On surface waves in generalized thermoelasticity

Knowles' representation theorem for harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space is extended to include harmonically time-dependent free processes for thermoelastic surface waves in generalized thermoelasticity of Lord and Shulman and of Green and Lindsay.r , , r , , .This work was done when author was unemployed.     

The CRONE Suspension: Management of the Dilemma Comfort-Road Holding

The CRONE suspension, French acronym of suspension à Comportement Robuste dOrdre Non Entier, results from a traditional suspension system whose spring and damper are replaced by a mechanical and hydropneumatic system defined by a fractional (so-called non-integer) order force–displacement transfer function. ling, frequency-domain robust control design methodology, optimal approach, stability analysis and performance are presented in this paper.     

Direct derivation of the two-shaft system balance conditions for the second order reciprocal inertia forces on the V-type eight cylinders internal combustion engines with a plane crankshaft

By employing the four shafts balance concept paper [1] has reported a balance regime for the second order reciprocal inertia forces on the V-type eight cylinder internal combustion engines with a plane crankshaft. Thereafter, paper [2] has acquired a two-shafts balance regime, but through a rather tedious roudabout degenerating manipulation. The present article has, but starting out directly from the two-shafts balance concept, successfully acquired the same results as those in paper [2]. In addition, we propose, herein, a third balance system which might be, in general, called the slipper balance regime.     

Calculating the parameters of an electron beam that drifts across a strong homogeneous magnetic field

Electron drift in specified fields has been examined in [1] and, as applied to a magnetron, in [2–4] with the averaging method. In [1,2], a first- and in [3,4] in a second-order approximation of the small parameter ) E/²L was used. Here and below, E and H=(c/) are the field strengths, L is the characteristic dimension of the field heterogeneity, is the charge-mass ratio of an electron (>0), and c is the velocity of light. An attempt to construct similar approximations for a drifting electron beam with allowance for the space-charge field, within the framework of the averaging method, involves considerable mathematical difficulties. This paper describes an attempt to solve the latter problem for a stationary monoenergetic beam that drifts under the influence of a plane electric field with potential (x,y) across a strong homogeneous magnetic field H_z H=const. Solutions are constructed by the method of successive approximations, in powers of the parameter =h/L, where h is the Larmor electron radius for narrow beams with a width on the order of 2h.I thank A. N. Ievlevu for assistance in the computational and graphical work, V. Ya. Kislov for a discussion of the results, and L. A. Vainshtein for suggesting the problem examined in §3 and for critical comments.     

Alloy separation of a binary mixture in a stressed elastic sphere

We consider the static state of a spherical isotropic binary elastic solid mixture whose boundary is given a uniform radial displacement. The elastic volumetric strain energy is given by the classical quadratic form from linear elasticity theory,% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakeaacaqGxbGaeyypa0ZaaSaa% aeaacaGGXaaabaGaaiOmaaaacaGG7bGaaCOUdiaacIcacaqGJbGaam% ykaiaadIcacaWG0bGaamOCaiaadwgacaWGPaWaaWbaaSqabeaacaGG% YaaaaOGaey4kaSIaaiOmaiabeY7aTjaacIcacaqGJbGaaiykaiaacY% hacaWGLbGaeyOeI0IaaiiiamaalaaabaGaamymaaqaaiaadodaaaGa% aiikaiaadshacaWGYbGaamyzaiaacMcacaWGXaGaaiiFamaaCaaale% qabaGaaiOmaaaakiaac2hacaGGUaaaaa!63E0!\[{\text{W}} = \frac{1}{2}\{ {\mathbf{\kappa }}({\text{c}})(tre)^2 + 2\mu ({\text{c}})|e - \frac{1}{3}(tre)1|^2 \} .\]Here, e is the infinitesimal strain tensor, c[0, 1] is the volumetric concentration of the mixture, and (·) and (·) are the (positive) bulk and shear material moduli, respectively, which are given functions of the concentration. As a function of c and e, the strain energy function is generally nonconvex. Thus, we consider the nonconvex problem of minimizing the potential energy of the body, among all spatial concentration and displacement fields, subject to a given boundary displacement and a fixed amount of component materials. Assuming spherical symmetry, we find that the two component materials must be separated in the optimal state of minimum potential energy. The harder material forms the central core of the sphere, and the softer material is segregated into a surrounding shell. This behavior is remindful of a general notion in metallurgy that in the casting of materials the harder material tends to migrate toward the center.Partial support of the NSF under grant MSS-9024637 and Alliant Techsystems Inc. is gratefully acknowledged.Professor R. Bartelletti of the Università di Pisa is gratefully acknowledged.     

