Dynamic stress from a cylindrical inclusion buried in a functionally graded piezoelectric material layer under electro-elastic waves

This paper presents a theoretical method to investigate the multiple scattering of electro-elastic waves and dynamic stress around a subsurface cylindrical inclusion in a functionally graded piezoelectric material layer bonded to homogeneous piezoelectric materials. The analytical solutions of wave fields are expressed by employing wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions around the inclusion. The image method is used to satisfy the mechanical and electrically short conditions at the free surface of the structure. Through the numerical solutions of dynamic stress concentration factors around the inclusion, it is found that when the cylindrical inclusion possesses higher rigidity and greater piezoelectric constant than the two phases of functionally graded materials, the dynamic stress around the inclusion increases greatly. When the distance between the surface of the structure and the inclusion is smaller, the effect of the properties of the inclusion becomes greater. When the cylindrical inclusion possesses lower rigidity and smaller piezoelectric constant than the two phases of functionally graded materials, the maximum dynamic stress shows little difference; however, the variation of the distribution of the dynamic stress around the inclusion is greater. The effect of the properties of the inclusion on the dynamic stress around the inclusion is greater than that on the electric field. The effects of wave frequency on the dynamic stress and electric field are also examined.     

功能梯度材料亚表面缺陷的热波多重散射问题

基于热传导波动模型,采用波函数展开法,研究了半无限功能梯度材料亚表面球形缺陷的热波多重散射.给出了热波散射的一般解.温度波由调制光束在材料表面激发,球形缺陷表面的边界条件为绝热,非均匀参数为指数函数变化.分析了结构几何参数和物理参数对温度分布的影响,并给出了温度变化的数值结果.本研究可为功能梯度材料的分析研究、物理反问题和红外热波成像等提供理论基础和参考数据.     

Dynamic stress from a subsurface cylindrical inclusion in a functionally graded material layer under anti-plane shear waves

The multiple scattering of shear waves and dynamic stress resulting from a subsurface cylindrical inclusion in a functionally graded material (FGM) layer bonded to homogeneous materials are investigated, and the analytical solution of this problem is derived. Image method is used to satisfy the traction free boundary condition of the FGM layer. The analytical solutions of wave fields around the actual and image inclusions are expressed by employing wave functions expansion method, and the expanded mode coefficients are determined by satisfying the continuous boundary conditions around the inclusions. Through the numerical solutions of dynamic stress concentration factors (DSCFs) around the inclusion, the effects of the position of the inclusion in the material layer, the properties of the inclusion, and the properties of the two phases of composites on the DSCFs are analyzed. Analyses show that when the cylindrical inclusion is stiffer than the two phases of FGMs, the dynamic stress around the inclusion increases greatly. When the distance between the surface of the structure and the inclusion is smaller, the effect of the properties of the inclusion becomes greater. When the cylindrical inclusion is softer than the two phases of FGMs, the maximum dynamic stress shows little difference; however, the variation of the distribution of the dynamic stress around the inclusion is greater.     

Scattering of Thermal Waves and Non-steady Effective Thermal Conductivity of Unidirectional Fibrous Composites with Functionally Graded Interface

In this paper, a thermal wave method is applied to investigate the non-steady effective thermal conductivity of unidirectional fibrous composites with a functionally graded interface, and the analytical solution of the problem is obtained. The Fourier heat conduction law is applied to analyze the propagation of thermal waves in the fibrous composite. The scattering and refraction of thermal waves by a cylindrical fiber with an inhomogeneous interface layer in the matrix are analyzed, and the results of the single scattering problem are applied to the composite medium. The wave fields in different material layers are expressed by using the wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions of the layers. The theory of Waterman and Truell is employed to obtain the effective propagating wave number and the non-steady effective thermal conductivity of composites. As an example, the effects of a graded interface on the effective thermal conductivity of composites are graphically illustrated and analyzed. Analysis shows that the non-steady effective thermal conductivity under higher frequencies is quite different from the steady thermal conductivity. In the region of intermediate and high frequencies, the effect of the properties of the interface on the effective thermal conductivity is greater. Comparisons with the steady thermal conductivity obtained from other methods are also presented.     

