第1期 王学滨等：单轴压缩条件下普通混凝土柱的峰后非线性尺寸效应 29。 关系，自然使本构关系平缓，再由这一本构关系预测 al compression fracture energy of concrete.J Wuhan Univ Tech- 较高的试样的应力应变曲线，自然使预测的峰后 nol Mater Sci Ed.2003.18(4):89 [4 Wang X B.Analytical solution of complete stress strain curve in 应力一应变曲线比实验结果平缓. uniaxial com pression based on gradient-dependent pasticityEu 在软化阶段后期，实测曲线与预测的峰后应力 rock 2005-Impact of Human Activity on the Geologial En viron- 应变曲线尚有一定程度差别的另一方面的原因是， ment.London:Taylor Francis Gmup,2005:661 在软化阶段的后期，在剪切带的中心位置出现了一 [5 Wang X B.Prediction of height effect,plastic defomation and 定长度的真实剪切裂纹，剪切裂纹之间的摩擦力将 fracture energy for high-strength concwete by linear shear soften- 起到重要的作用（这一现象本文的连续介质模型尚 ing constit utive weltion based on energy conservation method. Mag Concr Res,2007,595):341 不能描述). [6 De Borst R.Muhlhaus H B.Gradent-dependent plasticity:for 总之，在混凝土单轴压缩条件下，本文已经将遭 multion and algorithmic aspects Int J Numer Methods Eng, 受到剪切带破坏形式的单一剪切破坏试样的峰后应 1992.35(3):521 力一应变曲线斜率的解析解推广为非线性情形.在 [7 Zhou G L Tham L G,Lee P KK,et al.A phenomenological 峰值应力之前，采用Scot模型描述混凝土试样均 constitutive model for mocks w ith shear failure mode.Int/Numer Anal Methods Gomech,2001,25(4):391 匀变形阶段的非线性的本构关系；在应变软化阶段， [8 Labuz J F,Bioki L.Chss I vs class II stahiity:a demonstra- 为了建立宏观应力一应变曲线峰后斜率的表达式， tion of size effect.Int J Rock Mech Min Sci,1991,28(2/3): 使用了剪切带的非线性应变软化本构关系（式(3）)、 199 弹性体弹性卸载本构关系（式(5）)、弹性体静力平衡 [9 Schryer H L.Analytical solutions for nonlinear strain-gradient 条件（式(7）和一些几何关系（式(5）、式(6)及 softening and localizat ion.J Appl Mech,1990.57(3):522. 10 Muhlhaus H B.Vardoulkis I.The thickness of shear bands in 式(9).剪切带的非线性应变软化本构关系由导出 granular materials.Cotechnique.1987.37(3):271 的最短的普通混凝土试样应力一应变曲线峰后斜率 11]Alshibli K A,Sture S.Sand shear band thickness measurements 反算.利用得到的峰后本构关系，对其他较长的普 by digital imaging techniques.J Comput Civil Eng,1999.13 通混凝土试样（由于试样高度过大而难于实验）的应 (2):103 力一应变曲线进行了预测. [12 Okubo S,Nishimatsu Y.Uniaxial compression testing using a linear combination of stress and strain as the control variable. 3结论 Int J Rock Mech Min Sci.1985.22(5):323 13]Wang X B.Volme change of beterogeneous quasi-brittle mate- 利用本文方法预测的峰后应力一应变曲线依赖 rials in uriaxial compression.J Wuhan Univ Techndl Mater Sci 于试样的高度.试样越高，应力一应变曲线的峰后分 Ed,2006.21(3):162 支越陡峭.对于较高的试样，在应变软化阶段的后 14 Wang X B.Analysis of the post-peak Poisson's rat io of rock 期，预测的峰后应力一应变曲线比实验结果稍显平 specimens in uniaxial compression.Eng Mech,2006.23(4): 多 缓，这意味着端面约束效应（当试样高度较矮时，其 (王学滨。岩样单轴压缩蜂后泊松比理论研究.工程力学， 作用比较显著)将使峰后应力一应变曲线变得不陡 2006.23(4):99) 峭. [15]Wang X B.Size effect of shear fracture energy of concrete in 估算的剪切带内部的平均塑性剪切应变远大于 uniaxial comn pression basedon gradient-dependent plasticity.Key 在单轴压缩条件下测得的轴向应变的最大值.这意 Eng Mater,2006.312:299 [16 Wang X B.Local and global damages of quasi britth material in 味着若峰后应力一应变曲线被视为一个本构关系， uniaxial com pression basedon gradient-dependent plasticity.Key 则混凝土柱的峰后延性及极限变形将被极大地低 Eng Ma1e,2005,293/294:719 估. 17]Wang X B.Unified amlytical stress strain curve for quasibrit tle geomaterial in uniaxial tension,direct shear and uniaxial com- 参考文献 pression.J Cent South Univ Technol,2006,13(1):99 [ Ottosen N S.Themodynamic consequences of strain softering in 18 Wang X B.Unified instability criterion for quasibrittle materia tension.J Eng Mech,1986.112(11):1152 in uniaxial tension,direct shear and uniaxial compression subject- [2]Jansen D C.Shah S P.Effect of length on compressive strain ed to shear failure Euro-C Conference 2006-Computational softening of concrete.J Eng Mech,1997,123(1):25 Modelling of Concrete Structures.London:Taybr Fmancis [3]Wang X B.Pan YS.Effect of relative stress on post-peak uniaxi- Gmup,2006:467 (下转第47页)关系, 自然使本构关系平缓, 再由这一本构关系预测 较高的试样的应力-应变曲线, 自然使预测的峰后 应力-应变曲线比实验结果平缓. 在软化阶段后期, 实测曲线与预测的峰后应力- 应变曲线尚有一定程度差别的另一方面的原因是, 在软化阶段的后期, 在剪切带的中心位置出现了一 定长度的真实剪切裂纹, 剪切裂纹之间的摩擦力将 起到重要的作用(这一现象本文的连续介质模型尚 不能描述) . 