功能梯度材料运动裂纹及弹性波作用下的断裂行为研究
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摘要
功能梯度作为一种可设计的复合材料,在航空航天领域有着重要的应用价值和广阔的应用前景,是高温工作条件下最有应用前景的复合材料。随着功能梯度材料广泛地应用于航空航天领域,对功能梯度材料中力学问题的研究意义越来越重要。功能梯度材料是非均匀体,其非均匀性对材料的力学性能有很大的影响。但是,有关功能梯度材料断裂力学的研究还很不够,尤其是有关功能梯度材料运动裂纹和弹性波作用于功能梯度材料裂纹的问题。本文对以上几个难点问题上进行了深入探讨,得到了比较有意义的结论。
本文所做的主要工作有:
1.研究无限大功能梯度材料反平面剪切型运动裂纹问题。利用其材料剪切模量和密度的指数模型,通过Fourier积分变换导出无限大功能梯度材料反平面运动裂纹问题的对偶积分方程,利用Jacobi多项式将位移展开成级数形式,并采用Schmidt数值方法计算出了裂纹尖端动应力强度因子的半解析解,分析了裂纹运动速度、梯度参数和正交异性系数对动态应力强度因子的影响。
2.研究层板结构界面运动裂纹问题。分别考虑了均匀层板和功能梯度涂层—半无限大基体结构模型,利用不同界面的连接条件,将问题中所有各量用单一未知函数表达,用积分变换方法将运动裂纹问题化为对偶积分方程,并采用Schmidt数值方法得到此问题得半解析解。分析了裂纹运动速度、梯度参数和层板厚度对裂纹尖端应力场的影响。
3.研究无限长条功能梯度材料的运动裂纹问题,对应力场进行了半解析求解,并分析了裂纹运动速度和材料梯度参数对裂纹尖端应力场的影响。研究结果表明:对于无限长条功能梯度材料的约束边界条件下的运动裂纹,其动应力强度因子随着裂纹运动速度的增加而降低。对于无限长条功能梯度材料的自由边界运动裂纹问题,裂纹尖端应力的最大值随着裂纹运动速度的增加而增加。裂纹尖端应力始终随着材料梯度参数的增加而降低。
4.研究弹性波入射情况下功能梯度材料中反平面裂纹的动力学问题。利用无限大板任意一点承受的时间简谐的位移函数,以及散射波与入射波有相同的时间简谐因子,建立裂纹尖端的散射模型。采用积分变换方法得到对偶积分方程,然后对积分核进行偶部分和奇部分处理,通过Schmidt数值方法,得到了与裂纹相互作用下的散射波位移、应力表达式,研究了无限大板周围的动应力集中情况和动应力强度因子,并给出具体的解,讨论了入射波数、入射角度和材料梯度参数等因素对此问题的影响情况。
5.研究P波和SV波在功能梯度材料裂纹上的绕射,无论P或SV波单独或同时作用,裂纹散射的波都同时包含膨胀波与剪切波,情况比较复杂,包含了I型裂纹和II型动态裂纹问题。通过Fourier积分变换和微分算子矩阵法导出此问题的对偶积分方程,求解分析了入射波的入射角、材料的梯度参数、入射波的波数和材料的泊松比对动应力强度因子的影响。
本文的工作可以为功能梯度材料及其结构的动态断裂行为进行分析和评价提供理论根据,它们的解提供了裂纹尖端位移场与应力场的有价值的结论。
FGM which is considered to be the most promising composite material under the high temperature working condition, as a kind of designable composite material, has an important application and potential prospect in the areas of astronautics and aeronautics. With Functionally Graded Materials widely used in the astronautic and aeronautic fields, the mechanics problems research in Functionally Graded Materials become more and more important. FGMs are non homogeneous solid and the nonhomogeneality of FGMs has a great influence on their mechanical behavior. However, the studies on the dynamic fracture mechanics of FGMs are not enough, especially for the problems on moving and expanding cracks. We thoroughly explored several of these difficult issues mentioned and obtained some very important results.
The main contributions of this dissertation can be read as follow:
1. The stress at the crack tip on moving crack in an infinite body for FGM subjected to shear loading are studied. The dual integral equation of antiplane moving problem through Fourier transform with the help of the exponent model of the shear modulus and density is obtained. The displacement is expanded into series form using Jacobi Polynomial by Schmidt method, then the semi-analytic and numerical solutions of dynamic stress intensity factor are attained. and the Influences of the crack velocity, graded parameter, and orthotropic coefficient on the stress at crack tip are considered.
