Complex Characteristic Roots Research Articles

Hyperbolic conservation laws for continuous two-phase flow without mass exchange

It is well known that classic two-phase flow equation systems have complex characteristic roots and, therefore, constitute an ill-posed initial-value problem. Here we suggest that ill-posedness is due to working with two different material derivatives for the phases, which have varying velocities, but employing the same position vector for both operators. There follows an analysis of the conditions required for a global treatment of both phases, but using only one material derivative for both phases, now coherent with only one position vector. Consequently, new global mass- and momentum-conservation equations for a two-phase flow without mass exchange are derived by strictly following the classic Reynolds’ transport theorem. The new global mass-conservation equation proposed would only be valid if the ‘zero-net-mass-flux’ condition, another independent equation, was fulfilled. We also found that the new equation system is well-posed, i.e. its two characteristic roots are real if a new relation between velocities and densities is satisfied. Finally, we have highlighted the strong connections of new conservation laws with classic treatments, and also shown that minor modifications of the current equation system would turn it into a hyperbolic one, thus easing the computational solution of this complex problem.

Finite and transversely isotropic elastic cylinders under compression with end constraint induced by friction

This paper derives an exact solution for the non-uniform stress and displacement fields within a finite, transversely isotropic, and linear elastic cylinder under compression with a kind of radial constraint induced by friction between the end surfaces of the cylinder and the loading platens. The main feature of the present work is the introduction of a general solution form for Lekhnitskii’s stress function such that the governing equation and all end and curved boundary conditions of the cylinder are satisfied exactly. Two different solutions were obtained corresponding to the real or complex characteristic roots of the governing equation, depending on the combination of the elastic material constants. The solution by Watanabe [Watanabe, S., 1996. Elastic analysis of axi-symmetric finite cylinder constrained radial displacement on the loading end. Structural Engineering/Earthquake Engineering JSCE 13, 175s–185s] for isotropic cylinders under compression test can be recovered as a special case. Our numerical results show that both the non-uniform stress distribution and the difference between the apparent and the true Young’s moduli of the cylinder are very sensitive to the anisotropy of Young’s moduli, Poisson’s ratios and shear moduli. A more distinct bulging shape of the cylinder is expected when anisotropy in shear modulus is strong, the cylinder is relatively short, and the end constraint is large. The bulging shape, however, does not depend strongly on anisotropy of either Poisson’s ratio or Young’s modulus.

Propagation characteristics of thermoelastic waves in piezoelectric (6 mm class) rotating plate

Abstract The propagation of Lamb waves in a homogeneous, transversely isotropic (6 mm class), piezothermoelastic plate rotating with uniform angular velocity about normal to its boundary has been investigated. The generalized (non-classical) theories of thermoelasticity in contrast to Sharma and Pal [Sharma, J.N., Pal, M., 2004. Lamb wave propagation in transversely isotropic piezothermoelastic plate. J. Sound Vib. 270, 587–610] have been used to investigate the problem. The surfaces of the plate are subjected to stress free, thermally insulated/isothermal and electrically shorted boundary conditions. Secular equations for wave propagation modes in the plate are derived from a coupled system of governing partial differential equations of linear piezothermoelasticity. After obtaining the complex characteristic roots with the help of Descartes' algorithm, the transcendental secular equations have been solved by functional iteration numerical technique to compute phase velocity and attenuation coefficient. Finally, in order to illustrate the analytical development, numerical solution of secular equations is carried out for PZT-5A piezo-thermoelastic material. The corresponding simulated results of various physical quantities such as phase velocity, attenuation coefficients, specific loss factor of energy dissipation, thermo-mechanical coupling factor and relative frequency shifts have been presented graphically for both rotating and non-rotating plates for comparison purpose. There is a scope for extension of the present work to other classes of piezo/pyroelectric crystals. The study will be useful in design and construction of gyroscope, rotation sensors, temperature sensors and other pyro/piezoelectric surface acoustic wave (SAW) devices.

Straight and Circular Crested Lamb Waves in Generalized Piezothermoelastic Plates

This paper concentrates on the study of propagation of straight and circularly crested Lamb waves in homogeneous, transversely isotropic, piezothermoelastic plates subjected to: stress- and charge-free, thermally insulated/isothermal and stress free, thermally insulated/isothermaland electrically shorted boundary conditions in the context of generalized theories of thermoelasticity. The motion of purely transverse shear horizontal (SH) modes get decoupled from rest of the motion and do not interact with other fields. Secular equations for wave propagation modes in the plate are derived from a coupled system of governing partial differential equations of linear piezothermoelasticity. It is shown that the Rayleigh–Lamb secular equation also governs circular crested piezothermoelastic waves in a plate. Although the frequency wave number relationship holds whether the waves are straight or circularly crested the displacement, stress, electric and temperature fields vary according to Bessel functions rather than trigonometric one as far as the radial coordinate is concerned. At short wavelength limits, the secular equations reduce to Rayleigh surface wave frequency equation. The amplitudes of displacements, temperature change and electrical potential during vibrations of the plate have also been obtained. Finally, in order to illustrate and verify the analytical developments numerical solution of secular equations corresponding to stressfree, thermally insulated, open and closed circuit piezoelectric plates is carried out for cadmium-selenide (6 mm class) material. After obtaining the complex characteristic roots with the help of DesCartes' algorithm, the transcendental secular equations have been solved by functional iteration numerical technique to compute dispersion curves and attenuation coefficients. These are then presented graphically in order to illustrate and compare the results in the context of coupled and generalized theories of thermoelasticity.

