Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems

An n degree-of-freedom Hamiltonian system with r(1<r<n) independent first integrals which are in involution is called partially integrable Hamiltonian system and a partially integrable Hamiltonian system subject to light dampings and weak stochastic excitations is called quasi partially integrable Hamiltonian system. In the present paper, the averaged Itô and Fokker-Planck-Kolmogorov (FPK) equations for quasi partially integrable Hamiltonian systems in both cases of non-resonance and resonance are derived. It is shown that the number of averaged Itô equations and the dimension of the averaged FPK equation of a quasi partially integrable Hamiltonian system is equal to the number of independent first integrals in involution plus the number of resonant relations of the associated Hamiltonian system. The technique to obtain the exact stationary solution of the averaged FPK equation is presented. The largest Lyapunov exponent of the averaged system is formulated, based on which the stochastic stability and bifurcation of original quasi partially integrable Hamiltonian systems can be determined. Examples are given to illustrate the applications of the proposed stochastic averaging method for quasi partially integrable Hamiltonian systems in response prediction and stability decision and the results are verified by using digital simulation.     

Analytic formulas of energy release rates for delamination using a global–local method

This article presents analytic solutions of energy release rates of a cracked laminate by using a global–local method. Deformations of a cracked laminate subjected to pure bending moments are analyzed and then a new mode partition equation is proposed. By using this partition equation, closed-form solutions of energy release rates G_I and G_II are derived by using a global method. For a cracked laminate subjected to axial forces and bending moments, a local method based on the crack-tip force model is used and the unknown coefficient is solved by combining the present analytic solutions for the bending moment loading case. Numerical results of the mode mixity predicted by the present closed-form formulations correlate well with those numerically calibrated on the basis of the singular field and crack-tip force models.     

Complete solution of the stability problem for elastica of Euler's column

The stability of postcritical equilibrium forms of a simply supported column loaded with an axial force is analyzed. Investigating the sign of the second variation of the column's total energy, we obtain the Sturm-Liouville boundary-value problem, which is solved numerically. The stability conditions are formulated in terms of eigenvalues of the problem. The complete solution to the column plane elastica is given. The ranges of the compressive force corresponding to stable equilibrium configurations of the column are established.     

Stochastic Stabilization of Quasi-Partially Integrable Hamiltonian Systems by Using Lyapunov Exponent

A procedure for designing a feedback control to asymptoticallystabilize, with probability one, a quasi-partially integrableHamiltonian system is proposed. First, the averaged stochasticdifferential equations for controlled r first integrals are derived fromthe equations of motion of a given system by using the stochasticaveraging method for quasi-partially integrable Hamiltonian systems.Second, a dynamical programming equation for the ergodic control problemof the averaged system with undetermined cost function is establishedbased on the dynamical programming principle. The optimal control law isderived from minimizing the dynamical programming equation with respectto control. Third, the asymptotic stability with probability one of theoptimally controlled system is analyzed by evaluating the maximalLyapunov exponent of the completely averaged Itô equations for the rfirst integrals. Finally, the cost function and optimal control forces aredetermined by the requirements of stabilizing the system. An example isworked out in detail to illustrate the application of the proposedprocedure and the effect of optimal control on the stability of thesystem.     

Approximation of the third two-point moments of the velocity field in isotropic turbulence

Different methods for approximating the third two-point moments of the velocity fields entering in the von Kármán-Howarth equation are compared using the known experimental results of A. Townsend and R. Stewart, together with the model form of the energy spectrum of turbulence, which makes it possible to approximate the experimentally determined second moments of the velocity field. The latter moments are used for calculating the third two-point moments following the methods proposed by various authors. The calculated results are compared with the experimental data which makes it possible to quantitatively estimate the approximation accuracy. For several models the second-order structure function is calculated from the Kolmogorov equation for the inertial interval. In this case, for the models of K. Hasselmann and Yu.M. Lytkin the power-law dependence D _LL (r) ～ r ^2/3 expected for the inertial interval is obtained for the structure function D _LL (r).     

