压缩作用下岩体裂纹起裂扩展规律及失稳特性的研究
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摘要
受压条件下岩体裂纹的扩展问题是岩石断裂力学研究的重点课题之一。众多的试验表明:在单轴或低围压作用下岩体裂纹破坏几乎都是由于翼型裂纹(初始裂纹的Ⅰ型扩展)的扩展失稳造成的;对多裂纹体而言,翼型裂纹也是多裂纹相互贯通和岩桥破坏的重要原因,因此研究翼型裂纹的扩展规律对研究节理岩体的破坏模式和破坏机制具有重要的意义。本文采用理论、试验和数值模拟的方法对压应力作用下岩体中两种初始裂纹-张开裂纹和闭合裂纹的Ⅰ型扩展规律进行了详细的研究,同时研究了压缩作用下张开型裂纹闭合规律及其对起裂的影响,具体内容如下:
(一)张开型裂纹的闭合、起裂规律研究
1.应用复变函数和保角变换的方法,研究了张开型裂纹在压缩作用下裂纹面的变形规律,给出了准静态加载条件下变形后裂纹面构型的参数方程,并给出了未知参数的表达式,用于分析加载过程中张开型裂纹面形状的变化。
2.在研究裂纹面变形的基础上,建立了张开型裂纹面闭合的几何模型,把裂纹面的临界闭合载荷归结于一个一元二次方程的解,并分析了方程解的条件。结果表明:裂纹在压缩作用下,或者全部闭合,或者全不闭合,不存在着中间状态,且若不考虑裂纹面的起裂破坏问题,总能找到一个应力值,使得张开型裂纹发生闭合。由于该闭合条件建立在几何模型的基础上,依据是裂纹面的构型方程,不直接涉及到加载过程,而以往的准则往往只能计算等比加载条件下的张开型裂纹的临界闭合载荷大小,所以该准则使用范围更广。
3.在假设裂纹长度远大于裂纹宽度的基础上,对以上的闭合准则进行了进一步的简化,建立了简化闭合准则,并分析了简化闭合准则计算张开型裂纹临界闭合载荷的误差,结果表明简化准则可以有效地计算张开型裂纹的临界闭合载荷值。相比于简化前的准则,简化准则能直接体现临界闭合应力与裂纹几何特征和加载特征的关系,且计算更加简单。
4.采用有限元(ABAQUS)的方法对裂纹面的闭合过程进行了数值模拟,并采用数值方法计算了裂纹面的闭合载荷,数值结果表明,裂纹面在一定的载荷下,或者完全闭合或者完全不闭合,验证了以上的闭合规律,同时提出了采用数值方法求解裂纹闭合载荷的方法,并采用数值方法计算出的闭合载荷大小和理论计算结果进行了比较,两者吻合的很好,验证了以上理论推导的正确性。
5.基于以上结论,进一步研究了裂纹面的闭合变形对张开型裂纹起裂规律的影响。在起裂判断中,首先要分析裂纹在起裂前是否发生闭合,为此引入了一个新的参数—闭合系数,并应用这个系数建立了一个准则,用于判断张开型裂纹起裂时是否已经发生了闭合。对没有闭合的裂纹,采用张开型裂纹进行起裂分析,不考虑裂纹面的相互作用力,对发生闭合的裂纹,采用闭合裂纹进行起裂分析,这时要考虑裂纹面的相互作用力。同时考虑到张开型裂纹的变形和闭合对裂纹尖端应力分布的影响,对张开型裂纹的传统应力强度因子进行修正,以反映其闭合变形特征对张开型裂纹起裂的影响。
6.考虑到裂纹几何形状对裂纹起裂规律的影响,文中应用了陈篪提出的真实裂纹模型对张开型裂纹进行断裂分析。首先采用复变函数和保角变换的方法求出了裂纹面的周边应力分布,以此为依据得出了裂纹的应力强度因子,对张开型裂纹进行了起裂分析。结合试验采用陈篪裂纹模型和椭圆型裂纹模型对张开型裂纹进起裂分析,发现陈篪裂纹模型的分析结果更符合试验结果,为张开型裂纹的起裂分析提供了一个新的思路。
(二)张开型初始裂纹的翼型裂纹扩展和失稳研究
1.采用PYTHON语言对ABAQUS进行了二次开发,实现了有限元模拟裂纹扩展过程中网格的重新剖分功能,采用最大拉应力准则计算了翼型裂纹的扩展角,并编写了复合应力强度因子计算的子程序,在整个数值模拟过程中,计算了翼型裂纹路径上各点的Ⅰ/Ⅱ型应力强度因子、扩展角、复合应力强度因子等,并将翼型裂纹的这些参量写入指定的数据文件。成功地模拟了压缩作用下翼型裂纹的扩展过程。
2.采用数值和试验相结合的方法研究了翼型裂纹的渐近扩展过程,发现翼型裂纹的扩展路径有明显的渐近性质,其渐近线为过初始裂纹中心点、平行于最大压主应力的一条直线。基于翼型裂纹路径的这个特点,采用双曲线参数方程近似表示翼型裂纹路径,该双曲线方程的未知参数为初始裂纹的起裂角、初始裂纹加载角、初始裂纹长度,参数物理意义明确,便于应用,并采用了数值和试验的方法对文中提出的双曲线路径进行了验证。
3.根据翼型裂纹的渐近特点,建立了“张开型—曲线翼型裂纹模型”,用于分析受压条件下张开型初始裂纹的翼型裂纹扩展失稳规律。由于该模型的翼型裂纹扩展路径预知,不需要对有限元网格进行重新划分,可以采用有限元直接计算路径上的应力强度因子,即采用少数的几个点能得出裂纹的扩展载荷与翼型裂纹长度的对应关系,简化了有限元计算。最后采用该模型对翼型裂纹的扩展和失稳进行了分析,将模型的计算结果和数值及试验结果对比,发现三者吻合的很好,这表明了双曲线翼型裂纹模型的有效性。
