Surface effects in the deformation of an anisotropic elastic material with nano-sized elliptical hole

We examine the effect of surface energy on an anisotropic elastic material weakened by an elliptical hole. A closed-form, full-field solution is derived using the standard Stroh formalism. In particular, explicit expressions for the hoop stress, normal, in-plane tangential and out-of-plane displacement components along the edge of the hole are obtained. These expressions clearly demonstrate the effect of elastic anisotropy of the bulk material on the corresponding field variables. When the material becomes isotropic, the hoop stress agrees with the well-known result in the literature while both the in-plane tangential and out-of-plane displacements vanish and the normal displacement is constant along the entire boundary of the elliptical hole.     

FURTHER STUDIES ON STROH-TYPE FORMALISMS FOR ANISOTROPIC PLATES WITH BENDING-EXTENSION COUPLING

Stroh-type formalisms for anisotropic thin plates in literature are reviewed and dis- cussed,and two kinds of hybrid Stroh-type formalisms are compared.It is seen that the two Stroh- type formalisms are essentially equivalent.With simple transfer relations,they can be expressed each other.In addition,with properly defined notation systems,the two Stroh-type formalisms can also be written in unified forms,which will be convenient in applications.     

Edge waves in asymmetrically laminated plates

We study edge waves propagating along the edge of an asymmetrically laminated elastic plate for which the out-of-plane component of displacement is coupled with the in-plane components. A Stroh-like formulation is used to show that such a plate can support at most two edge waves. An efficient method for computing the edge-wave speeds is proposed and explained through examples.     

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.     

Transient Waves on the Free Surface of a Viscoelastic Anisotropic Half-Space

The Stroh formalism is employed to discuss the existence of transient surface waves on a viscoelastic anisotropic hall-space. The compatibility conditions, obtained using the integral formulation of Lothe and Barnett [13, 14], are examined on the basis of an asymptotic expansion of the viscoelastic kernel and a separation of space variables. Some previous results on elastic media are extended to viscoelasticity, exploiting the consequences of the second law of thermodynamics. It is found that all the allowed transient surface modes take the form of inhomogeneous plane waves whose amplitude exponentially decays along the propagation direction on the surface. Special solutions are derived explicitly for one-component surface waves where transient modes are admitted also in those cases in which stationary waves cannot occur. Mathematics Subject Classifications (2000) 74D05, 74J15.     

Green's function in plane anisotropic bimaterials with imperfect interface

^** Email: lsudak{at}ucalgary.ca The fundamental solutions or Green's functions for 2D or 3Danisotropic media with imperfect interface remain a challengingproblem. In this paper, a general method is presented for therigorous solution for the 2D Green's function in an anisotropicelastic bimaterial subject to a line force or a line dislocation.Most significant is the fact that the bonding along the bimaterialinterface is considered to be homogeneous imperfect. Specifically,the tractions are continuous but the displacements are discontinuousand proportional, in terms of interface stiffness parameters,to their respective traction components. Using complex variabletechniques, the basic boundary-value problem for two analyticvector functions is reduced to a coupled linear first-orderdifferential equation for a single analytic vector functiondefined in the lower half space. The coupled linear differentialequation for the single analytic vector function can be subsequentlydecoupled into three independent linear first-order differentialequations for three newly defined analytic functions. Closed-formsolutions for the 2D Green's function are derived in terms ofthe exponential integral. Unlike previous works which involvesome sort of inverse transform method to obtain the physicalquantities from the transform domain, the key feature of thepresent method is that the physical quantities can be readilycalculated without the need to perform any inverse transformoperations.     

Perturbation Formula for Phase Velocity of Rayleigh Waves in Prestressed Anisotropic Media

Herein we consider Rayleigh waves propagating along the free surface of a macroscopically homogeneous, anisotropic, prestressed half-space. We adopt the formulation of linear elasticity with initial stress and assume that the deviation of the prestressed anisotropic medium from a comparative ‘unperturbed’, unstressed and isotropic state, as formally caused by the initial stress and by the anisotropic part of the incremental elasticity tensor, be small. No assumption, however, is made on the material anisotropy of the incremental elasticity tensor. With the help of the Stroh formalism, we derive a first-order perturbation formula for the shift of phase velocity of Rayleigh waves from its comparative isotropic value. Our perturbation formula does not agree totally with that which was derived some years ago by Delsanto and Clark, and we provide another argument as further support for our version of the formula. According to our first-order formula, the anisotropy-induced velocity shifts of Rayleigh waves, taken in totality of all propagation directions on the free surface, carry information only on 13 elastic constants of the anisotropic part of the incremental elasticity tensor. The remaining eight elastic constants are those which would become zero if were monoclinic with the two-fold symmetry axis normal to the free surface of the material half-space in question.     

具有表面效应的压电半空间中的表面波

纳米科技的快速发展使压电纳米结构在纳米机电系统中得到广泛应用,形成了诸如纳米压电电子学等新的研究方向.与传统的宏观压电材料相比,在纳米尺度下压电材料往往呈现出不同的力学特性,而造成这种差异的原因之一便是表面效应.本文基于Stroh公式、Barnett-Lothe积分矩阵及表面阻抗矩阵,研究计入表面效应的任意各向异性压电半空间中的表面波传播问题,导出了频散方程.针对横观各向同性压电材料,假设矢状平面平行于材料各向同性面,发现Rayleigh表面波和B-G波解耦,并得到各自的显式频散方程.结果表明,Rayleigh表面波的波速小于偏振方向垂直于表面的体波,而B-G波的波速小于偏振方向垂直于矢状平面的体波.以PZT-5H材料为例,用数值方法考察表面残余应力和电学边界条件对表面波频散特性的影响发现:表面残余应力只对第一阶Rayleigh波有明显的影响;电学开路情形的B-G波比电学闭路情形的B-G波传播快.本文工作可为纳米表面声波器件的设计或压电纳米结构的无损检测提供理论依据.     

Stroh formalism in evaluation of 3D Green’s function in thermomagnetoelectroelastic anisotropic medium

The paper presents studies on the Green’s function for thermomagnetoelectroelastic medium and its reduction to the contour integral. Based on the previous studies the thermomagnetoelectroelastic Green’s function is presented as a surface integral over a half-sphere. The latter is then reduced to the double integral, which inner integral is evaluated explicitly using the complex variable calculus and the Stroh formalism. Thus, the Green’s function is reduced to the contour integral. Since the latter is evaluated over the period of the integrand, the paper proposes to use trapezoid rule for its numerical evaluation with exponential convergence. Several numerical examples are presented, which shows efficiency of the proposed approach for evaluation of Green’s function in thermomagnetoelectroelastic anisotropic solids.     

Surface and interfacial waves in anisotropic elastic quasicrystals

We present the Stroh formalism for two-dimensional subsonic steady-state motion of anisotropic quasicrystals. Using this new formalism and a series of identities and properties which follow, we investigate subsonic surface and interfacial waves in anisotropic quasicrystals. Our results suggest that there exist at most three subsonic surface wave speeds. This interesting observation is quite different from the unique surface wave speed known for anisotropic crystals. The degenerate case of decagonal quasicrystalline materials is discussed in detail.