Transient Heat Conduction in a Functionally Graded Cylindrical Panel Based on the Dual Phase Lag Theory

The transient heat conduction in a functionally graded cylindrical panel is investigated based on the dual phase lag (DPL) theory in this article. Except for the phase lags which are assumed to be constant, all the other material properties of the panel are assumed to change continuously along the radial direction according to a power-law formulation with different non-homogeneity indices. The heat conduction equations based on the DPL theory in the cylindrical coordinate system are written in a general form which are then used for the analyses of four different geometries: (1) a hollow cylinder of an infinite length; (2) a hollow cylinder of a finite length; (3) a cylindrical panel of an infinite length; and (4) a cylindrical panel of a finite length. Using the Laplace transform, the analytical solutions for temperature and heat flux are obtained in the Laplace domain. The solutions are then converted into the time domain by employing the fast Laplace inversion technique. The exact expressions for the radial thermal wave speed are obtained for the four different geometries. The numerical results are displayed to reveal the effect of different approximations of the DPL theory on the temperature distribution for various non-homogeneity indices. The results are verified with those reported in the literature.     

Dynamic stress in a semi-infinite solid with a cylindrical nano-inhomogeneity considering nanoscale microstructure

In this paper, the dynamic stress around a cylindrical nano-inhomogeneity embedded in a semi-infinite solid under anti-plane shear waves is investigated. The surface/interface stress effects around the nano-inhomogeneity and at the straight edge of the semi-infinite solid are both considered. The boundary condition at the straight edge of the semi-infinite solid with surface/interface effects is satisfied by the image method. The incident, scattered and refracted displacement fields in the nano-sized composites are expressed by employing the wave function expansion method. The addition theorem for a cylindrical wave function is applied to accomplish the superposition of wave fields in the two semi-infinite solids. Analyzes show that the effect of interface properties, especially that at the straight edge, on the dynamic stress is significant, and the effect increases noticeably due to the nanoscale of the structure. The incident frequency and angle of waves and the shear modulus ratio of the nano-inhomogeneity to matrix also show a pronounced effect on the dynamic stress distribution if the semi-infinite solid shrinks to nanoscale.     

Multiple scattering of shear waves and dynamic stress from a circular cavity buried in a semi-infinite slab of functionally graded materials

Based on the theory of elastodynamics and employing image method, the multiple scattering and dynamic stress in a semi-infinite slab of functionally graded materials with a circular cavity are investigated. The analytical solution of this problem is derived, and the numerical solutions of the dynamic stress concentration factor around the cavity are also presented. The effects of the distance between the cavity and the boundaries of the semi-infinite slab, the incident wave number and the non-homogeneity parameter of materials on the dynamic stress concentration factors are analyzed. Analyses show that the dynamic stress around the cavity increases with increasing non-homogeneity parameter of materials and incident wave number. The boundaries of the semi-infinite slab have great effect on both the maximum dynamic stress and the distribution of dynamic stress around the circular cavity, and the effect increases with increasing incident wave number.     

The effect of thermal radiation on the propagation of temperature waves in a semitransparent medium

The propagation of temperature waves arising as a result of periodic external thermal stimulation is investigated in a plane layer of semitransparent absorbing and radiating medium without scattering. The classification of temperature waves on the basis of two dimensionless parameters is suggested. It is rigorously demonstrated that not more than two temperature waves may simultaneously exist in a gray semi-infinite medium. The relative contribution of radiation to complex heat transfer is estimated. The system of equations of radiative-conductive heat transfer is reduced to a single integral equation on the boundary. The effect of reflection on the boundary is discussed.     

半无限功能梯度材料结构中圆孔对弹性波的多重散射

基于弹性波多体散射理论,采用波函数展开法,研究了半无限指数梯度材料中圆孔对弹性波的多重散射和动应力集中,得到了问题的解析解,给出了圆孔动应力集中系数的数值解,分析了圆孔与边界的距离、入射波波数以及材料的非均匀参数对圆孔周围动应力集中系数的影响。分析表明:梯度材料的非均匀参数小于零时对最大动应力影响较小,但是对动应力在圆周的分布有较大影响;大于零时对最大动应力和动应力在圆周的分布影响都很大,特别是在圆孔与边界的距离较小时影响更大。     

Transverse surface waves on a piezoelectric material carrying a functionally graded layer of finite thickness

The propagation behavior of transverse surface waves (Love waves) in a piezoelectric half space of polarized ceramics carrying a functionally graded material layer is studied from the three-dimensional equations of linear piezoelectricity. The Wentzel–Kramers–Brillouin (WKB) asymptotic technique is adopted for the theoretical derivations of analytical solutions in the functionally graded layer. The dispersion relations of Love wave in the structure are obtained for both electrically open and shorted cases. Firstly, these solutions are used to study effect of the gradient coefficients on the dispersive relations and phase velocities of Love wave propagation. Then influence of the gradient coefficients on the electromechanical coupling factor is discussed in detail. The results reported are meaningful for the design of surface acoustic wave (SAW) devices with high performance.     

