Showing papers on "Displacement field published in 1978"
TL;DR: In this article, the steady flow of a suspension of closely fitting, neutrally buoyant, incompressible and elastic spheres through a circular cylindrical tube is investigated under the assumption that lubrication theory is valid in the fluid region.
Abstract: The steady flow of a suspension of closely fitting, neutrally buoyant, incompressible and elastic spheres through a circular cylindrical tube is investigated under the assumption that lubrication theory is valid in the fluid region. A series solution giving the displacement field of an elastic incompressible sphere under axisymmetrically distributed surface tractions is developed. It is found that, for closely fitting particles, flow properties of the suspension are strongly dependent on the shear modulus of the elastic material and the velocity of the particle.
65 citations
TL;DR: In this article, a Timoshenko beam finite element is presented which has three nodes and two degrees of freedom per node, namely the values of the lateral deflection and the cross-sectional rotation.
59 citations
TL;DR: In this article, the surface displacements due to Rayleigh waves scattered by a circular cylindrical cavity in a half-space are considered. But the authors only consider the displacement field in terms of multipolar potentials.
Abstract: Diffraction of two dimensional elastic waves by a cylindrical cavity in a half‐space is considered in this paper. The paper is divided into two parts. In the first part we use an integral representation for the scattered displacement field to derive its asymptotic behavior at large distances from the cavity. We also give a formal derivation of the representation of the displacement field in terms of multipolar potentials. In the second part we use a method of matched asymptotic expansions to calculate the scattered field. Numerical results are presented for the surface displacements due to Rayleigh waves scattered by a circular cylindrical cavity.
37 citations
TL;DR: In this paper, the displacement field of dissolved hydrogen in Ta can be treated as a linear superposition of single-defect fields even at concentrations as high as c(H/Ta)=0.111.
Abstract: The diffuse scattered X-ray intensity close to several reciprocal lattice points (hh0) was measured at room temperature in single-crystal Ta-H alloys. The displacement field of dissolved hydrogen in Ta can be treated as a linear superposition of single-defect fields even at concentrations as high as c(H/Ta)=0.111. In order to determine the strength and the symmetry of the double force tensor, Pij, describing the displacement field, the authors calculated a static Debye-Waller factor, lowering both the Bragg and the diffuse intensity. After evaluating the static Debye-Waller factor, which is a function of the defect concentration, the displacement field and the scattering vector, Pij= delta ij(3.36+or-0.16) eV was determined. The cubic symmetry and the strength of Pij are in good agreement with results which have been obtained at lower hydrogen concentrations.
34 citations
TL;DR: In this paper, a finite element method for incompressible deformation is formulated from the virtual work equation based on deviatoric quantities, where the incompressibility constraint is imposed on the admissible displacement field by direct elimination of nodal displacements.
34 citations
TL;DR: In this paper, a Fourier transform was used to obtain the displacement field of edge dislocation that is moving in a finite-width strip having clamped boundaries, which can be compared with those from atomistic models.
Abstract: A Fourier-transform technique is presented for obtaining the displacement field of an edge dislocation that is moving in a finite-width strip having clamped boundaries. Known results are reproduced by this technique, and new solutions are obtained, which can be compared with those from atomistic models.
31 citations
TL;DR: In this paper, a Fourier transform was used to obtain the displacement field for edge dislocation, which can be compared with those from atomistic models of edge dislocations.
Abstract: A Fourier‐transform method is used to obtain the displacement field for an edge dislocation. The method reproduces known results and produces new solutions that can be compared with those from atomistic models of edge dislocations.
30 citations
Journal Article•
TL;DR: In this article, the crust is believed to form dominantly by cataclasis at low temperatures and/or effective confining pressure, by pressure solution at intermediate temperatures, and by dislocation creep at high temperatures, each flow mechanism gives rise to distinctive microscopic and small scale structures.
Abstract: Deformation of the crust is believed to occur dominantly by cataclasis at low temperatures and/or effective confining pressure, by pressure solution at intermediate temperatures, and by dislocation creep at high temperatures. Each flow mechanism gives rise to distinctive microscopic and small scale structures.
