bluuebunny/crossref_metadata_2025 huggingface dataset:
Unstable vibrations and buckling of rotating flexible rods
Authors: Lakin W. D. , Nachman A.
Published: December 2016

We consider a group of fourth-order boundary-value problems associated with the small vibrations or buckling of a uniform flexible rod which is clamped at one end and rotates in a plane perpendicular to the axis of rotation. The vibrations may be in any plane relative to the plane of rotation and the rod is off-clamped, i.e. the axis of rotation does not pass through the rod’s clamped end. The governing equation for the vibrations involves a small parameter for rapid rotation and must be treated by singular perturbation methods. Further, a turning point of the equation always coincides with a boundary point. Both free and unstable vibrations are examined, and a stability boundary is obtained. Results for the unstable vibrations predict the unexpected existence of a time-independent buckled state in non-transverse planes when the rod is wholly under tension. The general buckling problem in the transverse plane is also considered.
Hydromagnetic flow and heat transfer between two horizontal plates, the lower plate being a stretching sheet
Authors: Borkakoti A. K. , Bharali A.
Published: December 2016

Hydromagnetic flow and heat transfer between two horizontal parallel plates, where the lower one is a stretching sheet and the upper one is a porous solid plate are studied in the presence of a transverse magnetic field. Fluid velocities and temperature distributions are obtained and plotted graphically.
A mathematical model of solar flares
Authors: Heyvaerts Jean , Lasry Jean-Michel , Schatzman Michelle , Witomski Patrick
Published: December 2016

The phenomenon of solar flares is modeled assuming that the magnetic field is force-free and that its evolution is quasi-static. This model is simplified so as to be tractable and yields a semi-linear elliptic equation in a halfplane depending on a parameter

λ
\lambda

which describes the time evolution. It is proved that there are (at least) two branches of solutions which have distinct asymptotic behaviors at infinity. The upper branch exists for all

λ
>
0

\lambda > 0

, but the lower branch exists only on a finite interval

[

0
,

λ
c

]

\left [ {0,{\lambda ^c}} \right ]

. As stable solutions must have the same asymptotic behavior as the lower branch of solutions, and as this is impossible after

λ
c

{\lambda ^c}

, we contend that no stable solution exists after

λ
c

{\lambda ^c}

and that a solar flare is thus triggered.
A nonlinear diffusion equation arising in the study of soft tissue
Authors: Holmes Mark H.
Published: December 2016

A nonlinear diffusion equation which describes the deformation of a soft fluid-filled tissue is studied. The nonlinearity in the problem arises from the permeability of the elastic phase which for a number of tissues, such as articular cartilage, is strongly dependent on the strain. Moreover, for most tissues the exact dependence is not known, and so the functional dependence of the permeability on the strain is not determined until after the problem is solved. The approach uses perturbation methods for the diffusive boundary layers that occur in the problem and similarity solutions to solve the reduced problems. Once the solution is obtained, the permeability function is determined and some of the limitations of the present model of soft tissue are discussed.
A variational principle for surface waves in magnetohydrodynamics
Authors: Shivamoggi Bhimsen K.
Published: December 2016

A variational principle for the motion of the interface between an infinitely conducting fluid and a vacuum magnetic field is given. This variational principle not only gives the Laplace equation governing the velocity potential and the magnetic-field potential, but also provides all the boundary conditions appropriate to the interface between an infinitely conducting fluid and a vacuum magnetic field. However, unlike the hydrodynamical case, this variational principle does not have the simple physical interpretation of the stationarity of the fluid pressure + magnetic field pressure.
On existence of solution of the Dirichlet problem of fourth order partial differential equations with variable coefficients
Authors: Balasundaram S. , Bhattacharyya P. K.
Published: December 2016

Sufficient conditions for the existence and uniqueness of the solution of the Dirichlet problem of fourth order elliptic partial differential equations with variable coefficients have been derived. In a number of examples of practical interest, the easy applicability of these results has been shown.
Optimal temperature paths for thermorheologically simple viscoelastic materials with constant Poisson’s ratio are canonical
Authors: Gurtin Morton E. , Murphy Lea F.
Published: December 2016

In this note we discuss the thermal stress problem for a thermorheologically-simple linearly-viscoelastic body, subjected to a spatially-uniform temperature field and homogeneous boundary conditions, assuming that Poisson’s ratio is constant and inertia negligible. In particular, we consider the following optimization problem: of all temperature paths

θ
(
t
)
,
0
≤
t
≤

t
f

\theta (t), 0 \le t \le {t_f}

, which belong to a given function class, is there one which renders a given stress measure a minimum at time

t
f

{t_f}

. We show that a resulting optimal path

θ

(
t
)

\theta \left ( t \right )

(if it exists) is canonical:

θ

(
t
)

\theta \left ( t \right )

is independent of the shape of the body and of the particular homogeneous boundary conditions.
Buckling problems in finite plane elasticity-harmonic materials
Authors: Wu Chien H. , Cao Guang Zhong
Published: December 2016

Bucklings of biaxially deformed annular, rectangular and arbitrary regions are considered. It is found that for many different configurations the buckling conditions are governed by the same equation

χ
=
0

\chi = 0

, where

χ
\chi

is merely a material function. Furthermore, the buckling solutions are completely unrelated to the buckling loads.
A similar flow between two rotating disks
Authors: Hamza E. A. , MacDonald D. A.
Published: December 2016

