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Article(id=1322595348774761227, tenantId=1146029695717560320, journalId=1225396423026438145, issueId=1322595047657296718, articleNumber=null, orderNo=null, doi=null, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1712592000000, receivedDateStr=2024-04-09, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793260903781, onlineDateStr=2026-10-29, pubDate=1742832000000, pubDateStr=2025-03-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793260903781, onlineIssueDateStr=2026-10-29, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793260903781, creator=13701087609, updateTime=1793260903781, updator=13701087609, issue=Issue{id=1322595047657296718, tenantId=1146029695717560320, journalId=1225396423026438145, year='2025', volume='45', issue='3', pageStart='1', pageEnd='288', issueExtLink='null', onlineDate='null', pubDate='1742832000000', pubDateStr='2025-03-25', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793260831989, creator='13701087609', updateTime=1793260831989, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=119, endPage=127, ext={EN=ArticleExt(id=1322595348980282124, articleId=1322595348774761227, tenantId=1146029695717560320, journalId=1225396423026438145, language=EN, title=Damage Mechanical Properties and Energy Evolution of Hard Rock Under Cyclic Loading and Unloading Effects with Different Upper Limit Stresses, columnId=null, journalTitle=Mining Research and Development, columnName=null, runingTitle=null, highlight=null, articleAbstract=

The underground rock body is often affected by external factors, after repeated cycles of loading and unloading, it will lead to the increase of internal cracks in rock and the change of local stress environment. Therefore, the triaxial graded cyclic loading and unloading tests with different upper limit stresses were carried out on sandstone, and the damage mechanical properties and the laws of energy evolution of sandstone were obtained. The test results show that under the conditions of graded cyclic loading and unloading, the stress-strain curve of sandstone is characterized by hysteresis loops and downward bending, and the hysteresis loops are gradually enlarged with the increase of stress. Under the cyclic loading and unloading with different upper limit stresses, as the number of cycles increasing, the axial elastic strain and axial plastic strain of sandstone show linear relationships with the number of cycles, while the sudden drop of axial plastic strain indicates that the internal pore cracks of sandstone are compacted rapidly. Meanwhile, the input energy and elastic energy curves are very similar during the cycling process, showing an upward trend of nearly exponential curve. Dissipation energy grows relatively slowly in the initial cycling stage, which indicates that the original cracks in the sandstone are compressed and closed; while in the mid-cycling stage, the dissipation energy shows a nearly linear growth trend, which indicates that the rock samples are in the stage of stable expansion of the cracks. The dissipation energy ratio shows a sudden decrease in the initial cycling stage; while in the mid-cycling stage, it maintains an approximate horizontal trend. In addition, under the graded cyclic loading and unloading, there is a linear relationship between the degree of damage to sandstone and the number of cycles, indicating that the number of load cycles has a significant impact on the degree of damage to sandstone.

, authors=Jianping HAO1, Chunliang DONG2, 3, Mengyuan YAO1, authorsList=Jianping HAO, Chunliang DONG, Mengyuan YAO, authorCompany=null, correspAuthors=null, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, fund=null), CN=ArticleExt(id=1322595352633520935, articleId=1322595348774761227, tenantId=1146029695717560320, journalId=1225396423026438145, language=CN, title=不同上限应力循环加卸载作用下硬岩损伤力学特性与能量演化研究, columnId=1226462295417598973, journalTitle=矿业研究与开发, columnName=矿山安全与环保, runingTitle=null, highlight=null, articleAbstract=

地下岩体常受外部因素影响，反复的循环加卸载会导致岩石内部裂纹不断增多，局部应力环境亦发生变化。因此，对砂岩进行了不同上限应力值的三轴分级循环加卸载试验，分析得到了不同上限应力值下砂岩的损伤力学特性以及能量演化规律。试验结果表明：在分级循环加卸载的条件下，砂岩的应力–应变曲线呈现出滞回环和向下弯曲的特征，并且随着应力的增大，滞回环逐渐扩大。在不同上限应力的循环加卸载作用下，随着循环次数的增加，砂岩的轴向弹性应变和轴向塑性应变与循环次数呈线性关系，而轴向塑性应变发生骤降则表明砂岩内部孔隙裂缝被快速压实。同时，在循环过程中输入能与弹性能曲线非常相似，呈现出近似指数曲线的上升趋势。耗散能在初始循环阶段增长相对缓慢，这表明砂岩原有裂纹发生压缩与闭合；而在循环中期，耗散能呈现出近似线性的增长趋势，这表示岩样正处于裂纹稳定扩展阶段。耗能比在初始循环阶段呈现骤降的趋势，而在循环中期阶段耗能比曲线则保持近似水平的趋势。此外，在分级循环加卸载条件下，砂岩的损伤度与循环次数之间呈现线性关系，说明荷载循环次数对砂岩的损伤度有较大的影响。

