bluuebunny/crossref_metadata_2025 huggingface dataset:
Normality and convexity of the yield surface in nonlinear plasticity
Authors: Fosdick Roger , Volkmann Eric
Published: December 2016

An elementary proof of the normality condition is given. In addition, the convexity-concavity character of the yield surface is found to be related to the changes in elastic stiffness that take place during plastic loading.
The anisotropic elastic semi-infinite strip
Authors: Wang M. Z. , Ting T. C. T. , Yan Gong Pu
Published: December 2016

It is known that the stresses of an isotropic elastic semi-infinite strip decay exponentially at large distance

x
1

{x_1}

from the end

x
1

=
0

{x_1} = 0

if the sides

x
2

=
±
1

{x_2} = \pm 1

are traction free and the loading at

x
1

=
0

{x_1} = 0

is in self-equilibrium. We study the associated problem for a general anisotropic elastic strip. Eight different side conditions at

x
2

=
±
1

{x_2} = \pm 1

and eight different end conditions at

x
1

=
0

{x_1} = 0

are considered. With the Stroh formalism, all these different side and end conditions are encompassed in one simple formulation. It is shown that, for certain side conditions, the loading at

x
1

=
0

{x_1} = 0

need not be in self-equilibrium. The decay factor for the strip of monoclinic materials with the plane of symmetry at

x
3

=
0

{x_3} = 0

and with the sides

x
2

=
±
1

{x_2} = \pm 1

being traction free is derived, and it has a remarkably simple expression. Numerical calculations of the smallest decay factor are presented.
The completeness theorem for Rossby normal modes of a stably stratified flat ocean with an arbitrary form of side boundary
Authors: Masuda Akira
Published: December 2016

A mid-latitude flat ocean on a

β
\beta

-plane has characteristic oscillations called Rossby normal modes, where the motion is governed by the quasigeostrophic vorticity equation. Although the relevant eigenvalue problem differs from the usual one of Hilbert-Schmidt type, a variational proof is obtained that the Rossby normal modes constitute a complete orthonormal set for a basin with an arbitrary profile of stable density stratification and an arbitrary form of side boundary. In particular, for each fixed vertical mode, the set of the horizontal modes is complete and orthonormal in a two-dimensional Hilbert space. General solutions are expressed in terms of Rossby normal modes, not only to the initial-value problem, but also to the response problem of the closed basin.
Asymptotic solution for nonlinear chemical vapor deposition problems
Authors: Cassis B. , Tikhomirov O. , Wagner B. A.
Published: December 2016

The problem of chemical vapor deposition involving reaction kinetics of any order

n
n

at a heated substrate is considered. The deposition process is then described by a convective diffusion equation, coupled to nonlinear boundary conditions, describing the chemical reaction taking place at the heated substrate, where the nonlinearity is given in terms of the order

n
n

of the reaction kinetics. We derive boundary layer equations and use a combination of perturbation and similarity methods to find the deposition rate along the susceptor.
A potential theory for monotone multivalued operators
Authors: Romano G. , Rosati L. , Marotti de Sciarra F. , Bisegna P.
Published: December 2016

The concept of cyclic monotonicity of a multivalued map has been introduced by R. T. Rockafellar with reference to the subdifferential operator of a convex functional. He observed that cyclic monotonicity could be viewed heuristically as a discrete substitute for the classical condition of conservativity, i.e., the vanishing of all the circuital integrals of a vector field. In the present paper a potential theory for monotone multivalued operators is developed, and, in this context, an answer to Rockafellar’s conjecture is provided. It is first proved that the integral of a monotone multivalued map along lines and polylines can be properly defined. This result allows us to introduce the concept of conservativity of a monotone multivalued map and to state its relation with cyclic monotonicity. Further, as a generalization of a classical result of integral calculus, it is proved that the potential of the subdifferential of a convex functional coincides, to within an additive constant, with the restriction of the functional on the domain of its subdifferential map. It is then shown that any conservative monotone graph admits a pair of proper convex potentials which meet a complementarity relation. Finally, sufficient conditions are given under which the complementary and the Fenchel’s conjugate of the potential associated with a conservative maximal monotone graph do coincide.
Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions
Authors: Allaire Grégoire , Kohn Robert V.
Published: December 2016

This paper is concerned with two-dimensional, linearly elastic, composite materials made by mixing two isotropic components. For given volume fractions and average strain, we establish explicit optimal upper and lower bounds on the effective energy quadratic form. There are two different approaches to this problem, one based on the “Hashin-Shtrikman variational principle” and the other on the “translation method". We implement both. The Hashin-Shtrikman principle applies only when the component materials are “well-ordered", i.e., when the smaller shear and bulk moduli belong to the same material. The translation method, however, requires no such hypothesis. As a consequence, our optimal bounds are valid even when the component materials are not well-ordered. Analogous results have previously been obtained by Gibianski and Cherkaev in the context of the plate equation.
On the classification of solutions to the zero-surface-tension model for Hele-Shaw free boundary flows
Authors: Hohlov Y. E. , Howison S. D.
Published: December 2016

We discuss the classification of solutions to the zero-surface-tension model for Hele-Shaw flows in bounded and unbounded regions with suction and injection. We use results from the theory of univalent functions to derive estimates for certain geometric properties of the fluid region in the injection case.
The eigenvalue problem for a class of long, thin elastic structures with periodic geometry
Authors: Miller Robert E.
Published: December 2016

