State dependent linear moduli in ferroelectrics

A novel experimental method is used to measure the evolution of the linear elastic, dielectric and piezoelectric moduli of a soft ferroelectric ceramic during loading. The applied loading states are combinations of uniaxial compressive stress and electric field. Short pulses of electric field and stress are used to increment the remanent strain and polarization state of the material, while the rates of change of electric displacement and strain during unloading are used to assess the moduli. The remanent quantities are treated as state variables, with a view to expressing the moduli as functions of the material state. The piezoelectric moduli are found to vary approximately linearly with polarization, regardless of the remanent strain state, whilst the dielectric moduli and elastic compliances show more complex behaviour. A simple model of the state dependence of the moduli, based on varying the volume fractions of six crystal variants in the tetragonal system, is used to interpret the results.     

Material multipole mechanics of elastic dielectric composites

The overall mechanical and electrical behaviors of elastic dielectric composites are investigated with the aid of the concept of material multipoles. In particular, by introducing a statistical continuum material multipole theory, the effects of the electric-elastic interaction and the microstructure (size, shape, orientation,...) of inhomogeneous particles on the overall behaviors of the composites can be obtained. A basic solution for an ellipsoidal elastic inhomogeneity with electric polarization in an infinite elastic dielectric medium is first given, which shows that classical Eshelby ’s elastic solution is modified by the presence of electric-elastic interaction. The overall macroscopic constitutive relations and their overall macroscopic material parameters accounting for electroelastic interaction effect are then derived for the elastic dielectric composites. Some quantitative calculations on the problems with statistical anisotropy, the shape effect and the electric-elastic interaction are finally given for dilute composites.     

A nonlinear field theory of deformable dielectrics

Two difficulties have long troubled the field theory of dielectric solids. First, when two electric charges are placed inside a dielectric solid, the force between them is not a measurable quantity. Second, when a dielectric solid deforms, the true electric field and true electric displacement are not work conjugates. These difficulties are circumvented in a new formulation of the theory in this paper. Imagine that each material particle in a dielectric is attached with a weight and a battery, and prescribe a field of virtual displacement and a field of virtual voltage. Associated with the virtual work done by the weights and inertia, define the nominal stress as the conjugate to the gradient of the virtual displacement. Associated with the virtual work done by the batteries, define the nominal electric displacement as the conjugate to the gradient of virtual voltage. The approach does not start with Newton's laws of mechanics and Maxwell-Faraday theory of electrostatics, but produces them as consequences. The definitions lead to familiar and decoupled field equations. Electromechanical coupling enters the theory through material laws. In the limiting case of a fluid dielectric, the theory recovers the Maxwell stress. The approach is developed for finite deformation, and is applicable to both elastic and inelastic dielectrics. As applications of the theory, we discuss material laws for elastic dielectrics, and study infinitesimal fields superimposed upon a given field, including phenomena such as vibration, wave propagation, and bifurcation.     

Crack in functionally graded piezoelectric strip bonded to elastic surface layers under electromechanical loading

Solved is the problem of a crack in a functionally graded piezoelectric material (FGPM) bonded to two elastic surface layers. It is assumed that the elastic stiffness, piezoelectric constant, and dielectric permittivity of the FGPM vary continuously along the thickness of the strip. The outside layers are under antiplane mechanical loading and in-plane electric loading. The solution involves solving singular integral equations by application of the Gauss–Jacobi integration formula. Numerical calculations are carried out to obtain the energy density factors. Their variations with the geometric, loading and material parameters are shown graphically.     

具有线性介电常数的Ogden型介电弹性体的本构关系和机电稳定性

应用多材料常数的Ogden弹性应变能函数分析了介电弹性体的力学行为,研究了介电弹性体的机电稳定性.数值结果表明,通过对材料系数(如材料常数比和电致伸缩系数等)的恰当调节可以使得介电弹性体材料或介电弹性体结构更趋稳定.这些有益于深入理解介电弹性体的机电稳定性行为,进而设计恰当的介电弹性体器件.     

The seismic pulse in a semi-infinite medium

Summary In a foregoing paper the present author developed methods for studying the transient field from a vertical electric antenna placed in the vicinity of the plane boundary of two semi-infinite dielectric media.As the theory involved is applicable to the comparable elastodynamic pulse problem the present paper deals with the field from a buried transient longitudinal source in an elastic half space.The method appears to be relatively simple and is also applicable to the more general problem in which two elastic semi-infinite solids are separated by a plane boundary.     

Tresca-type plastic materials in the theory of hypoelasticity II. Optical constitutive equations and birefringence in simple shear

Behavior of a Tresca type plastic dielectric is investigated theoretically from a continuum mechanical point of view. The optical constitutive equations are defined as special cases of a hypo-elastic dielectric of grade two. The singularity condition of the constitutive equations satisfies the Tresca yield criterion. The index deviator tensor is proportional to the stress deviator tensor and, then, the birefringence and the extinction angle are expressed by the stress deviator. Their numerical variations with the angle of shear in simple shear deformation are shown.     