, authors=郝建平1, 董春亮2, 3, 姚梦圆1, authorsList=郝建平, 董春亮, 姚梦圆, authorCompany=null, correspAuthors=null, authorNote=

郝建平（1997—），男，安徽安庆人，硕士研究生，主要从事岩石力学理论和应用研究。E-mail:

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郝建平（1997—），男，安徽安庆人，硕士研究生，主要从事岩石力学理论和应用研究。E-mail:

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郝建平（1997—），男，安徽安庆人，硕士研究生，主要从事岩石力学理论和应用研究。E-mail:

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Energy analysis on damage of coal under cyclical triaxial loading and unloading conditions[J]. Journal of Coal Society, 2014, 39(2):245-252., articleTitle=Energy analysis on damage of coal under cyclical triaxial loading and unloading conditions, refAbstract=null)], funds=[Fund(id=1322595942050676908, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322595348774761227, awardId=52074006, language=CN, fundingSource=国家自然科学基金项目(52074006), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1322595937038483573, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322595348774761227, xref=1., ext=[AuthorCompanyExt(id=1322595937055260790, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322595348774761227, companyId=1322595937038483573, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1.School of Civil Engineering and Architecture, Anhui University of Science and Technology, Huainan, Anhui 232001, China), AuthorCompanyExt(id=1322595937072038007, 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articleId=1322595348774761227, language=EN, label=Fig.7, caption=Elastic energy, cumulative dissipation energy and damage degree of sandstone specimens under different numbers of cyclic loading and unloading, figureFileSmall=UEGlWQ3djImNvV9Z8Wr+Qw==, figureFileBig=ZickES+SWE7dmksEXsc/VQ==, tableContent=null), ArticleFig(id=1322595941467668645, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322595348774761227, language=CN, label=图7, caption=不同的循环加卸载次数下砂岩的弹性能、累积耗散能和损伤度, figureFileSmall=UEGlWQ3djImNvV9Z8Wr+Qw==, figureFileBig=ZickES+SWE7dmksEXsc/VQ==, tableContent=null), ArticleFig(id=1322595941534777510, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322595348774761227, language=EN, label=Table 1, caption=

Basic physical properties of sandstone specimens and experiment schemes

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样本编号直径/mm高度/mm纵波波速/(m/s)试验方案
R-150.0099.953 448常规三轴压缩
R-250.0299.973 390
R-349.96100.003 333
R-0.450.0099.963 440三轴循环加卸载
R-0.649.97100.023 448
R-0.849.95100.043 500
), ArticleFig(id=1322595941627052199, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322595348774761227, language=CN, label=表1, caption=

砂岩试件的基本物理性质及试验方案

, figureFileSmall=null, figureFileBig=null, tableContent=
样本编号直径/mm高度/mm纵波波速/(m/s)试验方案
R-150.0099.953 448常规三轴压缩
R-250.0299.973 390
R-349.96100.003 333
R-0.450.0099.963 440三轴循环加卸载
R-0.649.97100.023 448
R-0.849.95100.043 500
), ArticleFig(id=1322595941702549672, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322595348774761227, language=EN, label=Table 2, caption=

Calculated results of energy density of the sandstone under different upper limit stresses

