周期性多相材料热动力时空多尺度分析
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摘要
在许多先进的工程应用领域中，如航空、航天、生物医疗和电子设备以及超导线圈等，具有周期性微结构的多相材料得到了广泛使用，而此时材料设计需要微观和宏观层次的相关信息，这对材料设计提出了一个很大的挑战。多尺度力学模拟技术提供了一个非常有效的方法，来理解不同的微结构材料性质对非均匀材料整体和局部响应的影响，这对材料设计具有重要的指导意义。
另外，对具有周期性微结构的多相材料中波的传播，以及由材料分界面上存在的反射和折射作用引起的弥散、衰减现象的研究也是一个非常重要的问题，而此类问题具有明显的时间和空间多尺度特征，其主要包括微波和脉冲激光技术等应用领域。但对该问题的多尺度研究还不多见，因此对在具有周期性微结构的多相材料中的波传播进行多尺度分析是一个值得进一步研究的问题。
本文的主要工作是采用时—空间多尺度高阶渐近均匀化分析方法，研究具有周期性微结构的多相材料在热冲击和极端动力载荷作用下的非Fourier热传导及结构热动力响应问题。为了更深入的说明问题，首先从解决一维问题入手，然后再向多维问题扩展，具体内容如下：
通过建立一个广义的热动力波动函数场控制方程，描述一维具有周期性微结构的多相材料中波传播的问题。采用时—空间多尺度高阶渐近均匀化分析方法，通过引入多个空间和时间尺度，对波动函数场进行渐近展开，获得了不同阶次的时—空间多尺度波动函数场控制方程。由高阶均匀化理论在空间和时间尺度上进行均匀化求解，并通过近似合并整理，获得最高阶空间导数项为空间四阶导数的高阶非局部波动函数场控制方程。为了避免其有限元离散过程中C~1连续性和边界条件的要求，通过把空间四阶导数近似成为一个时间二阶和空间二阶混合的导数项，使其有限元离散过程中只需要C~0连续和较少的边界条件，最终得到了一个新的近似高阶非局部波动函数场控制方程。
基于高阶非局部波动函数场模型，建立了一维非Fourier热传导高阶非局部模型。通过把精细的有限元解作为参考解，与本文提出的非Fourier热传导高阶非局部模型、经典均匀化模型的解进行比较，并讨论了不同的参数和工况条件下对数值结果的影响，指出了本文提出的非Fourier热传导高阶非局部模型能够很好地模拟在热冲击载荷作用下的具有周期性微结构的多相材料中热波动以及弥散现象，体现出内尺度参数的特征。而经典均匀化模型只能处理脉冲激励波长大于材料特征长度的情况，且不能解决脉冲激励波长接近或小于材料特征长度的问题，所以不能模拟该现象。
根据高阶非局部波动函数场模型，建立了一维结构热动力高阶非局部模型。通过对在热冲击和极端动力载荷作用下的具有周期性微结构的一维多相细长杆进行研究，数值结果显示本文提出的结构热动力高阶非局部模型能够在花费较少的计算时间情况下模拟出高频动力响应及动力波的弥散现象，并与精细的有限元参考解取得了很好的一致。验证了本文所提出的结构热动力高阶非局部模型的正确性和有效性。
通过采用时—空间多尺度高阶渐近均匀化分析方法，对多维非Fourier热传导问题进行研究。并以二维问题为例，对具有不同的单胞微结构形状的多相材料中非Fourier热传导问题进行数值模拟分析，结果进一步证明了由高阶多尺度均匀化理论获得的多维高阶非局部模型的正确性和有效性。但这里的多维问题并不是一维问题的简单推广，其主要区别在于，一维问题是通过解析方式获得等效均匀化系数，而二维问题需要采用数值方法求解。
Currently, multiphase materials with periodic microstructures have been widely used inmany advanced engineering applications such as aerospace, aircraft, biomedical, electronicequipment, superconducting coil and so forth. However, the design and use of materials withmicrostructures necessarily involve the related information including both microscopic andmacroscopic scales. This is a major challenge for the materials design. Multiscale modelingtechniques offer an efficient approach for understanding of how different microstructuralproperties affect the average and local response of multiphase materials. Therefore, it isbeneficial to the materials design.
Moreover, it is very important subject to study dispersion and attenuation phenomena ofwave propagation in periodic multiphase materials due to successive reflection and refractionwaves from the material interfaces, which may obviously have multiscale characteristics inboth time and space. Possible applications of this research include many domains such asmicrowaves and laser with very high frequency and extremely short duration etc.. However,seldom multiscale methods can be found for studying this problem, so it is believed that wavepropagation in multiphase materials with periodic microstructures deserves furtherinvestigations.
