State-of-the-art: computational model for soil-interface-structure system
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摘要: 土与结构的相互作用是水利和土木工程的一个关键共性问题,其分析是揭示工程中结构损伤破坏机理的重要环节。将土-界面-结构作为一个体系进行整体分析是精细研究土-结构相互作用的基础,但在大型工程应用时面临着一系列难题,包括：土-结构接触边界约束,土-结构接触力学特性,以及土-界面-结构体系建模和分析等。本文对这些问题的相关研究进行了综述和总结,并对今后的主要发展方向提出了建议。Abstract: The interaction between soil and structure is a common problem in geotechnical engineering, which plays a key role in assessing the damage behavior of structure. The computational model for soil-interface-structure system is the foundation of the detailed study of the interaction between soil and structure, but it has to solve a series of problems, including soil-structure contact constraint and contact judgment, soil-structure mechanical behavior and constitutive model, and grid model for soil-interface-structure system, etc. The progress of the computational model for soil-interface-structure system is summarized, and a suggestion of its main developing trend is put forward.
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0. 引言
在岩土工程模型试验中,土压力测量是一个重要内容,目前土压力的获取多用微型土压力盒[1-2],虽然尺寸相比早期产品可以做到很小（28 mm直径×10 mm厚）,但其自身局限性使其在小型模型试验中存在不适用的问题,例如测量曲率较大的弯曲表面土压力等。
土压力盒（SPC）的误差主要来自两个方面[3]：第一,介质和土压力盒的相互作用改变了介质原有的应力场,导致应力集中或产生土拱效应；第二,土压力盒（SPC）和被测量结构的形状、刚度和受力方式不匹配导致较大误差,其中以测量大曲率弯曲表面时最显著。为了使第一种误差保持在一个可接受的范围内,必须使隔膜模量要尽量大[4-5],但隔膜模量越大,灵敏度越小,由隔膜模量引起的测量误差与灵敏度之间是一个不可调和的矛盾[6]。第二种误差与土压力盒（SPC）外形尺寸有关,Weller等[7]认为只有土压力盒外形尺寸保持一定比例才能获得较准确的结果。除此之外,由于土压力盒（SPC）的刚性特征,在埋设时要求有一个相对大的空间以避免两个相邻测量点边界之间的影响[8-9],Selig[3]认为这个距离一般是3D～5D（D=土压力盒直径）,这势必会妨碍试验中输出更密集的数据。
在岩土工程领域经常需要测量弯曲表面土压力,例如桩土相互作用和埋地管道室内试验,直接测量土压力是最理想的办法。从应变得到弯矩,然后再拟合弯矩求二阶导的方法也可以得到土抗力[10-11],但这样做的准确度取决于应变测量误差、拟合误差、求导误差[12-13],在某些情况下一阶导的相对误差已经达到55%[14]。因此在模型试验中,急需一种受自身形状局限性小,与土体受力方式相同,能够与曲面完全贴合的土压力测量装置。为叙述方便,下文中分别用FSR和SPC代替薄膜传感器和土压力盒。
