Stress singularities near the apex of a cylindrically polarized piezoelectric wedge

Summary In this paper, the stress singularities for a cylindrically polarized piezoelectric wedge are investigated. The characteristic equations are derived analytically by using the extended Lekhnitskii formulation. The piezoelectric material (PZT-4) is polarized in the radial, circular or axial direction, respectively. Similar to the rectilinearly polarized piezoelectric problem, the inplane and antiplane stress fields are uncoupled. The results show the variations of the singularity orders with the changes of the wedge angle, material constants, polarized direction, and the boundary conditions.     

The free jets as boundary-layer flows induced by continuous stretching surfaces

This paper proves that the free laminar jets of the classical hydrodynamics may be identified with certain boundary-layer flows induced by continuous surfaces immersed in quiescent incompressible fluids and stretched with well-defined velocities. In this sense: (i) Schlichting's round jet of momentum flow coincides with the axisymmetric flow induced by a thin continuous wire issuing from a small orifice at x=0 and stretching along the x-axis with velocity U _w(x) = 3J˙/(8πρνx), and (ii) the Schlichting–Bickley plane jet of momentum flow J˙ coincides with the boundary-layer flow induced by an impermeable plane wall issuing from a long slit (of length l) and stretching with velocity U _w(x)= [{3J˙ ²/(32νρ² l ² x)}]^1/3.     

Higher order fields for damaged nonlinear antiplane shear notch, crack and inclusion problems

Damaged nonlinear antiplane shear problems with a variety of singularities are studied analytically. A deformation plasticity theory coupled with damage is employed in analysis. The effect of microscopic damage is considered in terms of continuum damage mechanics approach. An exact solution for the general damaged nonlinear singular antiplane shear problem is derived in the stress plane by means of a hodograph transformation, then corresponding higher order asymptotic solutions are obtained by reversing the stress plane solution to the physical plane. As example, traction free sharp notch and crack, rigid sharp wedge and flat inclusion, and mixed boundary sharp notch problems are investigated, respectively. Consequently, higher order fields are obtained, in which analytical expressions of the dominant and second order singularity exponents and angular distribution functions of the near tip fields are derived. Effects of the damage and hardening exponents of materials and the geometric angle of notch/wedge on the near tip quantities are discussed in detail. It is found that damage leads to a weaker dominant singularity of stress, but to little stronger singularities of the dominant and second order terms of strain compared to that for undamaged material. It is also seen that damage has important effect on the angular distribution functions of the near tip stress and strain fields. As special cases, higher order analytical solutions of the crack and rigid flat inclusion tip fields are obtained, respectively, by reducing the notch/wedge tip solutions. Effects of damage and hardening exponents on the dominant and second order terms in the solutions of the crack and inclusion tip fields are discussed.     

Singular electro-mechanical fields near the apex of a piezoelectric bonded wedge under antiplane shear

This paper presents the explicit forms of singular electro-mechanical field in a piezoelectric bonded wedge subjected to antiplane shear loads. Based on the complex potential function associated with eigenfunction expansion method, the eigenvalue equations are derived analytically. Contrary to the anisotropic elastic bonded wedge, the results of this problem show that the singularity orders are single-root and may be complex. The stress intensity factors of electrical and mechanical fields are dependent. However, when the wedge angles are equal (α=β), the orders become real and double-root. The real stress intensity factors of electrical and mechanical fields are then independent. The angular functions have been validated when they are compared with the results of several degenerated cases in open literatures.     

A unified approach to closed-form solutions of moving heat-source problems

A closed-form model for the computation of temperature distribution in an infinitely extended isotropic body with a time-dependent moving-heat sources is discussed. The temperature solutions are presented for the sources of the forms: (i) ₀₁(t)=₀ exp(−λt), (ii) ₀₂(t) =₀(t/t ^*)exp(−λt), and ₀₃(t)=₀[1+a cos(ωt)], where λ and ω are real parameters and t ^* characterizes the limiting time. The reduced (or dimensionless) temperature solutions are presented in terms of the generalized representation of an incomplete gamma function Γ(α,x;b) and its decomposition C _Γ and S _Γ. The solutions are presented for moving, -point, -line, and -plane heat sources. It is also demonstrated that the present analysis covers the classical temperature solutions of a constant strength source under quasi-steady state situations. Received on 13 June 1997     

On the singularities of the thermo-electro-elastic fields near the apex of a piezoelectric bonded wedge

Based on the generalized Lekhnitskii formulation and Mellin transform, the thermo-electro-elastic fields of a piezoelectric bonded wedge are investigated in this paper. From the potential theory in a wedge-shaped region, a general form of the temperature change is proposed as a particular solution in the generalized Lekhnitskii formulation. The emphasis is on the singular behavior near the apex of the piezoelectric bonded wedge, including singularity orders and angular functions, which can be computed numerically. The interface between two materials can be either perfectly bonded, namely type A, so that the continuity of electric displacements holds, or a thin electrode, namely type B, so that the electric potential is grounded. Case studies of PZT-5H/PZT-4 and graphite-epoxy/PZT-4 bonded wedges reveal that, in most cases, the type B continuity condition has more severe singularities than type A due to the mixed boundary point of the electrostatics at the apex of the wedge. The results of this study show that the reduction or disappearance of singularity orders is possible through the appropriate selection of poling/fiber orientations and wedge angles.     

