混凝土裂缝扩展全过程的新G_R阻力曲线理论和能量转化分析
|
摘要
对混凝土乃至钢筋混凝土结构来说,其受力破坏过程与裂缝的蔓延滋生有着密切的关系,是一个裂缝扩展问题。断裂力学作为专门研究带裂缝固体强度及裂缝扩展规律的科学,它应该是一种有效且合理的分析混凝土结构响应的工具。然而,由于混凝土的不均匀性和非完全线性行为,其断裂破坏的过程并不像理想脆性材料简单,表现为裂缝起裂,稳定扩展,失稳扩展三个阶段。构成混凝土的骨料组份所发挥的粘聚咬合作用,造成了裂尖非线性断裂过程区的出现及发展,被认为是混凝土非线性断裂破坏主要的物理力学机理。结合这些断裂特征,迄今以应力强度因子为工具已建立了一些典型的断裂模型,很好地描述了混凝土的裂缝稳定扩展。然而,以能量为出发点定量分析整个断裂全过程材料的裂缝阻止能力以及能量的相互转化关系的工作却远远不够。鉴于此不足,本文尝试从能量的观点来进一步理解混凝土整个断裂过程所表现的断裂行为。
1.裂缝扩展阻力曲线很好地描述了随裂缝扩展材料裂缝抵制力的变化,是判定裂缝在其任意扩展时刻稳定与否的一种断裂准则。本文首先回顾了Irwin在上世纪六十年代提出的传统裂缝扩展阻力曲线概念以及在此概念基础上发展的两种确定R阻力曲线的半解析半经验方法,然后结合混凝土断裂的现象学观测,在两个基本假设的基础上从复合材料的观点构建了描述裂缝扩展阻力变化的G_R阻力曲线模型。在这个模型中,认为裂缝的扩展阻力由两部分组成:一部分是基体水泥凝胶材料的贡献,称为起裂韧度;另一部分是骨料咬合作用的贡献,称为骨料粘聚韧度。对于前者,根据Griffith脆性断裂理论,其大小是一常数,认为接近于水泥净浆材料的断裂韧度。但对于后者,随着裂缝的扩展是变化的。接着根据Hillerborg虚拟裂缝模型,本文给出了局部断裂能的计算公式及物理力学解释。在此基础上,使用Reinhardt非线性软化本构,根据等效裂缝长度a及其特征长度a_(w0)的大小,分两种情况着重推导了骨料粘聚韧度计算的解析公式,从而给出了材料整个断裂过程中裂缝扩展阻力的表达式。
2.浇注了最大尺寸分别为2000mm×500mm×200mm的三点弯曲梁和1440mm×1200mm×200mm的楔入劈拉试件共七个立方米的混凝土断裂试件,并对完好无损试件完成了断裂试验。根据本文构建的G_R阻力曲线模型,针对试验的两种几何形式,详细给出了确定任意裂缝扩展时刻裂缝扩展阻力的步骤和相应的计算公式。使用试验获得的P-CMOD曲线,得到了G_R阻力曲线和动力曲线,观察了G_R阻力曲线的基本特征。通过比较荷载裂缝嘴张开位移曲线,解释了它们相交点的物理意义。并讨论了G_R阻力曲线的尺寸效应和形状效应问题以及与使用软化本构曲线的相关性。
3.软化曲线描述了裂尖非线性断裂过程区的物理力学行为,不同的软化曲线对骨料粘聚性能的定量不尽相同,由此而确定的G_R阻力曲线也有所差异。为此,本文采用广泛使用的双线性软化曲线和非线性软化曲线,在假定Hillerborg定义的断裂能相同的情况下,以数值模拟的三点弯曲梁为研究对象,计算了G_R阻力曲线,研究了不同软化曲线对其结果的影响。
4.类似于理想脆性材料,混凝土的断裂过程也是一个能量相互转化的过程。在这个断裂过程中,外力功主要以弹性应变能和断裂表面能两种方式所消耗。在以往的研究中,大多针对的是试件断裂的最后时刻,在这个时刻外力功被假设完全被断裂窄条所吸收,而对裂缝发展的其它时刻却很少涉及。鉴于此,本文使用试验的楔入劈拉试件结果,研究了裂缝扩展的整个过程中弹性应变能和断裂表面能的变化规律,从而进一步深入地理解了混凝土的断裂行为。同时,计算了裂缝扩展任意时刻形成单位面积裂缝所需要的能量,对获得的结果进行了分析讨论。
For concrete or reinforced concrete structures,their failure process when subjected to external load is intimately associated with the appearance and propagation of crack and hence is regarded as a process of crack development.Fracture mechanics,where the strength of cracked solid and the general rules of crack development in cracked solid are major-cared, should be an effective and reasonable tool to analyze the mechanical response of cracked structures.However,owing to the heterogeneity of concrete and non-linear behavior exhibited in concrete,different from the ideal brittle materials,the fracture process of concrete material is complex and shows the typical three stages:crack initiation,crack stable propagation and unstable propagation.It had been commonly accepted that the bridging cohesive property of aggregate reinforcements is primary governing mechanism which results in the presence and development of fracture process zone(FPZ) and thereby resulting non-linear fracture behavior. So far,after considering the influence of fracture process zone on concrete fracture behavior, many fracture models applicable for concrete had been proposed in last several years. However,most of models were based on stress intensity factor approach not energy approach. It is very reason that the researches on the crack resistance determined using energy approach as well as studies regarding the transformed relationship between different energy consumption patterns during the whole fracture process in concrete are rather lack.As a consequence,in this paper the energy approach is attempted with the major purpose of obtaining further deeper understanding of fracture behaviors exhibited in concrete.
1.Crack extension resistance curve better describes the change of crack resistance capacity of material with crack development and is used as a criterion to judge whether crack is stable at certain crack propagation.In this paper,firstly,a state-of-the-art retrospection on the conception of traditional crack extension resistance curve proposed by Irwin in 1960s and two different semi-analytical and semi-experimental methods to construct R-curve are made.Then,combining with the phenomenology observations from fracture experiments of concrete,and based on the two basic assumptions,new G_R crack extension resistance curve model which describes the change of crack resistance with crack propagation is constructed from the view of the composite materials.In this model, the crack extension resistance is composed of two parts.One part is contribution produced by hardening cement-gel matrix and is called mitiation fracture toughness. Another part is contribution caused by bridging cohesive action of aggregates and is named as cohesive toughness.For the front,according to the Griffith's brittle fracture theory,the value is a constant and is assumed to be equal to the fracture toughness of cement paste.While for the latter,the value is changeable.To determine it,the calculation equation of local fracture energy is firstly deduced based on the Hillerborg's fictitious crack model.After this,using Reinhardt's non-linear softening curve,the two different equations for evaluating cohesive toughness are obtained respectively corresponding to two different cases defined in terms of the values of crack length a and crack characteristic length a_(w0).Finally,the equations for crack extension resistance at any time during the whole fracture process are obtained.
2.A total of 7m~3 concrete fracture specimens are cast.Of which,two typical geometries,i.e. three-point bending beam(TPB) and wedge-splitting(WS),are used.The maximum dimension for TPB and WS is 2000mm×500mm×200mm and 1440mm×1200mm×200mm,respectively.Fracture tests are performed on the intact concrete specimens.Then, according to the proposed G_R crack extension resistance curve model,the detailed calculation steps and corresponding calculation equations are further introduced for two geometries,respectively.Using the obtained P-CMOD curve,G_R crack extension resistance curve and crack driving curve are determined for every tested specimen.The basic features shown on G_R crack extension resistance curve are pointed out.Moreover, the implied physical meaning behind the intersection points of crack resistance curve and crack driving curve is clarified by comparing them with the P-CMOD curve.At last,the size effect and geometry effect of new G_R crack extension resistance curve together with the correlation between it and used softening curve are discussed.