Elastic structures consisting of many thin elements arranged periodically (such as grids and trusses) are common in applications. Using standard numerical techniques such as splines in attempting to analyze these structures leads to serious difficulties due to the complicated geometry. Instead, one can use methods of asymptotic analysis to derive a “simple” problem whose solution approximates that of the original problem. In this paper we begin with a linearized elastic system on a three-dimensional domain with two of the dimensions small relative to the third and derive a one-dimensional eigenvalue problem by letting a small parameter tend to zero. The resulting equation has coefficients which vary periodically with the spatial variable, so we let the period tend to zero to obtain the “homogenized” equation which has constant coefficients whose values can be easily calculated once the geometry of the structure is specified. We illustrate with several examples.
Optimal lower bounds on the elastic energy of a composite made from two non-well-ordered isotropic materials
Authors: Allaire Grégoire , Kohn Robert V.
Published: December 2016

This paper is a continuation of our previous work [AK] concerning optimal bounds on the effective behavior of a mixture of two linearly elastic materials. While in [AK] we restricted our attention to the case of two well-ordered components, here we focus on the case of two non-well-ordered and isotropic ones, i.e., the case when the smaller shear and bulk moduli do not belong to the same material. For given volume fractions and average strain, we establish an optimal lower bound on the effective energy quadratic form. We give two proofs of this result: one based on the Hashin-Shtrikman-Walpole variational principle, the other on the translation method.
A dispersive continuum model of jointed media
Authors: Shkoller S. , Maewal A. , Hegemier G. A.
Published: December 2016

The problem of formulating higher-order continuum models for a jointed medium is considered for propagation of waves whose wavelengths are much larger than the cell width. A multiscale asymptotic approach is used to derive exact solutions for the microstructure in this large wavelength limit. A mixed variational principle is then invoked to obtain the homogenized model. This model, which incorporates dispersive wave phenomena, yields results which agree well with the exact solution for the dispersion of harmonic waves propagating through the medium.
On the solution of the equation 𝑢_{𝑡}+𝑢ⁿ𝑢ₓ+𝐻(𝑥,𝑡,𝑢)=0
Authors: Joseph K. T. , Sachdev P. L.
Published: December 2016

We consider the equation

u
t

+

u
n

u
x

+
H

(
x
,
t
,
u
)

=
0

{u_t} + {u^n}{u_x} + H\left ( x, t, u \right ) = 0

and derive a transformation relating it to

u
t

+

u
n

u
x

=
0

{u_t} + {u^n}{u_x} = 0

. Special cases of the equation appearing in applications are discussed. Initial value problems and asymptotic behaviour of the solution are studied.
Some estimates for the maximum shear stress in plane, isotropic elasticity
Authors: Aron M.
Published: December 2016

Upper and lower bounds for the maximum shear stress in a configuration corresponding to a purely distortional deformation originating from a given undistorted (ground) state are obtained in the framework of plane, isotropic, nonlinear elasticity. The bounds are shown to be expressible in terms of the deformation and the boundary traction that is required for maintaining the purely dilatational deformation in the ground state.
Nonoccurrence of stability switching in systems of differential equations with distributed delays
Authors: Kuang Yang
Published: December 2016

This paper deals with stability aspects of delay differential equations with general distributed delays. The objective is to show that, frequently, general distributed delays are not harder to handle than discrete delays. This is accomplished by treating two-dimensional systems of differential delay equations with distributed delays via several different approaches. All of these approaches are general, effective, and easy to apply.
Two-dimensional cellular burner-stabilized flames
Authors: Kuske R. , Matkowsky B. J.
Published: December 2016

We consider the behavior of a premixed flame anchored on a flat burner. For Lewis numbers

L
>

L
∗

>
1

L > {L^*} > 1

, one-dimensional stationary spatially periodic solutions corresponding to stationary one-dimensional cellular flames (rolls) bifurcate from the basic solution which corresponds to a steady planar flame. We derive and analyze an equation for the evolution of the amplitude of the roll solution just beyond the critical Lewis number

L
∗

{L^*}

. That is, we consider the case of supercritical bifurcation

(

L
>

L
∗

)

\left ( {L > {L^*}} \right )

and determine the ranges of wave numbers of perturbations corresponding to both the Eckhaus instability (to longitudinal perturbations) and the zigzag instability (to transverse perturbations) of the bifurcating solution. We determine these ranges in terms of the flow rate

m
∈

(
0
,
1
)

m \in \left ( 0, 1 \right )

and the scaled heat loss to the burner

K
>
2

/

e

K > 2/e

. For wave numbers

k
>
0.25

k > 0.25

we find that the zigzag instability occurs for all allowed values of

K
K

and for

m
m

bounded away from 1 and 0. As

k
k

increases, the range of values of

m
m

and

K
K

for which this instability occurs decreases. For

k
≥
0.4

k \ge 0.4

the zigzag instability no longer occurs for any allowed value of

m
m

and

K
K

. For each value of

L
L

there is a minimum value

m
=

m
∗

(
L
)

m = {m_*}\left ( L \right )

above which the Eckhaus instability does not occur. As

L
L

approaches

L
∗

,

m
∗

(
L
)

{L^*}, {m_*}\left ( L \right )

increases.
Mathematical study of a nonlinear neuron multi-dendritic model
Authors: Colli Pierluigi
Published: December 2016

