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Showing papers in "Quarterly of Applied Mathematics in 1985"


Journal ArticleDOI
TL;DR: In this article, the authors presented a model for thin plates with rapidly varying thickness, distinguishing between thickness variation on a length scale longer than ('a 1'), on the order of ''a = 1'), or shorter than ''a 1'' the mean thickness.
Abstract: : A recent paper presented a model for thin plates with rapidly varying thickness, distinguishing between thickness variation on a length scale longer than ('a 1'), on the order of ('a = 1'), or shorter than ('a 1') the mean thickness. We review the model here, and identify the 'a 1' case as an asymptotic limit of the case 'a = 1' case, showing that the model correctly represents the solution of the equations of linear elasticity on the three-dimensional plate domain, asymptotically as the mean thickness tends to zero. (Author)

123 citations


Journal ArticleDOI
TL;DR: In this article, the methode quadratique par morceaux utilisees par Gerasoulis, is used to resoudre des equations integrales singulieres de premiere espece.
Abstract: La technique etend la methode quadratique par morceaux utilisees par Gerasoulis, pour resoudre des equations integrales singulieres de premiere espece. On deduit des formules de quadrature en termes de series infinies et on les utilise pour reduire l'equation integrale a un ensemble d'equations algebriques lineaires

47 citations


Journal ArticleDOI
TL;DR: In this article, an approximate linear stability analysis of a one-dimensional rigid-thermoviscoplastic model, based on data taken from dynamic torsion experiments on thin-walled tubes of mild steel, shows that shear band formation in this situation can be interpreted as a bifurcation from a homogeneous simple shearing deformation which occurs at the peak in the homogeneous stess vs. strain curve.
Abstract: The formation of adiabatic shear bands in ductile metals under dynamic loading conditions is generally thought to result from a material instability, which is associated with a peak in the curve of engineering plastic flow stress vs. engineering shear strain. This instability arises from the effect of thermal softening, caused by irreversible adiabatic heating, which counteracts the tendency of the material to harden with increasing plastic strain. An approximate linear stability analysis of a one-dimensional rigid-thermoviscoplastic model, based on data taken from dynamic torsion experiments on thin-walled tubes of mild steel, shows that shear band formation in this situation can be interpreted as a bifurcation from a homogeneous simple shearing deformation which occurs at the peak in the homogeneous stess vs. strain curve. The asymptotic method of multiple scales is used to show that the growth rate of small perturbations on the homogeneous deformation is controlled by the ratio of the slope of the homogeneous stress vs. strain curve to the material viscosity, i.e., the rate of change of the plastic flow stress with respect to the strain-rate. In addition, it is shown that this growth rate is essentially independent of wavelength in any small perturbation. Numerical methods are usedmore » to show that this growth rate beyond the bifurcation point may not be sufficiently large for the model to account for the experimental data, and some suggestions are made on how to modify the constitutive equation so that it better fits the experimental data.« less

45 citations


Journal ArticleDOI
TL;DR: For an elastic body containing periodically distributed voids, several effective techniques are presented which can be used to obtain the effective elastic moduli with any desired degree of accuracy as discussed by the authors, including the effects of void geometry as well as void interactions.
Abstract: For an elastic body containing periodically distributed voids, several effective techniques are presented which can be used to obtain the effective elastic moduli with any desired degree of accuracy. The results include the effects of void geometry as well as void interactions. For a body containing spherical voids, numerical results are presented and compared with those obtained by other methods.

37 citations


Journal ArticleDOI
TL;DR: In this paper, a formulation and comportement qualitatif des problemes de Stefan hyperboliques (processus de changement de phase regis par un transfert de chaleur hyperbolique) is presented.
Abstract: Questions et remarques relatives a la formulation et au comportement qualitatif des problemes de Stefan hyperboliques (processus de changement de phase regis par un transfert de chaleur hyperbolique). Correction d'une erreur dans la condition d'interface, solution explicite d'un probleme a une phase. Formulation en termes d'enthalpie d'un probleme a deux phases, et experiences numeriques

36 citations


Journal ArticleDOI
TL;DR: Etude du point d'arret dans le probleme d'un fluide leger tombant sur un fluide plus lourd au repos as discussed by the authors.
Abstract: Etude du point d'arret dans le probleme d'un fluide leger tombant sur un fluide plus lourd au repos. Solutions de similitude pour les regions inferieure et superieure. Etude du transfert de chaleur par convection

35 citations


Journal ArticleDOI
TL;DR: In this article, a macroscopic mathematical model is constructed describing the evolution of the phases of a binary alloy or mixture undergoing solidification under the action of simultaneous conduction of heat and diffusion of solute.
Abstract: A macroscopic mathematical model is constructed describing the evolution of the phases of a binary alloy or mixture undergoing solidification under the action of simultaneous conduction of heat and diffusion of solute. The formulation is global, in the form of a pair of conservation laws valid over the whole region occupied by the alloy in a weak (distributional) sense. Thus it is especially convenient for numerical solution since it does not require tracking of the interface, which, in fact, may develop into a ''mushy zone''.

