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扰动冲击下弱胶结红砂岩的能量耗散与分形特征

张慧梅, 陈世官, 王磊, 程树范, 杨更社, 申艳军

张慧梅, 陈世官, 王磊, 程树范, 杨更社, 申艳军. 扰动冲击下弱胶结红砂岩的能量耗散与分形特征[J]. 岩土工程学报, 2022, 44(4): 622-631. DOI: 10.11779/CJGE202204004
引用本文: 张慧梅, 陈世官, 王磊, 程树范, 杨更社, 申艳军. 扰动冲击下弱胶结红砂岩的能量耗散与分形特征[J]. 岩土工程学报, 2022, 44(4): 622-631. DOI: 10.11779/CJGE202204004
ZHANG Hui-mei, CHEN Shi-guan, WANG Lei, CHENG Shu-fan, YANG Geng-she, SHEN Yan-jun. Energy dissipation and fractal characteristics of weakly cemented red sandstone under disturbance impact[J]. Chinese Journal of Geotechnical Engineering, 2022, 44(4): 622-631. DOI: 10.11779/CJGE202204004
Citation: ZHANG Hui-mei, CHEN Shi-guan, WANG Lei, CHENG Shu-fan, YANG Geng-she, SHEN Yan-jun. Energy dissipation and fractal characteristics of weakly cemented red sandstone under disturbance impact[J]. Chinese Journal of Geotechnical Engineering, 2022, 44(4): 622-631. DOI: 10.11779/CJGE202204004

扰动冲击下弱胶结红砂岩的能量耗散与分形特征  English Version

基金项目: 

国家自然科学基金项目 12172280

国家自然科学基金项目 42077274

国家自然科学基金项目 41907259

陕西省自然科学基金重点项目 2020JZ-53

详细信息
    作者简介:

    张慧梅(1968—),女,山西大同人,现任教授、博士生导师,主要从事岩土工程稳定性评价及岩石力学理论与应用。E-mail:zhanghuimei68@163.com

    通讯作者:

    陈世官, E-mail: 18992178070@163.com

  • 中图分类号: TU458.3

Energy dissipation and fractal characteristics of weakly cemented red sandstone under disturbance impact

  • 摘要: 为探索动态扰动后西部矿区软岩夹层的能量耗散规律和破坏模式,利用分离式霍普金森压杆装置对弱胶结红砂岩进行动态冲击破坏试验,分析该类红砂岩在受到不同加载速率、不同次数扰动冲击以及是否扰动的条件下,试样在相同加载速率破坏性冲击过程中的能量耗散与分形特征。试验结果表明:在不同速率扰动冲击作用下,随着扰动冲击次数的增加反射能递增而透射能和耗散能呈减小趋势,其中较高速率扰动冲击下试样的反射能高于低速率扰动冲击,耗散能则相反,且低速率扰动冲击下试样的耗散能、能量耗散率和能量耗散密度高于较高速率扰动冲击,表明低速率扰动冲击下试样的能量利用率更高;在破坏性冲击试验中,随着扰动冲击次数的增加,低速率扰动后试样的破碎程度相较于未扰动与高速率扰动更为严重,对应分形维数Db低速率扰动 > 未扰动 > 高速率扰动,表明分形维数与扰动冲击次数呈正相关,与扰动冲击速率呈负相关;在相同扰动冲击次数下,低速率扰动试样的Db所对应累积耗散能和耗散能密度高于较高速率扰动试样,而对应的累积反射能则相反。
    Abstract: To explore the energy dissipation law and failure mode of soft rock interlayer in western mining areas of China after dynamic disturbance, the dynamic impact failure tests on weakly cemented red sandstone are carried out by using the separated Hopkinson compression bar device. Under the impact of this red sandstone under different loading rates, different times of disturbance and whether there is the disturbance or not, the energy dissipation and fractal characteristics of the samples during the same loading rate impact failure are analyzed. The experimental results show that under different disturbance impact rates, with the increase of disturbance impact times, the reflection energy increases, while the transmission energy and dissipation energy decrease. The reflection energy of the samples under the impact of high-speed disturbance is higher than that of the low-speed disturbance impact, while the dissipative energy is the opposite. Moreover, the dissipative energy of the samples under the impact of low-speed rate disturbance is opposite. The energy dissipation rate and energy dissipation density are higher than those of the high-speed disturbance impact, which indicates that the energy utilization rate of the samples is higher under the impact of low-speed disturbance. In the impact failure tests, with the increase of the number of disturbance impact, the fragmentation degree of the sample after the low-speed rate disturbance is more serious than that of the undisturbed and high-speed rate disturbance. The low-speed rate disturbance of fractal dimension Db > undisturbed > high-speed rate disturbance shows that the fractal dimension is positively correlated with the number of disturbance shocks. The results show that the impact rate is negatively correlated with the disturbance. Under the same number of disturbance impact, the cumulative dissipation energy and energy density of Db of the low-speed rate-disturbed samples are higher than those of the high-speed disturbed samples, while the cumulative reflection energy is opposite.
  • 随着城市轨道交通的快速发展,列车荷载引起的环境振动问题受到越来越多的关注。不少学者对移动荷载作用下地基的动力响应开展了研究。Eason[1]通过积分变换法求得了速度小于瑞丽波速的移动点荷载作用下均质弹性半空间位移、应力响应的数值解。Hung等[2]研究了亚音速、跨音速及超音速3种不同速度范围内的移动荷载作用下黏弹性半空间的动力响应规律,并分析了移动荷载形状分布及加振频率的影响。De等[3]采用传递反射矩阵法求得了成层弹性半空间在移动荷载的动力格林函数。自Biot[4-5]提出饱和多孔介质的动力控制方程后,多数学者都基于该控制方程对饱和土体动力响应进行研究。Jin等[6]利用梯形求积公式获得了振动荷载作用下半无限多孔饱和固体中的应力和孔隙水压力数值解。胡安峰等[7]利用Fourier积分变换求解了下卧基岩饱和半平面在移动线荷载作用下的动力响应。Xu[8]等研究了饱和成层土体上无限长Euler-Bernoulli梁在移动荷载作用下的动力响应。Cai等[9]研究了移动矩形荷载作用下饱和土体的动力响应。