Brittle deformation with grain fracture leading to a reduction of particle diameter is characteristic of cataclastic flow.
Pressure solution produces grain shape fabrics by intercrystalline diffusion assisted by the presence of water. Grains may change shape at constant mass, or decrease in mass (and therefore in size) by long range diffusion: mass is then not locally conserved. Reduction of grain diameter leads to increased rates of deformation (strain softening).
Distinctive spaced cleavage zones form by pressure solution in which mineral species are redistributed due to different rates of deformation: the displacement field is discontinuous and deformation non-isochemical. Tectonic veins associated with pressure solution structures probably form by local mass transport; thus brittle and ductile mechanical behaviour coexist.
Dislocation creep produces grain shape fabrics by intracrystalline deformation. and may cause grain size reduction by subgrain formation and recrystallization. Preferred crystallogra-phic orientations can arise from dislocation glide. Mass is conserved and deformation is believed to be essentially isochemical. Small scale structures formed by dislocation creep are ductile, with a continuous displacement field.
20 citations
TL;DR: In this paper, the authors explored the use of a modified static solution as an approximation to the displacement field of a penny-shape crack in the long-wavelength regime and then studied numerically how well it connects with the low-freqency limit of the diffraction regime.
Abstract: The elastic‐wave scattering by a flat crack can be represented by an integral expression involving displacement and strain on the surface of the crack. We have explored the use of a modified static solution as an approximation to the displacement field of a penny‐shape crack in the long‐wavelength regime and then studied numerically how well it connects with the low‐freqency limit of the diffraction regime. Comparisons between this and several other existing approximations are made. We conclude that this quasistatic approximation is useful practically in both the long‐wavelength and beginning diffraction regime.
19 citations
TL;DR: In this paper, a pretwisted cantilever plate is treated as a thin shallow shell and its potential and kinetic energies in torsional vibration are determined by assuming an appropriate displacement field.
Abstract: A pretwisted cantilever plate is treated as a thin shallow shell. Its potential and kinetic energies in torsional vibration are determined by assuming an appropriate displacement field. Applying Hamilton’s principle, the problem is reduced to a fourth-order ordinary differential equation with constant coefficients, which is solved to obtain the first four torsional frequencies of vibration. Plates of aspect ratios varying from 1.0 to 8.0 are analyzed with pretwist angles varying from 0 to 90 deg. Results of the present analysis are compared with existing theoretical and experimental results.
16 citations
TL;DR: In this article, the authors describe spontaneous failure as a generalized phase transition in a prestressed continuum, with instability and transition zone growth being driven by the energy contributions from the relaxation of stress in the surrounding medium.
Abstract: Spontaneous failure in a solid medium is described as a localized transition of the material from one physical state to another, characterized in part by contrasting rheological properties and density. Such a process is viewed as a local disordering of the relatively ordered structure of the solid due to any variety of causes, such as massive microfracturing or shear melting, and can be confined to a very thin zone, but nevertheless of finite volume such that a volumetric transition energy can be defined. This leads to the description of failure as a generalized phase transition in a prestressed continuum, with instability and transition zone growth being driven by the energy contributions from the relaxation of stress in the surrounding medium. Direct application of mass, momentum and energy conservation to such a generalized phase transition leads to ‘jump’ conditions specified on the growing boundary surface of the transition zone, that relate the rupture growth to discontinuous changes in the dynamic field variables across the failure zone boundary. These field discontinuities are, in turn, related to the localized changes in physical properties induced by failure. Dynamical conditions for rapid spontaneous failure growth in a stressed medium are investigated in some detail, and we find that the failure boundary growth can be simply expressed in terms of energy ‘failure condition’ and a dynamic growth condition specifying the rupture velocity. These results imply that the integral energy change associated with earthquakes is in the range 10^4−10^6 erg/g. Further the failure growth rate is shown to be expressible in terms of the rheological properties of the material before and after failure. For shear melting resulting in a low viscosity fluid, for example, the rupture velocity will be near the shear velocity of the original material. A general Green's function solution for the radiation due to stress relaxation in the medium surrounding the growing failure zone is given and provides the basis for detailed computations of the strain or displacement field changes due to spontaneous failure processes. In particular, it is shown that the jump conditions for the growing transition zone boundary appear naturally as surface integral terms over the boundary. Since these boundary conditions contain the failure rate explicitly, then these terms include effects that have not been represented in previous integral representations of the radiation field resulting from failure. Further, we show that the formal Green's integral representation for the dynamical wave field can be used with known, simple Green's functions to generate approximate solutions for complex failure processes occurring in media with inhomogeneous material properties and prestress.