, figureFileSmall=null, figureFileBig=null, tableContent=
围压/MPa样本编号循环次数上限应力/MPa输入能/(μJ·mm–3)弹性能/(μJ·mm–3)耗散能/(μJ·mm–3)耗能比/%
4R-0.4127.003.481.701.7851.15
234.646.394.182.2134.59
342.2810.927.962.9627.11
449.9216.5912.733.8523.21
557.5623.1018.374.7320.48
4R-0.6127.003.331.481.8655.86
234.646.063.712.3538.78
342.289.987.282.7027.05
449.9215.6011.983.6223.21
557.5621.9617.754.2119.17
665.2129.3924.015.3818.31
772.8537.8731.616.2616.53
880.4947.5839.659.1319.19
4R-0.8127.003.361.061.9257.14
234.645.963.352.6243.96
342.2810.186.753.4332.81
449.9215.6211.074.5529.13
557.5622.2917.574.7221.18
665.2129.4323.585.8419.84
772.8537.7831.046.7317.81
880.4947.3239.178.1417.20
988.1357.8448.249.6016.60
1095.7769.8858.3411.5316.50
11103.4183.1869.3013.8816.69
12111.0598.0581.4616.5816.91
), ArticleFig(id=1322595941782241449, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322595348774761227, language=CN, label=表2, caption=

不同上限应力作用下砂岩能量密度计算结果

, figureFileSmall=null, figureFileBig=null, tableContent=
围压/MPa样本编号循环次数上限应力/MPa输入能/(μJ·mm–3)弹性能/(μJ·mm–3)耗散能/(μJ·mm–3)耗能比/%
4R-0.4127.003.481.701.7851.15
234.646.394.182.2134.59
342.2810.927.962.9627.11
449.9216.5912.733.8523.21
557.5623.1018.374.7320.48
4R-0.6127.003.331.481.8655.86
234.646.063.712.3538.78
342.289.987.282.7027.05
449.9215.6011.983.6223.21
557.5621.9617.754.2119.17
665.2129.3924.015.3818.31
772.8537.8731.616.2616.53
880.4947.5839.659.1319.19
4R-0.8127.003.361.061.9257.14
234.645.963.352.6243.96
342.2810.186.753.4332.81
449.9215.6211.074.5529.13
557.5622.2917.574.7221.18
665.2129.4323.585.8419.84
772.8537.7831.046.7317.81
880.4947.3239.178.1417.20
988.1357.8448.249.6016.60
1095.7769.8858.3411.5316.50
11103.4183.1869.3013.8816.69
12111.0598.0581.4616.5816.91
), ArticleFig(id=1322595941849350314, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322595348774761227, language=EN, label=Table 3, caption=

Damage to sandstone specimens under different upper limit stresses

, figureFileSmall=null, figureFileBig=null, tableContent=
围压/MPa样本编号上限应力/MPa循环次数ΔEd/(μJ·mm–3)损伤度D
4R-0.457.5656.150.35
4R-0.680.49811.990.56
4R-0.8111.051223.010.73
), ArticleFig(id=1322595941908070571, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322595348774761227, language=CN, label=表3, caption=

不同上限应力作用下砂岩的损伤

, figureFileSmall=null, figureFileBig=null, tableContent=
围压/MPa样本编号上限应力/MPa循环次数ΔEd/(μJ·mm–3)损伤度D
4R-0.457.5656.150.35
4R-0.680.49811.990.56
4R-0.8111.051223.010.73
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不同上限应力循环加卸载作用下硬岩损伤力学特性与能量演化研究
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郝建平 1 , 董春亮 2, 3 , 姚梦圆 1
矿业研究与开发 | 矿山安全与环保 2025,45(3): 119-127
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矿业研究与开发 |矿山安全与环保 2025 , 45 (3) : 119 -127
不同上限应力循环加卸载作用下硬岩损伤力学特性与能量演化研究
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郝建平1 , 董春亮2, 3, 姚梦圆1
作者信息
  • 1.安徽理工大学 土木建筑学院，安徽　淮南市　232001
  • 2.安徽理工大学 力学与光电物理学院，安徽　淮南市　232001
  • 3.安徽理工大学 深部煤矿采动响应与灾害防控国家重点实验室，安徽　淮南市　232001
作者简介:

郝建平（1997—），男，安徽安庆人，硕士研究生，主要从事岩石力学理论和应用研究。E-mail:

Damage Mechanical Properties and Energy Evolution of Hard Rock Under Cyclic Loading and Unloading Effects with Different Upper Limit Stresses
Jianping HAO1 , Chunliang DONG2, 3, Mengyuan YAO1
Affiliations
  • 1.School of Civil Engineering and Architecture, Anhui University of Science and Technology, Huainan, Anhui 232001, China
  • 2.School of Mechanics and Optoelectronic Physics, Anhui University of Science and Technology, Huainan, Anhui 232001, China
  • 3.State Key Laboratory of Response to Mining Activity and Disaster Prevention and Control of Deep Coal Mining, Anhui University of Science and Technology, Huainan, Anhui 232001, China
出版时间: 2025-03-25
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地下岩体常受外部因素影响，反复的循环加卸载会导致岩石内部裂纹不断增多，局部应力环境亦发生变化。因此，对砂岩进行了不同上限应力值的三轴分级循环加卸载试验，分析得到了不同上限应力值下砂岩的损伤力学特性以及能量演化规律。试验结果表明：在分级循环加卸载的条件下，砂岩的应力–应变曲线呈现出滞回环和向下弯曲的特征，并且随着应力的增大，滞回环逐渐扩大。在不同上限应力的循环加卸载作用下，随着循环次数的增加，砂岩的轴向弹性应变和轴向塑性应变与循环次数呈线性关系，而轴向塑性应变发生骤降则表明砂岩内部孔隙裂缝被快速压实。同时，在循环过程中输入能与弹性能曲线非常相似，呈现出近似指数曲线的上升趋势。耗散能在初始循环阶段增长相对缓慢，这表明砂岩原有裂纹发生压缩与闭合；而在循环中期，耗散能呈现出近似线性的增长趋势，这表示岩样正处于裂纹稳定扩展阶段。耗能比在初始循环阶段呈现骤降的趋势，而在循环中期阶段耗能比曲线则保持近似水平的趋势。此外，在分级循环加卸载条件下，砂岩的损伤度与循环次数之间呈现线性关系，说明荷载循环次数对砂岩的损伤度有较大的影响。

砂岩  /  力学特性  /  能量演化  /  分级循环加卸载  /  上限应力

The underground rock body is often affected by external factors, after repeated cycles of loading and unloading, it will lead to the increase of internal cracks in rock and the change of local stress environment. Therefore, the triaxial graded cyclic loading and unloading tests with different upper limit stresses were carried out on sandstone, and the damage mechanical properties and the laws of energy evolution of sandstone were obtained. The test results show that under the conditions of graded cyclic loading and unloading, the stress-strain curve of sandstone is characterized by hysteresis loops and downward bending, and the hysteresis loops are gradually enlarged with the increase of stress. Under the cyclic loading and unloading with different upper limit stresses, as the number of cycles increasing, the axial elastic strain and axial plastic strain of sandstone show linear relationships with the number of cycles, while the sudden drop of axial plastic strain indicates that the internal pore cracks of sandstone are compacted rapidly. Meanwhile, the input energy and elastic energy curves are very similar during the cycling process, showing an upward trend of nearly exponential curve. Dissipation energy grows relatively slowly in the initial cycling stage, which indicates that the original cracks in the sandstone are compressed and closed; while in the mid-cycling stage, the dissipation energy shows a nearly linear growth trend, which indicates that the rock samples are in the stage of stable expansion of the cracks. The dissipation energy ratio shows a sudden decrease in the initial cycling stage; while in the mid-cycling stage, it maintains an approximate horizontal trend. In addition, under the graded cyclic loading and unloading, there is a linear relationship between the degree of damage to sandstone and the number of cycles, indicating that the number of load cycles has a significant impact on the degree of damage to sandstone.