    上述研究中,移动荷载大多作用在地基表面,而对地铁等地下构筑物来说,移动荷载作用在地表以下一定深度处。Pak[10]采用势函数的方法求解了弹性半空间内部点源荷载作用下的动力响应。Senjunctichai等[11]求得了二维均质多孔弹性半空间在内部荷载激励作用下的动力格林函数。陈胜立等[12]通过Hankel变换变换方法,求解了埋置点源荷载的轴对称Lame问题。杜秦文等[13]采用Hankel积分变换法,得到了埋置点源简谐荷载作用下Gibson土体的动力Green函数。上述文献中的埋置荷载多假定为固定位置动荷载。Metrikine等[14]通过建立一个二维弹性层-梁耦合的简化模型,对隧道中列车运行引起的地表振动规律进行了研究。Yuan等 [15]在Metrikine等[14]的基础上,求解了饱和地基-梁模型在移动荷载作用下的振动解析解。上述两个模型均为二维模型,忽略了垂直于荷载移动方向的地基振动。Forrest等[16]通过建立筒中筒(PIP)模型,在柱坐标系下对移动点荷载作用下圆形衬砌隧道埋置于弹性全空间中的动力响应情况进行了求解,但未考虑地表边界的影响。本文建立了三维黏弹性半空间地基模型,并通过引入势函数及利用三维傅里叶积分变换及逆变换方法,在笛卡尔坐标系下求解了埋置移动点荷载作用下黏弹性半空间的动力响应积分形式解。最后,通过数值算例分析了埋置荷载移动速度,埋置深度等因素对地表振动传播分布及衰减规律的影响。

    均质各向同性弹性体的控制方程以张量形式可表示为(不计体力):

    μui,jj+(λ+μ)uj,ji=ρ¨ui
    (1)

    式中:ui为弹性体的位移,i=x,y,zρ为弹性介质的质量密度;λμ为弹性体的Lame常数;(..)为对时间t的二阶导数。

    根据Helmholtz分解定理,引入标量势函数ϕ,矢量势函数ψj=(ψ1, ψ2,ψ3)。对位移场作如下分解:

    ui=ϕ,i+eijkψk,j
    (2)

    将式(2)代入式(1)可得如下关于势函数的表达式:

    2ϕ1cP22ϕt2=0 
    (3a)
    2Ψj1cS22Ψjt2=0
    (3b)

    式中:cPcS分别为弹性半空间体的压缩波与剪切波波速:cP=λ+2μρcS=μρ2为Laplace算子,2=2x2+2y2+2z2

    假定在直角坐标系中矢量势函数满足如下条件:

    μψj,j=0
    (3c)

    定义对xyt的三重Fourier变换及逆变换分别为

    ˆ¯¯g(k1,k2,z,ω)=+++g(x,y,z,t)eik1xeik2yeiωtdxdydt 
    (4a)
    g(x,y,z,t)=1(2π)3+++ˆ¯¯g(k1,k2,z,ω)eik1xeik2yeiωtdk1dk2dω
    (4b)

