考虑冻融劣化效应的混凝土单轴压缩统计损伤模型

白卫峰; 牛东旭; 管俊峰; 苑晨阳

doi:10.6052/j.issn.1000-4750.2022.01.0039

考虑冻融劣化效应的混凝土单轴压缩统计损伤模型

doi: 10.6052/j.issn.1000-4750.2022.01.0039

1.
华北水利水电大学水利学院，郑州 450046
2.
华北水利水电大学土木与交通学院，郑州 450046

基金项目: 国家自然科学基金项目(51679092，52179132)；河南省高等学校青年骨干教师培养计划项目(2021GGJS074)

详细信息

作者简介:
白卫峰(1982−)，男，河南人，教授，博士，博导，主要从事混凝土损伤力学研究(E-mail: yf9906@163.com)

牛东旭(1996−)，男，河北人，硕士生，主要从事混凝土损伤力学研究(E-mail: 657460765@qq.com)

苑晨阳(1988−)，男，河南人，讲师，博士，主要从事混凝土结构数值模拟研究(E-mail: yuanchenyang@ncwu.edu.cn)

通讯作者:
管俊峰(1980−)，男，河南人，教授，博士，博导，主要从事混凝土断裂力学研究(E-mail: junfengguan@ncwu.edu.cn)

中图分类号: TU528.01
计量
- 文章访问数: 233
- HTML全文浏览量: 90
- PDF下载量: 62
- 被引次数: 0
出版历程
- 收稿日期: 2022-01-07
- 修回日期: 2022-06-18
- 网络出版日期: 2022-07-14
- 刊出日期: 2023-09-06

THE STATISTICAL DAMAGE MODEL OF CONCRETE UNDER UNIAXIAL COMPRESSION CONSIDERING FREEZE-THAW DETERIORATION EFFECT

1.
School of Water Conservancy, North China University of Water Resources and Electric Power, Zhengzhou 450046, China
2.
School of Civil Engineering and Communication, North China University of Water Resources and Electric Power, Zhengzhou 450046, China

摘要

摘要: 该文建立了考虑冻融劣化效应的混凝土单轴压缩统计损伤本构模型，将整个压缩过程分为均匀损伤和局部破坏两个阶段，考虑细观屈服和断裂两种损伤模式。在冻融环境下，混凝土内部孔隙结构形态和微结构的力学特征会产生显著的变化，可由初始弹性模量E反映；在进一步承受压缩荷载过程中，微结构内部微裂纹萌生/扩展的形态、路径和数量也会由于初始冻融劣化的影响发生相应的改变，可由损伤参数ε_a、ε_h、ε_b和H表征。假设不同冻融循环次数情况下，混凝土微结构力学性能和细观损伤过程的演变趋势服从某种规律性，将上述5个特征参数定义为冻融循环次数N的函数。为验证模型的合理性，该文开展了混凝土单轴压缩试验，获得冻融循环次数N为0次~150次时对应的单轴压缩应力-应变全曲线，同时分析了文献中的5组试验数据。结果表明：预测曲线与试验曲线吻合较好，模型中的特征参数随着冻融循环次数的增加表现出明显的规律性。该模型为冻融环境下混凝土损伤机制分析和预测提供了有效工具。
- 混凝土 /
- 单轴压缩 /
- 损伤机制 /
- 本构模型 /
- 冻融循环
Abstract: A statistical damage constitutive model of concrete under uniaxial compression is proposed to consider the impact of freeze-thaw. The entire compression process is divided into two stages: uniform damage and local failure, considering two mesoscopic damage modes of yield and fracture. Under the freeze-thaw environment, the internal pore structure and the mechanical characteristics of the microstructure will change significantly, which can be reflected by the initial elastic modulus E. In the process of further bearing the compressive load, the shape, path and quantity of microcracks initiation and propagation in the microstructure will also change due to the influence of the initial freeze-thaw degradation, which could be characterized by damage parameters ε_a, ε_h, ε_b and H. Assuming that under different number of freeze-thaw cycles, the evolution trend of the mechanical properties of the microstructure and the meso-damage process obey a certain regularity, thusly define the above 5 characteristic parameters as functions of the number of freeze-thaw cyclesN. To verify the rationality of the model, the uniaxial compression test was carried out, and the stress-strain full curves of concrete were obtained when the number of freeze-thaw cycles N changed from 0 to 150. Meanwhile, five groups of experimental data in the literature were also analyzed. The results show that: prediction curves are in a good agreement with the test curves, and the characteristic parameters in the model show obvious regularity as the number of freeze-thaw cycles increases. This model provides an effective tool for both the analysis and predicting damage mechanism of concrete in the freeze-thaw environment.
- concrete /
- uniaxial compression /
- damage mechanism /
- constitutive model /
- freeze-thaw

