Nonstationary pandom vibpation analysis of linear elastic structures with finite element method

At present, the finite element method is an efficient method for analyzing structural dynamic problems. When the physical quantities such as displacements and stresses are resolved in the spectra and the dynamic matrices are obtained in spectral resolving form, the relative equations cannot be solved by the vibration mode resolving method as usual. For solving such problems, a general method is put forward in this paper. The excitations considered with respect to nonstationary processes are as follows: P(t)={P_i(t)},P_i(t)=a_i(t)P_i(t), a_i(t) is a time function already known. We make Fourier transformation for the discretized equations obtained by finite element method, and by utilizing the behaviour of orthogonal increment of spectral quantities in random process^[1], some formulas of relations about the spectra of excitation and response are derived. The cross power spectral denisty matrices of responses can be found by these formulas, then the structrual safety analysis can be made. When a_i(t)=l (i= 1,2,…n), the. method stated in this paper will be reduced to that which is used in the special case of stationary process.     

Stability of Degenerate Stationary Waves for Viscous Gases

This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous gases in the half space. We discuss the following two cases: (1) viscous conservation laws and (2) damped wave equations with nonlinear convection. In each case, we prove that the solution converges to the corresponding degenerate stationary wave at the rate t ^−α/4 as t → ∞, provided that the initial perturbation is in the weighted space L²_a=L²(\mathbb R₊; (1+x)^a dx){L^2_\alpha=L^2({\mathbb R}_+;\,(1+x)^\alpha dx)} . This convergence rate t ^−α/4 is weaker than the one for the non-degenerate case and requires the restriction α < α_*(q), where α_*(q) is the critical value depending only on the degeneracy exponent q. Such a restriction is reasonable because the corresponding linearized operator for viscous conservation laws cannot be dissipative in L²_a{L^2_\alpha} for α > α^*(q) with another critical value α^*(q). Our stability analysis is based on the space–time weighted energy method in which the spatial weight is chosen as a function of the degenerate stationary wave.     

Symmetry and secondary bifurcation of elastic structures

We consider non-linear bifurcation problems for elastic structures modeled by the operator equation F[w;α]=0 where F:X×R^k→Y,X,Y are Banach spaces and XY. We focus attention on problems whose bifurcation equations are of the form

f_i(α₁,α₂;λ,μ)=(a_iμ+b_iλ)α_i+p_iα_i³+q_iα_i∑_j=1,j≠i^kα_j+1²+α_ih_i(λ,μ;α₁,α₂,…α_k) i=1,2,…k

which emanates from bifurcation problems for which the linearization of F is Fredholm operators of index 0. Under the assumption of F being odd we prove an important theorem of existence of secondary bifurcation. Under this same assumption we prove a symmetry condition for the reduced equations and consequently we got an existence result for secondary bifurcation. We also include a stability analysis of the bifurcating solutions.     

Effect of hydrodynamic forces on the pressure-difference equation

Because of the influence of hydrodynamic forces, the difference in macroscopic pressure which exists, at static equilibrium, between two immiscible phases located in a porous medium may be different from that which pertains during flow. In this paper, the concept of relative pressure difference, together with a new pressure-difference equation, is used to investigate the impact that the hydrodynamic forces have on the difference in macroscopic pressure which pertains when two immiscible fluids flow simultaneously through a homogeneous, water-wet porous medium. This investigation reveals that, in general, the equation defining the difference in pressure between two flowing phases must include a term which takes proper account of the hydrodynamic effects. Moreover, it is pointed out that, while neglect of the hydrodynamic effects introduces only a small amount of error when the two fluids are flowing cocurrently, such neglect is not permissible during steady-state, countercurrent flow. This is because failure to include the impact of the hydrodynamic effects in the latter case makes it impossible to explain the pressure behaviour observed in steady-state, countercurrent flow. Finally, the results of this investigation are used as a basis for arguing that, during steady-state, countercurrent flow, saturation is uniform, as is the case of steady-state, cocurrent flow.Roman Letters a parameter in Equation (18) - k absolute permeability, m² - k _i effective permeability to phasei;i=1, 2, m² - k _ij generalized effective permeability for phasei;i, j=1, 2, m² - p _d p ₂–p ₁=difference in macroscopic pressure between two flowing phases, N/m² - p _i pressure for phasei;i=1, 2, N/m² - p _h hydrodynamic contribution to difference in macroscopic pressure which exists during flow, N/m² - P _c macroscopic static capillary pressure, N/m² - R ₁₂ function defined by Equation (18) - S _i saturation of phasei;i=1, 2 - S _n normalized saturation of phase 1 - t time, s - u _i flux of phasei;i=1, 2m³/m²/s - x distance in direction of flow, m Greek Letters _R relative pressure difference - _i k _i / _i =mobility of phasei;i=1, 2m²/Pa·s - _ij k _ij / _j =generalized mobility of phasei;i, j=1, 2m²/Pa·s - _i viscosity of phasei;i=1, 2, Pa·s - porosity     