An experimental investigation of crack-path directional stability

Using the criterion that a crack will extend perpendicular to the maximum circumferential stress,σ _θ, we show that the directional stability of crack growth is governed by the location of microcrack initiation ahead of the crack tip. At distances greater than a geometrical radiusr _o, the maximum value ofσ _θ deviates from the position of symmetry. Thus, if we assume that the physical processes involved in fracture lead to crack initiation at a distancer _c ahead of the crack tip, the criterion for directional stability isr _o>r _c. Experimental and theoretical values ofr _o verify that, asr _o becomes small, the crack's directional stability deteriorates. Observing that a lengthwise compressive stress increasesr _o, a center-cracked specimen was developed which allows the application of controlled lengthwise compression independently of the opening-mode load. A detailed photoelastic analysis of the specimen has provided the value ofr _o as a function of the crack length. The value ofr _o is then compared with the expected microcrack initiation distances in ductile fracture. By applying sufficient lengthwise compression, we are able to make the crack grow straight and obtain numerous data points from this specimen which would otherwise be directionally unstable. The results indicate that, as the total lengthwise tensile stress at the crack tip increases, the fracture toughness also increases. Using this information we can then adjustK _Ic for zero lengthwise loading and obtain a geometry independent fracture toughness.     

Capillary Pressure Curve for Liquid Menisci in a Cubic Assembly of Spherical Particles Below Irreducible Saturation

The capillary pressure–saturation relationship, P _c(S _w), is an essential element in modeling two-phase flow in porous media (PM). In most practical cases of interest, this relationship, for a given PM, is obtained experimentally, due to the irregular shape of the void space. We present the P _c(S _w) curve obtained by basic considerations, albeit for a particular class of regular PM. We analyze the characteristics of the various segments of the capillary pressure curve. The main features are the behavior of the P _c(S _w) curve as the wetting-fluid saturation approaches zero, and as this saturation is increased beyond a certain critical value. We show that under certain conditions (contact angle, distance between spheres, and saturation), the value of the capillary pressure may change sign.     

Stability of ribbed cylindrical shell under local edge loads

We examine a hinged cylindrical shell (l, r, h are the length, radius, wall thickness) that is regularly stiffened by stringers (k is the number of stringers; F, I, I^t are the cross-section area and the moments of inertia in bending and torsion) and frames (K₁ is the number of frames; I₁, I ₁ ^t are the moments of inertia in bending in the plane of the ring and in torsion). Compatibility of the deformation of the skin and the ribs is satisfied with respect to deflections and angles of rotation, the eccentricity of the placement of the ribs relative to the middle surface of the skin and the discreteness of arrangement of the stiffeners are not considered. To the edge of the shells there are applied cyclically K₂ local axial loads, which are represented in the form of uniformly distributed normal stresses P₂ on elementary regions (areas) of arc length b. The shell is additionally loaded by an axial force that is applied in the form of uniformly distributed over the end normal stresses P, and also by internal pressure of intensity q, applied to the side surface. The stability problem is solved in the linear formulation for the moment-free precritical state; the energy method in the Timoshenko form is used.S. P. Timoshenko Institute of Mechanics of the Ukraine Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 1, pp. 45–50, 1994.     

3-D kinematical conservation laws (KCL): Evolution of a surface in – in particular propagation of a nonlinear wavefront

3-D KCL are equations of evolution of a propagating surface (or a wavefront) Ω_t in 3-space dimensions and were first derived by Giles, Prasad and Ravindran in 1995 assuming the motion of the surface to be isotropic. Here we discuss various properties of these 3-D KCL. These are the most general equations in conservation form, governing the evolution of Ω_t with singularities which we call kinks and which are curves across which the normal n to Ω_t and amplitude w on Ω_t are discontinuous. From KCL we derive a system of six differential equations and show that the KCL system is equivalent to the ray equations of Ω_t. The six independent equations and an energy transport equation (for small amplitude waves in a polytropic gas) involving an amplitude w (which is related to the normal velocity m of Ω_t) form a completely determined system of seven equations. We have determined eigenvalues of the system by a very novel method and find that the system has two distinct nonzero eigenvalues and five zero eigenvalues and the dimension of the eigenspace associated with the multiple eigenvalue 0 is only 4. For an appropriately defined m, the two nonzero eigenvalues are real when m>1 and pure imaginary when m<1. Finally we give some examples of evolution of weakly nonlinear wavefronts.     

