Showing papers on "Displacement field published in 1972"

A new representation of the strain field associated with the cube‐edge dislocation in a model of a α‐iron

Abstract: The displacement field around an edge dislocation core is examined in an attempt to quantitatively modify the linear elastic Volterra solution. The approach derives from a computer simulation to obtain the equilibrium atomic positions in a dislocated bcc crystallite using Johnson's potential to model α‐iron. Unlike many previous calculations of this nature which employ rigid boundary conditions throughout the relaxation, the elastic media bounding the crystallite is here allowed to adjust in response to atomic readjustments within the crystallite. The procedure used to achieve flexible boundaries and its application to this problem is discussed in detail. This scheme permits linear elastic displacements at distances relatively far from the dislocation center. The difference between the computed strain field and the Volterra prescription results from nonlinear effects in the core, and is found to correspond to a net expansion of 0.25b2 per unit dislocation length. This is in accord with findings of experim...

An atomistic study of cracks in diamond-structure crystals

Abstract: The structuxe of atomically sharp equilibrium cracks in diamond, silicon and germanium is calculated. The treatment considers a long plane crack formed by bond rupture across the (111) cleavage plane, critically loaded in tension. Within a small 'core' region immediately surrounding the crack tip the interatomic interactions are represented by a potential function specially constructed to match macroscopic fracture parameters. Anisotropic linear elastic theory is invoked to provide boundary conditions for the core region, and a first approximation for lattice-point displacements within. The core atoms are then relaxed to a configuration of minimum potential energy by computer. The results indicate that continuum theory is capable of giving remarkably accurate predictions of the crack-tip displacement field, except within about three atom spacings from the tip, despite marked nonlinearity in the interatomic force function. These results are discussed in terms of existing continuum models of crack-tip structure: in particular, Barenblatt's model of a cusp-shaped tip region is found to be inapplicable to diamond-structure crystals. The crack-tip geometry is better pictured as a narrow slit terminated by a single line of bonds close to the rupture point. Brief reference is made to the possible extension of the treatment to other classes of highly brittle solid, especially glassy materials, and to the relevance of the results to some fracture problems of practical importance. The fracture of an ideally brittle solid is essentially an atomic process, in which cohesive bonds are ruptured at the tip of the growing crack. Yet traditionally the rnathematical treatment of the mechanics of fracture propagation has been developed almost exclusively from continuum concepts. The chief reason for this lies in the interest of simplicity, a proper description of the atomic configuration at a crack tip requiring seemingly formidable analysis in terms of a suitable structural model for the given solid. The continuum approach, based on linear elasticity theory, has in fact proved adequate in many fracture-mechanics problems: in particular, the growth of a semi-brittle crack in most 'engineering materials' can be described in terms of a macroscopic 'plastic zone' encasing the tip. Many mechanical properties, on the other hand, are highly sensitive to events occurring over distances no greater than a few interatomic spacings. For instance, the energetics of dislocations in plastic crystals, particularly covalently-bonded crystals, may depend largely on the atomic structure of the dislocation core. The ideally brittle crack provides a similar case, the crack front advancing one atomic

An improved method for obtaining the general-displacement field from a holographic interferogram

Abstract: A simple and reliable technique is proposed for predicting the cartesian components of a general-displacement field from the usual double-exposure holographic fringes. An overdetermined set of simultaneous equations is developed at each point of interest and a least-squares solution provides the three displacement components. The reliability of this technique was tested by varying the degree of overdeterminacy of the set of equations. The three-dimensional displacement field of a beam under pure bending was determined holographically and compared with that from a closed-form theoretical solution. Finally, a highly skewed marine-propeller-blade model under uniform air pressure was analyzed holographically and the results were correlated with those from a finite-element analysis.

Elastic displacement field of point defects in anisotropic cubic crystals

Abstract: The elastic displacement field of point defects in cubic crystals is calculated for weak anisotropy by second order perturbation theory and by a variational procedure. The results are compared with numerical calculations for Cu. Further analytical approximations are given for the volume change in an infinite crystal and for the interaction energy of two point defects.

Effect of earth layering on earthquake displacement fields

Abstract: The displacement fields due to a very long vertical strike-slip fault are calculated for earth models consisting of two layers over a half-space. It is shown that if zones of low rigidity are present beneath the Earth9s surface, they will result in an amplification of the displacement field observed at the Earth9s surface. The amount of amplification depends both on the structure and the distance from the fault. If thin soft layers exist in the upper mantle, it is shown that they will have a very strong effect on the observed displacements. It may eventually be possible to use static observations at the surface to detect such layers.

Diffraction contrast from coherent precipitates in elastically-anisotropic materials

Abstract: A series representation is modified for the computation of the displacement field around a coherent precipitate of any shape in an elastically-anisotropic cubic crystal. The shape effects are compared with the much greater influences of the anisotropic elasticity. The strain field contrast of a spherical particle in an elastically-anisotropic matrix of cubic symmetry has been computed and agrees with the electron microscope images of coherent Co-precipitates in a Cu-foil. By comparison of the computations with the micrographs the mismatch of the lattice parameters can be determined.

