Orientation-dependent piezoelectric Eshelby S-tensor for a lamellar structure in a transversely isotropic medium

Summary.  This paper provides explicit expressions for the orientation-dependent piezoelectric Eshelby S-tensor with a lamellar structure in a transversely isotropic medium. Many piezoelectric or ferroelectric materials prefer to form a lamellar structure during a change of nano-scale microstructure and domain switch; such a lamellar morphology is energetically more favorable than other shapes in a diffusionless process. According to the above observation, this paper seeks to derive the Eshelby tensor in a coupled field with the consideration of lamellar structure. In general, Eshelby's tensor can be obtained through a Green's function technique, but it is usually given in an integral form. Due to the geometric simplicity of a lamellar structure, a simple explicit form of piezoelectric S-tensor can be obtained directly from the interfacial continuous conditions. In a transversely isotropic medium, the angle between the global symmetric axis and one of the local parallel to the surface of the lamellar inclusion is the key geometrical factor in the construction of the S-tensor. Two special cases with 90° and 180° orientation that bear significant relevance to domain switch in ferroelectric ceramics are discussed in detail. Received September 2, 2002; revised December 3, 2002 Published online: May 8, 2003 This work was supported by the National Science Foundation, Surface Engineering and Materials Design Program, under grant CMS-0093808.     

Axisymmetric crack problem for a hollow cylinder imbedded in a dissimilar medium

The analytical solution for the linear elastic problem of flat annular crack in a transversely isotropic hollow cylinder imbedded in a transversely isotropic medium is considered. The hollow cylinder is assumed to be perfectly bonded to the surrounding medium. This structure, which can represent a cylindrical coating-substrate system, is subjected to uniform crack surface pressure. Because of the geometry and the loading, the problem is axisymmetric. The z = 0 plane on which the crack lies, is also a plane of symmetry. The composite media consisting of the hollow cylinder and the surrounding medium extends to infinity in z and r directions. The mixed boundary value problem is formulated in terms of the unknown derivative of the crack surface displacement by using Fourier and Hankel transforms. By extending the crack to the inner surface and to the interface, the cases of surface crack and crack terminating at the interface are obtained. Asymptotic analyses are performed to derive the generalized Cauchy kernel and associated stress singularities. The resulting singular integral equation is solved numerically. Stress intensity factors for various crack configurations, crack opening displacements and stresses along the interface and on z = 0 plane are presented for sample material combinations and geometric parameters.     

On the axisymmetric Mindlin's problem for a semi-space of granular material

Summary The methods of images and Hankel transforms are used to construct solution to an axisymmetric boundary value problem of a semi-space of transversely isotropic (granular) material due to a point force applied at a distanceh beneath its stress free plane boundaryz=0. Exact closed form expressions are determined for the components of displacements and stresses throughout the interior of the granular semi-space. The solution is then used to derive the surface displacements due to a uniformly distributed force over a circle of radius a with centre at (0, 0, –h) in the planez=–h of the semi-space. By a suitable choice of material constants and through a limit process as ₁, ₂ approach 1, the granular semi-space becomes isotropic and the corresponding results derived in this particular case agree with those presented in [14].With 2 Figures     

Edge cracks in a transversely isotropic hollow cylinder

The analytical solution for the linear elastic, axisymmetric problem of inner and outer edge cracks in a transversely isotropic infinitely long hollow cylinder is considered. The z = 0 plane on which the crack lies is a plane of symmetry. The loading is uniform crack surface pressure. The mixed boundary value problem is reduced to a singular integral equation where the unknown is the derivative of the crack surface displacement. An asymptotic analysis is done to derive the generalized Cauchy kernel associated with edge cracks. It is shown that the stress intensity factor is a function of three material parameters. The singular integral equation is solved numerically. Stress intensity factors are presented for various values of material and geometric parameters.     