When viscous incompressible fluid is contained between two parallel disks which, at time

t
t

, are spaced a distance

H

1
−
α
t

H\sqrt {1 - \alpha t}

apart and are rotating with angular velocities proportional to

Ω
1

(
1
−
α
t

)

−
1

{\Omega _1}{(1 - \alpha t)^{ - 1}}

the governing Navier-Stokes equations reduce to a set of ordinary differential equations. We present approximate solutions to these equations for a range of values of the three linearly independent parameters which influence the fluid motion. Special attention is given to the normal forces and the torques which the fluid exerts on the rotating surfaces.
The low-frequency theory of elastic wave scattering
Authors: Dassios George , Kiriaki Kiriakie
Published: December 2016

An incident longitudinal, or transverse, plane wave is scattered by a bounded region immersed in an infinite isotropic and homogeneous elastic medium. The region could be either a rigid scatterer or a cavity. Integral representations for the total displacement field, as well as for the introduced spherical scattering amplitudes are given explicitely in a compact form. Representations for the scattering cross-section whenever the incident wave is a longitudinal or a transverse wave are also provided. Using Papkovich potentials and low-frequency techniques the scattering problems are reduced to an iterative sequence of potential problems which can be solved successively in terms of expansions in appropriate harmonic functions. In each one of the four cases (longitudinal and transverse incidence on rigid scatterer and cavity) the corresponding exterior boundary value problems that specify the approximations as well as the analytic expressions for the scattering amplitudes and the scattering cross-section are given explicitly. The leading low-frequency term of the scattering cross-section for a rigid scatterer is independent of the wave number while for the case of a cavity it is proportional to the fourth power of the wave number. The low-frequency limit of the displacement field which corresponds to the static problem when the scatterer is a cavity, does not depend on the geometrical characteristics of the scatterer and it is always a constant.
Stability conditions for linear nonautonomous delay differential equations
Authors: Busenberg Stavros N. , Cooke Kenneth L.
Published: December 2016

We derive new sufficient conditions for uniform asymptotic stability of the zero solution of linear non-autonomous delay differential equations. The equations considered include scalar equations of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x prime left-parenthesis t right-parenthesis equals minus c left-parenthesis t right-parenthesis x left-parenthesis t right-parenthesis plus sigma-summation Underscript i equals 1 Overscript n Endscripts b Subscript i Baseline left-parenthesis t right-parenthesis x left-parenthesis t minus upper T Subscript i Baseline right-parenthesis">

x
′

(
t
)

=
−
c

(
t
)

x

(
t
)

+

∑

i
=
1

n

b
i

(
t
)

x

(

t
−

T
i

)

x’\left ( t \right ) = - c\left ( t \right )x\left ( t \right ) + \sum \limits _{i = 1}^n {{b_i}\left ( t \right )x\left ( {t - {T_i}} \right )}

\]
where

c

(
t
)

c\left ( t \right )

,

b
i

(
t
)

{b_i}\left ( t \right )

are continuous for

t
≥
0

t \ge 0

and

T
i

{T_i}

is a positive number

(
i
=
1
,
2
,
.
.
.
,
n
)

(i = 1, 2,...,n)

, and also systems of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x prime left-parenthesis t right-parenthesis equals upper B left-parenthesis t right-parenthesis x left-parenthesis t minus upper T right-parenthesis minus upper C left-parenthesis t right-parenthesis x left-parenthesis t right-parenthesis">

x
′

(
t
)
=
B
(
t
)
x
(
t
−
T
)
−
C
(
t
)
x
(
t
)

x’(t) = B(t)x(t - T) - C(t)x(t)

\]
where

B
(
t
)

B(t)

) and

C
(
t
)

C(t)

are

n
×
n

n \times n

matrices. The results are found by using the method of Lyapunov functionals.
On asymptotic solutions of boundary value problems defined on thin domains
Authors: Young Gerald W. , Davis Stephen H.
Published: December 2016

The solution of the Poisson equation subject to Dirichlet conditions is examined asymptotically on thin domains. The evolution of the structure of the solution is followed as the shape of the domain changes. It is found that the “end wall” boundary layers present when the domain is rectangular, shrink and weaken as the endwalls become less sloped and vanish when the domain slope is uniformly bounded. Such structural changes are important in certain viscous flows containing moving contact lines.
A note on mean square maximum and minimum principles in dynamic linear viscoelasticity
Authors: Day W. A.
Published: December 2016

The purpose of this note is to point out that, in certain circumstances, the mean square values of the displacement, the strain, and the stress in a viscoelastic slab which is undergoing steady trigonometric (but not necessarily periodic) oscillation satisfy maximum or minimum principles.
Upper bounds for the means of eigenvalues of random boundary value problems with weakly correlated coefficients
Authors: Boyce William E. , Xia Ning Mao
Published: December 2016

This paper concerns eigenvalue problems for second-order random differential equations with weakly correlated coefficients. The random problem and the mean (deterministic) problem are embedded in a parametrized problem whose eigenvalues are expanded in a power series in the parameter. This expansion leads, via the variational characterization of the eigenvalues, to computationally accessible upper bounds for the mean values of the eigenvalues of the original problem.
Some general results in the kinematics of axisymmetrical deformation of shells of revolution
Authors: Naghdi P. M. , Vongsarnpigoon L.
Published: December 2016

