Showing papers in "International Journal of Engineering Science in 1963"

Abstract: A theory of the electromagnetic field in moving, finitely deformed elastic dielectrics is developed. The equations for a weak field superimposed on a strong field and an arbitrary state of finite deformation are deduced. It is shown how certain electro- and magneto-optical effects and the retardation of a sound wave by a strong magnetic field in a dielectric can be explained and correlated.

Abstract: After plastic bending or torsion a body contains excess dislocations of a definite sign which give rise to a grain boundary type lattice curvature. In Section 2 the microscopical fluctuations of the appertaining residual stress field are described macroscopically by the introduction of a torque stress tensor which is defined by torques acting through an area element. A material constant can be derived which gives the ratio of the torque stress to the dislocation density or lattice-curvature. The characteristic quantity which goes into this constant is the glide plane distance of the dislocations. Finite torque stresses also arise as reactions against ordinary lattice-bending or torsion (without participation of dislocations) if the range of the atomic forces which try to resist the deformation is finite. The introduction of such torque stresses represents the first step in a transition from a local to a non-local elasticity theory. This is shown in Section 3. In the discussion of Section 4 two possibilities are mentioned to measure the new material constants experimentally. Finally the equilibrium conditions for force and torque stresses are satisfied identically by the introduction of a stress function tensor. This leads to a formulation of a system of simultaneous differential equations which is basic for the actual determination of the residual stress state.

Abstract: The Maxwell-Faraday theory of electrostatics is formulated in such a manner as to pave the way for a formulation of the electrostatics of deformable bodies. A variational principle is introduced leading to the basic field equations, the boundary conditions and the constitutive equations of an elastic dielectric subject to large deformations and polarizations. Various forms of the constitutive equations for the stress and the local electric field are proposed. For isotropic dielectrics, specific forms are obtained. The resultant equations are applied to the uniform extension of an internally charged hollow circular cylinder. An approximation to the constitutive equations is found for small strain and polarization.

Abstract: The coupling of electromagnetic and elastic waves is considered from the standpoint of linear elasticity and a linearized electromagnetic theory. The problem of plane waves traveling through a uniform magnetostatic field is considered and couplings of the waves are studied. An investigation of the same problem for a uniform electrostatic field shows that the usual plane waves propagate without any change in their phase velocities but that the mechanical waves are accompanied by small fluctuating electromagnetic fields. The problem of the vibration of a free infinite elastic plate in a large magnetostatic field is examined under the assumption that the resulting electromagnetic fields are quasistationary. Frequency equations are obtained for both symmetric and antisymmetric vibrations and the damping caused by the field for both the first two symmetric and antisymmetric modes is obtained as a linear correction to the usual free plate frequencies.

Abstract: A solution for the infinite solid is obtained by the method of dual integral equations. Results are expressed mainly in closed forms, singular round the edge of the crack. The stress components in the plane of the crack are shown in curves.

Abstract: This paper is concerned with a nonlinear theory of elastic shells with small deformations whose material response is nonlinear. The developments are carried out under the Love-Kirchhoff hypothesis. General constitutive equations are derived in which the geometrical properties (due to deformation) and the material characteristics are separable. Through this separability, it is shown how to extract constitutive equations of predetermined types. Particular examples are followed by a discussion of the membrane theory.

Abstract: This paper contains an analysis of the distribution of stress in a long circular cylinder of elastic material when it is deformed by the application of pressure to the inner surfaces of a penny-shaped crack situated with its centre on the axis of the cylinder and its plane perpendicular to that axis. It is assumed that the cylindrical surface is free from shear and is supported in such a way that the radial component of the displacement vector vanishes on the surface. By making a suitable representation of the stress function for the problem, the problem is reduced to the solution of a Fredholm integral equation of the second kind. Expressions for the various quantities of physical interest are derived for small values of the ratio of the radius of the crack to that of the cylinder by finding an iterative solution of this equation. For values of this ratio near unity the integral equation has been solved numerically using a high-speed computer and the relevant quantities calculated.

Abstract: A velocity tensor and a density tensor of a continuous distribution of dislocations are derived by considering the dynamic behavior of dislocations. The stress and displacement fields are discussed by the use of Green's functions. Application of the theory is demonstrated by an example where the number of positive dislocations is equal to the number of negative dislocations everywhere, but the velocities of the two sets are in the opposite direction.