4.裂纹体失稳载荷边界效应的分析,总体而言边界尺寸对翼型裂纹的扩展路径影响较小,而对翼型裂纹的扩展载荷(或应力强度因子)影响较大。对单轴作用下有限板而言,由于边界效应的影响,其应力强度因子趋近于一个正值,而无限大板的应力强度因子则逐渐趋向于0。所以对翼型裂纹扩展分析时,要注意边界效应的影响。
(三)闭合型初始裂纹的翼型裂纹的扩展、失稳分析
1.采用ABAQUS二次开发对翼型裂纹的扩展过程进行了数值模拟,同样发现翼型裂纹的扩展路径具有的渐近性质:其渐近线为平行于最大压主应力的某条直线。与张开型初始裂纹的翼型裂纹路径的渐近线不同,闭合型初始裂纹的翼型裂纹路径的渐近线不一定过初始裂纹的中心点,其渐近线的位置和裂纹面的摩擦系数相关,当摩擦系数为0时,渐近线过初始裂纹的中心点,当摩擦系数不为0时,其渐近线不过初始裂纹的中心线。并通过理论推导得出了闭合型初始裂纹的翼型裂纹路径的渐近线方程。
2.基于翼型裂纹路径的这个特点,仍采用双曲线参数方程近似表示了翼型裂纹路径,其未知参数为初始裂纹长度、初始裂纹角、初始裂纹面摩擦系数等,便于求解,并采用数值模拟的方法验证了所得双曲线方程表示翼型裂纹路径的可行性,在此基础上提出了“闭合型一曲线翼型裂纹模型”,用于分析闭合型初始裂纹的扩展和失稳特性,并采用试验和数值模拟的方法对该模型进行了验证。
3.采用双曲线翼型裂纹模型和传统的直线型翼型裂纹模型分别对试验中的翼型裂纹扩展进行了分析,发现两者计算结果有较大的差别,并和试验结果及数值模拟的结果进行了对比分析,发现文中的曲线翼型裂纹模型和试验吻合的更好,而传统的直线型翼型裂纹模型和试验相差较大,这表明了双曲线裂纹模型的有效性。
4.简要地分析了文中所提出的“张开型—翼型裂纹模型”和“闭合型—翼型裂纹模型”的差别,结果显示张开型初始裂纹的翼型裂纹扩展路径和相同条件下闭合型初始裂纹的翼型裂纹扩展路径及扩展过程中翼型裂纹应力强度因子差别较大,所以在研究裂纹的起裂扩展时,要分析初始裂纹起裂扩张过程中存在的状态(即判断是张开型初始裂纹还是闭合型初始裂纹),以确定采用哪一种翼型裂纹模型分析,同时也说明了对张开型裂纹闭合准则研究的必要性。
It is an important subject of rock fracture mechanics to study the law of the flaw initiation, propagation and failure in rock masses under compression. Numerous studies indicate that growth of wing cracks(I mode crack) is main reason for rock masses with the single crack or multiple cracks under uniaxial compression or under biaxial compression with low confining pressure. So laws of the wing crack initiation, propagation and failure have important significance for the study on the failure mode and failure mechanism of the rock masses with crack under compression. In this paper, the theoretical method, numerical methods and testing methods are combined to study the initiation, propagation and failure laws of the wing cracks for both open cracks and closing cracks under compression, besides the closing law for open crack and the effect of crack closing in the open crack initiation are also included. And the detail researches are explained as followings:
(I)Study on the closing law and initiation characteristics of open crack under compression.
