Showing papers on "Continuum mechanics published in 1977"

A model for viscoelastic fluid behavior which allows non-affine deformation

Abstract: A continuum theory of viscoelasticity is developed which allows non-affine deformation, defined in an appropriate manner. The constitutive equation is a generalization of that obtained from molecular theory with the addition of one scalar parameter which becomes important for large deformations. The theory is applied to simple shear flows, the scalar parameter being determined to match certain experimental data. The theory shows good agreement with all data examined. The paper concludes with the development of a general non-affine thermodynamic theory.

Mathematics applied to continuum mechanics

Abstract: Foreword to the Classics Edition Preface Conventions Part I. Geometrical Prerequisites for Three-Dimensional Continuum Mechanics: 1. Vectors, determinants, and motivation for tensors 2. Cartesian tensors Part II. Problems in Continuum Mechanics: 3. Viscous fluids 4. Foundations in elasticity 5. Some examples of static oroblems in elasticity 6. Introduction to dynamic problems in elasticity Part III. Water Waves: 7. Formulation of the theory of surface waves in an inviscid fluid 8. Solution in the linear theory 9. Group speed and group velocity 10. Nonlinear effects Part IV. Variational Methods and Extremum Principles: 11. Calculus of variations 12. Characterization of Eigenvalues and equilibrium states as extrema Bibliography Hints and answers Index.

Description and origin of banded deformation structures. II. Rheology and the growth of banded perturbations

Abstract: This paper offers a generalized mechanical explanation for the origin and development of bandlike deformation structures such as shear zones, mylonite zones, kink bands, 'pressure-solution' seams, extension gashes, and similar folds.Methods of continuum mechanics are used to examine permissible variations in strain rate, stress, and rheological properties across a region containing ideal banded perturbations. For bands to develop, the rheological properties must vary across the banding. The physical basis for this variation is a corresponding variation in microstructure or chemical composition, influenced in turn by finite deformation, stress, and temperature. Many rocks are likely to soften or harden during progressive deformation and these changes may be enhanced by thermal or other agents. Deformation softening (including strain softening and rotation softening) is a cause of instability and has two effects: first, the deformation tends to accelerate under constant stress; second, the deformation tends...

Superfluid mechanics for a high density of vortex lines

Abstract: There are two well known theories to describe the motion and thermodynamics of superfluids when a large number of quantized vortex lines are present and when the phenomena under study are on scales large compared with the vortex line spacing. These works have been criticised on the grounds that their governing equations for the smoothly varying, spatially averaged, fields do not satisfy the accepted invariance principles basic to modern continuum mechanics. This paper demonstrates one way in which such theories can arise from a properly invariant continuum approach and indicates the presence of hitherto unconsidered terms that bring them closer to the generally accepted microscopic picture. The resulting theory has applications both to rotating helium II in the laboratory, and to rotating neutron stars (pulsars).

Continuum Mechanics at the Atomic Scale.

Abstract: : The recent theories of nonlocal continuum mechanics are summarized and applied to the problems of plane waves, crack tip, screw dislocations, and secondary flow in rectangular pipes. The power and potential of these theories are demonstrated by predicting various critical physical phenomena in the range from the global to the atomic scales. (Author)

A Theory of Composites Modeled as Interpenetrating Solid Continua

Abstract: The differential equations and boundary conditions describing the behavior of a finitely deformable, heat-conducting composite material are derived by means of a systematic application of the laws of continuum mechanics to a well-defined macroscopic model consisting of interpenetrating solid continua. Each continuum represents one identifiable constituent of the N-constituent composite. The influence of viscous dissipation is included in the general treatment. Although the motion of the combined composite continuum may be arbitrarily large, the relative displacement of the individual constituents is required to be infinitesimal in order that the composite not rupture. The linear version of the equations in the absence of heat conduction and viscosity is exhibited in detail for the case of the two-constituent composite. The linear equations are written for both the isotropic and transversely isotropic material symmetries. Plane wave solutions in the isotropic case reveal the existence of high-frequency (optical type) branches as well as the ordinary low-frequency (acoustic type) branches, and all waves are dispersive. For the linear isotropic equations both static and dynamic potential representations are obtained, each of which is shown to be complete. The solutions for both the concentrated ordinary body force and relative body force are obtained from the static potential representation.