A mathematical model of potential spreading along neuron dendrites is proposed to describe the synaptic transmissions in the so-called cerebellar granule cells, which consist of a nearly spherical soma emitting a finite number of dendrites. The model accounts for the nonlinear dependence of the NMDA receptors, located at the virtual ends of any dendrite, upon the voltage. The corresponding initial-boundary value problem is formulated in the framework of Sobolev spaces. Existence and uniqueness of a weak solution are proved along with regularity results ensuring that the solution is classical.
Thermomechanical evolution of a microstructure
Authors: Hoffmann Karl-Heinz , Roubíček Tomáš
Published: December 2016

A nonisothermal microstructure evolution model, governed by a Helmholtz free energy which need not be convex as a function of deformations, is formulated by using a convexified geometry proposed already in [13]. A multidimensional but scalar case is treated. It is shown that, as a special case, this model includes the usual nonlinear thermo-visco-elasticity. In the case of an actual appearance of a microstructure, the existence of a weak solution to a partial linearized model is shown by a semi-implicit time discretization.
Solitary waves in Poiseuille flow of a rotating fluid
Authors: Yih Chia-Shun
Published: December 2016

Solutions are obtained for various modes of solitary waves in a strongly rotating fluid in Poiseuille flow. The nonlinear Bragg-Hawthorne equation is the basis of the theory, and the method of solution can be carried to any order of approximation. There are two sequences of modes, each infinite in number. One consists of modes travelling with the flow, called P modes, and the other consists of modes travelling against it, called N modes. Generally, the displacements of streamlines from their asymptotic positions are in opposite directions for two corresponding modes of the two sequences. In particular, the streamlines of the first mode travelling with the flow are displaced radially outward from their asymptotic positions, and those of the first mode travelling against the flow are displaced radially inward. In other words, the first P mode is a bulging wave, whereas the first N mode is a pinching or necking one.
Localization of age-dependent anti-crowding populations
Authors: Hernández Gastón E.
Published: December 2016

In this work we prove the existence of solutions and study the localization and nonlocalization of the population in the Gurtin-MacCamy model with age-dependence and diffusion
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column StartFraction partial-differential rho Over partial-differential t EndFraction plus StartFraction partial-differential rho Over partial-differential a EndFraction 2nd Column a m p semicolon equals left-parenthesis rho u Subscript x Baseline right-parenthesis Subscript x Baseline minus mu left-parenthesis a comma u right-parenthesis rho comma 2nd Row 1st Column rho left-parenthesis x comma t comma 0 right-parenthesis 2nd Column a m p semicolon equals integral Subscript 0 Superscript normal infinity Baseline beta left-parenthesis a comma u right-parenthesis rho left-parenthesis x comma t comma a right-parenthesis d a comma 3rd Row 1st Column rho left-parenthesis x comma 0 comma a right-parenthesis 2nd Column a m p semicolon equals rho 0 left-parenthesis x comma a right-parenthesis greater-than-or-equal-to 0 comma EndLayout">

∂
ρ

∂
t

+

∂
ρ

∂
a

a
m
p
;
=

(
ρ

u
x

)
x

−
μ
(
a
,
u
)
ρ
,

ρ
(
x
,
t
,
0
)

a
m
p
;
=

∫
0
∞

β
(
a
,
u
)
ρ
(
x
,
t
,
a
)
d
a

,

ρ
(
x
,
0
,
a
)

a
m
p
;
=

ρ
0

(
x
,
a
)
≥
0
,

\begin{align*} \frac {{\partial \rho }}{{\partial t}} + \frac {{\partial \rho }}{{\partial a}} &= {(\rho {u_x})_x} - \mu (a, u)\rho , \\ \rho (x, t, 0) &= \int _0^\infty {\beta (a, u)\rho (x, t, a) da},\\ \rho (x, 0, a) &= {\rho _0}(x, a) \ge 0, \end{align*}

where
\[

u
(
x
,
t
)
=

∫
0
∞

ρ
(
x
,
t
,
a
)
d
a

u(x, t) = \int _0^\infty {\rho (x, t, a) da}

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta left-parenthesis a comma u right-parenthesis equals beta left-parenthesis u right-parenthesis sigma-summation Underscript k equals 1 Overscript n Endscripts b Subscript k Baseline a Superscript k Baseline e Superscript minus alpha a Baseline comma mu left-parenthesis a comma u right-parenthesis equals mu left-parenthesis u right-parenthesis period">

β
(
a
,
u
)
=
β
(
u
)

∑

k
=
1

n

b
k

a
k

e

−
α
a

,

μ
(
a
,
u
)
=
μ
(
u
)

.

\beta (a, u) = \beta (u)\sum \limits _{k = 1}^n {{b_k}{a^k}{e^{ - \alpha a}}, \qquad \mu (a, u) = \mu (u)}.

