正交各向异性材料V型切口及界面问题的应力奇异性研究
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摘要
本文基于正交各向异性材料线弹性平面问题的基本方程,通过引入位移函数,推导出了正交各向异性平面问题的位移法通解。根据该通解,借助坐标转换关系和特征展开方法,得到了正交各向异性材料奇异点附近的位移场和奇异应力场,当两个材料特征参数均等于1时,该位移场和奇异应力场即退化为各向同性材料相应的场。在此研究基础上,取得了以下研究成果:
(1)对正交各向异性材料的对称V型切口尖端的应力奇异性进行了理论分析,得到了切口尖端的应力奇异性特征方程及其附近的位移场和奇异应力场。结果表明,与各向同性材料不同,正交各向异性材料V型切口尖端的应力奇异性次数不仅与切口的张角有关,还与材料的弹性常数有关。
(2)对正交各向异性材料的任意V型切口尖端的应力奇异性进行了理论分析,给出了在多种边界条件下切口尖端的应力奇异性特征方程及其附近的位移场和奇异应力场。
(3)就材料特征参数的三种匹配形式,对具有任意结合角的正交各向异性双材料界面角点进行了理论分析,得到了对称和反对称变形模态下界面角点的应力奇异性特征方程及其附近的位移场和奇异应力场。该解也可用于正交各向异性/各向同性以及各向同性/各向同性的双材料界面角点问题。
(4)研究了加固工程中一种常见几何形式的正交各向异性/各向同性双材料界面端的应力奇异性,得到了界面端的应力奇异性特征方程及其附近的位移场和奇异应力场,讨论了正交各向异性材料的端部结合角及其弹性常数对界面端应力奇异性的影响。
(5)提出了一种利用常规的数值分析结果确定单应力奇异性和双应力奇异性问题的应力奇异性次数及其相应的应力强度因子的数值计算方法,给出了数值计算公式。该方法具有通用性好、求解精度高、便于工程应用等特点。
(6)为了检验上述理论分析结果的正确性,应用有限元分析软件MSC. Patran & Nastran分别对正交各向异性材料的V型切口、界面角点和界面端等问题的应力奇异性进行了数值求解,并与理论分析结果进行了比较,两者吻合较好。
In this paper, based on the fundamental equations of the two-dimensional linear elasticity for orthotropic materials, the general solution of the displacement method for orthotropic materials is derived by introducing the displacement function. According to this general solution, and employing the coordinate transformation and the eigenequation expansion method, displacement and singular stress fields near the singular point in orthotropic materials are presented in closed form expressions. When the two characteristic parameters of orthotropic materials are all equal to 1, the above displacement and singular stress fields are degenerated into the corresponding fields in isotropic materials. From these presented expressions, the fruits of the study in this paper are obtained as follows:
(1) A symmetric V-notch in orthotropic materials is analyzed theoretically for the stress singularity at the notch tip. The eigenequations about the stress singularity, and the displacement and singular stress fields near the notch tip are acquired directly. It is found that the orders of the stress singularity at the notch tip in orthotropic materials are related not only to the angle of the V-notch, but also to the material properties, which being different to isotropic materials.
(2) An asymmetric V-notch in orthotropic materials is analyzed theoretically for the stress singularity at the notch tip. The eigenequations about the stress singularity, and displacement and singular stress fields near the notch tip under various boundary conditions are presented explicitly.
(3) For three match cases of the characteristic parameters of orthotropic materials, an interface corner of orthotropic bi-materials with an arbitrary joining angle is analyzed theoretically. The eigenequations about the stress singularity, and the displacement and singular stress fields near the interface corner are also explicitly established for the symmetric and anti-symmetric deformation modes, respectively. The results can be also applied to the interface corners in orthotropic/isotropic bi-materials and isotropic/isotropic bi-materials.
(4) An interface edge of the orthotropic/isotropic bi-materials with an arbitrary wedge angle for the orthotropic material, which is a common geometry in the strengthened engineering, is analyzed theoretically. The eigenequation as well as the displacement and singular stress fields near the interface edge are derived directly. The relations between the singular stresses near the interface edge of the orthotropic/isotropic bi-material and the wedge angle as well as the material property of the orthotropic material are discussed in details.
(5) A simple and effective numerical approach is developed to calculate the orders of the stress singularity and the related stress intensity factors for one and two stress singularities by using the results of an ordinary numerical analysis, and the associated formulae are presented for the numerical calculation. The approach can be used in engineering analysis conveniently, and the stress intensity factors evaluated by this method are very accurate.