1. The law of the deformation of the open crack surface under compression is studied. In the paper the methods of complex function and conformal transformation are combined to construct the parameter function, and the parameters of the function are solved. Through the parameter function, the geometry shapes of the deforming crack surface under compression can be described easily.
2. Based on the study about deformation of the open crack surface under compression, the geometric model for analysis of the open crack closure is established. Through the geometric model, two conclusions are drawn as follows: First the closing law of the open crack is obtained which agrees with the general viewpoints of rock mechanics - an open crack exists either completely closed or completely open under compressive loading in rock masses. Next the criterion for a crack closure is defined which is expressed by the deformation parameters. By use of this criterion the critical closing stress of an open crack can be determined by a simple quadratic equation easily. Besides this criterion doesn't deal with the assumption of the load pattern, so it can be applied under any compressive loads, which extends the applied range of the traditional closing criterions of the open crack. At the same time, it is very convenient to be applied because of its simple mathematical form and definite physical meanings of its unknown parameters.
3. Based on the simplified hypotheses that the thickness of open cracks is far smaller than their length, a simplified formula is established to determine closing stress of an open crack. Comparisons between the results calculated by the simplified formula and original theory are made. It shows that only small error happens by using the simplified formula, consequently, the simplified formula is feasible.
4. Referring to the experiments data, the finite element method (ABAQUS) is used to simulate the closing process of the open crack under compression. It reveals that the numerical results support the theoretical viewpoints: an open crack exists only in the state of completely closed or completely open under compressive loading. Through the analysis on the displacement curves of the nodes of the finite element model, a new numerical method is presented to determine the closing stress of an open crack. Finally, the closing stresses solved separately by the numerical method, the previous theoretical formula and the simplified formula are compared, showing that the results obtained by three methods are identical, which approve the validity of the proposed conclusion in the paper.
5. Base on the closing law of open crack under compression, the further study on the effect of the open crack closure in the initiation characters of the open crack. Before the analysis of the initiation of the open crack under compression, it is pre-requisite to ensure if the open crack is closed before its initiation. Therefore, a closure criterion of an open crack is established to determine existence state of crack before initiation. If the crack is closed before initiation, the closed crack model must be chosen and the interaction force along the crack surface should be considered for the fracture analysis. If the crack keeps open before initiation, the open model should be used, and deformation of the open crack should be considered. Based on the above study on the deformation and closing law of open crack, the traditional SIFs are modified in order to reflect the impact of the deformation or closure of the crack on the initiation characters of the open crack.
6. Considering the effect of the open crack's geometric shape in the crack initiation, Chenchi crack model is used in the study of the crack fracture in the rock masses under compression. Then the methods of complex function and conformal transformation are combined to construct the stress distribution around crack surface. Based on the above conclusion the stress intensity factor of crack is calculated, and the fracture criterion of the crack model is built to analyze the initiation compression stress and angle of crack which agree with the experiments. The Further comparison between the results of the ellipse model and the Chenchi model is done, and the results show the Chenchi model can reflect the initiation property of the crack open more accurately.
(II)Study on growth and failure characters of the wing cracks from open crack
1. The second development of ABAQUS is implemented to automaticly remesh the finite element meshes. Besides the maximum circumferential stress fracture criterion is used for the second development of ABAQUS to determine the cracking directions of wing cracks. Then the wing crack growth is simulated by the finite element method, by which the propagation paths and the mixed-mode stress-intensity factors of wing cracks were also obtained.