Sandstone  /  Mechanical property  /  Energy evolution  /  Graded cyclic loading and unloading  /  Upper limit stress
郝建平, 董春亮, 姚梦圆. 不同上限应力循环加卸载作用下硬岩损伤力学特性与能量演化研究. 矿业研究与开发, 2025 , 45 (3) : 119 -127 .
Jianping HAO, Chunliang DONG, Mengyuan YAO. Damage Mechanical Properties and Energy Evolution of Hard Rock Under Cyclic Loading and Unloading Effects with Different Upper Limit Stresses[J]. Mining Research and Development, 2025 , 45 (3) : 119 -127 .
正文
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0　引言
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在地下硐室开挖、隧道掘进和矿物开采过程中[1]，岩体经历由原岩应力状态到应力重分布再到后续开挖和采动的复杂过程，始终处于加载、卸载和再加载的循环加卸载应力状态，内部裂隙萌生的同时伴随着能量的积聚、耗散和突然释放。循环加卸载导致岩体变形破坏和力学性能劣化的本质是损伤累积和能量转化的结果[2-4]。研究循环加卸载下岩石的力学性能劣化和能量演化规律，可以更深层次地揭示岩体变形破坏的机制，为地下岩体工程的稳定安全提供理论依据。
目前，循环加卸载条件下岩石的力学损伤特性与能量演化规律已经受到了广泛的关注与研究。王天佐等[5]通过对红砂岩进行单轴恒下限循环加卸载和变下限循环加卸载试验，研究了不同循环加卸载路径下红砂岩的损伤演化规律。刘向御等[6]通过对粉砂岩进行单轴等幅循环、逐级一次循环加卸载试验，揭示了循环荷载下地下工程围岩体孔裂隙扩展及破坏机制。唐胡丹等[7]通过对石灰岩进行单轴不等幅循环加卸载试验，探索了岩体空间微裂隙扩展规律和渐进破坏过程中的非线性机制。郭启文等[8]通过对矽卡岩试样进行不同加载速率的单轴恒下限循环加卸载和变下限循环加卸载试验，对比分析了不同加载速率两种循环加卸载方式下试样的变形特征、能量演化规律以及破坏特性。袁和川等[9]通过对饱水黄砂岩进行分级等幅循环加卸载试验，揭示分级循环加卸载下饱水砂岩的变形破坏规律。王瑞红等[10]为研究节理岩体疲劳荷载条件下累积损伤宏细观力学特征及破坏机制，开展节理砂岩不同上限应力比等幅循环加卸载试验。徐颖等[3]通过对泥岩进行逐渐增加荷载等级的循环加卸载试验，深入探究了泥岩的能量演化和损伤特性。孟庆彬等[11]为探究围压对受载岩样能量演化特征的影响规律，开展了6种高围压下岩样的三轴峰前、后循环加卸载试验，揭示了高围压下受载岩样的能量演化过程及分配规律。EBERHARDT等[12]通过进行岩石的单轴等幅循环加卸载试验，详细研究了岩石在破裂过程中的损伤力学特性以及破裂准则。ZHENG等[13]通过对岩石进行变下限分级等幅循环加卸载试验，研究了循环加卸载作用下岩石不连续的能量演化规律。ZHU等[14]通过对非均匀岩体进行单轴等幅循环加卸载试验，研究了循环载荷作用下缺陷对岩体疲劳损伤和强度的影响。MENG等[15]通过对不同高径比的岩石进行单轴分级循环加卸载试验，分析了高径比对岩石试样单轴抗压强度、变形参数和破坏方式的影响。
综上所述，关于不同载荷作用下三轴分级循环加卸载导致的硬岩损伤力学特性的研究尚显不足。因此，本文采用不同上限应力分级循环加卸载的方法，系统地分析了硬岩的损伤力学特性与能量演化规律。研究成果不仅可为工程施工中地质灾害防治和安全施工提供重要依据，同时也可为优化工程设计提供有力支撑。
1　试验材料及方案
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1.1　试验材料