3.Softening curve gives a description of physical mechanical behavior occurred in non-linear fracture process zone ahead of crack tip.It can be said that the bridging cohesive properties determined by the different softening curve are distinguished.Hence, the calculated G_R crack extension resistance curve will show some differences,too.In this paper,the bilinear softening curves and non-linear softening curve are adopted.Also, the fracture energy determined from them according to the definition by Hillerborg is assumed to be a constant in all used softening curves.In such case,G_R crack extension resistance curves for seven numerical simulated TPB are computed.The impact of different softening curves on the obtained G_R crack extension resistance curves is highlighted.
4.Similar to ideal brittle materials,the fracture process of concrete is a process of energy transformation,too.During this process,the work done by external load is consumed in two prime manners,i.e.,elastic strain energy and fracture surface energy.However,last extensive studies put more interests in the finally completely fracturing time at which the external work caused by applied load is assumed to totally flow into fracture narrow zone. But,for others any crack propagation time,the related studies are seriously few.Because of this,in the present paper,the emphasis of the work is put on the whole fracture process. Then,two kinds of energy consumption of external force work,namely,elastic strain energy and fracture surface energy at any picked crack propagation are determined using the data obtained from wedge-splitting experiments.The calculated results provide new insights into the understanding of concrete fracture behavior.Meanwhile,the energy needed for creating unit area crack propagation at any crack propagation is computed,too. The corresponding discussions are made.
1.裂缝扩展阻力曲线很好地描述了随裂缝扩展材料裂缝抵制力的变化,是判定裂缝在其任意扩展时刻稳定与否的一种断裂准则。本文首先回顾了Irwin在上世纪六十年代提出的传统裂缝扩展阻力曲线概念以及在此概念基础上发展的两种确定R阻力曲线的半解析半经验方法,然后结合混凝土断裂的现象学观测,在两个基本假设的基础上从复合材料的观点构建了描述裂缝扩展阻力变化的G_R阻力曲线模型。在这个模型中,认为裂缝的扩展阻力由两部分组成:一部分是基体水泥凝胶材料的贡献,称为起裂韧度;另一部分是骨料咬合作用的贡献,称为骨料粘聚韧度。对于前者,根据Griffith脆性断裂理论,其大小是一常数,认为接近于水泥净浆材料的断裂韧度。但对于后者,随着裂缝的扩展是变化的。接着根据Hillerborg虚拟裂缝模型,本文给出了局部断裂能的计算公式及物理力学解释。在此基础上,使用Reinhardt非线性软化本构,根据等效裂缝长度a及其特征长度a_(w0)的大小,分两种情况着重推导了骨料粘聚韧度计算的解析公式,从而给出了材料整个断裂过程中裂缝扩展阻力的表达式。
2.浇注了最大尺寸分别为2000mm×500mm×200mm的三点弯曲梁和1440mm×1200mm×200mm的楔入劈拉试件共七个立方米的混凝土断裂试件,并对完好无损试件完成了断裂试验。根据本文构建的G_R阻力曲线模型,针对试验的两种几何形式,详细给出了确定任意裂缝扩展时刻裂缝扩展阻力的步骤和相应的计算公式。使用试验获得的P-CMOD曲线,得到了G_R阻力曲线和动力曲线,观察了G_R阻力曲线的基本特征。通过比较荷载裂缝嘴张开位移曲线,解释了它们相交点的物理意义。并讨论了G_R阻力曲线的尺寸效应和形状效应问题以及与使用软化本构曲线的相关性。
3.软化曲线描述了裂尖非线性断裂过程区的物理力学行为,不同的软化曲线对骨料粘聚性能的定量不尽相同,由此而确定的G_R阻力曲线也有所差异。为此,本文采用广泛使用的双线性软化曲线和非线性软化曲线,在假定Hillerborg定义的断裂能相同的情况下,以数值模拟的三点弯曲梁为研究对象,计算了G_R阻力曲线,研究了不同软化曲线对其结果的影响。
4.类似于理想脆性材料,混凝土的断裂过程也是一个能量相互转化的过程。在这个断裂过程中,外力功主要以弹性应变能和断裂表面能两种方式所消耗。在以往的研究中,大多针对的是试件断裂的最后时刻,在这个时刻外力功被假设完全被断裂窄条所吸收,而对裂缝发展的其它时刻却很少涉及。鉴于此,本文使用试验的楔入劈拉试件结果,研究了裂缝扩展的整个过程中弹性应变能和断裂表面能的变化规律,从而进一步深入地理解了混凝土的断裂行为。同时,计算了裂缝扩展任意时刻形成单位面积裂缝所需要的能量,对获得的结果进行了分析讨论。
For concrete or reinforced concrete structures,their failure process when subjected to external load is intimately associated with the appearance and propagation of crack and hence is regarded as a process of crack development.Fracture mechanics,where the strength of cracked solid and the general rules of crack development in cracked solid are major-cared, should be an effective and reasonable tool to analyze the mechanical response of cracked structures.However,owing to the heterogeneity of concrete and non-linear behavior exhibited in concrete,different from the ideal brittle materials,the fracture process of concrete material is complex and shows the typical three stages:crack initiation,crack stable propagation and unstable propagation.It had been commonly accepted that the bridging cohesive property of aggregate reinforcements is primary governing mechanism which results in the presence and development of fracture process zone(FPZ) and thereby resulting non-linear fracture behavior. So far,after considering the influence of fracture process zone on concrete fracture behavior, many fracture models applicable for concrete had been proposed in last several years. However,most of models were based on stress intensity factor approach not energy approach. It is very reason that the researches on the crack resistance determined using energy approach as well as studies regarding the transformed relationship between different energy consumption patterns during the whole fracture process in concrete are rather lack.As a consequence,in this paper the energy approach is attempted with the major purpose of obtaining further deeper understanding of fracture behaviors exhibited in concrete.
1.Crack extension resistance curve better describes the change of crack resistance capacity of material with crack development and is used as a criterion to judge whether crack is stable at certain crack propagation.In this paper,firstly,a state-of-the-art retrospection on the conception of traditional crack extension resistance curve proposed by Irwin in 1960s and two different semi-analytical and semi-experimental methods to construct R-curve are made.Then,combining with the phenomenology observations from fracture experiments of concrete,and based on the two basic assumptions,new G_R crack extension resistance curve model which describes the change of crack resistance with crack propagation is constructed from the view of the composite materials.In this model, the crack extension resistance is composed of two parts.One part is contribution produced by hardening cement-gel matrix and is called mitiation fracture toughness. Another part is contribution caused by bridging cohesive action of aggregates and is named as cohesive toughness.For the front,according to the Griffith's brittle fracture theory,the value is a constant and is assumed to be equal to the fracture toughness of cement paste.While for the latter,the value is changeable.To determine it,the calculation equation of local fracture energy is firstly deduced based on the Hillerborg's fictitious crack model.After this,using Reinhardt's non-linear softening curve,the two different equations for evaluating cohesive toughness are obtained respectively corresponding to two different cases defined in terms of the values of crack length a and crack characteristic length a_(w0).Finally,the equations for crack extension resistance at any time during the whole fracture process are obtained.