This paper deals with some kinematical aspects of shells of revolution whose torsionless axisymmetrical deformation may be finite or one with small strain accompanied by large or moderately large rotation. The results are summarized in the form of three theorems.
Laplace type integrals: transformation to standard form and uniform asymptotic expansions
Authors: Temme N. M.
Published: December 2016

Integrals are considered which can be transformed into the Laplace integral
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript lamda Baseline left-parenthesis z right-parenthesis equals StartFraction 1 Over normal upper Gamma left-parenthesis lamda right-parenthesis EndFraction integral Subscript 0 Superscript normal infinity Baseline t Superscript lamda minus 1 Baseline e Superscript minus z t Baseline f left-parenthesis t right-parenthesis d t">

F
λ

(
z
)

=

1

Γ

(
λ
)

∫
0
∞

t

λ
−
1

e

−
z
t

f

(
t
)

d
t

{F_\lambda } \left ( z \right ) = \frac {1}{{\Gamma \left ( \lambda \right )}}\int _0^\infty {{t^{\lambda - 1}}{e^{ - zt}}f\left ( t \right )dt}

\]
, where

f
f

is holomorphic,

z
z

is a large parameter,

μ
=
λ

/

z

\mu = \lambda /z

is a uniformity parameter,

μ
≥
0

\mu \ge 0

. A uniform asymptotic expansion is given with error bounds for the remainders. Applications are given for special functions, with a detailed analysis for a ratio of gamma functions. Further applications are mentioned for Bessel functions and parabolic cylinder functions. Analogue results are given for loop integrals in the complex plane.
On the optimal design of trusses under one loading condition
Authors: Levy Robert
Published: December 2016

The optimal design of singly loaded trusses required to satisfy allowable stress criteria has the popular structural properties of a statically determinate configuration and fully stressed members. Linear programming and the duality theorem were used to attain these properties. This paper formulates the truss problem as a nonlinear programming problem, derives the optimality criteria via the Lagrangian and through proper physical interpretation of the Lagrange multipliers demonstrates the validity of the above results.
Some aspects of the French flexible bag wave-energy device
Authors: Shaw D. C.
Published: December 2016

French (1977) has suggested a channel form wave energy absorber in which power is removed from the wave front by flexible bags along the channel walls. He has given a simple theoretical model for such a device which has, however, several drawbacks which we attempt to remedy. Several two-dimensional models of the channel are examined in which one or more of the dimensions are many wavelengths long. In particular, it is possible to apply realistic boundary conditions at the channel walls and obtain relationships between the wall stiffness

μ
\mu

and the decay rate of the wave front,

σ
\sigma

. Two main methods are used; the variational method developed by Evans and Morris (1972) and the Wiener-Hopf method, as modified by Jones (1952).
Nonlinear electrohydrodynamic instability of a jet
Authors: Kant R. , Malik S. K.
Published: December 2016

A weakly nonlinear theory of an electrohydrodynamic jet held together by capillary forces is investigated. The evolution of two-dimensional wave packets on the surface of the jet is shown to be governed by a nonlinear Schrödinger equation. It is found that the wave train of constant amplitude is unstable against modulation. The instability sets in at higher wave numbers as compared to the case of an unelectrified jet.
On the stability of linear nonconservative systems
Authors: Kliem W. , Pommer Chr.
Published: December 2016

For a system of linear second-order differential equations, a stability criterion is derived which gives a simple relation between eigenvalues of two of the coefficient matrices and an estimate of the lower bound

|
λ
|

min

{\left | \lambda \right |_{\min }}

of the eigenvalue for the nonlinear eigenvalue problem of the total system. Estimations of

|
λ
|

min

{\left | \lambda \right |_{\min }}

are given, and applications of the stability criterion are shown by numerical examples.
Existence of solutions in a population dynamics problem
Authors: Hernández Gastón E.
Published: December 2016

In this paper we show the existence of a solution for the Gurtin—MacCamy model in population dynamics with age dependence and diffusion. We also discuss the behavior of this solution.
Third-order solutions of Burgers’ equation
Authors: Lardner R. W.
Published: December 2016

Burgers’ equation with small dissipation coefficient and general initial conditions is considered. The first three terms are calculated in both the inner (i.e., close to the shock) and outer (away from the shock) expansions. It is shown that these two expansions can be matched and that this third-order matching essentially completes the determination of the secònd-order inner solution but leaves an undetermined function in the third-order solution. It is also shown that the second-order inner solution can be determined completely, without use of the third-order inner solution, by use of an integral conservation property.
Backward energy-flux for spherically symmetric scalar waves
Authors: Synge J. L.
Published: December 2016

I consider spherically symmetric scalar waves of two types: explosive waves in an infinite medium generated by a spherically symmetric source, and implosive waves in which the medium is contained inside a spherically symmetric source. The explosive waves travel outward, the implosive waves inward. Although there are exceptional cases, in general there are, at any time, shells in which the energy-flux is opposed to the sense of propagation of the waves.
Parameter identifiability under approximation
Authors: Kunisch K. , White L. W.
Published: December 2016