TL;DR: In this article, the displacement field around a finite circular edge dislocation loop lying parallel to a stress-free surface has been calculated, and the results indicate that when the loop is located close to the free surface, the displacement fields are extremely asymmetric across the loop plane.
Abstract: The displacement field around a finite circular edge dislocation loop lying parallel to a stress‐free surface has been calculated. The results indicate that when the loop is located close to the free surface the displacement field is extremely asymmetric across the loop plane. This influences the elastic interaction between the loop and point defects. The free‐surface effect is significant for loops located within about five times the loop radius from the free surface.
TL;DR: In this paper, Harkrider noted the presence of noncausal, nonpropagating near-field terms in the cylindrical shear wave potentials for dislocation sources.
Abstract: Harkrider noted the presence of noncausal, nonpropagating near-field terms in the cylindrical shear wave potentials for dislocation sources. He stated that these noncausal terms drop out when the total displacement field is calculated using both SV and SH shear wave potentials. When calculating realistic far-field radial and tangential displacement time histories, one must be careful in how the higher order distance terms are truncated in order to avoid the presence of noncausal terms in the time histories. Examples of this effect for tangential displacements due to a 90° dip-slip dislocation source as well as a 90° vertical strike-slip source are given to demonstrate the numerical problem.
TL;DR: In this article, the bending strain resulting from the resonant modal deformation of vibrating plate-like structures is calculated by computing the second derivative of the displacement series, where the terms of the series consist of the clamped-free or free-free eigenfunctions of a simple beam.
Abstract: A technique for computing the bending strain resulting from the resonant modal deformation of vibrating plate-like structures is described. Interferometric fringes obtained by time-average holography are used as the basis for generating a mathematically continuous series approximation of a plate-like structure's normal displacement. The terms of the series consist of the clamped-free or free-free eigenfunctions of a simple beam. The bending strain is then obtained by computing the second derivative of the displacement series. The coefficients of the terms of the displacement series are computed for a given segment of a cantilevered plate-like structure based upon the holographic-fringe values lying along the same plate segment. A linear least-squares-solution routine is used to solve for the series coefficients, called modal weighting coefficients, in terms of the normal displacement values obtained from the holographic-fringe value. A ‘best fit’ solution is thus obtained for the plate displacement. This least-squares approach in conjunction with the fact the beam-series functions exactly satisfy the plate's geometric boundary conditions and approximately satisfy the plate's natural boundary conditions, results in a displacement series that yields quite accurate displacement and bending strain values.
TL;DR: The atomic structure of a screw dislocation core in aluminum is calculated by the modified lattice-statics method developed in the preceding paper as discussed by the authors, which includes anharmonic as well as harmonic forces and permits relaxation of the atoms in all three dimensions.
Abstract: The atomic structure of a 〈110〉 screw dislocation core in aluminum is calculated by the modified lattice‐statics method developed in the preceding paper. The method includes anharmonic as well as harmonic forces and permits relaxation of the atoms in all three dimensions. All forces used in the present calculations were derived from a first‐principles interatomic pair potential obtained via pseudopotential theory. Several significant differences from the ordinary lattice statics results are noted, including the displacement field, Peierl’s energy barrier, and the equilibrium core‐center location.
TL;DR: In this article, the effect of initial axial stress on the torsional rigidity of straight uniform rods is well-known, and a system of linear equations governing the small deformations of an initially stressed, curved, twisted composite rod is obtained through the use of virtual work.