本次试验所采用的砂岩是从某矿场获取的岩样，并选取了具有较好完整性和均质性的砂岩作为研究对象。试验使用的砂岩试件为50 mm×100 mm的圆柱体，砂岩的两端面平行度在0.05 mm以内，表面平面度在0.02 mm以内。为了减少砂岩均匀性对试验的影响，采用了非金属超声波检测仪对砂岩进行纵向波速测量，以剔除离散性较大的试样，选择纵波波速在3 330~3 500 m/s范围内的砂岩试件开展试验。砂岩试件的基本物理性质见表1。
1.2　试验设备及方案
本次试验采用安徽理工大学自制的ZST-1500煤岩动静载荷自适应多场耦合试验系统开展常规三轴试验与分级循环加卸载试验。试验前制定了恒围压卸载应力砂岩的逐级增加荷载的循环加卸载试验方案。按轴向应力控制加载、卸载，平均抗压强度为135 MPa，其加载、卸载速率均设定为1.5 kN/s，以每级2 MPa的增量使砂岩的上限应力值σmax分别达到最大应力σc的40%、60%和80%；同时为防止试样与试验机压头在卸载阶段脱离接触，设计最低下限应力为最大应力的10%。不同上限应力循环加卸载应力路径如图1所示。
2　试验结果与分析
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2.1　循环加卸载应力–应变曲线
本研究的主要目标是探讨在不同上限应力循环加卸载的条件下，砂岩的损伤力学特性以及能量演化的规律。由于论文篇幅的限制，图2仅展示了最大上限应力值为0.8σc的砂岩应力–应变曲线。
根据图2可知，砂岩的应力–应变曲线呈现出明显的滞回环，这些滞回环并不完全重合，并且在循环加卸载作用下产生不可逆转的残余塑性永久变形。在相同下限应力时，砂岩的循环加卸载应力–应变曲线不重合，导致出现明显的滞回环，滞回环的面积随上限应力的增大呈现递增趋势，且滞回环在循环加卸载过程中不断向右侧移动，说明随着上限应力的增加，每次循环加卸载的能量损耗值在增大，且砂岩内部结构发生破坏以及累积损伤劣化等情况。滞回环区域的大小可以反映出单次循环加卸载过程中能量损耗值、阻尼特性、动弹性模量等的变化规律，可以为研究砂岩动态力学特性提供有效的依据[16]。
由图3还可知，在循环的早期加载阶段，同一加载、卸载周期内，砂岩的应力与应变之间呈现出线性增长的关系。然而，随着加载阶段的深入至后期加载阶段，砂岩的应力–应变曲线开始呈现出向下弯曲的趋势。此外，还可以发现，在低应力水平下，加载-卸载循环路径之间的间隙明显小于高应力水平时的间隙，且随着循环加卸载次数的增加，这些间隙逐渐扩大。这些现象的产生主要归因于砂岩在三轴分级循环压缩下的应力–应变曲线与常规三轴压缩应力–应变曲线之间存在一定的相似性，并且砂岩的刚度在此过程中发生了退化[17]。
2.2　变形特征分析
由于砂岩属于弹塑性材料，在循环加卸载作用时，会产生轴向应变、轴向弹性应变和轴向残余塑性应变。每次循环加卸载后，砂岩将产生新的轴向应变ε1、轴向弹性应变εe及轴向残余塑性应变εd，它们之间的关系[18]为：
根据式(1)可得出轴向应变、弹性应变和残余塑性应变与循环加卸载次数的关系，如图3所示。由图3(a)可知，随着循环加卸载次数的增加，砂岩的轴向弹性应变与循环次数呈现显著相关的线性关系，不同循环次数的轴向弹性应变的回归决定系数R2均大于0.998 72。当循环次数分别达到5, 8，12次时，轴向弹性应变分别为0.095, 0.151, 0.206。由图3(b)可知，随着循环次数的增加，轴向残余塑性应变与循环次数之间呈现出“L”形变化趋势。在第1次循环加卸载作用下，轴向残余塑性应变出现了骤降，其变化幅度与后续各次循环加卸载下所产生的轴向塑性应变的差值较大，主要原因是砂岩在纯天然状态下，其内部存在一定量的孔隙裂缝，在第1次循环加卸载作用下，砂岩内部的孔隙裂缝被压缩，导致其轴向残余塑性应变发生突降[19]。形成“L”形变化趋势的主要原因为在经历过第1循环加卸载的作用下，砂岩内部孔隙已经被压密，岩石只有产生新裂纹时应变才能继续发展，当循环次数达到第2次之后，轴向残余塑性应变几乎相等，则说明岩石内部的损伤在缓慢增加。由图3(b)可知，当砂岩的循环加卸载次数达到5, 8，12次时，砂岩轴向残余塑性应变没有出现骤增的趋势，因此其内部在缓慢地产生新裂纹。
3　能量演化分析