Third-order elastic,piezoelectric, and dielectric constants

The definitions of the third-order elastic, piezoelectric, and dielectric constants and the properties of the associated tensors are discussed. Based on the energy conservation and coordinate transformation, the relations among the third-order constants are obtained. Furthermore, the relations among the third-order elastic, piezoelectric, and dielectric constants of the seven crystal systems and isotropic materials are listed in detail.These third-order constants relations play an important role in solving nonlinear problems of elastic and piezoelectric materials. It is further found that all third-order piezoelectric constants are 0 for 15 kinds of point groups, while all third-order dielectric constants are0 for 16 kinds of point groups as well as isotropic material. The reason is that some of the point groups are centrally symmetric, and the other point groups are high symmetry.These results provide the foundation to measure these constants, to choose material, and to research nonlinear problems. Moreover, these results are helpful not only for the study of nonlinear elastic and piezoelectric problems, but also for the research on flexoelectric effects and size effects.     

Theory of scattering from multilayered bodies of arbitrary shape

Green 's functions and boundary integral equation methods are used to derive a matrix set of equations for scattering from a multilayered homogeneous elastic body embedded in an infinite elastic material. The surfaces separating the layers have arbitrary shape. The formalism for the three-layer material is derived in detail and generalized to N-layers. A matrix factorization method (MFM) is shown to considerably simplify the computational problem. The relation to the problems of acoustic waves in fluids and electromagnetic waves in a dielectric material is briefly indicated.     

Simple examples of non linear truss behavior

Nonlinear material models are defined to represent elastic/perfectly-plastic and elastic/buckling behavior. A simple three-bar truss is used to demonstrate that under a monotonically increasing prescribed displacement the truss may exhibit reverse stressing or non-uniqueness, and that when two different control displacements are applied the principal of superposition does not hold.     

Elastic and Dielectric Properties of Active Ag/BaTiO<Subscript>3</Subscript> Composites

This study examines the elastic and dielectric properties of active composites consisting of barium titanate (BaTiO₃) and silver (Ag) constituents using experimental and numerical approaches. The elastic constants including Young’s modulus, shear modulus and Poisson’s ratio were measured by resonant ultrasound spectroscopy (RUS), a nondestructive dynamic technique, while a dielectric (impedance) spectroscopy was used to measure the relative permittivity and dielectric loss at different frequencies. The dielectric tests were also conducted at temperature ranges from ?50 to 200 °C where the two phase transformations of barium titanate at around 0 °C and 120 °C were examined. The experimental results in this study were compared to data available in the literature. In addition to the experimental work, a numerical method is also considered in order to study the effects of blending silver into barium titanate on the effective elastic and dielectric properties of the composite and the local field fluctuations. For this purpose, two micromechanics models describing the detailed composite microstructures were constructed. The first model is based on two dimensional (2D) images of realistic microstructures obtained by the scanning electronic microscopy (SEM), while the second model is based on randomly generated three-dimensional (3D) microstructures with spherical particles. The effects of loading direction, porosity, particle shape and dispersion were examined using the micromechanics models. Numerical predictions of the effective elastic and dielectric constants were compared to the experiment results.     

含非完整界面的功能梯度压电材料Ⅲ型裂纹问题

本文研究了含非完整界面的功能梯度压电复合材料的Ⅲ型裂纹问题．此裂纹垂直于非完整界面,采用弹簧型力电耦合界面模型模拟非完整界面．界面两侧材料的性质,如弹性模量、压电常数和介电常数均假定呈指数函数形式且沿着裂纹方向变化．运用积分变换法将裂纹面条件转换为奇异积分方程,并使用Gauss-Chebyshev方法对其进行数值求解．根据算例结果讨论了一些退化问题并分析了裂纹尖端强度因子与材料的非均匀系数和非完整界面参数的关系．     

Analysis of a permeable interface crack in elastic dielectric/piezoelectric bimaterials

A permeable interface crack between elastic dielectric material and piezoelectric material is studied based on the extended Stroh’s formalism. Motivated by strong engineering demands to design new composite materials, the authors perform numerical analysis of interface crack tip singularities and the crack tip energy release rates for 35 types of dissimilar bimaterials, respectively, which are constructed by five kinds of elastic dielectric materials: Epoxy, Polymer, Al₂O₃, SiC, and Si₃N₄ and seven kinds of practical piezoelectric ceramics: PZT-4, BaTiO₃, PZT-5H, PZT-6B, PZT-7A, P-7, and PZT-PIC 151, respectively. The elastic dielectric material with much smaller permittivity than commercial piezoelectric ceramics is treated as a special transversely isotropic piezoelectric material with extremely small piezoelectricity. The present investigation shows that the structure of the singular field near the permeable interface crack tip consists of three singularities: and , which is quite different from that in the impermeable interface crack. It can be concluded that different far field loading cases have significant influence on the near-tip fracture behaviors of the permeable interface crack. Based on the present theoretical treatment and numerical analysis, the electric field induced crack growth is well explained, which provides a better understanding of the failure mechanism induced from interface crack growth in elastic dielectric/piezoelectric bimaterials. The project supported by the National Natural Science Foundation of China (10572110), Doctor Foundation of the Chinese Education Ministry and Doctorate Foundation of Xi’an Jiaotong University. The English text was polished by Yunming Chen.     