\]
Necessary and sufficient conditions for the oscillations of a multiplicative delay logistic equation
Authors: Grace S. R. , Győri I. , Lalli B. S.
Published: December 2016

Necessary and sufficient conditions are presented for the oscillation of all positive solutions of the multiplicative delay logistic equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction d upper N left-parenthesis t right-parenthesis Over d t EndFraction equals r left-parenthesis t right-parenthesis upper N left-parenthesis t right-parenthesis left-parenthesis 1 minus product Underscript j equals 1 Overscript m Endscripts left-parenthesis StartFraction upper N left-parenthesis g Subscript j Baseline left-parenthesis t right-parenthesis Over upper K EndFraction right-parenthesis right-parenthesis">

d
N

(
t
)

d
t

=
r

(
t
)

N

(
t
)

(

1
−

∏

j
=
1

m

(

N

(

g
j

(
t
)

K

)

)

\frac {{dN\left ( t \right )}}{{dt}} = r\left ( t \right )N\left ( t \right ) \left ( {1 - \prod \limits _{j = 1}^m { \left ( {\frac {{N\left ( {g_j}\left ( t \right ) \right .}}{K}} \right ) } } \right )

\]
about the positive equilibrium

K
K

. The cases when

r

(
t
)

=
r

r\left ( t \right ) = r

and

g
j

(
t
)

=
t
−

τ
j

{g_j}\left ( t \right ) = t - {\tau _j}

or

g
j

(
t
)

=

[

t
−

k
j

]

{g_j}\left ( t \right ) = \left [ {t - {k_j}} \right ]

, [•] denoting the greatest integer function, where

r
,

τ
j

r, {\tau _j}

, and

k
j

{k_j}

are positive constants,

j
=
1
,
2
,
.
.
.
,
m

j = 1, 2,...,m

, are also included.
Statistical tests of fit in estimation problems for structured population modeling
Authors: Fitzpatrick Ben G.
Published: December 2016

We present a statistical model of the type of data used in many applications of least squares estimation techniques to structured population problems. By viewing the data as a random sample from the size density, we may apply various statistical goodness of fit tests assess dynamic models for the data. We examine in particular classical

χ
2

{\chi ^2}

tests and a Cramér-von Mises test, based on empirical probability distributions.
Dynamics of reaction-diffusion equations with nonlocal boundary conditions
Authors: Pao C. V.
Published: December 2016

The purpose of this paper is to investigate the existence, uniqueness, and dynamics of a nonlinear reaction-diffusion equation with a nonlocal boundary condition which is motivated by a model problem arising from quasi-static thermoelasticity. The method of upper and lower solutions is used to obtain some existence-comparison results for both the time-dependent problem and its corresponding steady-state problem. A sufficient condition for the uniqueness of a steady-state solution is given. The comparison and uniqueness results are used to show the dynamical behavior of time-dependent solutions as well as their monotone convergence to a steady-state solution. Also given is a necessary and sufficient condition for the convergence of time-dependent solutions in relation to a steady-state solution which is not explicitly known. These results lead to the global stability of a steady-state solution for some special cases, including the model problem from thermoelasticity.
Improved bicharacteristic schemes for two-dimensional elastodynamic equations
Authors: Lin X. , Ballmann J.
Published: December 2016

Several authors have used explicit numerical schemes of bicharacteristics to solve the system of hyperbolic partial differential equations describing the two-dimensional propagation of stress waves in elastic solids. However, their numerical approaches differ slightly, which results in different limits of the CFL number for stable solutions and in different defects of numerical accuracy near singular points like, e.g., a numerically caused cutting trace emanating from a crack tip. After reviewing the different approaches, some techniques are presented to set up stable explicit schemes with CFL number up to the limiting values 1 for the fastest mode. Finally, the schemes are applied to the crack problem of a shock loaded body, where the main reasons for the appearance of a cutting trace become apparent.
Interface problems in elastoviscoplasticity
Authors: Carstensen Carsten , Stephan Ernst P.
Published: December 2016

This paper is concerned with three-dimensional interface (or transmission) problems in solid mechanics that consist of time-dependent nonlinear problems in a bounded Lipschitz domain and the homogeneous linear elasticity problem in an unbounded exterior domain. The exterior part of the interface problem is rewritten with integral operators on the interface boundary using the Poincaré-Steklov operator. This coupling approach uses the Calderón projector.
Temporary range of validity for some mathematical models which involve the heat-diffusion equation
Authors: Berrone Lucio R.
Published: December 2016

We study different initial and boundary value problems for the one-dimensional heat-diffusion equation, with the purpose of establishing conditions on initial and boundary data that ensure the solution is, during a certain interval of time, bounded by two predetermined constants. When the constants represent phase-change temperatures of a material medium, the desired conditions can be physically interpreted by saying that the medium does not undergo a phase-change during a certain lapse of time; i.e., the heat-conduction model will preserve its validity in the meantime. The tools employed to tackle this kind of problem consist in reducing the initial and boundary value problems to a Volterra integral equation. For these equations there exist simple methods to estimate their solutions. We improve some results that appeared in [8].
Analytic solutions of the vector Burgers’ equation
Authors: Nerney Steven , Schmahl Edward J. , Musielak Z. E.
Published: December 2016

The well-known analytical solution of Burgers’ equation is extended to curvilinear coordinate systems in three dimensions by a method that is much simpler and more suitable to practical applications than that previously used [22], The results obtained are applied to incompressible flow with cylindrical symmetry, and also to the decay of an initially linearly increasing wind.
Existence and stability of business cycles
Authors: Dohtani Akitaka
Published: December 2016

We provide several sufficient conditions for the existence and the stability of limit cycles in two-dimensional differential equation systems. We also apply our results to business cycle models in the Keynesian tradition.
A reaction-diffusion system arising in population genetics
Authors: Ruan W. H.
Published: December 2016

