A spectral resolving method for analyzing linear random vibrations with variable parameters

This paper is a development of ref. [1]. Consider the following random equation: Z(t)+2βZ(t)+ω₀²Z(t)=(a⁰+a¹Z(t))I(t)+c in which excitation I(t) and response Z(y) are both random processes, and it is proposed that they are mutually independent. Suppose that a(t) is a known function of time and I(t) is a stationary random process. In this paper, the spectral resolving form of the random equation stated above, the numerical solving method and the solutions in some special cases are considered.     

Rheology of polydisperse polymers: relationship between intermolecular interactions and molecular weight distribution

An expression of the relaxation function of linear polydisperse polymers is proposed in terms of intermolecular couplings of reptative chains. The relaxation times of each molecular weight are assumed to be shifted according to a tube renewal mechanism accounting for the diffusion of the surrounding chains. The subsequent shift is applied to the relaxation function of each molecular weight obtained from an analytical expression of the complex compliance J ^*(). Therefore the complex shear modulus G ^*() is derived from the overall relaxation function using the probability density accounting for the molecular weight distribution and four species-dependent parameters: a front factor A for zero-shear viscosity, plateau modulus G _N ⁰ , activation energy E and characteristic temperature T . All the main features of the theology of polydisperse polymers are described by the proposed model.     

The third-normal stress difference in entangled melts: Quantitative stress-optical measurements in oscillatory shear

Stress-optical measurements are used to quantitatively determine the third-normal stress difference (N ₃ = N ₁ + N ₂) in three entangled polymer melts during small amplitude (<15%) oscillatory shear over a wide dynamic range. The results are presented in terms of the three material functions that describe N ₃ in oscillatory shear: the real and imaginary parts of its complex amplitude ₃ ^* = ₃ - i ₃ , and its displacement ₃ ^d . The results confirm that these functions are related to the dynamic modulus by ² ₃ ^* ()=(1-)[G ^*())– G ^*(2)] and ² ₃ ^d ()=(1- )G() as predicted by many constitutive equations, where = –N ₂/N ₁. The value of (1-) is found to be 0.69±0.07 for poly(ethylene-propylene) and 0.76±0.07 for polyisoprene. This corresponds to –N ₂/N ₁ = 0.31 and 0.24±0.07, close to the prediction of the reptation model when the independent alignment approximation is used, i.e., –N ₂/N ₁ = 2/7 – 0.28.     

Stability of spherical Couette flow in thick layers when the inner sphere revolves

A natural generalization of cylindrical Couette flow is the flow of a viscous incompressible liquid between two concentric spheres rotating about the same axis with different angular velocities. As has often been noted, spherical Couette flow is, despite the apparent similarity, considerably more complex than cylindrical flow. It consists of differential rotation about the axis and one- or two-eddy circulation (depending on the ratio between the angular velocities of the two spheres = ₂/₁) in the meridional plane and depends significantly on the Reynolds number Re = ₁r ₁ ² and the relative thickness of the layer = (r₂-r₁)/r₁ (₁, ₂ and r₁, r₂ are the angular velocities and radii of the inner and outer spheres, respectively. The investigation of spherical Gouette flow and its stability has begun relatively recently (within the last 10 years) and has evidently been stimulated by applied problems associated with astro- and geophysics. Because of the great computational difficulties encountered in investigating such flow theoretically, experimental investigations have yielded more extensive and interesting results [1–8], although all the published results refer to the case of rotation of one inner sphere ( = 0).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 9–15, March–April, 1978.It remains to thank S. A. Shcherbakov for help in organizing automatic input of the signals to the BÉSM-6 computer.     

On the Ellipticity of the Middle Surface of a Shell and its Application to the Asymptotic Analysis of ‘Membrane’ Shells

We consider a surface S = (), where ² is a bounded, connected, open set with a smooth boundary and : ³ is a smooth map; let () denote the components of the two-dimensional linearized strain tensor of S and let 0 with length 0 > 0. We assume the the norm ,|| ()||0, in the space V0() = { H¹() × H¹() × L²(); = 0 on 0 } is equivalent to the usual product norm on this space. We then establish that this assumption implies that the surface S is uniformly elliptic and that we necessarily have 0 = .     