3D point force solution for a permeable penny-shaped crack embedded in an infinite transversely isotropic piezoelectric medium

This paper considers the non-axisymmetric three-dimensional problem of a penny-shaped crack with permeable electric conditions imposed on the crack surfaces, subjected to a pair of point normal forces applied symmetrically with respect to the crack plane. The crack is embedded in an infinite transversely isotropic piezoelectric body with the crack face perpendicular to the axis of material symmetry. Applying the symmetry of the problem under consideration then leads to a mixed–mixed boundary value problem of a half-space, for which potential theory method is employed for the purpose of analysis. The cases of equal eigenvalues are also discussed. Although the treatment differs from that for an impermeable crack reported in literature, the resulting governing equation still has a familiar structure. For the case of a point force, exact expressions for the full-space electro-elastic field are derived in terms of elementary functions with explicit stress and electric displacement intensity factors presented. The exact solution for a uniform loading is also given.     

Axisymmetric elasticity solutions for a uniformly loaded annular plate of transversely isotropic functionally graded materials

Summary The axisymmetric problem of a functionally graded, transversely isotropic, annular plate subject to a uniform transverse load is considered. A direct displacement method is developed that the non-zero displacement components are expressed in terms of suitable combinations of power and logarithmic functions of r, the radial coordinate, with coefficients being undetermined functions of z, the axial coordinate. The governing equations as well as the corresponding boundary conditions for the undetermined functions are deduced from the equilibrium equations and the boundary conditions of the annular plate, respectively. Through a step-by-step integration scheme along with the consideration of boundary conditions at the upper and lower surfaces, the z-dependent functions are determined in explicit form, and certain integral constants are then determined completely from the remaining boundary conditions. Thus, analytical elasticity solutions for the plate with different cylindrical boundary conditions are presented. As a promising feature, the developed method is applicable when the five material constants of a transversely isotropic material vary along the thickness arbitrarily and independently. A numerical example is finally given to show the effect of the material inhomogeneity on the elastic field in the annular plate.     

Thermomechanical analysis of elastic–plastic fibrous composites comprising an inhomogeneous interphase

An analytical approach is presented to investigate thermomechanical response of composites consisting of a transversely isotropic fiber, an inhomogeneous interphase and an elastic–plastic matrix. Using the existing cubic variation to describe the continuous change of the material properties of the interphase and dividing the interphase into a number of subdomains, the continuously varying material properties of the interphase are approximated by the constant ones of these subdomains, and the deformations and stresses of the interphase are described with the same formulae as those of transversely isotropic fibers. The analytical expressions of elastic–plastic deformations and stresses of the matrix are obtained from the basic equations of axisymmetric problems in elasticity, the assumption of generalized plane strain, the linear strain–hardening stress–plastic strain relation, Tresca’s yield condition, the associated flow rule and impressibility of plastic deformation. The boundary conditions of the composites and the continuities of the radial displacement and stress between different components are used to determine all the unknown constants and the obtained analytical solution is applied to thermomechanical analysis of the composites. The effects of the inhomogeneity of the interphase, and the plasticity and material properties of the matrix on the thermomechanical response of the composites are investigated.     

Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media Part I: Hypersingular integral equation and theoretical analysis

This paper deals with some basic linear elastic fracture problems for an arbitrary-shaped planar crack in a three-dimensional infinite transversely isotropic piezoelectric media. The finite-part integral concept is used to derive hypersingular integral equations for the crack from the point force and charge solutions with distinct eigenvalues s _i(i=1,2,3) of an infinite transversely isotropic piezoelectric media. Investigations on the singularities and the singular stress fields and electric displacement fields in the vicinity of the crack are made by the dominant-part analysis of the two-dimensional integrals. Thereafter the stress and electric displacement intensity factor K-fields and the energy release rate G are exactly obtained by using the definitions of stress and electric displacement intensity factors and the principle of virtual work, respectively. The hypersingular integral equations under axially symmetric mechanical and electric loadings are solved analytically for the case of a penny-shaped crack.     

