Showing papers on "Continuum mechanics published in 1982"

Mathematical Modeling of Two-Phase Flow

Abstract: : A continuum mechanics approach to two-phase flow is reviewed. An averaging procedure is discussed and applied to the exact equations of motion. Constitutive equations are supplied and discussed for the stresses, pressure differences and the interfacial force. The nature of the resulting equations is studied. (Author)

Formulas for determining local properties in molecular‐dynamics simulations: Shock waves

Abstract: Formulas are given that relate the mass, momentum, and energy densities and the momentum and heat fluxes to the masses, positions, and velocities of the individual particles making up a system. The formulas are similar to those of Irving and Kirkwood, but have forms that are easily implemented in molecular‐dynamics simulations. Even when simulating very inhomogeneous phenomena such as shock waves, the densities and fluxes exactly satisfy conservation laws for mass, momentum, and energy. Corrections to the virial formula for the pressure and to the related formulas for the stress tensor and heat flux are obtained. The relationship of the formulas given to those used by others is discussed.

Deformation of Ideal Granular Materials

Abstract: The formulation of the equations which govern the deformation and flow of granular materials is an outstanding problem in continuum mechanics. This article concentrates on one model for the mechanical behaviour of granular materials, termed the “double-shearing model”, in which deformation is assumed to occur by simultaneous shearing on the two families of characteristic curves of the stress equations. We review the formulation of the equations of the double-shearing model for plane deformations, and describe the known exact solutions of these equations. Further developments of the plane strain theory are then given in which the velocity equations are expressed in a new form from which some general results are derived. In the last part of the article the theory is extended to include problems in three dimensions. The three-dimensional theory is applied to some special cases of nonplanar deformations, and in particular we formulate a new theory of axially symmetric deformations of granular materials, and apply this theory to the solution of some boundary value problems.

Suspension Bridge Vibration: Continuum Formulation

Abstract: The methodology of the free vibration analysis of suspension bridges with horizontal decks is refined and extended, utilizing a continuum approach to include both the coupled vertical-torsional vibration and the effect of cross-sectional distortion. Variational principles are used to obtain the coupled equations of motion in their most general, and nonlinear form. The general equations are linearized, eliminating the coupling effect, and solved for a specific bridge; the resulting natural frequencies and modes of vertical and torsional vibrations are compared with those calculated using the finite element approach and with those estimated from low-amplitude full-scale ambient vibration tests which were conducted on the bridge. In addition, the general, nonlinear equations of motion governing the lateral vibration of suspension bridges are presented and the linearized forms are obtained; a numerical example and a comparison among the analytical, numerical and full-scale test results are also presented.

Direct solution of plasticity problems in soils by the method of characteristics

Abstract: The method of characteristics is well established as a direct method of solving for the stresses in plasticity deforming soils under conditions of plane strain. The application of similar methods to problems of axial symmetry, and to soils with non-homogeneous properties is described and some illustrative examples given. The method of characteristics may also be used for the solution of the plastic displacement equations, although the exact form which these equations should take is still a matter of controversy. Illustrations are given of displacement calculations using the simplifying assumptions either of an associated flow rule (which is unrealistic for a frictional material) or using a fixed rotation term (which, although it has no physical justification, leads to realistic displacement fields). Refs.

The Displacement Finite Element Method in Nonlinear Buckling Analysis of Shells

Abstract: The paper compiles the current status of the finite element method in linear and nonlinear buckling analysis of shells. The classical concept via shell theory, the degeneration method, continuum mechanics based and corotationalformulations used in the displacement approach and the corresponding incremental stiffness expression are briefly described. Some comments on the problem of non-uniqueness and stability of the solution and their practical evaluation are given. A classification of displacement dependent pressure loads is presented discussing the symmetry of the problem. The main characteristics of the different classes of shell elements are outlined. Besides flat and curved elements derived from shell theory the survey concentrates on degenerated elements. A detailed review on the main solution strategies in nonlinear shell analyses is presented. Among these are quasi-Newton methods combined with line search and iteration techniques in the displacement and load space. Finally selected numerical examples are described applying isoparametric degenerated elements to bifurcation buckling and nonlinear collapse analyses of shells.

Design Sensitivity Analysis in Structural Mechanics. III. Effects of Shape Variation

Abstract: The dependence of static response and eigenvalues on the shape of plates and plane elastic solids is characterized. Shape of elastic bodies is taken as the design variable. The material derivative idea of continuum mechanics is used to obtain expressions for directional derivatives of displacement fields and eigenvalues with respect to a transformation function that defines a shape variation. The result is used to obtain explicit and computable expressions for variations of integral functionals that arise in structural optimization problems.

On Material Frame-Indifference

Abstract: Noll’s concept of space-time is used to show how material frame-indifference may be regarded as imposing the same restrictions upon response functions in continuum mechanics as does invariance under superposed rigid motions. Related observations are made.

