Showing papers on "Continuum mechanics published in 1979"

Uniqueness of solutions to hyperbolic conservation laws

Abstract: : It is known that conservative systems of differential equations which result from continuum mechanics (e.g. the equations of shallow water waves, fluid dynamics, magneto-fluid dynamics and certain elasticity problems) do not have unique solutions. Thus the problem arises of proving that systems of this type have only one physically meaningful solution. This report shows that there exists at most one solution satisfying an entropy condition which generalizes the second law of thermodynamics.

Polymer Fluid Mechanics

Abstract: Publisher Summary Polymeric liquids consists of high-molecular weight polymers or “macromolecules'’ in the form of polymer solutions or polymer melts. This chapter provides one of the most ubiquitous examples of the viscoelastic or non-Newtonian liquid and discusses certain developments in the mechanics of viscoelastic fluids, as typified by polymeric liquids. The term “mechanics” embodies both the underlying rheology, in the form of mechanical constitutive theories or models, and dynamics, as represented by the phenomena of fluid motion and the governing field equations of continuum mechanics. The chapter focuses on developments in the continuum and microstructural theory of polymer-fluid rheology and, to a lesser extent, in the fluid dynamics. The emphasis is on continuum mechanics, and, except insofar as they relate grossly to the characteristic departures from classical (Newtonian) fluid behavior, there is little said about the chemical structure and molecular physics of polymers. The objectives are to trace the progress in the selected avenues of research and to summarize and evaluate certain ideas and methods that appear to be in need of refinement or are most promising for further development. Several extensions of existing methods and some original results are presented.

Determination of strain distribution in elastically bent materials by X-ray intensity measurement

Abstract: A method for quantitative determination of local curvature in elastically bent perfect crystals is described. The method is based on X-ray intensity measurements, and a comparison of experimentally determined values with those derived from diffraction theory gives satisfactory agreement. The method was applied to determine the strain gradient and strain concentration in the vicinity of the notch of an elastically bent crystal. The experimental results were compared with those derived from a similar model based on continuum mechanics. Possible applications of the X-ray method are discussed to obtain experimental solutions to strain analyses which, when approached by continuum mechanics, pose formidable mathematical obstacles.

On the Onset of Breakup in Inviscid and Viscous Jets

Abstract: : This paper is concerned with the instability of inviscid and viscous jets utilizing the basic equations of the one-dimensional direct theory of a fluid jet based on the concept of a Cosserat (or a directed) curve. First, a system of differential equations is derived for small motions superposed on uniform flow of an inviscid straight circular jet which can twist along its axis. Periodic wave solutions are then obtained for this system of linear equations; and, with reference to a description of growth in the unstable mode, the comparison of the resulting dispersion relation is found to agree extremely well with the classical (three-dimensional) results of Rayleigh. Next, constitutive equations are obtained for a viscous elliptical jet and these are used to discuss both the symmetric and the anti-symmetric small disturbances in the shape of the free surface of a circular jet. Through a comparison with available three-dimensional numerical results, the solution obtained is shown to be an improvement over an existing approximate solution of the problem. (Author)

Quasi‐static earthquake mechanics

Abstract: Viewed as a boundary value problem of continuum mechanics, the subject of earthquake mechanics reduces to the study of (1) constitutive properties (stress-deformation laws) and geometry of fault zones and surroundings, and (2) remotely applied boundary conditions. In the purely quasi-static formulations described here, a precise criterion for instability leading to dynamic rupture is also needed. The present consensus supports the concept of an earthquake cycle in which the somewhat arbitrarily chosen successive stages are (1) slow increase of stress due to remote loading, (2) the onset, propagation, and cessation of earthquake rupture, and (3) post-seismic adjustment. Although no single rigorous model yet exists for the complete cycle because of simplifying constitutive assumptions, there are detailed models for each stage.

Continuous crack growth or quantized growth steps

Abstract: The assertion that a non-vanishing Griffith energy release rate requires an r −1 type singularity at the tip of a crack for the energy intensity, i.e. the product of stress and strain, is examined. When the existence of such a singularity is denied on physical grounds continuum mechanics energy balance considerations suggest that initial unstable crack extension is by a discrete growth step of characteristic size Δa.

Statistical Theory of Effective Viscosity in a Random Suspension

Abstract: A general theory is presented to derive the coarse-grained equations governing the macroscopic behaviour of slow viscous flow in suspensions from the standpoint of statistical continuum mechanics. A system of equations governing the macroscopic behaviour of viscous flow in suspensions can be obtained using formal perturbation methods and coarse-graining procedures. The coarse-grained equations lead to an expression for the dependence of the effective viscosity of suspensions on the concentration of particles φ. In the dilute limit our result reduces to Einstein's theory of dilute suspensions, and the deviation from Batchelor's formula is not significant to the second order in φ. Even for more concentrated suspensions, the result is in good agreement with experimental data obtained by Vand.

