Transient motion of an anisotropic elastic half-space due to a buried line source

A method to deal with the two-dimensional transient problem of a line force or dislocation in an anisotropic elastic half-space is developed. The proposed formulation is similar to Stroh’s formalism for anisotropic elastostatics in that the two-dimensional anisotropic elastodynamic problem is cast into a six-dimensional eigenvalue problem and the solution is expressed in terms of the eigenvalues and eigenvectors. An analytic solution is obtained without performing integral transforms. Numerical examples are presented for a silicon half-space subjected to a line force or dislocation.     

Stress singularity due to traction discontinuity on an anisotropic elastic half-plane

In a half-plane problem with x₁ paralleling with the straight boundary and x₂ pointing into the medium, the stress components on the boundary whose acting plane is perpendicular to x₁ direction may be denoted by t₁ = [σ₁₁, σ₁₂, σ₁₃]^T. Stress components σ₁₁ and σ₁₃ are of more interests since σ₁₂ is completely determined by the boundary conditions. For isotropic materials, it is known that under uniform normal loading σ₁₁ is constant in the loaded region and vanishes in the unloaded part. Under uniform shear loading, σ₁₁ will have a logarithmic singularity at the end points of shear loading. In this paper, the behavior of the stress components σ₁₁ and σ₁₃ induced by traction-discontinuity on general anisotropic elastic surfaces is studied. By analyzing the problem of uniform tractions applied on the half-plane boundary over a finite loaded region, exact expressions of the stress components σ₁₁ and σ₁₃ are obtained which reveal that these components consist of in general a constant term and a logarithmic term in the loaded region, while only a logarithmic term exists in unloaded region. Whether the constant term or the logarithmic term will appear or not completely depends on what values of the elements of matrices Ω and Γ will take for a material under consideration. Elements for both matrices are expressed explicitly in terms of elastic stiffness. Results for monoclinic and orthotropic materials are all deduced. The isotropic material is a special case of the present results.     

Subinterface crack in an anisotropic piezoelectric bimaterial

In this paper, the problem of a subinterface crack in an anisotropic piezoelectric bimaterial is analyzed. A system of singular integral equations is formulated for general anisotropic piezoelectric bimaterial with kernel functions expressed in complex form. For commonly used transversely isotropic piezoelectric materials, the kernel functions are given in real forms. By considering special properties of one of the bimaterial, various real kernel functions for half-plane problems with mechanical traction-free or displacement-fixed boundary conditions combined with different electric boundary conditions are obtained. Investigations of half-plane piezoelectric solids show that, particularly for the mechanical traction-free problem, the evaluations of the mechanical stress intensity factors (electric displacement intensity factor) under mechanical loadings (electric displacement loading) for coupled mechanical and electric problems may be evaluated directly by considering the corresponding decoupled elastic (electric) problem irrespective of what electric boundary condition is applied on the boundary. However, for the piezoelectric bimaterial problem, purely elastic bimaterial analysis or purely electric bimaterial analysis is inadequate for the determination of the generalized stress intensity factors. Instead, both elastic and electric properties of the bimaterial’s constants should be simultaneously taken into account for better accuracy of the generalized stress intensity factors.     

Transient thermoelastic waves in an anisotropic hollow cylinder due to localized heating

This paper presents a theoretical study of transient ultrasonic guided waves generated by concentrated heating of the outer surface of an infinite anisotropic hollow circular cylinder. Generalized thermoelastic theory proposed by Lord and Shulman is adopted to model the dynamic thermoelastic behavior of the cylinder. The concentrated heat source model used is to represent heating due to a pulsed laser beam, which is focused on the outer surface of the cylinder. A semi-analytical finite element (SAFE) method is employed to evaluate guided wave modes in the cylinder. Using integral transform techniques, the modal wave forms are obtained in frequency and wave number domains. Time histories of the propagating modes are then calculated by applying inverse Fourier transformation in the time domain. Numerical results showing the dispersion curves for the group velocities of the propagating modes and transient radial displacements are presented. For this purpose it is assumed that the cylinder is made of transversely isotropic silicon nitride (Si₃N₄). Attention is focused on the propagation characteristics of longitudinal and flexural modes separately.     

