Showing papers on "Displacement field published in 1976"

Moment Tensors and other Phenomenological Descriptions of Seismic Sources—I. Continuous Displacements

Abstract: Summary. In most seismic calculations the real physical stress tensor S is replaced by a linearized isentropic elastic model stress tensor 9. Backus & Mulcahy showed how the stress glut r = 9- S gives rise to all the usual phenomenological descriptions of earthquake sources, the equivalent forces, the moment tensor densities, the stress-free strain and the seismic moment tensors of various types and degrees. In our earlier paper the displacement field s was supposed to be continuously differentiable, so discontinuous faulting could not be discussed. The present work extends our earlier paper to cover the possibility that 9 is a singular generalized function, so that s can be discontinuous or singular. Even when s is smooth, generalized functions simplify the notation. Surface and volume forces, real or equivalent, can be combined in a single distribution; boundary conditions are automatically included in the field equations; and representing a localized source by certain moment tensors of low degree amounts to approximating the source by a point source with the same moments of low degree. It is shown that all the results in our earlier paper can be extended to discontinuous s and singular 9. New results, not obtainable in our earlier paper, include these: a simple description of the most general seismic point source; a proof that every tensor with the obvious appropriate index symmetries is a forcemoment or glut-moment tensor of a seismic point source; a catalogue of point sources with no seismic effects; a new, direct derivation of the classical phenomenological descriptions of ideal fault sources which Burridge & Knopoff, Dahlen and Walton obtained from the reciprocity theorem and the impulse response; a discussion of how, unlike ideal faults, a real fault might have a seismic moment tensor with non-zero trace; and a description of a source type, the simple surface source, which may be a more accurate phenomenological description of real faults than are the classical ideal fault descriptions. It can be shown that a simple surface source with fault surface Z is uniquely determined by the motion it produces if it is developable or if the Gaussian curvature of Z never vanishes, but not if Z contains a piece of a plane.

Singularity methods for elastostatics

Abstract: By distributing the concentrated singularities such as a Kelvin doublet along the axis of symmetry we describe the displacement field, in an elastic medium, for various modes of rotation and translation for a rigid prolate and oblate spheroid. The limiting cases of a sphere, a slender body and a thin circular disk are also discussed. All the solutions are presented in a closed form.

Huang diffuse scattering of X-rays from the displacement field of hydrogen in niobium

Abstract: The symmetry and strength of the displacement field of hydrogen interstitially dissolved in niobium has been determined by means of Huang diffuse scattering of X-rays (Mo K alpha 1). The measurements were performed at room temperature in the alpha phase of NbHx. Detailed measurements of the scattered X-ray intensity close to the (330) reciprocal lattice point yielded the following results: (i) The scattered intensity increased linearly with hydrogen concentration as predicted for isolated point defects; (ii) the displacement field has cubic symmetry, (iii) the double force tensor describing the long ranged displacement field is Pij= delta ij(3.37+or-0.1) eV.

Surface and interfacial waves of arbitrary form in isotropic elastic media

Abstract: A method due to Friedlander of accommodating disturbances of arbitrary form into the theory of surface waves in a semi-infinite isotropic elastic body is extended and shown to yield a simple closed form solution for the displacement field. An analogous treatment of interfacial waves of arbitrary form at a plane contact discontinuity separating different isotropic elastic materials is also given.