Interface crack problem in elastic dielectric/piezoelectric bimaterials

By considering an isotropic elastic dielectric material as a transversely isotropic piezoelectric material with little piezoelectricity, the interface crack problem in elastic/piezoelectric bimaterials is treated in this paper based on Stroh's complex potential theory (1958) with the impermeable crack model. In order to obtain universal results, Numerical results of the near tip stress field and the electric field for 35 kinds of dissimilar bimaterials constructed by five kinds of elastic dielectric materials, namely Epoxy, Polymer, Al₂O₃, SiC and Si₃N₄, and seven kinds of piezoelectric ceramics, namely PZT-4, BaTiO₃, PZT-5H, PZT-6B, PZT-7A, P-7, and PZT-PIC151, are presented. It is concluded that all the combinations lead to the same results: in which the first crack tip singularity parameter does not vanish whereas the second parameter always vanishes. From the physical point of view, an interface crack in such a dissimilar material shows a similar oscillating singularity as that in dissimilar elastic bimaterials. Moreover, by using a maximization value technique, the regular inverse square root singularity r ^–1/2 of the stress and the electric field at the crack tip can be realized, although, theoretically, an interface crack in such bimaterials possesses the well-known oscillating singularity r ^{–1/2± i}. Of great significance is that, in the absence of mechanical loadings, a purely electric loading can induce relative large model I or II stress intensity factor for a interface crack in some elastic/piezoelectric bimaterials, which implies that the electric-induced failure may be realized in such bimaterials.     

Torsional loading of an elastic transversely isotropic nonhomogeneous semi-infinite solid

Summary The problem of torsion of a transversely isotropic nonhomogeneous and elastic semiinfinite solid due to certain distributions of shearing forces prescribed on the plane boundary has been considered. The elastic properties of the material are continuous function of position. The shear moduli of the material are chosen asC _ii (z) = _ii cosh² (kz), i=4, 6 where _ii andk are constants. The aim of this paper is to determine the torsional deformation and shear stresses in the semi-infinite solid.With 1 Figure     

Unique real‐variable expressions of displacement and traction fundamental solutions covering all transversely isotropic elastic materials for 3D BEM

A general, efficient and robust boundary element method (BEM) formulation for the numerical solution of three‐dimensional linear elastic problems in transversely isotropic solids is developed in the present work. The BEM formulation is based on the closed‐form real‐variable expressions of the fundamental solution in displacements U_ik and in tractions T_ik, originated by a unit point force, valid for any combination of material properties and for any orientation of the radius vector between the source and field points. A compact expression of this kind for U_ik was introduced by Ting and Lee (Q. J. Mech. Appl. Math. 1997; 50 :407–426) in terms of the Stroh eigenvalues on the oblique plane normal to the radius vector. Working from this expression of U_ik, and after a revision of their final formula, a new approach (based on the application of the rotational symmetry of the material) for deducing the derivative kernel U_{ik, j} and the corresponding stress kernel Σ_ijk and traction kernel T_ik has been developed in the present work. These expressions of U_ik, U_{ik, j}, Σ_ijk and T_ik do not suffer from the difficulties of some previous expressions, obtained by other authors in different ways, with complex‐valued functions appearing for some combinations of material parameters and/or with division by zero for the radius vector at the rotational‐symmetry axis. The expressions of U_ik, U_{ik, j}, Σ_ijk and T_ik have been presented in a form suitable for an efficient computational implementation. The correctness of these expressions and of their implementation in a three‐dimensional collocational BEM code has been tested numerically by solving problems with known analytical solutions for different classes of transversely isotropic materials. Copyright © 2007 John Wiley & Sons, Ltd.     

Stability analysis of an explicit finite element scheme for plane wave motions in elastic solids