2. The asymptotic failure characteristics of the wing cracks are studied by the combination of the testing method and the numerical simulation method, it is found that under compression, the wing cracks of the open main flaws start from the tip of the main crack and grow along the curve path. With the length increasing, the wing crack gradually approximate to the line which passes the middle point of the main crack and is parallel to the direction of maximum main stress. It is the important geometric feature of the wing crack paths. According to the geometric characteristics of the wing crack paths, a hyperbolic equation is set up to describe the curve paths of wing cracks approximately. In the equation, the unknown parameters are determined by the crack initial angle, the crack length and the angle between the direction of the maximum main stress and the crack surface. This equation of wing cracks is simple and its physical meaning is clear which is convenient for application.
3. According to Geometric Characteristics of the wing crack paths, the cure wing crack model- hyperbolic-wing crack model is built to analyze the extension and failure laws of wing crack from the open crack under compression. In the paper, the comparative analysis on the paths by the hyperbolic equation, numerical simulation and the experiments is made, and the results show the paths by the hyperbolic equation are in concordance with those by experiments, which prove that the hyperbolic equation in the paper can be used to describe the propagation paths of wing cracks under compression. Furthermore, the stress intensity factors along the hyperbolic path are calculated by ABAQUS, and the extending loads of the wing cracks are analyzed. Through comparing with the experiment results and numerical simulation results, it is found that the extending loads by the hyperbolic wing cracks in the paper fit better with by the experiments and numerical simulation, which shows validity of the results in the paper.
4. The effect of the boundary size on the extending characteristics of wing cracks under uniaxial compression. In general, the boundary size of the infinite plate has little effect on the wing crack paths, but has great impact on the extension loads (SIFs) along the wing crack path. For a crack in a finite plate, its SIFs along the wing crack path tend to a fixed positive value which depends on the size of the infinite plate. But for a crack in an infinite plate, its SIFs along the wing crack path will tend to zero value. That's to say, for a crack in an infinite plate, its wing crack will extend stably for ever with the load increase. However for a crack in a finite plate, its wing crack will extend unstably and the crack will burst when the load arrives at a certain value which can be observed in testing. So for the wing cracking in an infinite plate, it is notable that the boundary effect should be considered for the mixed-mode stress-intensity factors of the wing cracks.
(III)Study on growth and failure characters of the wing cracks from closing crack
1. The numerical simulation is implemented to study laws of the wing crack extension from the closing crack by the second development of ABAQUS. The asymptotic behaviors of the wing crack paths of the closing cracks are also found: the wing cracks of the open main flaws start from the tip of the main crack and grow along the curve path. With the length increasing, the wing crack gradually approximate to the line which passes a certain point of the main crack and is parallel to the direction of maximum main stress. The asymptotic lines of cure wing cracks of the closing crack are determined by theoretical analysis method. Compared with the wing crack of the open crack, the asymptotic lines of the wing crack of the closing crack don't always pass through the center point of the main crack.
2. According to the geometric characteristics of the wing crack paths, a hyperbolic equation is set up to describe the curve paths of wing cracks approximately. In the equation, the unknown parameters are determined by the initial crack angle, the initial crack length, the friction coefficient of the initial crack surfaces, and the angle between the direction of the maximum main stress and the crack surface. This equation of wing cracks is simple and its physical meaning is clear which is convenient for application. According to Geometric Characteristics of the wing crack paths, the cure wing crack model- hyperbolic-wing crack model is built to analyze the extension and failure laws of wing crack from the closing crack under compression. In the paper, the stress intensity factors along the hyperbolic path are calculated by ABAQUS, and the extending loads of the wing cracks are analyzed. Through comparing with the experiment results and numerical simulation results, it is found that the extending loads by the hyperbolic wing cracks in the paper fit better with by the experiments and numerical simulation, which shows validity of the results in the paper.
3. In order to show validity of the cure wing crack model in the paper, a comparative Analysis of the cure wing crack model and the traditional linear wing crack model is made. First the analysis of the extension loads of the wing crack in the testing is done respectively by the above two models. Then analysis results respectively by the above two models are compared with the testing result and the numerical computation results, and the comparison results show: the extension loads determined by the curve wing crack model coincide well with the results from the testing and numerical computation, but the extension loads determined by the traditional linear wing crack model have larger difference from the ones by the testing and numerical computation. The above comparative analysis results support the validity of the curve wing crack model proposed by the paper.