Postbuckling of pressure-loaded FGM hybrid cylindrical shells in thermal environments

A postbuckling analysis is presented for a functionally graded cylindrical shell with piezoelectric actuators subjected to lateral or hydrostatic pressure combined with electric loads in thermal environments. Heat conduction and temperature-dependent material properties are both taken into account. The temperature field considered is assumed to be a uniform distribution over the shell surface and varied in the thickness direction and the electric field considered only has non-zero-valued component E_Z. The material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and the material properties of both FGM and piezoelectric layers are assumed to be temperature-dependent. The governing equations are based on a higher order shear deformation theory with a von Kármán–Donnell-type of kinematic nonlinearity. A boundary layer theory of shell buckling is extended to the case of FGM hybrid laminated cylindrical shells of finite length. A singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of pressure-loaded, perfect and imperfect, FGM cylindrical shells with fully covered piezoelectric actuators under different sets of thermal and electric loading conditions. The results reveal that temperature dependency, temperature change and volume fraction distribution have a significant effect on the buckling pressure and postbuckling behavior of FGM hybrid cylindrical shells. In contrast, the control voltage only has a very small effect on the buckling pressure and postbuckling behavior of FGM hybrid cylindrical shells.     

考虑热对材料参数影响的FGM梁热后屈曲特性研究

功能梯度梁热后屈曲特性研究对于推进功能梯度材料在航天器热防护设计中的应用有着重要意义。基于经典梁的几何非线性理论和物理中面的概念,建立了热载荷作用下功能梯度梁的运动微分方程,通过化简得到一个仅关于挠度的四阶微分-积分方程,并与固支边界条件构成特征值问题,分析研究了功能梯度梁的热后屈曲及在此基础上的振动问题。首先证明了通过哈密顿原理推导的运动方程是轴线可伸长理论的近似形式。接着考虑热对材料物性参数的影响,并对梁的长细比、功能梯度指数和温度比作了详细分析,研究了这些参数对热后屈曲路径和后屈曲振动的影响规律。结果表明:只有在长细比较大时才可以不考虑温度对材料物性参数的影响,否则误差较大;长细比、功能梯度指数和温度比的增大会增大无量纲热屈曲载荷,同时使屈曲路径和频率-载荷曲线向热载荷增大的方向移动。     

Love waves in a smart functionally graded piezoelectric composite structure

The propagation of Love waves in a smart functionally graded piezoelectric structure is analyzed by applying elastic wave theory. There is an additional functionally graded layer between the piezoelectric layer and the substrate in this smart structure. When the piezoelectric and dielectric constants vary individually in a functionally graded layer, the asymptotic solutions of Love waves are obtained by applying the WKB method and solving the fourth order differential equation with variable coefficients. The effects of gradient variation on the phase velocity and the coupled electromechanical factor are discussed in detail. The analysis shows that the number of vibration modes is greater than that in the non-graded layer structure, and the coupled electromechanical factor increases with the increase of piezoelectric constant graded variation. Presented results are useful for the improvement of properties of surface acoustic wave (SAW) devices.     

Measurement of the thermal diffusivity coefficient of materials

A theoretical substantiation of the measuring procedure for the thermal diffusivity coefficient of substances is given. In doing this, the two-dimensional problem of heat conduction for a cylindrical semi-infinite specimen and the circular modulated heat flux is considered. The coefficient sought is estimated from the phase shift between temperature fluctuations on the flat surface of the speciment exposed to the heat flux and the fluctuations of this flux.Translated from Inzhenerno-fizicheskii Zhurnal, Vol. 64, No. 1, pp. 73–76, January, 1993.     

Scattering of vertical shear waves by a cluster of nanosized cylindrical holes with surface effect

This study considers the multiple scattering of vertical shear waves (SV-waves) by a cluster of nanosized cylindrical holes. When the radius of the holes shrinks to nanometers, surface effects play an important role in their mechanical performance, and the surface elasticity theory is adopted to analyze diffraction phenomena. By using the displacement potential method and wave functions expansion method, the scattering fields around the holes are derived. Both the dynamic stress concentration around the holes and the scattering cross section are calculated to illustrate the effect of surface effects on the multiple scattering of an SV-wave.     

