The multi-parameter inversion of elastic wave in half-space

The multi-parameter inverse scattering problem of elastic wave equation with single frequency is investigated within Born approximation. By use of a wideband measuring scheme in which both transmitters and receivers scan over the half-space surface, the formula of the scattering field of elastic wave is derived. Four types of mode conversion of elastic wave (P→P,P→S,S→P,S→S) are separated from the scattering field. These components contain sufficient information for us to reconstruct the configurations of the density and Lamé parameters of the medium. The inverse formulas have the form of filtered back-propagation as in the acoustic diffraction tomography. Computer simulations are also obtained. Supported by Foundation of Ph.D Program of the State Education Commission of China.     

The determination of elastic parameters of a half-space using a monochromatic vibration field at the surface

The paper presents a method to determine Lamé parameters λ, μ and density ϱ in a layered half-space, using monochromatic vibrations of its surface, excited by a harmonic source which is assumed to be known. The equations governing the vibrations are reduced to the Sturm-Liouville problem in scalar form for Love-type displacements, and in matrix form for the Rayleigh type. The scalar and matrix potentials of the Sturm-Liouville equations can be recovered from the corresponding impedance. Explicit formulas are given to construct the potentials by amplitudes and wavenumbers of normal (progressive) modes and attenuated standing waves. The potentials then are used to determine the elastic parameters and the density. The method can also be used for the acoustic equation.     

Effect of rigid boundary on propagation of torsional surface waves in porous elastic layer

The paper presents the effect of a rigid boundary on the propagation of torsional surface waves in a porous elastic layer over a porous elastic half-space using the mechanics of the medium derived by Cowin and Nunziato (Cowin, S. C. and Nunziato, J. W. Linear elastic materials with voids. Journal of Elasticity, 13(2), 125–147 (1983)). The velocity equation is derived, and the results are discussed. It is observed that there may be two torsional surface wave fronts in the medium whereas three wave fronts of torsional surface waves in the absence of the rigid boundary plane given by Dey et al. (Dey, S., Gupta, S., Gupta, A. K., Kar, S. K., and De, P. K. Propagation of torsional surface waves in an elastic layer with void pores over an elastic half-space with void pores. Tamkang Journal of Science and Engineering, 6(4), 241–249 (2003)). The results also reveal that in the porous layer, the Love wave is also available along with the torsional surface waves. It is remarkable that the phase speed of the Love wave in a porous layer with a rigid surface is different from that in a porous layer with a free surface. The torsional waves are observed to be dispersive in nature, and the velocity decreases as the oscillation frequency increases.     

EPSILON-ALGORITHM AND ETA-ALGORITHM OF GENERALIZED INVERSE FUNCTION-VALUED PAD APPROXIMANTS USING FOR SOLUTION OF INTEGRAL EQUATIONS

Two efficient recursive algorithms epsilon-algorithm and eta-algorithm are introduced to compute the generalized inverse function-valued Padé approximants. The approximants were used to accelerate the convergence of the power series with function-valued coefficients and to estimate characteristic value of the integral equations. Famous Wynn identities of the Padé approximants is also established by means of the connection of two algorithms. Foundation items: the National Natural Science Foundation of China (10271074) Biographies: LI Chun-jing (1958 −) GU Chuan-qing (1955 −)     

ON THE CRITICAL POINTS OF THE MAP Fp:X→‖AXB-C‖pp

Suppose A,B and C are the bounded linear operators on a Hilbert space H, when A has a generalized inverse A^- such that (AA^-)^*=AA^- and B has a generalized inverse B^- such that (B^-B)^*=B^-B,the general characteristic forms for the critical points of the map F_p:X^→‖AXB-C‖_p^p(1p=2. Similarly, the same question has been discussed for several operators.     

Pressure pulse on the boundary of an elastic inhomogeneous half-plane

The problem about the motion of a pressure pulse at constant velocity along the boundary of an elastic homogeneous half-plane has been examined in [1–3]. The problem was considered as stationary in [1, 2], while in [3] it was solved by using a Laplace time transformation. An analogous problem is considered in this paper for an elastic half-plane with variable Lamé parameters and density of the medium.     

Inverse scattering for stratified elastic media

The trace method is applied to recover the Lamé parameters λ and μ and the mass density ϱ of a stratified elastic medium.     

