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Article(id=1241768037230121091, tenantId=1146029695717560320, journalId=1235980550691926019, issueId=1241768035548205179, articleNumber=null, orderNo=null, doi=10.3969/j.issn.0253-6099.2024.01.006, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1693238400000, receivedDateStr=2023-08-29, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1773990171009, onlineDateStr=2026-03-20, pubDate=1706716800000, pubDateStr=2024-02-01, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1773990171009, onlineIssueDateStr=2026-03-20, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1773990171009, creator=13701087609, updateTime=1773990171009, updator=13701087609, issue=Issue{id=1241768035548205179, tenantId=1146029695717560320, journalId=1235980550691926019, year='2024', volume='44', issue='1', pageStart='1', pageEnd='178', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1773990170609, creator=13701087609, updateTime=1773993209826, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241780783011140021, tenantId=1146029695717560320, journalId=1235980550691926019, issueId=1241768035548205179, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241780783015334326, tenantId=1146029695717560320, journalId=1235980550691926019, issueId=1241768035548205179, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=25, endPage=30, ext={EN=ArticleExt(id=1241768037624385671, articleId=1241768037230121091, tenantId=1146029695717560320, journalId=1235980550691926019, language=EN, title=Energy Transformation and Time Effectiveness of Dynamical System for Breaking Weakened Mineral Rock, columnId=1236276106018484431, journalTitle=Mining and Metallurgical Engineering, columnName=MINING, runingTitle=null, highlight=null, articleAbstract=

Based on typical nonlinear negative exponent constitutive relations and catastrophe characteristic of weakened mineral rock, the variation of energy transformation and time effectiveness with mechanism stiffness in catastrophic and non-catastrophic breakup were analyzed. The results show that the catastrophic breakup induced by low mechanism stiffness will dramatically reduce the energy transformation efficiency of dynamical system and time effectiveness of rock breakage. From the perspective of energy utilization and rock-breaking efficiency, it is not appropriate to equip a dynamical system for rock breakage with low mechanism stiffness. Finally, an appropriate range of mechanism stiffness is provided for reference in design.

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基于弱化性质矿岩典型的非线性负指数本构关系和灾变特性，分析了灾变和非灾变两种破碎动力学模式下能量转化和时效特性随机构刚度的变化情况。结果表明，低机构刚度诱发的灾变破碎模式会急剧降低破碎动力系统的能量传递效率和破岩时效。从能量利用和破岩功效角度考虑，破碎动力系统不宜采用低机构刚度；给出了机构刚度设计适宜的选用范围。

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汪学清（1971—），男，山东成武人，博士（后），副教授，主要研究方向为岩石力学、矿山建设。E-mail：

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汪学清（1971—），男，山东成武人，博士（后），副教授，主要研究方向为岩石力学、矿山建设。E-mail：

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汪学清（1971—），男，山东成武人，博士（后），副教授，主要研究方向为岩石力学、矿山建设。E-mail：

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参数名称类别灾变区间（0≤≤0.135）非灾变区间（0.135<≤0.400）
能量传递效率ηt值域0～0.9270.907～0.966
平均变化率
破岩时效值域0～0.750.731～0.889
平均变化率
), ArticleFig(id=1241779826714022303, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241768037230121091, language=CN, label=表1, caption=

能量传递效率及破岩时效值域和区间平均变化率的比较

, figureFileSmall=null, figureFileBig=null, tableContent=
参数名称类别灾变区间（0≤≤0.135）非灾变区间（0.135<≤0.400）
能量传递效率ηt值域0～0.9270.907～0.966
平均变化率
破岩时效值域0～0.750.731～0.889
平均变化率
), ArticleFig(id=1241779826793714080, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241768037230121091, language=EN, label=null, caption=null, figureFileSmall=null, figureFileBig=null, tableContent=
类别参数名称计算式计算结果备注
机构特性无量纲刚度式（6）0.221>0.135，处于非灾变区间
传递效率破碎载荷式（6）0.149非灾变破碎
矿岩破碎能量式（14）0.801
机构释放能量式（15）～（16）0.050非灾变破碎
施加能量式（12）0.851
能量传递效率ηt式（19）0.941
破岩时效机构残余变形式（18）0.674非灾变破碎
总变形式（17）3.674非灾变破碎
破岩时效式（20）0.818
), ArticleFig(id=1241779826877600161, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241768037230121091, language=CN, label=表2, caption=