Logarithmic singularity of an elastic composite wedge subjected to out-of-the-plane extensional strain

When an elastic composite wedge is not under a plane strain deformation, an out-of-the-plane extensional strain exists. The singularity analysis for the stresses at the apex of the composite wedge reduces to a system of non-homogeneous linear equations. When the composite wedge consists of two anisotropic elastic materials, it is shown that the stresses have the (ln r) term for all combinations of wedge angles with few exceptions. The same is true when the materials are isotropic except that the (ln r) term may appear in the form of r(ln r) in the displacements only. For these isotropic composite wedges therefore the stresses are bounded, though not continuous, at the apex. However, there are isotropic composite wedges for which the stress singularity is logarithmic. Conditions are given for isotropic composite wedges for which the stresses are (a) uniform, (b) non-uniform but bounded and (c) logarithmic. Unlike the r^−λ singularity, the existence of the (ln r) term does not depend on the complete boundary conditions.     

The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation ${{\Delta^2} u=\frac{\lambda}{(1-u)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation D² u=\fracl(1-u)²{{\Delta^2} u=\frac{\lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball B ì \mathbbR^N{B\subset{\mathbb{R}}^N}, under Dirichlet boundary conditions u=?_n u=0{u=\partial_\nu u=0} on ?B{\partial B}. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u _λ with 0 < u _λ < 1 exists for l ? (0,l^*){\lambda\in (0,\lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution u_l^*{u_{\lambda^*}} is regular (sup_B u_l^* < 1 ){({\rm sup}_B u_{\lambda^*} <1 )} provided N \leqq 8{N \leqq 8} while u_l^*{u_{\lambda^*}} is singular (sup_B u_l^* = 1){({\rm sup}_B u_{\lambda^*} =1)} for N \geqq 9{N \geqq 9}, in which case 1-C₀|x|^4/3 \leqq u_l^* (x) \leqq 1-|x|^4/3{1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}} on the unit ball, where C₀:=(\fracl^*[`(l)])<sup>\frac</sup>13{C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}} and [`(l)]: = \frac89(N-\frac23)(N- \frac83){\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}.     

Steady rise of a small spherical gas bubble along the axis of a cylindrical pipe at high Reynolds number

Steady irrotational flow of inviscid liquid of density ρ_l around a spherical gas bubble which lies on the axis of a cylindrical pipe is investigated using the analysis of Smythe (Phys. Fluids 4 (1961) 756). The bubble radius b=qa is assumed small compared to the pipe radius a, and the interfacial tension between gas and liquid is γ. Far from the bubble, in the frame in which the bubble is at rest, the liquid velocity along the pipe is v₀, whereas the liquid velocity at points on the wall closest to the bubble is U_zw=v₀(1+1.776q³+⋯). The decrease in wall pressure as the bubble passes is therefore Δp=1.776ρ_lv₀²q³. When the Weber number W=2bv₀²ρ_l/γ is small, the bubble deforms into an oblate spheroid with aspect ratio χ=1+9W(1+1.59q³)/64. If the fluid viscosity μ is non-zero, and the Reynolds number Re=2v₀ρ_lb/μ is large, a viscous boundary layer develops on the walls of the pipe. This decays algebraically with distance downstream of the bubble, and an exponentially decaying similarity solution is found upstream. The drag D on the bubble is D=12πμv₀b(1−2.21Re^−1/2)(1+1.59q³)+7.66μv₀bRe^1/2q^9/2, larger than that given by Moore (J. Fluid Mech. 16 (1963) 161) for motion in unbounded fluid. At high Reynolds numbers the dissipation within the viscous boundary layers might dominate dissipation in the potential flow away from the pipe walls, but such high Reynolds numbers would not be achieved by a spherical air bubble rising in clean water under terrestrial gravity.     