Transient elastic wave propagation in a spherically symmetric bimaterial medium modeling the thorax

The transient response resulting from an impact wave on an elastic bimaterial, made out of a “hard” medium and a “soft” medium, welded at a spherical interface, have been investigated by using an integral transform technique. This technique permits isolation of the pressure and shear waves contributions to the wave field. The method of solution makes use of the generalized ray/Cagniard-de Hoop (GR/CdH) method associated with a “flattening approximation” (FA) technique, similar to the Earth flattening transformation used in geophysics. The GR/CdH method and the FA technique are briefly presented, together with their numerical implementations. The FA has proved to be useful in geophysical application, however, as far as the authors know, it has never been investigated for other applications. For the purpose of this paper, numerous tests of the method have been performed in order to check that the FA is appropriate to compute transient responses in the special case presented here. We could determine appropriate values for some parameters involved in the FA. This paper follows Grimal et al. [Int. J. Solid Struct. 39 (2002) 5345] in which we investigated the same bimaterial with a plane––instead of spherical––interface. Numerical examples are concerned with the propagation of an impact wave in the thorax modeled as a bimaterial (thoracic wall-lung). In addition to the effects of the weak coupling of the two media already observed in our previous study, we found that, for interface curvatures characteristic of those measured in the thorax, focalization of energy is manifest.     

Modeling core-spreading of interface dislocation and its elastic response in anisotropic bimaterial

Interfacial dislocation may have a spreading core corresponding to a weak shear resistance of interfaces. In this paper, a conic model is proposed to mimic the spreading core of interfacial dislocation in anisotropic bimaterials. By the Stroh formalism and Green's function, the analytical expressions of the elastic fields are deduced for such a dislocation. Taking Cu/Nb bimaterial as an example, it is demonstrated that the accuracy and efficiency of the method are well validated by the interface conditions, a spreading core can greatly reduce the stress intensity near the interfacial dislocation compared with the compact core, and the elastic fields near the spreading core region are significantly different from the condensed core, while they are less sensitive to a field point that is 1.5times the core width away from the center of the spreading core.     

Flexural-gravity wave resistances due to a surface-moving line source on a fluid covered by a thin elastic plate

Analytical solutions for the flexural-gravity wave resistances due to a line source steadily moving on the surface of an infinitely deep fluid are investigated within the framework of the linear potential theory. The homogenous fluid, covered by a thin elastic plate, is assumed to be incompressible and inviscid, and the motion to be irrotational. The solution in integral form for the wave resistance is obtained by means of the Fourier transform and the explicitly analytical solutions are derived with the aid of the residue theorem. The dispersion relation shows that there is a minimal phase speed c_min, a threshold for the existence of the wave resistance. No wave is generated when the moving speed of the source V is less than c_min while the wave resistances firstly increase to their peak values and then decrease when V ? c_min. The effects of the flexural rigidity and the inertia of the plate are studied.     

Buoyant plume above a line source of heat in an anisotropic porous medium

Buoyancy-induced convection arising from a horizontal line heat source embedded in an anisotropic porous medium is investigated analytically. The porous medium is anisotropic is permeability with its principal axes oriented in a direction that is oblique to the gravity vector. Assuming the boundary layer approximation, closed-form exact similarity solutions for both flow and temperature fields are presented and compared with those of isotropic case. Scale analysis is applied to predict the order of magnitudes involved in the boundary layer regime for which the conditions of validity are obtained. Effects of both anisotropic parameters (K* and %) and Rayleigh number Ra_L are observed to be strongly significant. It is demonstrated that a minimum (maximum) intensity of the thermal convective plume above the line source of heat can be obtained if the porous matrix is oriented with its principal axis with higher permeability parallel (perpendicular) to the vertical direction.     

Contact zone assessment for a fast growing interface crack in an anisotropic bimaterial

Summary A plane strain problem for a crack with a frictionless contact zone at the leading crack tip expanding stationary along the interface of two anisotropic half-spaces with a subsonic speed under the action of various loadings is considered. The cases of finite and infinite-length interface cracks under the action of a moving concentrated loading at its faces are considered. A finite-length crack for a uniform mixed-mode loading at infinity is considered as well. The associated combined Dirichlet-Riemann boundary value problems are formulated and solved exactly for all above-mentioned cases. The expressions for stresses and the derivatives of the displacement jumps at the interface are presented in a closed analytical form for an arbitrary contact zone length. Transcendental equations are obtained for the determination of the real contact zone length, and the associated closed form asymptotic formulas are found for small values of this parameter. It is found that independently of the types of the crack and loading, an increase of the crack tip speed leads to an increase of the real contact zone length and the correspondent stress intensity factor. The latter increase significantly for an interface crack tip speed approaching the Ragleigh wave speed.     