This paper is concerned with a nonweakly coupled system of parabolic equations that models the reaction-diffusion process of a population of continuously reproducing diploids with two alleles at one locus of genes. We assume that the fitnesses of genotypes are density dependent and the boundary condition is Dirichlet type with the boundary data given in the form such that the system has a unique uniform steady-state solution. The existence of nonuniform steady-state solutions is examined in relation with the stability of uniform steady-state solutions. In several cases we show that a nonuniform steady-state solution exists if the uniform one is unstable. We also investigate the stability and attractivity of the uniform steady-state solution in various cases. Some results of global convergence of solutions to a uniform steady-state solution are also given.
Growth rates for blow-up solutions of nonlinear Volterra equations
Authors: Roberts Catherine A. , Olmstead W. E.
Published: December 2016

An investigation is made of the blow-up growth property of the solution to certain nonlinear Volterra integral equations which model explosive behavior in a diffusive medium. The basic results provide the asymptotic form of the blow-up solution for a large class of kernels as well as various nonlinearities based on examples from solid combustion and adiabatic shear band formation.
Single phase energy minimizers for materials with nonlocal spatial dependence
Authors: Fosdick Roger L. , Mason Darren E.
Published: December 2016

We consider a one-dimensional body and assume that the total stored energy functional depends not only on the local strain field but also on the spatial average of the strain field over the body, weighted with an influence kernel. We investigate the problem of minimizing the total stored energy subject to given end displacements. The general existence theory for this problem is reviewed. Then, we narrow our study and concentrate on certain fundamental aspects of nonlocal spatial dependence by restricting our consideration to the case of a convex local energy and an exponential-type influence function for the nonlocal part. We find explicit solutions and show their characteristic properties as a function of the parameter that measures the extent of influence in the nonlocal kernel. We then study in detail the behavior that results when the total stored energy functional loses its coercivity. In this case, issues concerning the local and global stability of extremal fields are considered.
Linear stability analysis of reaction fronts in liquids
Authors: Garbey M. , Taik A. , Volpert V.
Published: December 2016

The propagation of a reaction front for liquid-to-solid reaction is studied. The model includes the heat equation, an equation for the concentration of the liquid reactant, and the equations of liquid motion under the Boussinesq approximation. The linear stability of the reaction front is studied, and conditions for cellular and oscillatory instability are determined.
Bending energy of highly elastic membranes. II
Authors: Hilgers M. G. , Pipkin A. C.
Published: December 2016

The strain energy per unit area for a deformed sheet of elastic material is estimated by representing the deformation as a power series in the thickness variable. The membrane energy is the lowest-order approximation obtained in this way. Strain-gradient and bending energies appear in the next order of approximation. Neither the membrane energy nor the higher-order approximation satisfy the Legendre-Hadamard material stability condition if the stress is compressive in some direction, so the theories based on either of these approximations can lead to problems with no stable solution. An energy function that does satisfy the material stability conditions is obtained by omitting the strain-gradient term, provided that certain longitudinal moduli are positive. A modified form for which existence of solutions can be guaranteed is proposed.
On the accurate computation of the prolate spheroidal radial functions of the second kind
Authors: Do-Nhat T. , MacPhie R. H.
Published: December 2016

The series expansion of the prolate radial functions of the second kind, expressed in terms of the spherical Neumann functions, converges very slowly when the spheroid’s surface coordinate

ξ
\xi

approaches 1 (thin spheroids). In this paper an analytical series expansion in powers of

(

ξ
2

−
1

)

\left ( {{\xi ^2} - 1} \right )

is obtained to facilitate the convergence. Then, by using the Wronskian test, it is shown that this newly developed expansion has been computed with a double precision accuracy.
Some quasi-static problems with elastic and viscous boundary conditions in linear viscoelasticity
Authors: Bosello Carlo Alberto , Gentili Giorgio
Published: December 2016

We study the quasi-static behaviour of a linearly viscoelastic body which is subject to boundary forces respectively of elastic type and of viscous type. The ensuing problems exhibit dynamic boundary conditions. We impose on the memory kernel only those restrictions deriving from thermodynamics and, making use of the Fourier transform method, we show existence and uniqueness of the solution to each problem.
Global existence and singularity formation in solutions of a modified Fourier law
Authors: Saxton K.
Published: December 2016

The aim of this paper is the analysis of formation of singularities from smooth initial data, for rigid conductors. The modified Fourier law used asserts that the governing second-order equation is hyperbolic with first-order dissipation. Global existence for small initial data is also examined.
On the computation of Infeld’s function used in evaluating the admittance of prolate spheroidal dipole antennas
Authors: Do-Nhat T. , MacPhie R. H.
Published: December 2016

The Infeld function expressed in terms of the outgoing prolate spheroidal radial wave function and its derivative, and employed in the expression of the input self-admittance of prolate spheroidal antennas, has accurately been calculated by using a newly developed asymptotic expression for large degree

n
n

. This asymptotic power series has been derived by using a perturbation method with a perturbation parameter

ϵ
=
1

/

(

λ

1
n

−

h
2

)

\epsilon = 1/\left ( {\lambda _{1n}} - {h^2} \right )

, where

λ

1
n

{\lambda _{1n}}

is the spheroid’s eigenvalue for the given parameter

h
h

of the spheroidal wave function.
Conservation laws and canonical forms in the Stroh formalism of anisotropic elasticity
Authors: Hatfield Gary A.
Published: December 2016

We find and classify all first-order conservation laws in the Stroh formalism. All possible non-semisimple degeneracies are considered. The laws are found to depend on three arbitrary analytic functions. In some instances, there is an “extra” law which is quadratic in