The problem of injectivity of the parameter-to-state map is discussed for Galerkin approximations of a linear parabolic equation. A necessary and sufficient condition is derived and illustrated by means of simple examples. Finally, output least squares identifiability of the Galerkin approximations is discussed.
A note on the strongest fixed-fixed column
Authors: Myers M. K. , Spillers W. R.
Published: December 2016

It is argued that the subsequent solutions generated in response to the 1962 Tadjbakhsh-Keller shape for the optimal fixed-fixed column are smooth approximations of that singular shape.
On a ratio theorem for a class of nonlinear boundary value problems
Authors: Rogers C. , Bracken A. J.
Published: December 2016

A generalization of Ussing’s flux ratio theorem is established as a consequence of a symmetry property of a Green’s function. Bäcklund transformations are then used to construct ratio theorems for certain nonlinear boundary value problems.
Transient waves produced by a moving pressure distribution
Authors: Cole S. L.
Published: December 2016

This paper provides a theoretical description of the 3-dimensional flow produced by a surface pressure distribution moving in a channel at speeds such that the Froude number based on depth is near 1. The transient flow begins in a fully 3-dimensional manner, then focuses away from the distribution, due to the side walls, into predominantly 2-dimensional “solitary” or “cnoidal” waves. Two sets of these waves are generated continuously by the distribution; one set runs upstream while the other extends downstream.
Discrete approximation of the Poisson-Vlasov system
Authors: Schaeffer Jack
Published: December 2016

Solutions of the Poisson-Vlasov system of equations that have spherical symmetry are considered. A numerical scheme which exploits the symmetry is proposed and is shown to converge in a pointwise sense. Prior convergence results due to Neunzert conclude convergence in only a weak topology although they require no symmetry.
A boundary integral equation for deformations of an elastic body with an arc crack
Authors: Ang W. T.
Published: December 2016

A solution for a class of two-dimensional elasticity problems concerning an isotropic body with an arc crack in its interior is obtained in terms of an integral taken around the exterior boundary of the body. A numerical procedure for solving this integral equation is outlined and numerical results for a test problem are given.
An application of the multivariate Lagrange-Bürmann expansion in mathematical geodesy
Authors: Henrici P. , Wilkens G. R.
Published: December 2016

In the simplified model of geodesy where the earth is conceived as a rotational ellipsoid, if the eccentricity of the ellipsoid is to be determined from gravity measurements, an equation of the form

y
=
x
−
z
h

(
x
)

y = x - zh\left ( x \right )

is to be solved for

x
x

, where

y
y

and

z
z

are small parameters whose values can be measured and

h
h

is a known function. We obtain the expansion of

x
x

in powers of

y
y

and

z
z

by means of the general Lagrange—Bürmann formula.
The unsteady boundary layer on a cylinder due to small transverse oscillations
Authors: Wilson D. E.
Published: December 2016

This paper deals with the response of an incompressible laminar boundary layer on a circular cylinder due to small periodic velocity fluctuations. These fluctuations may arise from the motion of the cylinder or be contained within the free stream. A perturbation solution employing the method of matched asymptotic expansions is given for the case of high reduced frequencies. These results are compared to a numerical solution valid for any frequency. In addition, the solution is compared to the classical result of Glauert [7] valid only at the stagnation point.
A rigorous justification of the Reynolds equation
Authors: Cimatti Giovanni
Published: December 2016

A small parameter technique is used to derive Reynolds’ lubrication equation from the Stokes equation. The error associated with the approximation is estimated in suitable norms.
Second-order closure models for rotating turbulent flows
Authors: Speziale Charles G.
Published: December 2016

The physical properties of the commonly used second-order closure models are examined theoretically for rotating turbulent flows. Comparisons are made with results which are a rigorous consequence of the Navier—Stokes equations for the problem of fully-developed turbulent channel flow in a rapidly rotating framework. It is demonstrated that all existing second-order closures yield spurious physical results for this test problem of rotating channel flow. In fact, the results obtained are shown to be substantially more unphysical than those obtained from the simpler

K
−
ϵ

K - \epsilon

and

K
−
l

K - l

models. Modifications in the basic structure of these second-order closure models are proposed which can alleviate this problem.
Caustics associated with the asymptotic solution of the diffusion equation
Authors: Anderson Steven P. , Gorman Arthur D.
Published: December 2016

The Lagrange manifold technique for finding asymptotic solutions of wave equations at turning points is adapted to diffusion-type equations.
The winding of a relaxed elastic line on a cylinder
Authors: Manning Gerald S.
Published: December 2016

We analyze several cases of the boundary-value problem that determines the path of a relaxed elastic line on a cylinder. A short line with oblique initial tangent deviates from the corresponding helix toward the direction of the axis of the cylinder. A line initially directed along the base circle continues to wind circumferentially if the ratio of its length to the radius of the cylinder does not exceed the critical value

π

/

2

3

/

2

\pi /{2^{3/2}}

; longer lines deviate from the circle. Infinitely long elastic lines wind in proportion to the logarithm of their arc lengths, as distinct from the direct proportion of a helix. Possible implications for biological structures are discussed.
Oscillations of the sunflower equation
Authors: Kulenović M. R. S. , Ladas G.
Published: December 2016