Approximate estimates of the high‐temperature creep of structural elements

The creep and longterm strength of structural elements under temperatures much higher than operational temperatures are studied. The actual nonuniform stress–strain state is reduced to a fictitious uniform state by averaging the specific dissipated power over the volume of the body. This allows one to estimate the intensity of creep and longterm strength of structural elements in terms of volumeaveraged energy parameters by using methods of ideal plasticity and considering statically possible stress fields and kinematically possible velocity and strain rate fields.     

A rotating hot-wire technique for spatial sampling of disturbed and manipulated duct flows

The documentation and control of flow disturbances downstream of various open inlet contractions was the primary focus with which to evaluate a spatial sampling technique. An X-wire probe was rotated about the center of a cylindrical test section at a radius equal to one-half that of the test section. This provided quasi-instantaneous multi-point measurements of the streamwise and azimuthal components of the velocity to investigate the temporal and spatial characteristics of the flowfield downstream of various contractions. The extent to which a particular contraction is effective in controlling ingested flow disturbances was investigated by artificially introducing disturbances upstream of the contractions. Spatial as well as temporal mappings of various quantities are presented for the streamwise and azimuthal components of the velocity. It was found that the control of upstream disturbances is highly dependent on the inlet contraction; for example, reduction of blade passing frequency noise in the ground testing of jet engines should be achieved with the proper choice of inlet configurations.List of symbols K _uv correlation coefficient= - P percentage of time that an azimuthal fluctuating velocity derivative dv/d is found - U streamwise velocity component U=U (, t) - V azimuthal or tangential velocity component due to flow and probe rotation V=V (, t) - mean value of streamwise velocity component - U _m resultant velocity from and - mean value of azimuthal velocity component induced by rotation - u fluctuating streamwise component of velocity u=u(, t) - v fluctuating azimuthal component of velocity v = v (, t) - u phase-averaged fluctuating streamwise component of velocity u=u(0) - v phase-averaged fluctuating azimuthal component of velocity v=v() - û average of phase-averaged fluctuating streamwise component of velocity (u()) over cases I-1, II-1 and III-1 û = û() - average of phase-averaged fluctuating azimuthal component of velocity (v()) over cases I-1, II-1 and III-1 - u fluctuating streamwise component of velocity corrected for non-uniformity of probe rotation and/or phase-related vibration u = u(0, t) - v fluctuating azimuthal component of velocity corrected for non-uniformity or probe rotation and/or phase-related vibration v=v (, t) - u ² rms value of corrected fluctuating streamwise component of velocity - rms value of corrected fluctuating azimuthal component of velocity - phase or azimuthal position of X-probe     

Extended Korn's inequalities and the associated best possible constants

We are concerned with the coerciveness of the strain energy E(u) (in linear elasticity) associated with a displacement vector u on the Sobolev space H¹ () or its subspaces, a domain in ⁿ representing an isotropic elastic body—certain specific cases are called Korn's inequalities. Sufficient (and necessary) conditions on the Lamé moduli for E(·) to be coercive (or uniformly positive) on such spaces are given, and the associated best possible constants are obtained for some cases.     