The Damped-Driven 2D Navier–Stokes System on Large Elongated Domains

The Navier–Stokes system with damping, which is motivated by Stommel–Charney model of ocean circulation, is considered in a large elongated periodic rectangular domain with area of the order α⁻¹, as α → 0. We obtain estimates for the dimension of the global attractor that are sharp as both α → 0 and ν → 0, where ν is the viscosity coefficient. This work was supported in part by the US Civilian Research and Development Foundation, grant no. RUM1-2654-MO-05 (A.A.I. and E.S.T.). The work of A.A.I. was supported in part by the Russian Foundation for Fundamental Research, grants no. 06-001-0096 and no. 05-01-429, and by the RAS Programme no. 1 ‘Modern problems of theoretical mathematics’. The work of E.S.T. was supported in part by the NSF, grant no. DMS-0204794, the MAOF Fellowship of the Israeli Council of Higher Education, and by the BSF, grant no. 200423.     

Direct and inverse inequalities for the isotropic Lamé system with variable coefficients

We consider a Cauchy–Dirichlet problem for the isotropic Lamé system with variable coefficients. We find an estimate for the L² -norm of the surface traction in terms of the initial data and the body force. Then we show that, in absence of body forces, the elastic energy can be controlled by the L² -norm of the surface traction exerted on a suitable sub-boundary, provided that the time interval is sufficiently large. These inequalities are basic for the applicability of the so-called HUM (Hilbert Uniqueness Method) and they can also be used to solve an inverse source problem for the Lamé system.     

Uniformly valid asymptotic solutions of the nonlinear unsymmetrical bending for orthotropic rectangular thin plate of four clamped edges with variable thickness

By using “the method of modified two-variable”,“the method of mixing perturbation ”and introducing four small parameters, the problem of the nonlinear unsymmetrical bending for orthotropic rectangular thin plate with linear variable thickness is studied. And the uniformly valid asymptotic solution of Nth- order for ε1 and Mth- order for ε2 of the deflection functions and stress function are obtained.     

Problems of the Flamant–Boussinesq and Kelvin Type in Dipolar Gradient Elasticity

This work studies the response of bodies governed by dipolar gradient elasticity to concentrated loads. Two-dimensional configurations in the form of either a half-space (Flamant–Boussinesq type problem) or a full-space (Kelvin type problem) are treated and the concentrated loads are taken as line forces. Our main concern is to determine possible deviations from the predictions of plane-strain/plane-stress classical linear elastostatics when a more refined theory is employed to attack the problems. Of special importance is the behavior of the new solutions near to the point of application of the loads where pathological singularities and discontinuities exist in the classical solutions. The use of the theory of gradient elasticity is intended here to model material microstructure and incorporate size effects into stress analysis in a manner that the classical theory cannot afford. A simple but yet rigorous version of the generalized elasticity theories of Toupin (Arch. Ration. Mech. Anal. 11:385–414, 1962) and Mindlin (Arch. Ration. Mech. Anal. 16:51–78, 1964) is employed that involves an isotropic linear response and only one material constant (the so-called gradient coefficient) additional to the standard Lamé constants (Georgiadis et al., J. Elast. 74:17–45, 2004). This theory, which can be viewed as a first-step extension of the classical elasticity theory, assumes a strain-energy density function, which besides its dependence upon the standard strain terms, depends also on strain gradients. The solution method is based on integral transforms and is exact. The present results show departure from the ones of the classical elasticity solutions (Flamant–Boussinesq and Kelvin plane-strain solutions). Indeed, continuous and bounded displacements are predicted at the points of application of the loads. Such a behavior of the displacement fields is, of course, more natural than the singular behavior present in the classical solutions.      

Poincaré-cartan integral invariants of Birkhoffian systems

Based on modern differential geometry, the symplectic structure of a Birkhoffian system which is an extension of conservative and nonconservative systems is analyzed. An integral invariant of Poincaré-Cartan’s type is constructed for Birkhoffian systems. Finally, one-dimensional damped vibration is taken as an illustrative example and an integral invariant of Poincaré’s type is found.     

On the Nonexistence of Some Special Eigenfunctions for the Dirichlet Laplacian and the Lamé System

Let Ω be a bounded Lipschitz domain in ℝ ⁿ with n ≥ 3. We prove that the Dirichlet Laplacian does not admit any eigenfunction of the form u(x) =ϕ(x′)+ψ(x _n) with x′=(x1, ..., x _n−1). The result is sharp since there are 2-d polygonal domains in which this kind of eigenfunctions does exist. These special eigenfunctions for the Dirichlet Laplacian are related to the existence of uniaxial eigenvibrations for the Lamé system with Dirichlet boundary conditions. Thus, as a corollary of this result, we deduce that there is no bounded Lipschitz domain in 3-d for which the Lamé system with Dirichlet boundary conditions admits uniaxial eigenvibrations. This revised version was published online in August 2006 with corrections to the Cover Date.     