Forced Convection Heat Transfer from a Heated Cylinder in an Axial Background Flow in a Porous Medium: Three Exactly Solvable Cases

Assuming a background flow of velocity U = U(x) in the axial direction x of a circular cylinder with surface temperature distribution T _w = T _w (x) in a saturated porous medium, for the temperature boundary layer occurring on the cylinder three exactly solvable cases are identified. The functions {U(x), T _w (x)} associated with these cases are given explicitly, and the corresponding exact solutions are expressed in terms of the modified Bessel function K ₀ (z), the incomplete Gamma function Γ (a, z) and the confluent hypergeometric function U(a, b, z), respectively. The correlation between the Nusselt number and the Péclet number as well as the curvature effects on the heat transfer are discussed in all these cases in detail. Some “universal” features of the exponential surface temperature distribution are also pointed out.     

State of stress at the vertex of a quarter-infinite crack in a half-space

Spherical coordinates are r, θ, φ. The half-space extends in θ < π/2. The crack occurs along φ = 0. The region to be investigated is the solid space-triangle (or cone) between the three planes θ = π/2, φ = +0 and φ = 2π ? 0, which planes are to be taken stress-free.In this space-angle a state of stress is considered in terms of the cartesian stress components σ_xx = r^λ?_xx(λ, θ, φ); σ_xy = r^λ(λ, θ, φ); etc. Possible values λ are determined from a characteristic (or eigenvalue) equation, expressing the condition that a determinant of infinite order is equal to zero. The root of λ which gives the most serious state of stress in the vertex region (the region r → 0) is the root closest to the limiting value Re λ > ?3/2. Knowledge of this state of stress, or at least of this value of λ is essential in the determination of the three-dimensional state of stress around a crack in a plate for distances of order of the plate thickenss.Along the front of the quarter-infinite crack (z-axis) the so called stress-intensity factor behaves like z^λ+½ (z → 0) and thus tends to zero, respectively to infinity, accordingly to Re λ being >?½ or <?½. But in the region z → 0 the notion stress-intensity factor loses its meaning. The required state of stress passes into the well-known state of plane strain around a crack tip if Poisson's ratio (v) tends to zero. The computed state of stress for the incompressible medium (v = ½) is confirmed by experiment.     

Localized bulging in an inflated cylindrical tube of arbitrary thickness – the effect of bending stiffness

We study localized bulging of a cylindrical hyperelastic tube of arbitrary thickness when it is subjected to the combined action of inflation and axial extension. It is shown that with the internal pressure P and resultant axial force F viewed as functions of the azimuthal stretch on the inner surface and the axial stretch, the bifurcation condition for the initiation of a localized bulge is that the Jacobian of the vector function $(P,F)$ should vanish. This is established using the dynamical systems theory by first computing the eigenvalues of a certain eigenvalue problem governing incremental deformations, and then deriving the bifurcation condition explicitly. The bifurcation condition is valid for all loading conditions, and in the special case of fixed resultant axial force it gives the expected result that the initiation pressure for localized bulging is precisely the maximum pressure in uniform inflation. It is shown that even if localized bulging cannot take place when the axial force is fixed, it is still possible if the axial stretch is fixed instead. The explicit bifurcation condition also provides a means to quantify precisely the effect of bending stiffness on the initiation pressure. It is shown that the (approximate) membrane theory gives good predictions for the initiation pressure, with a relative error less than 5%, for thickness/radius ratios up to 0.67. A two-term asymptotic bifurcation condition for localized bulging that incorporates the effect of bending stiffness is proposed, and is shown to be capable of giving extremely accurate predictions for the initiation pressure for thickness/radius ratios up to as large as 1.2.     