2.A total of 7m~3 concrete fracture specimens are cast.Of which,two typical geometries,i.e. three-point bending beam(TPB) and wedge-splitting(WS),are used.The maximum dimension for TPB and WS is 2000mm×500mm×200mm and 1440mm×1200mm×200mm,respectively.Fracture tests are performed on the intact concrete specimens.Then, according to the proposed G_R crack extension resistance curve model,the detailed calculation steps and corresponding calculation equations are further introduced for two geometries,respectively.Using the obtained P-CMOD curve,G_R crack extension resistance curve and crack driving curve are determined for every tested specimen.The basic features shown on G_R crack extension resistance curve are pointed out.Moreover, the implied physical meaning behind the intersection points of crack resistance curve and crack driving curve is clarified by comparing them with the P-CMOD curve.At last,the size effect and geometry effect of new G_R crack extension resistance curve together with the correlation between it and used softening curve are discussed.
3.Softening curve gives a description of physical mechanical behavior occurred in non-linear fracture process zone ahead of crack tip.It can be said that the bridging cohesive properties determined by the different softening curve are distinguished.Hence, the calculated G_R crack extension resistance curve will show some differences,too.In this paper,the bilinear softening curves and non-linear softening curve are adopted.Also, the fracture energy determined from them according to the definition by Hillerborg is assumed to be a constant in all used softening curves.In such case,G_R crack extension resistance curves for seven numerical simulated TPB are computed.The impact of different softening curves on the obtained G_R crack extension resistance curves is highlighted.
4.Similar to ideal brittle materials,the fracture process of concrete is a process of energy transformation,too.During this process,the work done by external load is consumed in two prime manners,i.e.,elastic strain energy and fracture surface energy.However,last extensive studies put more interests in the finally completely fracturing time at which the external work caused by applied load is assumed to totally flow into fracture narrow zone. But,for others any crack propagation time,the related studies are seriously few.Because of this,in the present paper,the emphasis of the work is put on the whole fracture process. Then,two kinds of energy consumption of external force work,namely,elastic strain energy and fracture surface energy at any picked crack propagation are determined using the data obtained from wedge-splitting experiments.The calculated results provide new insights into the understanding of concrete fracture behavior.Meanwhile,the energy needed for creating unit area crack propagation at any crack propagation is computed,too. The corresponding discussions are made.
引文
[1]潘家铮,断裂力学在水工结构中的应用。水利学报,1980,5:.
[2]于骁中,居襄,曹建国,单支墩大头坝劈头裂缝稳定性分析。水利学报,1981,4:.
[3]梁文浩,谯常忻,涂传林,柘溪大头坝裂缝问题研究述评。水利学报,1982,6:.
[4]赵中极,重力坝坝面和坝水平裂缝断裂力学分析。水利学报,1982,6:.
[5]涂传林等,柘溪大坝加固后垂直裂缝稳定性分析。水力发电学报,1985,3:.
[6]徐世娘。混凝土断裂韧性的试验及分析。水利学报,1982,6:61-66。
[7]徐世娘。混凝土脆性破坏的断裂统计理论。冶金建筑,1982,11:22-29。
[8]徐世娘。混凝土断裂韧度的概率统计分析。水利学报,1984,10:51-58。
[9]赵国藩,徐世烺。混凝土损伤和断裂的机理,国家自然科学基金资助项目总结报告。大连理工大学,1987,1-99,
[10]徐世烺,赵国藩。混凝土断裂韧度的概率模型研究。土木工程学报,1988,21(4):9-23,
[11]徐世烺。混凝土断裂机理,大连理工大学博士学位论文。大连理工大学,1-155。
[12]混凝土裂缝的评定技术,七五国重点科技攻关17-2-1 FLCL,2项目总结报告。大连理工大学,1989,1-295,赵国藩、徐世烺、王凤翼、高泉。
[13]徐世烺,赵国藩。混凝土断裂韧度的尺寸效应,水利水电科学基金资助项目总结报告。大连理工大学,1990,1-105,
[14]徐世烺,赵国藩。混凝土断裂力学研究.大连理工大学出版社.1991年4月.
[15]徐世烺,赵国藩。混凝土巨型试件断裂韧度和高混凝土坝裂缝评定的断裂韧度准则。土木工程学报,1991,24(2):1-9.
[16]徐世烺、赵国藩。混凝土裂缝的稳定扩展过程与临界裂缝尖端张开位移。水利学报,1989,4:33-44,
[17]徐世烺,赵国藩.光弹性贴片法研究混凝土裂缝扩展过程.水力发电学报,1991,3:8-18.
[18]徐世烺,赵国藩,黄承逵,刘毅,王凤翼,靳国礼。混凝土大型试件断裂能和缝端应变场。水利学报,1991,1:17-25.
[19]丁骁中主编.岩石和混凝土断裂力学.中南工业大学出版社.1991年12月.
[20]徐世烺,赵国藩。混凝土结构裂缝扩展的双K断裂准则。土木工程学报,1992,25(2):32-38.
[21]赵国藩、徐世烺、王凤翼。大骨料全级配混凝土断裂韧度和断裂能研究。工程力学,1996年增刊.
[22]Griffth A A.The phenomena of rupture and flow in solids.Phil.Trans.Roy.Soc.series A221,1921,163-168.
[23]Orowan E O.In Fatigue and fracture of metals.Murray,W.M.Ed.,Wilcy,New York,1950,139-167.
[24]Irwin G R.Analysis of stresses and strains near the end of a crack traversing a plate.Journal of Applied Mechanics,1957,24:361-364.
[25]Rice J R.A path independent integral and the approximate analysis of strain concentration by notches and cracks.Journal of Applied Mechanics.1968,35:379-386.
[26]Hutchinson J W.Singular behavior at the end of a tensile crack in a hardening material.Journal of Mechanics and Physics of solids,1968,1:13-31.
[27]Rice J R,Rosengren G R.Plane strain deformation near a crack tip in a power-hardening material,Journal of Mechanics and Physics of solids,1968,1:1-12.
[28]Dugdule D S.Journal of Mechanics and Physics of solids,1960,8:100-104.
[29]Kaplan M F.Crack propagation and the fracture of concrete.Journal of the American Concrete Institute,1961,58(5):591-610.
[30]Mindess S.The fracture procress zone in concrete.In Toughening Mechanisms in Quasi-brittle materials,Edited by Shah S P.Kluwer Academic,Dordrecht,The Netherlands,1991,1-15.
[31]Kachanov M.Interaction of a crack with some microcrack systems.In Fracture Toughness and Fracture energy of Concrete,edited by Wittmann F H.Elsevier Science Publishers B V,The Netherlands,1986:3-10.
[32]Krstulovic O N.Fracture process zone presence and behavior in mortar specimens.ACI Materials Journal,1993,90(6):618-626.
[33]Shum K M,Hutchinson J W.On Toughening by microcracks,Mechanics of Materials,1990,9:83-91.
[34]Van Mier J G M.Mode I fracture of concrete:Discontinuous crack growth and crack interface grain bridging.Cement and Concrete Research,1991,21(1):1-15.
[35]Mindess S.Fracture Process Zone Detection.Fracture Mechanics Test Method for Concrete,Report of Technical Committee 89-FMT,RILEM,Chapman and Hall,1991.
[36]Shah S P,Ouyang C.Measurement and Modeling of fracture processes in concrete,in Materials Science of concrete,edited by Scalny J,Amercian Ceramic Society,Westerville OH,1992,243-270.
[37]Castro-Montero A,Shah S P,Miller R A.Strain field measurement in fracture process zone.Journal of Engineering Mechanics,ASCE,1990,116(11):2463-2484.
[38]Ouyang C,Landis E,Shah S P.Damange assessment in concrete using quantitative acoustic emission.Journal of Engineering Mechanics,ASCE,1990,117(11):2681-2698.
[39]Landis E,Shah S P.Recovery of microcrack parameters in mortar using quantitative acoustic emission.Journal of Nondestructive Evaluation,1993,12(4):219-232.