FSR（force sensel resistance）是一种压阻式柔性传感器,由两层聚合物薄膜中间夹含有导电物质（炭黑/金属粉末/石墨烯/碳纳米管等）的基体高分子材料构成[15]。高分子材料是导电材料的载体,具有柔性且不导电。导电材料提供载流子,当未受到法向压力时因载流子密度小而使得整个FSR具有极大的电阻效应（≥1 Mohm）。在传感器正面受法向荷载后,基体材料承载载流子发生变形,使载流子间距发生变化形成导电通道使得FSR电阻降低[15],因此通过标定,建立P-R-V（压力-电阻-输出电压）的关系可以测量压力。Talesnick[16]认为准确测试土压力有两个先决条件：首先测试装置的嵌入方式不能影响土的应力场；其次测量装置应对荷载的方式应该与其嵌入材料应对荷载的方式相同。FSR因其厚度只有1 mm左右且具有柔性,能满足上述两个条件。在测量弯曲表面土压力时,能够与测量面紧密贴合共同变形共同受力,与刚性土压力盒相比具有明显的优势。
Paikowsky等[17]最先将FSR引入小型岩土工程试验,介绍了标定方法并对加载速率的影响进行了探究。Palmer等[18]研究了FSR标定时单点和多点标定的区别,探究了蠕变效应的影响,并创新性地提出了一种剪力作用下的保护方式,认为FSR测量的误差在10%以内。Gao等[19]提出了一考虑蠕变效应的阵列式FSR标定方法,显著提高了长期静力测量时FSR的标定速度。Paikowsky等[20]用不同粒径的玻璃珠,初步研究了粒径对阵列式FSR的影响,并发现随着粒径的增大,单元传感器的原始数据输出降低。张紫涛等[21]提出了测量静力时直接建立初始数字信号输出值与施加压强值间函数关系的标定方法,并初步探究了薄膜压力传感器在静力和动力土工试验中的适用性。廖波等[22]研制了一种灵敏度更高的双层结构FSR,并通过模型试验检测了测量土压力的性能。
在FSR试验方面,Kenarsari等[23]用FSR测量了一定深度处,特殊花纹轮胎的土压力分布。Suleiman等[24]在受到侧向土体位移的被动桩试验中,用阵列式FSR测量了桩上土压力分布、变化过程和真实土压力大小,他认为FSR的测量误差在4%～8%之间。Ahmed等[25]测量了埋地管道在静力和重复荷载下土压力分布和变化趋势,探究了聚苯乙烯土工泡沫层的工作机理。Palmer等[18]用FSR测量了模型埋地管道在发生水平位移时表面的土压力分布,他认为切向力只占到总阻力的6%,法向土抗力起主要作用。
基体材料承载载流子发生变形是FSR电阻变化的基础,称为负压阻现象,当FSR正面法向受压时形成导电通道,使电阻降低。FSR弯曲到一定程度也会产生负压阻现象,贴在弯曲表面时,FSR不可避免的产生挠度,使初始电阻值减小从而导致总量程变小,同时可能影响测量性能。以上众多学者虽然在测量土压力时使用了FSR,但都没有考虑挠度带来的压阻效应影响,本文将对此效应在测量时产生的影响进行探究,并设计试验对比了两种传感器测量曲面土压力的能力。
1. 电路设计与FSR标定
单点式FSR的测量电路有多种搭建方法,但总体思路是将电阻信号转换为电压信号,经过放大和AD转换输入上位机处理,图1（a）是数据采集流程图（b）为本文搭建的电路。
像土压力盒一样,FSR使用前也必须逐一标定,FSR使用者需要自己标定获得P-R-V（电阻-电压-压力）曲线,在数据处理时调用。本文采用机械夹持标定,图2是标定装置示意图,将下部钢板换为承载砂土的刚性盒可以考虑砂土环境的影响。
本文对边长L=40 mm的正方形单点式FSR进行了标定。标定时将FSR贴在5 mm厚的钢板中心,钢板边缘距离传感器边缘大于10 mm,两者之间夹有厚度1 mm左右的橡胶垫片使之受荷均匀。采用Geocomp三轴仪上的轴力控制系统施加压力。采用多点标定,分级加载,初始荷载1 N,1～25 N之间增量5 N,50 N以上增量50 N,直至FSR过载或电路饱和。