Densification and hysteresis of sand under cyclic shear

Abstract: The accumulation of inelastic strain due to the irreversible rearrangement of grain configurations associated with deviatoric strains is characterized by a nondecreasing material variable, termed the rearrangement measure; this variable, in turn, forms the basis of an intrinsic time scale and other related variables termed the densification measure and distortion measure. The shear modulus is identified to be a function of the mean normal stress and the second invariant of the strain deviator. In contrast with empirical methods, the material behavior is described herein by a constitutive law that satisfies all invariance requirements of continuum mechanics; thus, this law should, in principle, be generally applicable, including the cases of nonsinusoidal loadings with varying amplitudes, general multiaxial stress states, and nonproportional stress component histories. In addition, the law automatically exhibits hysteretic damping and is fully continuous, i.e., it contains no inequalities, such as those used in plasticity to distinguish unloading.

On boundary conditions for polar materials

Abstract: We discuss some restrictions on boundary conditions in continuum mechanics arising from the requirement that the basic balance laws, invariance conditions and entropy inequality must hold for the material outside of the bounding surface. We illustrate our assertions for polar materials, but the ideas we use are applicable to all continuum theories.

Continuum models for static and dynamic analysis of repetitive lattices

Abstract: A simple, rational approach is presented for developing continuum models for large repetitive lattice structures subjected to static and dynamic loadings. The procedure involves introducing kinematic assumptions to reduce the dimensionality of the lattice, and equating the strain and kinetic energies of the continuum model to those of the original lattice structure. The proposed procedure is applied to obtain effective elastic and dynamic characteristics of continuum plate models for double-layered grids. Numerical results are presented of stress and free vibration problems for two double-layered grids. These problems demonstrate the high accuracy of the continuum plate models developed. Also, an assessment is made of different approximations to these models.

Theory of interpenetrating elastic mixtures

Abstract: Composite materials, impregnated porous media, suspensions, etc. constitute heterogeneous mixtures with inhomogeneities whose characteristic dimensions are much larger than the dimensions of molecules and, therefore, the equations of continuum mechanics are applicable within each component phase. The description of their behavior [3, 4] has been based on the concept of "homogenizability" of a nonhomogeneous medium, by way of introducing parameters averaged over elementary macrovolumes much larger than thesize of these inhomogeneities. As a result, the fundamental equations of continuum mechanics have been formulated in a space the points of which are elementary mixture macrovolumes containing each phase proportionately to its overall volume fraction.

Rotational Dynamics of a Deformable Medium

Abstract: The aim of the present investigation has been to derive from the fundamental Cauchy's first law of continuum mechanics the explicit form of the Eulerian general equation which governs the three-axial generalized rotation about the centre of mass of a self-gravitating deformable finite material continuum, viscolinear (i.e., Newtonian) or not, consisting of compressible fluid of arbitrary viscosity, in an external field of force. The generalized rotation is a superposition of the so-called rigid-body (i.e., time dependent only) rotation of the continuum plus a nonrigidbody (i.e., position-time dependent) rotation of its configurations.

Rheology and Kinetic Theory of Polymeric Liquids

Abstract: Previous reviews of kinetic theory of macromolecules in the Annual Review of Physical Chemistry have been concerned primarily with (a) equilibrium properties, and departures from equilibrium that include only linear viscoelastic phenomena; (b) very simple flows, such as simple shear flows of the form Vx = k(t)y, where k(t) is often taken to be a small-amplitude sinusoidal function; (c) the shear stress 'yx in shear flows (as opposed to the normal stresses 'XX' Tyy, Tzz); and (d) molecular models of polymers that incorporate linear or Hookean springs. However, they have generally ignored the relation with modern continuum mechanics. These previous reviews have certainly reflected the chief concerns and accomplishments of polymer chemistry investigations during the �st two decades, and it is generally acknowledged that many unresolved problems remain in these areas. In this review the four aspects listed above are deemphasized and attention is focused on the following : (a) nonlinear viscoelastic properties, such as shear-rate-dependent viscosity and normal stress coefficients in steady shear flow and their time-dependent counter­ parts, (b) large-deformation flows including viscometric flows, elongational flows, radial flows, and time-dependent flows, (c) the complete stress tensor expressions from kinetic theory and the measurement of components of the stress tensor other than 'yx in steady shear flow, (d) molecular models involving nonlinear springs or constraints that may be capable of better describing the observed rheological phe­ nomena in terms of internal structural changes, and (e) the use of modern continuum mechanics to provide a framework for presenting kinetic theory results, analyzing rheological experiments, and solving polymer fluid dynamics problems. Interest in these areas has been steadily growing and reflects the observation of new rheo­ logical phenomena, the advances in rheological instrumentation, and the increased activity in computer modeling of polymer processing operations.

