Resolved shear stress intensity coefficient and fatigue crack growth in large crystals

Fatigue crack growth is caused primarily by shear decohesion due to dislocation motion in the crack tip region. The resolved shear stress, which drives dislocation in a crystal, is strongly orientation dependent, and therefore, the cyclic plastic deformation of the shear decohesion process is highly anisotropic.The crack planes are often inclined to the loading axis both in the inplane orientation and in the thickness direction. This inclination induces all three modes of the crack tip stress field, K_I, K_II, and K_III.Fatigue crack growth in large-grain Al 7029 aluminum alloy was studied. The crack tip stress fields of the test specimens are calculated with the finite element method. The values of K_I, K_II, and K_III are evaluated. The orientation of the crystal at a crack tip was determined with the Laue X-ray method. The crystal orientation and the calculated crack tip stress fields are used to compute the resolved shear stress intensity of each of the twelve slip systems of the crystal at the crack tip. The resolved shear stress field of a slip system is linearly proportional to the resolved shear stress intensity coefficient, RSSIC.The values of RSSIC thus evaluated are used to analyze the orientations of the crack plane and to correlate with the shear fatigue crack growth rate.     

Lyapunov exponent and stochastic stability of quasi-non-integrable Hamiltonian systems

An n degree-of-freedom (DOF) non-integrable Hamiltonian system subject to light damping and weak stochastic excitation is called quasi-non-integrable Hamiltonian system. In the present paper, the stochastic averaging of quasi-non-integrable Hamiltonian systems is briefly reviewed. A new norm in terms of the square root of Hamiltonian is introduced in the definitions of stochastic stability and Lyapunov exponent and the formulas for the Lyapunov exponent are derived from the averaged Itô equations of the Hamiltonian and of the square root of Hamiltonian. It is inferred that the Lyapunov exponent so obtained is the first approximation of the largest Lyapunov exponent of the original quasi-non-integrable Hamiltonian systems and the necessary and sufficient condition for the asymptotic stability with probability one of the trivial solution of the original systems can be obtained approximately by letting the Lyapunov exponent to be negative. This inference is confirmed by comparing the stability conditions obtained from negative Lyapunov exponent and by examining the sample behaviors of averaged Hamiltonian or the square root of averaged Hamiltonian at trivial boundary for two examples. It is also verified by the largest Lyapunov exponent obtained using small noise expansion for the second example.     

First-Passage Time of Duffing Oscillator under Combined Harmonic and White-Noise Excitations

The first-passage time of Duffing oscillator under combined harmonic andwhite-noise excitations is studied. The equation of motion of the system is firstreduced to a set of averaged Itô stochastic differential equations by using thestochastic averaging method. Then, a backward Kolmogorov equation governing theconditional reliability function and a set of generalized Pontryagin equationsgoverning the conditional moments of first-passage time are established. Finally, theconditional reliability function, and the conditional probability density and momentsof first-passage time are obtained by solving the backward Kolmogorov equation andgeneralized Pontryagin equations with suitable initial and boundary conditions.Numerical results for two resonant cases with several sets of parameter values areobtained and the analytical results are verified by using those from digital simulation.     

Feedback minimization of first-passage failure of quasi non-integrable Hamiltonian systems

The non-linear stochastic optimal control of quasi non-integrable Hamiltonian systems for minimizing their first-passage failure is investigated. A controlled quasi non-integrable Hamiltonian system is reduced to an one-dimensional controlled diffusion process of averaged Hamiltonian by using the stochastic averaging method for quasi non-integrable Hamiltonian systems. The dynamical programming equations and their associated boundary and final time conditions for the problems of maximization of reliability and of maximization of mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The dynamical programming equations for maximum reliability problem and for maximum mean first-passage time problem are finalized and their relationships to the backward Kolmogorov equation for the reliability function and the Pontryagin equation for mean first-passage time, respectively, are pointed out. The boundary condition at zero Hamiltonian is discussed. Two examples are worked out to illustrate the application and effectiveness of the proposed procedure.     

GENERALIZED FINITE SPECTRAL METHOD FOR 1D BURGERS AND KDV EQUATIONS

A generalized finite spectral method is proposed. The method is of high-order accuracy. To attain high accuracy in time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme was used. To avoid numerical oscillations caused by the dispersion term in the KdV equation, two numerical techniques were introduced to improve the numerical stability. The Legendre, Chebyshev and Her-mite polynomials were used as the basis functions. The proposed numerical scheme is validated by applications to the Burgers equation (nonlinear convection- diffusion problem) and KdV equation (single solitary and 2-solitary wave problems), where analytical solutions are available for comparison. Numerical results agree very well with the corresponding analytical solutions in all cases.     

