Showing papers on "Continuum mechanics published in 1981"

An introduction to continuum mechanics

Abstract: Preface. Acknowledgments. Tensor Algebra. Tensor Analysis. Kinematics. Mass. Momentum. Force. Constitutive Assumptions. Inviscid Fluids. Change in Observer. Invariance of Material Response. Newtonian Fluids. The NavierStokes Equations. Finite Elasticity. Linear Elasticity. Appendix. References. Hints for Selected Exercises. Index.

On the local form of the second law of thermodynamics in continuum mechanics

Abstract: The Clausius-Duhem inequality a(r, t) > 0, a widely adopted axiom in continuum mechanics, leads to the conclusion that for many materials the entropy s cannot depend on gradients like the temperature gradient g and the velocity gradient e. But this is at variance with the received view (since Gibbs) that entropy is a function of thermodynamic state, however detailed that state description may be. Gradients, and even higher derivatives of macroscopic variables, may be included as state variables (although only on macroscopic time scales shorter than or comparable with their natural relaxation times), and the fundamental property of entropy is its convexity—the more detailed the specification of state, the smaller is the corresponding value of entropy. The entropy of a perfect monatomic gas is evaluated via the maximum principle, on the assumption that g and e are state coordinates, and it is found that s does depend on g • g and e : e, contrary to statements appearing in many textbooks on continuum mechanics. The source of the error in these works is shown to lie in applying 0 to relations involving second derivatives. The correct form of the Clausius-Duhem inequality contains only first-order derivatives; that is, it must be confined to linear constitutive relations. In this form the inequality is a consequence of the convexity of s, which is a somewhat more general manifestation of the second law.

On turbulent Reynolds stress closure and modern continuum mechanics

Abstract: The consistency of the problem of Reynolds stress closure in turbulence with the fundamental principles of modern continuum mechanics is examined. It is shown that Reynolds stress closure is completely consistent with the principles of determinism and material frame-indifference of modern continuum mechanics. The consequences that this has on turbulence modeling are briefly discussed.

On Eulerian and Lagrangean Objectivity in Continuum Mechanics

Abstract: In continuum mechanics the commonly—used definition of objectivity (or frame-indifference) of a tensor field does not distinguish between Eulerian, Lagrangean and two—point tensor fields. This paper highlights the distinction and provides a definition of objectivity which reflects the different transformation rules for Eulerian, Lagrangean and twopoint tensor fields under an observer transformation. The notion of induced objectivity is introduced and its implications examined.

Constitutive equations for meeting elevated-temperature-design needs

Abstract: Constitutive equations for representing the inelastic behavior of structural alloys at temperatures in the creep regime are discussed from the viewpoint of advances made over the past decade. An emphasis is placed on the progress that has been made in meeting the needs of the program whose design process is based in part on a design-by-inelastic-analysis approach. In particular, the constitutive equations that have been put into place for current use in design analyses are discussed along with some material behavior background information. Equations representing short-term plastic and long-term creep behaviors are considered. Trends towards establishing improved equations for use in the future are also described. Progress relating to fundamentals of continuum mechanics, physical modeling, phenomenological modeling, and implementation is addressed.

A gravitational model for a matter-free torsion ball

Abstract: A classical model of a gravitational field is constructed in terms of two distinct geometries. It is formulated in terms of a purely geometric action with distinct densities in different space-time domains. An explicit state spherically symmetric solution for each geometry is found and matching properties across a world tube generated by a closed space-like submanifold are discussed in terms of the associated second fundamental forms. The interior geometry has bounded curvature and torsion. The exterior region has a Schwarzschild geometry. The model simulates distinct phases of the gravitational field and is motivated by E. Cartan's (1922) analogy of torsion effects with dislocation phenomena in continuum mechanics.