25 citations



Journal ArticleDOI
TL;DR: In this article, the linear stability of two-dimensional boundary layer flow of an incompresible viscous fluid over a flat deformable sheet is investigated when the sheet is stretched in its own plane with an outward velocity proportional to the distance from a point on it.
Abstract: The linear stability of two-dimensional boundary layer flow of an incompresible viscous fluid over a flat deformable sheet is investigated when the sheet is stretched in its own plane with an outward velocity proportional to the distance from a point on it. Using Galerkin's method the stability equations are solved for three-dimensional disturbances periodic in a direction normal to the plane of the basic flow and it is shown that the flow is stable.

20 citations


Journal ArticleDOI
TL;DR: On considere des integrales qui peuvent se transformer en la transformee de Laplace as mentioned in this paper, on donne des applications aux fonctions speciales, on donné des applications.
Abstract: On considere des integrales qui peuvent se transformer en la transformee de Laplace. On donne des applications aux fonctions speciales

20 citations


Journal ArticleDOI
TL;DR: In this paper, the authors considered the boundary layer diffusion problem with boundary conditions t/(0) = 0, LU'(V) = t/(l) and the thermal conductivity is conventionally set to be one.
Abstract: This equation can be thought of as describing steady-state temperature distribution U{x) in a (one-dimensional, in our case) conducting medium in which a local voltage drop V(x) is mantained; a steady current makes each point into a heat source. The electrical resistivity at x is o(x)(U(x))r, where V2(x)/a(x) = K(x), and thermal conductivity is conventionally set to be one. This is a particular case of a problem treated in [1] for the transient case, [2, 3], and [4]. Equation (1) is subject to the usual boundary conditions. Analogies between one-dimensional diffusion and boundary layer problems in fluid dynamics are well known. In Sec. 4 we show how this electrical analogy can be used in the analysis of diffusion problems. In Sec. 2 we set up the two problems with the various boundary conditions we shall consider. In Sec. 3 we discuss the bibliography on the two problems, and the hypotheses made on the function K(x) in (1), related to the diffusivity in the diffusion context. In Sec. 4 we give, in I, a unified uniqueness proof for problems (2), (2.1)—(2.3) below; we give, in II and III, estimates for the values «(0) in (2.2) and tj, in (2.3), and in IV, we include an existence proof for (1) with boundary conditions t/(0) = 0, LU'(V) = t/(l) (cf. (4), (4.3)).

Journal ArticleDOI
TL;DR: On deduit des conditions suffisantes pour que toutes les solutions nonnegatives nonconstantes des equations de la forme dx(t)/dt=x(t){a-b∫ − ∫ −∞ t k(t-s)x(s)ds}soient oscillatoires autour de leurs etats stationnaires positifs respectifs.
Abstract: On deduit des conditions suffisantes pour que toutes les solutions nonnegatives nonconstantes des equations de la forme dx(t)/dt=x(t){a-Σ j=1 n bjx(t-τ j )} et dx(t)/dt=x(t){a-b∫ − ∫ −∞ t k(t-s)x(s)ds}soient oscillatoires autour de leurs etats stationnaires positifs respectifs

Journal ArticleDOI
TL;DR: In this article, the conical cross flow surface of a delta wing was artificially constructed using a smooth sonic surface and an efficient numerical method to calculate the flow field to obtain cones with smooth cross flow.
Abstract: In order to have a high level of maneuverability, supersonic delta wings should have a cross flow that is free of embedded shock waves. The conical cross flow sonic surface differs from that of plane transonic flow in many aspects. Well-known properties such as the monotone law are not true for conical cross flow sonic surfaces. By using a local analysis of the cross flow sonic line, relevant conditions for smooth cross flow are obtained. A technique to artificially construct a smooth sonic surface and an efficient numerical method to calculate the flow field are used to obtain cones with smooth cross flow.