On the three-dimensional vibrations of a hollow elastic torus of annular cross-section

This paper offers a three-dimensional elasticity-based variational Ritz procedure to examine the natural vibrations of an elastic hollow torus with annular cross-section. The associated energy functional minimized in the Ritz procedure is formulated using toroidal coordinates (r,q, j)({r,\theta , \varphi}) comprised of the usual polar coordinates (r, θ) originating at each circular cross-sectional center and a circumferential coordinate j{\varphi} around the torus originating at the torus center. As an enhancement to conventional use of algebraic–trigonometric polynomials trial series in related solid body vibration studies in the associated literature, the assumed torus displacement, u, v and w in the r, θ and j{\varphi } toroidal directions, respectively, are approximated in this work as a triplicate product of Chebyshev polynomials in r and the periodic trigonometric functions in the θ and j{\varphi} directions along with a set of generalized coefficients. Upon invoking the stationary condition of the Lagrangian energy functional for the elastic torus with respected to these generalized coefficients, the usual characteristic frequency equations of natural vibrations of the elastic torus are derived. Upper bound convergence of the first seven non-dimensional frequency parameters accurate to at least five significant figures is achieved by using only ten terms of the trial torus displacement functions. Non-dimensional frequencies of elastic hollow tori are examined showing the effects of varying torus radius ratio and cross-sectional radius ratio.     

Explicit solitary-wave solutions to generalized Pochhammer-Chree equations

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Equivalent lagrangians and the solution of some classes of non-linear equations

Invasion percolation was studied on three-dimensional regular lattices of various node numbers. A new model has been developed to obtain the pore-size distribution from capillary pressure measurements. The new model is superior to the conventional percolation model, since it takes into account the physical trapping of the wetting phase. The irreducible wetting phase saturation includes the film of the wall of the pores, the dead-end pore volume, and the main contribution by pores isolated from the outlet of the medium by the nonwetting phase. This has been related to the node number and the sample 3dimensions. Over 100 capillary pressure curves of consolidated media have been collected. Good agreement was obtained between this data set out and our invasion percolation predictions using node numbers of 6–13, as reported by Mishra and Sharma. The pore-throat size distribution function estimated by our new model is broader than from the conventional percolation and the capillary tube models.Nomenclature c constant - D pore throat diameter [m] - D _max maximum pore diameter [m] - f(D) correlation function of pore throat size and pore body size - L a parameter representing the dimension of a sample - n node number - p pressure [N/m²] - S _n the nonwetting phase saturation - x random number ranging from 0 to 1.0 - X ^a X _t ^a /X/ _t - X _e ^a X _t ^a –X _t ⁱ - X ⁱ X _t ⁱ /X _t ^a - X _nw fraction of pore volume occupied by the injected phase - X _t fraction of pores larger thanD - X _t ^a total accessibility of pores larger thanD - X _t ⁱ total isolation of pores larger thanD - contact angle - interfacial tension [N/m] - (D) pore throat size distribution     