The finite deflection equations of anisotropic laminated shallow shells

In this paper, basing on ref. [1] we improved and extended that which is concerned with a view of investigating the finite deflection equations of anisotropic laminated shallow shells subjected to static loads, dynamic loads and thermal loads. We have considered the most general bending-stretching couplings and the shear deformations in the thickness direction, and derived the equilibrium equations, boundary conditions and initial conditions. The differential equations expressed in terms of generalized displacements u₀, ₀ and are obtained. From them, we could solve the problems of stress analysis, deformation, stability and vibration. For some commonly encountered cases, we derived the simplified equations and methods.     

Normal forms for random diffeomorphisms

Given a dynamical system (,, ,) and a random diffeomorphism (): ^{d d} with fixed point at x=0. The normal form problem is to construct a smooth near-identity nonlinear random coordinate transformation h() to make the random diffeomorphism ()=h()^–1() h() as simple as possible, preferably linear. The linearization D(, 0)=:A() generates a matrix cocycle for which the multiplicative ergodic theorem holds, providing us with stochastic analogues of eigenvalues (Lyapunov exponents) and eigenspaces. Now the development runs pretty much parallel to the deterministic one, the difference being that the appearance of turns all problems into infinite-dimensional ones. In particular, the range of the homological operator is in general not closed, making the conceptof-normal form necessary. The stochastic versions of resonance and averaging are developed. The case of simple Lyapunov spectrum is treated in detail.     

Considerations on self- and parametrically excited vibrational systems

Summary In order to enhance our ability to predict and detect the behaviour characteristics of a parametrically excited vibratory system, one has to grasp the fundamental features concerning the mode interactions between self-excitation and parametric excitation. In this regard, two types of nonlinear models are proposed, and the correlation and the interaction between both excitations are discussed. These models differ from those obtained earlier. Taking into account the effect of exciting terms y ⁿ cos (2t) and y ^n–1y × cos (2t) (n=1, 3) where y represents a deflection and is the frequency, the features of these models are considered in comparison with those in previous works. Resonance phenomena of orders 1 and 1/2 and the behaviour in the neighborhood of resonances are mainly investigated by the averaging method.

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Einige Bemerkungen zu selbst- und parametererregten Schwingungssystemen

Übersicht Zur Vorhersage und Charakterisierung des Verhaltens parametererregter Schwingungssysteme ist es hilfreich, die Wechselwirkungen zwischen Selbsterregung und Parametererregung zu untersuchen. Zu diesem Zweck werden am Beispiel zweier nichtlinearer Modelle diese Wechselwirkungen studiert. Dabei werden Erregerterme der Form y ⁿ cos (2t) und y ^n–1y cos (2t) (n=1, 3) betrachtet, wobei die Erregerfrequenz ist. Die typischen Eigenschaften der untersuchten nichtlinearen Systeme werden mit Resultaten früherer Arbeiten verglichen. Die Resonanzen der Ordnungen 1 und 1/2 und das Verhalten der Systeme in der Nachbarschaft dieser Resonanzen werden untersucht, wobei die Mittelwertmethode Anwendung findet.

     

The accuracy of some formulae for the interconversion of creep and stress-relaxation data

The accuracy of the approximation formulaeJ (t) ~ 1/G (t) andd lnJ (t)/d lnt ~ —d lnG (t)/d lnt, which interconnect stress-relaxation modulusG (t) and creep complianceJ (t) and their double logarithmic rates are investigated. For glassy polymers, the errors in the first formula are less than 1–2%, and in the second, they are generally in the order of a few percent, too. Similar estimates can also be found for the real parts of the analogous complex functionsJ ^* () andG ^* ().     