Energy and entropy of interacting dislocations

Abstract: The energy and entropy of interacting edge dislocations have been calculated by atomistic simulations, with the use of piecewise-linear forces in a two-dimensional triangular lattice. We conclude that the interaction energy between small groups of dislocations is well described by continuum mechanics for separations greater than a few lattice spacings. Our calculations enable us to make a precise determination of the core energy, which is an essential parameter in determining dislocation multiplication rates. We find also that continuum mechanics gives an accurate representation of the interaction of a dislocation pair with a homogeneous elastic stress field. The vibrational contribution to the entropy of such a pair is small, about 0.3k.

Determination of notch-tip plasticity by X-ray diffraction and comparison to continuum mechanics analysis

Abstract: Dislocation-free silicon crystals of ({\bar 2 \bar 1 \bar 1}) orientation with a hyperbolic notch, subjected to tensile deformation at 1073 K, were used as model material for the analysis of the induced plastic zone. The results obtained by X-ray topography and X-ray rocking-curve measurements were compared to theoretical calculations and predictions based on continuum mechanics. Good agreement between experiment and theory was obtained regarding the shape of the plastic zone, the contribution of the active slip systems to the size of the plastic zone and the direction of the maximum plastic strain trajectory in the zone. Discrepancies between experiment and theory regarding the symmetry relation of the plastic zone lobes and the dislocation density near the notch tip were attributed to the interactions and resulting work-hardening. These aspects were not taken into account in calculations of continuum mechanics.

Microstructure and Mechanisms of Fracture

Abstract: Fracture of alloys can be treated on the level of the atomic bond, of micro-structure, and of continuum mechanics. Microstructural aspects of formation, as well as subcritical and critical growth of cracks are discussed, for static and cyclic loading conditions. Four principle considerations are useful to derive models for quantitative relations between microstructure and fracture properties: 1 Homogeneous or localized plastic strain. 2 Partial properties of microstructural components and their morphology in two- or multiphase alloys. 3 Ratios of fracture mechanical (plastic zone size, crack opening displacement) and microstructural (grain size, particle spacing) dimensions. 4 Effective crack extension forces as modified by strain induced martensitic transformation, crack branching, or pre-existing micro-cracks, and inclusions.

Theory of Invariants in Creep Mechanics of Anisotropic Materials

Abstract: Many mathematicians have studied the theory of algebraic invariants in detail. Many results can be found, for instance, in [13, 14, 29]. Very extensive accounts of algebraic invariant theory from the point of view of its application to modern continuum mechanics are presented, for example, in [17, 26, 28].

Development of an Analytical Model for Large Space Structures.

Abstract: : A methodology is presented for modeling large truss-type structures based on the concept of equivalent continuum. The equivalent effective elastic and dynamic properties of the continuum model in terms of the material and geometric properties of the truss are derived in a simple and straightforward manner. The accuracy of the model is demonstrated in a free vibration problem, when the results are compared to those obtained from the conventional finite element method. Both simply supported and free-free boundary conditions are considered. In addition, the assumptions made in obtaining the continuum solution are discussed. Numerical results clearly indicate the potential of this approach for modeling large repetitive truss-type structures. (Author)

Determination of the strain concentration factors around holes and inclusions in crystals by X-ray topography

Abstract: fringe topog- raphy and were quantitatively evaluated by X-ray intensity measurements of transverse-oscillation top- ographs. Improvement of strain measurements were obtained by considering the contribution of anom- alous transmission to the intensity measurements and by deposition of an appropriate metal film on the developed topograph to heighten the fluorescence of the silver grains in the emulsion. The strain gradient emanating from a bent specimen containing a hole was experimentally determined and the results were com- pared to calculations based on continuum mechanics having closed-form solution. Good agreement between experiment and theory was obtained. The dependence of strain interaction on interflaw distance was ex- perimentally demonstrated for specimens containing two holes. Introduction It has been previously shown that the X-ray intensities reflected from an elastically bent, perfect crystal are linear functions of the curvatures of the reflecting planes (Kalman & Weissmann, 1979). Owing to the action of a stress raiser, induced externally by an applied moment, the intensities become enhanced and this enhancement is a manifestation of the strain gradient developed by the stress raiser. Based on this principle the strain gradients emanating from stress raisers such as U-shaped notches were determined by X-ray intensity measurements and the results were compared to calculations based on continuum me- chanics (Kalman & Weissmann, 1979; Kalman, Chaudhuri, Weng & Weissmann, 1980). Satisfactory *Racah Institute of Physics, Hebrew University of Jerusalem, Israel. tTo whom all correspondence should be addressed. agreement between experiment and theory was obtained. Encouraged by these results this X-ray method was applied in the present study to measure the elastic strain distribution induced by holes and by inclusions subjected to the out-of-plane bending mode. The theoretical determination of strains induced under such conditions pose formidable problems in macro- mechanics. For a brittle material, or within the scope of elasticity, the stress field due to an elliptical hole under tension was first analyzed by Inglis 11913). More systematic treatments on an elliptical hole and on two hyperbolic notches were later provided by Neuber (1958, 1961) under both tension and bending. Using the stress functions in a plane elasticity and the technique of complex variables, Westergaard (1939) obtained the stress field near a sharp crack, which eventually gave rise to the concept of stress intensity factors under various modes of loading (Irwin, 1957). For a crack with an arbitrary shape, Muskhelishvili (1953) developed a standard technique, using the conformal mapping to map the crack shape into a unit circle and then to transform the known results of a unit circle back to the real space. All of these methods, however, consider only the existence of a single crack; the interaction between cracks or holes, therefore, cannot be accounted for precisely. Ling (1948) first studied successfully the stress field of a plate con- taining two identical circular holes under tension and gave explicit results to account for the crack in- teraction. Furthermore, Weiss, Takimoto & Nash (1967) employed Weiss's simplified version of Neuber's solution to study the crack interaction under tension, and also compared the calculations with their photoelastic measurements. Though several other at- tempts have been made to study crack interactions, they were generally not as successful as in the case of a single crack because the presence of a second crack or a second hole imposes an additional boundary con- dition for the stress functions to satisfy. This makes the principle of superposition from the solution of a single crack inaccurate and the actual solution is much more complicated. The problem of stress concentration due to a second 0021-8898/82/040423-07501.00 © 1982 International Union of Crystallography