The Mechanics and Physics of Adhesion of Elastic Solids

Abstract: The paper discusses a number of important questions raised in the theory of adhesion between elastic bodies under the action of normal forces. the relation between the microscopic quantities such as the interaction energies of attractive and repulsive forces, the equilibrium separations of atoms and the macroscopic quantities used in the continuum mechanics like the area of contact, the work of adhesion is examined, at first, by taking the interaction between rigid bodies as an example. This leads us to the concept of a thermodynamical, effective contact area between the interacting solids. the effective contact area of deformable bodies is calculated for the case of a Hertzian contact. It is shown that the inclusion of the term “work of adhesion” in the analysis based upon continuum theory of contacts is justifiable within the limits of the theory. The rest of the paper deals with the theory of adhesion in the framework of continuum mechanics and briefly reviews the recent work inspired by the various new developments in the fields of fracture mechanics. the relatively straight-forward extension of Johnson's theory of adhesion to more general axisymmetric shapes is indicated. the most important limitation of the previous work is that the choice of materials is severely restricted. the present work extends the theory to include the practically interesting case of contact between dissimilar elastic materials. in this case, the constraint imposed by adhesion on the tangential displacements within the contact area gives rise to shear stresses. the role of the shear stresses in the mechanics of adhesion is found to be significant.

Numerical implementation of inelastic time-dependent and time-independent, finite-strain constitutive equations in structural mechanics

Abstract: A number of complex issues are resolved in a way that allows the incorporation of finite strain, inelastic material behavior into the piecewise numerical construction of solutions in solid mechanics. Without recourse to extensive continuum mechanics preliminaries, an elementary time independent plasticity model, an elementary time dependent creep model, and a viscoelastic model are introduced as examples of constitutive equations which are routinely used in engineering calculations. The constitutive equations are all suitable for problems involving large deformations and finite strains. The plasticity and creep models are in rate form and use the symmetric part of the velocity gradient or the stretching to compute the co-rotational time derivative of the Cauchy stress. The viscoelastic model computes the current value of the Cauchy stress from a hereditary integral of a materially invariant form of the stretching history. The current configuration is selected for evaluation of equilibrium as opposed to either the reference configuration or the last established equilibrium configuration. The process of strain incrementation is examined in some depth. The stretching, evaluated at the mid-interval and multiplied by the time step, is identified as the appropriate finite strain increment to use with the selected forms of the constitutive equations. Discussed ismore » the conversion of rotation rates based on the spin into incremental orthogonal rotations. These rotations are used to update stresses and state variables due to rigid body rotation during the load increment. Comments and references to the literature are directed at numerical integration of the constitutive equations with an emphasis on doing this accurately, if not exactly, for any time step and stretching. This material taken collectively provides an approach to numerical implementation which is marked by its simplicity.« less

The Equilibrium Statistical Mechanics of Self-gravitating Systems

Abstract: The partition function and one- and two-particle distribution functions are calculated fora spherically symmetric self-gravitating system using a method which is exact except for terms of relative order N -1. The results are in agreement with those found in the continuum approximation. First approximations to the correction terms are evaluated with particular emphasis on the form of the pair distribution as compared with the product of two one-particle distributions.

Finite Element Analysis of Magnetoelastic Plate Problems.

Abstract: : Magnetic forces acting on conducting metal structures are significant in the design of such devices as fusion reactors, magnetohydrodynamic generators, magnetically levitated vehicles, magnetic forming devices, and various electric machines. The magnetic loads on these structures may be pulsed or cyclical and arise from the interaction between the induced eddy currents in the conductor and the externally applied magnetic fields. Moreover, the velocity and finite deformation of the conducting structure may interact with the electromagnetic fields to yield a coupled problem. This thesis presents an integrated study of magnetoelasticity problems for thin nonferrous conducting plates. Three phases of the study have been emphasized: theoretical modelling of interaction problems, finite element eddy current calculations for rigid conducting plates, and finite element numerical studies of coupled magnetoelastic problems. Experimental results by others and analytical solutions for limiting cases have been used to verify the numerical results obtained at each stage of the investigation. The Fortran programs developed for each part of the study are also described. The electrodynamics and continuum mechanics bases of the problem presented in the modelling part of the study.