因FSR的组成材料具有黏弹性,因此受到时间效应影响。Palmer等[18]对FSR的黏弹特性进行了详细研究,他将影响分为两个阶段,即过渡（trans）阶段和蠕变（creep）阶段,认为这两个阶段可以在60～120 s内完成并给出了经验公式。不同型号的FSR这一时间不同,但有相同的表现形式。本文在标定中发现,在不同压力下20 s内读数可达稳定,加载30～50 s左右时可获得较准确结果,160 s时因蠕变效应导致误差在10%左右,与Palmer等[18]结论相类似,综上以30～50 s的平均值作为输出。图3（a）和3（b）为边长L=40 mmFSR的标定结果,图3（c）为FSR输出值与实际施加荷载的偏差曲线。对同一FSR标定3次,可以看出重复性较好,电阻随荷载增大呈幂函数变化,电压输出线性程度高,两者R2相同都为0.99881。图中550 N的荷载已经超出了最大量程,拟合时将此点去掉。
图3（c）是测量值与实际值的偏差曲线,将输出的压力值除以施加的实际值得到偏差值。从图中可以看到,首次加压的FSR在低压力和高压力段偏差较大,尤其是低压力段最高偏差达到20%,这与Paikowsky等[17]和Palmer[18]的结论一致。经过加卸载和充分静置恢复后,FSR在低压力段偏差大幅度减小,除高压力段以外,其余点偏差都在5%以内。综上本文的夹持标定设备能够较好的对FSR进行标定,同时作者认为,在使用前应对FSR预压,施加压力应大于目标压力,这样可以减小低量程段的测量偏差。
2. 挠度对FSR测量的影响分析
测量弯曲表面土压力时,FSR与曲面紧贴,挠度会改变电阻值,这种改变只是影响量程还是同时影响测量能力需要进行探究。
为了解挠度对FSR的影响,本文选取了两个生产商的边长L=40 mm和直径D=40 mm的正方形单点式FSR作为研究对象,对其在挠度影响下的电阻变化以及挠度对测量的影响进行了初步探究。
试验以曲率作为变量,将FSR黏贴在直径100,76,65,54,42,36,27,23 mm的圆柱表面,对应圆曲率为0.2,0.26,0.31,0.37,0.48,0.56,0.87（1/cm）。为减小分离时对FSR的影响,首先将FSR黏贴于纸带上,再将纸带与圆柱表面贴紧,只将纸带尾部固定,如图4（a）所示。每次试验前将FSR整平,静置60 min让其完全恢复。试验读数和数据处理与标定时相同。
图4（b）是半对数坐标下的初始电阻随曲率变化的曲线。两者规律一致,现以L=40 mm的FSR为例,在圆柱直径100 mm即曲率0.2时电阻超过1
MΩ ,与平铺时电阻基本相同,可以认为曲率小于0.2不会引起初始阻值的变化,因此这点并未在图中表示。FSR初始电阻值随曲率增大而成幂函数减小,相关性系数R2=0.99839。初始电阻下降速度很快,这也意味着FSR量程的快速减小,在曲率为0.87时对应圆柱直径23 mm,阻值10.7kΩ ,结合图3（a）来看对应荷载180 N,相对于总量程500 N来说量程减小36%,但仍能满足绝大部分测量需求。挠度改变FSR初始阻值有两种可能的方式,第一种是载流子重新分布,造成局部集中或稀疏,这种情况会使得带有挠度的FSR标定曲线改变,与无挠度的标定曲线相比,同一电阻或电压的输出代表不同压力,需要在同等挠度下重新标定。第二种是挠度单纯造成了基体材料受压,原理与FSR法向受压相同,这种情况不会改变测量曲线,不用重新标定。为探究挠度是否改变了测量曲线,作者将FSR黏贴于曲率0.31（D=65 mm）和0.56（D=27 mm）的半圆形钢柱表面,用相同曲率的半圆盖与钢柱组成标定装置,将重新标定的曲线与无挠度下标定的曲线相比,探究挠度是否会对标定结果产生影响,重新标定装置如图5所示。
图6 (a)是L=40 mm的FSR在无挠度和曲率0.31下的标定曲线,每种曲率标定两次用以评判可重复性。需要指出的是,图6（a）曲线起始点横坐标为5.2 N,并未从0开始,是为了减小低量程段电阻幂指数急剧变化给中高量程段带来的差别弱化效果,突出加载后曲线的区别。