Journal ArticleDOI
TL;DR: In this article, it was shown that there exists a term r1/2(ln r) besides the familiar terms r~ 1/2 and r2/2 for a crack with curved free boundaries.
Abstract: When the boundaries of an elastic wedge are straight lines, the asymptotic solution near the apex r = 0 of the wedge is simply a series of eigenfunctions of the form rxf(0. A) in which (r, 6) is the polar coordinate with origin at the wedge apex and A is the eigenvalue. When the wedge boundaries are curved, the eigenvalues remain the same but the curvatures of the boundaries change the form of the eigenfunctions. The eigenfunction associated with a A contains not only the term rx, but also rx + 1, rx + 2,... In some cases it also contains the term rx + 1(ln r ). Therefore, the second and higher order terms of asymptotic solution are not simply the second and next eigenfunctions. Examples are given for the first few terms of asymptotic solution for wedges with wedge angle v and 2u. The latter corresponds to a crack with curved free boundaries and we show that there exists a term r1/2(ln r) besides the familiar terms r~1/2.

Journal ArticleDOI
TL;DR: In this article, the determination numerique des orbites des systemes dynamiques is studied. André et al. evaluate de la vitesse de convergence vers les courbes invariantes.
Abstract: Etude de la determination numerique des orbites des systemes dynamiques. Etablissement de l'existence de courbes invariantes basee sur l'utilisation d'une methode multipas convergente, fortement stable. Evaluation de la vitesse de convergence vers les courbes invariantes

Journal ArticleDOI
TL;DR: In this article, the approche de Galerkin is used to demontrer des theoremes sur le decouplage, la symetrie et similitudes reliees aux diagrammes de stabilite.
Abstract: On utilise une approche de Galerkin pour demontrer des theoremes sur le decouplage, la symetrie et similitudes reliees aux diagrammes de stabilite. On donne une classification des fonctions excitations possibles

Journal ArticleDOI
TL;DR: In this article, the Stokes flow inside a torus is approximated by an eigenfunction expansion in toroidal coordinates, and the ratio of volume flow carried by the torus to that carried by a straight tube is computed as a function of the vessel radius: coil radius ratio.
Abstract: The hydrodynamic resistance of a buckled microvessel in the form of a tightly would helix is approximated by studing the Stokes flow inside a torus. The unidirectional flow is driven by a constant tangential pressure gradient. The solution is obtained by an eigenfunction expansion in toroidal coordinates. The ratio of volume flow carried by the torus to that carried by a straight tube is computed as a function of the vessel radius: coil radius ratio. An asymptotic expansion for this flux ratio is also obtained. The results show that the resistance of a moderately curved vessel is slightly less than the resistance of a straight one, whereas the resistance of a greatly curved vessel is at most 3% greater than the straight one.

Journal ArticleDOI
TL;DR: On deduit des conditions suffisantes for qu'un systeme matriciel vectoriel de la forme: d n X(t)/dt n +(−1) n−1 [P(t)X(t−τ 1 (t))+Q(t)-X(T+τ 2 (t)).
Abstract: On deduit des conditions suffisantes pour qu'un systeme matriciel vectoriel de la forme: d n X(t)/dt n +(−1) n−1 [P(t)X(t−τ 1 (t))+Q(t)X(t+τ 2 (t))]=0 soit non oscillatoire

Journal ArticleDOI
TL;DR: In this paper, 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.
Abstract: 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.

Journal ArticleDOI
TL;DR: In this article, an exact dynamic 2D solution for a concentrated force near a stationary semi-infinite crack in an unbounded plane can be used in the transient analysis of wave-scattering problems.
Abstract: An exact dynamic 2D solution for a concentrated force near a stationary semi-infinite crack in an unbounded plane can be used in the transient analysis of wave-scattering problems. Direct approaches to obtaining the solution, however, are complicated by the existence of a characteristic length. A less direct approach is used here which circumvents these complications. As an example, the dynamic stress intensity factors are derived and studied for their behavior w.r.t. time and concentrated force-crack edge orientation.