[40]Swartz S E,Go C G.Validity of compliance calibration to cracked concrete beams in bending tests.Experimental Mechanics,1984,24(2):129-134.
[41]Mindess S.The cracking and fracture of concrete:an annotated bibliography 1982-1985.Fracture Mechanics of Concrete(edited by Wittmann F H).Elsevier Science Publishers B V,The Netherlands,1986:629-694.
[42]Swartz S E,Refai T M E.Influence of size on opening mode fracture parameters for precracked concrete beams in bending.Proceedings of ESM-RILEM International conference on fracture of concrete and rock.Shah S P and Swartz S E(eds),Houston,Texas,1987:242-254.
[43]Kan Y C,Swartz S E.Influence of curing conditions and width on the fracture of concrete beams,in Proceeding of the 1989 spring conference on experimentals mechanics,Cambirdge MA,1989,196-201.
[44]Shah S P.Experimental methods for determining fracture process zone and fracture parameters.Engineering Fracture Mechanics,1990,35(1):3-14.
[45]Lawn Brian R.Fracture of brittle solids.U K,Cambridge University Press,1993.
[46]Hillerborg A.A Analysis of Crack Formation and Crack Growth in Concrete by means of Fracture Mechanics and Finite Elements.Cement and Concrete Research,1976,6:773-782.
[47]Hillerborg A.The theoretical basis of method to determine the fracture energy Gr of concrete.Materials and Structures,1985,18(106):291-296.
[48]Bazant Z P.Crack band theory for fracture of concrete.RILEM.Materials and Structures,1983,16(93):155-177.
[49]Bazant Z P.Size effect in blunt fracture:concrete,rock,metal.Journal of Engineering Mechanics,ASCE,1984,110(4):518-535.
[50]RILEM Technical Committee 50-FMC,Determination of the Fracture Energy of Mortar and Concrete by Means of Tree-Point Bend Tests of Notched Beams,Proposed RILEM Draft Recommendations.RILEM,Maters.& Structs.,1985,Vol.18,No.106,285-296.
[51]Jenq Y S,Shah S P.Two parameter fracture model for concrete.Journal of Engineering Mechanics,ASCE,1985,111(10):1227-1241.
[52]Jenq Y S,Shah S P.A fracture toughness criterion for concrete.Engineering Fracture Mechanics,1985,21(5):1055-1069.
[53]Bazant Z P,Kazemi M T.Size dependence of concrete fracture energy determined by Rilem work-of-fracture method.International Journal of Fracture,1991,51:121-138.
[54]Bazant Z P,Kazemi M T.Determination of fracture energy,process zone length and brittleness number from size effect,with application to rock and concrete.International Journal of Fracture,1990,44:111-131.
[55]Bazant Z P.Fracture energy of heterogeneous materials and similitude,in Fracture of Concrete and Rock,edited by Shah SP,Swartz S E.Springer-Verlag,New York,1986,121-150.
[56]Karihaloo B L,Nallathambi.Effective crack model for the determination of fracture toughness K(?)~S of concrete.Enigneering Fracture Mechanics,1990,35(4/5):637-645.
[57]Karihaloo B L,AbdallaHM,Xiao Q Z.Size effect in concrete beams.Engineering Fracture Mechanics,2003,70:979-993.
[58]Karihaloo B L,Nallathambi.An improved effective crack model for the determination of fracture toughness of concrete.Cement and Concrete Research,1989,19:603-610.
[59]Swartz S E,Go C G.Validity of compliance calibration to cracked concrete beams in bending tests.Experimental Mechanics,1984,24(2):129-134.
[60]Swartz S E,Refai T H E.Influence of size on opening mode fracture parameters for precracked concrete beams in bending.Proceedings of ESM-RILEM International conference on fracture of concrete and rock.Shah S P and Swartz S E(eds),Houston,Texas,1987:242-254.
[61]RILEM Technical Committee 89-FMT,Determination of Fracture Parameters(KIcs and CTODc) of Plain Concrete Using Tree-Point Bend Tests,Proposed RILEM Draft Recommen-dations.Ibid.,23(138),1990,457-460.
[62]Planas J,Elices M.Fracture criteria for concrete:Mathematical applications and experimental validation.Engineering Fracture Mechanics,1990,35(3):87-94.
[63]Elices M,Planas J.Fracture mechanics parameters of concrete.Advanced Cement Based Materials,1996,4,116-127.
[64]Ouyang C,Shah S P.Geometry-dependent R-curve for quasi-brittle materials.Journal of the American Ceramic Society,1991,74(11):2831-2836.
[65]Ouyang C,Mobasher B,Shah S P.A R-curve approach for fracture of quasi-brittle materials.Enigneering Fracture Mechanics,1990,37(4):901—913.
[66]John R,Shah S P,Jenq Y S.A fracture mechanics model to predict the rate sensitivity of mode I fracture of concrete.Cement and Concrete Research,1987,17(2):249-262.
[67]RILEM-Draft-Recommandation(50-FCM),Fracture mechanics of concrete-test methods.Size-effect method for determining fracture energy and process zone size of concrete.Materilas and Structure,1990,23:461-465.
[68]Bazant Z P.Current Trends in Concrete Fracture Research,Kluwer Academic Publishers,Dordrecht,1991.
[69]Bazant Z P,Kim J K.Size Effect in Shear Failure of Longitudinal Reinforced Beams.ACI Structural Journal,1984,81(5):456-468.
[70]Xu Shilang,Reinhardt H W.Determination of double-K criterion for crack propagation in quasi-brittle fracture.Part I:Experimental investigation of crack propagation.International Journal of Fracture,1999,98:111-149.
[71]Xu Shilang,Reinhardt H W.Determination of double-K criterion for crack propagation in quasi-brittle fracture.Part Ⅱ:Analytical evaluating and practical measuring methods for three-point bending notched beams.International Journal of Fracture,1999,98:151-177.
[72]Xu Shilang,Reinhardt H W.Determination of double-K criterion for crack propagation in quasi-brittle fracture Part Ⅲ:Compact tension specimens and wedge splitting specimens.International Journal of Fracture,1999,98:179-193.
[73]Xu Shilang,Reinhardt H W.A simplified method for determining double-K fracture parameters for three-point bending tests.International Journal of Fracture,2000,104:181-208.
[74]Xu Shilang,Reinhardt H W.Determination of double-K facture parameters in standard three-point bending notched beams.Fracture Mechanics of Concrete Structures,Proceedings FRAMCOS-3,Aedificatio Publishers,Germany,1998,1:431-440.
[75]Reinhardt H W,Shilang Xu.Crack extension resistance based on the cohesive force in concrete.Engineering Fracture Mecjanics 1999,64,563-587.
[76]Shilang Xu,Reinhardt,H W.Crack extension resistance and fracture properties of quasi-brittle softening material like concrete based on the complete process of fracture.International Journal of Fracture,1998 92(1):71-99.
[77]徐世烺。混凝土双K断裂参数计算理论及规范化测试方法。第七届全国岩石混凝土断裂损伤和强度学术讨论会大会特邀报告,武汉,2001年10月,(见三峡大学学报,2002,24(1):1-8.
[78]赵志方、徐世烺。用于确定双K断裂参数的混凝土软化本构曲线。清华大学学报(自然科学版).2000,40(S1):110-113.
[79]赵志方、徐世烺。混凝土软化本构曲线形状对双K断裂参数的影响。土木工程学报,2001,34(5):29-34.
[80]徐世烺,周厚贵,高洪波,赵守阳。各种级配大坝混凝土双K断裂参数实验研究。土木工程学报,2006,39(11):64-76.
[81]徐世烺,张秀芳,郑爽。小骨料混凝土双K断裂参数的实验测定。水利学报,2006,37(5):26-36.