Instability of a plane axisymmetric flow with a shear discontinuity in the velocity profile

It is proposed to investigate the stability of a plane axisymmetric flow with an angular velocity profile (r) such that the angular velocity is constant when r < r_O – L and r > r_O + L but varies monotonically from ₁ to ₂ near the point r_O, the thickness of the transition zone being small L r_O, whereas the change in velocity is not small ¦₂ – ₁¦ ₂, ₁. Obviously, as L O short-wave disturbances with respect to the azimuthal coordinate (k=m/r_O 1/r_O) will be unstable with a growth rate-close to the Kelvin—Helmholtz growth rate. In the case L=O (i.e., for a profile with a shear-discontinuity) we find the instability growth rate _O and show that where the thickness of the discontinuity L is finite (but small) the growth rate does not differ from _O up to terms proportional to kL 1 and 1/m 1. Using this example it is possible to investigate the effect of rotation on the flow stability. It is important to note that stabilization (or destabilization) of the flow in question by rotation occurs only for three-dimensional or axisymmetric perturbations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 111–114, January–February, 1985.     

Uniformly convergent difference method for the convection-diffusion singular perturbation problem in a curved boundary region

In this paper we construct a difference scheme for the convection-diffusion singular perturbation problem in a convex curved boundary region, and discuss the uniform convergence of its solution. We have proved that the order of uniform convergence of its solution isO (h +^/2) (0<<1/2), where h, are the mesh steps in the space and time directions respectively.     

Thermal entrance region heat transfer for MHD laminar flow in parallel-plate channels with unequal wall temperatures

The problem of thermal entry heat transfer for Hartmann flow in parallel-plate channels with uniform but unequal wall temperatures considering viscous dissipation, Joule heating and axial conduction effects is approached by the eigenfunction expansion method. The series expansion coefficients for the nonorthogonal eigenfunctions are obtained by using a method for nonorthogonal series described by Kantorovich and Krylov [21]. Numerical results are obtained for the case with entrance condition parameter _o=1 and open circuit condition K=1. The parametric values of Ha=0, 2, 6, 10 and Br=0, –1 are considered for Hartmann and Brinkman numbers, respectively.

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Zusammenfassung Das Problem der Wärmeübertragung im thermischen Einlauf einer Hartmannströmung im ebenen Spalt mit einheitlichen, aber ungleichen Wandtemperaturen wurde unter Berücksichtigung viskoser Dissipation, Joulescher Heizung und axialer Wärmeleitung mit Hilfe einer Entwicklung nach Eigenfunktionen behandelt. Die Koeffizienten der Entwicklung für nichtorthogonale Eigenfunctionen wurde nach einer Methode für nichtorthogonale Reihen nach Kantorovicz und Krylow [21] berechnet. Numerische Ergebnisse werden für den Eintrittsparameter _o=1 und die Bedingung für den offenen Stromkreis K=1 erhalten. Die Parameterwerte Ha=0, 2, 6, 10 und Br=0, –1 werden für die jeweiligen Werte der Hartmann- und der Brinckman-Zahl betrachtet.

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Nomenclature a one-half of channel height - ¯B,B₀ magnetic field Induction vector and magnitude of applied magnetic field - Br Brinkman number, _f U_m ²/(k_c) - C_n,D_n coefficients in the series expansion of _e, see eq. 16 - c_p specific heat at constant pressure - ,E₀ electric field intensity vector and component - E_n,O_n even and odd eigenfunctions - Ha Hartmann number, (/_f)^1/2 B_o a - h₁,h₂ local heat transfer coefficients at lower and upper plates - ¯J,J_y electric current density vector and component - K external loading parameter, E_o/(B_o U_m) - k thermal conductivity - Nu₁, Nu₂ local Nusselt numbers, h₁,a/k and h₂a/k, respectively - P fluid pressure - Pe Peclet number, PrRe - Pr Prandtl number, C_{p f}/k - q₁,q₂ rates of heat transfer per unit area,^–k(T/Z)Z=–a'^–k(T/Z) Z=a respectively - Re Reynolds number, U_ma/u_f - T,T₀,T₁,T₂ fluid temperature, uniform entrance temperature, uniform but different lower and upper plate temperatures, respectively - T_b,T_m bulk temperature and (T₁+T₂)/2 - U,U_m,u axial, mean and dimensionless velocities, respectively - ¯V velocity vector - X,Z axial and transverse coordinates - x,z dimensionless coordinates - _n,_n even and odd eigenvalues - ,₀,_b dimensionless fluid, entrance and bulk temperatures, respectively - _c,_e,_f characteristic temperature difference (T₂-T_m), and dimensionless fluid temperatures, defined by eq. (10) - _e,_f magnetic permeability and viscosity of fluid - fluid density - electric conductivity - viscous dissipation function - (1-)/2     