Consider the delay differential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove y With two-dots left-parenthesis t right-parenthesis plus alpha ModifyingAbove y With dot left-parenthesis t right-parenthesis plus beta f left-parenthesis y left-parenthesis t minus r right-parenthesis right-parenthesis equals 0 comma left-parenthesis asterisk right-parenthesis">

y
¨

(
t
)

+
α

y
˙

(
t
)

+
β
f

(

y

(

t
−
r

)

)

=
0
,

(
∗
)

\ddot y\left ( t \right ) + \alpha \dot y\left ( t \right ) + \beta f\left ( {y\left ( {t - r} \right )} \right ) = 0, \qquad \left ( * \right )

\]
where

α
,
β

\alpha , \beta

, and

r
r

are positive constants and

f
f

is a continuous function such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u f left-parenthesis u right-parenthesis greater-than 0 f o r u element-of left-bracket negative upper A comma upper B right-bracket comma u not-equals 0 comma a n d limit Underscript u right-arrow 0 Endscripts StartFraction f left-parenthesis u right-parenthesis Over u EndFraction equals 1 comma">

u
f

(
u
)

>
0

f
o
r
u
∈

[

−
A
,
B

]

,
u
≠
0
,
a
n
d

lim

u
→
0

f

(
u
)

u

=
1
,

uf\left ( u \right ) > 0 \qquad for u \in \left [ { - A, B} \right ], u \ne 0, and \lim \limits _{u \to 0} \frac {{f\left ( u \right )}}{u} = 1,

\]
where

A
A

and

B
B

are positive numbers. When

f

(
u
)

=
sin
⁡
u
,

(
∗
)

f\left ( u \right ) = \sin u, \left ( * \right )

is the so-called “sunflower” equation, which describes the motion of the tip of the sunflower plant.
Expansion flows on walls with nonequilibrium condensation
Authors: Clarke Joseph H. , Delale Can F.
Published: December 2016

The streamtube method for supersonic expansion flows on walls with nonequilibrium condensation is developed incorporating the radius dependence on the droplet growth rate. In the presence of an embedded, frozen, oblique shock wave arising from supercritical heat addition from condensation, the method is supplemented by the derived oblique shock relations, and the oblique shock location is determined by employing a shock fitting technique originally introduced by Barschdorff [11]. Some advantages of the proposed streamtube method are explored in comparison with the numerical method of characteristics.
Laminar boundary layer flow over a cone with suction or injection
Authors: Watanabe Takashi
Published: December 2016

The effect of uniform suction or injection on the flow of an incompressible laminar boundary layer over a cone was theoretically investigated. The boundary layer equations along a cone are transformed into nonsimilar ones, and the numerical calculations of the resulting equations are performed by the difference-differential method for various values of the suction/injection parameter. The neutral stability curves are then presented for values of the cone angle and the suction/injection parameter. The results are given for the velocity profiles, coefficient of skin friction, displacement thickness, and the critical Reynolds numbers.
The boundary layer due to a moving heated line on a horizontal surface
Authors: Wang C. Y.
Published: December 2016

A line heat source lies on an adiabatic horizontal surface. The governing equations under Boussinesq approximation show nonexistence of horizontal boundary layers if the source is still. Boundary layer solutions exist only when the source is moving laterally on the bottom surface with a certain minimum speed. Perturbation solutions for weak heat input agree well with exact numerical integration. The velocity and temperature profiles show similarity. Nonexistence and nonuniqueness are found.
Oscillation and nonoscillation in a nonautonomous delay-logistic equation
Authors: Zhang B. G. , Gopalsamy K.
Published: December 2016

Sufficient conditions are obtained for the delay-logistic equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove x With dot left-parenthesis t right-parenthesis equals r left-parenthesis t right-parenthesis x left-parenthesis t right-parenthesis left-bracket 1 minus x left-parenthesis t minus tau left-parenthesis t right-parenthesis right-parenthesis slash upper K right-bracket">

x
˙

(
t
)

=

r

(
t
)

x

(
t
)

[

1
−
x

(

t
−
τ

(
t
)

)

/

K

]

\dot x\left ( t \right ) = \\ r\left ( t \right )x\left ( t \right )\left [ {1 - x\left ( {t - \tau \left ( t \right )} \right )/K} \right ]

to be respectively oscillatory and nonoscillatory.
The asymptotic behavior of classical solutions to the mixed initial-boundary value problem in finite thermo-viscoelasticity
Authors: Beevers C. E. , Šilhavý M.
Published: December 2016

In this paper we consider the asymptotic stability of a class of solutions to the mixed initial-boundary value problem in nonlinear thermo-viscoelasticity. The continuum model is a viscoelastic material of rate type with the thermal conduction obeying Fourier’s law. The work in this article generalizes in two ways the results obtained by the present authors in a previous paper [1], The results in this present paper are valid for nonisothermal conditions and for a genuinely nonlinear viscous stress.
Finite groups of gravity waves
Authors: Yih Chia-Shun
Published: December 2016

Groups of gravity waves of finite length created in deep, originally quiescent water by an oscillating or moving surface pressure are constructed by superposition of the Cauchy-Poisson solution. This construction gives substance and reassurance to the concept of “wave packets” progressing with group velocity associated with the individual waves in the packets, a concept so important to water-wave research. The effects of viscosity are taken into account, thereby not only justifying the extensively used but completely artificial damping factor initiated by Lamb (1916, see Lamb 1945, p. 413), but also showing the hitherto largely unexplored spatial damping of waves.
Nonisothermal dynamic phase transitions
Authors: Grinfeld Michael
Published: December 2016