从电阻曲线来看,3次标定在95%的置信度下不可区分,重复性较好。从电压输出来看,首次挠度标定曲线与无挠度时基本没有区别,所有数据点相对差值均在5%以内。第二次标定时,250 N对应的数据点相对误差12.5%,其他数据均在8%以内,笔者认为这是由于FSR产生不可恢复损伤造成的,使用中应加强保护。综上当曲率为0.31时,挠度并没改变FSR的测量性能,不用在挠度作用下重新标定。图6（a）是曲率0.56时的标定曲线,可以看到,电压输出呈现非线性,3条线间相对差值很大,首次标定中FSR受到较大的不可恢复损耗,使得带挠度的两条标定曲线区别很大。综上,对本文选用的FSR来说挠度小于0.31时测量不受影响,可以调用无挠度标定曲线,但曲率大于0.56时,从图6（b）来看FSR已不具备测量能力。作者认为,FSR挠度小时可以认为其作用方式与法向压力相同,当挠度过大时,基体材料的变形与挤压使导电物质在一定范围内发生重分布,从根本上改变了FSR的压阻特性。同时从挠度作用下的标定曲线的重复性可以看出,测量曲面时需要对FSR加强保护。
3. FSR测量圆筒内砂土K0的探讨
本试验目的在于比较FSR和SPC测量曲面土压力的能力,探究FSR是否受到“嵌入”效应影响,能否准确输出曲面土压力。试验采用堆载的方式模拟上覆土,可以有效减小模型尺寸,试验装置如图3所示。
试验中的SPC尺寸为D（直径）=28 mm、T（厚度）=10 mm,这种尺寸在室内模型试验中经常用到,容易在狭小的空间内布置,适用于小型试验。SPC-1和FSR-1用薄双面胶贴在地面上,用以测量竖向土压力,SPC-1距离圆筒内边距离大于3D,可以认为不受圆筒边界效应的影响[3]。
在筒壁高10 mm的位置对称布置SPC-2、SPC-3和FSR-2,并使它们的底边高度与SPC-1传感表面齐平,如图7所示,目的是为了测量侧向土压力,由此可以推算K0。SPC-2固定在筒壁表面,外形整体暴露,SPC-3镶嵌在筒体中,传感表面与筒内壁齐平。SPC的T/D=0.36>0.2,因此SPC-1和SPC-2将产生“嵌入”效应,SPC隔膜变形下陷可能会导致局部土拱效应。测量水平向土压力的传感器（FSR-2 SPC-2/3）受理方向与土体主要压缩变形方向不一致,可能引入误差。综上,测量误差应是以上3种影响因素共同作用造成的,但影响能力不同。
图8是加载装置示意图。亚克力圆筒内径300 mm外径320 mm,高1000 mm,曲率0.067。由前文可知0.067的曲率不会对贴在表面的FSR测量能力产生影响,但弯曲表面外加相对狭小的传感器布置环境,足以给传统的刚性SPC带来测量困难。
试验开始时,首先在桶底铺筑一层厚度约20 mm的砂,然后再开始记录,铺砂土的目的是保护传感器并防止移位。采用分层填筑的方法,每层200 mm,在距离填筑面高1 m处用“砂雨”法将砂均匀倒入筒中,通过控制重量,倾倒高度和速度的方法控制密度。砂土级配曲线如图9所示,Cu=6.03,Cc=1.40 D50=0.588,含水率为0%。填筑平均单位密度为16.9 kN/m3,Dr=55.2%,属于中密砂。在相同密度下进行直剪试验得到内摩擦角
ϕ =42°。亚克力筒填满后,在砂土表面铺钢板,用单块10 kg的砝码进行堆载。在圆筒对称方向固定两块砝码,起到稳定圆筒的作用,并且可以防止堆载中砂土从筒底挤出。
采用分级堆载的方式,每级增加20 kg砝码,最后一次增加30 kg,总重430 kg。试验中每1 s记录一组数据,数据处理与标定时类似,考虑堆载荷载向下传递的反应时间,FSR读数最长在30 s内可达稳定,将40～70 s内的数据平均值作为这一级荷载的输出值,后续结果证明这样做误差较小。
试验中分别进行了瞬时卸载和逐级卸载。瞬时卸载是在1 s内卸去部分荷载,逐级卸载则每次卸载20 kg,待到读数稳定后卸载下一级,将平均值作为数据输出。
4. 试验结果及数据分析