Exact Results for the Homogenization of Elastic Fiber-Reinforced Solids at Finite Strain

This work is concerned with the homogenization of solids reinforced by aligned parallel continuous fibers or weakened by aligned parallel cylindrical pores and undergoing large deformations. By alternatively exploiting the nominal and material formulations of the corresponding homogenization problem and by applying the implicit function theorem, it is shown that locally homogeneous deformations can be produced in such inhomogeneous materials and form a differentiable manifold. For every macroscopic strain associated to a locally homogeneous deformation field, the effective nominal or material stress–strain relation is exactly determined and connections are also exactly established between the effective nominal and material elastic tangent moduli. These results are microstructure-independent in the sense that they hold irrespectively of the transverse geometry and distribution of the fibers or pores. A porous medium consisting of a compressible Mooney–Rivlin material with cylindrical pores is studied in detail to illustrate the general results. This work was the first time presented at the Euromech Colloqium 464 on “Fiber-reinforced Solids: Constitutive Laws and Instabilities”, September 28–October 1, 2004, Cantabria, Spain.     

A critical review of dispersion in packed beds

The phenomenon of dispersion (transverse and longitudinal) in packed beds is summarized and reviewed for a great deal of information from the literature. Dispersion plays an important part, for example, in contaminant transport in ground water flows, in miscible displacement of oil and gas and in reactant and product transport in packed bed reactors. There are several variables that must be considered, in the analysis of dispersion in packed beds, like the length of the packed column, viscosity and density of the fluid, ratio of column diameter to particle diameter, ratio of column length to particle diameter, particle size distribution, particle shape, effect of fluid velocity and effect of temperature (or Schmidt number). Empirical correlations are presented for the prediction of the dispersion coefficients (D _T and D _L) over the entire range of practical values of Sc and Pe_m, and works on transverse and longitudinal dispersion of non-Newtonian fluids in packed beds are also considered.     

Jacobi's multiplier for Poincaré's equations

In this paper we use Poincaré's equations in group variables to describe the motion of a holonomic mechanical system and to determine Jacobi's multiplier for the equations of motion.     

Experimental investigation of flow and heat transfer for the ethanol-water solution and FC-72 in rectangular microchannels

Rapid development of super scale integration circuit (IC) provides unprecedented challenge to thermal control for aviation electronic equipments. To solve the problem of cooling electronic chips and devices for aircraft avionics, this paper experimentally investigated the characteristics of single-phase forced convection heat transfer and flow resistance in rectangular microchannels with two liquid coolants. One was 30% of ethanol–water solution, the most commonly used coolant in aviation. The other was FC-72, the latest coolant for electronic equipments. Based on the experimental data collected and those available in the open literature, comparisons and analyses were carried out to evaluate the influences of liquid velocity, supercooling temperature, microchannel structures and wall temperature etc. on the heat transfer behaviors. And the correlations of flow resistance and heat transfer characteristics were provided for the ethanol–water solution and FC-72 respectively. The results indicate transition from laminar to turbulent flow occurs at the Reynolds number of 750–1,250 for FC-72, and the behaviors of flow and heat transfer in rectangular microchannels strongly depend on the kind of coolant and geometric configuration of microchannels.     

Strain energy density of a circular cavity buried in semi-infinite functionally graded materials subjected to shear waves

In this paper, based on the theory of multiple scattering of elastic waves, employing wave functions expansion method, multiple scattering and strain energy density in semi-infinite functional graded materials with a circular cavity are investigated, the analytical solution of the problem is derived, and the numerical solution of the strain energy density factors around the cavity is also presented. The effects of the distance between the cavity and the edge of the materials, the wave number and the non-homogeneous parameter of materials on strain energy density factors are analyzed. From analysis, it can be seen that when the non-homogeneous parameter of materials is less than zero, it has less influence on the maximum strain energy density factor around the cavity; however, it has greater influence on the distribution of strain energy density factors around the cavity. When the non-homogeneous parameter of materials is greater than zero, it has greater influence on both the maximum strain energy density factor and the distribution of strain energy density factor around the cavity, especially in the case that the distance between the cavity and the edge is comparatively little.