Potentials and displacements for two theoretical seismic sources

Abstract: Theoretical, P, SV, and SH displacement potentials and displacements for a double couple or point shear dislocation source and for a ‘mixed quadrupole’ source at any arbitrary orientation in an isotropic homogeneous elastic space are expressed as multiple integral and derivative operations on the source history in the time domain and their algebraic equivalent in the frequency domain. These sources have the same angle orientation functions, which are given explicitly. The double couple and ‘mixed quadrupole’ are both quadrupole sources but, unlike the double couple, the P and S waves from a ‘mixed quadrupole’ have different source histories. Analytic displacements are obtained using as examples the Ohnaka shear dislocation history for a double couple and the Randall and Archambeau tectonic release histories for ‘mixed quadrupole’ sources. The displacement fields are investigated numerically, in order to establish a criterion for estimating the minimum range for applying far-field theory results to the total displacement field. The chosen criterion is the ratio of the far-field peak amplitude, which is a function of source rise or duration time, to the static displacement, which is a near-field phenomenon. The proposed criterion is found to be conservative as to the minimum range for the farfield, predicted (1/R) dependence of the total field peak amplitude, but quite satisfactory for time domain estimates of moment and corner frequency based on far-field theory.

Matrix Analysis of Structure-Foundation Interaction

Abstract: For the unified analysis of skeletal structures and foundations, a one-dimensional finite element for the foundation is formulated. This is achieved by assuming the same displacement field for the foundation as that used for the beam. The foundation is assumed to be of the Winkler type but it offers resistance not only to normal forces but to shear and torsional ones. The new foundation stiffness matrix is easily incorporated into existing structural analysis computer programs. Numerical examples are given. They include single members on or in an elastic medium and general skeletal structures partially or fully buried underground. Accurate and efficient solutions are obtained.

A simple cubic displacement element for plate bending

Abstract: Early attempts to construct a triangular finite element for plate bending problems from a compatible cubic displacement field are not entirely satisfactory. The present paper shows how an accurate plate element can be achieved using independent cubic polynomial assumptions for the internal and boundary displacements in conjunction with a modified potential energy principle. This approach yields a simple algebraic formulation with favourable connection quantities at the element vertices which will appeal to practical users of the conventional finite element displacement method. Moreover, in Appendix I it is shown that the cubic element is identical to a previous hybrid stress element with linear internal bending and twisting moments and cubic boundary displacements. The stresses obtained from the former hybrid finite element solution therefore satisfy the strain compatibility conditions exactly. This remarkable result has an important significance in the theory of hybrid finite elements.

Calculation of ground motion in a three-dimensional model of the 1966 Parkfield earthquake

Abstract: Near-field ground displacements are calculated from an earthquake source in a homogeneous, elastic half-space. An analytical formulation of the problem is presented that requires no physical approximations except at the source. A model of the source is constructed by retaining the essential kinematic character of the faulting process. A computer program is developed to calculate ground motion from an assumed model of the 1966 Parkfield, California earthquake. Favorable agreement is obtained between the theoretically computed ground displacements and those derived from the recorded accelerations. The relative contributions of the body waves and surface waves to the displacement field are examined. The results indicate that a significant portion of near-field motion may consist of surface waves, especially in the vertical component of the ground motion.

Spectral representation of the Love wave operator

Abstract: Summary This paper is concerned with the spectral representation of the two-dimensional Love wave operator, associated with the propagation of monochromatic SH waves in a laterally uniform layered strip or half-space. A method is presented for obtaining a complete set of proper or improper eigenfunctions belonging to the operator, in terms of which the displacement field may be represented generally. The method is illustrated by means of two-layer models of an infinite strip, overlying another infinite strip or a half-space, with constant rigidity and density within each layer.

Camera with digital control device

Abstract: The present invention relates to a camera with a displacement means for controlling the exposure amount as the function of the mechanical displacement position. The displacement means are so designed that a brush is slid over a comb-shaped conductive pattern mechanically linked with the displacement means so as to convert the displacement position of the displacement means into a pulse number. Further, when the pulse number reaches a certain determined value, the displacement means is stopped in such a manner that the pulse number corresponds with the exposure amount and characterized in that, by determining the position at which each conductor of the comb-shaped conductive pattern is provided, relative to the position at which the displacement means stops in accordance with the displacement speed of the displacement means, the displacement amount of the displacement means due to the time delay since the pulse number reaches a certain determined amount till the displacement means actually stops is compensated so that the pulse number always corresponds with the displacement position in a precise way.

