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2D integral formulae and equations for thermoelectroelastic bimaterial with thermally insulated interface
Abstract
The paper presents a complex variable approach for obtaining of the integral formulae and integral equations for plane thermoelectroelasticity of an anisotropic bimaterial with thermally insulated interface. Obtained relations do not contain domain integrals and incorporate only physical boundary functions such as temperature, heat flux, extended displacement and traction, which are the main advances of these relations.
This paper utilizes the Stroh formalism and the complex variable approach to derive the integral formulae and boundary integral equations of anisotropic thermoelectroelasticity for a bimaterial solid with Kapitza-type interface. Obtained integral formulae and boundary integral equations do not contain domain integrals, thus, the boundary element approach based on them does not require any additional procedures accounting for the stationary temperature field acting in the solid. All kernels of the boundary integral equations are written explicitly in a closed form. Verification for limiting values of thermal resistance of the interface is provided. Obtained boundary integral equations are incorporated into the boundary element analysis procedure. Several problems are considered, which shows the influence of thermal resistance of the bimaterial interface on fields’ intensity at the tips of electrically permeable and impermeable cracks.
By means of the extended version of the Stroh formalism for uncoupled thermo-anisotropic elasticity, two-dimensional Green’s
function solutions in terms of exponential integrals are derived for the thermoelastic problem of a line heat source and a
temperature dislocation near an imperfect interface between two different anisotropic half-planes with different thermo-mechanical
properties. The imperfect interface investigated here is modeled as a generalized spring layer with vanishing thickness: (1)
the normal heat flux is continuous at the interface, whereas the temperature field undergoes a discontinuity which is proportional
to the normal heat flux; (2) the tractions are continuous across the interface, whereas the displacements undergo jumps which
are proportional to the interface tractions. This kind of imperfect interface can be termed a thermally weakly conducting
and mechanically compliant interface. In the Appendix we also present the isothermal Green’s functions in anisotropic bimaterials
with an elastically stiff interface to demonstrate the basic ingredients in the analyses of a stiff interface.
Green's Function and Boundary Elements of Multifield Materials contains a comprehensive treatment of multifield materials under coupled thermal, magnetic, electric, and mechanical loads. Its easy-to-understand text clarifies some of the most advanced techniques for deriving Green's function and the related boundary element formulation of magnetoelectroelastic materials: Radon transform, potential function approach, Fourier transform. Our hope in preparing this book is to attract interested readers and researchers to a new field that continues to provide fascinating and technologically important challenges. You will benefit from the authors' thorough coverage of general principles for each topic, followed by detailed mathematical derivation and worked examples as well as tables and figures where appropriate. * In-depth explanations of the concept of Greens function * Coupled thermo-magneto-electro-elastic analysis * Detailed mathematical derivation for Greens functions.
The paper presents a rigorous and straightforward approach for obtaining the 2D boundary integral equations for a thermoelectroelastic half-space containing holes, cracks and thin foreign inclusions. It starts from the Cauchy integral formula and Stroh orthogonality relations to obtain the integral formulae for the Stroh complex functions, which are piecewise-analytic in the complex half-plane with holes and opened mathematical cuts. Further application of the Stroh formalism allows derivation of the Somigliana type integral formulae and boundary integral equations for a thermoelectroelastic half-space. The kernels of these equations correspond to the fundamental solutions of heat transfer, electroelasticity and thermoelectroelasticity for a half-space. It is shown that the difference between the obtained fundamental solution of thermoelectroelasticity and those presented in literature is due to the fact, that present solution additionally accounts for extended displacement and stress continuity conditions, thus, it is physically correct. Obtained integral equations are introduced into the boundary element approach. Numerical examples validate derived boundary integral equations, show their efficiency and accuracy.
This paper develops Somigliana type boundary integral equations for 2D thermoelectroelasticity of anisotropic solids with cracks and thin inclusions. Two approaches for obtaining of these equations are proposed, which validate each other. Derived boundary integral equations contain domain integrals only if the body forces or distributed heat sources are present, which is advantageous comparing to the existing ones. Closed-form expressions are obtained for all kernels. A model of a thin pyroelectric inclusion is obtained, which can be also used for the analysis of solids with impermeable, permeable and semi-permeable cracks, and cracks with an imperfect thermal contact of their faces. The paper considers both finite and infinite solids. In the latter case it is proved, that in contrast with the anisotropic thermoelasticity, the uniform heat flux can produce nonzero stress and electric displacement in the unnotched pyroelectric medium due to the tertiary pyroelectric effect. Obtained boundary integral equations and inclusion models are introduced into the computational algorithm of the boundary element method. The numerical analysis of sample and new problems proved the validity of the developed approach, and allowed to obtain some new results.