Abstract: This note contains an analysis of the distribution of stress in a long circular cylinder of elastic material when it is deformed by the application of pressure to the inner surfaces of a penny-shaped crack situated symmetrically at the centre of the cylinder. It is assumed that the cylindrical surface is free from Stress. The equations of the classical theory of elasticity are solved in terms of an unknown function which is then shown to be the solution of a Fredholm integral equation of the second kind previously derived by W.D. Collins. The solutions of this equation for constant pressure and for various values of the radius of the crack to that of the cylinder, derived using a high-speed computer, are discussed and quantities of physical interest calculated. The calculations were repeated for the case of a variable pressure following a parabolic law and they are also reported.

Abstract: The present paper is concerned with the formulation of non-linear theories of thermo-visco-elasticity. The general theory presented embodies the non-linear generalization of Voigt, Maxwell and other classes of viscoelastic solids and fluids. The fundamental equations for the thermoelastic solid, the thermoviscous fluid and other special classes of materials are obtained from the general theory by specialization. The theory is applied to the problem of rectilinear laminar flow between two heat reservoirs for three special classes of thermo-viscoelastic materials. Some results of matrix algebra pertinent to the development of the theory are presented in the appendix.

Abstract: The problem considered is that of producing combined elastic and electro-magnetic waves in an elastic semi-space by means of a thermal shock acting on the boundary of the semi-space. It is assumed that the elastic semi-space is adjacent to a vacuum. Finite electric conductivity is assumed for the body-displacement currents being rejected. In the case of a real conductor two cases are considered ; the case of an arbitrary magnetic field and considerable conductivity and that of an arbitrary conductivity and weak initial field. This enables us to introduce small parameters and obtain a first approximation solution sufficiently accurate for the practice. The mechanical and electro-magnetic waves in the body produced by the action of a thermal shock and electro-magnetic waves radiating into the vacuum are determined. This solution differs from the solution for a perfect conductor [1] by some additional terms of a diffusion character. If the displacement currents were taken into consideration some discontinuity waves would appear with the velocity of propagation of the order of the light velocity, in addition to modified discontinuity waves propagating with a velocity of the order of the sound velocity in the body.

Abstract: This paper is concerned with a new derivation of the general equations of the linear theory of elastic shells under the Kirchhoff-Love hypothesis. The entire boundary-value problem of shell theory is recast in terms of new variables for the strain measures as well as the stress and couple resultants. Particular attention is paid to an exact derivation of the constitutive equations and their first approximations which meet all invariance requirements. The natural boundary conditions for stress and couple resultants and all field equations consisting of compatibility, equilibrium and the constitutive equations (or their first approximations) involve only symmetric tensors and are, moreover, remarkably free of the anti-symmetric parts of both the middle surface strains and the couple resultants.

Abstract: ‘Plasticity’ is to assume an arbitrary torsion and a Riemann-Christoffel curvature tensor associated with the material manifold and so to ‘tear’ it. If it reduces the curvature to nil, the tearing is ‘perfect’. A perfect tearing is not always realized and can be only ‘virtual’. The criterion for yielding can be reached by statistical summary as a variation over a finite region from the equation of geodesic deviation which includes the curvature tensor. The details being immaterial, the views can be restricted to the variation of the metric tensor and an extension of the three-dimensional analogue of Einstein's field equation in general relativity theory is derived, the analogue of the material-energy tensor being given a thermodynamical interpretation. The Einsteinian assumption is shown to be a special case of general possibilities. The approach can be made in terms of Finsler's as well as of simpler non-Riemannian geometry.

Abstract: Equilibrium conditions for internal stresses are derived for a continuum in a linearly connected space with torsion and a corresponding stress space is defined. This stress space is a generalization of the stress space studied by Minagawa in [1], [2], [3].

Abstract: The major aims of this paper are 1. (1) to formulate a few general principles, of use in the applications of thermodynamics to heterogeneous systems, where the properties of impermeability, rigidity and thermal insulation play an important part. 2. (2) to develop a macroscopic theory of irreversible processes applicable to systems which are not near to equilibrium. For these purposes, in addition to the conservation laws and Caratheodory's Principle, one further principle is needed, which may be stated as follows: ‘an unperturbed system tends to approach a state of equilibrium, which is completely determined by the extensive parameters in the equilibrium state’. This principle is applied to several fundamental questions concerning thermodynamical equilibrium. In the final section, a general formalism is developed for the description of irreversible processes, without requiring the usual (approximate) linear relations between fluxes and ‘forces’. Though this formalism is based on statistical considerations, it is independent of any molecular hypothesis.