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3.1　能量演化规律
基于热力学第一定律，砂岩在循环加卸载的作用下会产生变形。假设这个物理过程不与外界交换热量，那么外力所做的功将完全被岩石吸收[20]。在这种情境下，外力所产生的总输入能U主要由可释放的弹性应变能和耗散能构成。
根据岩体弹性能的可逆性、滞回环面积与耗散能之间的关系，纪洪广等[21]、HUANG等[22]提出了使用循环加卸载应力–应变曲线计算砂岩弹性能和耗散能的方法。图4为第i次分级循环加卸载的应力–应变曲线，可用来描述输入能、弹性能及耗散能之间的关系。在砂岩经历单次循环加卸载作用下，BC加载阶段产生的轴向应变为ε1，CD卸载阶段产生的可恢复轴向弹性应变为εe，BD不可恢复的轴向残余塑性应变为εd，卸载时产生的环向应变为ε3。从能量角度分析，输入能为加载曲线BC与最小应力水平围成的面积，可恢复弹性应变能为曲线DC与最小应力水平围成的面积和环应变所做功之和，第i次循环加卸载的耗散能为两者之差。它们之间关系如下：
式中：U、Ue、Ud分别为输入能、弹性能、耗散能；σ1为轴向压力；σ3为围压力。
根据式(2)至式(4)计算得到不同上限应力值分级循环加卸载作用下砂岩的能量密度，见表2。3种能量密度、耗能比与循环次数之间的变化关系如图5所示。
图5(a)展示了输入能与循环次数之间的演化关系。在初期，输入能的增加比较慢，但随着循环次数的增加，输入能量的增加呈现出一种近似指数型的变化趋势。图5(b)描绘了弹性能与循环次数之间的演化关系。弹性能是指岩石在恢复形变时所释放出的能量，可反映岩石的储能能力。由图5(b)可以看出，弹性能的增加趋势与输入能的变化趋势十分接近。岩体压缩过程中的耗散能是指岩体内部裂隙的压缩、生成和扩展，从而使岩体的内部结构发生破坏所消耗的能量。由图5(c)中的曲线可以看出，在初期的循环过程中，耗散能的增加较慢，说明在较低的应力状态下，早期的裂缝以压合为主。在循环中期，耗散能量接近于线性增加，表明岩石进入了稳定的裂缝发展状态。图5(d)展示了耗能比与循环次数之间的演化关系。试验结果表明：能量耗散系数在初始周期内呈陡降趋势；循环中期，能量耗散系数基本维持在某一个水平上。
3.2　压缩阶段划分
杨小彬等[23]引入了耗能比的概念，耗能比是指单次循环加卸载过程中耗散能与输入能的比例关系，耗能比的计算公式如下：
式中：ωi、Uei、Ui分别为第i次循环加卸载作用下的耗能比、弹性能、输入能。
耗能比的大小能够在一定程度上反映岩石内部损伤的变化情况。耗能比曲线通常呈现出“勺”形特征，并可分为5个阶段：第Ⅰ阶段为压密阶段，第Ⅱ阶段为弹性阶段，第Ⅲ阶段为稳定破裂发展阶段，第Ⅳ阶段为不稳定破裂发展阶段，第Ⅴ阶段为峰后破坏阶段。由于本次试验采用分级循环加卸载方式，并未导致砂岩发生明显的破坏，因此在循环次数达到12次时，耗能比曲线并未出现第Ⅳ阶段的急剧上升以及第Ⅴ阶段的勺柄，表明试验中砂岩处于稳定破裂发展阶段。在分级循环加卸载的初始阶段，耗能比达到最大值，约为0.50，并且第一次循环与第二次循环之间的耗能比差值最大。
不同上限应力循环加卸载作用下损伤砂岩各破坏阶段的输入能、弹性能、耗散能、耗能比的变化趋势如图6所示。第Ⅰ阶段，即压密阶段(OA段)，此时以耗散能为主，同时促进新裂缝的生成和扩展。与此同时，大多数的弹性能会被存储在砂岩中。致密化后，砂岩进入第Ⅱ阶段，即弹性阶段（AB段）。此时，砂岩中的裂缝扩展速度较慢。在弹性阶段，耗散能的消耗很小，耗能比的下降趋于平缓，砂岩吸收的能量主要转变成了弹性能。当砂岩进入第Ⅲ阶段，即稳定破裂发展阶段（BC段）时，岩体进入弹塑性变形期，裂隙继续扩展，引起岩体内部结构破坏。耗散能的迅速增加，说明砂岩中裂缝的扩展速度加快，进而导致其内部结构破坏。
4　损伤表征
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通过对砂岩进行不同上限应力循环加卸载试验，可以发现不同的上限应力和加卸载循环次数会对砂岩造成不同程度的损伤。耗散能对砂岩产生塑性应变和内部发育裂隙起着重要的作用，使试件发生不可逆变形。随着循环加卸载上限应力的增大，砂岩消耗的耗散能不断增加，因此耗散能的大小可以反映试件的损伤程度。当外载荷提供的总能量大于砂岩材料的本构能时，材料单元则发生破坏，因此从能量耗散角度来定义砂岩内部损伤更具有意义[24]，可以通过砂岩应力与耗散能之间的变化来定义其内部损伤关系[25]：