Journal ArticleDOI
TL;DR: On considere des problemes aux valeurs propres pour des equations differentielles aleatoires d'ordre 2 a coefficients faiblement correles as discussed by the authors.
Abstract: On considere des problemes aux valeurs propres pour des equations differentielles aleatoires d'ordre 2 a coefficients faiblement correles

Journal ArticleDOI
TL;DR: In this article, the Stokes equations are written in orthogonal curvilinear coordinates and the centerline curvature is specified as a function of arc length, where the primary small parameter is the slenderness ratio e, which is the ratio of vessel radius to vessel length or wavelength.
Abstract: Steady slow viscous flow is considered inside a vessel with circular cross section. The centerline curvature is specified as a function of arc length. The Stokes equations are written in orthogonal curvilinear coordinates. The primary small parameter is the slenderness ratio e, which is the ratio of vessel radius to vessel length or wavelength. The product of centerline curvature and vessel length is assumed to be of order unity. A transverse drift appears at 0(e2) that is proportional to the rate of change of curvature. Contours of axial velocity show a primary peak shifted toward the inside wall and a secondary peak grows toward the outside wall as curvature is increased. The flux ratio or relative hydrodynamic conductance is calculated to 0(e4) and includes the effect of variable curvature. The present calculations tend to indicate that the sinusoidal mode of buckled micro-vessel could offer substantially more resistance to flow than the helical buckled mode.