Abstract: A system of linear equations governing the small deformations of an initially stressed, curved, twisted composite rod is obtained through the use of the principle of virtual work. Apart from the extensions! and bending deformations, transverse shear as well as warping of the rod cross section are incorporated into the assumed displacement field. The resulting equations are applicable for transversely isotropic rods for a wide range of values of the ratio of the shear modulus to Young's modulus. Special cases of the governing equations are also presented for plane-curved symmetric rods. N this paper, we consider the effect of initial stresses on the small deformations of a curved, twisted rod. The rod is assumed to be transversely isotropic with a heterogeneous cross section. The effect of initial axial stress on the torsional rigidity of straight uniform rods is well-known.1'2 An examination of the formula in Ref. 1 for the effective torsional rigidity reveals that the effect of axial stress is particularly significant whenever the shear modulus of the material is small compared to Young's modulus. Corresponding to this range of values of the shear modulus, the shear deformation of the rod may be significant. In the following treatment, we obtain the effective torsional rigidity for space curved rods in the presence of shear deformation as part of our results. The analysis of the initial stress problem is based on the linearized three-dimensional initial stress problem, and the associated variational principles discussed in Ref. 2. In conjunction with the principle of virtual work, we assume a displacement field incorporating transverse shear deformation and warping of the cross section. In the subsequent considerations, approximations are introduced on the basis of the thinness of the rod to simplify the resulting relations. Inertia forces, associated with the small deformations, are included through the use of D'Alembert's principle. A system of equations for the treatment of symmetric, plane-curved rods is obtained by specializing the general relations. These equations readily uncouple into two sets— one describing the in-plane deformations, the other describing the out-of-plane deformations of the rod. Explicit forms of these relations are also stated for the case of negligible transverse shear deformation. An analysis of superposed small displacements on finite deformations of rods has been previously given by Green et al.3 The constitutive relations obtained in Ref. 3, on the basis of thermodynamically consistent strain energy functions, introduce unnecessary complexity in an approximate treatment of practical rod problems, such as the dynamic stability
01 Jan 1978
TL;DR: In this paper, a method to evaluate the reliability and to predict the lifetime of rigid-plastic structures is developed when the dynamic loading is a stochastic function of time alone.
Abstract: A method to evaluate the reliability and to predict the lifetime of rigid-plastic structures is developed when the dynamic loading is a stochastic function of time alone. The method uses the mode approximation assumption relative to the deformed shape of the structure and supposes high reliability of the structure against plastic collapse under static loading. Failure of the structure is understood to be the first crossing of the level of admissible displacements. The random displacement field is determined by numerical simulation limited to the generation of a two-dimensional random variable. The method is illustrated by means of an example of a simply supported plate uniformly subjected to a stochastically varying pressure. The influence of the statistical parameters of stochastic loading on the reliability and on the lifetime of the structure is discussed.
TL;DR: In this paper, a model of the displacement field along the meridian is constructed by means of piecewise cubic Hermite polynomials, and three different weighted residual methods, i.e., subdomain collocation, Galerkin and moments, are used to generate element stiffness matrices and load vectors.
Abstract: The shell thickness and the shell reinforcement both may vary along the meridian of the shell, but they must remain constant around the circumference of the shell. The shell may be subjected to pressure and to thermal effects that may be nonsymmetrically distributed around the shell circumference and vary along the shell meridian. The analysis is based upon thin shell theory that utilizes the Love-Kirchhoff approximation. By means of Fourier expansions in the circumferential coordinate, the partial differential equations of equilibrium of the shell are reduced to ordinary differential equations in the meridional coordinate. For each Fourier harmonic, numerical solutions of the reduced equations are obtained by means of finite element weighted residual methods. A finite element model of the displacement field along the meridian is constructed by means of piecewise cubic Hermite polynomials. Three different weighted residual methods, i.e., subdomain collocation, Galerkin, and moments, are used to generate element stiffness matrices and load vectors.
01 Jan 1978
TL;DR: In this paper, Park and Hellen's virtual crack extension method was used to predict the energy release rates for crack growth in three mutually perpendicular directions, and an example of a circular crack in a half-space was presented.