Investigation of chemically reacting and radiating supersonic flow in channels

The two-dimensional time dependent Navier-Stokes equations are used to investigate supersonic flows undergoing finite rate chemical reaction and radiation interaction for a hydrogen-air system. The explicit multi-stage finite volume technique of Jameson is used to advance the governing equations in time until convergence is achieved. The chemistry source term in the species equation is treated implicitly to alleviate the stiffness associated with fast reactions. The multidimensional radiative transfer equations for a nongray model are provided for general configuration, and then reduced for a planar geometry. Both pseudo-gray and nongray models are used to represent the absorption-emission characteristics of the participating species.The supersonic inviscid and viscous, nonreacting flows are solved by employing the finite volume technique of Jameson and the unsplit finite difference scheme of MacCormack to determine a convenient numerical procedure for the present study. The specific problem considered is of the flow in a channel with a 10° compression-expansion ramp. The calculated results are compared with the results of an upwind scheme and no significant differences are noted. The problem of chemically reacting and radiating flows are solved for the flow of premixed hydrogen-air through a channel with parallel boundaries, and a channel with a compression corner. Results obtained for specific conditions indicate that the radiative interaction can have a significant influence on the entire flowfield.Nomenclature A band absorptance (m^–1) - A _o band width parameter (m^–1) - C _j concentration of thejth species (kg mol/m³) - C _o correlation parameter ((N/m²)^–1m^–1) - C _p constant pressure specific heat (J/kgK) - e Planck's function (J/m²S) - E total internal energy (J/kg) - f _j mass fraction of thejth species - h static enthalpy of mixture (J/kg) - H total enthalpy (J/kg) - I identity matrix - I _v spectral intensity (J/m s) - I _bv spectral Planck function - k thermal conductivity (J/m sK) - K _b backward rate constant - K _f forward rate constant - I unit vector in the direction of - M _j molecular weight of thejth species (kg/kg mol) - P pressure (N/m²) - P _j partial pressure of thejth species (N/m²) - P _e equivalent broadening pressure ratio - Pr Prandtl number - P _w a point on the wall - q _R total radiative heat flux (J/m² s) - spectral radiative heat flux (J/m³ s) - R gas constant (J/KgK) - r _w distance between the pointsP andP _w(m) - S integrated band intensity ((N/m²)^–1/m^–2) - S integrated band intensity ((N/m²)^–1 m^–2) - T temperature (K) - u, v velocity inx andy direction (m/s) - production rate of thejth species (kg/m³ s) - x, y physical coordinate - z dummy variable in they direction Greek symbols ratio of specific heats - t _ch chemistry time step (s) - t _f fluid-dynamic time step (s) - absorption coefficient (m^–1) - ,_v spectral absorption coefficient (m^–1) - _p Planck mean absorption coefficient - second coefficient of viscosity, wavelength (m) - dynamic viscosity (laminar flow) (kg/m s) - , computational coordinates - density (kg/m³) - Stefan-Boltzmann constant (erg/s cm² K³) - shear stress (W/m²) - equivalence ratio - wave number (m^–1) - _c frequency at the band center     

Calculation of a wall-stabilized argon arc with account for radiative energy transfer

It was shown experimentally in [1, 2] and in a study by E. I. Asinovskii and A. V. Kirillin reported at the Scientific Technical Conference of the High-Temperature Scientific Research Institute held in 1964 that the heat transfer mechanism in a wall-stabilized argon arc was not purely purely conductive at gas temperatures greater than 11 000° K. Asinovskii and Kirillin also showed that radiative energy transfer is the reason why similarity is lost when the current-voltage characteristics are constructed in dimensionless form. The radiation of an argon arc has been studied experimentally by a number of authors [3–5], Dautov [6] calculated an argon arc without allowing for radiation.In this article an argon arc stabilized by the cooled duct walls is calculated with account for radiation using theoretically computed relationships describing the transport properties of argon plasma. A large portion of the radiated energy pertains to spectral lines whose role has been studied by L. M. Biberman. The authors have used I. T. Yakubov's data on argon radiation published in the journal Optics and Spectroscopy. A method of calculation and data on argon plasma radiation are also given in [7].Reference [8] deals with the problem of the role of radiation in an arc burning in nitrogen. In particular, the above-mentioned loss of similarity follows from the results of this work. However, the relationships used in this article to describe the transport properties of nitrogen plasma were obtained experimentally in [9].Notation r₀ arc radius (cm) - r variablesradius (cm) - T temperature (°K) - heat transfer coefficient (ergcm^–1sec^–1deg^–1) - E electric field intensity (g^1/2cm^–1/2sec^–1) - electrical conductivity (sec^–1) - q₁ heat flux density due to conduction - q₂ heat flux density due to radiation - u divergence of radiative energy flux density in the transparent part of the spectrum (ergcm^–3sec^–1) - u₂ same for part of the spectrum where reabsorption is taken taken into account - m₀ atomic mass - m_e electronic mass - Stefan-Boltzmann constant - h Planck constant - k Boltzmann constant - e electronic charge - p pressure - degree of ionization - N_e electron concentration (cm^–3) - n₀ neutral atom concentration - Q_0e electron-neutral collision cross section - Q_ie electron-ion collision cross section (cm²) - ₀ line center frequency (sec^–1) - ₊ line halfwidth (distance from line center to contour for ) due to effects giving dispersion contour - k _v absorption coefficient (cm^–1) - energy radiated by a hemispherical volume - emissivity of hemispherical volume - radius of hemispherical volume - S line intensity The authorS thank I. T. Yakubov for allowing them to use his data on arc plasma radiation.     

