Showing papers in "Archive for Rational Mechanics and Analysis in 1966"

Nonlinear stability of the Boussinesq equations by the method of energy

Abstract: Energy method applied to stability of flows governed by nonlinear Boussinesq equations, establishing universal stability region in Reynolds-Rayleigh number plane

On Saint-Venant's principle in the two-dimensional linear theory of elasticity

Abstract: This paper presents results which provide a formulation and proof of a version of Saint-Venant's principle appropriate to the plane strain and generalized plane stress solutions of the equations of the linear theory of elastic equilibrium.

Variational problems of minimal surface type II. Boundary value problems for the minimal surface equation

Abstract: in a convex domain D, and taking on assigned continuous values on the boundary of D. This problem was solved by RAD6 in 1930, on the basis of the existence theorem for the parametric problem of least area. It should be observed that the restriction to convex domains is necessary in order that solutions should exist corresponding to arbitrarily given boundary data. On the other hand, if we consider the data itself, the strict requirement of continuity is certainly stronger than necessary for the problem to be well posed. NITSCnE has in fact recently shown that the Dirichlet problem remains well posed even when the data is discontinuous on a set of vanishing linear measure. In his work it was, however, necessary for the data to be uniformly bounded. One may nevertheless envisage situations where infinite boundary values are admitted on arcs of the boundary. Although this sort of phenomenon is certainly impossible for linear equations, for the nonlinear minimal surface equation it can occur quite naturally. We offer in evidence the famous solution of H.F. SCrIERK, discovered in 1834,

A numerical method for the exterior dirichlet problem for the reduced wave equation.

Abstract: : A digital computer technique for approximating the solution of the exterior Dirichlet problem for the reduced wave equation is described and discussed. (Author)

Dynamical theory of hyperelastic rods

Abstract: A rod is regarded as a one-dimensional mathematical model of a three-dimensional body. The exact field equations governing the motion of a hyperelastic rod are derived from the general three-dimensional theory. Then, by a suitable restriction on the number of displacement variables, a hierarchy of approximating theories is established. Because such theories are generated by a kinematic hypothesis, a precise, quantitative idea of the nature of the simplifying assumptions is furnished. An analysis of the structure of these approximating theories yields three distinct approaches by which they may be interpreted. Finally, constraints and their connection with other approximate theories are investigated. In particular, classical nonlinear theories and theories for planar motion are developed in this context of constrained theories.

Waves in materials with memory. v. on the amplitude of acceleration waves and mild discontinuities.

Abstract: : This paper studies one-dimensional acceleration waves and higher-order waves in general nonlinear materials with memory. Explicit expressions are derived for the time-dependence of the amplitude of such waves without assuming the material ahead of the waves to be undistorted or homogeneous. (Author)

A fading memory hypothesis which suffices for chain rules

Abstract: where f ( t ) is the value of some state function, such as the specific entropy, at the (present) instant t, and where Ft( . ) is the history of the deformation gradient, 'gt(.) is the history of the temperature up to the instant t, and g is the present temperature gradient at the material point. The functionals f, called the response funetionals, characterize the material. Clearly the arguments of the response functionals can be replaced by the set

Minimum energy characterizations of Saint-Venant's solution to the relaxed Saint-Venant problem

Abstract: The singular importance, in both theory and practice, of SAINT-VENANT'S celebrated memoirs [1], [2] on what has long since become known as SAINTVENANT'S problem, requires no emphasis. Indeed, a comprehensive bibliography of the vast and varied literature to which the work contained in [1], [2] has given impetus would multiply the length of this rather limited study. With a view toward describing the aim of the present investigation we recall first that SAINT-VENANT'S problem consists in the determination of the stresses and deformations in an elastic cylinder (or prism), which - in the absence of body forces - is subjected to surface tractions arbitrarily prescribed over its ends and which is free from lateral loading. In this formulation the problem, even within the linearized equilibrium theory of homogeneous and isotropic elastic solids, has remained one of undiminished notoriety. SAINT-VENANT'S treatment of the foregoing problem rests on a relaxed formulation in which the detailed assignment of the terminal tractions is abandoned in favor of prescribing merely the appropriate stress resultants.