Extensions of the Prager-Shield theory of optimal plastic design

Abstract: The Prager-Shield associated displacement field method for optimal plastic design is extended to multi-component specific cost functions and multiple load conditions, and a lower bound theorem based on kinematic requirements only is introduced. Since any statically admissible stress field results in an upper bound, the proposed theorem provides a simple method for establishing bounds on the optimal cost. By a simple substitution of parameters into the general equations presented, the optimality criteria can be obtained for particular design problems. Examples of optimal fibre-reinforced plates are given.

Displacement field around a shear dislocation loop

Abstract: The Green's function method has been applied to derive analytical expressions for the displacement field around a circular shear dislocation loop in an infinite isotropic medium. Previously, solutions have been available only for a prismatic dislocation loop with an axially symmetrical stress distribution. The displacement field is obtained in terms of tabulated elliptic integrals. Numerical results are presented graphically by means of contour plots.

Meridian seeking instrument

Abstract: A gyroscope unit having a horizontal spin axis hangs from a suspension band within a container. The container is rotatably supported about a vertical axis. A transducer generates a control signal representative of the angular displacement between the gyroscope unit and the container about the vertical axis from a reference position. The container is rotated about the vertical axis responsive to the control signal to reduce the angular displacement from the reference position. A torque about the vertical axis that opposes the angular displacement is directly applied to the gyroscope unit responsive to the control signal. The directly applied torque is a nonlinear function of the angular displacement, being zero below a threshold value of angular displacement and having a predetermined gradient above the threshold value.

Determination of three orthogonal displacement components from one double exposure hologram

Abstract: Another method for the interpretation of double-exposure holograms is presented It is shown that by a filtering procedure carried out in the plane of the real image, fringe patterns can be formed that also give information about the displacement undergone by the surface of the object An interpretation for those patternis presented fo rsmall out-or plane displacements presented for small out-of-plane displacements A method for determining the three orthogonal displacement components for those patterns and the virtual image is outlined

X-Ray Scattering from the Displacement Field of Point Defects

Abstract: We measured diffuse x-ray scattering from the displacement field of point defects in KBr single crystals. From these results absolute values for the double-force tensors of the anion vacancy and the interstitial anion can be derived. From the symmetry of the displacement field a model for the interstitial anion is proposed.

Stress Analysis of an Adhesive Lap Joint Subjected to Tension, Shear Force and Bending Moments

Abstract: : A stress analysis of the lap joint is presented treating the problem as one of plane strain. The joint is subjected to a general loading, consisting of tension, shear force and bending moments. The variation in the material properties and thickness of the two adherends is considered. The displacement field in the adhesive layer is expressed in series form and the compatibility condition at the interface is used to express the displacement field in the adherends. The potential energy of the joint is calculated and minimized to obtain linear, ordinary differential equations and boundary conditions. The differential equations are solved on the computer. Photoelasticity is used to confirm the theory. Two specimens of lap joint using a 1/4 in. layer of a photoelastic plastic simulating the adhesive, are tested photoelastically. The agreement between the theoretical and the experimental results is found to be good. Design recommendations for the lap joint are made based on the results of this investigation.

An investigation of a plane strain continuous penetration problem

Abstract: The Paper presents the results of a series of experiments in which both rough and smooth faced, long, symmetrical 60° wedges were driven vertically, at a constant rate, into uniform beds of sand prepared at different porosities. The apparatus, instrumentation and sample preparation are described and the measured values of penetration resistance, normal and tangential forces and stresses on the wedge face and the friction angles developed are presented and discussed in detail. Continuous penetration problems of this kind are unique in that the displacement field grows at a constant rate whilst maintaining complete geometrical similarity unless instability arises due to strain softening of the deforming material. The test results show that instability does not occur in loose sand but develops as the sand bed porosity is decreased, demonstrating quite conclusively the inadequacy of analyses which take no account of the kinematic aspect of the system being investigated. Cette communication donne les resultats...

Large Deformations of Rigid-Viscoplastic Structures under Impulsive and Pressure Loading

Abstract: It has recently been shown that by imposing a time-independent kine-matically admissible displacement field on a rigid-plastic structure the approximate determination of its response to pressure or impulsive loading can be reduced to the investigation of an equivalent one-degree-of-freedom system. In the present paper this approximation is generalized to structures of rigid-viscoplastic material undergoing moderately large deformations. The analysis leads to a nonlinear second-order differential equation the solution of which is obtained by using numerical methods. The investigation of a circular plate shows that this approximate method leads to simple calculations and its accuracy is satisfactory. As expected, the numerical results confirm that strain-rate sensitivity and geometry changes have an important influence on the maximum permanent deflections. In addition, a comparative study shows that the usual approximation according to which the pressure pulse is replaced by an equivalent impulse g...