    将式(3a)~(3c)进行式(4a)中的Fourier积分变换可得:

    d2ˆ¯¯ϕdz2BP2ˆ¯¯ϕ=0  
    (5a)
    d2^¯¯Ψjdz2BS2^¯¯Ψj=0 
    (5b)
    d^¯¯Ψ3dz=(ik1^¯¯Ψ1+ik2^¯¯Ψ2)
    (5c)

    式中:BP2=k12+k22kP2BS2=k12+k22kS2kP=ωcPkS=ωcS。其中BpBs的实部大于零。

    由弹性介质本构方程及傅里叶变换公式可得波数-频率域内位移及应力响应分量表达式为

    ˆ¯¯ux=ik1ˆ¯¯ϕ+ik2ˆ¯¯ψ3ˆ¯¯ψ2z 
    (6a)
    ˆ¯¯uy=ik2ˆ¯¯ϕik1ˆ¯¯ψ3+ˆ¯¯ψ1z 
    (6b)
    ˆ¯¯uz=ˆ¯¯ϕz+ik1ˆ¯¯ψ2ik2ˆ¯¯ψ1 
    (6c)
    ˆ¯¯σzz=λ(2ˆ¯¯ϕz2k12ˆ¯¯ϕk22ˆ¯¯ϕ)+2μ(ik1ˆ¯¯ψ2zik2ˆ¯¯ψ1z+2ˆ¯¯ϕz2) 
    (6d)
    ˆ¯¯τxz=μ(2ik1ˆ¯¯ϕz+k1k2ˆ¯¯ψ1k12ˆ¯¯ψ22ˆ¯¯ψ2z2+ik2ˆ¯¯ψ3z) 
    (6e)
    ˆ¯¯τzy=μ(2ik2ˆ¯¯ϕz+k22ˆ¯¯ψ1+2ˆ¯¯ψ1z2k1k2ˆ¯¯ψ2ik1ˆ¯¯ψ3z) 
    (6f)

    图 1给出了本文的理论研究模型示意图。沿x轴正方向匀速移动的竖向点荷载作用在黏弹性半空间体表面以下深度z=h处的平面上,移动荷载表达式为F=peiω0tδ(xct)δ(y),其在变换域内荷载表达式为ˆ¯¯F=p2πδ(ω0+ω+ck1)。其中p为荷载幅值,c为荷载移动速度,ω0为荷载自振频率,δ为狄拉克函数。考虑荷载由x轴的负无穷处向正无穷处移动,当t=0时,荷载恰好移动到坐标原点处。

    图  1  埋置移动荷载作用于弹性半空间内部示意图
    Figure  1.  Diagram of elastic half-space with embedded moving loads

    根据波的辐射特性,将弹性半空间体分为上下两个区域:区域1(0zh)与区域2(h+z+),如图 1所示。则由式(3a)、(3b)可解得变换域内势函数可表示为

    0zh时,

    ˆ¯¯Φ=a1eBPz+a2eBPz ˆ¯¯Ψ1=a3eBSz+a4eBSz ˆ¯¯Ψ2=a5eBSz+a6eBSz ˆ¯¯Ψ3=ieBSzBS(a3k1+a5k2)+ieBSzBS(a4k1+a6k2)}
    (7a)

    h+z+时,

    ˆ¯¯Φ=a7eBPz ˆ¯¯Ψ1=a8eBSz ˆ¯¯Ψ2=a9eBSz ˆ¯¯Ψ3=ieBSzBS(a8k1+a9k2)}
    (7b)

    式中:aj为未知常数,j=19

    图 1可看出,半空间表面的边界条件及荷载作用平面处的连续条件可表示为

    z=0时,

    σzz(x,y,0)=0 τxz(x,y,0)=0 τyz(x,y,0)=0}
    (8)

    z=h时,

    σzz(x,y,h+)σzz(x,y,h)=F 
    (9a)
    τxz(x,y,h+)τxz(x,y,h)=0 
    (9b)
    τyz(x,y,h+)τyz(x,y,h)=0 
    (9c)
    uzz(x,y,h+)uzz(x,y,h)=0 
    (9d)
    uxz(x,y,h+)uxz(x,y,h)=0 
    (9e)
    uyz(x,y,h+)uyz(x,y,h)=0
    (9f)