The primary research work in this thesis is to study non-Fourier heat conduction andthermal dynamic response in multiphase materials with periodic microstructures underextreme heat impulse and dynamic impact load, which use a spatial and temporal multiscalehigh-order asymptotic homogenization method. To further illustration, one-dimensional caseis studied in this thesis firstly, and the work is extended to multi-dimensional cases. Specificdetails are as follows:
A general field equation of thermo-dynamic wave propagation is developed to describeboth the heat transfer and the mechanical behaviour in the periodic multiphase materials undervarious temperature shock and impact load. Amplified spatial and reduced temporal scales are,respectively, introduced by a spatial-temporal multiscale high-order asymptotichomogenization method, and a multiple scale asymptotic expansion is employed toapproximate the transient function field. Different orders of function field equation includingvarious spatial and temporal multiscales are derived. By combining different orders ofhomogenized equations, the high-order nonlocal function field equation with the fourth-orderspatial derivative term is obtained. To avoid the requirement of C~1-continuous finite element formulation and boundary conditions in numerical implementation, the fourth-order spatialderivative term can be approximated by a mixed second-order derivative in space and time.So the C~0-continuous and few boundary conditions are used only. Finally, an approximatedhigher-order nonlocal function field equation is formulated.
Based on the higher-order nonlocal function field equation, one-dimensional higher-ordernonlocal equation of non-Fourier heat conduction is developed. The higher-order nonlocalmodel and the classical homogenization model are compared with the fine finite elementsolution. The effects of differenet parameters and conditions are discussed. The results showthat the higher-order nonlocal model of non-Fourier heat conduction can be used to simulatethe dispersion and attenuation phenomena of heat wave propagation in periodic multiphasematerials under extreme heat impulse, and which shows the influence of the internal lengthscale parameter. However, the classical homogenization model is valid only when theexciation wavelength is larger than the characteristic length, while it is invalid when theexciation wavelength is smaller than or near the characteristic length. So the classicalhomogenization model can not simulate these phenomena.
According to the higher-order nonlocal function field equation, one-dimensionalhigher-order nonlocal thermo-dynamic equation is developed. A one-dimensional slender barwith a periodic microstructure under heat impulse and dynamic impact load is studied. Thenumerical results show the solutions obtained by the proposed higher-order nonlocal model ofthermo-dynamics are generally in reasonable agreement with the fine finite element solutionand can be available for little computational cost to simulate the high frequency dynamicresponse and dispersion phenomena of dynamic wave propagation. It is demonstrated that thethe nonlocal model proposed is effective and valid.
Mutli-dimensional problem of non-Fourier heat conduction is studied by a spatial andtemporal mutiscale high-order asymptotic homogenization method. The two-dimensionalnumerical examples are computed to analyse non-Fourier heat conduction in the differentmicrostructure of multiphase materials. The results further demonstrate thatmutli-dimensional higher-order nonlocal model obtained higher-order mathematicalhomogenization theory is valid and effective. However, in this work the higher-order nonlocalmodel is not simply extended from one-dimensional case to mutli-dimensional case. The maindifference is that the homogenized coefficients are obtained by the analytical approach inone-dimensional problem, while as for two-dimensional problem these coefficients are solvedby the numerical method.
另外，对具有周期性微结构的多相材料中波的传播，以及由材料分界面上存在的反射和折射作用引起的弥散、衰减现象的研究也是一个非常重要的问题，而此类问题具有明显的时间和空间多尺度特征，其主要包括微波和脉冲激光技术等应用领域。但对该问题的多尺度研究还不多见，因此对在具有周期性微结构的多相材料中的波传播进行多尺度分析是一个值得进一步研究的问题。
本文的主要工作是采用时—空间多尺度高阶渐近均匀化分析方法，研究具有周期性微结构的多相材料在热冲击和极端动力载荷作用下的非Fourier热传导及结构热动力响应问题。为了更深入的说明问题，首先从解决一维问题入手，然后再向多维问题扩展，具体内容如下：
通过建立一个广义的热动力波动函数场控制方程，描述一维具有周期性微结构的多相材料中波传播的问题。采用时—空间多尺度高阶渐近均匀化分析方法，通过引入多个空间和时间尺度，对波动函数场进行渐近展开，获得了不同阶次的时—空间多尺度波动函数场控制方程。由高阶均匀化理论在空间和时间尺度上进行均匀化求解，并通过近似合并整理，获得最高阶空间导数项为空间四阶导数的高阶非局部波动函数场控制方程。为了避免其有限元离散过程中C~1连续性和边界条件的要求，通过把空间四阶导数近似成为一个时间二阶和空间二阶混合的导数项，使其有限元离散过程中只需要C~0连续和较少的边界条件，最终得到了一个新的近似高阶非局部波动函数场控制方程。
基于高阶非局部波动函数场模型，建立了一维非Fourier热传导高阶非局部模型。通过把精细的有限元解作为参考解，与本文提出的非Fourier热传导高阶非局部模型、经典均匀化模型的解进行比较，并讨论了不同的参数和工况条件下对数值结果的影响，指出了本文提出的非Fourier热传导高阶非局部模型能够很好地模拟在热冲击载荷作用下的具有周期性微结构的多相材料中热波动以及弥散现象，体现出内尺度参数的特征。而经典均匀化模型只能处理脉冲激励波长大于材料特征长度的情况，且不能解决脉冲激励波长接近或小于材料特征长度的问题，所以不能模拟该现象。
根据高阶非局部波动函数场模型，建立了一维结构热动力高阶非局部模型。通过对在热冲击和极端动力载荷作用下的具有周期性微结构的一维多相细长杆进行研究，数值结果显示本文提出的结构热动力高阶非局部模型能够在花费较少的计算时间情况下模拟出高频动力响应及动力波的弥散现象，并与精细的有限元参考解取得了很好的一致。验证了本文所提出的结构热动力高阶非局部模型的正确性和有效性。
通过采用时—空间多尺度高阶渐近均匀化分析方法，对多维非Fourier热传导问题进行研究。并以二维问题为例，对具有不同的单胞微结构形状的多相材料中非Fourier热传导问题进行数值模拟分析，结果进一步证明了由高阶多尺度均匀化理论获得的多维高阶非局部模型的正确性和有效性。但这里的多维问题并不是一维问题的简单推广，其主要区别在于，一维问题是通过解析方式获得等效均匀化系数，而二维问题需要采用数值方法求解。
Currently, multiphase materials with periodic microstructures have been widely used inmany advanced engineering applications such as aerospace, aircraft, biomedical, electronicequipment, superconducting coil and so forth. However, the design and use of materials withmicrostructures necessarily involve the related information including both microscopic andmacroscopic scales. This is a major challenge for the materials design. Multiscale modelingtechniques offer an efficient approach for understanding of how different microstructuralproperties affect the average and local response of multiphase materials. Therefore, it isbeneficial to the materials design.