The effect of oblique functional gradation to transient thermal stresses in the functionally graded infinite strip

Transient thermal stresses in the strip with boundaries oblique to the functionally graded direction are studied theoretically. The transient temperature and the transient thermal stresses are derived by the use of the variable separation and the stress function method. The material properties are assumed to be exponential functions of the position along the functionally graded direction. The prescribed surface heat flux is given for temperature condition, and the initial temperature is assumed to be zero over the strip. The strip is free of surface traction for mechanical boundary condition. The numerical calculations are carried out for ZrO₂ /Ti-6Al-4V functionally graded materials. The numerical results of temperature and thermal stresses are illustrated with the lapse of time for certain oblique angles.     

非理想界面功能梯度压电/压磁层状半空间结构中的SH波

研究非理想界面下功能梯度压电/压磁层状半空间结构中SH波的传播特性。界面性能由“弹簧”模型表征, 假设功能梯度压电层材料性能沿层厚度方向指数变化, 其表面为电学开路。推导了频散方程, 并结合数值算例分析了界面性能、功能梯度压电层的梯度变化和厚度对相速度的影响。研究结果对功能梯度压电/压磁复合材料在声波器件中的应用提供了理论依据。     

Size-dependent effective dynamic properties of unidirectional nanocomposites with interface energy effects

This article studies the size effect on wave propagation characteristics of plane longitudinal and transverse elastic waves in a two-phase nanocomposite consisting of transversely isotropic and unidirectionally oriented identical cylindrical nanofibers embedded in a transversely isotropic homogeneous matrix. The surface elasticity theory is employed to incorporate the interfacial stress effects. The effect of random distribution of nanofibers in the composite medium is taken into account via a generalized self-consistent multiple scattering model. The phase velocities and attenuations of longitudinal and shear waves along with the associated dynamic effective elastic constants are calculated for a wide range of frequencies and fiber concentrations. The numerical results reveal that interface elasticity at nanometer length scales can significantly alter the overall dynamic mechanical properties of nanofiber-reinforced composites. Limiting cases are considered and excellent agreements with solutions available in the literature have been obtained.     

Non-destructive Detection of a Circular Cavity in a Finite Functionally Graded Material Layer Using Anti-plane Shear Waves

A semi-analytical method is proposed to investigate the non-destructive detection of a circular cavity buried in a functionally graded material layer bonded to homogeneous materials, and the multiple scattering effect of shear waves is described accurately. The image method is used to satisfy the traction free boundary condition at the edge of the functionally graded material layer. The analytical solutions of wave fields are expressed by employing wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions at the edge and around the cavity. The analytical and numerical solutions of dynamic stress concentration factors around the cavity are presented. The effects of the position of the cavity in the material layer, the incident wave number, and the properties of the two phases of materials on the dynamic stress concentration factors are analyzed. Analyses show that when the buried depth of the cavity and the thickness of the layer are relatively small, the properties of the two phases of materials have great effect on the distribution of dynamic stress around the cavity. In the region of higher frequency, the effects of the position of the cavity and the properties of the two phases of materials on the maximum dynamic stress are greater.     

热流温度场下功能梯度板的热问题研究

功能梯度材料因其内部组分沿着空间位置连续变化,能有效缓解热应力集中等现象,在高超音速飞行器的热防护系统设计中具有良好的应用前景。以金属-陶瓷功能梯度板为研究对象,探讨在不同热环境下功能梯度板热传导、热变形和热应力的变化规律。首先,基于功能梯度材料的幂律分布模型,分析了线性温度场、正弦温度场、热流温度场和非线性温度场四种热环境对功能梯度材料物理特性的影响。其次,基于有限元分层建模思想,通过Python编程建立了功能梯度板的有限元模型。仿真分析了在热流温度场下功能梯度板在不同板厚、陶瓷体积分数指数和热流密度等参数条件下热传导、热变形和热应力的变化规律。最后,探讨了基于参数调控的改善功能梯度板热防护性能的方法。结果表明：用功能梯度材料设计飞行器热防护板时,陶瓷体积分数指数应小于5.0;在热流温度场下,功能梯度板的厚度为7.2~9.0 mm、陶瓷体积分数指数为1.0~2.5时,可实现功能梯度板的轻质和高效隔热;热流密度一定时,调整飞行时间对确保飞行器安全至关重要;热流密度应限制在5~10 mW/mm²以控制功能梯度板的变形。研究结论对于热流温度场下金属-陶瓷功能梯度板的热防护设计具有参考价值。