Showing papers on "Continuum mechanics published in 1975"

Finite element formulations for large deformation dynamic analysis

Abstract: SUMMARY Starting from continuum mechanics principles, finite element incremental formulations for non-linear static and dynamic analysis are reviewed and derived. The aim in this paper is a consistent summary, comparison, and evaluation of the formulations which have been implemented in the search for the most effective procedure. The general formulations include large displacements, large strains and material non-linearities. For specific static and dynamic analyses in this paper, elastic, hyperelastic (rubber-like) and hypoelastic elastic-plastic materials are considered. The numerical solution of the continuum mechanics equations is achieved using isoparametric finite element discretization. The specific matrices which need be calculated in the formulations are presented and discussed. To demonstrate the applicability and the important differences in the formulations, the solution of static and dynamic problems involving large displacements and large strains are presented.

Constitutive equations in suspension mechanics. Part 1. General formulation

Abstract: Neither the phenomenological nor the structural approach to the determination of constitutive equations has yet shown itself to be capable of producing useful and predictive descriptions of the majority of technologically important complex fluids. In the present paper we explore the suggestion that significant progress can be made when these two complementary approaches to rheology are combined. For this initial study we restrict our attention to materials which can be modelled as a suspension of particles in a Newtonian fluid, thereby including most polymer solutions while excluding polymer melts. By applying phenomenological techniques to the basic formulation of suspension mechanics we are able to deduce a common simplified constitutive model for all suspension-like materials and to reveal its physical origin. The present analysis demonstrates that the constitutive model of Hand (1962), involving a single second-order tensor, is not sufficiently general for a rigorous description of the majority of suspension-like materials. Consideration of the constitutive forms for the limiting cases of near-equilibrium and strongly non-equilibrium microstructure suggests, however, that Hand's model may provide a reasonable approximation to the exact constitutive behaviour which is useful over the whole range of flow strengths.

Electrodynamics for moving elastic solids and viscous fluids

Abstract: A general theory of electrodynamics of moving and deformable media is developed on the basis of the Chu formulation of the Maxwell equations and the general principles of continuum mechanics. From the two-dipole model for polarization and magnetization and the concept of the Lorentz force on electric and magnetic dipoles, the body force, the body couple, and the energy supply of electromagnetic origin are determined explicitly in terms of the dectromagnetic variables. These kinetic quantities appear in the balance equations for the mechanics of deformable bodies, which are coupled to the Maxwell equations. Boundary conditions and constitutive equations which satisfy the principle of objectivity are obtained for polarizable and magnetizable elastic solids and viscous fluids.

On the theory of heat-conducting micropolar fluids with microtemperatures

Abstract: Microthermopolar fluids are introduced as a subclass of microthermofluids. In this subclass micromotion consists of rigid rotation only. The local balance laws and the linear constitutive equations are extracted from the theory of microthermofluids. Restriction regarding the material coefficients appearing in the constitutive equations are deduced by thermodynamical considerations. The problem of Poiseuille flow between two parallel plates is solved.

Theories and Principles of Viscosity

Abstract: Viscosity is a transport phenomenon. Viscosity is the transport of momentum due to a velocity gradient. The beginning of the phenomenological studies of viscosity goes back to the ancient Greeks and later the Romans characteristically applied what they had learned in practical ingenious ways [1]. Modern theories of viscosity of liquids are based on continuum mechanics and molecular theory.

On the theory of boundary conditions

Abstract: The question which boundary conditions are appropriate for the given differential equations of a continuum theory, is treated by the method of nonequilibrium thermodynamics. In the first part the previous investigation1) which had been confined to ordinary hydro(aero-)dynamics is generalized by taking into account higher derivatives in the continuum constitutive laws. This kind of generalization becomes important when a rarefied gas with boundary is treated phenomenologically (wall influence, slip-flow regime). In the second part an even more general scheme is discussed. The underlying applications now are rarefied polyatomic gases within walls, perhaps in an external field. So, the starting point is a general set of transport-relaxation equations with more variables than the hydrodynamical ones. By considering the corresponding entropy production, especially its part due to the boundary, it is again possible to set up constitutive laws, i.e., matching or boundary conditions, at an interface or a surface.

Anisotropic and nonlinear viscoelastic behaviour in solid polymers

Abstract: Two aspects of the mechanical behaviour of solid polymers are reviewed: (i) anisotropic mechanical behaviour of oriented polymers, and (ii) nonlinear viscoelastic behaviour. In the case of anisotropic behaviour, the discussion includes a comprehensive account of progress to date in obtaining experimental data, together with recent attempts at understanding the behaviour in structural terms. The discussion of nonlinear viscoelastic behaviour, on the other hand, is almost entirely at a phenomenological level. Here it is attempted to bring together the many different approaches and to show their common relationships, particularly between the engineering formulations and those based on continuum mechanics.