4. In order to show the difference between the 'initial open flaw-curve wing crack model' and the 'initial closing flaw-curve wing crack model' proposed in the paper, a comparative Analysis of the two models is made. According to the initial open flaw and the initial closing flaw, though they have the same flaw angle, the same flaw length and the same friction coefficients, the paths of their wing cracks make a great difference, so do the extending loads for their wing cracks. Therefore, before the study of the extending law of the initial flaw under compression, it is needed to distinguish the initial closing flaw and the initial open flaw by use of the closing law proposed in the paper, which shows the necessity of the closing law for the open crack.
(一)张开型裂纹的闭合、起裂规律研究
1.应用复变函数和保角变换的方法,研究了张开型裂纹在压缩作用下裂纹面的变形规律,给出了准静态加载条件下变形后裂纹面构型的参数方程,并给出了未知参数的表达式,用于分析加载过程中张开型裂纹面形状的变化。
2.在研究裂纹面变形的基础上,建立了张开型裂纹面闭合的几何模型,把裂纹面的临界闭合载荷归结于一个一元二次方程的解,并分析了方程解的条件。结果表明:裂纹在压缩作用下,或者全部闭合,或者全不闭合,不存在着中间状态,且若不考虑裂纹面的起裂破坏问题,总能找到一个应力值,使得张开型裂纹发生闭合。由于该闭合条件建立在几何模型的基础上,依据是裂纹面的构型方程,不直接涉及到加载过程,而以往的准则往往只能计算等比加载条件下的张开型裂纹的临界闭合载荷大小,所以该准则使用范围更广。
3.在假设裂纹长度远大于裂纹宽度的基础上,对以上的闭合准则进行了进一步的简化,建立了简化闭合准则,并分析了简化闭合准则计算张开型裂纹临界闭合载荷的误差,结果表明简化准则可以有效地计算张开型裂纹的临界闭合载荷值。相比于简化前的准则,简化准则能直接体现临界闭合应力与裂纹几何特征和加载特征的关系,且计算更加简单。
4.采用有限元(ABAQUS)的方法对裂纹面的闭合过程进行了数值模拟,并采用数值方法计算了裂纹面的闭合载荷,数值结果表明,裂纹面在一定的载荷下,或者完全闭合或者完全不闭合,验证了以上的闭合规律,同时提出了采用数值方法求解裂纹闭合载荷的方法,并采用数值方法计算出的闭合载荷大小和理论计算结果进行了比较,两者吻合的很好,验证了以上理论推导的正确性。
5.基于以上结论,进一步研究了裂纹面的闭合变形对张开型裂纹起裂规律的影响。在起裂判断中,首先要分析裂纹在起裂前是否发生闭合,为此引入了一个新的参数—闭合系数,并应用这个系数建立了一个准则,用于判断张开型裂纹起裂时是否已经发生了闭合。对没有闭合的裂纹,采用张开型裂纹进行起裂分析,不考虑裂纹面的相互作用力,对发生闭合的裂纹,采用闭合裂纹进行起裂分析,这时要考虑裂纹面的相互作用力。同时考虑到张开型裂纹的变形和闭合对裂纹尖端应力分布的影响,对张开型裂纹的传统应力强度因子进行修正,以反映其闭合变形特征对张开型裂纹起裂的影响。
6.考虑到裂纹几何形状对裂纹起裂规律的影响,文中应用了陈篪提出的真实裂纹模型对张开型裂纹进行断裂分析。首先采用复变函数和保角变换的方法求出了裂纹面的周边应力分布,以此为依据得出了裂纹的应力强度因子,对张开型裂纹进行了起裂分析。结合试验采用陈篪裂纹模型和椭圆型裂纹模型对张开型裂纹进起裂分析,发现陈篪裂纹模型的分析结果更符合试验结果,为张开型裂纹的起裂分析提供了一个新的思路。
(二)张开型初始裂纹的翼型裂纹扩展和失稳研究
1.采用PYTHON语言对ABAQUS进行了二次开发,实现了有限元模拟裂纹扩展过程中网格的重新剖分功能,采用最大拉应力准则计算了翼型裂纹的扩展角,并编写了复合应力强度因子计算的子程序,在整个数值模拟过程中,计算了翼型裂纹路径上各点的Ⅰ/Ⅱ型应力强度因子、扩展角、复合应力强度因子等,并将翼型裂纹的这些参量写入指定的数据文件。成功地模拟了压缩作用下翼型裂纹的扩展过程。
2.采用数值和试验相结合的方法研究了翼型裂纹的渐近扩展过程,发现翼型裂纹的扩展路径有明显的渐近性质,其渐近线为过初始裂纹中心点、平行于最大压主应力的一条直线。基于翼型裂纹路径的这个特点,采用双曲线参数方程近似表示翼型裂纹路径,该双曲线方程的未知参数为初始裂纹的起裂角、初始裂纹加载角、初始裂纹长度,参数物理意义明确,便于应用,并采用了数值和试验的方法对文中提出的双曲线路径进行了验证。
