无网格GALERKIN方法及裂纹扩展数值模拟研究
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摘要
本文以裂隙岩体为研究背景,利用无网格Galerkin方法和偶应力理论下的无网格Galerkin方法,根据本文提出的最大周向拉应力裂纹扩展准则,对二维裂纹扩展和三维裂纹扩展问题进行了数值模拟研究。
首先推导了二维无网格Galerkin方法、偶应力理论下的无网格Galerkin方法及三维无网格Galerkin方法的基本公式。然后根据最小势能原理,导出了弹性体的离散方程。以经典的二维和三维弹性力学问题为算例研究了无网格方法的程序实施和几个重要参数的合理配置。采用罚参数法来释放位移约束,引入边界条件;基于节点控制的思想,确定动态影响半径,那么在裂纹前沿节点加密区域适当减少了最小二乘的计算量,可以适当提高无网格方法的计算效率。
利用空间解析几何理论,建立了裂纹前沿局部坐标系,计算了局部坐标和全局坐标的变换矩阵,导出了局部坐标中周向正应力与全局坐标中应力分量的变换公式,为裂纹曲面(线)的描述和裂纹扩展数值模拟做好了充分的数学准备工作。
在研究裂纹扩展时,采用可视准则,体现裂纹对节点之间、节点与高斯积分点之间的隔离作用。
选择合适的裂纹扩展准则,确定裂纹扩展条件、扩展方向和扩展步长是研究裂纹扩展的关键。考虑到目前常用的二维裂纹扩展准则存在缺陷和没有三维裂纹扩展准则,在其它学者的思想基础上,本文提出了一个新的裂纹扩展准则——最大周向拉应力裂纹扩展准则。该准则不仅给出了裂纹扩展方向、扩展条件,还给出了确定扩展步长的依据,摒弃了计算应力强度因子带来的误差和麻烦,尤其是三维裂纹问题,克服了靠经验给定扩展步长的弊端。
结合本文提出的最大周向拉应力裂纹扩展准则,考虑到三维裂纹的扩展曲面可能出现弯折、扭曲现象,把裂纹扩展曲面分成若干个片断面,并将每个裂纹片断面划分成三角形表示,实现了裂纹曲面的数学描述。
在裂纹扩展的数值模拟中,为了识别裂纹形态和提高模拟精度,在裂纹面两侧增加了附加节点,并对处在裂纹扩展段或扩展片断面上的节点实施了劈裂技术,分别置于两侧的裂纹表面上。
研究无水压平面裂纹扩展数值模拟,并与其它数值计算方法的结果和相关的试验结果进行了对比,得到了一致的结论,确保了最大周向拉应力裂纹扩展准则的有效性。在此基础上,分别研究了裂隙水压和偶应力对裂纹扩展的影响,得到了一些新结论。
由于三维裂纹扩展问题的复杂性,目前对三维裂纹扩展数值模拟的研究成果较少。用传统的计算方法仅对币形裂纹的Ⅰ型扩展进行了数值模拟。本文以三维无网格Galerkin方法为基础,采用最大周向拉应力裂纹扩展准则,首先对椭圆形裂纹的Ⅰ型扩展进行了数值模拟,数值模拟结果与试验结果一致,再一次证实了三维无网格Galerkin方法程序实施技术的正确性、最大周向拉应力裂纹扩展准则的正确性和三维裂纹扩展曲面数学描述的有效性。在此基础上,对三维表面斜置半圆形裂纹的扩展进行了数值模拟,了解了复杂三维裂纹不同扩展阶段的扩展方式、扩展曲面的形态,以及空间方位对三维裂纹扩展模式的影响,填补了国内在三维复杂裂纹扩展数值模拟方面的空白。
通过三维裂纹的数值模拟可以清楚地了解三维裂纹扩展的全貌,弥补了三维裂隙断裂试验与理论研究的不足,为进一步研究三维裂隙断裂的本质提供一定的参考价值。
The study background of this paper is the crack rock mass. A new crack propagation criterions, the maximum circumferential tensile stress criterions, are presented in this paper. The numerical simulation of two-dimensional(2-D) and three-dimensional(3-D) crack propagation are studied baseed on element free Galerkin (EFG) method and 2-D EFG method based on couple stresses theory.
Basic formulae of 2-D EFG method and 2-D EFG meothod based on couple stresses theory and 3-D EFG method are derived. Afterwards, discrete equations for elastic body are derived by use of the minimus potential energy principle. By use of classical 2-D and 3-D problems, the implement of 2-D and 3-D EFG method are discussed. And the matching of several important parameters in 2-D and 3-D EFG method are discussed, too. The displacement restriction is released by use of a penalty parameter. So the essential boundary conditions are introduced into EFG methods. Based on the idea of controlling nodal number, alterable influence radii can be defined. The least square calculating estimate is cut down in dense nodal area and calculating efficiency of EFG methods is enhanced properly.
