Showing papers on "Displacement field published in 1981"

Finite element analysis of deformation of strain‐softening materials

Abstract: A finite element analysis of strain-softening materials is presented in which the shear band of prescribed thickness is assumed to exist within elements where maximal stress intensity is reached. The incremental stiffness matrix of the element is derived including shear band deformation. Examples presented in the Paper demonstrate that the load-displacement curve and the displacement field are not sensitive to the mesh size used in the solution.

Global invertibility of Sobolev functions and the interpenetration of matter

Abstract: A global inverse function theorem is established for mappings u: Ω → ℝn, Ω ⊂ ℝn bounded and open, belonging to the Sobolev space W1.p(Ω), p > n. The theorem is applied to the pure displacement boundary value problem of nonlinear elastostatics, the conclusion being that there is no interpenetration of matter for the energy-minimizing displacement field.

Growth as A Finite Displacement Field

Abstract: The growth of an animal or plant may be described by a time-dependent displacement field. When cells proliferate throughout a tissue, growth may be regarded as a spatial distribution of mass sources. To allow for anisotropic growth, a growth rate tensor may be defined which is similar to a strain rate tensor. Some tissues, notably bones, horns, and seashells, grow by accretion on certain surfaces which move as new tissue is produced. Such growth is modeled by a surface distribution of mass sources. At each point of the surface, a vector is assigned whose magnitude and direction specify the rate and direction of growth. Growth can lead to internal stresses that may be relieved by stress relaxation or additional growth. The vocabulary of distributed dislocations may be adapted to describe growth which produces internal stresses. A psychological aspect of growth concerns the identification of invariant aspects of a growth displacement field which are recognized as maturation of form.

The determination of displacement fields from geodetic data along a strike slip fault

Abstract: Observations of angles or distances between stations of a geodetic network are commonly used to infer information about the movement of the surface of the earth. The absence of any observations external to the network leads to an ambiguous displacement field. Existing techniques for eliminating this ambiguity are all unsatisfactory in some respect. The best technique, an ‘inner coordinate’ solution, is not appropriate for networks located in a strike slip fault environment. The inner coordinate solution zeros the rotation of all stations about their center of mass. Along a strike slip fault like the San Andreas, however, motion normal to the fault is less likely than motion parallel to the fault. The solution presented here, an ‘outer coordinate’ solution, finds the rotation of the network that minimizes the components of displacement normal to the fault. Since motion along a strike slip fault is generally expected to be parallel to the fault, the displacements obtained with the outer coordinate solution are more reasonable than those obtained with other techniques. Examination of a trilateration network near San Francisco Bay demonstrates the large effect that the choice of adjustment technique can have on the inferred relative motion of the two sides of the fault. The inner coordinate solution gave a rate of about 1 mm/yr, whereas the preferred outer coordinate solution rate was 36 mm/yr.

Rotational modes in a slowly and uniformly rotating star

Abstract: A perturbation method is derived forr-modes in a slowly and uniformly rotating star. In contrast to previous studies, the perturbation of the gravitational potential is included in the perturbation method. On the assumption that the effects of the centrifugal force are taken into account in the equilibrium model up to the second order in the angular velocity, an eigenvalue problem of sixth-order in the radial coordinate is derived that allows one to determine the zeroth-order toroidal displacement field and the third-order term in the expansion of the eigenfrequency. Furthermore, another eigenvalue problem is derived that governs the first-order toroidal displacement field and the fourth-order term in the expansion of the eigenfrequency. This second eigenvalue problem is also of the sixth-order in the radial coordinate. It is shown that the third-order term in the expansion of the eigenfrequency is real, and that the fourth-order term is zero.

Locking of thin plate/shell elements

Abstract: Often, finite element solutions of thin plate/shell elements become very stiff and the displacement field solutions diverge from those predicted by Kirchhoff's theory. This phenomenon is known as the locking phenomenon. A theoretical fomulation demonstrating its existence is developed, and results of finite element analysis of a single element and mesh are discussed. This leads to a sufficient and necessary criterion which must be satisfied to avoid the locking phenomenon.

Strain field around a tunnel in stiff soil

Abstract: An experimental tunnel excavated through a stiff silty clay (overconsolidated glacial till) has been extensively instrumented. The instruments provided data on a complete three-dimensional displacement field around the tunnel. From in-situ measured displacements, displacement gradients and thus strains have been calculated. The derived maximum shear strain field has been compared with stress-strain testing data for the till and used as a failure criterion to investigate the process of mobilization of shear strength and development of failure zones around the tunnel. The deduced mechanism of mobilization of shear strength provides an insight into the arching process around the tunnel and explains the low pressures observed on lining of tunnels in till.

Representation of Seismic Sources

Abstract: On 26 November, 1849 G. G. Stokes, then Fellow of Pembroke College and Lucasian Professor of Mathematics at the University of Cambridge, read his paper on the dynamical theory of diffraction. As a model for a light source in the luminiferous ether, he chose a tangential force in an infinite elastic solid. Drawing on earlier results of Poisson (see Bibliography, Chapter 1) he obtained an exact solution for the displacement field caused by a single force in an infinite elastic medium. Without knowing it, he had conceived the first mathematical model of an earthquake.