In this work we prove existence of travelling wave solutions to the Korteweg theory of capillarity regularization of conservation laws governing the motion of a van der Waals fluid. These solutions connect states in different phases and thus can be interpreted as dynamic phase transitions.
The three-dimensional stress intensity factor due to the motion of a load on the faces of a crack
Authors: Ramirez Jean-Claude
Published: December 2016

The dynamic stress intensity factor history for a half plane crack in an otherwise unbounded elastic body, with the crack faces subjected to a traction distribution consisting of a pair of point loads that move in a direction perpendicular to the crack edge, is considered. The exact expression for the mode I stress intensity factor as a function of time for any point along the crack edge is obtained by extending a procedure recently introduced by Freund [1]. The method of solution is based on integral transform methods and the theory of analytic functions of a complex variable. Some features of the solution are discussed and graphical results for various point load speeds are presented.
Viscoelastic surface waves on a standard linear solid
Authors: Currie P. K.
Published: December 2016

We investigate the number of surface waves on a standard linear viscoelastic solid. It is shown that two or one surface modes are possible, depending on the frequency and the material parameters. For materials on which two waves are possible, they will generally both be of equal significance over a wide range of frequencies.
Plane-strain problem of two coplanar cracks in an initially stressed neo-Hookean elastic layer
Authors: Dhaliwal Ranjit S. , Singh Brij Mohan
Published: December 2016

We consider the problem of determining the stress intensity factors and the crack energy in an infinitely long strip of an initially stressed neo-Hookean elastic material containing two coplanar Griffith cracks. We assume that the cracks are opened by a constant internal pressure and that the edges of the strip are either rigidly fixed or stress-free. By the use of Fourier transforms we reduce the problem to solving a set of triple integral equations with cosine kernel and a weight function. These equations are reduced to a Fredholm integral equation of the second kind by using finite Hilbert transform technique. Analytical expressions up to the order

δ

−
10

{\delta ^{ - 10}}

are derived for the stress intensity factors and the crack energy, where

2
δ

2\delta

denotes the width of the strip and

δ
\delta

is much greater than 1. Numerical values of the stress intensity factors and the crack energy are graphed to display the effect of initial stress.
On the apsidal limits of a rolling missile
Authors: Rath P. C. , Namboodiri A. V.
Published: December 2016

It is proved that a rolling missile whose initial angular oscillations are nonlinear will have the same librations as its equivalent common top provided that

q
>
0

q > 0

and

Z
1

+

Z
4

≤
−
2

{Z_1} + {Z_4} \le - 2

where

q
q

is certain aerodynamic parameter contained in the nonlinear overturning moment of the missile and

Z
1

{Z_1}

,

Z
4

{Z_4}

are respectively the least negative and the largest positive zeros of a certain quartic polynomial.
Viscoelastic Rayleigh waves II
Authors: Currie P. K. , O’Leary P. M.
Published: December 2016

The number of possible surface waves on a viscoelastic half-space is investigated numerically as a function of the material parameters. The results are applied to simple theoretical models of viscoelastic behavior and also to experimental data. It is concluded that two surface waves are possible on some rocks and on nearly all solid polymers.
Symmetry-axis elastic waves for transversely isotropic media
Authors: Payton Robert G.
Published: December 2016

An unbounded, linear transversely isotropic elastic solid is excited by a suddenly applied point body force of arbitrary orientation. Simple closed-form expressions are found for the various displacement components on the symmetry axis as a function of time and distance from the source. Applications are given for 13 hexagonal crystals.
Stability under step loading of infinitely long columns with localized imperfections
Authors: Amazigo John C. , Lockhart Deborah F.
Published: December 2016

The dynamic buckling of a long column with small dimple imperfections resting on a nonlinear foundation and subjected to axial step-loading is studied using a formal multi-variable perturbation expansion. Simple asymptotic formulas are obtained for the dynamic buckling load and lateral deflection in terms of the Fourier transform of the imperfection. It is found that the static and dynamic buckling loads are equal.
Atmospheric penetration of exhaust plumes under rarefied flow conditions
Authors: Baum Howard R. , Germeles Apostolos E.
Published: December 2016

The interaction of a low-density atmosphere with the exhaust gases produced by the steady firing of a rocket motor moving at hypersonic speed is studied using the kinetic theory of gases. The Krook collision model is employed in conjunction with a simple representation of the exhaust gas distribution to derive analytical and numerical solutions for the atmospheric gas density as it penetrates the rocket plume. The atmosphere gas is composed of an initially unscattered beam, which is attenuated as it collides with the much denser exhaust plume, and a scattered component, which is convected away from the motor by the expanding jet. These processes are illustrated by the computed results for both an axially symmetric and a three-dimensional flow.
The Sanders-Koiter shell equations can be reduced to two coupled equations for all minimal midsurfaces
Authors: Latta G. E. , Simmonds J. G.
Published: December 2016