Abstract: Resume La determination de la diffusibilite d'un certain milieu isotrope est reduite a la resolution d'une simple equation differentielle lineaire de type elliptique par le choix approprie de conditions aux limites surspecifiees.

Abstract: Steel Discs of 12 in. outside diameter were centrally clamped while various combinations of transverse loads were applied at the outer edge. Measured stiffnesses obtained from an initially stress-free disc were found to agree with theoretical values. Further discs were subjected to hammering treatments to introduce internal stresses. After stiffness measurements were made, strain gauges were attached for measuring the internal stresses. A strain energy method was used for calculating the effect of these stresses on stiffness. Fair agreement was found between experimental and calculated values.

Abstract: This paper contains a detailed analysis of the axisymmetric deformation of elastic/plastic ‡ circular annular plates in the state of plane stress under combined pressure and couple (due to circumferential shear) at the inner boundary. With the use of Tresca's loading function and its associated flow rules, the results obtained include complete elastic/perfectly plastic solutions during both loading and unloading, and complete workhardening solutions during loading with isotropic and with kinematic hardening. The major portion of the paper deals with the unloading problem which requires two new solutions (in addition to the usual elastic unloading solution) in order to cover the full range of load programs including those with one load parameter increasing and the other decreasing. Several novel features of this combined load problem are discussed and illustrated through numerical examples.

Abstract: The method of Green's function is applied to the quasi-static thermoelastic theory of shallow shells with heat conduction equations included. Solution formulae are derived for middle-surface displacements and stress and temperature resultants in terms of initial and edge temperatures, internal heat sources, ambient temperatures at the upper and lower surfaces, and surface tractions. Equations are given for the Green's functions appearing in the solution formulae. Extension to more general shell theory is discussed. By way of example, the method is applied to thermoelastic problems for two classes of shallow shells. Also, the effect of transverse shear deformation is examined with reference to a shallow spherical shell.

Abstract: A novel method for solving linear boundary-value problems is outlined through the use of the linear-programming computational algorithm. The method is equivalent to obtaining an approximate solution by a Chebyshev fit wherein the maximum error is minimized. In this work the method is used to solve the first biharmonic problem ▽ 4 u = c in a rectangular two-dimensional region subject to u = 0 and δu / δn = 0 on the boundary of the region. The advantages of the present method over the collocation method (wherein a set of linear equations are solved uniquely for a set of parameters) seem to be the following: (1) an estimate of the error in the approximate solution is given by the objective function of the linear program: (2) for a fixed number of parameters in the approximate solution, the present method gives the best value, in some sense, of these parameters.

Abstract: The motion of a single charged particle in magnetic and electric fields B and E is described by the basic non-relativistic equation of motion in which radiation damping is neglected. With emphasis on vector methods, data are obtained for the drift-free reference case of motion in a circular helix, and then the drift velocity perpendicular to B due to electric field perturbation of the circular case is used to obtain the familiar gravitational and curved magnetic line drifts. Application of the gravitational drift to the plasma physics Rayleigh-Taylor instability is examined critically in the light of information which can be obtained from the macroscopic equation of motion for a plasma in a magnetic field. For a spatially perturbed magnetic field a guiding centre treatment using the standard orthogonal curvilinear co-ordinate system of differential geometry yields, after time averaging, the first-order Alfven and curvature drifts, the latter being in agreement with the result obtained less rigorously from the electric drift velocity. The magnetic field geometry is described in terms of an extended set of Frenet-Serret partial derivatives which involve six scalar quantities, five of which are independent when a scalar magnetic potential exists. After time averaging there also exists a first order velocity component of unusual form parallel to B . For the usually encountered case of B irrotational, this component vanishes. The three equations of motion for the components of the particle velocity are also obtained in terms of this curvilinear co-ordinate system, and when time averaged the equation of motion for the particle velocity component parallel to B enables the spatial adiabatic invariance of a spiralling particle's magnetic moment to be rapidly established. The case of B dependent on the time, which can lead to radial compression of a plasma, is discussed from drift and non-drift view points, and the difference between Larmor's frequency and the gyrofrequency is emphasized. A simple microscopic treatment of the polarization drift is also included. Finally the historical and mathematical significance of adiabatic invariance is discussed, with particular reference to the magnetic moment in plasma physics.