Vorticity and Regularity for Viscous Incompressible Flows under the Dirichlet Boundary Condition. Results and Related Open Problems

In reference [7] it is proved that the solution of the evolution Navier–Stokes equations in the whole of R ³ must be smooth if the direction of the vorticity is Lipschitz continuous with respect to the space variables. In reference [5] the authors improve the above result by showing that Lipschitz continuity may be replaced by 1/2-H?lder continuity. A central point in the proofs is to estimate the integral of the term (ω · ∇)u · ω, where u is the velocity and ω = ∇ × u is the vorticity. In reference [4] we extend the main estimates on the above integral term to solutions under the slip boundary condition in the half-space R ₊³. This allows an immediate extension to this problem of the 1/2-H?lder sufficient condition. The aim of these notes is to show that under the non-slip boundary condition the above integral term may be estimated as well in a similar, even simpler, way. Nevertheless, without further hypotheses, we are not able now to extend to the non slip (or adherence) boundary condition the 1/2-H?lder sufficient condition. This is not due to the “nonlinear" term (ω · ∇)u · ω but to a boundary integral which is due to the combination of viscosity and adherence to the boundary. On the other hand, by appealing to the properties of Green functions, we are able to consider here a regular, arbitrary open set Ω.      

The effective permeability of a heterogeneous porous medium

The effective (single-phase) permeability of an (infinite) heterogeneous porous medium is studied using a formalism of Green's functions. We give formal expressions for it in the form of a series expansion involving the microscopic random-permeability field many-body correlation functions of higher and higher order.The particular case of a log-normal medium of infinite extent is studied using field-theoretical methods. Using partial series resummation techniques, we derivea formula up to all orders in the local correlations which was first reckoned by many authors by means of a first-order calculation. The formula — which remains an approximation — works whatever the dimensionality of the space, and gives the following simple estimate for the effective permeability in 3 D:K _eff=k ^1/33. The method is general and the approximations can be systematically improved on when more complex situations are studied.Roman Letters D number of dimensions of the space in which the flow takes place - f(r) body force field,N - f(q) Fourier-transformed body-force field, Nm³ - G ₀(r, r) Green's function of the Laplace operator, m^–1 - g(k,r, r) velocity propagator before averaging, m^–1 - G(r, r) velocity propagator after averaging, m^–1 - j(r) a scalar dimensionless field - k(r) local value of the permeability at point r, m² - K _eff effective permeability - K _g geometric average of the local permeability, m² - l typical size of the averaging volume, m - L characteristic length of the porous medium or of the reservoir, m - L(r, r) projection operator, m^–2 - M(r, r) scattering operator, m^–3 - p(r) local value of the pressure, Nm^–2 - p(k,r, r) pressure propagator before averaging, m^–1 - P(r, r) pressure propagator after averaging, m^–1 - r position vector, m - r modulus of vectorr, m - unit vector pointing in the direction ofr - q Fourier wave vector, m^–1 - q modulus of the Fourier wave-vectorq, m^–1 - unit vector pointing in the direction ofq - projector over vector - 1 unit tensor - X(r) a local random variable - ¯X(r) volume averaged local random variable - X (r) ensemble averaged local random variable - V large-scale averaging volume, m³ - Z(j) generating functional of a random field - Z(r,j) modified generating functional of a random field - Z normalization factor Greek Letters ₀ average value of the logarithm of the permeability - (r) fluctuation of the logarithm of permeability at pointr - viscosity of the fluid, Nt/m² - (r–r) two-point correlation function of the fluctuations of the logarithm of the permeability - _k correlation length of the permeability correlation function, m - _u correlation length of the velocity correlation function, m     