Moreover, it is very important subject to study dispersion and attenuation phenomena ofwave propagation in periodic multiphase materials due to successive reflection and refractionwaves from the material interfaces, which may obviously have multiscale characteristics inboth time and space. Possible applications of this research include many domains such asmicrowaves and laser with very high frequency and extremely short duration etc.. However,seldom multiscale methods can be found for studying this problem, so it is believed that wavepropagation in multiphase materials with periodic microstructures deserves furtherinvestigations.
The primary research work in this thesis is to study non-Fourier heat conduction andthermal dynamic response in multiphase materials with periodic microstructures underextreme heat impulse and dynamic impact load, which use a spatial and temporal multiscalehigh-order asymptotic homogenization method. To further illustration, one-dimensional caseis studied in this thesis firstly, and the work is extended to multi-dimensional cases. Specificdetails are as follows:
A general field equation of thermo-dynamic wave propagation is developed to describeboth the heat transfer and the mechanical behaviour in the periodic multiphase materials undervarious temperature shock and impact load. Amplified spatial and reduced temporal scales are,respectively, introduced by a spatial-temporal multiscale high-order asymptotichomogenization method, and a multiple scale asymptotic expansion is employed toapproximate the transient function field. Different orders of function field equation includingvarious spatial and temporal multiscales are derived. By combining different orders ofhomogenized equations, the high-order nonlocal function field equation with the fourth-orderspatial derivative term is obtained. To avoid the requirement of C~1-continuous finite element formulation and boundary conditions in numerical implementation, the fourth-order spatialderivative term can be approximated by a mixed second-order derivative in space and time.So the C~0-continuous and few boundary conditions are used only. Finally, an approximatedhigher-order nonlocal function field equation is formulated.
Based on the higher-order nonlocal function field equation, one-dimensional higher-ordernonlocal equation of non-Fourier heat conduction is developed. The higher-order nonlocalmodel and the classical homogenization model are compared with the fine finite elementsolution. The effects of differenet parameters and conditions are discussed. The results showthat the higher-order nonlocal model of non-Fourier heat conduction can be used to simulatethe dispersion and attenuation phenomena of heat wave propagation in periodic multiphasematerials under extreme heat impulse, and which shows the influence of the internal lengthscale parameter. However, the classical homogenization model is valid only when theexciation wavelength is larger than the characteristic length, while it is invalid when theexciation wavelength is smaller than or near the characteristic length. So the classicalhomogenization model can not simulate these phenomena.
According to the higher-order nonlocal function field equation, one-dimensionalhigher-order nonlocal thermo-dynamic equation is developed. A one-dimensional slender barwith a periodic microstructure under heat impulse and dynamic impact load is studied. Thenumerical results show the solutions obtained by the proposed higher-order nonlocal model ofthermo-dynamics are generally in reasonable agreement with the fine finite element solutionand can be available for little computational cost to simulate the high frequency dynamicresponse and dispersion phenomena of dynamic wave propagation. It is demonstrated that thethe nonlocal model proposed is effective and valid.
Mutli-dimensional problem of non-Fourier heat conduction is studied by a spatial andtemporal mutiscale high-order asymptotic homogenization method. The two-dimensionalnumerical examples are computed to analyse non-Fourier heat conduction in the differentmicrostructure of multiphase materials. The results further demonstrate thatmutli-dimensional higher-order nonlocal model obtained higher-order mathematicalhomogenization theory is valid and effective. However, in this work the higher-order nonlocalmodel is not simply extended from one-dimensional case to mutli-dimensional case. The maindifference is that the homogenized coefficients are obtained by the analytical approach inone-dimensional problem, while as for two-dimensional problem these coefficients are solvedby the numerical method.
引文
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