Interpretation of capillary pressure curves using invasion percolation theory

Invasion percolation was studied on three-dimensional regular lattices of various node numbers. A new model has been developed to obtain the pore-size distribution from capillary pressure measurements. The new model is superior to the conventional percolation model, since it takes into account the physical trapping of the wetting phase. The irreducible wetting phase saturation includes the film of the wall of the pores, the dead-end pore volume, and the main contribution by pores isolated from the outlet of the medium by the nonwetting phase. This has been related to the node number and the sample 3dimensions. Over 100 capillary pressure curves of consolidated media have been collected. Good agreement was obtained between this data set out and our invasion percolation predictions using node numbers of 6–13, as reported by Mishra and Sharma. The pore-throat size distribution function estimated by our new model is broader than from the conventional percolation and the capillary tube models.Nomenclature c constant - D pore throat diameter [m] - D _max maximum pore diameter [m] - f(D) correlation function of pore throat size and pore body size - L a parameter representing the dimension of a sample - n node number - p pressure [N/m²] - S _n the nonwetting phase saturation - x random number ranging from 0 to 1.0 - X ^a X _t ^a /X/ _t - X _e ^a X _t ^a –X _t ⁱ - X ⁱ X _t ⁱ /X _t ^a - X _nw fraction of pore volume occupied by the injected phase - X _t fraction of pores larger thanD - X _t ^a total accessibility of pores larger thanD - X _t ⁱ total isolation of pores larger thanD - contact angle - interfacial tension [N/m] - (D) pore throat size distribution     

Similarities Between Rotation and Stratification Effects on Homogeneous Shear Flow

Linear theory is applied to examine rotation and buoyancy effects on homogeneous turbulent shear flows with given vertical velocity shear, S=d/dx ₃. In the rotating shear case (where the rotation vector is perpendicular to the plane of the mean flow, Ω _i =Ωδ _{i 2}), general solutions for the Fourier components of the fluctuating velocity are proposed. These solutions are compared with those proposed in the literature for the Fourier components of the fluctuating velocity and density in the case of a homogeneous stratified shear flow with vertical density gradient, S _ρ=d/dx ₃. It is shown that from the normal mode stability stand point the Bradshaw parameter B=2Ω/S(1+2Ω/S) (in the rotating shear case) and the Richardson number R _i (in the statified shear case) play similar roles in identifying the stability for all the wave components except in the case where Ω·k=0, for which rotation has no effects on the flow. Analysis of the long-time behavior of the non-dimensional spectral density of energy, e ^g , is carried out. In the stable case, e ^g has decaying oscillations or undergoes a power law decay in time. Analytical solutions for the streamwise two-dimensional energy ℰ _ii ¹/2 (i.e. the limit at k ₁=0 of the one-dimensional energy spectra) are proposed. At large time, ℰ _ii ¹(t)/ℰ _ii ¹(0) oscillates around the value (3R _i +1)/(4R _i ) except at R _i =1 it stays constant in time. Similar behavior for ℰ _ii ¹(t)/ℰ _ii ¹(0) is also observed in the rotating shear case (ℰ _ii ¹(t)/ℰ _ii ¹(0) oscillates around the value (1+4B)/(4B)). Due to the behavior of the dimensionless spectral density of energy in both flow cases, the turbulent kinetic energy, /2, the production rate, ?, and the rate due to the buoyancy forces, ℬ, are split into two parts, , ?=?₁+?₂, ℬ=ℬ₁+ℬ₂ (in the stratified shear case, both ?₁ and ℬ₁ vanish when R _i >?, while in the rotating shear case one has ℬ=0). It is shown that when rotation is “cyclonic” (i.e. Ω/S>0), part reaches maximum magnitudes at St ≈2, independent of the B value, and the first time to a zero crossing of ?₂ occurs at this particular value. When rotation is “anticyclonic” (i.e. Ω/S<0) one finds St ≈1.6 instead of St ≈2. In the stratified shear case, both ?₂ and ℬ₂ cross zero at Nt=St ≈2, and part reaches maximum magnitudes at this particular value. These results and in particular those for the turbulent kinetic energy are compared with previous direct numerical simulation (DNS) results in homogeneous stratified shear flows. Received 30 July 2001 and accepted 19 February 2002     

On the nonlinear slewing dynamics and control of the Space Station based Mobile Servicing System