Differential geometry of the coincidence site lattice

Abstract: The tensor properties of simple internal surfaces, such as two-phase interfaces and grain boundaries, have been studied in detail. In particular, these tensor quantities have been defined with respect to the original crystal lattice as well as to a common coincidence site lattice that is a characteristic of the boundary. Such lattices allow a given type of distortion to be represented in either a Riemannian or a non-Riemannian (dislocated) space. This particular generality provides a powerful method of anlysing problems in continuum mechanics.

Formulation of Constitutive Laws for Deformation During Irradiation

Abstract: Constitutive laws for radiation-induced deformation are derived based on a phenomenological approach which involves only simple principles of continuum mechanics and avoids detailed mechanistic assumptions. This approach is applied to materials which are initially in an isotropic condition that serves as a reference state. As specific examples of this approach, we consider in detail three forms of constitutive laws. In the first case it is assumed that the deformation rate depends on the current stress as the only tensor variable. For the second case we postulate a dependence on the current stress tensor and the strain tensor as produced during the previous radiation-induced deformation. Finally, the third case deals with a dependence on stress and strain as accumulated during cold-working. In both the latter cases, the material is rendered anisotropic with respect to the reference state. It is shown for the last two cases that the phenomenological approach provides a proper formulation and a clear distinction of such phenomena as stress-affected swelling, irradiation creep, radiation-induced anisotropic growth, and creep-swelling interaction. It also supplies the superposition rules for these phenomena when the material is subject to triaxial stresses which change and redistribute with time. The constitutive laws as obtained from the continuum approach are not completely suitable for modeling structural materials. Since they are polycrystalline in nature, the constitutive properties may vary from grain to grain, and it becomes necessary to derive a macroscopic constitutive law for the aggregate. This is accomplished by considering an aggregate of many viscoelastic elements linked in parallel. The resulting constitutive law for the aggregate depends on the strain history. Upon sudden load changes, the aggregate exhibits anelastic transient creep. It is shown that the magnitude of the anelastic strain is related to the range of variability for the constitutive properties of the grains.

Coarse-grained equations of sound-wave propagation in two-phase random media. Renormalisation of physical properties

Abstract: A general theory is presented to derive the coarse-grained equations governing the macroscopic behaviour of sound-wave propagation in suspensions from the standpoint of statistical continuum mechanics The expressions for effective physical properties are obtained The resulting general expression for the effective sound-wave velocity is applied to acoustic wave propagations in bubbly fluid It is shown that the result for the sound-wave velocity agrees well with experiment It is found that the basic equation for wave propagation in continuous random media does not apply to wave propagation in discrete random media (or suspensions) The expressions for the effective viscosity and thermal conductivity agree with those for the effective viscosity and thermal conductivity in a dilute suspension in the dilute limit and are applied to more concentrated suspensions

Basic Principles of Continuum Mechanics

Abstract: Continuum mechanics is the branch of classical mechanics concerned with motions of deformable material bodies. The mathematical model for such a body is called a body manifold which is an oriented 3-dimensional differentiable manifold endowed with global coordinate systems. In this chapter we develop the basic principles governing the motions of body manifolds.

Application of the Infinitesimal Operators of Translation and Rotation : No.2. Extension of the Technique to Continuum Mechanics

Abstract: The infinitesimal operators translation and rotation are useful to derive the basic dynamic quantities and corresponding conservation equations fro the energy from and its conservation equation. this technique is applied effectively to continuum mechanics. The general commutation rules between the infinitesimal operators of rheonomic coordinate transformations and the differential with respect to time and space are formulated.

Some Melt Flow Processes

Abstract: In chapter 4, the fundamental continuum mechanics equations governing the flow of polymer melts were established. These may now be used to analyse the types of flows occurring in practical processing operations. In addition to aiding the understanding of observed flow behaviour, such analyses are often capable of providing rational means of designing processing equipment to meet particular performance requirements.

Some Basic Concepts of Differential Geometry and Continuum Mechanics

Abstract: In this book, we shall frequently encounter vectors in the space of elementary geometry as well as linear transformations, also called tensors, applied onto them. Therefore, at the very beginning, we feel it advisable to sum up the reasonings we shall be using all along. They relate, on the one hand, to the geometry of three-dimensional space, particularly on curved surfaces, and, on the other hand, to the kinematics of deformation.