    观察式(7)~(9)可看到,边界条件个数与连续条件个数之和等于未知数个数,故可进行求解。

    对边界条件与连续条件进行傅里叶变换后,将式(6a)~(6d)代入,可得到一组关于未知系数aj的线性方程组:Aijaj=fi,其中矩阵Aijfi的表达式见附录1。求得未知系数aj后,代入位移表达式, 即可得到变换域内的位移响应解ˆ¯¯ux,ˆ¯¯uy,ˆ¯¯uz。由Fourier逆变换即可求得稳态响应的表达式:

    {ux,uy,uz}=1(2π)3++ + δ(ω0+ω+ck1)ˆ¯¯ux,ˆ¯¯uy,ˆ¯¯uz                              eik1xeik2yeiωtdk1dk2dω
    (10)

    为考虑土体的黏滞性,引入如下黏滞阻尼模型:λ* = λ(1 + 2iβ)μ* = μ(1 + 2iβ),其中β为土体介质的黏滞阻尼比。由于式(10)中的动力响应解为无穷积分形式,且其被积函数为一个复杂的振荡函数,故采用IFFT方法进行求解。

    h=0时,本文解可退化为移动荷载作用在黏弹性半空间表面的动力响应解。Hung等[2]研究了黏弹性半空间在不同类型的地表移动荷载作用下的动力响应。土体参数取值密度ρ=2000 kg/m3,泊松比ν=0.25,阻尼比β=0.02,横波波速cs=100 m/s,纵波波速cp=173.2 m/s,表面波波速cR=92 m/s。定义观察点(x,y,z)=(0,0,z0)(其中z0=1 m)处的无量纲化竖向位移V=2πμz0uz/p,无量纲化水平纵向位移W=2πμz0ux/p图 2为荷载移动速度c=50 m/s时,本文位移响应退化解与Hung等[2]结果的对比图,可见两者吻合得较好。

    图  2  本文退化解与Hung等[2]的对比图
    Figure  2.  Comparison between degenerated solutions and results of Hung et al[2]

    定义无量纲位移ux=uxλl0/puz=uzλl0/p,其中l0为特征长度(l0=10 m)。土体Lame常数λ=2×107 N/m2μ=2×107 N/m2,土体密度ρ= 2000 kg/m3,黏滞阻尼比β=0.02,移动荷载幅值p=1 N。下面分析埋置移动点荷载作用下黏弹性半空间地表的振动情况。

    (1)地表竖向位移响应分布

    引入移动坐标系xt=xct图 3给出了不同速度的埋置移动荷载作用下,地表无量纲竖向位移uz在移动坐标系下的曲线分布图,其中y=0,移动荷载埋深分别取h为10,15,20 m。由图 3可看到,地表竖向位移响应关于xt=0基本呈轴对称分布,且荷载埋深越深,振动沿水平方向衰减越快,振动影响范围越小,地表振动幅值越小。此外,对比图 3(a)(b)可发现,荷载速度越大,地表最大竖向位移幅值也越大。

    图  3  地表竖向位移uz
    Figure  3.  Vertical displacements of surface uz

    (2)地表纵向位移响应分布

    在埋置移动荷载作用下,地表不仅会产生竖直方向的振动,也会产生水平方向的动力响应。图 4为当荷载埋深h=10 m时,地表x方向的纵向水平位移响应沿移动坐标xt的分布图。由图 4可看到埋置移动点荷载作用下,地表纵向水平位移响应曲线关于移动坐标轴呈反对称分布,且垂直于荷载移动方向,距离荷载作用位置越远,即y值越大,纵向水平位移越小;对比图 4(a)(b)可看出,随荷载速度增大,纵向水平位移幅值降低。同时,对比图 34可看出,同一埋置荷载作用下,地表土体竖向位移响应幅值远大于纵向水平位移响应幅值,这与实测地面振动规律吻合[17]

    图  4  地表纵向位移ux
    Figure  4.  Longitudinal displacements of surface ux

    (3)地表位移响应的频谱分析

    由式(10)可得原点(0,0,0)处的竖向位移响应表达式为

    uz(0,0,0,t)=1(2π)2+ + ˆ¯¯uz(k1,k2,0,ctω0)                     eik1(xctω0)eik2ydk1dk2
    (11)

    则该点处的竖向振动频谱可表示为

    uz(f)= + uz(0,0,0,t)ei2πftdt       =12π+ˆ¯¯uz(2πf02πfc,k2,0,2πf)dk2
    (12)

    同理可得原点处的水平位移频谱表达式ux(f)