Abstract: Publisher Summary This chapter discusses virtual crack extension methods for combined tensile and shear loading. The virtual crack extension methods offer several advantages over other methods of applying finite element techniques of linear elastic fracture mechanics (LEFM). The methods of Parks and Hellen are both designed to compute the energy release rate from the displacement field before crack growth predicted by the finite element or other method and the change of stiffness during growth. The method of virtual crack extension can be used to predict the energy release rates for crack growth in three mutually perpendicular directions. The chapter presents an example of a circular crack in a half-space
TL;DR: Two standard integral forms of the cylindrical shear potentials for point quadrupole seismic sources, a frequency domain k-integral and a time domain Cagniardde Hoop path p-Integral, are shown not to be the frequency-time domain inverse of each other.
Abstract: Two standard integral forms of the cylindrical shear potentials for point quadrupole seismic sources, a frequency domain k-integral and a time domain Cagniardde Hoop path p-integral, are shown not to be the frequency-time domain inverse of each other. Their relationship is derived and they are shown to be seismically equivalent in that they yield the same displacement field.
TL;DR: In this paper, the authors show that deformationpotential coupling should lead to a local, crater-type lattice distortion centered at the electron-hole-drop (EHD), and that surface restraints on this displacement field generate attractive forces between the drop and the crystal surface.
TL;DR: In this article, the dispersion relation and displacement field of acoustical or optical surface vibration modes associated with extremal points in the surfaces of constant frequency of the corresponding, infinitely extended crystal were derived.
Abstract: The effective-mass approximation enables one to obtain the energies and wave functions of impurity levels, in a slowly varying perturbing potential, associated with extrema in the electronic band structure of the unperturbed crystal. We have developed an analogous method for obtaining the dispersion relation and displacement field of acoustical or optical surface vibration modes associated with extremal points in the surfaces of constant frequency of the corresponding, infinitely extended crystal. By factoring out the rapidly varying part of the displacement field corresponding to each of the normal modes associated with such an extremal point, a set of partial difference equations is obtained for the slowly varying amplitudes. These equations are converted into a set of coupled partial differential equations which explicitly incorporate the stress-free boundary conditions at the crystal surface. The solution of these equations yields the surface-mode dispersion relation and displacement field. The method is illustrated by applying it to one- and three-dimensional semi-infinite models.
TL;DR: In this paper, a reciprocity theorem is derived and the representation theorem for the displacement field is obtained in terms of the Green's function for the primary configuration in a dynamic state and no assumption is made regarding the smallness of time derivatives of the displacement fields.
Abstract: A dynamical theory, regarding the superposition of small deformations upon large ones, is applied to the study of the prestress effects in an isotropic medium. The primary configuration is in a dynamic state and no assumption is made regarding the smallness of time derivatives of the displacement field. In this framework, a reciprocity theorem is derived and the representation theorem for the displacement field is obtained in terms of the Green's function. Body force equivalents are derived in terms of the discontinuity of the displacement across the fault surface. The ray theory is briefly reviewed and applied to the study of seismic-wave propagation in homogeneous isotropic media. The prestressed configuration in proximity of the fault surface is treated as a perturbation on seismic waves and its effects are found to be of first order in the perturbation at the origin. The wave front equation and the nodal-surface equation for compressional waves are derived and both are found to suffer significant changes due to the perturbation.
01 Dec 1978
TL;DR: In this article, a system of equations is developed which relates the general displacement field from usual and image-plane double exposure holograms of objects illuminated by collimated or point light sources.
Abstract: : A system of equations is developed which relates the general displacement field from usual and image-plane double exposure holograms of objects illuminated by collimated or point light sources. By defining a coordinate system in the direction of the holographic plate and requiring that the object be several plate diameters from the plate, certain simplifications in the system of equations can be made to decouple the displacement components. As a result, the in-plane components can be independently determined from the apparent fringe shift observed from the holographic plate. The out-of-plane component can then be solved from a linear equation using the in-plane components. All three components of displacement can be simultaneously determined from a single film plate, regardless of the proportion of in- to out-of-plane components. The method is validated by comparisons with speckle results and theoretical analyses. (Author)
TL;DR: In this paper, an explicit result for the two-dimensional Green function for an orthotropic body containing a crack along an orthogonal plane and with its tip pointing in an orthographic direction is presented.