[82]张秀芳,徐世烺,高洪波。混凝土楔入劈拉试件的双K断裂参数叠加计算及其边界效应。大连理工大学学报,2006,46(3):868-874.
[83]DL/T5332-2005水工混凝土断裂试验规程[S].北京:中国电力出版社,2006.
[84]尹双增.断裂·损伤·理论及应用.清华大学出版社,1992.
[85]Reinhardt H W,Cornelissen H A W,Hordijik Dirk A.Tensile tests and failure analysis of concrete.Journal of Structure Engineering,ASCE,1986,112,2462-2477.
[86]Lawn Brian R.Fracture of brittle solids.U K,Cambridge University Press,1993.
[87]Mindess S.The cracking and fracture of concrete:an annotated bibliography 1928-1981.Fracture Mechanics of Concrete(edited by Wittmann F H).Elsevier Science Publishers B V,The Netherlands,1983:539-561.
[88]Ouyang C,Mobasher B,Shah S P.A R-curve approach for fracture of quasi-brittle materials.Enigneering Fracture Mechanics,1990,37(4):901-913.
[89]Liang R Y,Li Yuan Neng.Study of size effect in concrete using fictitious crack model.Journal of Engineering Mechanics,1991,117(7):1631-1651.
[90]Malvar L J,Fourney M E.A three dimensional application of the smeared crack approach.Engineering Fracture Mechanics,1990,35(3):251-260.
[91]张君.混凝土材料断裂力学参数的测定方法与评价.北京:清华大学.2005.
[92]Nallathambi P,Karihaloo B L.Determination of specimen-size independent fracture toughness of plain concrete.Magazine of Concrete Research,1986,38(135):67-76.
[93]Kleinschrodt H D,Winkler H.The influence of the maximum aggregate size and the size of specimen on fracture mechanics parameters.Fracture toughness and fracture energy of concrete(edited by Wittmann F H).Elsevier Science Publishers B V,Amsterdam,The Netherlands,1987.391-402.
[94]Bazant Z P,Kim,Pfeiffer P A.Determination of fracture energy from size effect and brittleness number.ACI Mater J 1987,84:463-480.
[95]Bazant Z P.Size effect in blunt fracture:concrete,rock,metal.Journal engineering mechanics,ASCE,1984,110(4):518-535.
[96]Murakami.Stress intensity factors handbook.Pergamon Press,London,1987.
[97]徐世烺.混凝土断裂机理.大连理工大学博士学位论文,1988.
[98]Jenq Y S,Shah S P.Nonlinear fracture parameters for cement based composites:theory and experiments.In application of fracture mechanics to cementitious composites(edited by S P Shah) 1985,319-359.
[99]徐世烺.混凝土断裂机理.大连理工大学博士学位论文,1988.
[100]Hillemeier B,Hilsdorf H K.Fracture mechanics studies on concrete compounds.Cement and Concrete Research,1977,7,523-536.
[101]Br(u|¨)hwiler E,Wittmann F H.The wedge splitting test,a new method of Performing stable fracture mechanics tests.Engineering Fracture Mechanics,1990,35(3):117-125.
[102]焦辉.混凝土Ⅰ型裂缝断裂性能试验研究.大连理工大学硕士学位论文,1990.
[103]Zhao Guofan,Jiao Hui,Xu Shilang.Study on Fracture Behavior with Wedge Splitting Test Method.Fracture Process in Concrete,Rock and Ceramics(edited by J.G.M.van Mier,J.G.Rots and A.Bakker),E & FN Sport,An Imprint of Chapman & Hall,London,1991.789-798.
[104]Zhao Guofan,Jiao Hui,Xu Shilang.Study of Fracture Toughness and Fracture Energy by Means of Wedge Splitting Test Specimens.Brittle Matrix Composites 3(edited by A M Brandt),Elsevier Applied Science,The Netherlands,1991.
[105]niroshi Tada,Paris P C,Irwin G R.The stress analysis of cracks handbook,New York,ASME Press,2000.
[106]Gopalaratnam V S,Shah S P,Batson G B,Criswell M E,Ramakrishnan V and Wecharatana M.Fracture toughness of fiber reinforced concrete,ACI Materials Journal,1991,88:339-353.
[107]Gopalaratnam V S,Gettu R,Carmona S,Jamet D.Charaterization of the toughness of fiber reinforced concrete using the load-CMOD response,in proceedings Framcos-2,Edited by Wittmann F H,AEDIFICATIO Publishers,Frriburg 1995,769-782.
[108]Barr B,Gettu R,AlOraimi S K A,Bryars L S,Toughness measurement-the need to think again.Cement and Concrete Composites,1996,18:281-297.
[109]Barr B,M K Lee,Dupont D,Erdem E,Schaerlaekens S,Schnutegen B,Stang H and Vandewalle L,Round-robin analysis of the RILEM TC 162-TDF beam-bending test: Part2-Approximation of δ from the CMOD response.Materials and Structures,2003,36:621-630.
[110]Navalurkar R K,Hsu T T C,Kim S K,Wecharatana M,True fracture energy of concrete.ACI Materials Journal,1999,96 213-227.
[111]陈篪,蔡其巩,王仁智.《工程断裂力学》(上册).国防工业出版社,1978.
[112]Li V C,Chan C M,Leung C K Y.Experimental determiantion of the tension-softening relations for cementitious composites.Cement and Concrete Research,1987,17:441-452.
[113]Nanakorn P,Horii H.Back analysis of tension-softening relationships of concrete.Journal of Materials,Concrete Structure Pavements,JSCE,1996,32(544):256-275.
[114]Kitsutaka Y.Fracture parameters by polylinear tension-softening relationship of concrete.Journal of Engineering Mechanics,1997,123(5):444-450.
[115]Zhang Jun,Victor C L.Simulation of crack propagation in fiber-reinforced concrete by fracture mechanics.Cement and concrete Research,2004,34,333-339.
[116]Zhang Jun,Liu Qian.Determination of concrete fracture parameters from a three-point bending test.rsinghua Science and Technology,2003,8(6):726-733.
[117]Zhang Jun,Liu Qian,Lin Wang.Effect of coarse aggregate size on relationship between stress and crack opening in normal and high strength concretes.Journal of Materials Science and Technology,2005,5(21).
[118]张君,于林,孙明,刘骞.粗细骨料比例和水泥石强度对混凝土断裂参数的影响.工程力学,2004,21(1):136-142.
[119]Petersson P E.Crack Growth and Development of Fracture Zones in Plain Concrete and Similar Materials.Division of Building Materials,Lund Institute of Technology,Report TVBM-1006,1981.
[120]CEB-Comite Euro-lnternational du Beton-EB-FIP Model Code 1990,Bulletin D'Information No.213/214,Lausanne.
[121]Xu S L,Reinhardt H W.Analytical Solution of the Fictitious Crack and Evaluation of the Crack Extension Resistance for a Griffith Crack.Fracture Mechanics of Concrete Structures,Proceedings FRAMCOS-3,aedificaio Publishers,Germany,1:409-420.
[122]Xu S L.Determination of Parameters in the Bilinear,Reinhardt's and Exponentially Nonlinear Softening Curves and Their Physical Meanings.Werkstoffe und Werkstoffpruefung im bauwesen,Hamburg,Libri BOD,1999:410-424.
[123]Liaw B M,Jeang F L,Du J J,Hawkins N M,Kobayashi A S.Improved Nonlinear Model for Concrete Fracture.Journal of Engineering Mechanics,ASCE,1990,116(2):429-445.
[124]Gopalartnam V S,Shah S P.Softening Response of Plain Concrete in Direct Tension.ACI Journal,82(3):310-323.