3.根据翼型裂纹的渐近特点,建立了“张开型—曲线翼型裂纹模型”,用于分析受压条件下张开型初始裂纹的翼型裂纹扩展失稳规律。由于该模型的翼型裂纹扩展路径预知,不需要对有限元网格进行重新划分,可以采用有限元直接计算路径上的应力强度因子,即采用少数的几个点能得出裂纹的扩展载荷与翼型裂纹长度的对应关系,简化了有限元计算。最后采用该模型对翼型裂纹的扩展和失稳进行了分析,将模型的计算结果和数值及试验结果对比,发现三者吻合的很好,这表明了双曲线翼型裂纹模型的有效性。
4.裂纹体失稳载荷边界效应的分析,总体而言边界尺寸对翼型裂纹的扩展路径影响较小,而对翼型裂纹的扩展载荷(或应力强度因子)影响较大。对单轴作用下有限板而言,由于边界效应的影响,其应力强度因子趋近于一个正值,而无限大板的应力强度因子则逐渐趋向于0。所以对翼型裂纹扩展分析时,要注意边界效应的影响。
(三)闭合型初始裂纹的翼型裂纹的扩展、失稳分析
1.采用ABAQUS二次开发对翼型裂纹的扩展过程进行了数值模拟,同样发现翼型裂纹的扩展路径具有的渐近性质:其渐近线为平行于最大压主应力的某条直线。与张开型初始裂纹的翼型裂纹路径的渐近线不同,闭合型初始裂纹的翼型裂纹路径的渐近线不一定过初始裂纹的中心点,其渐近线的位置和裂纹面的摩擦系数相关,当摩擦系数为0时,渐近线过初始裂纹的中心点,当摩擦系数不为0时,其渐近线不过初始裂纹的中心线。并通过理论推导得出了闭合型初始裂纹的翼型裂纹路径的渐近线方程。
2.基于翼型裂纹路径的这个特点,仍采用双曲线参数方程近似表示了翼型裂纹路径,其未知参数为初始裂纹长度、初始裂纹角、初始裂纹面摩擦系数等,便于求解,并采用数值模拟的方法验证了所得双曲线方程表示翼型裂纹路径的可行性,在此基础上提出了“闭合型一曲线翼型裂纹模型”,用于分析闭合型初始裂纹的扩展和失稳特性,并采用试验和数值模拟的方法对该模型进行了验证。
3.采用双曲线翼型裂纹模型和传统的直线型翼型裂纹模型分别对试验中的翼型裂纹扩展进行了分析,发现两者计算结果有较大的差别,并和试验结果及数值模拟的结果进行了对比分析,发现文中的曲线翼型裂纹模型和试验吻合的更好,而传统的直线型翼型裂纹模型和试验相差较大,这表明了双曲线裂纹模型的有效性。
4.简要地分析了文中所提出的“张开型—翼型裂纹模型”和“闭合型—翼型裂纹模型”的差别,结果显示张开型初始裂纹的翼型裂纹扩展路径和相同条件下闭合型初始裂纹的翼型裂纹扩展路径及扩展过程中翼型裂纹应力强度因子差别较大,所以在研究裂纹的起裂扩展时,要分析初始裂纹起裂扩张过程中存在的状态(即判断是张开型初始裂纹还是闭合型初始裂纹),以确定采用哪一种翼型裂纹模型分析,同时也说明了对张开型裂纹闭合准则研究的必要性。
It is an important subject of rock fracture mechanics to study the law of the flaw initiation, propagation and failure in rock masses under compression. Numerous studies indicate that growth of wing cracks(I mode crack) is main reason for rock masses with the single crack or multiple cracks under uniaxial compression or under biaxial compression with low confining pressure. So laws of the wing crack initiation, propagation and failure have important significance for the study on the failure mode and failure mechanism of the rock masses with crack under compression. In this paper, the theoretical method, numerical methods and testing methods are combined to study the initiation, propagation and failure laws of the wing cracks for both open cracks and closing cracks under compression, besides the closing law for open crack and the effect of crack closing in the open crack initiation are also included. And the detail researches are explained as followings:
(I)Study on the closing law and initiation characteristics of open crack under compression.