For an elastically isotropic shell of constant thickness with an analytic midsurface of zero mean curvature, it is shown that the linear Sanders—Koiter equations can be reduced exactly to two coupled fourth-order partial differential equations, involving as unknowns a stress function and the rotation of the midsurface about the normal.
Statistical estimation of a crack damage parameter in manufacturing processes
Authors: Wiggins Alvin D.
Published: December 2016

A stochastic model of the production of cracks in certain types of linear manufacturing processes is constructed. Maximum likelihood is used to estimate a parameter, representing crack damage, of the resulting probability law.
On the differential equation of a rapidly rotating slender rod
Authors: Lakin W. D.
Published: December 2016

In this work we consider a boundary-value problem arising from the transverse vibrations of a slender, finite, uniform rod which rotates with constant angular velocity about an axis through the rod’s fixed end. The relevant dimensionless parameter is assumed to lie in a range corresponding to rapid rotation. The differential equation in this problem is fourth-order, linear, and takes its distinctive character from the simple turning point where the coefficient of the second derivative term vanishes. A significant feature is that the turning point is also a boundary point and hence outer expansions alone are not adequate for formation of a characteristic equation. Approximations valid at and away from the turning point are obtained and related through the method of matched asymptotic expansions. Outer expansions are required to be “complete” in the sense of Olver, and approximations are found for the Stokes multipliers which describe the analytic continuations of these expansions across Stokes lines in the complex plane. A consistent approximation to the characteristic equation is obtained, and the limiting behavior of the spectrum is derived analytically.
Poincare-Lighthill and linear-time-scales methods for linear perturbation problems
Authors: Klimas A. , Murphy F. X. , Sandri G.
Published: December 2016

For a class of multidimensional linear perturbation problems of considerable significance in applications, the Poincaré—Lighthill technique is shown to give first-order expansion terms which grow unbounded relative to the leading term (secular behavior), while the method of linear time scales leads to well-behaved expansion terms. A solvable example is introduced for comparison with the exact solution.
Asymptotic solutions for shells with general boundary curves
Authors: Logan T. R.
Published: December 2016

The influence of arbitrary edge loads on the stresses and deformations of thin, elastic shells with general boundaries is studied by means of asymptotic expansions of a general tensor equation. Expansions are made in terms of an exponential or an Airy function and a series in powers of a small-thickness parameter. Most of the steps in the procedure are effected by using the dyadic form of the tensors. Solutions are obtained that are valid in the large, with no restrictions on the loading or on the boundary geometry.
Nonlinear multiple-scale solution of a cylindrical shell
Authors: Stafford James R. , Hsu Adolf T.
Published: December 2016

A multiple-scale perturbation technique is used with a two-parameter expansion to study the asymptotic solution of Reissner’s axisymmetric finite-deformation equations for a circular cylindrical shell with an edge-bending moment load. Beyond the assumptions of Reissner’s differential equations, it is assumed that (1) the rotations of a shell element are finite but not excessively large, (2) thickness variations in the differential equations are of order one and (3) the boundary-layer behavior is of the linear bending type to a first approximation. An asymptotic solution is then found which is uniformly valid in that it contains boundary-layer effects and corrections for extending the analysis into the shell’s interior. Upon considering certain limits, it is observed that the solution contains well-established linear and nonlinear approximations to the solution.
On variational methods in finite and incremental elastic deformation problems with discontinuous fields
Authors: Nemat-Nasser S.
Published: December 2016

With a view toward a numerical solution by means of the finite-element method, we give here a variational statement for large elastic deformations at finite strains which involves independent variation of the displacement, the (nonsymmetric first Piola-Kirchhoff) stress, and the deformation-gradient fields, and which includes both the boundary and the jump conditions. Then we present, for small deformations superimposed on the large, three variational statements, each involving three independent fields and each including both the boundary and the jump conditions. These statements are such that the first variation of the corresponding functional yields the field equations which characterize the equilibrium of the finitely-deformed state considered and also the field equations that pertain to the incremental deformations. Several specializations of these results are discussed. By way of illustration, finally, we present a finite-element formulation of the large deformation problem, using three independent fields, where each field is approximated by a piecewise-linear function within each element.
On the weakly damped harmonic oscillator
Authors: Baum Howard R.
Published: December 2016

The initial-value problem for the differential equation
\[

u

(
t
)

+
ϵ

[

a
+

(

u
′

)

2

]

u
′

+
u
=
0

u\left ( t \right ) + \epsilon \left [ {a + {{\left ( {u’} \right )}^2}} \right ]u’ + u = 0

\]
is studied under the assumption that

a
a

and

ϵ
\epsilon

are positive constants and

ϵ
\epsilon

is small. Perturbation methods are used to obtain a first approximation to the solution that is uniformly valid in

t
t

. For any non-zero

a
a

the solution ultimately decays exponentially with a time scale

(

ϵ
a

)

−
1

{\left ( { \epsilon a} \right )^{ - 1}}

. For sufficiently small

a
a

, however, the damping is dominated by the cubic damping term for times of order

ϵ

−
1

{ \epsilon ^{ - 1}}

and is thus algebraic in character. The frequency of the oscillations is reduced by an amount

1
8

(

ϵ
a

)

2

\frac {1}{8}{\left ( { \epsilon a} \right )^2}

and is unaffected by the cubic damping to the order of approximation considered.
Further observations on a problem of constant surface heating of a variable-conductivity halfspace
Authors: Cooper Leonard Y.
Published: December 2016