Journal ArticleDOI
TL;DR: Naghdi and Vongsarnpigoon as discussed by the authors used a Cosserat surface with a single director to study the kinematic properties of shells of revolution and showed that the deformation of a shell may be finite or one with small strain accompanied by large or moderately large rotation.
Abstract: 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. 1. Preliminary remark. The purpose of this paper is to state and prove certain purely kinematical results for torsionless axisymmetrical deformation of shells of revolution. The deformation of the shell may be finite or one with small strain accompanied by large or moderately large rotation. Since the results obtained are purely kinematical, they hold for all materials and are not necessarily limited to elastic shells. We utilize a direct formulation of the theory of shells based on a 2-dimensional continuum model as a Cosserat surface with a single director [1,2], but the corresponding kinematical developments can be also effected from an appropriate approximation for the position vector in the 3-dimensional theory (see, e.g., [2, Eq. (2.25)]). For readers' convenience, we recall that a Cosserat surface is a body comprising a material surface embedded in a Euclidean 3-space, together with a single deformable vector field—called director—attached to every material point of Sf. The director, which is not necessarily along the unit normal to the surface surface Sf, has in particular the property that it remains unaltered in length under superposed rigid body motions.1 Clearly, a Cosserat surface is not just a 2-dimensional surface; but is, in fact, endowed with some structure in the form of an additional primitive kinematical vector field. The magnitude of the director along the normals to the reference surface may be regarded as •Received June 21, 1983. The results reported here we obtained in the course of research supported by the U.S. Office of Naval Research under Contract N00014-75-C-0148, Project NR 064-436 with the University of California, Berkeley. 1 As in the paper of Naghdi [2], the director here is chosen to have the physical dimension of length. This differs from an earlier choice in Naghdi [1], where the director was assumed to be dimensionless. For further discussion on this point, see Naghdi [2, Sec. 3]. ©1985 Brown University 24 P. M. NAGHDI AND L. VONGSARNPIGOON representing the thickness of the shell-like body 38 and similarly the material surface Zfol may be identified with the reference surface (e.g., the middle surface) in the shell-like body. Related remarks pertaining to the identification of various constitutive results, the assigned fields and the inertia coefficients in the theory of a Cosserat surface are made in Naghdi [1] and [2, Sec. 5]. Also, following the procedure of Casey and Naghdi [3], in the kinematical development of a Cosserat surface we remove the local rotation and the relative displacement at one point of the surface ^resulting in a configuration k* so as to render all kinematical and associated kinetical results invariant under arbitrary finite superposed rigid body motion (for details, see Naghdi and Vongsarnpigoon [4, Sec. 4-5] and [5, Sec. 2]). By way of additional background and in the context of a 3-dimensional theory, consider a relative displacement2 u* = r* R* and the relative displacement gradient H* = F* I = Gradu*, where r* and R* are, respectively, the position vectors of a material point in the deformed and undeformed configurations, F* is the deformation gradient tensor and I is the identity tensor. Recall that through the polar decomposition theorem, we have F* = R*U*, where R* is the (local) rotation tensor and U* is the right stretch tensor which are related to the relative strain tensor E* through the expression 2E* = (U*)2 — I = p*rF* I. If H* is infinitesimal, then the strain E* and the rotation R* are also infinitesimal. But if only E* is infinitesimal of order e the relative deformation u* is not necessarily small and the rotation tensor R* may be large or moderately large of order e1/2 (see Naghdi and Vongsarnpigoon [4]). Keeping the above background in mind, consider now a motion of a Cosserat surface # characterized by the two functions r and d corresponding to the position vector of a material point on the material surface Sfand the director at taht point. The deformation graident F and director gradient G can be defined in terms of suitable components of Gradr and Gradd (see Eqs. (Al)12 of Appendix A) and these, in turn, give rise to the strain tensor 2E = FrF — I and the curvature tensor F7G defined on the material surface ^in the current configuration. Further, through the polar decomposition theorem, we again have F = RU, where R is the (local) rotation tensor and U the right stretch tensor. Given F, the position vector r and the director d in the deformed configuration k can be determined (see (Al)t) and thus the deformation is known. A deformation of a shell-like body is said to be infinitesimal if the deformation gradient F differs only infinitesimally from the unit tensor I and, consequently, the strain tensor E and rotation tensor R are both infinitesimal. However, if E is small, the displacement vector u = r — rR, where rR is the position vector in the reference configuration, is not necessarily infinitesimal; and the deformation gradient tensor F could differ significantly from I and may be accompanied by large or moderately large R. Our main results are summarized as three theorems in Sec. 2 and the proofs are supplied in Sec. 3. Briefly, with reference to torsionless axisymmetrical deformation of a shell of revolution, we first show in Theorem 1 that the knowledge of the relative strain tensor at every point of the reference surface (e.g., the middle surface) is sufficient to 2 We attach an asterisk to the kinematical quantities associated with the 3-dimensional theory to differentiate them from corresponding quantities employed in the direct (2-dimensional) theory of a Cosserat surface. AXISYMMETRICAL DEFORMATION OF SHELLS OF REVOLUTION 2 5 determine the change in the second fundamental form and hence the deformed configuration of the shell. Furthermore, if the strain and its first and higher order gradients are assumed to be small, then by our Theorem 2 the only kind of deformation accompanied by large rotation (or large deflection)—excluding, of course, the rigid body translation—of a nonshallow shell is that which involves an inversion of the shell. Indeed, it is a consequence of Theorem 2 that if the possibility of an inversion of the (nonshallow) shell is not admitted, a deformation with small strain accompanied by large rotation can take place only by allowing the strain gradient to be large. On the other hand, for a shallow shell of revolution (which includes an initially flat circular plate as a special case), while a deformation with large rotation (or large deflection) is not possible when the strain and strain gradients are assumed to be small, according to Theorem 3 the case in which the shell is subjected to a deformation with moderate rotation is admissible. Throughout the paper, we use the usual summation convention over repeated indices with Greek indices taking the values 1, 2 and Latin indices having the ranges 1, 2, 3. Also, in the main text and the Appendix A, we employ both a coordinate-free notation and the component form of various results and equations. 2. Some background information. A summary of the main results. Let e, (/ = 1,2, 3) be a set of orthonormal base vectors associated with a system of rectangular Cartesian coordinates (x, y, z) and let 8" (a = 1,2) designate a convected coordinate system on the material surface yof a Cosserat surface e€. Further, with reference to a cyclindrical polar coordinate system (r, 8, z), let e, = er(8) and e0 = eg(8) be the unit base vectors defined by er = cos^! + sin#e2, e9 = -sintfej + cos 0e2. (2.1) Without loss in generality we may identify the convected coordinate 62 with the angle 8 of the cyclindrical polar coordinate system and write 81 = 8. (2.2) Let SfR, a two-dimensional region of space occupied by the material surface Sf of #in the reference configuration k0, be a surface of revolution with its position vector specified by tr = ro(01)*r + Zo(0l)*3(2.3) Then, from (2.3), the surface base vectors AQ associated with the convected coordinates 6" and the outward unit normal A 3 to £fR are calculated to be Ai = roer + zoe3> A2 = r0efl, A3 = (r0'e3 z'0er)/a0, (2.4) where prime denotes partial differentiation with respect to 81 and where a0 = ao(0') is defined by «o = (rof +(zo)2(2.5) The coefficients of the first and second fundamental forms of the material surface yin the reference configuration are ^11 = «0> ^22 = r0 ' ^12 = ^2 ^ •®11 = (r0Z0 ~ z0r0')/a0< ^22 ~ r0Z0/a0> ^12 = 026 p. M. NAGHDI AND L. VONGSARNPIGOON Let D denote the director at rR in the reference configuration k0 of the Cosserat surface and, for covenience, introduce the notation D„ = Aa, D3 = D (2.7) In what follows, whenever desirable, the notations D, = (Dx, D2, D3) and the set (2.7), i.e., (Aa,D), will be used interchangeably depending on the particular context. Since D are linearly independent, as in Naghdi [2, Sec. 3], we may also introduce a set of reciprocal vectors D' such that D,. ■ D' = 8{, (2.8) where 8/ is the Kronecker symbol in 3-space. Without loss in generality, we choose the director in the reference configuration to be along the normal to every point of so that D = Z)A3, D = D{dl). (2.9) It follows from the choice (2.9) and the definition of D' in (2.8) that D° = A", D3 = -^A3, (2.10) where AQ are the surface reciprocal base vectors to AQ. Let the Cosserat surface be deformed axisymmetrically without torsion such that the reference surface £fR becomes another surface of revolution with the position vector r = t(6a, t) in the current configuration k specified by r = r(0\ t)er + z(6\ t)e3. (2.11) Since the deformation is assumed to be torsionless, the meridians 62 = const, and the parallels 61 = const, of the undefo