A relatively general Lagrangian formulation for studying the nonlinear dynamics and control of space-craft with interconnected flexible members in a tree-type topology is developed. Versatility of the formulation is illustrated through a dynamical study of the Space Station based two-link Mobile Servicing System (MSS). The performance of the MSS undergoing inplane and out-of-plane slewing maneuvers is compared. Results indicate that, in absence of control, the maneuvers induce undesirable librational motion of the Space Station as well as vibration of the links. Nonlinear control, based on the Feedback Linearization Technique (FLT), appears promising. Quasi-Closed Loop Control (QCLC), a variation of the FLT, is applied to control the libration of the Space Station. Once the attitude of the Space Station is controlled, the performance of the MSS improves significantly. For a 5-minute maneuver of the MSS, the maximum control torque required is only 34.5 Nm.Nomenclature f _i ¹ , f _i,j ¹ fundamental frequency of bodies B _i and B _i,j, respectively - l _c, l _i, l _i,j length of bodies B _c, B _i, and B _i,j, respectively - m _c, m _i, m _i,j mass of bodies B _c, B _i, and B _i,j, respectively - % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaerbhv2BYDwAHbacfiGab8xCayaaraqefavySfgDP52BGWuAU9gD% 5bxzaGGbciaa+zgacaWFSaGaa8hiaiqa-fhagaqeaiaa-jhaaaa!4B1F!\[\bar qf, \bar qr\] vector representing flexible and rigid generalized coordinates - % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaerbhv2BYDwAHbacfiGaa8hkaiqa-fhagaqeaiaa-jhacaWFPaqe% favySfgDP52BGWuAU9gD5bxzaGGbciaa+rgaaaa!4A18!\[(\bar qr)d\] vector representing the desired rigid generalized coordinates - (I _xx)k, (I _yy)k, (I _zz)k principal inertia of body B _k about X _k, Y _k, and Z _k axes, respectively; ksc, i or i, j - K _p, K _v displacement and velocity gain matrices - N _q total number of generalized coordinates - % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaerbwvMCKfMBHbacfiGab8xuayaaraqefavySfgDP52BGWuAU9gD% 5bxzaGGbciaa+zgaieaacaqFSaGaa0hiaiqa-ffagaqeaGqaciaa8j% haaaa!4AEF!\[\bar Qf, \bar Qr\] control effort vectors for flexible and rigid coordinates, respectively - Q , Q , Q control effort for pitch, roll and yaw degree of freedom, respectively - _k ^y , _k ^z tip deflection of a beam type appendage (B _k) in the Y _k and Z _k directions, respectively.     

Direct numerical simulation of turbulent heat transfer in annuli: Effect of heat flux ratio

Fully developed turbulent flow and heat transfer in a concentric annular duct is investigated for the first time by using a direct numerical simulation (DNS) with isoflux conditions imposed at both walls. The Reynolds number based on the half-width between inner and outer walls, δ=(r₂-r₁)/2, and the laminar maximum velocity is Re_δ=3500. A Prandtl number Pr=0.71 and a radius ratio r^*=0.1 were retained. The main objective of this work is to examine the effect of the heat flux density ratio, q^*=q₁/q₂, on different thermal statistics (mean temperature profiles, root mean square (rms) of temperature fluctuations, turbulent heat fluxes, heat transfer, etc.). To validate the present DNS calculations, predictions of the flow and thermal fields with q^*=1 are compared to results recently reported in the archival literature. A good agreement with available DNS data is shown. The effect of heat flux ratio q^* on turbulent thermal statistics in annular duct with arbitrarily prescribed heat flux is discussed then. This investigation highlights that heat flux ratio has a marked influence on the thermal field. When q^* varies from 0 to 0.01, the rms of temperature fluctuations and the turbulent heat fluxes are more intense near the outer wall while changes in q^* from 1 to 100, lead to opposite trends.     

On the Influence of the Kinetic Energy¶on the Stability of Equilibria¶of Natural Lagrangian Systems

Natural Lagrangian systems (T,Π) on R ² described by the equation are considered, where is a positive definite quadratic form in and Π(q) has a critical point at 0. It is constructively proved that there exist a C ^∞ potential energy Π and two C ^∞ kinetic energies T and such that the equilibrium q(t)≡ 0 is stable for the system (T,Π) and unstable for the system . Equivalently, it is established that for C ^∞ natural systems the kinetic energy can influence the stability. In the analytic category this is not true. Accepted: October 20, 1999     