A solution to the problem of constant surface heating of an initially constant-temperature,

T
0
∗

T_0^*

, halfspace where the material in question has a temperature-dependent thermal conductivity is obtained. The thermal conductivity,

k
∗

{k^ * }

, is specifically given by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k Superscript asterisk Baseline equals k 0 Superscript asterisk Baseline exp left-bracket lamda left-parenthesis upper T Superscript asterisk Baseline minus upper T 0 Superscript asterisk Baseline right-parenthesis slash upper T 0 Superscript asterisk Baseline right-bracket">

k
∗

=

k
0
∗

exp
⁡

[

λ

(

T
∗

−

T
0
∗

)

/

T
0
∗

]

{k^ * } = k_0^ * \exp \left [ {\lambda \left ( {{T^ * } - T_0^ * } \right )/T_0^ * } \right ]

. The solution is valid for both heating and cooling of the material where

λ
\lambda

and

k
0
∗

k_0^ *

are arbitrary in magnitude, and

λ
\lambda

can be either positive or negative in sign.
Optimal structural design for given deflection under periodic loading
Authors: Plaut Raymond H.
Published: December 2016

This paper is concerned with the minimum-weight design of structures subjected to periodic loading, under the constraint of a prescribed deflection at a specified point of the structure. For elastic sandwich beams, the method of stationary mutual potential energy is extended to cover periodic loading and a sufficient condition for stationary weight is derived.
Temperature and heat transfer history of a solid body in a forced convection flow
Authors: Konopliv N. , Sparrow E. M.
Published: December 2016

Consideration is given to problems of unsteady forced convection heat transfer in the presence of either time-invariant or time-dependent surface temperatures. The transient is initiated when a solid body is exposed to a fluid having a temperature different from its own. In the first part of the paper, a solution method is developed for determining the surface heat transfer for the case of steady, uniform surface temperature. Then, attention is turned to the determination of the temperature history of non-internally-heated bodies of high thermal conductance, which lose heat by convection to the fluid environment. A numerical scheme for deducing results for the temperature history is described, while analytical expressions appropriate to the initial and quasi-steady stages of the transient are presented. Detailed consideration is given to the case of a sphere in a low Péclet number flow, for which an exact solution for the temperature history is worked out. The results from the numerical scheme are found to be in excellent agreement with those from the exact solution, while the expressions for the initial and quasi-steady stages, when taken together, serve to establish the general behavior of the solution over the entire transient period.
Effects of empirical dissipation terms in the solution of the undular bore
Authors: Byatt-Smith John G. B.
Published: December 2016

In this paper we consider the problem of a steady bore running downhill. The effects of dissipation are included empirically, using the Chézy law. The method of solution is based on an averaging technique which assumes that the uniform solution is slowly varying.
A note on dipolar inertia
Authors: Green A. E. , Naghdi P. M.
Published: December 2016

With reference to the spatial form of the kinetic energy in the theory of simple dipolar continuum mechanics, a suitable expression is obtained for dipolar inertia. This includes the case in which the coefficient of the kinetic energy due to velocity gradient (in spatial form) may be constant.
Motion of a charged particle in a field of plane-wave electromagnetic radiation
Authors: Synge J. L.
Published: December 2016

A charged particle is acted on by plane waves of electromagnetic radiation, in general not monochromatic. In the absence of radiation damping, the relativistic equations of motion are solved exactly by quadratures. Radiation damping is then treated as perturbation.
On the nonunique equilibrium states of a shallow arch subjected to a uniform lateral load
Authors: Kerr Arnold D. , El-Bayoumy Lotfi
Published: December 2016

Using the exact solutiom of a shallow elastic arch clamped at both ends and subjected to a uniform lateral load, it is shown that, in addition to the equilibrium states usually discussed in the literature, also other equilibrium states, which correspond to higher modes of deformation, do exist. It is shown that there exist only a finite number of equilibrium branches which correspond to these higher modes, the number depending upon the shallowness of the arch. A number of graphs are presented which demonstrate these findings. The paper concludes with the proof that all states of equilibrium which correspond to the branches of higher order are unstable.
Separation of streamlines for spatially periodic flow at zero Reynolds numbers
Authors: Ranger K. B.
Published: December 2016

The stream function is found for the creeping flow between two cylinders. The inner cylinder is approximately circular and is rotating with constant angular velocity. The outer cylinder is fixed and corrugated or spatially periodic. The boundary vorticity on the outer boundary is discussed in relation to separation of the streamlines, as a function of the parameters describing the boundary geometry.
On stress-strain relations for isotropic rigid perfectly plastic solids
Authors: Dais J. L.
Published: December 2016

Sufficient conditions under which principal directions of stress and strain rate must coincide are established rigorously. It is the coincidence of these directions which permits a proper interpretation of principal strain rate components in principal stress space.
A note on the potential flow past a lemniscate and a general method of Milne-Thomson
Authors: Miles J. W. , Backus G. E.
Published: December 2016

Basset’s (1885) solution for potential flow past a lemniscate is (a) disproved and (b) reproduced by a naive application of “a general method” of Milne-Thomson. The correct solution is also given. The result is of interest for Rayleigh scattering, as well as for potential theory.