    图 5给出了两种不同速度的常值移动荷载(h=10 m)作用下,地表原点处竖向位移与水平纵向位移响应的频谱图。可看到,当荷载速度较小时,位移频谱主要分布在频率较低的范围内,且竖向位移与水平位移几乎在同一频率达到峰值。当荷载速度变大时,位移频谱曲线的频率分布范围变大,位移幅值也增大,位移峰值所对应的频率也越大。

    图  5  地表位移频谱图
    Figure  5.  Displacement spectra of surface

    采用傅里叶变换及Helmholtz分解的方法对埋置移动荷载作用下弹性半空间地基的动力响应进行理论求解,并通过数值计算,分析了荷载埋深、速度等因素对地表位移响应的影响,得到3点结论。

    (1)移动荷载埋深越深,地表水平位移与竖向位移响应均越小;荷载移动速度越大,地表竖向位移响应幅值越大,纵向水平位移反而越小。

    (2)垂直于荷载移动方向,距离荷载作用位置越远,地表水平位移响应越小。

    (3)埋置荷载移动速度越大,位移响应频谱分布范围越大,位移幅值峰值对应的频率也越大。

  • 图  1   红砂岩试样

    Figure  1.   Red sandstone sample

    图  2   试验方案流程图

    Figure  2.   Flow chart of experimental scheme

    图  3   动态应力平衡验证

    Figure  3.   Verification of dynamic stress equilibrium

    图  4   动态冲击应变时程曲线

    Figure  4.   Curves of strain and time under impact

    图  5   不同加载速率扰动冲击下入射能量时程曲线

    Figure  5.   Time-history curves of incident energy under disturbance impact with different loading rates

    图  6   扰动冲击下砂岩各能量随冲击次数的变化关系

    Figure  6.   Relationship between energy of sandstone and impact times under disturbance impact

    图  7   不同加载速率循环扰动后试样损伤形态

    Figure  7.   Damage morphology of samples after cyclic disturbance at different loading rates

    图  8   扰动冲击下耗散能密度与冲击次数的关系

    Figure  8.   Relationship between dissipative energy density and impact times under disturbance impact

    图  9   扰动与未扰动试样冲击破坏后的破碎形态

    Figure  9.   Fracture morphology of disturbed and undisturbed samples after impact failure

    图  10   扰动与未扰动试样冲击破坏后的破碎块度分布

    Figure  10.   Fragmentation distribution of disturbed and undisturbed samples after impact failure

    图  11   SHPB冲击试验中lg[MR/MT]–lgR曲线

    Figure  11.   Curve of lg[MR/MT]–lgR in SHPB impact tests

    图  12   分形维数与各累积能量关系曲线

    Figure  12.   Relationship between fractal dimension and cumulative energy

    表  1   红砂岩基本物理力学参数

    Table  1   Basic physical and mechanical parameters of red sandstone

    密度/(g·cm-3) 纵波波速/(m·s-1) 单轴抗压强度/MPa 弹性模量/GPa 孔隙度/%
    1.874 1897 13.78 1.22 22.8
    下载: 导出CSV

    表  2   扰动作用下红砂岩冲击破碎块度的筛分结果

    Table  2   Screening results of impact fragmentation of red sandstone under disturbing action

    试件编号 筛分直径/mm 总质量/g Db
    0.080 0.160 0.315 0.630 1.250 2.500 5.000 10.000 15.000 30.000
    A1-Z 2.18 3.56 1.52 1.20 0.80 3.44 6.48 2.53 35.82 39.18 96.71 2.365
    A3-Z 2.78 4.41 1.33 1.20 0.72 2.65 7.46 3.33 33.96 37.35 95.19 2.403
    A5-Z 5.49 10.54 3.23 3.35 2.00 6.95 13.39 13.18 9.37 34.07 101.57 2.592
    C1-Z 7.84 5.34 1.38 1.35 0.70 3.35 13.89 24.45 36.46 0.00 94.76 2.586
    C3-Z 8.68 12.39 3.50 3.43 2.15 6.68 12.68 15.64 30.57 0.00 95.72 2.644
    C5-Z 9.40 17.37 4.56 3.35 1.80 7.91 15.73 16.52 16.6 0.00 93.24 2.673
    DZ 3.66 10.41 2.95 2.28 1.19 4.11 4.87 8.50 34.84 23.01 95.82 2.537
    注:试件编号说明,A和C分别表示较高速率扰动组和低速率扰动组,编号1,3,5表示需要进行扰动的次数,-Z表示每组所有扰动次数完成后的破坏性冲击试验,DZ表示对照组。
    下载: 导出CSV
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  • 收稿日期:  2021-06-09
  • 网络出版日期:  2022-09-22
  • 刊出日期:  2022-03-31

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