计算结果

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类别参数名称计算式计算结果备注
机构特性无量纲刚度式（6）0.221>0.135，处于非灾变区间
传递效率破碎载荷式（6）0.149非灾变破碎
矿岩破碎能量式（14）0.801
机构释放能量式（15）～（16）0.050非灾变破碎
施加能量式（12）0.851
能量传递效率ηt式（19）0.941
破岩时效机构残余变形式（18）0.674非灾变破碎
总变形式（17）3.674非灾变破碎
破岩时效式（20）0.818
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弱化矿岩破碎动力系统的能量转化和时效特性
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汪学清 1, 2 , 唐红平 2 , 王建华 2 , 邹煜 1
矿冶工程杂志 | 采矿 2024,44(1): 25-30
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矿冶工程杂志 | 采矿 2024, 44(1): 25-30
弱化矿岩破碎动力系统的能量转化和时效特性
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汪学清1, 2 , 唐红平2, 王建华2, 邹煜1
作者信息
  • 1.中国矿业大学（北京）力学与建筑工程学院，北京　100083
  • 2.深海矿产资源开发利用技术国家重点实验室，湖南　长沙　410012

    • 汪学清（1971—），男，山东成武人，博士（后），副教授，主要研究方向为岩石力学、矿山建设。E-mail：

Energy Transformation and Time Effectiveness of Dynamical System for Breaking Weakened Mineral Rock
Xueqing WANG1, 2 , Hongping TANG2, Jianhua WANG2, Yu ZOU1
Affiliations
  • 1.School of Mechanics & Civil Engineering, China University of Mining & Technology-Beijing, Beijing 100083, China
  • 2.State Key Laboratory of Exploitation and Utilization of Deep Sea Mineral Resources, Changsha 410012, Hunan, China
出版时间: 2024-02-01 doi: 10.3969/j.issn.0253-6099.2024.01.006
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摘要
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基于弱化性质矿岩典型的非线性负指数本构关系和灾变特性，分析了灾变和非灾变两种破碎动力学模式下能量转化和时效特性随机构刚度的变化情况。结果表明，低机构刚度诱发的灾变破碎模式会急剧降低破碎动力系统的能量传递效率和破岩时效。从能量利用和破岩功效角度考虑，破碎动力系统不宜采用低机构刚度；给出了机构刚度设计适宜的选用范围。

矿岩  /  弱化特性  /  破碎动力系统  /  能量传递  /  破岩时效

Based on typical nonlinear negative exponent constitutive relations and catastrophe characteristic of weakened mineral rock, the variation of energy transformation and time effectiveness with mechanism stiffness in catastrophic and non-catastrophic breakup were analyzed. The results show that the catastrophic breakup induced by low mechanism stiffness will dramatically reduce the energy transformation efficiency of dynamical system and time effectiveness of rock breakage. From the perspective of energy utilization and rock-breaking efficiency, it is not appropriate to equip a dynamical system for rock breakage with low mechanism stiffness. Finally, an appropriate range of mechanism stiffness is provided for reference in design.