Saint-Venant's principle in nonlinear plane elasticity with sufficiently small strains

A homogeneous, isotropic cylinder in an equilibrium state of plane strain, whose cross-section is a rectangle R : [0 < y ₁ < 2L; 0 < y ₂ < h] with h/L 1, is considered. There are no body forces and the long sides are stress free. At y ₁ = 0 and y ₁ = 2L, there are arbitrary loadings, each statically equivalent to a uniformly distributed tensile or compressive stress c. Within the theory of nonlinear elasticity and with the strains and strain gradients assumed to be sufficiently small (but with no such assumptions on the displacement gradients), it is proved that if (,=1,2) represents the Cauchy stress tensor and the Kronecker delta, then |–c₁₁| decays exponentially to zero in R with distance from the nearer end, and the decay constant depends only upon the material but is independent of L.     

Numerical analysis of axisymmetric buckling of conical shells

Nonlinear boundaryvalue problems of axisymmetric buckling of conical shells under a uniformly distributed normal pressure are solved by the shooting method. The problems are formulated for a system of six firstorder ordinary differential equations with independent rotation and displacement fields. Simply supported and clamped cases are considered. Branching solutions of the boundaryvalue problems are studied for different pressures and geometrical parameters of the shells. The nonmonotonic and discontinuous curves of equilibrium states obtained show that collapse, i.e., snapthrough instability is possible. For a simply supported shell, multivalued solutions are obtained for both external and internal pressure. For a clamped thinwalled shell, theoretical results are compared with experimental data.     