复合材料切口应力奇性指数计算

提出一种计算广义平面应交状态下复合材料切口应力奇性指数的新方法.在切口尖端的位移幂级数渐近展开式被引入正交各向异性材料的物理方程后,将用位移表示的应力分量代入切口端部柱状邻域的线弹性理论控制方程,切口应力奇性指数的计算被转化为常微分方程组特征值的求解.采用插值矩阵法求解该常微分方程组,可一次性地获取切口尖端多阶应力奇性指数.本法适合平面和反平面应力场耦合或解耦的情形,并可退化计算裂纹或各向同性材料切口的应力奇性指数.算例表明,所提方法对分析复合材料切口应力奇性指数是一种准确有效的手段.     

SINGULAR PERTURBATIONS FOR A CLASS OF BOUNDARY VALUE PROBLEMS OF HIGHER ORDER NONLINEAR DIFFERENTIAL EQUATIONS

ＳＩＮＧＵＬＡＲ　ＰＥＲＴＵＲＢＡＴＩＯＮＳ　ＦＯＲ　Ａ　ＣＬＡＳＳ　ＯＦ　ＢＯＵＮＤＡＲＹ　ＶＡＬＵＥ　ＰＲＯＢＬＥＭＳ　ＯＦ　ＨＩＧＨＥＲ　ＯＲＤＥＲ　ＮＯＮＬＩＮＥＡＲ　ＤＩＦＦＥＲＥＮＴＩＡＬ　ＥＱＵＡＴＩＯＮＳＳｈｉＹｕａｎｉｎｇ（史玉明）...     

Analysis of an Anisotropic Finite Wedge under Antiplane Deformation

The antiplane deformation of an anisotropic wedge with finite radius is considered in this paper within the classical linear theory of elasticity. The traction-free condition is imposed on the circular segment of the wedge. Three different cases of boundary conditions on the radial edges are considered, which are: traction-displacement, displacement-displacement and traction-traction. The solution to the governing differential equation of the problem is accomplished in the complex plane by relating the displacement field to a complex function. Several complex transformations are defined on this complex function and its first and second derivatives to formulate the problem in each of the three cases of the problem corresponding to the radial boundary conditions, separately. These transformations are then related to integral transforms which are complex analogies to the standard finite Mellin transforms of the first and second kinds. Closed form expressions are obtained for the displacement and stress fields in the entire domain. In all cases, explicit expressions for the strength of singularity are derived. These expressions show the dependence of the order of stress singularity on the wedge angle and material constants. In the displacement-displacement case, depending upon the applied displacement, a new type of stress singularity has been observed at the wedge apex. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the crack-tip stress singularity of interfacial cracks in transversely isotropic piezoelectric bimaterials

In this paper, characteristics of the interface crack-tip stress and electric displacement fields in transversely isotropic piezoelectric bimaterials are studied. The authors have proven, within the framework of the generalized Stroh formalism for piezoelectric bimaterials, that there is no coexistence of the parameters (oscillating) and κ (non-oscillating) in the interface crack-tip generalized stress field for all transversely isotropic piezoelectric bimaterials. This leads to the classification of piezoelectric bimaterials into one group that exhibits the oscillating property in the interface crack-tip generalized stress field and the other that does not. Fifteen (15) pair-combinations of six (6) piezoelectric materials PZT-4, PZT-5H, PZT-6B, PZT-7A, P-7, and BaTiO₃, which are commonly used in practice, are numerically analyzed in this study, and the results backup the above theoretical conclusions. Moreover, the associated eigenvectors for such material systems (with either =0 or κ=0) are also obtained numerically, and the result show that there still exist four linear independent associate eigenvectors for each bimaterial.     

Decay of Almost Periodic Solutions¶of Conservation Laws

We consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables, which are almost periodic in the space variables in a generalized sense introduced by Stepanoff and Wiener, which extends the original one of H. Bohr. We prove that if u(x,t) is such a solution whose inclusion intervals at time t, with respect to ?>0, satisfy l _epsiv;(t)/t→0 as t→∞, and such that the scaling sequence u ^T (x,t)=u(T x,T t) is pre-compact as t→∞ in L _loc ¹(? ^{d +1} ₊, then u(x,t) decays to its mean value \(\), which is independent of t, as t→∞. The decay considered here is in L ¹ _loc of the variable ξ≡x/t, which implies, as we show, that \(\) as t→∞, where M _x denotes taking the mean value with respect to x. In many cases we show that, if the initial data are almost periodic in the generalized sense, then so also are the solutions. We also show, in these cases, how to reduce the condition on the growth of the inclusion intervals l _?(t) with t, as t→∞, for fixed ? > 0, to a condition on the growth of l _?(0) with ?, as ?→ 0, which amounts to imposing restrictions only on the initial data. We show with a simple example the existence of almost periodic (non-periodic) functions whose inclusion intervals satisfy any prescribed growth condition as ?→ 0. The applications given here include inviscid and viscous scalar conservation laws in several space variables, some inviscid systems in chromatography and isentropic gas dynamics, as well as many viscous 2 × 2 systems such as those of nonlinear elasticity and Eulerian isentropic gas dynamics, with artificial viscosity, among others. In the case of the inviscid scalar equations and chromatography systems, the class of initial data for which decay results are proved includes, in particular, the L ^∞ generalized limit periodic functions. Our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws.     