2. The theoretical treatment of an interface moving crack is provided for laminated media. Two physical models, namely homogeneous bonded media and a functionally graded coating-substrate structure, are considered respectively. Using conditions of the welding surface of different media, we express all the quantities in terms of a single unknown function. Using method of integral transform, we formulate the moving crack problem as dual integral equations, then the semi-analytic of dynamic stress intensity factor are attained by Schmidt method. The influences of parameters such as crack velocity, graded parameter and laminated height on dynamic stress intensity factor are studied.
3. The stress at the crack tip on moving crack in an infinite length strip for FGM subjected to shear loading are studied and the Influences of the crack velocity and graded parameter on the stress at crack tip are considered. The results show that stress intensity factor at the crack tip decrease with increasing crack velocity for the clamped boundary problems of moving crack in an infinite length strip, but for the problem of free boundary, the maximum value of stress at the crack tip increase with increasing crack velocity. Stress field at the crack tip decrease with increasing graded parameter.
4. The problem of elastic waves scattering and dynamic stress concentration by FGM plane with cracks of any limited lengths near the gap is investigated. Due to the same time factor of scattering wave and incident wave, the scattering model of the crack tip can be constructed by making use of the displacment function of harmonic load on any point of the infinite plane. With the use of the integral transform, the dual integral equation for determining the external forces can be abtained, then have some process on the even and odd term of the integral kernel, the expression of displacement and stress is established while the interaction of infinite plate with cracks is studied with Schmidt method. Dynamic stress concentration near the plate is studied, and dynamic stress intensity factor at crack tip is discussed. Some examples and results are given. The influences of wave number, incident angles of electic wave, and graded parameter are discussed.
5. Consider the propagation of P-wave and SV-wave, produced by the action of oscillating compressional and shear forces, which vary harmonically in time and are applied in the xy-plane. These input waves are diffracted at the crack in FGM. The indicates that the waves scattered by the crack are composed of both compression and shear waves even if the incident wave may only be of one type, either the P- or SV-waves. By semi-analytic solutions the influences of wave number, incident angles of electic wave, graded parameter and Poisson’s ratio are discussed.
This work can provide a foundation for the optimization design and property evaluation of Functionally Graded Materials in theory. The results provide valuable conclusion on the displacement field and the stress field of a crack tip.
本文所做的主要工作有:
1.研究无限大功能梯度材料反平面剪切型运动裂纹问题。利用其材料剪切模量和密度的指数模型,通过Fourier积分变换导出无限大功能梯度材料反平面运动裂纹问题的对偶积分方程,利用Jacobi多项式将位移展开成级数形式,并采用Schmidt数值方法计算出了裂纹尖端动应力强度因子的半解析解,分析了裂纹运动速度、梯度参数和正交异性系数对动态应力强度因子的影响。
2.研究层板结构界面运动裂纹问题。分别考虑了均匀层板和功能梯度涂层—半无限大基体结构模型,利用不同界面的连接条件,将问题中所有各量用单一未知函数表达,用积分变换方法将运动裂纹问题化为对偶积分方程,并采用Schmidt数值方法得到此问题得半解析解。分析了裂纹运动速度、梯度参数和层板厚度对裂纹尖端应力场的影响。
3.研究无限长条功能梯度材料的运动裂纹问题,对应力场进行了半解析求解,并分析了裂纹运动速度和材料梯度参数对裂纹尖端应力场的影响。研究结果表明:对于无限长条功能梯度材料的约束边界条件下的运动裂纹,其动应力强度因子随着裂纹运动速度的增加而降低。对于无限长条功能梯度材料的自由边界运动裂纹问题,裂纹尖端应力的最大值随着裂纹运动速度的增加而增加。裂纹尖端应力始终随着材料梯度参数的增加而降低。
4.研究弹性波入射情况下功能梯度材料中反平面裂纹的动力学问题。利用无限大板任意一点承受的时间简谐的位移函数,以及散射波与入射波有相同的时间简谐因子,建立裂纹尖端的散射模型。采用积分变换方法得到对偶积分方程,然后对积分核进行偶部分和奇部分处理,通过Schmidt数值方法,得到了与裂纹相互作用下的散射波位移、应力表达式,研究了无限大板周围的动应力集中情况和动应力强度因子,并给出具体的解,讨论了入射波数、入射角度和材料梯度参数等因素对此问题的影响情况。
5.研究P波和SV波在功能梯度材料裂纹上的绕射,无论P或SV波单独或同时作用,裂纹散射的波都同时包含膨胀波与剪切波,情况比较复杂,包含了I型裂纹和II型动态裂纹问题。通过Fourier积分变换和微分算子矩阵法导出此问题的对偶积分方程,求解分析了入射波的入射角、材料的梯度参数、入射波的波数和材料的泊松比对动应力强度因子的影响。
本文的工作可以为功能梯度材料及其结构的动态断裂行为进行分析和评价提供理论根据,它们的解提供了裂纹尖端位移场与应力场的有价值的结论。