(6) To verify the correctness of the abovementioned theoretical analyses, the stress singularities at the V-notch tips, interface corners, and interface edges of orthotropic materials are analyzed numerically by Patran & Nastran code (MSC Corporation), based on the displacement finite element method. The analytical results are compared with the related numerical ones, respectively; it is found that they coincide with each other very well.
(1)对正交各向异性材料的对称V型切口尖端的应力奇异性进行了理论分析,得到了切口尖端的应力奇异性特征方程及其附近的位移场和奇异应力场。结果表明,与各向同性材料不同,正交各向异性材料V型切口尖端的应力奇异性次数不仅与切口的张角有关,还与材料的弹性常数有关。
(2)对正交各向异性材料的任意V型切口尖端的应力奇异性进行了理论分析,给出了在多种边界条件下切口尖端的应力奇异性特征方程及其附近的位移场和奇异应力场。
(3)就材料特征参数的三种匹配形式,对具有任意结合角的正交各向异性双材料界面角点进行了理论分析,得到了对称和反对称变形模态下界面角点的应力奇异性特征方程及其附近的位移场和奇异应力场。该解也可用于正交各向异性/各向同性以及各向同性/各向同性的双材料界面角点问题。
(4)研究了加固工程中一种常见几何形式的正交各向异性/各向同性双材料界面端的应力奇异性,得到了界面端的应力奇异性特征方程及其附近的位移场和奇异应力场,讨论了正交各向异性材料的端部结合角及其弹性常数对界面端应力奇异性的影响。
(5)提出了一种利用常规的数值分析结果确定单应力奇异性和双应力奇异性问题的应力奇异性次数及其相应的应力强度因子的数值计算方法,给出了数值计算公式。该方法具有通用性好、求解精度高、便于工程应用等特点。
(6)为了检验上述理论分析结果的正确性,应用有限元分析软件MSC. Patran & Nastran分别对正交各向异性材料的V型切口、界面角点和界面端等问题的应力奇异性进行了数值求解,并与理论分析结果进行了比较,两者吻合较好。
In this paper, based on the fundamental equations of the two-dimensional linear elasticity for orthotropic materials, the general solution of the displacement method for orthotropic materials is derived by introducing the displacement function. According to this general solution, and employing the coordinate transformation and the eigenequation expansion method, displacement and singular stress fields near the singular point in orthotropic materials are presented in closed form expressions. When the two characteristic parameters of orthotropic materials are all equal to 1, the above displacement and singular stress fields are degenerated into the corresponding fields in isotropic materials. From these presented expressions, the fruits of the study in this paper are obtained as follows:
(1) A symmetric V-notch in orthotropic materials is analyzed theoretically for the stress singularity at the notch tip. The eigenequations about the stress singularity, and the displacement and singular stress fields near the notch tip are acquired directly. It is found that the orders of the stress singularity at the notch tip in orthotropic materials are related not only to the angle of the V-notch, but also to the material properties, which being different to isotropic materials.
(2) An asymmetric V-notch in orthotropic materials is analyzed theoretically for the stress singularity at the notch tip. The eigenequations about the stress singularity, and displacement and singular stress fields near the notch tip under various boundary conditions are presented explicitly.
(3) For three match cases of the characteristic parameters of orthotropic materials, an interface corner of orthotropic bi-materials with an arbitrary joining angle is analyzed theoretically. The eigenequations about the stress singularity, and the displacement and singular stress fields near the interface corner are also explicitly established for the symmetric and anti-symmetric deformation modes, respectively. The results can be also applied to the interface corners in orthotropic/isotropic bi-materials and isotropic/isotropic bi-materials.
(4) An interface edge of the orthotropic/isotropic bi-materials with an arbitrary wedge angle for the orthotropic material, which is a common geometry in the strengthened engineering, is analyzed theoretically. The eigenequation as well as the displacement and singular stress fields near the interface edge are derived directly. The relations between the singular stresses near the interface edge of the orthotropic/isotropic bi-material and the wedge angle as well as the material property of the orthotropic material are discussed in details.
(5) A simple and effective numerical approach is developed to calculate the orders of the stress singularity and the related stress intensity factors for one and two stress singularities by using the results of an ordinary numerical analysis, and the associated formulae are presented for the numerical calculation. The approach can be used in engineering analysis conveniently, and the stress intensity factors evaluated by this method are very accurate.
(6) To verify the correctness of the abovementioned theoretical analyses, the stress singularities at the V-notch tips, interface corners, and interface edges of orthotropic materials are analyzed numerically by Patran & Nastran code (MSC Corporation), based on the displacement finite element method. The analytical results are compared with the related numerical ones, respectively; it is found that they coincide with each other very well.
引文
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