Journal Article
TL;DR: In this paper, an abstract operator theoretic formulation of the eigenvalue problem is derived and spectral properties investigated, using spline-based Rayleigh-Ritz methods applied to elliptic differential operators and the approximation properties of interpolatory splines.
Abstract: Rayleigh-Ritz methods for the approximation of the natural modes for a class of vibration problems involving flexible beams with tip bodies using subspaces of piecewise polynomial spline functions are developed. An abstract operator theoretic formulation of the eigenvalue problem is derived and spectral properties investigated. The existing theory for spline-based Rayleigh-Ritz methods applied to elliptic differential operators and the approximation properties of interpolatory splines are useed to argue convergence and establish rates of convergence. An example and numerical results are discussed.

Journal ArticleDOI
TL;DR: In this paper, the solution of the Poisson equation subject to Dirichlet conditions is examined asymptotically on thin domains and the evolution of the structure of the solution is followed as the shape of the domain changes.
Abstract: 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.

Journal ArticleDOI
TL;DR: In this paper, the Dirichlet problem of incremental viscoelasticity is well posed and an existence and uniqueness theorem for these equations with Dirichlets boundary conditions is proved.
Abstract: Introduction. The equations governing small deformations superposed on a large deformation for a body composed of viscoelastic material were established by Pipkin and Rivlin [1], Recently Iesan [2] has presented equations for small thermoelastic deformations superposed on a general nonlinear thermomechanical deformation and an existence and uniqueness theorem for these equations was proven by Navarro and Quintanilla [3]. In this paper equations are developed which describe motions of a viscoelastic body which are incremental in the sense that they are close to a nonequilibrium nonlinear deformation of the body and an existence and uniqueness theorem for these equations with Dirichlet boundary conditions is proved. The paper is concluded with the remark that the Dirichlet problem of incremental viscoelasticity is well posed.

Journal ArticleDOI
TL;DR: In this article, the fluctuating velocity field of monomolecular films of arbitrary configuration is investigated when gravity waves propagate on the air-water interface, and a Dirichlet boundary value problem involving Helmholtz' equation for the divergence of the velocity field is obtained for the film.
Abstract: The fluctuating velocity field of monomolecular films of arbitrary configuration is investigated when gravity waves propagate on the air-water interface. The surfaceactive material is assumed to have visco-elastic properties and to be insoluble. Boundarylayer techniques are employed, and a Dirichlet boundary value problem, involving Helmholtz' equation for the divergence of the velocity field, is obtained for the film. Circular and rectangular films are considered explicitly, whilst an approximate method is given for slender films of arbitrary orientation. Application is made to viscous wavedamping.

Journal ArticleDOI
TL;DR: In this article, the shape of the wavefront induced by a line material imperfection in a large body which is being subjected to a homogeneous, time dependent antiplane shear deformation is investigated.
Abstract: The shape of the two dimensional wavefront induced by a line material imperfection in a large body which is being subjected to a homogeneous, time dependent antiplane shear deformation, is investigated. The body is composed of isotropic, incompressible, hyperelastic material and the constitutive relation is assumed to be such that depending on the value of one parameter, strong ellipticity fails at a strain level corresponding to the local maximum of the shear stress-strain curve. The wavefront shapes are compared when this occurs and when it does not.