mineral rock  /  weakened property  /  dynamical system for rock breakage  /  energy transformation  /  time effectiveness of rock breakage
汪学清, 唐红平, 王建华, 邹煜. 弱化矿岩破碎动力系统的能量转化和时效特性. 矿冶工程杂志, 2024 , 44 (1) : 25 -30 . DOI: 10.3969/j.issn.0253-6099.2024.01.006
Xueqing WANG, Hongping TANG, Jianhua WANG, Yu ZOU. Energy Transformation and Time Effectiveness of Dynamical System for Breaking Weakened Mineral Rock[J]. Mining and Metallurgical Engineering, 2024 , 44 (1) : 25 -30 . DOI: 10.3969/j.issn.0253-6099.2024.01.006
正文
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矿岩的弱化特性表现在矿岩达到极限荷载后并不完全破坏，仍具有一定的承载能力，要损伤到一定程度后才整体破坏。具有弱化性质的矿岩在加载过程中经常会出现失稳（灾变）现象，动力系统的失稳灾变不仅取决于矿岩特性，也与加载系统特性（刚度）相关[1-3]。在测试试样全应力-应变曲线试验过程中，为避免失稳灾害的发生，常采用提高试验机刚度的方法。破碎动力系统的能量传递和破碎时效是其关键指标，即尽量减少能量损失和缩短破岩时间，以最少的输入能量，实现更多、更快的矿岩破碎。破碎动力系统能量转化的前期研究工作大多简化为研究载荷-变形的线性关系[4-10]，这些研究不能真实地描述能量转换状况。本文基于弱化矿岩典型的本构关系和灾变特性，分析了灾变和非灾变两种破碎模式的能量转化和时效特性，给出了加载系统机构刚性对系统能量传递效率和破岩时效的影响。
1　弱化特性矿岩破碎系统的灾变和破碎模式
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1.1　矿岩的弱化特性和本构关系
矿岩的弱化特性可以用其本构关系来描述。
根据损伤力学[11]，破碎过程中弱化特性矿岩的非线性应力-应变关系一般可表达为：
式中σ为应力；ε为应变；D为损伤参量；E为弹性模量。由统计强度理论，视其损伤D（各微元强度）服从Weibull分布[12]，其本构特性可表达为载荷和变形的关系[13]：
可得其刚度为：
式中P为载荷；ur为变形值；u0为对应于最大载荷的变形值。当ur=0时，Kr=λ，即λ为矿岩的初始刚度；在ur=u0处，Kr=0；在卸载区间（Kr<0），ur=2u0处，矿岩有最大的负刚度，Kr=-0.135λ。
最大载荷Pm=e-1λu0=0.368λu0，对应矿岩变形的损伤能量为：
全损伤能量为：
采用无量纲表达：
实际中弱化特性矿岩变形到一定程度，其损伤参量D尚小于1时，已不能维持结构的平衡，即出现整体破裂。一般可取变形达峰值载荷变形的3倍、损伤能达全损伤能量的80%时构成损伤失效。在本构曲线上，构成整体破裂处称为破裂点，相应的载荷、变形称为破裂点载荷、破裂点变形。
1.2　矿岩破碎动力系统及失稳灾变
矿岩破碎动力系统简化力学模型如图1所示。其中ut为破碎动力系统的总变形；um、Mm和Km分别为破碎动力机构的变形、质量和刚度；ur、Mr和Kr分别为破碎矿岩的变形、质量和刚度；P为外加载荷。
破碎动力机构具有线性特性，其刚度Km为常数；矿岩具有非线性特性，其刚度Kr=f′（ur）；总变形为：
将系统视为一个二自由度振动系统（图1（a）），根据振动理论[14]作准稳态处理，在单位微元中有：
式中ω为加载频率，当外加载荷趋于静载（ω→0）时，分母末项Mrω2→0，即可忽略质量Mr，视其为准静态模型（图1（b））；左边的分子、分母同除以δt，给出通常灾变分析[15]中的变形速率关系为：
由于破碎机构刚度Km不为0，加载速率为常数，当破碎机构刚度Km与矿岩刚度Kr之和为0时，→∞，即矿岩变形速率突然增大，矿岩将产生瞬间爆裂破碎；同时，由于机构反应速率跟不上，此时将伴随着机构能量的大量释放（转化为矿岩爆裂的动能和机构的冲击振动），此即构成了动力系统的失稳灾变现象，其失稳判据为：
应当注意的是：矿岩的灾变只是变形速率的突变，其本构关系并未改变。由于Km>0，故Km+Kr=0只在Kr<0（卸载）情况下发生。采用无量纲表达，即发生在矿岩刚度由0到达其最大负刚度（-0.135）的存在域[1,2]区间，对应于矿岩的本构曲线，此区间为矿岩失稳灾变破碎的对应区间。
1.3　弱化矿岩破碎的力学模式