Time-dependent Green's functions for an anisotropic bimaterial with viscous interface

By virtue of the Stroh formalism, we derive the exact closed-form solutions for the time-dependent two-dimensional Green's functions due to a line force and line dislocation in an anisotropic bimaterial with a viscous interface. We first reduce the boundary value problem to two coupled homogeneous first-order partial differential equations, which can be solved using a decoupling technique. The full-field expressions of the time-dependent displacements and stresses due to the line force and line dislocation interacting with the viscous interface are obtained.     

Transient MHD stagnation flow of a non-Newtonian fluid due to impulsive motion from rest

The transient boundary layer flow and heat transfer of a viscous incompressible electrically conducting non-Newtonian power-law fluid in a stagnation region of a two-dimensional body in the presence of an applied magnetic field have been studied when the motion is induced impulsively from rest. The non-linear partial differential equations governing the flow and heat transfer have been solved by the homotopy analysis method and by an implicit finite-difference scheme. For some cases, analytical or approximate solutions have also been obtained. The special interest are the effects of the power-law index, magnetic parameter and the generalized Prandtl number on the surface shear stress and heat transfer rate. In all cases, there is a smooth transition from the transient state to steady state. The shear stress and heat transfer rate at the surface are found to be significantly influenced by the power-law index N except for large time and they show opposite behaviour for steady and unsteady flows. The magnetic field strongly affects the surface shear stress, but its effect on the surface heat transfer rate is comparatively weak except for large time. On the other hand, the generalized Prandtl number exerts strong influence on the surface heat transfer. The skin friction coefficient and the Nusselt number decrease rapidly in a small interval 0<t^*<1 and reach the steady-state values for t^*≥4.     

The dynamic behaviour of a piezoelectric actuator bonded to an anisotropic elastic medium

Surface-bonded piezoelectric actuators can be used to generate elastic waves for monitoring damages of composite materials. This paper provides an analytical and numerical study to simulate wave propagation in an anisotropic medium induced by surface-bonded piezocermic actuators under high-frequency electric loads. Based on a one-dimensional actuator model, the dynamic load transfer between a piezoceramic actuator and an anisotropic elastic medium under in-plane mechanical and electrical loading is obtained. The wave propagation induced by the surface-bonded actuator is also studied in detail by using Fourier transform technique and solving the resulting integral equations in terms of the interfacial shear stress. Typical examples are provided to show effects of the geometry, the material combination, the loading frequency and the material anisotropy of the composite upon the load transfer and the resulting wave propagation.     

On interface crack models with contact zones situated in an anisotropic bimaterial

Summary A boundary value problem for two semi-infinite anisotropic spaces with mixed boundary conditions at the interface is considered. Assuming that the displacements are independent of the coordinate x ₃, stresses and derivatives of displacement jumps are expressed via a sectionally holomorphic vector function. By means of these relations the problem for an interface crack with an artificial contact zone in an orthotropic bimaterial is reduced to a combined Dirichlet-Riemann problem which is solved analytically. As a particular case of this solution, the contact zone model (in Comninou's sense) is derived. A simple transcendental equation and an asymptotic formula for the determination of the real contact zone length are obtained. The classical interface crack model with oscillating singularities at the crack tips is derived from the obtained solution as well. Analytical relations between fracture mechanical parameters of different models are found, and recommendations concerning their implementation are given. The dependencies of the contact zone lengths on material properties and external load coefficients are illustrated in graphical form. The practical applicability of the obtained results is demonstrated by means of a FEM analysis of a finite-sized orthotropic bimaterial with an interface crack. Received 19 October 1998; accepted for publication 13 November 1998     

Finite deformations of an elastic anisotropic body

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 166–174, May–June, 1989.     

Reflection and diffraction of SH waves due to a line source in a heterogeneous medium

The field due to a line source of harmonic SH waves embedded in a semi infinite medium whose density and rigidity vary exponentially with depth is derived in the integral form. The displacement due to diffraction at any point in the shadow zone is obtained and, by the saddle point method of evaluation of the integral, the field at any point in the illuminated region is also found. Finally, geometrical interpretation is given to the different rays arriving in the illuminated as well as in the shadow zone.Nomenclature b shear wave velocity on the free surface - C wave velocity - H _v ⁽¹⁾ (p), H _v ⁽²⁾ (p) Hankel's function of the first and second kind respectively - k Fourier transform parameter with respect to x - v the displacement - fourier transform of v with respect to x - X grazing angle - , small positive constants - positive constant, –/2 - ₀ coefficient of rigidity at the free surface - coefficient of rigidity - _is values of _i at the saddle point (i=1, 2, 3, 4) - the density of the medium - ₀ the density of the medium at the free surface - /2 frequency of vibration