The extension of Beltrami's ellipsoid to anisotropic bodies

Summary The spectral decomposition of the compliance, stiffness, and failure tensors for transversely isotropic materials was studied and their characteristic values were calculated using the components of these fourth-rank tensors in a Cartesian frame defining the principal material directions. The spectrally decomposed compliance and stiffness or failure tensors for a transversely isotropic body (fiber-reinforced composite), and the eigenvalues derived from them define in a simple and efficient way the respective elastic eigenstates of the loading of the material. It has been shown that, for the general orthotropic or transversely isotropic body, these eigenstates consist of two double components, ₁ and ₂ which are shears (₂ being a simple shear and ₁, a superposition of simple and pure shears), and that they are associated with distortional components of energy. The remaining two eigenstates, with stress components ₃, and ₄, are the orthogonal supplements to the shear subspace of ₁ and ₂ and consist of an equilateral stress in the plane of isotropy, on which is superimposed a prescribed tension or compression along the symmetry axis of the material. The relationship between these superimposed loading modes is governed by another eigenquantity, the eigenangle .The spectral type of decomposition of the elastic stiffness or compliance tensors in elementary fourth-rank tensors thus serves as a means for the energy-orthogonal decomposition of the energy function. The advantage of this type of decomposition is that the elementary idempotent tensors to which the fourth-rank tensors are decomposed have the interesting property of defining energy-orthogonal stress states. That is, the stress-idempotent tensors are mutually orthogonal and at the same time collinear with their respective strain tensors, and therefore correspond to energy-orthogonal stress states, which are therefore independent of each other. Since the failure tensor is the limiting case for the respective _x, which are eigenstates of the compliance tensor S, this tensor also possesses the same remarkable property.An interesting geometric interpretation arises for the energy-orthogonal stress states if we consider the projections of _x in the principal₃D stress space. Then, the characteristic state ₂ vanishes, whereas stress states ₁, ₃ and ₄ are represented by three mutually orthogonal vectors, oriented as follows: The ₃ and ₄ lie on the principal diagonal plane (₃₁₂) with subtending angles equaling (–/2) and (-), respectively. On the positive principal ₃-axis, is the eigenangle of the orthotropic material, whereas the ₁-vector is normal to the (₃₁₂)-plane and lies on the deviatoric -plane. Vector ₂ is equal to zero.It was additionally conclusively proved that the four eigenvalues of the compliance, stiffness, and failure tensors for a transversely isotropic body, together with value of the eigenangle , constitute the five necessary and simplest parameters with which invariantly to describe either the elastic or the failure behavior of the body. The expressions for the _x-vector thus established represent an ellipsoid centered at the origin of the Cartesian frame, whose principal axes are the directions of the ₁-, ₃- and ₄-vectors. This ellipsoid is a generalization of the Beltrami ellipsoid for isotropic materials.Furthermore, in combination with extensive experimental evidence, this theory indicates that the eigenangle alone monoparametrically characterizes the degree of anisotropy for each transversely isotropic material. Thus, while the angle for isotropic materials is always equal to _i = 125.26° and constitutes a minimum, the angle || progressively increases within the interval 90–180° as the anisotropy of the material is increased. The anisotropy of the various materials, exemplified by their ratiosE _L/2G_L of the longitudinal elastic modulus to the double of the longitudinal shear modulus, increases rapidly tending asymptotically to very high values as the angle approaches its limits of 90 or 180°.     

A third order scheme in the vlasov theory and some consequences in the evolution of few body systems

Summary A set of implemented evolution equations, describing the coherent nonlinear interaction of plasma waves, based on the perturbation method, has been derived, taking into account initial value effects and third order nonlinearities in the modal amplitudes.The equations reduce, in the appropriate limit, to well known stochastic triplets of hydrodynamic type.It is argued that, the stochastization mechanism of the decay instability in strongly damped regime, may be interpreted as a Duffing-type behavior.

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Sommario Si presenta un sistema di equazioni di evoluzione descrivente l'interazione coerente nonlineare di onde di plasma, tenendo conto di effetti di condizioni iniziali e nonlinearità del terzo ordine nelle ampiezze modali. Le equazioni si riducono, nel limite appropriato, a ben noti tripletti stocastici di tipo idrodinamico.Si ipotizza che il processo di stocastizzazione della instabilità di decadimento, in regime di forte dissipazione, possa essere interpretato da un modello dinamico del tipo Duffing.

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Lyst of greek symbols omega (lower case)=frequency - delta (1.c.)=partial derivative - pi (1.c.)=greek pi (=3,14r. units) in (26) - Sigma (Capital case)=Summation symbol - Delta (C.c)=def. as in (14) - epsilon (1.c.) def. as in (10) - sigma (1.c.)=def. as in (36) - gamma (1.c.)=def. as in (37) - beta (1.c.)=def. as in (37) - ro, (1.c.)=def. as in (37) - alfa (1.c.)=def. in connection with _k in (60) - Gamma (C.c.)=def. as in (63) and (66), (67) - delta (1.c.)=def. as in (66) - psi (1.c.)=def. as in (75) - fi (1.c.)=def. in (71) in connection with ₀ - Fi (C.c.)=def. as in (75) - Omega (C.c.)=def. in (78)     

Buckled states of a spherical shell under uniform external pressure

A method is outlined to obtain the preferred buckled states of a (complete) spherical shell under uniform external pressure. The shell model investigated is that of the John equations, a system of six nonlinear partial differential equations. Methods in bifurcation theory and group representations are used to reduce the problem to a finite-dimensional problem whose solutions generate buckled states that are preferred in a certain least-energy sense. Asymptotic methods and Newton's method are used in some special cases to relate the preferred buckled states obtained by the above approach to actual buckled states observed in experimental studies.G. H. Knightly's research was supported in part by the U.S. National Science Foundation Grants No. MCS 77-04927 and No. MCS 79-03555; that of D. Sather, in part by U.S. National Science Foundation Grant No. MCS 78-02140 and in part by the Council on Research and Creative Work of the University of Colorado.     