1. The law of the deformation of the open crack surface under compression is studied. In the paper the methods of complex function and conformal transformation are combined to construct the parameter function, and the parameters of the function are solved. Through the parameter function, the geometry shapes of the deforming crack surface under compression can be described easily.
2. Based on the study about deformation of the open crack surface under compression, the geometric model for analysis of the open crack closure is established. Through the geometric model, two conclusions are drawn as follows: First the closing law of the open crack is obtained which agrees with the general viewpoints of rock mechanics - an open crack exists either completely closed or completely open under compressive loading in rock masses. Next the criterion for a crack closure is defined which is expressed by the deformation parameters. By use of this criterion the critical closing stress of an open crack can be determined by a simple quadratic equation easily. Besides this criterion doesn't deal with the assumption of the load pattern, so it can be applied under any compressive loads, which extends the applied range of the traditional closing criterions of the open crack. At the same time, it is very convenient to be applied because of its simple mathematical form and definite physical meanings of its unknown parameters.
3. Based on the simplified hypotheses that the thickness of open cracks is far smaller than their length, a simplified formula is established to determine closing stress of an open crack. Comparisons between the results calculated by the simplified formula and original theory are made. It shows that only small error happens by using the simplified formula, consequently, the simplified formula is feasible.
4. Referring to the experiments data, the finite element method (ABAQUS) is used to simulate the closing process of the open crack under compression. It reveals that the numerical results support the theoretical viewpoints: an open crack exists only in the state of completely closed or completely open under compressive loading. Through the analysis on the displacement curves of the nodes of the finite element model, a new numerical method is presented to determine the closing stress of an open crack. Finally, the closing stresses solved separately by the numerical method, the previous theoretical formula and the simplified formula are compared, showing that the results obtained by three methods are identical, which approve the validity of the proposed conclusion in the paper.
5. Base on the closing law of open crack under compression, the further study on the effect of the open crack closure in the initiation characters of the open crack. Before the analysis of the initiation of the open crack under compression, it is pre-requisite to ensure if the open crack is closed before its initiation. Therefore, a closure criterion of an open crack is established to determine existence state of crack before initiation. If the crack is closed before initiation, the closed crack model must be chosen and the interaction force along the crack surface should be considered for the fracture analysis. If the crack keeps open before initiation, the open model should be used, and deformation of the open crack should be considered. Based on the above study on the deformation and closing law of open crack, the traditional SIFs are modified in order to reflect the impact of the deformation or closure of the crack on the initiation characters of the open crack.
6. Considering the effect of the open crack's geometric shape in the crack initiation, Chenchi crack model is used in the study of the crack fracture in the rock masses under compression. Then the methods of complex function and conformal transformation are combined to construct the stress distribution around crack surface. Based on the above conclusion the stress intensity factor of crack is calculated, and the fracture criterion of the crack model is built to analyze the initiation compression stress and angle of crack which agree with the experiments. The Further comparison between the results of the ellipse model and the Chenchi model is done, and the results show the Chenchi model can reflect the initiation property of the crack open more accurately.
(II)Study on growth and failure characters of the wing cracks from open crack
1. The second development of ABAQUS is implemented to automaticly remesh the finite element meshes. Besides the maximum circumferential stress fracture criterion is used for the second development of ABAQUS to determine the cracking directions of wing cracks. Then the wing crack growth is simulated by the finite element method, by which the propagation paths and the mixed-mode stress-intensity factors of wing cracks were also obtained.
2. The asymptotic failure characteristics of the wing cracks are studied by the combination of the testing method and the numerical simulation method, it is found that under compression, the wing cracks of the open main flaws start from the tip of the main crack and grow along the curve path. With the length increasing, the wing crack gradually approximate to the line which passes the middle point of the main crack and is parallel to the direction of maximum main stress. It is the important geometric feature of the wing crack paths. According to the geometric characteristics of the wing crack paths, a hyperbolic equation is set up to describe the curve paths of wing cracks approximately. In the equation, the unknown parameters are determined by the crack initial angle, the crack length and the angle between the direction of the maximum main stress and the crack surface. This equation of wing cracks is simple and its physical meaning is clear which is convenient for application.