[125]Barr B,Lee M K.Modelling the strain-softening behaviors of plain concrete using a double-exponential model,Magazine of Concrete Research,2003,55(4):343-353.
[126]赵志方,徐世烺.试件尺寸对混凝土新K_R阻力曲线的影响,水利学报,2001,12:48-55.
[127]赵志方.混凝土软化本构关系和基于粘聚力的新K_R阻力曲线的研究,大连理工大学博士后研究工作报告,大连,大连理工大学,2000.
[128]Swartz S E,Refai T.In Proceedings of SEM/RILEM International conference on fracture of concrete and rock.Houston,Texas,1987,403-417.
[129]Swartz S E,Refai T.In Proceedings of International workshop on fracture toughness and fracture energy-test methods for concrete and rock,Sendal,Japan,1988.
[130]Yeou-Shang Jenq,Shah S P.Features of mechanics of quasi-brittle crack propagation in concrete.International Journal of Fracture,1991 51(1):103-120.
[2]于骁中,居襄,曹建国,单支墩大头坝劈头裂缝稳定性分析。水利学报,1981,4:.
[3]梁文浩,谯常忻,涂传林,柘溪大头坝裂缝问题研究述评。水利学报,1982,6:.
[4]赵中极,重力坝坝面和坝水平裂缝断裂力学分析。水利学报,1982,6:.
[5]涂传林等,柘溪大坝加固后垂直裂缝稳定性分析。水力发电学报,1985,3:.
[6]徐世娘。混凝土断裂韧性的试验及分析。水利学报,1982,6:61-66。
[7]徐世娘。混凝土脆性破坏的断裂统计理论。冶金建筑,1982,11:22-29。
[8]徐世娘。混凝土断裂韧度的概率统计分析。水利学报,1984,10:51-58。
[9]赵国藩,徐世烺。混凝土损伤和断裂的机理,国家自然科学基金资助项目总结报告。大连理工大学,1987,1-99,
[10]徐世烺,赵国藩。混凝土断裂韧度的概率模型研究。土木工程学报,1988,21(4):9-23,
[11]徐世烺。混凝土断裂机理,大连理工大学博士学位论文。大连理工大学,1-155。
[12]混凝土裂缝的评定技术,七五国重点科技攻关17-2-1 FLCL,2项目总结报告。大连理工大学,1989,1-295,赵国藩、徐世烺、王凤翼、高泉。
[13]徐世烺,赵国藩。混凝土断裂韧度的尺寸效应,水利水电科学基金资助项目总结报告。大连理工大学,1990,1-105,
[14]徐世烺,赵国藩。混凝土断裂力学研究.大连理工大学出版社.1991年4月.
[15]徐世烺,赵国藩。混凝土巨型试件断裂韧度和高混凝土坝裂缝评定的断裂韧度准则。土木工程学报,1991,24(2):1-9.
[16]徐世烺、赵国藩。混凝土裂缝的稳定扩展过程与临界裂缝尖端张开位移。水利学报,1989,4:33-44,
[17]徐世烺,赵国藩.光弹性贴片法研究混凝土裂缝扩展过程.水力发电学报,1991,3:8-18.
[18]徐世烺,赵国藩,黄承逵,刘毅,王凤翼,靳国礼。混凝土大型试件断裂能和缝端应变场。水利学报,1991,1:17-25.
[19]丁骁中主编.岩石和混凝土断裂力学.中南工业大学出版社.1991年12月.
[20]徐世烺,赵国藩。混凝土结构裂缝扩展的双K断裂准则。土木工程学报,1992,25(2):32-38.
[21]赵国藩、徐世烺、王凤翼。大骨料全级配混凝土断裂韧度和断裂能研究。工程力学,1996年增刊.
[22]Griffth A A.The phenomena of rupture and flow in solids.Phil.Trans.Roy.Soc.series A221,1921,163-168.
[23]Orowan E O.In Fatigue and fracture of metals.Murray,W.M.Ed.,Wilcy,New York,1950,139-167.
[24]Irwin G R.Analysis of stresses and strains near the end of a crack traversing a plate.Journal of Applied Mechanics,1957,24:361-364.
[25]Rice J R.A path independent integral and the approximate analysis of strain concentration by notches and cracks.Journal of Applied Mechanics.1968,35:379-386.
[26]Hutchinson J W.Singular behavior at the end of a tensile crack in a hardening material.Journal of Mechanics and Physics of solids,1968,1:13-31.
[27]Rice J R,Rosengren G R.Plane strain deformation near a crack tip in a power-hardening material,Journal of Mechanics and Physics of solids,1968,1:1-12.
[28]Dugdule D S.Journal of Mechanics and Physics of solids,1960,8:100-104.
[29]Kaplan M F.Crack propagation and the fracture of concrete.Journal of the American Concrete Institute,1961,58(5):591-610.
[30]Mindess S.The fracture procress zone in concrete.In Toughening Mechanisms in Quasi-brittle materials,Edited by Shah S P.Kluwer Academic,Dordrecht,The Netherlands,1991,1-15.
[31]Kachanov M.Interaction of a crack with some microcrack systems.In Fracture Toughness and Fracture energy of Concrete,edited by Wittmann F H.Elsevier Science Publishers B V,The Netherlands,1986:3-10.
[32]Krstulovic O N.Fracture process zone presence and behavior in mortar specimens.ACI Materials Journal,1993,90(6):618-626.
[33]Shum K M,Hutchinson J W.On Toughening by microcracks,Mechanics of Materials,1990,9:83-91.
[34]Van Mier J G M.Mode I fracture of concrete:Discontinuous crack growth and crack interface grain bridging.Cement and Concrete Research,1991,21(1):1-15.
[35]Mindess S.Fracture Process Zone Detection.Fracture Mechanics Test Method for Concrete,Report of Technical Committee 89-FMT,RILEM,Chapman and Hall,1991.
[36]Shah S P,Ouyang C.Measurement and Modeling of fracture processes in concrete,in Materials Science of concrete,edited by Scalny J,Amercian Ceramic Society,Westerville OH,1992,243-270.
[37]Castro-Montero A,Shah S P,Miller R A.Strain field measurement in fracture process zone.Journal of Engineering Mechanics,ASCE,1990,116(11):2463-2484.
[38]Ouyang C,Landis E,Shah S P.Damange assessment in concrete using quantitative acoustic emission.Journal of Engineering Mechanics,ASCE,1990,117(11):2681-2698.
[39]Landis E,Shah S P.Recovery of microcrack parameters in mortar using quantitative acoustic emission.Journal of Nondestructive Evaluation,1993,12(4):219-232.
[40]Swartz S E,Go C G.Validity of compliance calibration to cracked concrete beams in bending tests.Experimental Mechanics,1984,24(2):129-134.
[41]Mindess S.The cracking and fracture of concrete:an annotated bibliography 1982-1985.Fracture Mechanics of Concrete(edited by Wittmann F H).Elsevier Science Publishers B V,The Netherlands,1986:629-694.
[42]Swartz S E,Refai T M E.Influence of size on opening mode fracture parameters for precracked concrete beams in bending.Proceedings of ESM-RILEM International conference on fracture of concrete and rock.Shah S P and Swartz S E(eds),Houston,Texas,1987:242-254.
[43]Kan Y C,Swartz S E.Influence of curing conditions and width on the fracture of concrete beams,in Proceeding of the 1989 spring conference on experimentals mechanics,Cambirdge MA,1989,196-201.
[44]Shah S P.Experimental methods for determining fracture process zone and fracture parameters.Engineering Fracture Mechanics,1990,35(1):3-14.