Acceleration waves in constrained elastic materials

Well known results on the propagation and growth of acceleration waves in Cauchy elastic materials are extended to materials which suffer one or two internal constraints. It is proved, under certain restrictions, that acceleration waves will not propagate in a material which has three or more internal constraints. The great simplifications deriving from an assumption of hyperelasticity are indicated. The present results could be extended to materials other than simple elastic.     

Fully coupled, multi-axial, symmetric constitutive laws for polycrystalline ferroelectric ceramics

In this paper, a general form for multi-axial constitutive laws for ferroelectric ceramics is constructed. The foundation of the theory is an assumed form for the Helmholtz free energy of the material. Switching surfaces and associated flow rules are postulated in a modified stress and electric field space such that a positive dissipation rate during switching is guaranteed. The resulting tangent moduli relating increments of stress and electric field to increments of strain and electric displacement are symmetric since changes in the linear elastic, dielectric and piezoelectric properties of the material are included in the switching surface. Finally, parameters of the model are determined for two uncoupled cases, namely non-remanent straining ferroelectrics and purely ferroelastic switching, and then for the fully coupled ferroelectric case.     

功能梯度压电压磁材料粘结的Ⅲ型裂纹问题

利用奇异积分方程方法研究两个半无限大的功能梯度压电压磁材料粘结,在渗透和非 渗透边界条件下的III型裂纹问题. 首先通过积分变换构造出原问题的形式解,然 后利用边界条件通过积分变换与留数定理得到一组奇异积分方程, 最后利用Gauss-Chebyshev方法进行数值 求解,讨论材料参数、材料非均匀参数以及裂纹几何形状等对裂纹尖端应力 强度因子的影响. 从结果中可以看出,压电压磁复合材料中反平面问题的应力奇异性 形式与一般弹性材料中的反平面问题应力奇异形式相同,但材料梯度参数对功能梯度压电压 磁复合材料中的应力强度因子和电位移强度因子有很大的影响.     

Effective elastic properties of periodic composite medium

A new method is presented for calculating the bulk effective elastic stiffness tensor of a two-component composite with a periodic microstructure. The basic features of this method are similar to the one introduced by Bergman and Dunn (1992) for the dielectric problem. It is based on a Fourier representation of an integro-differential equation for the displacement field, which is used to produce a continued-fraction expansion for the elastic moduli. The method enabled us to include a much larger number of Fourier components than some previously proposed Fourier methods. Consequently our method provides the possibility of performing reliable calculations of the effective elastic tensor of periodic composites that are neither dilute nor low contrast, and are not restricted to arrays of nonoverlapping inclusions. We present results for a cubic array of nonoverlapping spheres, intended to serve as a test of quality, as well as results for a cubic array of overlapping spheres and a two dimensional hexagonal array of circles (a model for a fiber reinforced material) for comparison with previous work.     

Singular surfaces in a linear thermo-elastic dielectric material

Singular surfaces in a linear thermo-elastic dielectric material are considered, where the constitutive equations of the elastic dielectric proposed by Toupin and the heat equations with finite wave velocities are combined. There exist six types of singular surfaces including a stationary one. The velocities, the coupled fields and the variation of the amplitudes of the surfaces with respect to time are investigated. It is found that the amplitude of the mechanical transverse wave rotates during propagation and at the stationary surface the amplitude of the electric field periodically reverses in direction and the one of the polarization field rotates elliptically with the same period.     

A finite strain theory for elastic-plastic deformation

An elastic-plastic theory that is applicable when the elastic part of the strain is finite is proposed. A flow rule for an incompressible solid is obtained from Drucker's postulate [1]. Isothermal simple shear of a material which is neo-Hookean both before yielding and during elastic unloading after yielding is considered as an application of the theory. The problem is solved for two yield conditions and associated flow rules.     

Asymptotic analysis of an impermeable crack in an electrostrictive material subjected to electric loading

The simple asymptotic problem of an impermeable crack in an electrostrictive ceramic under electric loading is analyzed. Closed form solutions of elastic fields are obtained by using the complex function theory. It is found that the K_I-dominant region is very small compared to the electric saturation zone. A fracture parameter for an electrostrictive material subjected to electric loading is discussed. In order to investigate the influence of the transverse electric displacement on fracture behavior under the small-scale conditions, we also consider the modified boundary layer problem of a crack in an electrostrictive material. Analytic solutions of electric displacement fields for the asymptotic problem are obtained based on the nonlinear dielectric theory from a modified boundary layer analysis. The shape of the electric displacement saturation zone is shown to depend on the transverse electric displacement. Stress intensity factors induced by the electrostrictive strains are evaluated using the nonlinear solution of the electric displacements. It is found that the transverse electric displacement affects strongly the variation of the mode mixity.