矿岩破碎系统由外载施加能量，通过破碎机构传递给矿岩，转化为矿岩破碎能量，进行矿岩破碎。弱化矿岩破碎有灾变和非灾变两种破碎力学模式。
灾变破碎是指破碎动力系统在矿岩灾变失稳区间满足灾变条件时，矿岩自灾变点处变形速率突然增大，急速变形至破裂点形成的矿岩瞬间破碎，所对应的机构刚度0<≤0.135，期间施载速率跟不上，可以认为系统在灾变点失载（外加载荷→0），其机构变形能量除部分转化为矿岩灾变点后的变形能量外，构成能量的大量释放。
矿岩非灾变破碎是指矿岩在灾变失稳区间未构成灾变，迄后变形至破裂点形成矿岩破碎，其破碎产生的条件仅与矿岩变形是否达到破裂点变形相关；对应的机构刚度>0.135，处于大刚度区间，机构能量不会大量释放。
灾变破碎和非灾变破碎两种破碎模式均至破裂点完成破碎，其主要区别在于失载点能量转化方式和最终的机构释放能量不同。两种破碎模式特性的比较如图2所示。
2　弱化矿岩破碎系统的能量传递和破岩时效特性
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2.1　弱化矿岩破碎系统的能量和变形转化
由破碎系统的准静态力学模型，破碎系统施加能量（外力做功）由施载中的矿岩变形能量和机构变形能量两部分构成，在矿岩破碎时按两种力学模式转化。
灾变破碎：施加能量由灾变点的矿岩变形能量和机构变形能量构成，其中机构变形能量的一部分继后将释出作为灾变点后的矿岩变形能量，该能量与灾变点矿岩变形能量一起合为破裂点变形能量，之后转化为矿岩破碎能量，剩余部分转化为机构（对外）释放能量。
非灾变破碎：施加能量由矿岩破裂时的矿岩变形能量和机构变形能量构成，和分别转化为矿岩破碎能量和机构对外释放能量。
在能量转化过程中，灾变破碎和非灾变破碎中的施加能量最终均转化为矿岩破碎能量和机构（对外）释放能量。
矿岩破碎能量：
矿岩破碎能量即为矿岩变形至破裂点时产生（吸收）的变形能量：
矿岩破碎能量只取决于矿岩固有的本构特性。灾变破碎和非灾变破碎的区别在于灾变破碎时变形至灾变点后变形速率发生突变，而非灾变破碎时不会发生突变。两者均至破裂点完成破碎，其本构特性和破碎能量不会因灾变而改变。
机构释放能量：
机构释放能量为系统施加能量和矿岩破碎能量之差。灾变破碎时亦可表达灾变机构释放能量与灾变点后矿岩变形能量之差；非灾变破碎时机构释放能量即为破裂点机构变形能量：
机构释放能量为机构弹性变形在矿岩破碎时对外释放（逸散）的能量。由于产生灾变时机构刚度小于非灾变破碎时的机构刚度，且灾变点载荷高于破裂点载荷，一般情况下，灾变破碎的机构释放能量高于非灾变破碎的机构释放能量。
破碎动力系统的总变形由施载中矿岩变形和机构变形两部分构成。灾变破碎时，总变形为灾变点矿岩变形与机构变形之和，其中机构变形的一部分将释出补偿灾变点后的矿岩变形，合成为破裂点变形，剩余部分为机构残余变形；非灾变破碎时，总变形为破裂点矿岩变形与机构变形之和，其中矿岩变形即为破裂点变形，机构变形即为机构残余变形。两种破碎模式的总变形均为矿岩破裂点变形与机构残余变形之和：
机构变形：
机构残余变形：
在一般情况下，灾变时的机构残余变形大于非灾变时的机构残余变形。
灾变和非灾变破碎两种情况的载荷-变形特性及对应的能量关系如图3所示。
图3表明：
两种情况的矿岩破碎能量一致（=0.801），发生灾变不会改变矿岩破碎（吸收）的能量，但机构释放能量（灾变：0.142；非灾变：0.028）和总变形（灾变：4.491；非灾变：3.373）在灾变情况下显著增大。
在能量和变形转化过程中，两种破碎模式的主要区别在于：灾变破碎外力施载做功至灾变点失载，而后矿岩靠机构的位能释出快速变形至破裂点完成破碎；非灾变破碎外力施载直至破裂点，不靠机构的位能释出完成破碎。
2.2　能量传递效率和破岩时效
2.2.1　能量传递效率
破碎系统能量转化中，矿岩破碎能量为有用能量，机构释放能量为无用能量，其能量传递效率为矿岩破碎能量与破碎系统施加能量之比：
2.2.2　破岩时效
破岩时效是完成矿岩破碎施载时间利用率的测度。在恒定的加载速率（为常数）下，破碎时间取决（正比）于机构完成破碎施载所需之总变形，其中，矿岩破裂点变形为有效变形（对应于有效破碎时间），机构残余变形为无效变形（对应于无效破碎时间），则破岩时效为矿岩有效变形与总变形之比：
传递效率和破岩时效随机构在矿岩破碎时释放能量和残余变形的增大而降低，这是由于破碎时（无论灾变或非灾变情况）机构的释放能量和残余变形会形成无用功和无效破碎时间。
图4给出了破岩动力系统能量、传递效率（ηt）、总变形以及破岩时效随系统机构刚度在区间（0≤≤0.4）的变化。
以机构刚度=0.135为界分为左右两个区间。0≤≤0.135为动力系统灾变破碎区间，在灾变点处失载构成灾变破碎，矿岩灾变变形能量（0.264～0.594）和变形（1.0～2.0）随机构刚度增大而增大。矿岩破碎能量=0.801；破裂点变形=3，→0，机构释放能量、施加能量；机构残余变形、总变形均趋于∞；能量传递效率ηt和破岩时效均趋于0。随着机构刚度增大，、、、急速降低，ηt和迅速增大，直至=0.135。机构刚度进一步增大，进入>0.135非灾变破碎区间，构成非灾变破碎，矿岩破碎能量和破裂点变形不变，、、、随机构刚度增大进一步降低，ηt和相应增大。→∞时，，→0，ηt，→1。