The two sector overlapping generations model: A simple formulation

In this paper we provide a simple formulation of a two-sector overlapping generations model based on the social production function which characterizes the factor-price frontier associated with interior temporary equilibria. We first give a simple restriction for the existence of a long run equilibrium. Under gross substitutability, we show that such a steady state is locally indeterminate if the consumption good is sufficiently capital intensive, the elasticity of savings with respect to wage is low enough and the elasticity of savings with respect to the interest rate is large enough. We also exhibit a flip bifurcation giving rise to period-two cycles. When gross substitutability is violated, we prove that local indeterminacy may also occur when the investment good is capital intensive. However, when the consumption good is capital intensive, it requires the existence of at least two distinct steady state. Indeed, saddle-point stability is obtained as soon as uniqueness of the steady state holds. We also give conditions for the existence of complex characteristic roots and we exhibit a Hopf bifurcation giving rise to quasi-periodic cycles.

Analytic Solution for Finite Transversely Isotropic Circular Cylinders under the Axial Point Load Test

An exact solution for stress distributions within a finite transversely isotropic cylinder for the axial point load strength test (PLST) is analytically derived. Lekhnitskii’s stress function is first used to uncouple the equations of equilibrium. Two different kinds of solutions corresponding to the real and the complex characteristic roots of the governing equation of the stress function are derived. The solution type to be used for stress analysis depends on the anisotropic parameters of the cylinder. The solution for isotropic cylinders under the axial PLST is recovered as a special case. Numerical results show that the pattern of stress distribution along the line joining the point loads does not depend on the degree of anisotropy of the cylinder, but the magnitude of the stress distributions does. In particular, the local maximum tensile stress, which is located near the point loads, may be either larger or smaller than that of isotropic cylinders. In general, the maximum tensile stress inside the cylinder increases with the ratio of Young’s moduli, but decreases with both the ratio of Poisson’s ratio and the ratio of the shear moduli. If anisotropy of the cylinder is ignored, the point load strength index may be overestimated when the ratio of Young’s moduli is greater than one, or when the ratios of Poisson’s ratio or of the shear moduli is smaller than one.

Bootstrapping State-Space Models: Gaussian Maximum Likelihood Estimation and the Kalman Filter

Abstract The bootstrap is proposed as a method for assessing the precision of Gaussian maximum likelihood estimates of the parameters of linear state-space models. Our results also apply to autoregressive moving average models, since they are a special case of state-space models. It is shown that for a time-invariant, stable system, the bootstrap applied to the innovations yields asymptotically consistent standard errors. To investigate the performance of the bootstrap for finite sample lengths, simulation results are presented for a two-state model with 50 and 100 observations; two cases are investigated, one with real characteristic roots and one with complex characteristic roots. The bootstrap is then applied to two real data sets, one used in a test for efficient capital markets and one used to develop an autoregressive integrated moving average model for quarterly earnings data. We find the bootstrap to be of definite value over the conventional asymptotics.

Interaction of a pair of coplanar central cracks in an orthotropic strip of finite width

The interactive effects of a pair of coplanar central cracks in an orthotropic strip with finite width are investigated using the techniques of Fourier transforms and finite Hilbert transforms. The surfaces of the cracks are subjected to uniform pressures, while the edges of the strip are free from stresses. The solutions are largely written in terms of the sum and difference of the characteristic root so that the results can easily be seen as real-valued functions for both real and complex roots. Exact expressions for the stress-intensity factors at the inner and outer crack tips and for the crack shape are obtained as products of dimensional quantities and nondimensional functions in terms of material constants, strip width, and a varying distance between the cracks. The nondimensional functions are calculated numerically for various values of nondimensional parameters for three different materials that involve both real and complex characteristic roots. The interactive effects of the cracks on the stress-intensity factors and on the crack surface opening are described in terms of the distance between the cracks, the material anisotropy, and the strip width.

Symmetrical contact problem of a thick transversely isotropic plate

The symmetrical problem of the contact between a spherical indenter and a thick transversely isotropic plate is solved using the techniques of the Hankel transform. Solutions are written as the sums of the associated half-space solution and plate thickness effect terms. The normalized normal contact stress and the surface radial stress are obtained and calculated numerically for both composite materials and metallic substances. The example materials give both real and complex characteristic roots. A general method of calculation is described to determine the maximum tensile radial stress and the maximum compressive normal contact stress.

Ductile penny-shaped crack in a thick transversely isotropic plate

The Dugdale hypothesis is adapted to the problem of a penny-shaped crack contained in a thick transversely isotropic plate. The problem is solved using the techniques of Hankel transform. Exact expressions for the finite-stress condition and the crack shape function are obtained and calculated numerically for both composite materials and metallic substances. The normal stress is shown to be continuous at the value of the material yield stress along the outer circle of the inelastic zone. The example materials give both real and complex characteristic roots. The effects that the material anisotropy has on the inelastic zone size, the crack shape, and the crack opening displacement are clearly revealed in the numerical results presented. The effects of material anisotropy for the composites are compared to those for the metallic substances.