The contact problem of electroelasticity for a flat punch with parabolic cross-section

The solution of the problem of a rigid punch with a parabolic cross-section and flat base that is forced into an elastic piezoelectric ceramic half-space is derived in explicit form. The punch is somewhat displaced, being parallel to the isotropy plane that coincides with the boundary surface of the half-space. The symmetry axis coincides with the direction of the force lines of the field with the previous polarization. Formulas are derived to determine the stresses on the surface of the half-space under the punch and the components of the conjugate electric field for certain boundary conditions on the contact area. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 11, pp. 20–26, November, 1999.     

IMPULSIVE CONTROL FOR THE STABILIZATION AND SYNCHRONIZATION OF LUR＇ E SYSTEMS

An impulsive control scheme of the Lur‘e system and several theorems on stability of impulsive control systems was presented, these theorems were then used to find the conditions under which the Lur‘e system can be stabilized by using impulsive control with varying impulsive intervals. The parameters of Lur‘e system and impulsive control law are given, a theory of impulsive synchronization of two Lur‘e system is also presented. A numerical example is used to verify the theoretical result.     

The Boussinesq problem in dipolar gradient elasticity

The three-dimensional axisymmetric Boussinesq problem of an isotropic half-space subjected to a concentrated normal quasi-static load is studied within the framework of dipolar gradient elasticity involving linear constitutive relations and small strains. Our main concern is to determine possible deviations from the predictions of classical linear elastostatics when a more refined theory is employed to attack the problem. Of special importance is the behavior of the new solution near to the point of application of the load where pathological singularities exist in the classical solution. The use of the theory of gradient elasticity is intended here to model the response of materials with microstructure in a manner that the classical theory cannot afford. A linear version of this theory (as regards both kinematics and constitutive response) results by considering a linear isotropic expression for the strain-energy density that depends on strain gradient terms, in addition to the standard strain terms appearing in classical elasticity and by considering small strains. Through this formulation, a microstructural material constant is introduced, in addition to the standard Lamé constants. The solution method is based on integral transforms and is exact. The present results show significant departure from the predictions of classical elasticity. Indeed, continuous and bounded displacements are predicted at the points of application of the concentrated load. Such a behavior of the displacement field is, of course, more natural than the singular behavior exhibited in the classical solution.     

An automatic constraint violation stabilization method for differential/ algebraic equations of motion in multibody system dynamics

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Semi-weight function method on computation of stress intensity factors in dissimilar materials

Semi-weight function method is developed to solve the plane problem of two bonded dissimilar materials containing a crack along the bond. From equilibrium equation, stress and strain relationship, conditions of continuity across interface and free crack surface, the stress and displacement fields were obtained. The eigenvalue of these fields is lambda. Semi-weight functions were obtained as virtual displacement and stress fields with eigenvalue-lambda. Integral expression of fracture parameters, KⅠ and KⅡ, were obtained from reciprocal work theorem with semi-weight functions and approximate displacement and stress values on any integral path around crack tip. The calculation results of applications show that the semi-weight function method is a simple, convenient and high precision calculation method.     

Energy release or absorption due to simultaneous rotation of two nano voids in plane elastic materials as influenced by both surface effect and interacting effect

In this paper, the L-integral analysis for two nano-sized voids in plane elasticity under uniaxial loading is present. Three surface parameters are considered including the surface tension and two surface Lamé constants. Attention is focused on the mutual influence on the L-integral from both the surface effect at voids’ rims and the interacting effect between voids. A close-form expression of L-integral for multiple nano voids is obtained. Comparing with those in macro fracture mechanics, the L-integral shows some different features when the surface effect is taken into account. It is concluded that under tensile loading and due to the mutual influence, the L-integral might be either positive or negative, depending on the loading level. The numerical results show that the surface tension is the dominant one in surface parameters on impacting the L-integral. It is also concluded that the surface effect shields the energy release (positive L-integral value) while enhances the energy absorption (negative L-integral value). The two-state L-integral analysis is performed to clarify the way that the surface effect impacts the L-integral. It is concluded that the contribution to L-integral from the voids’ configuration could either be negative or positive, while that from the surface effect is always negative. Besides, the size dependence in the present problem is studied in detail.