Finite element analysis of automotive structures under crash loadings. volume ii. technical report

Abstract: A research project was undertaken to develop a finite element computer program for use in the dynamic analysis of vehicle structures, including sheet metal, in a crash environment. This research program involved the following major tasks: A technique was developed for the finite element analysis of the dynamic response of plate beam structures involving very large displacements and rotations and elastic-plastic material behavior. The principal feature of this technique involves the decomposition of the element displacement field into rigid body components and deformation components thus allowing the use of a small deflection formulation in the analysis.

Some inconsistencies of the finite element method as applied to inelastic response

Abstract: The inadequacy of a two noded beam-column element with a linear axial and a cubic transverse displacement field for inelastic analysis is demonstrated For complete equilibrium satisfaction in the linear elastic range a three noded beam-column element is shown to be consistent Next, the sensitivity of the inelastic response to numerical solutions of the inelastic response of a cantilever beam resulting from approximate integration of strain energy are brought out and finally, consequences of this on the nonlinear transient response of structures are considered

Dynamics and identification of flexible aircraft

Abstract: The equations of motion and a maximum likelihood parameter identification formulation are developed for a flexible aircraft. The various levels of approximation associated with the modal substitution representation of the elastic displacement field are discussed and illustrated when appropriate. The necessary extension of the parameter set of stability and control derivatives due to the aeroelastic effects is obtained.

Variational theorems for superimposed motions in elasticity, with application to beams

Abstract: Variational theorems are presented for a theory of small motions superimposed on large static deformations and governing equations for prestressed beams on the basis of 3-D theory of elastodynamics. First, the principle of virtual work is modified through Friedrichs's transformation so as to describe the initial stress problem of elastodynamics. Next, the modified principle together with a chosen displacement field is used to derive a set of 1-D macroscopic governing equations of prestressed beams. The resulting equations describe all the types of superimposed motions in elastic beams, and they include all the effects of transverse shear and normal strains, and the rotatory inertia. The instability of the governing equations is discussed briefly.

High Frequency Cylindrical and Spherical Elastic Waves in a Heterogenous Half-Space

Abstract: A high-frequency asymptotic representation of the displacement field in an inhomogeneous elastic half-space is secured for periodic normal line or point loading. It is assumed that the constitutive moduli of the medium depend solely, and analytically, on distance from the solid’s plane boundary. Our technique involves a transformation of the linearized elastodynamic equations in order to recast these as a first order system of ordinary differential equations, the utilization of an asymptotic theory of Birkhoff and inversion by means of the method of stationary phase and the geometrical theory of diffraction. If the solid is homogeneous, the leading term of our asymptotic expansion in inverse powers of frequency is identically equal to the known displacement vector. In a half-space whose wavespeeds decrease monotonically, the formation of shadow zones and the functional dependence of the field on the material parameters are discussed.

Analysis of the Displacement Field for Localized Disturbances in a Stratified Fluid with Shear

Abstract: : An initial-value problem is studied for two-dimensional density perturbations of finite amplitude in an incompressible, viscous, stratified fluid. A background shear flow is permitted. Certain characteristics of the net displacement field are demonstrated to be directly computable from the initial data. An inverse problem is studied, and it is shown how characteristics of the initial density perturbation can be deduced from knowledge of the initial and final shapes of the region containing the initial density perturbation.

Green's function for an infinite pre-stressed elastic medium

Abstract: The first terms of the asymptotic expansion of the time-harmonic Green's function for an infinite pre-stressed elastic medium have been obtained. The Green's function represents the vector displacement field generated by a time-harmonic source of finite extent. The complete displacement field due to the source is represented in terms of Fourier integrals that are evaluated asymptotically and yield explicit expressions for the displacement at points far from the source. The major feature of the asymptotic displacement field is the directional dependence or the geometric decay of the displacement amplitudes.