A continuum approach to the second-sound effect

Abstract: Standard techniques of continuum mechanics are used to describe the flow of certain microscopic excitations which are believed to give rise to the second-sound phenomena in some materials at low temperatures. Appropriate balance laws are formulated, and constitutive equations for an elastic solid are postulated. The propagation of small-amplitude second-sound waves is discussed and the results compared with those predicted by the theory of Lord and Shulman [1].

A Multipolar Approach to Liquid Helium II

Abstract: Using the general framework of multipolar continuum mechanics as developed byGreen andRivlin [1], we discuss a possible alternative to the two-fluid theories of liquid helium II. We show in particular that our theory predicts the existence of second-sound waves which are attenuated in the manner obseved both in the stationary fluid and also in the rotating fluid. Moreover we suggest a possible continuum analogue of the concept of quantised vortex lines.

Continuum thermodynamics based on a notion of rate of loss of information

Abstract: It is not known for sure how to formulate the second law so as to describe adequately the behaviour of bodies which deform and remember the past. Following the lead of COLEMAN & NOLL 1 it has become common practice in continuum mechanics to take as the second law the proposition that the Clausius-Duhem inequality holds in every process compatible with the balance of mass, momentum, moment of momentum and energy. Interpreted in that way the law leads to restrictions on constitutive relations, and COLEMAN & NOLL developed a method for determining those res t r i c t ions -a method which has been extended, elaborated and applied by many. 2 To take the second law to be the Clausius-Duhem inequality is to assume, among other things, that temperature and entropy are defined far from equilibrium. Those who do not find the truth of that assumption transparently obvious ask for an alternative approach and my intention is to propose one tentatively. The future behaviour of a body is determined by prescribing initial data, which may well include information about its past history, and also the action of the external world by way of surface traction and body forces and the supply of heat through conduction and radiation. The action of the external world may be such that only certain gross features of the initial data remain relevant to predicting the ultimate behaviour whereas the details become less and less relevant. Thus a homogeneous, rigid, heat conductor, which is thermally isolated and whose initial temperature field is prescribed, approaches equilibrium at a temperature determined solely by the total initial internal energy and the specific heat of the conductor; all features of the initial temperature other than the total energy ultimately become irrelevant. In statistical mechanics the increasing irrelevance of the details of the initial data is called, not strictly accurately, loss of information and I shall use the same terminology here.

Incremental Behaviour of an Elasto-Plastic Continuum.

Abstract: : A three dimensional incremental theory which describes the behaviour of an elasto-plastic continuum is proposed. At any instant of the deformation process a constitutive relation between infinitesimal quantities is constructed. By passing to the limit an exact rate theory is obtained. The deformation can be of any magnitude. For infinitesimal deformation the theory degenerates to the classical elasto-plastic theory. Tensors are considered as invariant objects rather than as a collection of components. By using full tensorial notation a simple invariant formation is obtained. The work contains a complete derivation of a rate theory; kinematics, equilibrium, boundary conditions, virtual principles and constitutive equations.

Elasticity theory. IV. The stress–strain relation

Abstract: The stress–strain relation is analyzed in terms of the gyration tensor where the sum is over all N atomic groups comprising the elastomeric material. All groups are taken for convenience to have the same mass. This tensor is linear in the strain, and furthermore it may be evaluated from a configuration integral. The definition of the strain by use of Sαβ gives new insight into the connection between statistical and continuum mechanics, and further clarifies the significance of “affine deformation.” Stress components are obtained as derivatives of the (Helmholtz) free energy A with respect to the invariants of Sαβ. The first invariant is just the radius of gyration S2 of the network, and the dependence of A on S2 alone provides a basis for analysis of stress–strain isotherms.

Continuum Physics Vol 2: Continuum Mechanics of Single Substance Bodies

Abstract: A Cemal Eringen (ed) London: Academic Press 1975 pp xv + 616 price £23.55 A mathematical treatment of continuum mechanics of simple materials, incorporating an axiomatic thermodynamic theory. Particular classes of solid and fluid are described and some exact solutions presented.

The Continuum Mechanics of Solid Polymers

Abstract: The field of continuum mechanics treats the response of continuous bodies to externally applied forces. Molecular structure is disregarded and the body is pictured as a continuum without gaps or empty spaces. These external forces can deform the body and set it into motion. The motion of a rigid body depends only on its mass and mass distribution, not on its composition. The deformation of the body, however, depends upon its internal constitution and is characterized by “constitutive equations” or “rheological equations of state.” These equations describe the relation between the stress and strain of the material.