FGM which is considered to be the most promising composite material under the high temperature working condition, as a kind of designable composite material, has an important application and potential prospect in the areas of astronautics and aeronautics. With Functionally Graded Materials widely used in the astronautic and aeronautic fields, the mechanics problems research in Functionally Graded Materials become more and more important. FGMs are non homogeneous solid and the nonhomogeneality of FGMs has a great influence on their mechanical behavior. However, the studies on the dynamic fracture mechanics of FGMs are not enough, especially for the problems on moving and expanding cracks. We thoroughly explored several of these difficult issues mentioned and obtained some very important results.
The main contributions of this dissertation can be read as follow:
1. The stress at the crack tip on moving crack in an infinite body for FGM subjected to shear loading are studied. The dual integral equation of antiplane moving problem through Fourier transform with the help of the exponent model of the shear modulus and density is obtained. The displacement is expanded into series form using Jacobi Polynomial by Schmidt method, then the semi-analytic and numerical solutions of dynamic stress intensity factor are attained. and the Influences of the crack velocity, graded parameter, and orthotropic coefficient on the stress at crack tip are considered.
2. The theoretical treatment of an interface moving crack is provided for laminated media. Two physical models, namely homogeneous bonded media and a functionally graded coating-substrate structure, are considered respectively. Using conditions of the welding surface of different media, we express all the quantities in terms of a single unknown function. Using method of integral transform, we formulate the moving crack problem as dual integral equations, then the semi-analytic of dynamic stress intensity factor are attained by Schmidt method. The influences of parameters such as crack velocity, graded parameter and laminated height on dynamic stress intensity factor are studied.
3. The stress at the crack tip on moving crack in an infinite length strip for FGM subjected to shear loading are studied and the Influences of the crack velocity and graded parameter on the stress at crack tip are considered. The results show that stress intensity factor at the crack tip decrease with increasing crack velocity for the clamped boundary problems of moving crack in an infinite length strip, but for the problem of free boundary, the maximum value of stress at the crack tip increase with increasing crack velocity. Stress field at the crack tip decrease with increasing graded parameter.
4. The problem of elastic waves scattering and dynamic stress concentration by FGM plane with cracks of any limited lengths near the gap is investigated. Due to the same time factor of scattering wave and incident wave, the scattering model of the crack tip can be constructed by making use of the displacment function of harmonic load on any point of the infinite plane. With the use of the integral transform, the dual integral equation for determining the external forces can be abtained, then have some process on the even and odd term of the integral kernel, the expression of displacement and stress is established while the interaction of infinite plate with cracks is studied with Schmidt method. Dynamic stress concentration near the plate is studied, and dynamic stress intensity factor at crack tip is discussed. Some examples and results are given. The influences of wave number, incident angles of electic wave, and graded parameter are discussed.
5. Consider the propagation of P-wave and SV-wave, produced by the action of oscillating compressional and shear forces, which vary harmonically in time and are applied in the xy-plane. These input waves are diffracted at the crack in FGM. The indicates that the waves scattered by the crack are composed of both compression and shear waves even if the incident wave may only be of one type, either the P- or SV-waves. By semi-analytic solutions the influences of wave number, incident angles of electic wave, graded parameter and Poisson’s ratio are discussed.
This work can provide a foundation for the optimization design and property evaluation of Functionally Graded Materials in theory. The results provide valuable conclusion on the displacement field and the stress field of a crack tip.
引文
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