灾变和非灾变破碎区间能量传递效率和破岩时效的变化值域及其区间平均变化率存在显著差异，详见表1。
非灾变区间的能量传递效率和破岩时效处于高值区，远高于灾变区间的能量传递效率和破岩时效，且其区间平均变化率较小，即在非灾变区间破碎系统具有高值且稳定的能量传递效率和破岩时效，机构刚度的变化对其影响不大。
能量和变形随机构刚度的变化具有很好的一致性，变化规律基本相同，能量参量与其所对应的变形参量之间显现相应的准线性关系。这是由于破碎载荷在灾变区间变化不大（=0.368～0.271）；在非灾变区间为一恒定值（==0.149）；而在恒力作用下，能量（外力做功）和变形之间存在线性（正比）关系（E=Pu）。事实上，在矿岩破碎中，破碎能量和变形为相对应的两个映像，前者反映了能量的转化关系，后者反映了变形转化关系。能量传递效率给出了破碎岩石能量在施加能量中占有的比率，破岩时效给出了破岩时间在施载时间中占有的比率，两者从破碎能量和时间转化两个侧面给出了破碎功效测度。
3　计算实例
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某矿岩单轴抗压强度σb为2.8 MPa，弹性模量E为5.4 GPa，初始刚度λ为6.8×107 N/m（截割破碎方式），机构刚度Km为1.5×107 N/m（液压加载方式），非灾变破碎时矿岩破碎变形。计算结果如表2所示。
计算结果表明，=Km/λ=0.221>0.135，处于非灾变破碎区间，能量传递效率ηt==0.941，破岩时效=0.818，可较好地实现矿岩破碎。
4　结语
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1）根据弱化性质矿岩的非线性本构特性，其破碎动力系统有灾变和非灾变两种破碎力学模式。前者发生于机构低刚度（0<≤0.135）的灾变失稳区间，在灾变点矿岩变形速率突然增大，伴随着机构能量的大量释放；后者发生于机构高刚度（>0.135）非灾变区间，没有机构能量的大量释放。
2）低机构刚度灾变破碎模式和高机构刚度非灾变破碎模式能量传递效率和破岩时效存在很大差异，前者显著低于后者。从能量利用和破岩功效角度考虑，破碎动力系统不宜采用低机构刚度。
3）在高机构刚度非灾变破碎区间，能量传递效率和破岩时效随机构刚度变化率很小，即对刚度变化的敏感度不高，故从简化结构、降低制造成本角度，不必追求过高的机构刚度，一般控制>0.2即可满足要求。
4）对矿岩初始刚度较大的情况，应采用较大的机构刚度，以降低系统能量释放、提高破碎功效。
5）从破碎能量和时效的角度分析了弱化矿岩破碎系统的动力特性，研究结果可为弱化矿岩破碎动力系统的参数选择和设计提供参考。
基金
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  • 国家重点实验室开放课题(2021-HYSDDYB-001)
  • 湖南省创新专项(2019SK2271)
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doi: 10.3969/j.issn.0253-6099.2024.01.006
  • 接收时间：2023-08-29
  • 首发时间：2026-03-20
  • 出版时间：2024-02-01
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  • 收稿日期：2023-08-29
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国家重点实验室开放课题(2021-HYSDDYB-001)
湖南省创新专项(2019SK2271)
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    1.中国矿业大学（北京）力学与建筑工程学院，北京　100083
    2.深海矿产资源开发利用技术国家重点实验室，湖南　长沙　410012
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2种不同金属材料的力学参数
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Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
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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