Based on space analytic geometry, the local coordinates at crack front is built up. The coordinates transform matrix, base vector transform, vector transform and displacement transform are established from the global coordinates to the local coordinates at crack front. The circumferential stress formulae are derived in the main normal plane of the local coordinates at crack front. So there have sufficient preparative work for the numerical simulation of crack propagation and surface or line of crack propagation.
In studying crack problem, the visibility criterion is adopted to figure isolation of crack between nodals and integral nodals.
Chosen a proper crack propagation criterion to determine condition, directions and steps of crack propagation is the key.
Shortcoming of 2-D crack propagation criterions and no criterions of 3-D crack propagation are considered. Based on other scholars work, a new criterions of crack propagation, namely the maximum circumferential tensile stress criterion for 2-D and 3-D crack are brought forward in this paper. The new crack propagation criterions give out not only crack propagation condition and directions, but also steps too. It avoids the error and trouble of calculating stress intensity factor, especially for 3-D crack propagation problems, and overcomes the disadvantage giving out crack propagation step depending on experience of calculator.
Surface of 3-D crack propagation maybe appear bending and twisting in crack propagation. It is divided into some snippets. Every snippet is triangulated. Mathematic description of crack surface is realized.
In numerical simulation of crack propagation, for identifying crack shape and improving simulation precision of crack propagation surface, append nodal points are added on surface at crack front. Some nodal point maybe lie in a crack snippet. It is splited to two points to be laid on the two side of crack surface.
The numerical simulation of 2-D crack propagation without cranny hydraulic pressure is studied by EFG methods. The results are identical with that of other computing method and test. The validity of the maximum circumferential tensile stress criterion is confirmed. The effect on crack propagation of cranny hydraulic pressure and couple stresses is studied. Some new results are found.
Because of the complexity of 3-D crack propagation, there are a few results about the numerical simulation of 3-D crack propagation at present. Only model- I propagation of penny crack is simulated by use of traditional numerical method. In this paper, model- I propagation of ellipse crack is simulated by EFG method. The results are identical with that of other computing method and test. Once more, the validity of the maximum circumferential tensile stress criterion and mathematic description of 3-D crack surface are proved. Based on the above, the crack propagation of 3-D surface inclined semi-circular cracks are simulated numerically. The manner in different stage and the surface shape of 3-D crack propagation are known. Different attitude of original crack produces great effect on 3-D crack propagation. The blankness of the numerical simulation of 3-D complicated crack propagation is filled up in homeland.
The full propagation process of 3-D crack is simulated. The simulation results make up some faults of the experimental results and theory research results. The simulation results have some reference value to further study 3-D crack fracture.
首先推导了二维无网格Galerkin方法、偶应力理论下的无网格Galerkin方法及三维无网格Galerkin方法的基本公式。然后根据最小势能原理,导出了弹性体的离散方程。以经典的二维和三维弹性力学问题为算例研究了无网格方法的程序实施和几个重要参数的合理配置。采用罚参数法来释放位移约束,引入边界条件;基于节点控制的思想,确定动态影响半径,那么在裂纹前沿节点加密区域适当减少了最小二乘的计算量,可以适当提高无网格方法的计算效率。
利用空间解析几何理论,建立了裂纹前沿局部坐标系,计算了局部坐标和全局坐标的变换矩阵,导出了局部坐标中周向正应力与全局坐标中应力分量的变换公式,为裂纹曲面(线)的描述和裂纹扩展数值模拟做好了充分的数学准备工作。
在研究裂纹扩展时,采用可视准则,体现裂纹对节点之间、节点与高斯积分点之间的隔离作用。
选择合适的裂纹扩展准则,确定裂纹扩展条件、扩展方向和扩展步长是研究裂纹扩展的关键。考虑到目前常用的二维裂纹扩展准则存在缺陷和没有三维裂纹扩展准则,在其它学者的思想基础上,本文提出了一个新的裂纹扩展准则——最大周向拉应力裂纹扩展准则。该准则不仅给出了裂纹扩展方向、扩展条件,还给出了确定扩展步长的依据,摒弃了计算应力强度因子带来的误差和麻烦,尤其是三维裂纹问题,克服了靠经验给定扩展步长的弊端。
结合本文提出的最大周向拉应力裂纹扩展准则,考虑到三维裂纹的扩展曲面可能出现弯折、扭曲现象,把裂纹扩展曲面分成若干个片断面,并将每个裂纹片断面划分成三角形表示,实现了裂纹曲面的数学描述。
在裂纹扩展的数值模拟中,为了识别裂纹形态和提高模拟精度,在裂纹面两侧增加了附加节点,并对处在裂纹扩展段或扩展片断面上的节点实施了劈裂技术,分别置于两侧的裂纹表面上。
研究无水压平面裂纹扩展数值模拟,并与其它数值计算方法的结果和相关的试验结果进行了对比,得到了一致的结论,确保了最大周向拉应力裂纹扩展准则的有效性。在此基础上,分别研究了裂隙水压和偶应力对裂纹扩展的影响,得到了一些新结论。
由于三维裂纹扩展问题的复杂性,目前对三维裂纹扩展数值模拟的研究成果较少。用传统的计算方法仅对币形裂纹的Ⅰ型扩展进行了数值模拟。本文以三维无网格Galerkin方法为基础,采用最大周向拉应力裂纹扩展准则,首先对椭圆形裂纹的Ⅰ型扩展进行了数值模拟,数值模拟结果与试验结果一致,再一次证实了三维无网格Galerkin方法程序实施技术的正确性、最大周向拉应力裂纹扩展准则的正确性和三维裂纹扩展曲面数学描述的有效性。在此基础上,对三维表面斜置半圆形裂纹的扩展进行了数值模拟,了解了复杂三维裂纹不同扩展阶段的扩展方式、扩展曲面的形态,以及空间方位对三维裂纹扩展模式的影响,填补了国内在三维复杂裂纹扩展数值模拟方面的空白。
通过三维裂纹的数值模拟可以清楚地了解三维裂纹扩展的全貌,弥补了三维裂隙断裂试验与理论研究的不足,为进一步研究三维裂隙断裂的本质提供一定的参考价值。