HTML全文

图 1 微结构的“劣化”和“强化”

Figure 1. "Deterioration" and "strengthening" of microstructure

下载: 全尺寸图片幻灯片

图 2 宏观本构行为与细观损伤演化过程的对应关系

Figure 2. Relationship between mesoscopic damage mechanism and macroscopic nonlinear stress-strain behavior

下载: 全尺寸图片幻灯片

图 3 冻融劣化效应对混凝土细观损伤机制的影响

Figure 3. Influence of freeze- thaw deterioration on meso damage mechanism of concrete

下载: 全尺寸图片幻灯片

图 4 名义应力-应变曲线

Figure 4. Nominal stress-strain curves

下载: 全尺寸图片幻灯片

图 5 有效应力-应变曲线

Figure 5. Effective stress-strain curves

下载: 全尺寸图片幻灯片

图 6 FIF_σ-N关系曲线

Figure 6. FIF_σ-N curve

下载: 全尺寸图片幻灯片

图 7 FIF_ε-N关系曲线

Figure 7. FIF_ε-N curve

下载: 全尺寸图片幻灯片

图 8 FIF_E-N关系曲线

Figure 8. FIF_E-N curve

下载: 全尺寸图片幻灯片

图 9 ε⁺-N关系曲线(GP-1)

Figure 9. ε⁺-N curves (GP-1)

下载: 全尺寸图片幻灯片

图 10 H-N关系曲线(GP-1)

Figure 10. H-N curve (GP-1)

下载: 全尺寸图片幻灯片

图 11 E_v演化曲线(GP-1)

Figure 11. Evolution curves ofE_v(GP-1)

下载: 全尺寸图片幻灯片

图 12 D_R演化曲线(GP-1)

Figure 12. Evolution curves of D_R(GP-1)

下载: 全尺寸图片幻灯片

图 13 FIF_a-N关系曲线

Figure 13. FIF_a-N curve

下载: 全尺寸图片幻灯片

图 14 FIF_h-N关系曲线

Figure 14. FIF_h-N curve

下载: 全尺寸图片幻灯片

图 15 FIF_b-N关系曲线

Figure 15. FIF_b-N curve

下载: 全尺寸图片幻灯片

图 16 FIF_H-N关系曲线

Figure 16. FIF_H-N curve

下载: 全尺寸图片幻灯片

图 17 σ_cr/σ_p-N关系曲线

Figure 17. σ_cr/σ_p-N curve

下载: 全尺寸图片幻灯片

图 18 σ_E,cr/σ_E,p-N关系曲线

Figure 18. σ_E,cr/σ_E,p-N curve

下载: 全尺寸图片幻灯片

图 19 ε_cr/ε_p-N关系曲线

Figure 19. ε_cr/ε_p-N curve

下载: 全尺寸图片幻灯片

表 1 基本信息表

Table 1. Basic Information

组别	来源	试件尺寸/(mm×mm×mm)	冻融循环次数N/次	N=0
组别	来源	试件尺寸/(mm×mm×mm)	冻融循环次数N/次	弹性模量E/(×10 GPa)	峰值应力σ_p/MPa	峰值应变ε_p/(×10⁻⁴)
GP-1	本文	100×100×100	0、50、75、100、125、150	2.6	−41.3	−21.3
GP-2	王家滨等^[39]	100×100×400	0、50、100、150、200	3.0	−35.2	−17.8
GP-3	邱继生等^[8]	100×100×300	0、50、100、150、200	2.6	−32.3	−18.4
GP-4	齐振麟^[40]	100×100×400	0、100、150、200	2.1	−29.6	−24.1
GP-5	马彬等^[41]	100×100×400	0、50、100、150、200	2.7	−34.5	−23.4
GP-6	段安等^[42]	100×100×400	0、75、100、125、150	2.9	−28.7	−15.2

