功能梯度材料相关的几个动静态问题分析及结构优化
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摘要
功能梯度材料作为一种特殊的复合材料,其材料的微观组成和性能随位置状态呈现连续变化,并且功能梯度材料可将性能各异的材料按照设计的意愿在结构内部非均匀、连续地合成新型材料,以适用于各种特殊的工程结构极限需要。现在,其用途已由原来的航天工业扩大到核能源、燃气涡轮发动机和土木工程等诸多领域。然而功能梯度材料以其优良的特性被广泛应用于各领域的同时,其材料的连续变化特性也给功能梯度材料的力学行为研究带来了一定的难题。本文从更符合实际的角度,不需要事先对梯度材料参数作特殊的假设,对任意梯度的功能梯度材料的相关动静态力学问题提出了一种积分方程解法,主要包括以下几个方面：
(1)对任意梯度变化的功能梯度轴对称结构,我们提出了一种新的简单有效的解析方法—积分方程方法,分别对功能梯度圆环、圆筒、空心圆球、以及转动圆盘等结构的纯弹性问题进行了研究。首先将所考虑的问题简化为Fredholm积分方程,继而通过数值求解积分方程给出其应力场和位移的分布情况。通过对一些典型的梯度变化情况进行计算,我们发现某些梯度可使环向应力的最大值出现在结构内部,而不是在结构表面,并能有效缓和结构内应力的分布。因此在实际结构优化设计中,可根据本文提出的方法以及不同的使用条件来对具体的梯度变化形式以及参数大小进行设计,从而使得结构在实际服役环境中更为安全。
(2)采用本文提出的积分方程方法对具有任意梯度变化的匀速转动极正交各向异性功能梯度圆环的弹性场进行了分析,重点讨论了正交各向异性参数和材料梯度参数对弹性场、尤其是环向应力的影响。结果表明,在不同边界条件下,极正交各向异性功能梯度圆环中的环向应力并非总大于径向应力,梯度的影响规律也并不一样,而这是和均匀材料结构所不同的地方。因此,在对功能梯度正交各向异性圆环结构优化设计时,需要对不同边界条件下的结构建立各自对应的破坏准则。此方法为功能梯度正交各向异性轴对称问题的优化设计提供了一定的参考依据。
(3)进一步,本文对功能梯度轴对称结构相关的耦合问题如热-力耦合、力-电耦合问题进行了分析。对材料性能按特殊的幂函数形式变化的情况,我们推导得到了相关问题的封闭解。而对任意梯度变化的材料性能,采用积分方程方法求得了结构内电弹性场的分布情况。结果表明,选择合适的材料以及梯度参数一方面可有效缓和应力分布,另一方面可使功能梯度压电器件达到其最好的性能,从而提高压电器件的可靠性和使用寿命。
(4)最后对均匀压电(和磁电弹)材料为基底及其表面为功能梯度压电涂层(和磁电弹涂层)结构在承受冲击载荷情况下裂尖处的瞬态响应进行了分析。考虑了两种不同的加载情况：实际的冲击载荷直接作用的材料表面的情况和载荷作用在裂纹表面的情况。通过积分变换法将所考虑的问题简化为对奇异积分方程的求解,并分别得到了四种常用的断裂参数。研究表明梯度参数对裂纹的动态响应因子有重大影响,尤其当冲击载荷直接作用于功能梯度涂层表面时。同时,本文对不同的动态断裂参数(动态应力、应变强度因子、能量释放率和能量密度因子)分别进行了计算和比较。对智能材料而言,由于与一些实验结果较为吻合,应变强度因子可作为有效的断裂判据。
As a class of new advanced composite materials, functionally graded materials whose material properties continuously vary along one or multi spatial directions can be tailored to satisfy particular engineering application by technologically manufacturing process. In recent years, functionally graded materials have been widely used in nuclear energy reactor, turbine motors, aerospace engineering, and civil engineering etc. However, its continuous material properties bring out some difficulties in analyzing their mechanical behaviors. From a more practical point of view, we do not need to assume the gradient to be special functions and the integral equation method is presented to treat static and dynamic mechanical problems of functionally graded materials with any gradient. This thesis mainly includes the following items:
(1) Firstly, for functionally graded axisymmetrical structures with arbitrarily varying material properties, a simple and efficient approach is put forward to study the elastic problem of functionally graded circular disk, cylinder, sphere and sandwich rotating disk. The associated elastic problem is reduced to a Fredholm integral equation. By solving the resulting equation, the distribution of the stress components and displacement can be determined. As examples, some typically varying material properties are considered. The obtained results indicate that change in the gradient of the functionally graded axisymmetrical structure does not induce a substantial variation of the radial stress, but strongly affects the circumferential stress. And for some special gradient parameters, the distribution curves of circumferential stress become more gently and maximum value of circumferential stress occurs at an interior position of the structure, not at the surfaces like homogeneous materials. Thus, in practical design, the method presented in this thesis may help engineering designers to choose appropriate gradients and materials to acquire an optimal state and to ensure the structure to be safer in the service-environment.