Flow patterns in polyethylene and polystyrene melts during extrusion through a die entry region: measurement and interpretation

Flow visualization experiments have been carried out on these melts flowing from a reservoir into a capillary die. The existence and magnitude of vortices at the die entrance have been determined over a range of extrusion conditions. The vortex size is interpreted in terms of the theory of viscoelastic fluid mechanics. It is found that the second-order fluid-perturbation solution cannot represent the observed experimental results. The data are correlated with (i) a Weissenberg number τ_ch$\text{V}\text{L}$ ≡ $\text{}\text{?}$gt($\text{γ}\text{?}$_w)$\text{γ}\text{?}$_w ≡ Ψ₁$\text{γ}\text{?}$_w/2η  (N₁)_w/ 2(σ₁₂)_w measured at the die wall and (ii) with the deformation-rate dependence of relaxation time. Interpretation of vortex formation and size in terms of elongational viscosity is offered.Several polystyrene and polyethylene melts have been rheologically characterized as part of this study with measurements of viscosity η and principal normal stress difference N₁. The zero shear viscosity η₀ of the polystyrenes varies with the 3.5 power of the weight-average molecular weight M_w while the principal normal stress difference coefficient Ψ₁ varies with the sixth power of M_w when evaluated at a shear rate of 1 sec^?1.     

Direct Numerical Simulation and Theory of a Wall-Bounded Flow with Zero Skin Friction

We study turbulent plane Couette-Poiseuille (CP) flows in which the conditions (relative wall velocity ΔU _w ≡ 2U _w , pressure gradient dP/dx and viscosity ν) are adjusted to produce zero mean skin friction on one of the walls, denoted by APG for adverse pressure gradient. The other wall, FPG for favorable pressure gradient, provides the friction velocity u _τ , and h is the half-height of the channel. This leads to a one-parameter family of one-dimensional flows of varying Reynolds number Re ≡ U _w h/ν. We apply three codes, and cover three Reynolds numbers stepping by a factor of two each time. The agreement between codes is very good, and the Reynolds-number range is sizable. The theoretical questions revolve around Reynolds-number independence in both the core region (free of local viscous effects) and the two wall regions. The core region follows Townsend’s hypothesis of universal behavior for the velocity and shear stress, when they are normalized with u _τ and h; on the other hand universality is not observed for all the Reynolds stresses, any more than it is in Poiseuille flow or boundary layers. The FPG wall region obeys the classical law of the wall, again for velocity and shear stress. For the APG wall region, Stratford conjectured universal behavior when normalized with the pressure gradient, leading to a square-root law for the velocity. The literature, also covering other flows with zero skin friction, is ambiguous. Our results are very consistent with both of Stratford’s conjectures, suggesting that at least in this idealized flow turbulence theory is successful like it was for the classical logarithmic law of the wall. We appear to know the constants of the law within a 10% bracket. On the other hand, that again does not extend to Reynolds stresses other than the shear stress, but these stresses are passive in the momentum equation.     

Stochastic stabilization of quasi non-integrable Hamiltonian systems

A procedure for designing a feedback control to asymptotically stabilize in probability a quasi non-integrable Hamiltonion system is proposed. First, an one-dimensional averaged Itô stochastic differential equation for controlled Hamiltonian is derived from given equations of motion of the system by using the stochastic averaging method for quasi non-integrable Hamiltonian systems. Second, a dynamical programming equation for an ergodic control problem with undetermined cost function is established based on the stochastic dynamical programming principle and solved to yield the optimal control law. Third, the asymptotic stability in probability of the system is analysed by examining the sample behaviors of the completely averaged Itô differential equation at its two boundaries. Finally, the cost function and the optimal control forces are determined by the requirement of stabilizing the system. Two examples are given to illustrate the application of the proposed procedure and the effect of control on the stability of the system.     