下载: 导出CSV

表 2 参数计算表

Table 2. Results for calculation parameter

组别	冻融循环次数N/次	弹性模量E/ (×10 GPa)	ε_a/ (×10⁻⁴)	ε_h/ (×10⁻⁴)	ε_b/ (×10⁻⁴)	H
GP-1	0	2.60	1.300	4.208	5.335	0.207
	50	1.20	2.653	5.902	7.752	0.243
	75	0.68	2.934	7.481	10.487	0.257
	100	0.40	4.039	8.926	13.197	0.264
	125	0.25	5.084	9.599	15.534	0.269
	150	0.21	5.237	10.052	16.891	0.285
GP-2	0	3.00	0.234	3.579	4.843	0.296
	50	2.00	0.349	4.902	5.556	0.315
	100	1.13	1.539	6.263	7.618	0.326
	150	0.90	1.659	7.701	8.322	0.334
	200	0.65	2.339	9.081	9.621	0.340
GP-3	0	2.60	0.264	3.009	5.292	0.239
	50	2.10	0.622	3.820	5.353	0.247
	100	1.60	1.246	4.509	5.418	0.254
	150	1.30	1.693	5.322	5.584	0.262
	200	0.90	2.504	6.288	6.649	0.271
GP-4	0	2.10	0.802	3.400	5.744	0.260
	100	1.05	3.541	5.515	8.254	0.333
	150	0.62	6.543	7.261	9.564	0.347
	200	0.43	7.308	10.000	11.130	0.350
GP-5	0	2.7	0.187	1.554	7.135	0.261
	50	1.73	0.213	3.710	8.350	0.284
	100	1.10	1.195	4.761	9.870	0.293
	150	0.76	1.647	8.095	11.350	0.315
	200	0.43	2.765	10.000	13.534	0.398
GP-6	0	2.90	0.758	1.801	4.569	0.271
	75	0.65	3.228	7.254	9.198	0.313
	100	0.35	6.634	9.026	12.134	0.324
	125	0.20	8.663	10.800	17.515	0.332
	150	0.12	9.193	11.230	18.368	0.342

下载: 导出CSV

参考文献(43)