Abstract: A rapid method for determining internal stresses in thin discs was examined. A radial cut was made in the disc and stresses were calculated from the amount by which the cut opened. Calculations were carried out for a disc having a bore diameter one fifth of the outside diameter, and were verified by loading a split disc of 36 in diameter. Good agreement was found between internal stress distributions in a 12 in diameter disc determined by the radial cut method and by the usual method of attaching strain gauges.

Abstract: The conditions governing the motion of an elastic/plastic loading interface propagating into the elastic region with an associated acceleration (and strain-rate) discontinuity are set up. It is shown that for the general class of elastic/plastic solids considered, the speed of propagation of the interface must lie in one of three ranges, speeds in the fastest range being associated with the “creation” of a plastic regime. Detailed consideration is given to an interface with its normal along a principal axis of the tensor whose components define the normal to the yield surface in strain-space. The speed of the interface is determined for arbitrary acceleration and strain-rate discontinuities, and the effects of incident acceleration waves are discussed.

Abstract: Zusammenfassung Es wird die Gibbs'sche Fundamentalbeziehung fur die Thermodynamik flussiger Stoffe fur den Fall erortert, in welchem electromagnetische Felder anwesend sind. Sodann werden die Bedingungen fur die thermodynamische Stabilitat formuliert. Thomsons Satz in der Elektrostatik wird aus dem zweiten Gesetz der Thermodynamik abgeleitet, welches gleichzeitig die Gleichgewichtsbedingungen fur das flussige Material zwischen den geladenen Korpern angibt.

Abstract: Using a relatively simple theory of anisotropic fluids, we explore the possibility of obtaining solutions which possess finite discontinuities on a surface. Examples of such singularities have been brought out by earlier researches. We obtain general conditions for the existence of such surfaces, indicating that they are very likely to be material surfaces.

Abstract: The rigid body motion of a cavity determines an irrotational motion of the liquid filling its inside. From the point of view of external dynamics, the fluid behaves as an equivalent solid. The purpose of the paper is to examine general properties of the equivalent inertia tensor and to calculate it explicitly in the case of a cylindrical fuel tank.

Abstract: The transients due to uniform heat addition to the initially steady, sub-true-sonic one-dimensional flow of an ideal, inviscid, perfectly conducting compressible fluid are determined by the use of a linearization based on small heat addition. Although the solution obtained ceases to be valid in the neighborhood of true-sonic flow, and this difficulty is due to the inadequacies of the linearization which approximates the C+ and C − characteristics by two rectilinear and parallel families of lines, a solution valid for all flow speeds may be obtained by making a more detailed examination of the negative characteristics. In the limiting case of zero magnetic field, the solution reduces exactly to the solution of the corresponding problem in conventional gas dynamics.

Abstract: In this paper, an attempt is made to develop a general theory of the motion of a compressible fluid by means of the geometry of the general space propounded by Cartan based on the concept of area. Assuming that the flow is irrotational and isentropic, it is shown in Section I, that the equation of motion of an inviscid compressible fluid is regarded as that of hypersurfaces in the Cartan space. Various hydrodynamical features of the compressible fluid motion including the “shock” phenomena are clarified by the metric properties of the space. The fluid motion in the general case, in which the above assumptions are not valid, is treated in Section 2. The physical quantities such as vorticity, entropy and the Croccian vector are expressed by means of the geometrical terminology of the Cartan space along lines parallel to the Section 1.

Abstract: The linearized analysis used to determine the perturbation produced when an initially uniform magnetohydrodynamic shock wave of arbitrary strength encounters a section of non-uniform cross-sectional area is singular when the flow behind the incident shock is initially close to the true sonic speed. A modified perturbation theory, based on small area variations and valid for all flow speeds, is derived, and it is seen that the mentioned singularity masks the formation of secondary shocks. The present theory includes results on the propagation of non-uniform gas dynamic shocks as a special case.

Abstract: The propagation of electromagnetic waves, along a circular cylindrical waveguide containing a holohedral isotropic material under the influence of applied static, electric and magnetic fields, is investigated. The constitutive equations employed are based upon those derived by Toupin and Rivlin [1] for such materials. The applied static fields are assumed to be uniform and directed along the axis of the waveguide. An exact solution of the problem is obtained and the speed of propagation and rate of rotation of the wave is investigated.