LES of turbulent flow in a concentric annulus with rotating outer wall

Fully-developed turbulent flow in a concentric annulus, r₁/r₂ = 0.5, Re_h = 12,500, with the outer wall rotating at a range of rotation rates N = U_θ,wall/U_b from 0.5 up to 4 is studied by large-eddy simulations. The focus is on the effects of moderate to very high rotation rates on the mean flow, turbulence statistics and eddy structure. For N up to ∼2, an increase in the rotation rate dampens progressively the turbulence near the rotating outer wall, while affecting only mildly the inner-wall region. At higher rotation rates this trend is reversed: for N = 2.8 close to the inner wall turbulence is dramatically reduced while the outer wall region remains turbulent with discernible helical vortices as the dominant turbulent structure. The turbulence parameters and eddy structures differ significantly for N = 2 and 2.8. This switch is attributed to the centrifuged turbulence (generated near the inner wall) prevailing over the axial inertial force as well as over the counteracting laminarizing effects of the rotating outer wall. At still higher rotation, N = 4, the flow gets laminarized but with distinct spiralling vortices akin to the Taylor–Couette rolls found between the two counter-rotating cylinders without axial flow, which is the limiting case when N approaches to infinity. The ratio of the centrifugal to axial inertial forces, Ta/Re² ∝ N² (where Ta is the Taylor number) is considered as a possible criterion for defining the conditions for the above regime change.     

The effect of variable properties on laminar boundary layer flow

The influence of temperature dependent fluid properties on laminar boundary layer flows is examined for wedge flows (Falkner-Skan). An asymptotic expansion for small heat transfer rates is applied under the boundary conditionsT _w= const andq _w=const. Linear deviations from the zero order solution (constant properties) can be given in a form known as the property ratio method. As a consequence this method is no longer an empirical one.     

Deformation of a Convex Hydrogel Shell by Parallel Plate and Central Compression

To characterize the materials parameters and deformation of a convex shell of axial symmetry, a hydrogel contact lens is mechanically deformed by two loading configurations: (a) compression between two parallel plates and (b) central load applied by a shaft with a spherical tip. A universal testing machine with nano-Newton and submicron resolutions is used to measure the applied force, F, as a function of vertical displacement of the plate/shaft, w ₀, while a homemade laser aided topography system records the in-situ deformed shell profile and the contact radius or central dimple, a. A nonlinear shell theory and an iterative finite difference method are used to account for the large elastic deformation, the central buckling for the central load compression, and the interrelationship between the measureable quantities (F, w ₀, a).     

Steady and transient shear flows of polystyrene solutions I: Concentration and molecular weight dependence of non-dimensional viscometric functions

Start up from rest and relaxation from steady shear flow experiments have been performed on monodisperse polystyrene solutions with molecular weight ranging from 1.3 × 10⁵ to 1.6 × 10⁶ and concentration c ranging from 5% to 40%. A method of reduced variables based on the use of a characteristic time τ_w is proposed. τ_w is defined as the product of zero shear viscosity with the steady state elastic compliance.Reduced steady and transient viscometric functions so obtained depend on the ratio M/M_e (where M_e is the entanglement molecular weight). Limiting forms are obtained when M/M_e ? 18. In steady flow, a simple correlation is found between shear and normal stresses.In stress relaxation experiments, independent of shear rate, the long-time behaviour can be characterised by a single relaxation time τ₁, which is identical for shear and normal stresses. τ₁ can be simply related to the zero shear rate viscosity and the limiting elastic compliance.     

A simple criterion for finite time stability with application to impacted buckling of elastic columns

The existing theories of finite-time stability depend on a prescribed bound on initial disturbances and a prescribed threshold for allowable responses. It remains a challenge to identify the critical value of loading parameter for finite time instability observed in experiments without the need of specifying any prescribed threshold for allowable responses. Based on an energy balance analysis of a simple dynamic system, this paper proposes a general criterion for finite time stability which indicates that finite time stability of a linear dynamic system with constant coefficients during a given time interval [0, t _f ] is guaranteed provided the product of its maximum growth rate (determined by the maximum eigen-root p₁ >0) and the duration t _f does not exceed 2, i.e., p₁t _f <2. The proposed criterion (p₁t _f =2) is applied to several problems of impacted buckling of elastic columns: (i) an elastic column impacted by a striking mass, (ii) longitudinal impact of an elastic column on a rigid wall, and (iii) an elastic column compressed at a constant speed (“Hoff problem”), in which the time-varying axial force is replaced approximately by its average value over the time duration. Comparison of critical parameters predicted by the proposed criterion with available experimental and simulation data shows that the proposed criterion is in robust reasonable agreement with the known data, which suggests that the proposed simple criterion (p₁t _f =2) can be used to estimate critical parameters for finite time stability of dynamic systems governed by linear equations with constant coefficients.