left( {\frac{{\dot \varepsilon }}{{{{\dot \varepsilon }_{\text{s}}}}}} \right)\left( {1 + 0.01R_V^{0.82}} \right)} \right) \\ B = 0.22\exp \left( {3.8 \times {{10}^{ - 7}}\left( {\frac{{\dot \varepsilon }}{{{{\dot \varepsilon }_{\text{s}}}}}} \right)\left( {1 + 0.002R_V^{0.82}} \right)} \right) \\ \end{gathered} \begin{gathered} 25{\text{ }}{{\text{s}}^{ - 1}} \leqslant \dot \varepsilon \leqslant 125{\text{ }}{{\text{s}}^{ - 1}} \\ 1.2{\text{% }} \leqslant V \leqslant 1.4\% \\ \end{gathered} Zhou et al[22] \begin{gathered} \sigma = {\sigma _{\text{m}}}\left( {1 - D} \right) = E\varepsilon \left( {1 - D} \right) \\ D = 1 - \exp \left[ { - {{\left( {\frac{\varepsilon }{{{F_0}}}} \right)}^m}} \right] \\ \end{gathered} \begin{gathered} {F_0} = 3.3578\ln \dot \varepsilon - 10.6562 \\ m = 1.2804\ln \dot \varepsilon - 1.8711 \\ \end{gathered} 19.8{\text{ }}{{\text{s}}^{ - 1}} \leqslant \dot \varepsilon \leqslant 281{\text{ }}{{\text{s}}^{ - 1}} Hou et al[23] \begin{gathered} \sigma = {\sigma _{\text{m}}}\left( {1 - {C_{\text{n}}}D} \right) = E\varepsilon \left( {1 - {C_{\text{n}}}D} \right) \\ D = 1 - \exp \left[ { - {{\left( {\frac{\varepsilon }{{{F_0}}}} \right)}^m}} \right] \\ \end{gathered} \begin{gathered} {F_0} = 0.0062 + 0.31{V^{1.7}} - 0.00167\left( {0.1 + V} \right)\ln \dot \varepsilon \\ m = - 0.56 + 3V + \left( {0.35 - 2V} \right)\ln \dot \varepsilon \\ {C_{\text{n}}} = 0.977 - 1.4V + \left( {0.004 + 0.25V} \right)\ln \dot \varepsilon \\ \end{gathered} \begin{gathered} \dot \varepsilon \leqslant 294{\text{ }}{{\text{s}}^{ - 1}} \\ V \leqslant 5\% \\ \end{gathered} Sun et al[25] \begin{gathered} \sigma = {\sigma _{\text{m}}}\left( {1 - D} \right) = E\varepsilon \left( {1 - D} \right) \\ \dot D = AD\left( {1 - D} \right) \\ \end{gathered} \begin{gathered} {D} = 0.35 - 8.2 \times {10^{ - 9} }\left( {\frac{ {\dot \varepsilon } }{ { { {\dot \varepsilon }_{\text{s} } } } } } \right) - \\ \qquad 4.4V + 1.3 \times {10^{ - 7} }\left( {\frac{ {\dot \varepsilon } }{ { { {\dot \varepsilon }_{\text{s} } } } } } \right)V \\ A = 234 + 1.3 \times {10^{ - 7} }\left( {\frac{ {\dot \varepsilon } }{ { { {\dot \varepsilon }_{\text{s} } } } } } \right) + \\ \qquad 1718V - 8.6 \times {10^{ - 5} }\left( {\frac{ {\dot \varepsilon } }{ { { {\dot \varepsilon }_{\text{s} } } } } } \right)V \\ \end{gathered} \begin{gathered} 53{\text{ }}{{\text{s}}^{ - 1}} \leqslant \dot \varepsilon \leqslant 152{{\text{s}}^{ - 1}} \\ V \leqslant 6.0\% \\ \end{gathered} Zhang et al[26] \begin{gathered} \sigma = \left( {1 - D} \right){\sigma _{\text{m}}} \\ {\sigma _{\text{m}}} = {\sigma _{\text{e}}}\left( \varepsilon \right) + {E_1}\int_0^t {\dot \varepsilon \exp \left( { - \frac{{t - \tau }}{{{\varphi _1}}}} \right){\rm{d}}\tau } + \\ \qquad {E_2}\int_0^t {\dot \varepsilon \exp \left( { - \frac{{t - \tau }}{{{\varphi _2}}}} \right){\rm{d}}\tau } \\ {\sigma _{\text{e}}}\left( \varepsilon \right) = E\varepsilon + \alpha {\varepsilon ^2} + \beta {\varepsilon ^3} \\ D = \left\{ \begin{gathered} 0{\text{ }}\qquad(\varepsilon \leqslant {\varepsilon ^{{\text{th}}}}) \\ 1 - \exp \left[ { - {{\left( {\frac{{\varepsilon - {\varepsilon ^{{\text{th}}}}}}{{{F_0}}}} \right)}^m}} \right]{\text{ }}\qquad(\varepsilon > {\varepsilon ^{{\text{th}}}}) \\ \end{gathered} \right. \\ \end{gathered} No fitting equation \begin{gathered} 27{\text{ }}{{\text{s}}^{ - 1}} \leqslant \dot \varepsilon \leqslant 94{\text{ }}{{\text{s}}^{ - 1}} \\ V \leqslant 0.2\% \\ \end{gathered} Wang et al[27] \begin{gathered} \sigma = \left( {1 - D} \right)\sigma \\ {\sigma _{\text{m}}} = {\sigma _{\text{e}}}\left( \varepsilon \right) + {E_1}\int_0^t {\dot \varepsilon \exp \left( { - \frac{{t - \tau }}{{{\varphi _1}}}} \right){\rm{d}}\tau } + \\ \qquad {E_2}\int_0^t {\dot \varepsilon \exp \left( { - \frac{{t - \tau }}{{{\varphi _2}}}} \right){\rm{d}}\tau } \\ D = \left\{ \begin{gathered} 0{\text{ }}\qquad(\varepsilon \leqslant {\varepsilon ^{{\text{th}}}} ) \\ {K_D}{{\dot \varepsilon }^{\lambda - 1}}{\left( {\varepsilon - {\varepsilon ^{{\text{th}}}}} \right)^\kappa }{\text{ }}\qquad(\varepsilon > {\varepsilon ^{{\text{th}}}} ) \\ \end{gathered} \right. \\ \end{gathered} No fitting equation {10^{ - 4}}{\text{ }}{{\text{s}}^{ - 1}} \leqslant \dot \varepsilon \leqslant {10^3}{\text{ }}{{\text{s}}^{ - 1}} Notes: y = {\sigma \mathord{\left/ {\vphantom {\sigma { {\sigma ^{\text{p} } } } } } \right. } { {\sigma ^{\text{p} } } }}, x = {\varepsilon \mathord{\left/ {\vphantom {\varepsilon { {\varepsilon ^{\text{p} } } } } } \right. } { {\varepsilon ^{\text{p} } } }}, \sigma —Stress, {\sigma ^{\text{p}}} —Peak stress, \varepsilon —Strain, {\varepsilon ^{\text{p}}} —Peak strain; RV—Reinforcing index of hybrid fibers; D—Damage parameter; t—Loading time; τ—Delay time of stress; E1, {\varphi _1} —Elastic modulus and relaxation time at a low strain rate and frequency; E2, {\varphi _2} —Elastic modulus and relaxation time at a high strain rate and frequency; α, β—Elastic constant; {K_D},{\text{ }}\lambda ,{\text{ }}\kappa —Material parameter; {\varepsilon ^{{\text{th}}}} —Strain threshold. 表 8 HyF/RAC修正模型参数
\alpha 拟合值Table 8 Fitted results of parameter
\alpha in modified constitutive models for HyF/RACMix \dot \varepsilon 10−5 s−1 10−4 s−1 10−3 s−1 5×10−3 s−1 Fitted value R2 Fitted value R2 Fitted value R2 Fitted value R2 A0 3.434 0.991 3.399 0.946 2.597 0.953 2.285 0.940 A1 1.882 0.978 1.175 0.998 1.699 0.986 1.352 0.973 A2 1.386 0.999 1.401 0.997 1.020 0.981 1.039 0.994 A3 1.442 0.997 1.912 0.992 1.607 0.965 1.650 0.974 A4 2.219 0.994 2.431 0.970 2.057 0.965 1.466 0.984 A5 2.600 0.998 2.633 0.984 2.288 0.971 1.757 0.992 B0 3.524 0.991 3.742 0.994 2.589 0.942 3.053 0.997 B1 1.440 0.976 1.125 0.983 1.225 0.997 1.873 0.996 B2 1.932 0.985 2.124 0.991 2.001 0.996 1.583 0.979 B3 2.026 0.994 1.636 0.999 2.274 0.988 2.297 0.984 B4 1.876 0.990 2.683 0.992 2.305 0.994 2.613 0.935 B5 2.457 0.994 3.036 0.997 2.873 0.940 2.496 0.984 C0 3.173 0.978 2.958 0.978 3.609 0.986 4.137 0.954 C1 1.552 0.993 1.657 0.976 2.280 0.976 2.749 0.993 C2 1.673 0.975 2.013 0.999 2.259 0.999 2.364 0.953 C3 2.059 0.998 2.149 0.999 2.278 0.998 2.732 0.996 C4 1.868 0.976 2.174 0.997 3.339 0.968 2.842 0.934 C5 2.218 0.971 2.859 0.990 3.483 0.964 3.394 0.993 -
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