 Expressions for critical timesteps are provided for an explicit finite element method for plane elastodynamic problems in isotropic, linear elastic solids. Both 4-node and 8-node quadrilateral elements are considered. The method involves solving for the eigenvalues directly from the eigenvalue problem at the element level. The characteristic polynomial is of order 8 for 4-node elements and 16 for 8-node elements. Due to the complexity of these equations, direct solution of these polynomials had not been attempted previously. The commonly used critical time-step estimates in the literature were obtained by reducing the characteristic equation for 4-node elements to a second-order equation involving only the normal strain modes of deformation. Furthermore, the results appear to be valid only for lumped-mass 4-node elements. In this paper, the characteristic equations are solved directly for the eigenvalues using <ty>Mathematica<ty> and critical time-step estimates are provided for both lumped and consistent mass matrix formulations. For lumped-mass method, both full and reduced integration are considered. In each case, the natural modes of deformation are obtained and it is shown that when Poisson's ratio is below a certain transition value, either shear-mode or hourglass mode of deformation dominates depending on the formulation. And when Poisson's ratio is above the transition value, in all the cases, the uniform normal strain mode dominates. Consequently, depending on Poisson's ratio the critical time-step also assumes two different expressions. The approach used in this work also has a definite pedagogical merit as the same approach is used in obtaining time-step estimates for simpler problems such as rod and beam elements. Received: 8 January 2002 / Accepted: 12 July 2002 The support of NSF under grant number DMI-9820880 is gratefully acknowledged.     

Eshelby inclusion of arbitrary shape in an anisotropic plane or half-plane

Summary.  Analytical solution for Eshelby's problem of an anisotropic non-elliptical inclusion remains a challenging problem. In this paper, a simple method is presented to obtain an analytical solution for Eshelby's problem of an inclusion of arbitrary shape within an anisotropic plane or half-plane of the same elastic constants. The method is based on an observation that the interface conditions for arbitrary inclusion-shape can be written in a decoupled form in which three unknown Stroh's functions are decoupled from each other. The solution is given in terms of three auxiliary functions constructed by three conformal mappings which map the exteriors of three image curves of the inclusion boundary, defined by three Stroh's variables, onto the exterior of the unit circle. With aid of these auxiliary functions, techniques of analytical continuation can be applied to the inclusion of any shape. The solution is given in the physical plane rather than in the image plane, and is exact provided that the expansion of every mapping function includes only a finite number of terms. On the other hand, if at least one of the mapping functions includes infinite terms, a truncated polynomial mapping should be used, and thus the method gives an approximate solution. A remarkable feature of the present method is that it gives elementary expressions for the internal stress field within an inclusion in an anisotropic entire plane. Elliptical and polygonal inclusions are used to illustrate the construction of the auxiliary functions and the details of the method. Received February 18, 2002; revised July 22, 2002 Published online: February 10, 2003 Acknowledgement The financial support of the Natural Science and Engineering Research Council of Canada is gratefully acknowledged.     

Tangential contact problem for a transversely isotropic elastic layer bonded to an elastic foundation

A system consisting of an elastic layer made of a transversely isotropic material bonded to an elastic half-space made of a different transversely isotropic material is considered. An arbitrary tangential displacement is prescribed over a domain S of the layer, while the rest of the layer’s surface is stress-free. The tangential contact problem consists of finding a complete field of stresses and displacements in this system. The generalized-images method developed by the author is used to get an elementary solution to the problem. It is also shown that an integral transform can be interpreted as a sum of generalized images. The case of a circular domain of contact is considered in detail. The results are valid for the case of isotropy as well.     

Stresses in a constrained transversely isotropic elastic solid caused by a moving dislocation

Summary An unbounded, transversely isotropic elastic solid whose elastic constants obey the constraint (c ₁₁–c ₄₄) (c ₃₃–c ₄₄)=(c ₁₃+c ₄₄)² is excited by a suddenly applied moving dislocation. Explicit expressions are obtained for the plane stress field induced by the (constant speed) dislocation moving parallel to the material symmetry axis. The speed of the dislocation may be sub-, tran-, or super-sonic with respect to the material wave speeds. An unexpected result is the existence of a special dislocation speed, in the transonic range, which causes the Mach head wave to be annihilated.With 2 Figures     

Piezoelectric study for a coated hole of quasi-polygonal shape in an infinite plate

This paper studies the piezoelectric problems for a coated hole of quasi-polygonal shape embedded in an infinite matrix subjected to electromechanical loadings. The electromechanical loadings considered in this work include a screw dislocation, a point force, a point charge, a far-field anti-plane shear and an in-plane electric field. Each component is assumed to be transversely isotropic medium belonging to a hexagonal crystal class 6 mm and poled in the x₃ direction. Based on the complex variable and analytical continuation method, the general expressions for the complex potentials can be derived in each medium. Numerical results are provided to show the effect of hole shape, the material combinations and the loading condition on the electro-elastic fields due to the presence of the coated film. The image force exerted on a dislocation, which can be used to probe the mobility of the screw dislocation, will be calculated by means of the generalized Peach-Koehler formula.     