TL;DR: In this paper, a new discrete plate bending element with the first order shape function is proposed, and their higher order derivatives are approximated by the finite difference expression which results in substantial simplification of energy integrals.
Abstract: Geometric non-linearlity depends on the first derivative of the displacement field, so that it is very advantageous to use the lower displacement field with respect to cost and computing time.In this paper, a new discrete plate bending element with the first order shape function is proposed, and their higher order derivatives are approximated by the finite difference expression which results in substantial simplification of energy integrals.Furthermore, in this element a number of displacement parameter of one node is 1/3 of a well-known plate bending element and therefore considereable reduction of computing time can be expected.From the results of numerical analysis it can be concluded that this model might be very powerful in non-linear in-plane as well as bending analysis of plates.
01 Jan 1978
TL;DR: The strain energy function developed by Veronda and Westmann was utilized in a finite element computer program to study the stress state and displacement field of an elliptical hole in the skin subjected to a tension field.
Abstract: Publisher Summary This chapter describes the finite element analysis of an elliptical hole in stretched skin. In this study, the strain energy function developed by Veronda and Westmann was utilized in a finite element computer program to study the stress state and displacement field of an elliptical hole in the skin subjected to a tension field. This finite element program is based on finite elasticity theory and has been successfully used by Oden and Key to investigate the deformed shape of an initially circular hole in a square rubbery sheet subjected to finite uniaxial stretching. To test the accuracy of the finite element program, a simple uniaxial tension model was run. The results indicate that the finite element method provides a powerful tool to investigate the stress states surrounding wounds in the skin. The results were based on an isotropic model and can only be considered valid in a relative sense. To increase the utility of the program, anisotropic effects must be included.
TL;DR: In this article, a qualitative assessment of the performance of several holographic interferometry techniques for the formation of full field fringe patterns representative of individual displacement components is made and areas where practical problems exist are indicated and an estimation of the accuracies of the techniques given.
Abstract: A qualitative assessment of the performance of several holographic interferometry techniques for the formation of full field fringe patterns representative of individual displacement components is made. Typical interferograms obtained from surfaces undergoing various types of displacements are shown and discussed. Areas where practical problems exist are indicated and an estimation of the accuracies of the techniques given.
TL;DR: In this paper, the displacement data are converted to a digital form and analyzed on a computer using curve fitting and a finite element model with displacement data as input yields a set of stress contours.
Abstract: This paper presents an experimental method for measuring the displacement fields in extremely soft materials and the subsequent analysis of the data to yield the stress field. The stresses in soft materials are difficult to measure by transducers because the measurement device can easily alter the quantity to be measured. In this study, strains in seat cushions and in the buttock model are indicated by lead markers and recorded on X-ray films. The method is different from previous grid methods in that the strains can be measured in any plane inside a three dimensional model. The lead-ball matrix gives a clear picture of the overall displacement field and areas of high strain. The displacement data are converted to a digital form and analyzed on a computer. Two separate aspects of the computer analysis are discussed in detail in this paper. First, by curve fitting, the hydrostatic pressure gradient is obtained. Second, a finite-element model with the displacement data as input yields a set of stress contours.
TL;DR: In this paper, a new approach to the description of wave processes in real crystals is described; in this approach, the equations of motion of a continuous medium are written as a system of first-order partial differential equations interms of the displacement vector.
Abstract: A new approach to the description of wave processes in real crystals is described; in this approach, the equations of motion of a continuous medium are written as a system of first-order partial differential equations interms of the displacement vector. Some consequences which do not follow directly from the initial equations of motion are analyzed. It is shown, in particular, that the boundary conditions for the displacement field may be obtained directly from the processed equations. The analogy with electromagnetic waves is briefly discussed.