[1]	张艺欣, 郑山锁, 裴培, 等. 钢筋混凝土柱冻融损伤模型研究[J]. 工程力学, 2019, 36(2): 78 − 86. doi: 10.6052/j.issn.1000-4750.2017.11.0791 ZHANG Yixin, ZHENG Shansuo, PEI Pei, et al. Research on the modelling method of reinforced concrete column subjected to freeze-thaw damage [J]. Engineering Mechanics, 2019, 36(2): 78 − 86. (in Chinese) doi: 10.6052/j.issn.1000-4750.2017.11.0791
[2]	时旭东, 李亚强, 李俊林, 等. 不同超低温温度区间冻融循环作用混凝土受压强度试验研究[J]. 工程力学, 2020, 37(4): 153 − 164. doi: 10.6052/j.issn.1000-4750.2019.07.0339 SHI Xudong, LI Yaqiang, LI Junlin, et al. Experimental study on the compressive strength of concrete undergoing freeze-thaw cycle actions with different ultralow temperature ranges [J]. Engineering Mechanics, 2020, 37(4): 153 − 164. (in Chinese) doi: 10.6052/j.issn.1000-4750.2019.07.0339
[3]	杨琦, 李克非. 引气混凝土在自然条件下的吸水模型研究[J]. 工程力学, 2022, 39(5): 159 − 166, 176. doi: 10.6052/j.issn.1000-4750.2021.03.0165 YANG Qi, LI Kefei. Study on water absorption model of air-entraining concrete under natural conditions [J]. Engineering Mechanics, 2022, 39(5): 159 − 166, 176. (in Chinese) doi: 10.6052/j.issn.1000-4750.2021.03.0165
[4]	DONG F Y, WANG H P, YU J T, et al. Effect of freeze–thaw cycling on mechanical properties of polyethylene fiber and steel fiber reinforced concrete [J]. Construction and Building Materials, 2021, 295(4): 123427.
[5]	ZHU X Y, CHEN X D, ZHANG N, et al. Experimental and numerical research on triaxial mechanical behavior of self-compacting concrete subjected to freeze-thaw damage [J]. Construction and Building Materials, 2021, 288: 123110. doi: 10.1016/j.conbuildmat.2021.123110
[6]	SIAMARDI K, SHABANI S. Evaluation the effect of micro-synthetic fiber on mechanical and freeze-thaw behavior of non-air-entrained roller compacted concrete pavement using response surface methodology [J]. Construction and Building Materials, 2021, 295(9): 123628.
[7]	段安, 钱稼茹. 受冻融环境混凝土的应力-应变全曲线试验研究[J]. 混凝土, 2008(8): 13 − 16. doi: 10.3969/j.issn.1002-3550.2008.08.005 DUAN An, QIAN Jiaru. Experimental study on complete compressive stress- strain curve of concrete subjected to freeze-thaw cycles [J]. Concrete, 2008(8): 13 − 16. (in Chinese) doi: 10.3969/j.issn.1002-3550.2008.08.005
[8]	邱继生, 王民煌, 关虓, 等. 煤矸石混凝土冻融损伤本构关系试验研究[J]. 科学技术与工程, 2018, 18(27): 216 − 222. doi: 10.3969/j.issn.1671-1815.2018.27.037 QIU Jisheng, WANG Minhuang, GUAN Xiao, et al. Experimental study on freezing-thawing damage constitutive relationship of coal gangue concrete [J]. Science Technology and Engineering, 2018, 18(27): 216 − 222. (in Chinese) doi: 10.3969/j.issn.1671-1815.2018.27.037
[9]	徐超. 纤维混杂效应对混凝土复合材料的力学及耐久性能的影响[J]. 功能材料, 2021, 52(1): 1202 − 1207. doi: 10.3969/j.issn.1001-9731.2021.01.030 XU Chao. Effect of fiber hybrid effect on mechanical and durability of concrete composites [J]. Journal of Functional Materials, 2021, 52(1): 1202 − 1207. (in Chinese) doi: 10.3969/j.issn.1001-9731.2021.01.030
[10]	张卫东, 董云, 彭宁波, 等. 冻融循环下透水再生混凝土力学性能损伤分析[J]. 建筑材料学报, 2020, 23(2): 292 − 296. ZHANG Weidong, DONG Yun, PENG Ningbo, et al. Analysis on mechanical properties of pervious recycled concrete by damage under freeze-thaw cycles [J]. Journal of Building Materials, 2020, 23(2): 292 − 296. (in Chinese)