Modelling of anisotropic elastic properties in alloy 718 built by electron beam melting

ABSTRACT

Owing to the inherent nature of the process, typically material produced via electron beam melting (EBM) has a columnar microstructure. As a result of that, the material will have anisotropic mechanical properties. In this work, anisotropic elastic properties of EBM built Alloy 718 samples at room temperature were investigated by using experiments and modelling work. Electron backscatter diffraction data from the sample microstructure was used to predict the Young’s modulus. The results showed that the model developed in the finite element software OOF2 was able to capture the anisotropy in the Young’s modulus. The samples showed transversely isotropic elastic properties having lowest Young’s modulus along build direction. In addition to that, complete transversely isotropic stiffness tensor of the sample was also calculated.

This paper is part of a thematic issue on Nuclear Materials.     

Reflection of piezothermoelastic waves from the charge and stress free boundary of a transversely isotropic half space

The present paper concentrates on the study of propagation and reflection characteristics of waves from the stress free, thermally insulated/isothermal boundary of a piezothermoelastic half space. The non-classical (generalized) theories of linear piezo-thermoelasticity have been employed to investigate the problem. In the two-dimensional model of the transversely isotropic piezothermoelastic medium, there are three types of plane waves quasi-longitudinal (QL), quasi-transverse (QT) and thermal wave (T-mode), whose velocities depend on the angle of incidence and frequency. These waves are dispersive in character and are also affected by piezoelectric as well as pyroelectric properties of the materials. The low and high frequency approximations for the speeds of propagation and the attenuation coefficients of these waves have been obtained. The quasi-longitudinal (QL), quasi-transverse (QT) and thermal wave (T-mode) incident cases at the stress free, thermally insulated or isothermal open circuit boundary of a transversely isotropic piezothermoelastic half space are considered to discuss the reflection characteristics of various waves. The amplitude ratios of reflected waves to that of incident one in each case have been obtained. The special cases of normal and grazing incidence are also derived and discussed. Finally, the numerical computations of reflection coefficients are carried out for cadmium Selenide (CdSe) material by using Gauss elimination procedure. In addition the phase velocities and attenuation coefficients are also computed along various directions of wave propagation. The obtained results in each case are presented graphically.     

Spherical indentation of a transversely isotropic elastic half-space reinforced with a thin layer

The frictionless spherical indentation test is considered for a transversely isotropic elastic half-space reinforced with a thin layer whose flexural stiffness is negligible compared to its tensile stiffness. It is assumed that the deformation of the reinforcing layer can be treated as the generalized plane stress state. Closed-form analytical approximate equations for the maximum contact pressure, contact radius, and contact force are presented. The isotropic case is considered in detail.     

Displacement fields around 1/2[111] screw dislocations in bcc metals and their defocus convergent-beam electron diffraction patterns

Displacement fields have been calculated around 1/2[111] screw dislocations with various types of core structures in bcc metals. Three types of cores are studied: two types of polarized cores with large and small extensions of the displacement fields and an isotropic core. The difference in the displacement along the [111] direction Δu_z from that for the elastic solution has been evaluated for each type of dislocation. In the outside of the core region, the Δu_z values are close to zero along the six directions and the regions with Δu_z>0 and Δu_z<0 are alternately arranged, lying between those directions. Appreciable difference in Δu_z has been detected between the polarized cores and the isotropic core up to large distance from the core region. The defocus convergent-beam electron diffraction patterns have been calculated for the dislocations with the incident beam parallel to the dislocation line. Winding and spiral features have been shown in the higher-order Laue zone (HOLZ) lines for the dislocated structures, which have been confirmed by a preliminary experiment. In addition, small shifts of the HOLZ lines have been shown by the calculation between the polarized cores and the isotropic one.