∇
u

\nabla u

. Separable and inseparable canonical forms for the stored energy function are given for each type of degeneracy and they are used to compute the conservation laws. The existence of a real Stroh eigenvector is found to be a necessary and sufficient condition for separability. The laws themselves are stated in terms of the Stroh eigenvectors.
On a class of limit states of frictional joints: formulation and existence theorem
Authors: Andersson Lars-Erik , Klarbring Anders
Published: December 2016

The model dealt with is a linear elastic body in frictional contact with a rigid support. Limit states of such an assemblage are characterized by deformations and forces such that a small perturbation may introduce a large change in configuration. The class of limit states considered here is specified by the possibility of superposing a time constant rigid body velocity field to a static deformation. The problem of finding such states (i.e., forces and static deformations) for a prescribed rigid body velocity is formulated, and for the case when the geometrically admissible rigid body displacements form a linear space an existence result is given. It is proved that under restrictions on the magnitude of the friction coefficient and in the case that an intuitively clear condition on the direction of the forces is satisfied, there exist a load multiplier and a corresponding static displacement.
Convergence of the two-phase Stefan problem to the one-phase problem
Authors: Stoth Barbara E.
Published: December 2016

We study the limit of the one-dimensional Stefan problem as the diffusivity coefficient of the solid phase approaches zero. We derive a weak formulation of the equilibrium condition for the resulting one-phase problem that allows jumps of the temperature across the interface. The weak formulation consists of a regularity condition that only enforces the usual equilibrium condition to hold from the liquid phase.
Equations of motion for constrained mechanical systems and the extended d’Alembert’s principle
Authors: Udwadia Firdaus E. , Kalaba Robert E. , Eun Hee-Chang
Published: December 2016

Starting from the principle of virtual work, this paper states and establishes an extended version of D’Alembert’s Principle. Using this extended principle and elementary linear algebra, it develops, from first principles, the explicit equation of motion for constrained mechanical systems. The results are compared with the authors’ previous results. The approach points to new ways of extending these results.
On the force between a dielectric cylinder in a constant electric field and a conducting half space
Authors: Dhondt Guido , Köhl Manfred
Published: December 2016

The force between an infinitely long dielectric cylinder in a constant electric field and a conducting half space is determined using the separation of variables technique on the Laplace equation in bipolar coordinates. The force is obtained as a series containing the relative distance between the cylinder and the half space as a parameter. This series is not uniformly convergent for the cylinder approaching the half space and the corresponding force cannot be obtained by performing the limit term per term. A special asymptotic analysis is presented leading to an analytic expression for this limiting value of the force.
Kinetic energy of highly elastic membranes
Authors: Hilgers M. G. , Pipkin A. C.
Published: December 2016

A theory of elastic sheets with bending stiffness has been proposed in which the strain energy density of the sheet includes a dependence on the second-order derivatives. To study the motion of such sheets, a kinetic energy is required that is accurate to the same order. This is obtained by representing the deformation as a power series in the thickness variable. The lowest-order approximation yields the standard membrane kinetic energy. The next order includes a velocity gradient term. A particularly simple physical interpretation for the additional term is obtained. Furthermore, the matrices involved in this term are shown to possess desirable properties, which can be utilized in future analysis.
An approximation theory for optimum sheets in unilateral contact
Authors: Petersson Joakim , Haslinger Jaroslav
Published: December 2016

In this paper we give an approximation theory for the optimum variable thickness sheet problem considered in [1] and [2]. This problem, which is a stiffness maximization of an elastic continuum in unilateral contact, admits complete material removal, i.e., the design variable is allowed to take zero values.
A uniqueness theorem for a Robin boundary value problem of physical geodesy
Authors: Otero Jesús
Published: December 2016

We get a uniqueness theorem for a Robin type boundary value problem for the Laplace equation arising in Physical Geodesy in the context of the gravimetric determination of the geoid. The boundary is an oblate ellipsoid of revolution and we have uniqueness of solutions provided that its eccentricity is (approximately) less than 0.526428.
The explicit Gibbs-Appell equation and generalized inverse forms
Authors: Udwadia F. E. , Kalaba R. E.
Published: December 2016

This paper develops an extended form of the Gibbs-Appell equation and shows that it is equivalent to the generalized inverse equation of motion. Both equations are shown to follow from Gauss’s principle. An example to highlight the two equivalent, though different, equations of motion is provided. Conceptual differences between the equations, and differences in their practical application to physical situations are discussed.
Linear stability of shock profiles for a rate-type viscoelastic system with relaxation
Authors: Luo Tao , Serre Denis
Published: December 2016

We investigate the linear stability of a shock profile for a rate-type viscoelastic system with relaxation. The linear stability of shock profiles for a general initial perturbation is proved by introducing new waves through time-asymptotic expansion and using energy estimates.
Effects due to shear flow on the diffusive-thermal instability of premixed gas flames
Authors: Kortsarts Y. , Kliakhandler I. , Shtilman L. , Sivashinsky G. I.
Published: December 2016

The diffusively unstable premixed gas flame is shown to be strongly affected by the background shear flow tangent to the flame interface. The pertinent flame-flow interaction alters both the flame stability limits and the character of its nonlinear evolution. The shear flow may expand the limits of the flame’s diffusive instability simultaneously inducing new nonlinear saturation and linear dispersion effects similar to those occurring in KdV systems.
A new model for acoustic-structure interaction and its exponential stability
Authors: Fahroo Fariba , Wang Chunming
Published: December 2016