Dislocation Shielding of a Crack in a Quasi Continuum Approximation

Abstract: : Earlier predictions of toughness by Thomson and Weertman are reviewed. The differing predictions of the two authors are shown to be due to different interpretations of the functional dependence of the size of the elastic region on intrinsic surface energy. An analysis of the quasi continuum model in terms of dislocation rearrangements is given, and the nature of the boundary condition on the elastic enclave boundary is discussed.

Analiza niestandardowa w mechanice newtonowskiej punktu materialnego

Abstract: The aim of the present paper is to apply and to investigate the non-standard analysis as the mathematical tool of Newtonian mechanics. Such approach introduces into mechanics non-Archimedean ordered fields and makes it possible to describe explicitly also infinite and infinitesimal values of the physical quantities. Generally speaking, the methods of the non-standard analysis in Newtonian mechanics of mass-point systems lead to the wider class of mathematical models of physical phenomena than that which can be obtained within the classical formulation of the discrete mechanics. For example, from the ,,non-standard'' formulation of the Newton's law of motion in discrete mechanics we can derive the fundamental ,,standard'' relations of the continuum mechanics. In this part of the paper the general introduction into the non-standard analysis as well as the fundamentals of Newtonian mechanics within non-standard analysis have been outlined. As an example of applications the new interpretation of the standard concept of constrains has been derived from the Newton's law of motion of certain non-standard unconstrained mass-point systems.

Impact and Penetration Problems.

Abstract: : Impact and Penetration phenomena are investigated by recently established nonlocal continuum theories Constitutive equations are developed for nonlocal elastic and plastic materials The problems of nonlocal line crack, half space under punch, plug formation and perforation of visco-plastic plates are solved The absence of singularities made it possible to introduce several fracture and penetration initiation criteria based on maximum stress hypotheses (Author)

Classical Statistical Mechanics

Abstract: Statistical mechanics is the bridge between molecular science and continuum mechanics. The input to statistical mechanics is a force law between particles. The particles can be atoms in a crystal, molecules in a gas or liquid, electrons in a plasma, amino acid units in a protein, elementary constituents in a complex polymer, etc. The forces between particles originate from Coulomb forces between electric charges and from magnetic dipole forces between magnetic moments. The classical force laws may be modified by the quantum mechanics (especially the Pauli exclusion principle) describing the particles. Normally the force laws are quite complicated and are not given by a simple analytical expression. Rather, they are given in one or more of the following four ways: (1) In terms of an unknown function, for which qualitative properties are postulated as laws of physics. (2) As a specific function, such as the Lennard-Jones potential or hard sphere potential. Such functions are chosen because they have representative features in common with the true force laws; for example, they may be asymptotically exact in some limiting region. (3) As a result of numerical calculation, based e.g. on the complicated exact force law and a Hartree-Fock approximation. (4) As a result of experimental measurement.

A variational approach to finite elasticity

Abstract: My main object is not to speak about well known interesting variational theorems in continuum mechanics (see, for instance, [1]) but to show that there exists a certain possibility of a systematic approach to finite elasticity by a variational procedure. The existing variational theorems are generally inadequate for the purpose, although interesting from a heuristic point of view. A systematic variational procedure allows a global approach to finite elasticity, that is an approach in which field and constitutive equations are written in integral form. It seems evident that even when only the field equations have such a form there is the greatest generality and meaning because the general equations of continuum mechanics have their natural origin in integral forms, while some regularity conditions must be satisfied for deducing differential field equations. Further, a global approach generally leads to improved convergence in numerical methods. I shall consider the basic problem of finite elasticity in the presence of unilateral surface constraints, treating either the classical case of non-polar continua or that of Cosserat continua with free rotations. For the latter case the results are not yet exhaustive or complete with the exception of the linear case, but I think it is useful to show some differences in the variational aspect between the two cases. For simplicity I shall consider only elastic continua without inner constraints.