[11]	GOU Y J, ZHANG L H , ZHANG H P, et al. Investigation of freeze-thaw mechanism for crumb rubber concrete by the online strain sensor [J]. Measurement, 2021, 174(8): 109080.
[12]	LIU M X, LIU D X, QIAO P Z, et al. Characterization of microstructural damage evolution of freeze-thawed shotcrete by an integrative micro-CT and nanoindentation statistical approach [J]. Cement and Concrete Composites, 2021, 117: 103909. doi: 10.1016/j.cemconcomp.2020.103909
[13]	SUZUKI T, NISHIMURA S, SHIMAMOTO Y, et al. Damage estimation of concrete canal due to freeze and thawed effects by acoustic emission and X-ray CT methods [J]. Construction and Building Materials, 2020, 245: 118343. doi: 10.1016/j.conbuildmat.2020.118343
[14]	LI B, MAO J, NAWA T, et al. Mesoscopic damage model of concrete subjected to freeze-thaw cycles using mercury intrusion porosimetry and differential scanning calorimetry (MIP-DSC) [J]. Construction and Building Materials, 2017, 147: 79 − 90. doi: 10.1016/j.conbuildmat.2017.04.136
[15]	解国梁, 申向东, 刘金云, 等. 玄武岩纤维再生混凝土抗冻性能及损伤劣化模型[J]. 复合材料科学与工程, 2021(4): 55 − 60. doi: 10.19936/j.cnki.2096-8000.20210428.008 XIE Guoliang, SHEN Xiangdong, LIU Jinyun, et al. Frost resistance and damage degradation model of basalt fiber regenerated concrete [J]. Composites Science and Engineering, 2021(4): 55 − 60. (in Chinese) doi: 10.19936/j.cnki.2096-8000.20210428.008
[16]	关虓, 牛狄涛, 肖前慧. 考虑残余强度修正的混凝土冻融损伤层及轴心受压模型研究[J]. 铁道学报, 2021, 43(3): 175 − 182. doi: 10.3969/j.issn.1001-8360.2021.03.022 GUAN Xiao, NIU Ditao, XIAO Qianhui. Study on concrete freezing-thawing damage layer considering residual strength correction and model under axial compression [J]. Journal of the China Railway Society, 2021, 43(3): 175 − 182. (in Chinese) doi: 10.3969/j.issn.1001-8360.2021.03.022
[17]	王青, 卫泽众, 徐港, 等. 盐冻融环境下钢筋混凝土黏结-滑移本构模型[J]. 建筑材料学报, 2017, 20(5): 692 − 699, 793. doi: 10.3969/j.issn.1007-9629.2017.05.006 WANG Qing, WEI Zezhong, XU Gang, et al. Bond-slip model for reinforced concrete under deicer-frost environment [J]. journal of Building Materials, 2017, 20(5): 692 − 699, 793. (in Chinese) doi: 10.3969/j.issn.1007-9629.2017.05.006
[18]	郑山锁, 赵鹏. 冻融循环条件下砌体受压损伤本构模型[J]. 工业建筑, 2015, 45(2): 15 − 18. ZHENG Shangsuo, ZHAO Peng. Constitutive relationship model for masonry in compression under action of freeze-thaw cycle [J]. Industrial Construction, 2015, 45(2): 15 − 18. (in Chinese)
[19]	白以龙, 汪海英, 夏蒙棼, 等. 固体的统计细观力学——连接多个耦合的时空尺度[J]. 力学进展, 2006, 36(2): 286 − 305. doi: 10.3321/j.issn:1000-0992.2006.02.012 BAI Yilong, WANG Haiying, XIA Mengfen, et al. Statistical mesomechanics of solid, linking coupled multiple space and time scales [J]. Advances in Mechanics, 2006, 36(2): 286 − 305. (in Chinese) doi: 10.3321/j.issn:1000-0992.2006.02.012
[20]	KRAJCINOVIC D, SILVA M A G. Statistical aspects of the continuous damage theory [J]. International Journal of Solids and Structures, 1982, 18: 551 − 562. doi: 10.1016/0020-7683(82)90039-7
[21]	白卫峰, 陈健云, 胡志强, 等. 准脆性材料单轴拉伸破坏全过程物理模型研究[J]. 岩石力学与工程学报, 2007, 26(4): 670 − 681. doi: 10.3321/j.issn:1000-6915.2007.04.003 BAI Weifeng, CHEN Jianyun, HU Zhiqiang, et al. Study on physical model of complete failure process of quasi-brittle materials in tension [J]. Journal of Rock Mechanics and Engineering, 2007, 26(4): 670 − 681. (in Chinese) doi: 10.3321/j.issn:1000-6915.2007.04.003