A new model for the interaction between the acoustic wave in an enclosed air cavity and the transversal motion of a flexible beam is proposed in this paper. This new boundary condition for the coupled wave and Euler-Bernoulli beam equations introduces sufficient damping of the energy of the system to gain uniform exponential stability. Careful physical justification of the boundary condition is based upon well-established theoretical results in acoustics. The estimate of the energy decay rate is obtained using a multiplier technique.
Conservation laws and symmetrization of the equations of incompressible inviscid fluids
Authors: Vulkov Lubin G.
Published: December 2016

Many physical processes are described by conservative quasilinear systems of equations that admit a supplementary independent conservation law. A typical example is the equations of incompressible fluid dynamics. If for basic equations the conservation laws of the momentum and the volume are chosen then the conservation law of the energy is an additional conservation law. In the present paper all conservation laws for incompressible fluid dynamics are derived and the symmetric forms of its equations are obtained. Godunov’s method of symmetrization is generalized for the case of an arbitrary quasilinear system of equations.
The singular wedge problem in the nonlinear elastostatic plane stress theory
Authors: Tarantino Angelo Marcello
Published: December 2016

A finite elastostatic analysis of the singular equilibrium field in the proximity of the apex of a wedge, with clamped-free radial edges and general far-field loading conditions, is performed. The problem is formulated for compressible hyperelastic sheets under a plane stress condition. An asymptotic procedure is proposed to compute the deformation and stress singular fields. Emphasis is placed on the investigation of the dependence of the order of singularity in the asymptotic Piola-Kirchhoff and Cauchy stresses on the wedge angles. The case of a half-plane bounded to a rigid substrate is studied in detail.
Vanishing specific heat for the classical solutions of a multidimensional Stefan problem with kinetic condition
Authors: Yi Fahuai , Liu Jinduo
Published: December 2016

In this paper we prove that the multidimensional Hele-Shaw problem with kinetic condition at the free boundary is the limit case of the Stefan problem with kinetic condition at the free boundary in the classical sense when the specific heat

ε
\varepsilon

goes to zero. The method is the use of a fixed point theorem; the key step is to construct a suitable function space in which we can get the existence and uniform estimates with respect to

ε
>
0

\varepsilon > 0

at the same time as for classical solutions of the multidimensional Stefan problem with kinetic condition at the free boundary. For the sake of simplicity, we only consider one-phase problems in three space dimensions, although the method used here is also applicable for two-phase problems and any space dimensions.
On compressible materials capable of sustaining axisymmetric shear deformations. III. Helical shear of isotropic hyperelastic materials
Authors: Beatty Millard F. , Jiang Qing
Published: December 2016

A helical shear deformation is a composition of non-universal, axisymmetric, anti-plane shear and rotational shear deformations, shear states that are separately controllable only in special kinds of compressible and incompressible, homogeneous and isotropic hyperelastic materials. For incompressible materials, it is only necessary to identify a specific material, such as a Mooney-Rivlin material, to determine the anti-plane and rotational shear displacement functions. For compressible materials, however, these shear deformations may not be separately possible in the same specified class of hyperelastic materials unless certain auxiliary conditions on the strain energy function are satisfied. We have recently presented simple algebraic conditions necessary and sufficient in order that both anti-plane shear and rotational shear deformations may be separately possible in the same material subclass. In this paper, under the same physical condition that the shear response function be positive, we present an essentially algebraic condition necessary and sufficient to determine whether a class of compressible, homogeneous and isotropic hyperelastic materials is capable of sustaining controllable, helical shear deformations. It is then proved that helical shear deformations are possible in a specified hyperelastic material if and only if that material can separately sustain both axisymmetric, anti-plane shear and rotational shear deformations. The simplicity of the result in applications is illustrated in a few examples.
Global asymptotics toward the rarefaction wave for solutions of viscous 𝑝-system with boundary effect
Authors: Matsumura Akitaka , Nishihara Kenji
Published: December 2016

The initial-boundary value problem on the half-line

R
+

⋅
=

(
0
,
∞
)

{R_ + } \cdot = \left ( 0, \infty \right )

for a system of barotropic viscous flow
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v Subscript t Baseline minus u Subscript x Baseline equals 0 comma u Subscript t Baseline plus p left-parenthesis v right-parenthesis Subscript x Baseline equals mu left-parenthesis StartFraction u Subscript x Baseline Over v EndFraction right-parenthesis Subscript x Baseline">

v
t

−

u
x

=
0
,

u
t

+
p

(
v
)

x

=
μ

(

u
x

v

)

x

{v_t} - {u_x} = 0, {u_t} + p{\left ( v \right )_x} = \mu {\left ( {\frac {{{u_x}}}{v}} \right )_x}

is investigated, where the pressure

p

(
v
)

=

v

−
γ

(
γ
≥
1
)

p\left ( v \right ) = {v^{ - \gamma }}\left ( \gamma \ge 1 \right )

for the specific volume

v
>
0

v > 0

. Note that the boundary value at

x
=
0

x = 0

is given only for the velocity

u
u

, say

u
−

{u_ - }

and that the initial data

(

v
0

,

u
0

)

(
x
)

\left ( {v_0}, {u_0} \right )\left ( x \right )

have the constant states

(

v
+

,

u
+

)

\left ( {v_ + }, {u_ + } \right )

at

x
=
+
∞

x = + \infty

with

v
0

(
x
)

>
0
,

v
+

>
0

{v_0}\left ( x \right ) > 0, {v_ + } > 0

. If

u
−

>

u
+

{u_ - } > {u_ + }

, then there is a unique

v
−

{v_ - }

such that

(

v
+

,

u
+

)