3. According to Geometric Characteristics of the wing crack paths, the cure wing crack model- hyperbolic-wing crack model is built to analyze the extension and failure laws of wing crack from the open crack under compression. In the paper, the comparative analysis on the paths by the hyperbolic equation, numerical simulation and the experiments is made, and the results show the paths by the hyperbolic equation are in concordance with those by experiments, which prove that the hyperbolic equation in the paper can be used to describe the propagation paths of wing cracks under compression. Furthermore, the stress intensity factors along the hyperbolic path are calculated by ABAQUS, and the extending loads of the wing cracks are analyzed. Through comparing with the experiment results and numerical simulation results, it is found that the extending loads by the hyperbolic wing cracks in the paper fit better with by the experiments and numerical simulation, which shows validity of the results in the paper.
4. The effect of the boundary size on the extending characteristics of wing cracks under uniaxial compression. In general, the boundary size of the infinite plate has little effect on the wing crack paths, but has great impact on the extension loads (SIFs) along the wing crack path. For a crack in a finite plate, its SIFs along the wing crack path tend to a fixed positive value which depends on the size of the infinite plate. But for a crack in an infinite plate, its SIFs along the wing crack path will tend to zero value. That's to say, for a crack in an infinite plate, its wing crack will extend stably for ever with the load increase. However for a crack in a finite plate, its wing crack will extend unstably and the crack will burst when the load arrives at a certain value which can be observed in testing. So for the wing cracking in an infinite plate, it is notable that the boundary effect should be considered for the mixed-mode stress-intensity factors of the wing cracks.
(III)Study on growth and failure characters of the wing cracks from closing crack
1. The numerical simulation is implemented to study laws of the wing crack extension from the closing crack by the second development of ABAQUS. The asymptotic behaviors of the wing crack paths of the closing cracks are also found: the wing cracks of the open main flaws start from the tip of the main crack and grow along the curve path. With the length increasing, the wing crack gradually approximate to the line which passes a certain point of the main crack and is parallel to the direction of maximum main stress. The asymptotic lines of cure wing cracks of the closing crack are determined by theoretical analysis method. Compared with the wing crack of the open crack, the asymptotic lines of the wing crack of the closing crack don't always pass through the center point of the main crack.
2. According to the geometric characteristics of the wing crack paths, a hyperbolic equation is set up to describe the curve paths of wing cracks approximately. In the equation, the unknown parameters are determined by the initial crack angle, the initial crack length, the friction coefficient of the initial crack surfaces, and the angle between the direction of the maximum main stress and the crack surface. This equation of wing cracks is simple and its physical meaning is clear which is convenient for application. According to Geometric Characteristics of the wing crack paths, the cure wing crack model- hyperbolic-wing crack model is built to analyze the extension and failure laws of wing crack from the closing crack under compression. In the paper, the stress intensity factors along the hyperbolic path are calculated by ABAQUS, and the extending loads of the wing cracks are analyzed. Through comparing with the experiment results and numerical simulation results, it is found that the extending loads by the hyperbolic wing cracks in the paper fit better with by the experiments and numerical simulation, which shows validity of the results in the paper.
3. In order to show validity of the cure wing crack model in the paper, a comparative Analysis of the cure wing crack model and the traditional linear wing crack model is made. First the analysis of the extension loads of the wing crack in the testing is done respectively by the above two models. Then analysis results respectively by the above two models are compared with the testing result and the numerical computation results, and the comparison results show: the extension loads determined by the curve wing crack model coincide well with the results from the testing and numerical computation, but the extension loads determined by the traditional linear wing crack model have larger difference from the ones by the testing and numerical computation. The above comparative analysis results support the validity of the curve wing crack model proposed by the paper.
4. In order to show the difference between the 'initial open flaw-curve wing crack model' and the 'initial closing flaw-curve wing crack model' proposed in the paper, a comparative Analysis of the two models is made. According to the initial open flaw and the initial closing flaw, though they have the same flaw angle, the same flaw length and the same friction coefficients, the paths of their wing cracks make a great difference, so do the extending loads for their wing cracks. Therefore, before the study of the extending law of the initial flaw under compression, it is needed to distinguish the initial closing flaw and the initial open flaw by use of the closing law proposed in the paper, which shows the necessity of the closing law for the open crack.
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