式中：D、ΔEd、Δσ分别为每次循环加卸载对应的岩石损伤度、耗散能增量、应力增量。
根据式（6）中反正切函数的特性，当ΔEd=0时，D=0；当ΔEd=+∞时，D=1，因此式(6)可以表征岩石的损伤程度。结合表2与式(6)可计算出砂岩在分级循环加卸载作用下的内部损伤情况。不同上限应力作用下砂岩的损伤值见表3。
不同的循环加卸载次数下砂岩的弹性能、累积耗散能和损伤度如图7所示。从图7可以看出，在不同上限应力的循环加卸载作用下，砂岩的弹性能、累积耗散能和损伤度均随着循环次数的增加而增加，说明循环加卸载的次数对砂岩的损伤具有重大影响。损伤度的取值范围在0~1之间，这符合损伤度的定义。当砂岩受到不同上限应力的循环加卸载作用时，其损伤度随循环次数的增加呈线性增长趋势。这是因为每次循环荷载作用下，耗散能的增量也是线性增加的。耗散能的消耗过程与砂岩损伤的持续累积相对应，这种损伤度的变化与应力历史以及实际工程中砂岩的损伤破坏情况是一致的。
5　结论
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（1）在不同上限应力循环加卸载的作用下，由于砂岩自身的损伤特性以及加载方式的特点，砂岩的应力–应变曲线出现滞回环、向下弯曲的现象；随着外载荷的作用，砂岩刚度发生了退化。
（2）在不同上限应力的循环加卸载的作用下，随着荷载循环次数的增加，砂岩的轴向弹性应变与循环次数之间表现出线性关系，而砂岩的残余塑性应变曲线则展现出“L”形的变化趋势。在首次循环加卸载作用之后，轴向塑性应变发生显著的骤降。这一现象归因于砂岩在纯天然状态下内部存在的孔隙裂缝被迅速压实。
（3）随着循环加卸载次数的增加，输入能与弹性能变化曲线非常相似，呈现出近似指数曲线的上升趋势；而耗散能在初始循环阶段增长相对缓慢，而在循环中期，耗散能呈现出近似线性的增长趋势；耗能比在初始循环阶段呈现骤降的趋势，而在循环中期阶段耗能比曲线保持近似水平的趋势。
（4）在不同上限应力循环加卸载的作用下，砂岩的损伤度与循环次数之间呈现出线性关系。这种线性关系主要归因于单次循环荷载作用下耗散能的增量也呈线性增加。这意味着随着荷载循环次数的增加，砂岩的损伤度会相应地线性增长，这与耗散能的线性增加是密切相关的。
基金
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  • 国家自然科学基金项目(52074006)
文献
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2025年第45卷第3期
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  • 接收时间：2024-04-09
  • 首发时间：2026-10-29
  • 出版时间：2025-03-25
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  • 收稿日期：2024-04-09
基金
国家自然科学基金项目(52074006)
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    1.安徽理工大学 土木建筑学院，安徽　淮南市　232001
    2.安徽理工大学 力学与光电物理学院，安徽　淮南市　232001
    3.安徽理工大学 深部煤矿采动响应与灾害防控国家重点实验室，安徽　淮南市　232001
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2种不同金属材料的力学参数
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鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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