The study background of this paper is the crack rock mass. A new crack propagation criterions, the maximum circumferential tensile stress criterions, are presented in this paper. The numerical simulation of two-dimensional(2-D) and three-dimensional(3-D) crack propagation are studied baseed on element free Galerkin (EFG) method and 2-D EFG method based on couple stresses theory.
Basic formulae of 2-D EFG method and 2-D EFG meothod based on couple stresses theory and 3-D EFG method are derived. Afterwards, discrete equations for elastic body are derived by use of the minimus potential energy principle. By use of classical 2-D and 3-D problems, the implement of 2-D and 3-D EFG method are discussed. And the matching of several important parameters in 2-D and 3-D EFG method are discussed, too. The displacement restriction is released by use of a penalty parameter. So the essential boundary conditions are introduced into EFG methods. Based on the idea of controlling nodal number, alterable influence radii can be defined. The least square calculating estimate is cut down in dense nodal area and calculating efficiency of EFG methods is enhanced properly.
Based on space analytic geometry, the local coordinates at crack front is built up. The coordinates transform matrix, base vector transform, vector transform and displacement transform are established from the global coordinates to the local coordinates at crack front. The circumferential stress formulae are derived in the main normal plane of the local coordinates at crack front. So there have sufficient preparative work for the numerical simulation of crack propagation and surface or line of crack propagation.
In studying crack problem, the visibility criterion is adopted to figure isolation of crack between nodals and integral nodals.
Chosen a proper crack propagation criterion to determine condition, directions and steps of crack propagation is the key.
Shortcoming of 2-D crack propagation criterions and no criterions of 3-D crack propagation are considered. Based on other scholars work, a new criterions of crack propagation, namely the maximum circumferential tensile stress criterion for 2-D and 3-D crack are brought forward in this paper. The new crack propagation criterions give out not only crack propagation condition and directions, but also steps too. It avoids the error and trouble of calculating stress intensity factor, especially for 3-D crack propagation problems, and overcomes the disadvantage giving out crack propagation step depending on experience of calculator.
Surface of 3-D crack propagation maybe appear bending and twisting in crack propagation. It is divided into some snippets. Every snippet is triangulated. Mathematic description of crack surface is realized.
In numerical simulation of crack propagation, for identifying crack shape and improving simulation precision of crack propagation surface, append nodal points are added on surface at crack front. Some nodal point maybe lie in a crack snippet. It is splited to two points to be laid on the two side of crack surface.
The numerical simulation of 2-D crack propagation without cranny hydraulic pressure is studied by EFG methods. The results are identical with that of other computing method and test. The validity of the maximum circumferential tensile stress criterion is confirmed. The effect on crack propagation of cranny hydraulic pressure and couple stresses is studied. Some new results are found.
Because of the complexity of 3-D crack propagation, there are a few results about the numerical simulation of 3-D crack propagation at present. Only model- I propagation of penny crack is simulated by use of traditional numerical method. In this paper, model- I propagation of ellipse crack is simulated by EFG method. The results are identical with that of other computing method and test. Once more, the validity of the maximum circumferential tensile stress criterion and mathematic description of 3-D crack surface are proved. Based on the above, the crack propagation of 3-D surface inclined semi-circular cracks are simulated numerically. The manner in different stage and the surface shape of 3-D crack propagation are known. Different attitude of original crack produces great effect on 3-D crack propagation. The blankness of the numerical simulation of 3-D complicated crack propagation is filled up in homeland.
The full propagation process of 3-D crack is simulated. The simulation results make up some faults of the experimental results and theory research results. The simulation results have some reference value to further study 3-D crack fracture.
引文
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