∈

R
2

(

v
−

,

u
−

)

\left ( {v_ + }, {u_ + } \right ) \in {R_2}\left ( {v_ - }, {u_ - } \right )

(the 2-rarefaction curve) and hence there exists the 2-rarefaction wave

(

v
2
R

,

u
2
R

)

(
x

/

t
)

\left ( v_2^R, u_2^R \right )\left ( x/t \right )

connecting

(

v
−

,

u
−

)

\left ( {v_ - }, {u_ - } \right )

with

(

v
+

,

u
+

)

\left ( {v_ + }, {u_ + } \right )

. Our assertion is that, if

u
−

>

u
+

{u_ - } > {u_ + }

, then there exists a global solution

(
v
,
u
)

(
t
,
x
)

\left ( v, u \right )\left ( t, x \right )

in

C
0

(

[
0
,
∞
)

;

H
1

(

R
+

)

)

{C^0}\left ( \left [ 0, \infty \right );{H^1}\left ( {R_ + } \right ) \right )

, which tends to the 2-rarefaction wave

(

v
2
R

,

u
2
R

)

(
x

/

t
)

|

x
≥
0

\left ( v_2^R, u_2^R \right )\left ( x/t \right )\left | {_{x \ge 0}} \right .

as

t
→
∞

t \to \infty

in the maximum norm, with no smallness condition on

|

u
+

−

u
−

|

\left | {{u_ + } - {u_ - }} \right |

and

‖

(

v
0

−

v
+

,

u
0

−

u
+

)

‖

H
1

{\left \| {\left ( {{v_0} - {v_ + }, {u_0} - {u_ + }} \right )} \right \|_{{H^1}}}

, nor restriction on

γ

(
≥
1
)

\gamma \left ( \ge 1 \right )

. A similar result to the corresponding Cauchy problem is also obtained. The proofs are given by an elementary

L

2

{L^{2}}

-energy method.
Unidirectional infiltration in deformable porous media: mathematical modeling and self-similar solution
Authors: Billi L. , Farina A.
Published: December 2016

This study is motivated by the necessity to develop models and to obtain mathematical results which can help in improving certain industrial processes used for manufacturing composite materials. We focus on unidirectional infiltration processes in deformable porous material and on the analytical study of the related mathematical problem. A key point in developing the model is the use of a set of Lagrangian coordinates fixed on the solid. This technique allows us to simplify the mathematical problem. In the Eulerian formalism, in fact, such a problem is a free boundary problem characterized by the presence of two time-dependent interfaces. The use of material coordinates fixed on the solid allows, vice-versa, to fix one boundary and so to obtain a nonlinear one-phase Stefan problem. The latter, then, has been studied from the analytical viewpoint. In particular, we have proved the existence of a solution exhibiting a self-similar form.
Models of low-speed flow for near-critical fluids with gravitational and capillary effects
Authors: Denny D. L. , Pego R. L.
Published: December 2016

We study low-speed flows of a highly compressible, single-phase fluid in the presence of gravity, for example, in a regime appropriate for modeling recent space-shuttle experiments on fluids near the liquid-vapor critical point. In the equations of motion, we include forces due to capillary stresses that arise from a contribution made by strong density gradients to the free energy. We derive formally simplified sets of equations in a low-speed limit analogous to the zero Mach number limit in combustion theory.
Equisum matrices and their permanence
Authors: Davis Philip J. , Najfeld Igor
Published: December 2016

In a square equisum matrix, all row and column sums are equal. In a rectangular equisum matrix, the common row sum is a rational multiple of the common column sum. This paper explores properties of equisum matrices, in particular, the preservation of the equisum condition under a variety of linear, nonlinear and pattern-maintaining transformations. A principal tool employed is a representation via the Fourier matrix or the circulant projectors associated with it.
Existence of classical solutions of the equations of motion for compressible fluids of second grade
Authors: Matušů-Nečasová Š.
Published: December 2016

The existence of the classical solutions has been proved for the equations of motion for compressible fluids of second grade in a bounded domain.
The birth of a cusp in the two-dimensional, undercooled Stefan problem
Authors: Herrero M. A. , Medina E. , Velázquez J. J. L.
Published: December 2016

This paper deals with the one-phase, undercooled Stefan problem, in space dimension

N
=
2

N = 2

. We show herein that planar, one-dimensional blow-up behaviours corresponding to the undercooling parameter

Δ
=
1

\Delta = 1

are unstable with respect to small, transversal perturbations. The solutions thus produced are shown to generically generate cusps in finite time, when they exhibit an undercooling

Δ
=
1
−
O

(
ϵ
)

>
1

\Delta = 1 - O\left ( \epsilon \right ) > 1

, where

0
>
ϵ
>>
1

0 > \epsilon >> 1

, and

ϵ
\epsilon

is a parameter that measures the strength of the perturbation. The asymptotic behaviour of solutions and interfaces near their cusps is also obtained. All results are derived by means of matched asymptotic expansions techniques.
Asymptotic analysis of torsional and stretching modes of thin rods
Authors: Irago H. , Kerdid N. , Viaño J. M.
Published: December 2016

In this article, we show that a class of high frequencies of the three-dimensional linearized elasticity system in a thin rod and their associated eigenfunctions converge in a precise sense, as the area of the cross section of the rod goes to zero. The limit model is a coupled one-dimensional problem giving the classical equations for torsion and stretching modes in rods.