[45]Lawn Brian R.Fracture of brittle solids.U K,Cambridge University Press,1993.
[46]Hillerborg A.A Analysis of Crack Formation and Crack Growth in Concrete by means of Fracture Mechanics and Finite Elements.Cement and Concrete Research,1976,6:773-782.
[47]Hillerborg A.The theoretical basis of method to determine the fracture energy Gr of concrete.Materials and Structures,1985,18(106):291-296.
[48]Bazant Z P.Crack band theory for fracture of concrete.RILEM.Materials and Structures,1983,16(93):155-177.
[49]Bazant Z P.Size effect in blunt fracture:concrete,rock,metal.Journal of Engineering Mechanics,ASCE,1984,110(4):518-535.
[50]RILEM Technical Committee 50-FMC,Determination of the Fracture Energy of Mortar and Concrete by Means of Tree-Point Bend Tests of Notched Beams,Proposed RILEM Draft Recommendations.RILEM,Maters.& Structs.,1985,Vol.18,No.106,285-296.
[51]Jenq Y S,Shah S P.Two parameter fracture model for concrete.Journal of Engineering Mechanics,ASCE,1985,111(10):1227-1241.
[52]Jenq Y S,Shah S P.A fracture toughness criterion for concrete.Engineering Fracture Mechanics,1985,21(5):1055-1069.
[53]Bazant Z P,Kazemi M T.Size dependence of concrete fracture energy determined by Rilem work-of-fracture method.International Journal of Fracture,1991,51:121-138.
[54]Bazant Z P,Kazemi M T.Determination of fracture energy,process zone length and brittleness number from size effect,with application to rock and concrete.International Journal of Fracture,1990,44:111-131.
[55]Bazant Z P.Fracture energy of heterogeneous materials and similitude,in Fracture of Concrete and Rock,edited by Shah SP,Swartz S E.Springer-Verlag,New York,1986,121-150.
[56]Karihaloo B L,Nallathambi.Effective crack model for the determination of fracture toughness K(?)~S of concrete.Enigneering Fracture Mechanics,1990,35(4/5):637-645.
[57]Karihaloo B L,AbdallaHM,Xiao Q Z.Size effect in concrete beams.Engineering Fracture Mechanics,2003,70:979-993.
[58]Karihaloo B L,Nallathambi.An improved effective crack model for the determination of fracture toughness of concrete.Cement and Concrete Research,1989,19:603-610.
[59]Swartz S E,Go C G.Validity of compliance calibration to cracked concrete beams in bending tests.Experimental Mechanics,1984,24(2):129-134.
[60]Swartz S E,Refai T H E.Influence of size on opening mode fracture parameters for precracked concrete beams in bending.Proceedings of ESM-RILEM International conference on fracture of concrete and rock.Shah S P and Swartz S E(eds),Houston,Texas,1987:242-254.
[61]RILEM Technical Committee 89-FMT,Determination of Fracture Parameters(KIcs and CTODc) of Plain Concrete Using Tree-Point Bend Tests,Proposed RILEM Draft Recommen-dations.Ibid.,23(138),1990,457-460.
[62]Planas J,Elices M.Fracture criteria for concrete:Mathematical applications and experimental validation.Engineering Fracture Mechanics,1990,35(3):87-94.
[63]Elices M,Planas J.Fracture mechanics parameters of concrete.Advanced Cement Based Materials,1996,4,116-127.
[64]Ouyang C,Shah S P.Geometry-dependent R-curve for quasi-brittle materials.Journal of the American Ceramic Society,1991,74(11):2831-2836.
[65]Ouyang C,Mobasher B,Shah S P.A R-curve approach for fracture of quasi-brittle materials.Enigneering Fracture Mechanics,1990,37(4):901—913.
[66]John R,Shah S P,Jenq Y S.A fracture mechanics model to predict the rate sensitivity of mode I fracture of concrete.Cement and Concrete Research,1987,17(2):249-262.
[67]RILEM-Draft-Recommandation(50-FCM),Fracture mechanics of concrete-test methods.Size-effect method for determining fracture energy and process zone size of concrete.Materilas and Structure,1990,23:461-465.
[68]Bazant Z P.Current Trends in Concrete Fracture Research,Kluwer Academic Publishers,Dordrecht,1991.
[69]Bazant Z P,Kim J K.Size Effect in Shear Failure of Longitudinal Reinforced Beams.ACI Structural Journal,1984,81(5):456-468.
[70]Xu Shilang,Reinhardt H W.Determination of double-K criterion for crack propagation in quasi-brittle fracture.Part I:Experimental investigation of crack propagation.International Journal of Fracture,1999,98:111-149.
[71]Xu Shilang,Reinhardt H W.Determination of double-K criterion for crack propagation in quasi-brittle fracture.Part Ⅱ:Analytical evaluating and practical measuring methods for three-point bending notched beams.International Journal of Fracture,1999,98:151-177.
[72]Xu Shilang,Reinhardt H W.Determination of double-K criterion for crack propagation in quasi-brittle fracture Part Ⅲ:Compact tension specimens and wedge splitting specimens.International Journal of Fracture,1999,98:179-193.
[73]Xu Shilang,Reinhardt H W.A simplified method for determining double-K fracture parameters for three-point bending tests.International Journal of Fracture,2000,104:181-208.
[74]Xu Shilang,Reinhardt H W.Determination of double-K facture parameters in standard three-point bending notched beams.Fracture Mechanics of Concrete Structures,Proceedings FRAMCOS-3,Aedificatio Publishers,Germany,1998,1:431-440.
[75]Reinhardt H W,Shilang Xu.Crack extension resistance based on the cohesive force in concrete.Engineering Fracture Mecjanics 1999,64,563-587.
[76]Shilang Xu,Reinhardt,H W.Crack extension resistance and fracture properties of quasi-brittle softening material like concrete based on the complete process of fracture.International Journal of Fracture,1998 92(1):71-99.
[77]徐世烺。混凝土双K断裂参数计算理论及规范化测试方法。第七届全国岩石混凝土断裂损伤和强度学术讨论会大会特邀报告,武汉,2001年10月,(见三峡大学学报,2002,24(1):1-8.
[78]赵志方、徐世烺。用于确定双K断裂参数的混凝土软化本构曲线。清华大学学报(自然科学版).2000,40(S1):110-113.
[79]赵志方、徐世烺。混凝土软化本构曲线形状对双K断裂参数的影响。土木工程学报,2001,34(5):29-34.
[80]徐世烺,周厚贵,高洪波,赵守阳。各种级配大坝混凝土双K断裂参数实验研究。土木工程学报,2006,39(11):64-76.
[81]徐世烺,张秀芳,郑爽。小骨料混凝土双K断裂参数的实验测定。水利学报,2006,37(5):26-36.
[82]张秀芳,徐世烺,高洪波。混凝土楔入劈拉试件的双K断裂参数叠加计算及其边界效应。大连理工大学学报,2006,46(3):868-874.
[83]DL/T5332-2005水工混凝土断裂试验规程[S].北京:中国电力出版社,2006.
[84]尹双增.断裂·损伤·理论及应用.清华大学出版社,1992.
[85]Reinhardt H W,Cornelissen H A W,Hordijik Dirk A.Tensile tests and failure analysis of concrete.Journal of Structure Engineering,ASCE,1986,112,2462-2477.
[86]Lawn Brian R.Fracture of brittle solids.U K,Cambridge University Press,1993.
[87]Mindess S.The cracking and fracture of concrete:an annotated bibliography 1928-1981.Fracture Mechanics of Concrete(edited by Wittmann F H).Elsevier Science Publishers B V,The Netherlands,1983:539-561.