A mathematical model of strain softening in simulating earthquake

Abstract: In this paper, we attempt to construct an unstable constitutive model for the fault medium by starting from the general constitutive equation of an elastic- plastic rock medium and considering the coefficient of internal friction and cohesion of the fault medium to vary with deformation. The constitutive relation thus obtained may be applied to fault of any occurrence and subjected to arbitrary stress conditions. To apply them in the boundary value problem of continuum mechanics dealing with complicated structure and non-uniform stress field, an instability criterion is also given. It is used to judge the onset of instability of the entire system comprising both the fault zone and the surrounding medium. Two simple examples are given to illustrate the usage of the model in finite element analysis.

An elastic-viscoplastic constitutive model for earth materials

Abstract: : This report describes the development of a three-dimensional elastic- viscoplastic work-hardening constitutive relationship for earth materials. The constitutive relationship is capable of reproducing the hysteretic behavior of the material under both hydrostatic and deviatoric states of stress; it also accounts for shear-induced volume change and the effect of superimposed hydrostatic stress on shearing response. The capability of the constitutive relationship for simulating the time-dependent response of earth materials is examined; an example fit for a clayey sand is given based on static laboratory triaxial shear and static and dynamic uniaxial strain test results.

Some Applications of the Boundary Element Method for Potential Problems

Abstract: The boundary element method is now firmly established as an important alternative technique to the prevailing numerical methods of analysis in continuum mechanics [1][2]. One of the most important types of applications is for the solution of a range of problems such as temperature diffusion, some types of fluid flow motion, flow in porous media and many others which can be written in function of a potential and whose governing equation is the Laplacian type. All these potential cases can generally be efficiently and economically analysed using boundary elements.

Mechanics of Composite Materials Recent Advances: Proceedings of the IUTAM (International Union of Theoretical and Applied Mechanics) Symposium on Mechanics of Composite Materials Held at Blacksburg, Virginia on August 16-19, 1982,

Abstract: : The main objectives of this discipline are to study by the methods of mechanics the properties of various composite systems such as matrices containing particles, fiber composites and others. The broader subject of composite materials encompasses both the science and technology of composite materials. The mechanics of composite materials is an organic part of this subject just as mechanics of structures is an organic part of structures. At the same time the mechanics of composite materials addresses questions which are fundamental to continuum mechanics since all continua model materials which have microstructure at some scale of magnitude. This symposium is concerned with a multitude of micromedia, matrices reinforced with whiskers or short fibers, unidirectional fiber composites, fiber composite laminates, etc. We are concerned with a multitude of properties such as: Elasticity, Thermal Expansion, Viscoelasticity and Vibration Damping, Plasticity, Nonlinear Behavior, Temperature Dependence of Mechanical Properties, Conductivity, Moisture Absorption, Static Strength, Fracture Mechanics, Fatigue Failure.

Symmetry Relations for Anisotropic Materials

Abstract: In continuum mechanics the description of physical properties of materials takes place in constitutive equations expressing relations between tensors which otherwise appear in some field equations. The possible symmetries that an anisotropic material may have will be reflected one way or another in the constitutive equations. The relations may be linear or non-linear tensor functions or functional of tensor arguments. In Sections 2–5 the case of linear relations will be discussed while non-linear relations are briefly mentioned in Section 6.

Mechanics of couple-stress fluid coatings

Abstract: The formal development of a theory of viscoelastic surface fluids with bending resistance - their kinematics, dynamics, and rheology are discussed. It is relevant to the mechanics of fluid drops and jets coated by a thin layer of immiscible fluid with rather general rheology. This approach unifies the hydrodynamics of two-dimensional fluids with the mechanics of an elastic shell in the spirit of a Cosserat continuum. There are three distinct facets to the formulation of surface continuum mechanics. Outlined are the important ideas and results associated with each: the kinematics of evolving surface geometries, the conservation laws governing the mechanics of surface continua, and the rheological equations of state governing the surface stress and moment tensors.

Pointwise velocity error estimates for an elliptic fluid jet

Abstract: Pointwise estimates are obtained for the perturbation velocity to a novel theory of fluid jet behaviour based on Cosserat continuum mechanics. The method employed is one of weighted energy.