Squeezing flows of Newtonian liquid films an analysis including fluid inertia

A theoretical study is made of the flow behavior of thin Newtonian liquid films being squeezed between two flat plates. Solutions to the problem are obtained by using a numerical method, which is found to be stable for all Reynolds numbers, aspect ratios, and grid sizes tested. Particular emphasis is placed on including in the analysis the inertial terms in the Navier-Stokes equations.Comparison of results from the numerical calculation with those from Ishizawa's perturbation solution is made. For the conditions considered here, it is found that the perturbation series is divergent, and that in general one must use a numerical technique to solve this problem.Nomenclature a half of the distance, or gap, between the two plates - a ₀ the value of a at time t=0 - adot da/dt - ä d² a/dt ² - d³ a/dt ³ - a ⁱ components of a contravariant acceleration vector - f unknown function of z ₀ and t defined in (6) - f ⁱ function defined in (9) f ¹=r ₀ g(z ₀, t) f ²= ₀ f ³=f(z ₀, t) - F force applied to the plates - g unknown function of z ₀ and t defined in (6) - g g/z ₀ - h grid dimension in the z ₀ direction (see Fig. 5) - Christoffel symbol - i, j, k, l indices - k grid dimension in the t direction (see Fig. 5) - r radial coordinate direction defined in Fig. 1 - r ₀ radial convected coordinate - R radius of the circular plates - t time - v _r fluid velocity in the r direction - v _z fluid velocity in the z direction - v fluid velocity in the direction - x ⁱ cylindrical coordinate x ¹=r x²= x³=z - z vertical coordinate direction defined in Fig. 1 - z ₀ vertical convected coordinate - tangential coordinate direction - ₀ tangential convected coordinate - viscosity - kinematic viscosity, / - ⁱ convected coordinate ¹=r₀ ²=₀ ³=z₀ - density     

Symmetry groups and the pseudo-Riemann spacetimes for mixed-hardening elastoplasticity

The constitutive postulations for mixed-hardening elastoplasticity are selected. Several homeomorphisms of irreversibility parameters are derived, among which X_a⁰ and X_c⁰ play respectively the roles of temporal components of the Minkowski and conformal spacetimes. An augmented vector X_a:=(YQ_a^t,YQ_a⁰)^t is constructed, whose governing equations in the plastic phase are found to be a linear system with a suitable rescaling proper time. The underlying structure of mixed-hardening elastoplasticity is a Minkowski spacetime $\text{M}{}^{\mathrm{n+1}}$ on which the proper orthochronous Lorentz group SO_o(n,1) left acts. Then, constructed is a Poincaré group ISO_o(n,1) on space X:=X_a+X_b, of which X_b reflects the kinematic hardening rule in the model. We also find that the space (Q_a^t,q₀^a) is a Robertson–Walker spacetime, which is conformal to X_a through a factor Y, and conformal to X_c:=(ρQ_a^t,ρQ_a⁰)^t through a factor ρ as given by ρ(q₀^a)=Y(q₀^a)/[1−2ρ₀Q_a⁰(0)+2ρ₀Y(q₀^a)Q_a⁰(q₀^a)]. In the conformal spacetime the internal symmetry is a conformal group.     

Impact energy absorption by a rate-dependent locking target

The impact by an elastic cylindrical piston on a thin plate-like target resting on a rigid foundation is considered. The relationship between force F acting on the target and displacement x is given by F=kx+q dx/dt provided dx/dt≥0 and 0≤x<d (k, q and d≥0). When x=d locking occurs, and F can assume any value ≥kd without increase in x. The displacement is assumed to be completely irreversible. The motion of the impactor is assumed to be governed by the elementary wave equation and, since the target is thin, wave motion in the target is neglected. The energy W=εFdx and its components W _k=kεx dx (the energy absorbed in a corresponding quasistatic process) and W _q=qε(dx/dt)² dt (the excessive energy because of the rate-dependence) are determined in terms of the impact energy as functions of non-dimensional parameters representing k, q and d. With the aid of diagrams, it is shown under what circumstances locking occurs, and under what circumstances W _k or W _q, or both, are large.     