The primary goal of this research is to show the fundamental features of an interface crack in metal/piezoelectric bimaterials via Stroh's theory [Phil. Mag. 7 (1958) 625]. Based on the previous works [Phil. Mag. 7 (1958) 625; J. Mech. Phys. Solids 40 (1992) 739; Int. J. Fract. 119 (2003) L41; Singularities and near-tip field intensity factors of piezoelectric interface cracks, J. Mech. Phys. Solids (in press)] and by considering a metal as a special piezoelectric material with extremely small piezoelectricity and extremely large permittivity (conductor), we obtain the two dominant parameters ε and κ for description of interface crack-tip singularity in such bimaterials. Numerical results show that almost all of such bimaterials have the feature that the first parameter ε vanishes whereas the second parameter κ remains non-zero. An interface crack in these bimaterials always possesses the stress singularity r−1/2±κ at the crack tip. From the physical point of view, this implies that an interface crack in such bimaterials shows a feature with non-oscillating singularity, which is far apart from previous results [Prik. Mat. Mekh. 39 (1975) 145; V.Z. Parton, B.A. Kudryavtsev, Electromagnetoelasticity, Gordon and Breach Science Publishers, New York, 1988; J. Mech. Phys. Solids. 51 (2003) 921] and our classical understanding in dissimilar elastic or anisotropic materials [Bull. Seism. Soc. Am. 49 (1959) 119; ASME J. Appl. Mech. 32 (1965) 400]. On the other hand, there is one exceptional metal/piezoelectric bimaterial with non-zero ε and vanishing κ, the oscillating stress singularity r−1/2±iε at the crack tip is reached in this bimaterial. Consequently, metal/piezoelectric bimaterials are categorized into two classes: one with non-zero κ and vanishing ε could be called as κ-class metal/piezoelectric bimaterials and the other one with non-zero ε and vanishing κ could be termed as ε-class metal/piezoelectric. Analysis of the crack-tip generalized stress field is performed. Of great significance is that: if a purely electric-induced interface crack growth occurs in metal/piezoelectric bimaterials, it is most likely due to the shear mode II fracture rather than the open mode I, and then a purely remote electrical loading enhances the interface crack extension in such bimaterials. Only when an external tensile loading is applied, could the mode I fracture play a dominant role.
This paper presents a boundary element formulation for the analysis of linear elastic fracture mechanics problems involving anisotropic bimaterials. The most important feature associated with the present formulation is that it is a single domain method, and yet it is accurate, efficient and versatile. In this formulation, the displacement integral equation is collocated on the uncracked boundary only, and the traction integral equation is collocated on one side of the crack surface only. The complete Green's functions for anisotropic bimaterials are also derived and implemented into the boundary integral formulation so that discretization along the interface can be avoided except for the interfacial crack part. A special crack-tip element is introduced to capture exactly the crack-tip behavior.Numerical examples are presented for the calculations of stress intensity factors for a straight crack with various locations in infinite bimaterials. It is found that very accurate results can be obtained by the proposed method even with relatively coarse discretization. Numerical results also show that material anisotropy can greatly affect the stress intensity factor.
A parallel crack near the interface of magnetoelectroelastic bimaterials is considered. The crack is modelled by using the continuously distributed edge dislocations, which are described by the density functions defined on the crack line. With the aid of the fundamental solution for the edge dislocation, the present problem is reduced to a system of singular integral equations, which can be numerically solved by using the Chebyshev numerical integration technique. Then, the stress intensity factor (SIF), the magnetic induction intensity factor (MIIF) and the electric displacement intensity fac-tor (EDIF) at the crack tips are evaluated. Using these fracture criteria, the cracking behaviour of magnetoelectroelastic bimaterials is investigated. Numerical examples demonstrate that the interface, mechanical load, magnetic load and material mismatch condition are all important factors affecting the fracture toughness of the magnetoelectroelastic bimaterials.