[22]	陈建云, 白卫峰. 考虑动态应变率效应的混凝土单轴拉伸统计损伤模型[J]. 岩石力学与工程学报, 2007, 26(8): 1603 − 1611. doi: 10.3321/j.issn:1000-6915.2007.08.010 CHEN Jianyun, BAI Weifeng. Statistical damage model of concrete under uniaxial tension considering dynamic strain-rate effect [J]. Journal of rock mechanics and Engineering, 2007, 26(8): 1603 − 1611. (in Chinese) doi: 10.3321/j.issn:1000-6915.2007.08.010
[23]	白卫峰, 管俊峰, 崔莹, 等. 混凝土双轴压-压细观统计损伤本构模型[J]. 四川大学学报(工程科学版), 2013, 45(6): 74 − 81. doi: 10.15961/j.jsuese.2013.06.004 BAI Weifeng, GUAN Junfeng, CUI Ying, et al. Statistical damage constitutive model for concrete under biaxial compression [J]. Journal of Sichuan University (Engineering Science Edition), 2013, 45(6): 74 − 81. (in Chinese) doi: 10.15961/j.jsuese.2013.06.004
[24]	白卫峰, 张树珺, 管俊峰, 等. 混凝土正交各向异性统计损伤本构模型研究[J]. 水利学报, 2014, 45(5): 607 − 618. doi: 10.13243/j.cnki.slxb.2014.05.012 BAI Weifeng, ZHANG Shujun, GUAN Junfeng, et al. Orthotropic statistical damage constitutive model for concrete [J]. Journal of Hydraulic Engineering, 2014, 45(5): 607 − 618. (in Chinese) doi: 10.13243/j.cnki.slxb.2014.05.012
[25]	白卫峰, 刘霖艾, 管俊峰, 等. 基于统计损伤理论的硫酸盐侵蚀混凝土本构模型研究[J]. 工程力学, 2019, 36(2): 66 − 77. doi: 10.6052/j.issn.1000-4750.2017.09.0734 BAI Weifeng, LIU Linai, GUAN Junfeng, et al. Study on constitutive model of sulfate attack concrete based on statistical damage theory [J]. Engineering Mechanics, 2019, 36(2): 66 − 77. (in Chinese) doi: 10.6052/j.issn.1000-4750.2017.09.0734
[26]	白卫峰, 韩浩田, 管俊峰, 等. 考虑高温劣化效应的混凝土统计损伤本构模型研究[J]. 应用基础与工程科学学报, 2020, 28(6): 1397 − 1409. BAI Weifeng, HAN Haotian, GUAN Junfeng, et al. Statistical damage model of concrete considering the effect of high temperature degradation [J]. Journal of Basic Science and Engineering, 2020, 28(6): 1397 − 1409. (in Chinese)
[27]	白卫峰. 混凝土损伤机理及饱和混凝土力学性能研究[D]. 大连: 大连理工大学, 2008: 220. BAI Weifeng. Study on damage mechanism of concrete and mechanical property of saturated concrete [D]. Dalian: Dalian University of Technology, 2008: 220. (in Chinese)
[28]	白卫峰, 李思蕾, 管俊峰, 等. 再生混凝土的单轴压缩动态力学性能[J]. 建筑材料学报, 2022, 25(5): 498 − 508. doi: 10.3969/j.issn.1007-9629.2022.05.009 BAI Weifeng, LI Silei, GUAN Junfeng, et al. Dynamic mechanical properties of recycled concrete under uniaxial compression [J]. Journal of Building Materials, 2022, 25(5): 498 − 508. (in Chinese) doi: 10.3969/j.issn.1007-9629.2022.05.009
[29]	郝圣旺. 非均匀脆性介质损伤演化的一维准静态线性失稳及其发展[J]. 燕山大学学报, 2011, 35(5): 459 − 464. doi: 10.3969/j.issn.1007-791X.2011.05.017 HAO Shengwang. One-dimension damage evolution instability and its growth of heterogeneous brittle material under quasi-static loading [J]. Journal of YanShan University, 2011, 35(5): 459 − 464. (in Chinese) doi: 10.3969/j.issn.1007-791X.2011.05.017
[30]	张晓君. 岩石损伤统计本构模型参数及其临界敏感性分析[J]. 采矿与安全工程学报, 2010, 27(1): 45 − 50. doi: 10.3969/j.issn.1673-3363.2010.01.009 ZHANG Xiaojun. Parameters of statistical damage constitutive model for rocks and its critical sensibility analysis [J]. Journal of Mining and Safety Engineering, 2010, 27(1): 45 − 50. (in Chinese) doi: 10.3969/j.issn.1673-3363.2010.01.009
[31]	VAN GEEL E. Concrete behaviour in multiaxial compression: Experimental research [D]. Eindhoven: Eindhoven University, 1998: 169.