GREEN''''S FUNCTIONS FOR 2D PROBLEMS OF ANISOTROPIC PIEZOELECTRIC MATERIAL WITH A STRIP REGION

By using Stroh's formalism and the conformal mapping technique, we derive the simple explicit elastic fields of a generalized line dislocation and a generalized line force in a general anisotropic piezoelectric strip with fixed surfaces, which are two fixed conductor electrodes. The solutions obtained are usually considered as Green's functions which play important roles in the boundary element methods. The Coulomb forces of the distributed charges along the region boundaries on the line chargeq atz ⁰ are analysed in detail. The results are valid not only for plane and antiplane problems but also for the coupled problems between inplane and outplane deformations.     

Separation zones in the flow near a small-angle convex corner

On the basis of an asymptotic analysis of the Navier-Stokes system of equations for large Reynolds numbers (Re → ∞), the plane incompressible fluid flow near a surface having a convex corner with a small angle 2θ* is investigated. It is shown that for θ* = O(Re^?1/4), in addition to the known solution that describes a separated flow completely localized in a thin “viscous” sublayer of the interaction region near the corner point, another solution corresponding to a flow with a developed separation zone is possible. For θ ₀ = Re^1/4 θ* = O(1), the longitudinal dimension of this zone varies from finite values up to values of the order of Re^?3/8. The nonuniqueness of the solution is established on a certain range of variation of the parameter θ ₀. The dependence of the drag coefficient on the angle θ* is found.     

Green's functions for a piezoelectric semi-infinite body with a fixed conductor surface

By using Stroh's formalism, simple explicit compact expressions of Green's functions for a piezoelectric semi-infinite body, with a fixed conductor surface electrode, subject to a singularity (i.e., a generalized line dislocation and a generalized line force at a point z°) are presented. Coulomb forces acted on the free line charge at z° due to the boundary polarization charges of the medium and the induction charges of the conductor together with the electromechanical coupling effects inside the region are analyzed in detail. The obtained results are valid not only for plane and antiplane problems but also for the coupled problems between inplane and outplane deformations.     

A stress analytical solution for Mode III crack within modified gradient elasticity

The strain gradient exists near a crack tip may significantly influence the near-tip stress field. In this paper, the strain gradient and the internal length scales are introduced into the basic equations of mode III crack by the modified gradient elasticity (MGE). By using a complex function approach, the analytical solution of stress fields for mode III crack problem is derived within MGE. When the internal length scales vanish, the stress fields can be simplified to the stress fields of classical linear elastic fracture mechanics. The results show that the singularity of the shear stress is made up of two parts, r^−1/2 part and r^−3/2 part, and the sign of the stress σ_yz changes. With the increase of l_x, the peak value of σ_yz decrease and its location moves farther from the fracture vertex. The influence of strain gradient for mode III crack problem cannot be ignored.     

Electrostatic stress analysis of an anisotropic piezoelectric half-plane under surface electromechanical loading

The generalized stress components on an anisotropic piezoelectric half-plane boundary under surface electromechanical loading are investigated. It is found that the behaviors of generalized stress components are related to matrices Γ and Ω, which have the same form as those for the purely elastostatic problem. Matrices Γ and Ω contain all the electro-mechanical coupling phenomena of the generalized stress components. All elements of matrices Γ and Ω are expressed explicitly in terms of generalized elastic stiffness for monoclinic piezoelectric materials with the plane of symmetry at x₃ = 0 and for transversely isotropic piezoelectric materials in which the coupled effects between the mechanical (electrical) deformations induced by electrical (mechanical) loadings are studied analytically. A numerical example of the electro-mechanical coupling behavior for PZT-4 is also given.     

Non-piezoactivity in piezoacoustics: basic properties and topological features

The existence conditions of zero electric fields E and zero electric displacements D are studied for bulk acoustic waves in piezoelectric crystals. General equations are derived for lines of zero electric fields, E(m)=0, and for specific points m ₀ of vanishing electric displacements, D(m ₀)=0, on the unit sphere of propagation directions m ²=1. The obtained equations are solved for a series of examples of particular crystal symmetry. It is shown that the vectors D _α (m) being generally orthogonal to the wave normal m are characterized by definite orientational singularities in the vicinity of m ₀ and can be described by the Poincaré indices n=0, ±1 or ±2. The algebraic expressions for the indices n are found both for unrestricted anisotropy and for a series of particular cases.