总之, 在混凝土单轴压缩条件下, 本文已经将遭 受到剪切带破坏形式的单一剪切破坏试样的峰后应 力-应变曲线斜率的解析解推广为非线性情形.在 峰值应力之前, 采用 Scott 模型描述混凝土试样均 匀变形阶段的非线性的本构关系;在应变软化阶段, 为了建立宏观应力-应变曲线峰后斜率的表达式, 使用了剪切带的非线性应变软化本构关系(式( 3) ) 、 弹性体弹性卸载本构关系(式( 5)) 、弹性体静力平衡 条件( 式 ( 7)) 和一些几何关系( 式( 5) 、式 ( 6) 及 式( 9)) .剪切带的非线性应变软化本构关系由导出 的最短的普通混凝土试样应力-应变曲线峰后斜率 反算.利用得到的峰后本构关系, 对其他较长的普 通混凝土试样( 由于试样高度过大而难于实验)的应 力-应变曲线进行了预测. 3 结论 利用本文方法预测的峰后应力-应变曲线依赖 于试样的高度.试样越高, 应力-应变曲线的峰后分 支越陡峭.对于较高的试样, 在应变软化阶段的后 期, 预测的峰后应力-应变曲线比实验结果稍显平 缓, 这意味着端面约束效应(当试样高度较矮时, 其 作用比较显著) 将使峰后应力-应变曲线变得不陡 峭. 估算的剪切带内部的平均塑性剪切应变远大于 在单轴压缩条件下测得的轴向应变的最大值 .这意 味着若峰后应力-应变曲线被视为一个本构关系, 则混凝土柱的峰后延性及极限变形将被极大地低 估. 参 考 文 献 [ 1] Ottosen N S.Thermodynamic consequences of strain softening in t ension.J Eng Mech , 1986, 112( 11) :1152 [ 2] Jansen D C, Shah S P .Effect of length on compressive strain softening of concrete.J E ng Mech , 1997, 123( 1) :25 [ 3] Wang X B, Pan Y S.Effect of relative stress on post-peak uniaxi￾al compression fracture energy of concrete .J Wuhan Univ Tech￾nol Mater Sci Ed , 2003, 18( 4) :89 [ 4] Wang X B .Analytical solution of complet e stress-strain curve in uniaxial com pression based on gradient-dependent plasticity ∥E u￾rock 2005-Impact o f Human Activit y on the Geologi ca l En viron￾men t .London:Taylor &Francis Group, 2005:661 [ 5] Wang X B.Prediction of height effect, plasti c def ormation and fractu re energy for high-strength concrete by linear shear soften￾ing constitutive relation based on energy conservation method. Mag Concr Res, 2007, 59( 5) :341 [ 6] De Borst R, Mǜ hlhaus H B .Gradient-dependent plasticity :f or￾mulation and algorithmic aspects.Int J Numer Methods Eng , 1992, 35( 3) :521 [ 7] Zhou G L, Tham L G, Lee P K K, et al.A phenomenological constitutive model f or rocks w ith shear f ailure mode.Int J Nu mer Anal Methods Geomech , 2001, 25( 4) :391 [ 8] Labuz J F, Biolzi L .C lass Ⅰ vs class Ⅱ stability :a demonstra￾tion of size effect .Int J Rock Mech Min Sci, 1991, 28 ( 2/ 3 ) : 199 [ 9] Sch reyer H L .Analytical solutions for nonlinear strain-gradient softening and localization.J Appl Mech , 1990, 57( 3) :522. [ 10] Mǜ hlhaus H B, Vardoulakis I .The thickness of shear bands in granular materials.Géotechnique, 1987, 37( 3) :271 [ 11] Alshibli K A, S tu re S.S and shear band thickness measurements by digital imaging techniques.J Comp ut Ci vil Eng , 1999, 13 ( 2) :103 [ 12] Okubo S , Nishimatsu Y .Uniaxial compression testing using a linear combination of stress and strain as the control variable . Int J Rock Mech Min S ci, 1985, 22( 5) :323 [ 13] Wang X B .Volume change of heterogeneous quasi-brittle mate￾rials in uniaxial compression.J Wuhan U niv Technol Mater Sci Ed , 2006, 21( 3) :162 [ 14] Wang X B .Analysis of the post-peak Poisson' s ratio of rock specimens in uniaxial compression.Eng Mech, 2006, 23 ( 4 ) : 99 ( 王学滨.岩样单轴压缩峰后泊松比理论研究.工程力学, 2006, 23( 4) :99) [ 15] Wang X B.Size eff ect of shear fracture energy of concret e in uniaxial com pression based on gradient-dependent plasticity .Key Eng Mat er, 2006, 312:299 [ 16] Wang X B.Local and global damages of quasi-brittle material in uniaxial com pression based on gradient-dependent plasticity .Key Eng Mat er, 2005, 293/ 294:719 [ 17] Wang X B .Unified analytical stress-strain curve for quasibrittle geomaterial in uniaxial t ension, direct shear and uniaxial com￾pression.J Cent Sou th Univ Technol, 2006, 13( 1) :99 [ 18] Wang X B.Unified instability criterion f or quasibrittle material in uniaxial tension, direct shear and uniaxial compression subject￾ed t o shear failure ∥Euro-C Con f erence 2006-Comp utationa l Modelling of Concrete S tructures.London:Taylor & Francis Group, 2006:467 ( 下转第 47 页) 第 1 期 王学滨等:单轴压缩条件下普通混凝土柱的峰后非线性尺寸效应 · 29 ·