Two-Dimensional Approximations of Three-Dimensional Models in Nonlinear Plate Theory

Abstract: The asymptotic expansion method, with the thickness as the parameter, is applied to the nonlinear, three-dimensional, equations for the equilibrium of elastic plates under suitable loads and appropriate boundary conditions. It is shown that the leading term of the expansion is a solution of a system of equations equivalent to a well-known two-dimensional nonlinear plate model, namely the von Karman equations. The existence of solutions of the two-dimensional problem is established in all cases (by contrast with the three-dimensional model, where no satisfactory existence theory is as yet available). It is also shown that the displacement and the stress corresponding to the leading term of the expansion have the specific form generally assumed a priori in the usual derivations of two-dimensional plate models. In particular, the displacement field is of Kirchhoff-Love type. This approach clarifies in particular the nature of the admissible three-dimensional boundary conditions for a given two-dimensional plate model. A discussion is also given regarding the class of admissible three-dimensional models.

The displacement field of a dislocation distribution

Abstract: Burgers' formula gives the displacement field due to a discrete dislocation line of arbitrary shape in an isotropic, linearly-elastic, infinitely extended, homogeneous body. This paper derives the analogous expression for a continuous distribution of dislocations in an anisotropic body. Special cases give the displacement due to a discrete dislocation line in anisotropic elasticity and due to a continuous dislocation distribution in isotropic elasticity. These expressions all represent generalizations of Burgers' original formula.

Elastic‐plastic analysis of plates by the hybrid‐stress model and initial‐stress approach

Abstract: Two alternative hybrid-stress-based functionals are examined for the incremental elastic-plastic static analysis of single layer plates. Material nonlinear effects are incorporated via the initial-stress approach so that an equivalent nodal force vector is defined and the stiffness remains constant throughout the incremental loading. The alternative functionals differ in the incremental stress which is assumed to satisfy equilibrium; in the first, it is the actual stress increment, and in the second it is the elastic stress increment. Results are presented for two example problems, and comparisons of the alternative functionals and plausible iteration schemes are given. The effects of variation of pertinent solution parameters are also shown. A 4-node hybrid-stress plate element based on a Mindlin-type displacement field is used for most cases; however, limited results are also presented using an 8-node plate element, thus permitting comparisons of the relative efficiencies of the two elements.

Some considerations of crazing of polymers: Two‐dimensional quasifracture of polymers

Abstract: In a tensile stress field, certain polymeric solids deform and develop quasifracture formations Each such formation is termed a craze Previous analyses by Knight and Lauterwasser and Kramer have treated a craze by introducing either an assumed or a measured displacement field for the calculation of the stress distribution around the craze profile In this paper the new phase of oriented polymer molecular bundles within the craze profile has been considered through the use of a linear modulus function for those bundles Both displacement and stress fields have been calculated This model developed for studying planar crack has its limitations when it is used for craze investigations The basic model treatment depends upon classical linear fracture mechanics which is not valid for materials with a zone of craze occurrence In addition, anisotropic and nonlinear as well as time dependent considerations are difficult to be incorporated into the model analysis

Black-white contrast on transmission electron microscope images of small dislocation loops in elastically anisotropic crystals

Abstract: The black-white contrast of small dislocation loops in elastically anisotropic crystals is calculated using the first-order perturbation solution for the dynamical two-beam diffraction condition. It is shown that under realistic conditions the contrast is given essentially by the projection of the displacement field of the dislocation loop onto the image plane. From the Fourier representation [gtilde](k) of the Green's function of the displacement field of point forces the projected displacement field of the loop is obtained by two-dimensional inversion of the Fourier transformation. The final result is applicable for arbitrary directions of the incident electron beam and implies as the critical step the determination of the six roots of the denominator of [gtilde](k) in the image plane. Examples referring to crystals of cubic symmetry demonstrate that the orientation and shape of the black-white contrast figure becomes more and more insensitive to the actual directions of the loop normal n and t...

On elastic wave calculations in a sphere using moment tensor sources

Abstract: Summary. Propagator matrix solutions to the elastic equations of motion in spherically symmetric, inhomogeneous media with moment tensor sources are recast into a simple and intuitively satisfying form which is applicable to both exact and approximate calculations. The transformed expression benefits from the analogous equations of normal mode excitation, while clearly distinguishing the finer partitions of the displacement field and the more flexible boundary conditions that body wave formulations protide. I believe that this new representation, because of its many advantages, should be favoured as the foundation for elastic wave calculations in a sphere.

Geometric Elastodynamics: Rays and Generalized Rays

Abstract: The separation of dependent and independent variables is of fundamental importance in the analytic determination of the fields associated with the partial differential equations of mathematical physics. In particular, the determination of the displacement field at any point of a heterogeneous elastic medium is greatly simplified if the vector wave equation of elasticity can be split into three equations; corresponding to the P, SV, and SH waves of seismology. Although the SH motion can be separated from the P and SV motions for all radially heterogeneous isotropic media, the P and SV motions, in general, can at most be reduced to a system of two coupled scalar equations. The vector wave equation is called decoupled if these two scalar equations can be transformed into two uncoupled equations, one corresponding to P waves and the other to SV waves.

Propagation of Displacement Pulses in Harmonic Crystals

Abstract: The time evolution of the displacement field due to a localized initial pulse is calculated for a harmonic crystal. It is analyzed in detail for some simple phonon dispersion relations and for propagation in symmetric directions. The front of the pulse usually travels with sound velocity, except the case of a concave dispersion relation – the velocity is then different (higher). A “thought” experiment in which the dispersion relation may be determined from the analysis of the time profile of the arriving signal is considered. [Russian Text Ignored].