Concentration dependence of the critical viscoelastic properties of gelatin at the gel point

Gelatin gel properties have been studied through the evolution of the storage [G()] and the loss [G()] moduli during gelation or melting near the gel point at several concentrations. The linear viscoelastic properties at the percolation threshold follow a power-law G()G() and correspond to the behavior described by a rheological constitutive equation known as the Gel Equation. The critical point is characterized by the relation: tan = G/G = cst = tan ( · /2) and it may be precisely located using the variations of tan versus the gelation or melting parameter (time or temperature) at several frequencies. The effect of concentration and of time-temperature gel history on its variations has been studied. On gelation, critical temperatures at each concentration were extrapolated to infinite gel times. On melting, critical temperatures were determined by heating step by step after a controlled period of aging. Phase diagrams [T = f(C)] were obtained for gelation and melting and the corresponding enthalpies were calculated using the Ferry-Eldridge relation. A detailed study of the variations of A with concentration and with gel history was carried out. The values of which were generally in the 0.60–0.72 range but could be as low as 0.20–0.30 in some experimental conditions, were compared with published and theoretical values.     

Approximate First Integrals of a Galaxy Model

First-order approximate first integrals (conserved quantities)of a Hamiltonian dynamical system with two degrees of freedomwhich arises in the modeling of central part of a deformed galaxy [1] havebeen obtained based on the approximate Noether symmetries for resonances₁=₂, ₁=2₂ and 2₁=3₂. Furthermore,KAM curves have been obtained analytically and they have been compared with thenumerical ones on the Poincaré surface of section.     

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°.     

Scaling mixing during miscible displacement in heterogeneous porous media

This paper discusses scaling of mixing during miscible flow in heterogeneous porous media. In large field systems dispersivity appears to depend on system length due to heterogeneities. Three types of scaling are discussed to investigate the heterogeneous effects. Dimensional analysis of mixing during flow through geometerically scaled heterogeneous models is illustrated using measured dispersion. Fractal analysis of mixing in statistically scaled heterogeneous porous media is discussed. Analog scaling of pressure transients in heterogeneous porous media is suggested as an in-situ method of estimating dispersion.Notation L Length - M mass - t time, (1) indicates dimensionless - a dispersivity (L) - V local velocity (L/t) - c concentration (l). - v velocity (L/t) - C₁ fluid compressibility (Lt²/M) - v time averaged velocity (LJt) - D dispersion VA) - W width (L) - D fractional dimension (1) - x coordinate (L) - d Euclidean dimension (1) - Y Y=In \-k (l) - \-d average particle size (L) - y coordinate (L) - g acceleration due to gravity (L/t2) - _c fractal cutoff (L) - \-k average permeability (L2) - viscosity (LM/t) - L length (L) - porosity (1) - L correlation scale (1/L) - density (N/L³) - N Number of sites (l) - ² variance (dimension depends on variable) - p pressure (W/t²L) - spectral exponent (l) - [R] randomnumber (1) - r radius (L) - t time (t)     

Representation of the rheological properties of polymer melts in terms of complex fluidity