Waves in an elastic plate with an irregular boundary

Abstract: Investigated is the effect of a surface irregularity on the propagation of waves in an isotropic, elastic plate, the method of perturbation being employed to determine the scattered field to the first order in a small parameter descriptive of the height of the irregularity. While the firstorder displacement field in the region under, and in the immediate vicinity of, the disturbed surface is very complicated, the far field in either direction, generated by either a symmetric or an antisymmetric incident wave, consists of a finite number of propagating waves, symmetric and antisymmetric, their number corresponding to the number of real roots of the frequency equation for the given frequency of incident wave.

Some variational principles pertaining to non-stationary heat conduction and coupled thermoelasticity

Abstract: In the first part of this paper, some restricted variational formulations for non-stationary and non-linear heat conduction are presented. The principles proposed are shown to be valid for isotropic as well as anisotropic solids with temperature dependent density, heat conductivity and specific heat. In the second part, two new variational principles for coupled thermoelasticity are proposed: these differ essentially from each other by the functions selected to be varied and by their range of application. The first principle is an extension of a general principle for purely dissipative processes, as presented by the authors in a previous paper, and uses, as functions to be varied, the temperature, the heat flux vector and the velocity field. This formulation is quite general in that the thermal and elastic coefficients may depend on the temperature. The second principle presented is inspired by the principle ofBiot: the functions to be varied are the temperature, the displacement field and the total heat flux vector ofBiot. The range of application of this formulation is limited to the case of constant thermal and elastic properties.

Study of effects of design details on structural response to acoustic excitation

Abstract: The normal mode vibration characteristics of one-dimensional and two-dimensional panel arrays were investigated analytically and experimentally. The finite element displacement method was used to formulate the structural models. The structural models include a stiffness and consistent mass matrix for thin-walled open-section beams not previously reported, and a modification of a rectangular plate bending element to include a fundamental interior mode for the element as a generalized coordinate. Provision for adding lumped masses to represent accelerometers is also included. For the one-dimensional panel, the normal mode stress resultants are obtained by integrating the equilibrium equations directly. For the two-dimensional panel arrays, the solution for the stress resultants in the cover sheet was attempted by introducing stress functions dependent upon the displacement field of the plate element. This approach, although incomplete, is a natural extension of the one-dimensional analysis for plate elements. The exerimental program which was used to verify the analysis is described and experimental results are compared with the analysis.

Pointwise displacement errors in linear shell theory resulting from errors in the stress-strain relations

Abstract: It is rigorously proved that relative errors of order e in the stress-strain relations of linear shell theory result in relative pointwise errors in the solution displacement field of order e.

A finite element model for shells based on the discrete Kirchhoff hypothesis

Abstract: A ring type finite element model based on the discrete Kirchhoff hypothesis is developed and its feasibility for use in shell analysis is illustrated by determining the linear axisymmetric deformation behaviour of a conical shell loaded by lateral pressure. The displacement field is represented by a linear function of the meridional co-ordinate. The effects of coupling between bending and extension are included and in this respect the analysis is an extension of earlier work by other investigators using this same model, where bending deformations only were considered. The results of the investigation indicate that the proposed model is capable of accurately described coupled bending and extension effects with a relatively small number of elements. In addition to the numerical results obtained several distinct advantages and disadvantages of this element are brought out. In particular, the element stiffness coefficients can be easily derived because of the simplicity of the assumed vector field. However, it appears that the price paid for such a simple vector field approximation is an undue amount of machine storage, even for a relatively small number of elements. Several suggestions are made for minimizing some of these difficulties.

Interaction Energy and Configuration of Ledges on (001) Copper Surfaces

Abstract: A computer model of the surface, consisting of a crystal with (001) terraces and either one or two ledges, has been constructed. Interaction between the atoms of the model has been described by means of a Morse potential. Given this force law and a previously developed zero-force atomic relaxation technique, it has been possible to establish the configuration of an isolated ledge, the attendant displacement field as well as the increase in energy which results upon introduction of a ledge on an otherwise perfect surface.

Radiation of seismic waves from a propagating source near the free surface

Abstract: The radiation of the "break-out phase" is studied. This phase is explained to be generated when a fracture initiated within the earth's interior reaches the earth's surface; when the fracture reaches a free surface an extremely large pulse can be generated. For an explanation of generation of the "break-out phase, " the displacement field when a propagating double-couple source reaches the free surface from depths in a semi-infinite homogeneous elastic medium is calculated numerically. Equations for such calculations are formulated. When the source propagates to the free surface, the effect of the free surface on the seismogram is found to be considerable.

Thick Shell and Oriented Surface Theories

Abstract: Equations of equilibrium and kinematic variables of a thick shell theory based on a power series expansion of the displacement field are compared with corresponding equations and variables of an oriented surface theory. The comparison indicates a possible physical interpretation in the light of thick shell theory for some of the kinematic and static quantities of Cosserat surface theory. Approximations of the energy potentials for the two theories are used to evaluate all coefficients of linear elastic oriented surfaces.