General Continuum Research Articles

A General Continuum Theory With Microstructure for Wave Propagation in Elastic Laminated Composites

A continuum theory with microstructure for wave propagation in laminated composites, proposed in previous works concerning propagation normal and parallel to the laminates, is extended herein to the general two-dimensional case. Continuum model construction is based upon an asymptotic scheme in which dominant signal wavelengths are assumed large compared to typical composite microdimensions. A hierarchy of models is defined by the order of truncation of the obtained asymptotic sequence. Particular attention is given to the lowest order dispersive theory. The phase velocity spectrum of the general theory is investigated for one-dimensional wave propagation at various propagation angles with respect to the laminates. Retention of all terms in the asymptotic sequence is found to yield the exact elasticity spectrum, while spectral collation of the lowest order dispersive theory with the first three modes of the exact theory gives excellent agreement.

Continuum descriptions of dislocations under stress reversals

A general continuum theory of dislocation motion is used to investigate the response of crystalline solids to cyclic straining in uniaxial tension and compression. For macroscopically homogeneous deformation under uniaxial stress a simple one-dimensional equation suffices to relate the plastic strain rate to dislocation flux. The material is characterized by evolutionary equations for multiplication of dislocations and for immobilization of moving dislocations. Some simple example materials are considered and it is shown by numerical calculation that these exhibit respectively a Bauschinger effect, isotropic hardening, and isotropic softening when subjected to a program of alternating strains at a constant rate.

Wave propagation in elastic laminates using a multi-continuum theory

A general continuum approach to the dynamical behavior of elastic composites proposed in an earlier paper is specialized to infinitesimal deformations of elastic laminates. A simple procedure is proposed to evaluate the constitutive parameters appropriate to a particular choice of constituents, which is easily generalized to other types of material behavior and composite structure. The resulting theory is then used to investigate the propagation of plane harmonic waves to furnish a basis for comparison with other dynamical treatments of elastic laminates.

Axioms for work and energy in general continua

Axioms for work and energy in general continua

Continuum theory of type-A smectic liquid crystals

A new general continuum theory for smectic liquid crystals of type A is presented. The method used can be applied equally well to type-C smectics and probably also, in some modified form, to smectics of the other types.

Calculation of mixed phases in continuum mechanics

A general procedure is developed to calculate a phase transformation consistently. The differential forms of the equations constraining a binary mixture to be in equilibrium are inverted to obtain volume and internal energy as independent variables. The set of finite difference equations may be incorporated into a general purpose continuum mechanics program.

Axioms for work and energy in general continua

Axioms for work and energy in general continua

A consistent finite element model for the two-dimensional continuum

The finite element approximation to the continuum problem is examined from the viewpoint of the principle of virtual work. It is shown that the usual nodal equilibrium equations for triangular elements are a consistent consequence of a piecewise constant strain field, thus guaranteeing that many results of general continuum theory can be directly applied to the finite element model, and also clarifying the relation between the two models.

Mechanical and Thermodynamic Considerations of an Assemblage of Homogeneous Elastic-Plastic States

Abstract The mechanics and the thermodynamics of plastic deformation are considered in terms of a general assemblage or continuum of elastic, perfectly plastic elements or states. Such models not only match the external mechanical behavior of real materials structures and continua, but they also afford a simple thermodynamic definability. A consideration of the internal behavior shows that the stress-free state has the maximum elastic range. Hardening in the sense of an increasing macroscopic elastic range is accompanied by a release of stored strain energy; the stress-free state always is restorable. This behavior is appropriate for real structures and continua. However, it is precisely these continuum characteristics which make the assemblages inappropriate models of material behavior. Barriers to continuing plastic deformation are required which do exist on the microscale, but lie outside of the scope of the most complex of these thermodynamically well-defined assemblages.

A continuum model for fibre-reinforced plastic materials

A simple, general continuum model is proposed for describing the plastic behaviour of a composite material consisting of a metal matrix reinforced by strong fibres. The model is that of an incompressible rigid/plastic continuum which is transversely isotropic—the single preferred direction at any point, and at all times, being the fibre-direction at that point—and which is inextensible in the preferred direction. The principal axes of anisotropy are therefore explicitly determined by the deformation history. The kinematics and general three-dimensional theory for the material are developed and then applied to two cases of plane strain and one of plane stress. The latter is employed in the analysis of previously published experimental data on the yielding of thin fibre-reinforced sheets; good agreement is obtained between the theoretical predictions and the experimental data.

Source Flux Densities at 153 MHz

This paper gives fiux density measurements of 312 sources at 153 MHz obtained from a general continuum survey made with the 210 ft radio telescope of the Australian National Radio Astronomy Observatory, Parkes, N.S.W. The beamwidth of the telescope at this frequency is 2� 18�. Some discussion of results for spectra is given and a comparison is made with other measurements of the same sources.

General continuum hypothesis and ramifications

General continuum hypothesis and ramifications

Universal infinite partially ordered sets

Introduction. The purpose of this paper is to construct for each infinite cardinal X an M-universal partially ordered set in which every partially ordered set of cardinality N can be embedded. An R-universal partially ordered set of cardinality 2H is constructed for every infinite cardinal N. Furthermore, an N-universal partially ordered set of cardinality N is constructed for every cardinal N which is the sum of denumerably many smaller cardinals. The existence for an arbitrary cardinal N of an N-universal partially ordered set of cardinality N is left an open question and appears to be a difficult problem. The ordering relation < in a partially ordered set is understood to be transitive, nonreflexive, and nonsymmetric. The Axiom of Choice will be assumed throughout this paper. The General Continuum Hypothesis will be explicitly mentioned when it is used.

Certain problems related to the cutting of a simply connected plane region by a continuum

1. If R is a simply connected plane region whose frontier F is bounded and C is a simple arc lying in R save for its end points, it is well known that C enjoys the following properties: (a) R R C consists of two simply connected regions R1 and R2; (b) C is a part of the frontiers of both R1 and R2; (c) F+C is the union of the frontiers of R1 and R2. Nothing could be simpler than this theorem, but when one desires to use it, certain questions arise. For instance, what precisely are the frontiers of R1 and R2? If F is the union of two continua H and K, can C be so drawn that its ends lie on two components of H. K? If so, will the frontiers of the two regions be C+H and C+K? These questions are readily answered in the case that F is a simple closed curve, but the answers in this case do not always remain correct when F is more complicated. In particular, a negative answer to the second question suggests the substitution of a more general continuum for C and this in turn leads to a re-examination of the truth of the resulting extension of the theorem. Such an extension has been made elsewheret and it is intended in the present article to examine further these questions and certain related topics. The first and third of the questions suggested above are discussed in ??3-6. The remainder of the paper deals with the situation arising when the arc C is replaced by a special kind of irreducible continuum and with related questions of accessibility. In particular, it is shown that under certain broad conditions the second and third of the above questions can be answered in the affirmative if the arc C is replaced by a slightly more general continuum. 2. Notation. In addition to the common notation of the aggregate theory, the following will be used. The whole plane is denoted throughout by Z. If F is a continuum, a component of Z F whose frontier is F is called a principal component; a component of Z F whose frontier is a proper part of F is called secondary. A