In dynamic rheological experiments melt behavior is usually expressed in terms of complex viscosity ^* () or complex modulusG ^* (). In contrast, we attempted to use the complex fluidity ^* () = 1/µ ^* () to represent this behavior. The main interest is to simplify the complex-plane diagram and to simplify the determination of fundamental parameters such as the Newtonian viscosity or the parameter of relaxation-time distribution when a Cole-Cole type distribution can be applied. ^* () complex shear viscosity - () real part of the complex viscosity - () imaginary part of the complex viscosity - G ^* () complex shear modulus - G() storage modulus in shear - G() loss modulus in shear - J ^* () complex shear compliance - J() storage compliance in shear - J() loss compliance in shear - shear strain - rate of strain - angular frequency (rad/s) - shear stress - loss angle - ^* () complex shear fluidity - () real part of the complex fluidity - () imaginary part of the complex fluidity - ₀ zero-viscosity - ₀ average relaxation time - h parameter of relaxation-time distribution     

Dynamics as a mechanism preventing the formation of finer and finer microstructure

We study the dynamics of pattern formation in the one-dimensional partial differential equation $$u_u - (W'(u_x ))_x - u_{xxt} + u = 0{\text{ (}}u = u(x,t),{\text{ }}x \in (0,1),{\text{ }}t > 0)$$ proposed recently by Ball, Holmes, James, Pego & Swart [BHJPS] as a mathematical “cartoon” for the dynamic formation of microstructures observed in various crystalline solids. Here W is a double-well potential like 1/4((u _x )² ?1)². What makes this equation interesting and unusual is that it possesses as a Lyapunov function a free energy (consisting of kinetic energy plus a nonconvex “elastic” energy, but no interfacial energy contribution) which does not attain a minimum but favours the formation of finer and finer phase mixtures: $$E[u,u_t ] = \int\limits_0^1 {(\frac{{u_t^2 }}{2} + W(u_x ) + \frac{{u^2 }}{2})dx.}$$ Our analysis of the dynamics confirms the following surprising and striking difference between statics and dynamics, conjectured in [BHJPS] on the basis of numerical simulations of Swart & Holmes [SH]:

?While minimizing the above energy predicts infinitely fine patterns (mathematically: weak but not strong convergence of all minimizing sequences (u _nv_n) of E[u,v] in the Sobolev space W ^{1 p}(0, 1)×L²(0,1)), solutions to the evolution equation of ball et al. typically develop patterns of small but finite length scale (mathematically: strong convergence in W ₁ ^p(0,1)×L²(0,1) of all solutions (u(t),u_t(t)) with low initial energy as time t → ∞).

Moreover, in order to understand the finer details of why the dynamics fails to mimic the behaviour of minimizing sequences and how solutions select their limiting pattern, we present a detailed analysis of the evolution of a restricted class of initial data — those where the strain field u _x has a transition layer structure; our analysis includes proofs that

?at low energy, the number of phases is in fact exactly preserved, that is, there is no nucleation or coarsening

?transition layers lock in and steepen exponentially fast, converging to discontinuous stationary sharp interfaces as time t → ∞

?the limiting patterns — while not minimizing energy globally — are ‘relative minimizers’ in the weak sense of the calculus of variations, that is, minimizers among all patterns which share the same strain interface positions.

     

A new technique for ultrasonic nondestructive evaluation of adhesive joints: Part I. Theory

In this paper we proposed a new technique for ultrasonic nondestructive evaluation (UNDE) of adhesively bonded joints. We report an exact solution to the problem of reflection and tarnsmission of a plane, time-harmonic, longitudinal wave through anN-layered medium. The solution is valid for perfectly elastic as well as linear-viscoelastic materials, and for isotropic as well as anisotropic materials (for example, fiber-reinforced composite) so long as the wavepropagation vector coincids with one of the material coordinates. The transfer function,H ^*() is defined as the trans-mitted (or reflected) field normalized with respect to the incident field. A closed-form solution forH ^*() for the case of an adhesive joint (consisting of two adherends joined by an adhesive layer) immersed in an elastic fluid is derived. A detailed analysis of the sensitivity ofH ^*() to the wave speed and thickness of the adherends and the adhesive is carried out. An experimental verification of the analysis is the subject of Part II of this paper.Paper was presented at the 1991 SEM Spring Confernce on Experimental Mechanics held in Milwaukee, WI on June 9–13.