Displacement pattern matching and boundary-element methods for elastic-plastic stress analysis

A two-dimensional hybrid experimental-numerical technique for elastic-plastic stress analysis is presented. This technique results from merging two relatively new technologies in engineering mechanics: boundary-element methods and image processing. A syntactic pattern recognition scheme, termed displacement pattern matching (DPM), determines the displacement boundary conditions which are used in an elasticplastic boundary-element method (EPBEM) code. The result is an automated stress-analysis tool.Displacement pattern matching is a process where displacements are measured by tracking an arbitrary array of black spots on a white specimen. The digitized images of the specimen are compared in a double-exposure format to determine displacements. Displacement pattern matching is a full-field technique, with spatial resolution on the order of. 1 pixels.Displacement pattern matching supplies the actual specimen displacement increments to the EPBEM code which is based on a von Mises, isotropic work-hardening constitutive model. Given these displacements and free surface conditions, EPBEM is able to incrementally calculate the internal state of stress at selected locations. Results obtained for a variety of geometries and loading conditions compared well with ANSYS finite-element solutions and selected published experimental solutions and therefore are encouraging.     

The wedge subjected to tractions: a paradox resolved

The classical two-dimensional solution provided by Lévy for the stress distribution in an elastic wedge, loaded by a uniform pressure on one face, becomes infinite when the opening angle 2 of the wedge satisfies the equation tan 2 = 2. Such pathological behavior prompted the investigation in this paper of the stresses and displacements that are induced by tractions of O(r ^–) as r0. The key point is to choose an Airy stress function which generates stresses capable of accommodating unrestricted loading. Fortunately conditions can be derived which pre-determine the form of the necessary Airy stress function. The results show that inhomogeneous boundary conditions can induce stresses of O(r ^–), O(r ^– ln r), or O(r ^– ln² r) as r0, depending on which conditions are satisfied. The stress function used by Williams is sufficient only if the induced stress and displacement behavior is of the power type. The wedge loaded by uniform antisymmetric shear tractions is shown in this paper to exhibit stresses of O(ln r) as r0 for the half-plane or crack geometry. At the critical opening angle 2, uniform antisymmetric normal and symmetric shear tractions also induce the above type of stress singularity. No anticipating such stresses, Lévy used an insufficiently general Airy stress function that led to the observed pathological behavior at 2.     

Deformed particle image pattern matching in Particle Image Velocimetry

The deformation of particle image patterns due to velocity gradients causes errors of velocity measurements and false velocity detections in PIV (Particle Image Velocimetry). A novel technique to overcome those limitations inherent in the conventional PIV by correcting the particle image pattern according to the local velocity gradients in two dimensional flows, i.e. u/x, u/y, v/x and v/y, is proposed and successfully applied to a water flow downstream of a backward facing step.     

Methods of Small and Large δ in the Nonlinear Dynamics – A Comparative Analysis

New asymptotic approaches for dynamical systems containing a power nonlinear term x ⁿ are proposed and analyzed. Two natural limiting cases are studied: n 1 + , 1 and n . In the firstcase, the 'small method' (SM)is used and its applicability for dynamical problems with the nonlinearterm sin as well as the usefulness of the SMfor the problem with small denominators are outlined. For n , a new asymptotic approach is proposed(conditionally we call it the 'large method' –LM). Error estimations lead to the followingconclusion: the LM may be used, even for smalln, whereas the SM has a narrow application area. Both of the discussed approaches overlap all values ofthe parameter n.