Potentials for tangent tensor fields on spheroids

Abstract: In three-dimensional Euclidean space let S be a closed simply connected, smooth surface (spheroid). Let \(\hat n\) be the outward unit normal to S, ▽Sthe surface gradient on S, IS the metric tensor on S, gij the four covariant components of IS (i,j = 1, 2), hijthe four covariant components of -\(\hat n\)xIS, and Dicovariant differentiation on S. It is well known that for any tangent vector field u on S there exist scalars ϕ and ψ on S, unique to within additive constants, such that \(u = abla _s \varphi - \hat n \times abla _s \psi \); the covariant components of u are \(u_i = D_i \varphi + h_i^j D_j \psi \). This theorem is very useful in the study of vector fields in spherical coordinates. The present paper gives an analogous theorem for real second-order tangent tensor fields F on S: for any such F there exist scalar fields H, L, M, N such that the covariant components of F are $$F_{ij} = H h{}_{ij} + Lg_{ij} + E_{ij} (M,N),$$ where $$E_{ij} (M,N) = ( - abla _s ^2 M)g_{ij} + 2D_i D_j M + (h_i ^k D_j + h_j ^k D_i )D_k N$$ It is shown that H and L are uniquely determined by F but that M and N are not. The set of complex scalar fields ℳ′ = M′+iN′ such that Eij(M′, N′)=0 is shown to constitute a four-dimensional complex linear space \(\mathfrak{W}\). The scalars M and N which help to generate a given F are uniquely determined by F and the condition that, for every ℳ′ in \(\mathfrak{W}\), $$\mathop \smallint \limits_s (M - iN)\mathcal{M}\prime dS = 0$$ The real linear space of second-order tangent tensor fields on S which have simultaneously the form E(M, 0) and the form E(0, N) is shown to have dimension zero on a sphere, dimension four on a non-spherical, intrinsically axisymmetric spheroid (a spheroid whose isometries form a compact, one parameter group), and dimension six on a spheroid which is not intrinsically axisymmetric. Applications of the representation theorem to tensor problems in spherical coordinates are briefly discussed.

Spectral theory for the wave equation with a potential term

Abstract: A spectral representation for the potentially-perturbed wave equation $$u_t {\text{ }}_t (x,t) - \Delta u(x,t) + q(x){\text{ }}u(x,t) = 0{\text{ }}(x \in E^3 , - \infty < t < \infty )$$ is given for a class of non-negative potentials q(x) which vanish at ∞. The proof depends for the most part on energy considerations.

On saint-venant's principle and the torsion of solids of revolution.

Abstract: Although a comprehensive review of the literature on SAINT-VENANT'S principle in the linearized equilibrium theory of elastic solids would serve a useful purpose, such a survey is clearly beyond the scope of these introductory remarks. The principle was originally introduced by SAINT-VENANT in connection with, and with limitation to, the problem of extension, torsion, and flexure of slender cylindrical or prismatic beams. Renewed interest in the intriguing theoretical questions posed by SAINT-VENANT'S principle was stimulated by VON MISES [1] ,(1945), who brought into focus the vagueness of the traditional universal statements of the principle, which go back to BOUSSINESQ [2]. Guided by BOUSSINESQ'S own efforts in support of the principle, and on the basis of two specific examples, YON MISES was led to interpret and amend the conventional statements in terms of assertions concerning the order of magnitude of the stresses at interior points of an elastic body under loads that are confined to several distinct portions of its boundary. The limit process underlying these assertions refers to the contraction of the regions of load-application to fixed points of the boundary and the prevailing order of magnitude of the internal stresses depends upon the nature of the individual load resultants. A mathematical formulation and proof of von Mises' version of SAINT-VENANT'S principle was supplied later by STERNBERG [3]* (1952).

On the inclusion of the complete symmetry group in the unimodular group

Abstract: : The complete symmetry of a group of a material is usually assumed to consist only of transformations which preserve the density of the material. In this paper it is shown that for a large class of materials this restriction is in fact a consequence of the Clausius-Duhen inequality. It is shown that elastic materials and a particular type of rate dependent materials fall into this class; the proof extends to quasi-elastic materials and quasi-Rivlin-Ericksen materials of all orders. Finally certain restriction which the Clausius-Duhen inequality places on the individual (stress, heat flux, entropy, etc.) Symmetry groups are established for an elastic material. (Author)