The equations of miscible flow with negligible molecular diffusion

A set of equations with generalized permeability functions has been proposed by de la Cruz and Spanos, Whitaker, and Kalaydjian to describe three-dimensional immiscible two-phase flow. We have employed the zero interfacial tension limit of these equations to model two phase miscible flow with negligible molecular diffusion. A solution to these equations is found; we find the generalized permeabilities to depend upon two empirically determined functions of saturation which we denote asA andB. This solution is also used to analyze how dispersion arises in miscible flow; in particular we show that the dispersion evolves at a constant rate. In turn this permits us to predict and understand the asymmetry and long tailing in breakthrough curves, and the scale and fluid velocity dependence of the longitudinal dispersion coefficient. Finally, we illustrate how an experimental breakthrough curve can be used to infer the saturation dependence of the underlying functionsA andB.Roman Letters A a surface area; cross-sectional area of a slim tube or core - A _1s pore scale area of interface between solid and fluid 1 - A ₁₂ pore scale area of interface between fluid 1 and fluid 2 - A(S ₁) fluid flow weighting function defined by Equation (3.21) - a _i ,b _a ,c _a ,d _i macro scale parameters,i=1...2 (Section 3); polynomial coefficients,i=1...N (Section 7) - B(S ₁) fluid flow weighting function defined by Equation (3.16) - c _e effluent concentration - c _i mass concentration fluidi=1...2 - c _fi fractional mass concentration of fluidi=1...2 - D dispersion tensor - D _m mechanical dispersion tensor - D ₀ molecular dispersion tensor - D _L longitudinal dispersion coefficient - D _T transverse dispersion coefficient - D _L ⁰ defined by Equation (6.21) - F(c _f2) defined by Equation (5.17) - f ₁(S ₁) fractional flow - g acceleration of gravity - j ₂ deviation mass flux of fluid 2 - K permeability of porous medium - K _ij generalized relative permeability function,i=1...2,j=1...2 - K _ri relative permeability functions,i=1...2 - L length of a slim tube or core - M _i total mass of fluidi=1...2 in volumeV - N number of points used to generate numerical curves - n unit normal to a surface - P pressure - P _i pressure in fluidi=1...2 - P _c capillary pressure - P ₁₂ macroscopic capillary pressure parameter - P(x) normal distribution function - q Darcy velocity of total fluid - q _i Darcy velocity of fluidi=1...2 - S _i saturation of fluidi=1...2 - S _L a low saturation value forS ₁ - S _H a high saturation value forS ₁ - u average intersitial fluid velocity - u _S isosaturation velocity - V volume used for volume averaging - V(c _f2) function defined by Equation (6.28) - V _e effluent volume - V _f fluid volume - V _i volume of fluidi=1...2 (Section 2); injected fluid volume - V _p pore volume of a slim tube or core - v macro scale fluid velocity - v _i macro scale velocity of fluidi=1...2 - _q (S ₁) isosaturation speed - _g (S ₁) component of isosaturation velocity due to gravity - w(S _L,S _H,t) width of a displacement front - w(t) overall width of a displacement front Greek Letters static interfacial tension - _ME macroscopic dispersivity - divergence operator - porosity - _i fraction of pore space occupied by fluidi=1...2 - (S ₁) effective viscosity of the fluid - _i viscosity of fluidi=1...2 - ₁₂ macroscopic fluid viscosity coupling parameter - macro scale fluid density - _i density of fluid i=1...2 - _q effective gravitational fluid density     

Micro/macro-crack growth due to creep–fatigue dependency on time–temperature material behavior

The implicit character of micro-structural degradation is determined by specifying the time history of crack growth caused by creep–fatigue interaction at high temperature. A dual scale micro/macro-equivalent crack growth model is used to illustrate the underlying principle of multiscaling which can be applied equally well to nano/micro. A series of dual scale models can be connected to formulate triple or quadruple scale models. Temperature and time-dependent thermo-mechanical material properties are developed to dictate the design time history of creep–fatigue cracking that can serve as the master curve for health monitoring.In contrast to the conventional procedure of problem/solution approach by specifying the time- and temperature-dependent material properties as a priori, the desired solution is then defined for a class of anticipated loadings. A scheme for matching the loading history with the damage evolution is then obtained. The results depend on the initial crack size and the extent of creep in proportion to fatigue damage. The path dependent nature of damage is demonstrated by showing the range of the pertinent parameters that control the final destruction of the material. A possible scenario of 20 yr of life span for the 38Cr2Mo2VA ultra-high strength steel is used to develop the evolution of the micro-structural degradation. Three micro/macro-parameters μ^*, d^* and σ^* are used to exhibit the time-dependent variation of the material, geometry and load effects. They are necessary to reflect the scale transitory behavior of creep–fatigue damage. Once the algorithm is developed, the material can be tailor made to match the behavior. That is a different life span of the same material would alter the time behavior of μ^*, d^* and σ^* and hence the micro-structural degradation history. The one-to-one correspondence of the material micro-structure degradation history with that of damage by cracking is the essence of path dependency. Numerical results and graphs are obtained to demonstrate how the inherently implicit material micro-structure parameters can be evaluated from the uniaxial bulk material properties at the macroscopic scale.The combined behavior of creep and fatigue can be exhibited by specifying the parameter ξ with reference to the initial defect size a₀. Large ξ (0.90 and 0.85) gives critical crack size a_cr = 11–14 mm (at t < 20 yr) for a₀ about 1.3 mm. For small ξ (0.05 and 0.15), there results critical a_cr = 6–7 mm (at t < 20 yr) for a₀ about 0.7–0.8 mm. The initial crack is estimated to increase its length by an order of magnitude before triggering global to the instability. This also applies ξ ≈ 0.5 where creep interacts severely with fatigue. Fine tuning of a_cr and a₀ can be made to meet the condition oft = 20 yr.Trade off among load, material and geometric parameters are quantified such that the optimum conditions can be determined for the desired life qualified by the initial–final defect sizes. The scenario assumed in this work is indicative of the capability of the methodology. The initial–final defect sizes can be varied by re-designing the time–temperature material specifications. To reiterate, the uniqueness of solution requires the end result to match with the initial conditions for a given problem. This basic requirement has been accomplished by the dual scale micro/macro-crack growth model for creep and fatigue.     