[88]Ouyang C,Mobasher B,Shah S P.A R-curve approach for fracture of quasi-brittle materials.Enigneering Fracture Mechanics,1990,37(4):901-913.
[89]Liang R Y,Li Yuan Neng.Study of size effect in concrete using fictitious crack model.Journal of Engineering Mechanics,1991,117(7):1631-1651.
[90]Malvar L J,Fourney M E.A three dimensional application of the smeared crack approach.Engineering Fracture Mechanics,1990,35(3):251-260.
[91]张君.混凝土材料断裂力学参数的测定方法与评价.北京:清华大学.2005.
[92]Nallathambi P,Karihaloo B L.Determination of specimen-size independent fracture toughness of plain concrete.Magazine of Concrete Research,1986,38(135):67-76.
[93]Kleinschrodt H D,Winkler H.The influence of the maximum aggregate size and the size of specimen on fracture mechanics parameters.Fracture toughness and fracture energy of concrete(edited by Wittmann F H).Elsevier Science Publishers B V,Amsterdam,The Netherlands,1987.391-402.
[94]Bazant Z P,Kim,Pfeiffer P A.Determination of fracture energy from size effect and brittleness number.ACI Mater J 1987,84:463-480.
[95]Bazant Z P.Size effect in blunt fracture:concrete,rock,metal.Journal engineering mechanics,ASCE,1984,110(4):518-535.
[96]Murakami.Stress intensity factors handbook.Pergamon Press,London,1987.
[97]徐世烺.混凝土断裂机理.大连理工大学博士学位论文,1988.
[98]Jenq Y S,Shah S P.Nonlinear fracture parameters for cement based composites:theory and experiments.In application of fracture mechanics to cementitious composites(edited by S P Shah) 1985,319-359.
[99]徐世烺.混凝土断裂机理.大连理工大学博士学位论文,1988.
[100]Hillemeier B,Hilsdorf H K.Fracture mechanics studies on concrete compounds.Cement and Concrete Research,1977,7,523-536.
[101]Br(u|¨)hwiler E,Wittmann F H.The wedge splitting test,a new method of Performing stable fracture mechanics tests.Engineering Fracture Mechanics,1990,35(3):117-125.
[102]焦辉.混凝土Ⅰ型裂缝断裂性能试验研究.大连理工大学硕士学位论文,1990.
[103]Zhao Guofan,Jiao Hui,Xu Shilang.Study on Fracture Behavior with Wedge Splitting Test Method.Fracture Process in Concrete,Rock and Ceramics(edited by J.G.M.van Mier,J.G.Rots and A.Bakker),E & FN Sport,An Imprint of Chapman & Hall,London,1991.789-798.
[104]Zhao Guofan,Jiao Hui,Xu Shilang.Study of Fracture Toughness and Fracture Energy by Means of Wedge Splitting Test Specimens.Brittle Matrix Composites 3(edited by A M Brandt),Elsevier Applied Science,The Netherlands,1991.
[105]niroshi Tada,Paris P C,Irwin G R.The stress analysis of cracks handbook,New York,ASME Press,2000.
[106]Gopalaratnam V S,Shah S P,Batson G B,Criswell M E,Ramakrishnan V and Wecharatana M.Fracture toughness of fiber reinforced concrete,ACI Materials Journal,1991,88:339-353.
[107]Gopalaratnam V S,Gettu R,Carmona S,Jamet D.Charaterization of the toughness of fiber reinforced concrete using the load-CMOD response,in proceedings Framcos-2,Edited by Wittmann F H,AEDIFICATIO Publishers,Frriburg 1995,769-782.
[108]Barr B,Gettu R,AlOraimi S K A,Bryars L S,Toughness measurement-the need to think again.Cement and Concrete Composites,1996,18:281-297.
[109]Barr B,M K Lee,Dupont D,Erdem E,Schaerlaekens S,Schnutegen B,Stang H and Vandewalle L,Round-robin analysis of the RILEM TC 162-TDF beam-bending test: Part2-Approximation of δ from the CMOD response.Materials and Structures,2003,36:621-630.
[110]Navalurkar R K,Hsu T T C,Kim S K,Wecharatana M,True fracture energy of concrete.ACI Materials Journal,1999,96 213-227.
[111]陈篪,蔡其巩,王仁智.《工程断裂力学》(上册).国防工业出版社,1978.
[112]Li V C,Chan C M,Leung C K Y.Experimental determiantion of the tension-softening relations for cementitious composites.Cement and Concrete Research,1987,17:441-452.
[113]Nanakorn P,Horii H.Back analysis of tension-softening relationships of concrete.Journal of Materials,Concrete Structure Pavements,JSCE,1996,32(544):256-275.
[114]Kitsutaka Y.Fracture parameters by polylinear tension-softening relationship of concrete.Journal of Engineering Mechanics,1997,123(5):444-450.
[115]Zhang Jun,Victor C L.Simulation of crack propagation in fiber-reinforced concrete by fracture mechanics.Cement and concrete Research,2004,34,333-339.
[116]Zhang Jun,Liu Qian.Determination of concrete fracture parameters from a three-point bending test.rsinghua Science and Technology,2003,8(6):726-733.
[117]Zhang Jun,Liu Qian,Lin Wang.Effect of coarse aggregate size on relationship between stress and crack opening in normal and high strength concretes.Journal of Materials Science and Technology,2005,5(21).
[118]张君,于林,孙明,刘骞.粗细骨料比例和水泥石强度对混凝土断裂参数的影响.工程力学,2004,21(1):136-142.
[119]Petersson P E.Crack Growth and Development of Fracture Zones in Plain Concrete and Similar Materials.Division of Building Materials,Lund Institute of Technology,Report TVBM-1006,1981.
[120]CEB-Comite Euro-lnternational du Beton-EB-FIP Model Code 1990,Bulletin D'Information No.213/214,Lausanne.
[121]Xu S L,Reinhardt H W.Analytical Solution of the Fictitious Crack and Evaluation of the Crack Extension Resistance for a Griffith Crack.Fracture Mechanics of Concrete Structures,Proceedings FRAMCOS-3,aedificaio Publishers,Germany,1:409-420.
[122]Xu S L.Determination of Parameters in the Bilinear,Reinhardt's and Exponentially Nonlinear Softening Curves and Their Physical Meanings.Werkstoffe und Werkstoffpruefung im bauwesen,Hamburg,Libri BOD,1999:410-424.
[123]Liaw B M,Jeang F L,Du J J,Hawkins N M,Kobayashi A S.Improved Nonlinear Model for Concrete Fracture.Journal of Engineering Mechanics,ASCE,1990,116(2):429-445.
[124]Gopalartnam V S,Shah S P.Softening Response of Plain Concrete in Direct Tension.ACI Journal,82(3):310-323.
[125]Barr B,Lee M K.Modelling the strain-softening behaviors of plain concrete using a double-exponential model,Magazine of Concrete Research,2003,55(4):343-353.
[126]赵志方,徐世烺.试件尺寸对混凝土新K_R阻力曲线的影响,水利学报,2001,12:48-55.
[127]赵志方.混凝土软化本构关系和基于粘聚力的新K_R阻力曲线的研究,大连理工大学博士后研究工作报告,大连,大连理工大学,2000.
[128]Swartz S E,Refai T.In Proceedings of SEM/RILEM International conference on fracture of concrete and rock.Houston,Texas,1987,403-417.
[129]Swartz S E,Refai T.In Proceedings of International workshop on fracture toughness and fracture energy-test methods for concrete and rock,Sendal,Japan,1988.
[130]Yeou-Shang Jenq,Shah S P.Features of mechanics of quasi-brittle crack propagation in concrete.International Journal of Fracture,1991 51(1):103-120.