Some effects in liquid helium II — a continuum approach

Abstract: The isotope helium 4 below 2.2°K, known as helium II, exhibits a number of curious mechanical and thermal macroscopic effects. In this paper a possible alternative to the two-fluid theory for helium II is reviewed. Standard techniques of continuum mechanics are used to describe the flow of certain microscopic excitations which are believed to give rise to the observed effects. Appropriate balance laws are formulated and constitutive equations are proposed. Some simple solutions of the resulting theory are discussed which give qualitative agreement with observed effects. These effects include Hall's hanging plate experiment, the fountain effect and smallamplitude second-sound waves.

Inertial motion of a continuum

Abstract: The Burgers equation ut+uux=νuxx represents pure inertial motion except for the effects of viscosity ν. If ν=0, this equation becomes u=0 and describes the inertial motion of a one‐dimensional continuum until the time of formation of discontinuities, or ’’shocks’’. The study of pure inertial motion is extended to an arbitrary number of dimensions. Starting from some initial state of motion each parcel of the continuum may or may not have the intrinsic ability to form a shock, this property being a function of the symmetrical part of the velocity gradient matrix in the vicinity of the parcel. This study determines the conditions for which a parcel will, in fact, form a shock, the time that is required, and the temporal development of the full velocity gradient matrix and of coordinate invariants of the flow such as the divergence and the vorticity.

On the fountain effect in liquid helium II

Abstract: The authors develop further a theory of generalised continuum mechanics which may provide an alternative to the two-fluid theories of liquid helium II. They review briefly the balance laws governing the flow of microscopic excitations in the fluid and postulate some simple constitutive equations which give a possible explanation of the force on a plate emitting heat and also the fountain effect.

Theoretical Foundations of Equations of State for the Terrestrial Planets

Abstract: Our understanding of the constitution and structure of the terrestrial planets consists of a lengthy chain of inference. The major links of the chain are near-surface sampling, seismological estimation of elastic parameters of the deeper layers, interpretation with the help of laboratory experiment at high pressure and temperature, and theoretical synthesis. This paper reviews the current status of the last link only of this chain, in interpretive fashion. A noninterpretive, but consider­ ably more exhaustive, listing of recent contributions has been given by Ahrens (1975). The present work represents considerable evolution of thought since a previous review (Thomsen 1971). As it is intended to inspire a similar evolution of the reader's ideas, the reasons for these changes of opinion merit a philosophical summarization at this point. A set of familiar concepts (velocity-density systematics, Voigt-Reuss-Hill averaging, finite strain extrapolation formulas, etc) have been extensively used to discuss the mantles of the terrestrial planets, and are in fact adequate where the rocks are familiar and the pressure and temperature are modest. However, in the lower mantles of Earth and Venus, the uncertainties in the elastic parameters due to scatter about the velocity-density systematic trends, coupled with the large compression and uncertain temperature, restrict the power of realistic conclusions. The crudity of these concepts, by which we estimate the effects of compression, offer a strange contrast to the elegance of the theory of lattice dynamics, by which we calculate the effects of temperature. The reason for this dissimilarity is that temperature affects density, elasticity, etc only through averages over the eigenfrequencies of vibration (which can be understood in terms of harmonic forces between atoms, plus a small perturbation). By contrast, the effects of compression require not only the energy eigenvalues, but also the electronic charge density, which in turn requires the full apparatus of quantum mechanics.