图10是FSR-1和SPC-1测量的竖向土压力,其中10（a）,（b）分别表示从填筑砂土开始到堆载完成和从堆载开始到逐级慢速卸载结束时的土压力变化过程。
亚克力圆筒内径只有30 cm,在填筑和堆载过程中伴随着竖向土拱的形成与发展,从10（a）中可以看出,在圆筒填筑到60 cm时土拱形成,并具备了承担荷载的能力,在这之前传感器输出值与填筑高度成线性关系,之后增幅骤减,60 cm是一个明显的拐点。A点对应堆载的开始,在这一过程中,土压力有两个台阶式的跃升,并伴随着压力曲线斜率的减小,作者认为这是随着堆载的增大,土拱破坏又重新形成的迭代过程,每一次迭代都意味着土拱承担荷载能力的增强,这也是压力曲线斜率减小的原因,这一点从图10（b）中也可以看出。相比于黏土,砂土回弹能力很小,且颗粒间的咬合使得堆载造成的压力不能完全释放,导致卸载完成后残余土压力为57.5%。两者慢速卸载曲线差别不大,FSR的材料黏弹性并没有造成影响。从图10中可以看出FSR与SPC最大相差5.2%,两者测量竖向土压力能力相当。FSR与SPC的输出数据相近,且SPC边缘距容器边大于3D,受边界影响小,从而可以推断试验选用的SPC隔膜变形没有引起的土拱效应或影响很小。
瞬时卸载时竖向和水平向传感器响应情况类似,现以水平向记录为例说明。图11表示瞬时卸载时水平向SPC-3和FSR-2所测量的土压力随时间的变化。
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图 11 瞬时卸载时水平向压力随时间变化图Figure 11. Time-dependent chart of horizontal pressure during sudden unloading此次试验中瞬时卸载分3次进行,第一次对应图A点,卸载270 kg,第二次对应B点卸载120 kg,第3次对应C点卸载40 kg,每次均在1 s内完成。从图中可以看到,卸载中FSR虽有滞后效应,但并未影响最终卸载效果,对于静力试验来说滞后反应时间可以接受。
图12是FSR-2、SPC-2、SPC-3所记录的水平土压力和相应的K0值。从图12（a）中可以看出外表完全暴露在砂中的SPC-2土压力远远小于FSR-2和SPC-3,笔者认为原因有两个：第一,由于尺寸限制,SPC形状影响系数大于0.2导致“嵌入”效应明显。第二,刚度不同造成SPC-2与土变形不协调,增大了误差。由于SPC-3镶嵌在筒壁上,FSR-2本身很薄,因此均避免了“嵌入”效应的影响,两者结果接近,最大相差9%。同时从FSR-2和SPC-3曲线上可以明显反应砂土拱的形成与发展。
不同时期多位学者给出了砂土K0公式,例如Jacy[26]和Brooker等[27]的修正公式,基于砂土一维压缩特性推导的Hendron[28]公式,针对砂性土统计得到Bolton[29]经验公式,基于弹性理论得到的Sharif等[30]公式等。将基于以上公式计算得到的K0值与本试验结果作比较,同时将《建筑边坡工程技术规范》中建议的K0= 0.34～0.45也标注于图12（b）中。从图中可以看到FSR-2和SPC-3得到的K0值与Jaky等的结果约为规范建议取值区间值,证明了测量的准确性。
试验表明,FSR由于具有柔性且超薄,在测量狭小空间内的曲面土压力时比SPC有着更高的准确性,避免了影响土的应力场,减小了“嵌入”效应的影响,且安装简单方便,适合小型模型试验使用。
5. 结论
（1）在使用FSR前,用大于测量目标值的压力预压,可以有效减小低量程范围的误差。
（2）较小的挠度只改变FSR量程,不改变标定曲线,但过大的挠度会改变标定曲线以致失去测量能力。
（3）当SPC受力形式与土体不同时,“嵌入”效应是产生较大误差的主要原因。
（4）FSR在测量曲面土压力时,不改变土的应力场,不受“嵌入”效应影响,较SPC准确度高,适用于狭小空间的曲面测量。
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