[32]	纪洪广, 贾立宏, 李造鼎. 混凝土材料在单轴拉伸时的声发射机理探讨[J]. 东北大学学报, 1995(6): 568 − 572. JI Hongguang, JIA Lihong, LI Zaoding. Approach to acoustic emission mechanism of concrete in uniaxial tension [J]. Journal of Northeastern University (Natural Science), 1995(6): 568 − 572. (in Chinese)
[33]	董毓利, 谢和平, 李玉寿. 砼受压全过程声发射特性及其损伤本构模型[J]. 力学与实践, 1995(4): 25 − 28. DONG Yuli, XIE Heping, LI Yushou. Acoustic emission characteristics and damage constitutive model of concrete under compression [J]. Mechanics in Engineering, 1995(4): 25 − 28. (in Chinese)
[34]	BAI Weifeng, CHEN Jianyun, LIN Gao, et al. The statistical damage constitutive model for concrete materials under uniaxial compression [J]. Journal of Harbin Institute of Technology, 2010, 17(3): 338 − 344.
[35]	白卫峰, 沈鋆鑫, 管俊峰, 等. 基于统计损伤理论的混凝土应力-应变行为[J]. 建筑材料学报, 2021, 24(3): 551 − 561. doi: 10.3969/j.issn.1007-9629.2021.03.015 BAI Weifeng, SHEN Junxin, GUAN Junfeng, et al. Stress-strain behavior of concrete based on statistical damage theory [J]. Journal of Building Materials, 2021, 24(3): 551 − 561. (in Chinese) doi: 10.3969/j.issn.1007-9629.2021.03.015
[36]	POWERS T C, HELMUTH R A. Theory of volume changes in hardened portland cement paste during freezing [J]. Highway Research Board Proceedings, 1953, 32: 285 − 297.
[37]	王立久, 袁大伟. 关于混凝土冻害机理的思考[J]. 低温建筑技术, 2007(5): 1 − 3. doi: 10.3969/j.issn.1001-6864.2007.05.001 WANG Lijiu, YUAN Dawei. The research of mechanism of freeze-thaw damage in concrete [J]. Low Temperature Architecture Technology, 2007(5): 1 − 3. (in Chinese) doi: 10.3969/j.issn.1001-6864.2007.05.001
[38]	李金玉, 曹建国, 徐文雨, 等. 混凝土冻融破坏机理的研究[J]. 水利学报, 1999(1): 42 − 50. doi: 10.3321/j.issn:0559-9350.1999.01.008 LI Jinyu, CAO Jianguo, XU Wenyu, et al. Study on the mechanism of concrete destruction under frost action [J]. Journal of Hydraulic Engineering, 1999(1): 42 − 50. (in Chinese) doi: 10.3321/j.issn:0559-9350.1999.01.008
[39]	王家滨, 牛狄涛, 袁斌. 冻融损伤喷射混凝土本构关系及微观结构[J]. 土木建筑与环境工程, 2016, 38(1): 30 − 39. WANG Jianbin, NIU Ditao, YUAN Bin. Constitutive relation and microstructure on shotcrete after freeze and thaw damage [J]. Journal of Civil and Environment Engineering, 2016, 38(1): 30 − 39. (in Chinese)
[40]	齐振麟. 冻融损伤后再生混凝土单调及重复荷载下本构关系研究[D]. 哈尔滨: 哈尔滨工业大学, 2016: 90. QI Zhenlin. Study on constitutive relation of recycled aggregate concrete damaged by freeze-thaw under monotonic and cyclic loading [D]. Harbin: Harbin Institute of Technology, 2016: 90. (in Chinese)
[41]	马彬, 叶英华, 孙洋. 基于损伤模型的盐蚀、冻融混凝土力学性能分析[J]. 建筑结构学报, 2009, 30(增刊 2): 298 − 302. MA Bin, YE Yinghua, SUN Yang. Mechanical performance analysis of salt-frozen or freeze-thaw concrete based on damage theory [J]. Journal of Building Structures, 2009, 30(Suppl 2): 298 − 302. (in Chinese)
[42]	段安. 受冻融混凝土本构关系研究和冻融过程数值模拟[D]. 北京: 清华大学, 2009: 129. DUAN An. Research on constitutive relationship of frozen-thawed concrete and mathematical modeling of freeze-thaw process [D]. Beijing: Tsinghua University, 2009: 129. (in Chinese)
[43]	XIAO J, ZHANG K, AKBARNEZHAD A. Variability of stress-strain relationship for recycled aggregate concrete under uniaxial compression loading [J]. Journal of Cleaner Production, 2018, 181: 753 − 771. doi: 10.1016/j.jclepro.2018.01.247