(2) With above-mentioned method we study the elastic problem of rotating functionally graded hollow polar orthotropic circular disks whose relevant material properties arbitrarily varying along the radial direction. Emphasis is placed on the influence of orthotropy and gradient on the elastic field in particular the circumferential stress. Numerical results are presented for two particular cases:free boundaries and clamped-free boundaries. The numerical results show that, for different boundary conditions, the circumferential stress is not always larger than radial stress, and the influences of gradient parameter on circumferential stress are different, too. So, during the design of functionally graded orthotropic rotating disk, for different boundary conditions, individual failure criterion needs to be established. The obtained results are helpful for the optimization on design of functionally graded orthotropic rotating disk for the purpose of optimal design.
(3) Furthermore, some coupling problems of functionally graded axisymmetrical structures are investigated such as thermoelastic and electroelastic problems. For the gradient of a power-law behavior, a closed form solution is derived. For general gradient, an integral equation method is suggested to reduce the problem to a Fredholm integral equation, and the response of the coupling field can be determined. The obtained results show that choosing appropriate materials and gradient parameters can make not only the stresses more gently, but also the functionally graded piezoelectric devices to achieve the best performance, and then enhance the reliability and service life of devices.
(4) The transient response of the dynamic fracture parameters for an interfacial crack of functionally graded piezoelectric or magnetoelectric material coated by a homogeneous piezoelectric or magnetoelectric material substrate on the surface is analyzed. Two different loading situations are mainly considered:one is a realistic situation when impacts are suddenly applied on the material surface; and the other is on the crack surfaces. With the integral transform method, the problem is reduced to solving singular integral equations, and four different dynamic fracture parameters are obtained. It is found that when applied electromechanical impacts are exerted on the material surface, the gradient index causes the transient response to be significantly amplified or reduced depending on negative or positive gradient index. At the same time, by comparing the four dynamic fracture parameters, dynamic stress intensity factor, strain intensity factor, energy release rate and energy density factor, it is found that the stain intensity factor seems to be an efficient fracture parameter for smart materials, due to its consistency with some experimental observations.
(1)对任意梯度变化的功能梯度轴对称结构,我们提出了一种新的简单有效的解析方法—积分方程方法,分别对功能梯度圆环、圆筒、空心圆球、以及转动圆盘等结构的纯弹性问题进行了研究。首先将所考虑的问题简化为Fredholm积分方程,继而通过数值求解积分方程给出其应力场和位移的分布情况。通过对一些典型的梯度变化情况进行计算,我们发现某些梯度可使环向应力的最大值出现在结构内部,而不是在结构表面,并能有效缓和结构内应力的分布。因此在实际结构优化设计中,可根据本文提出的方法以及不同的使用条件来对具体的梯度变化形式以及参数大小进行设计,从而使得结构在实际服役环境中更为安全。
(2)采用本文提出的积分方程方法对具有任意梯度变化的匀速转动极正交各向异性功能梯度圆环的弹性场进行了分析,重点讨论了正交各向异性参数和材料梯度参数对弹性场、尤其是环向应力的影响。结果表明,在不同边界条件下,极正交各向异性功能梯度圆环中的环向应力并非总大于径向应力,梯度的影响规律也并不一样,而这是和均匀材料结构所不同的地方。因此,在对功能梯度正交各向异性圆环结构优化设计时,需要对不同边界条件下的结构建立各自对应的破坏准则。此方法为功能梯度正交各向异性轴对称问题的优化设计提供了一定的参考依据。
(3)进一步,本文对功能梯度轴对称结构相关的耦合问题如热-力耦合、力-电耦合问题进行了分析。对材料性能按特殊的幂函数形式变化的情况,我们推导得到了相关问题的封闭解。而对任意梯度变化的材料性能,采用积分方程方法求得了结构内电弹性场的分布情况。结果表明,选择合适的材料以及梯度参数一方面可有效缓和应力分布,另一方面可使功能梯度压电器件达到其最好的性能,从而提高压电器件的可靠性和使用寿命。
(4)最后对均匀压电(和磁电弹)材料为基底及其表面为功能梯度压电涂层(和磁电弹涂层)结构在承受冲击载荷情况下裂尖处的瞬态响应进行了分析。考虑了两种不同的加载情况：实际的冲击载荷直接作用的材料表面的情况和载荷作用在裂纹表面的情况。通过积分变换法将所考虑的问题简化为对奇异积分方程的求解,并分别得到了四种常用的断裂参数。研究表明梯度参数对裂纹的动态响应因子有重大影响,尤其当冲击载荷直接作用于功能梯度涂层表面时。同时,本文对不同的动态断裂参数(动态应力、应变强度因子、能量释放率和能量密度因子)分别进行了计算和比较。对智能材料而言,由于与一些实验结果较为吻合,应变强度因子可作为有效的断裂判据。
As a class of new advanced composite materials, functionally graded materials whose material properties continuously vary along one or multi spatial directions can be tailored to satisfy particular engineering application by technologically manufacturing process. In recent years, functionally graded materials have been widely used in nuclear energy reactor, turbine motors, aerospace engineering, and civil engineering etc. However, its continuous material properties bring out some difficulties in analyzing their mechanical behaviors. From a more practical point of view, we do not need to assume the gradient to be special functions and the integral equation method is presented to treat static and dynamic mechanical problems of functionally graded materials with any gradient. This thesis mainly includes the following items:
(1) Firstly, for functionally graded axisymmetrical structures with arbitrarily varying material properties, a simple and efficient approach is put forward to study the elastic problem of functionally graded circular disk, cylinder, sphere and sandwich rotating disk. The associated elastic problem is reduced to a Fredholm integral equation. By solving the resulting equation, the distribution of the stress components and displacement can be determined. As examples, some typically varying material properties are considered. The obtained results indicate that change in the gradient of the functionally graded axisymmetrical structure does not induce a substantial variation of the radial stress, but strongly affects the circumferential stress. And for some special gradient parameters, the distribution curves of circumferential stress become more gently and maximum value of circumferential stress occurs at an interior position of the structure, not at the surfaces like homogeneous materials. Thus, in practical design, the method presented in this thesis may help engineering designers to choose appropriate gradients and materials to acquire an optimal state and to ensure the structure to be safer in the service-environment.
(2) With above-mentioned method we study the elastic problem of rotating functionally graded hollow polar orthotropic circular disks whose relevant material properties arbitrarily varying along the radial direction. Emphasis is placed on the influence of orthotropy and gradient on the elastic field in particular the circumferential stress. Numerical results are presented for two particular cases:free boundaries and clamped-free boundaries. The numerical results show that, for different boundary conditions, the circumferential stress is not always larger than radial stress, and the influences of gradient parameter on circumferential stress are different, too. So, during the design of functionally graded orthotropic rotating disk, for different boundary conditions, individual failure criterion needs to be established. The obtained results are helpful for the optimization on design of functionally graded orthotropic rotating disk for the purpose of optimal design.
(3) Furthermore, some coupling problems of functionally graded axisymmetrical structures are investigated such as thermoelastic and electroelastic problems. For the gradient of a power-law behavior, a closed form solution is derived. For general gradient, an integral equation method is suggested to reduce the problem to a Fredholm integral equation, and the response of the coupling field can be determined. The obtained results show that choosing appropriate materials and gradient parameters can make not only the stresses more gently, but also the functionally graded piezoelectric devices to achieve the best performance, and then enhance the reliability and service life of devices.
(4) The transient response of the dynamic fracture parameters for an interfacial crack of functionally graded piezoelectric or magnetoelectric material coated by a homogeneous piezoelectric or magnetoelectric material substrate on the surface is analyzed. Two different loading situations are mainly considered:one is a realistic situation when impacts are suddenly applied on the material surface; and the other is on the crack surfaces. With the integral transform method, the problem is reduced to solving singular integral equations, and four different dynamic fracture parameters are obtained. It is found that when applied electromechanical impacts are exerted on the material surface, the gradient index causes the transient response to be significantly amplified or reduced depending on negative or positive gradient index. At the same time, by comparing the four dynamic fracture parameters, dynamic stress intensity factor, strain intensity factor, energy release rate and energy density factor, it is found that the stain intensity factor seems to be an efficient fracture parameter for smart materials, due to its consistency with some experimental observations.
引文
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