In the Chapman-Enskog Expansion¶the Burnett Coefficients Satisfy¶the Universal Relation ω3+ω4+θ3= 0

The Chapman–Enskog expansion when applied to a gas of spherical molecules yields formal expressions for the stress deviator P and energy-flux vector q, P=μP ⁽¹⁾+μ² P ⁽²⁾+…, q=μq ⁽¹⁾+μ² q ⁽²⁾+…. The Burnett terms P ⁽²⁾, q ⁽²⁾ depend on 11 coefficients ω _i , 1≦i≦6, θ&; _i , 1≦i≦ 5. This paper shows that ω₃+ω₄+θ₃= 0.     

Thermodynamic analysis of the one-dimensional heat conduction with inhomogeneities in the heat flux direction

Starting from the results of Li, Prigogine and others about the one-dimensional heat conduction with constant temperature boundary conditions, the aim of this paper is to study, according to the methods and the purposes of the generalized thermodynamics, the more general case of one-dimensional heat conduction in systems, whose conductivity is function of both temperature and the coordinate in the heat flux direction and presents a finite number of discontinuities.

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Thermodynamische Analyse für eindimensionale Wärmeströmung mit Ungleichartigkeiten in der Wärmeflußleitung

Zusammenfassung Ausgehend von den Ergebnissen von Li, Prigogine und anderen, für eindimensionale Wärmeströmung mit konstanten Temperaturen an den Grenzen, versucht diese Arbeit, gemäß den Methoden und den Zielen der verallgemeinerten Thermodynamik, den allgemeineren Fall der eindimensionalen Wärmeströmung mit Ungleichartigkeiten zu examinieren.

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Nomenclature c volumetric specific heat - G discriminating parameter [G (t) =k ₂ (t)t ² orG _i (t)=k_i(t) t², when the separation of the variables for thermal conductivity can be done, or in general: G(x, t)=k(x,t)t ²] - J heat transfer rate (generalized flux) [W] - J ₀ heat flux [W/m²] - k thermal conductivity - k ₁ component of thermal conductivity depending upon the coordinate in the heat flux direction - k ₂ component of thermal conductivity depending upon temperature - k _i component of the thermal conductivity of a homogeneous layer (i) depending upon temperature - L ₁,L ₂ extreme coordinates - Lip Lipschitz's function - P entropy production rate - (P(T))_min temperature distribution in a system corresponding to the minimum of entropy production rate - p ₁ thermokinetic potential - (P ₁ (T))_min temperature distribution in a system corresponding to the minimum of thermokinetic potential - P ₂ generalized force potential - Q differential form - dQ total differential form - ¯R set of all real numbers - S(x) area of the isothermal surface corresponding to the coordinatex - T system temperature distribution - t absolute temperature - ¯T set of all the possible system temperature distributions - x symbol of Cartesian product - x coordinate in the heat flux direction - X local generalized force - B,Y ₀,Y ₀ ^*,Z sets (defined in this paper) - function representing the time evolution of the temperature distribution in a system - time - ₀ reference time interval