Showing papers on "Displacement field published in 1985"

Anomalous polarization of elastic waves in transversely isotropic media

Abstract: The behavior of transversely isotropic elastic media is analyzed from both the kinematic (slowness surface) and dynamic (particle displacement) point of view. The relations for the slowness surfaces and wave front surfaces are derived in polar coordinates. Examination of the eigenvectors of the displacement equations of motion gives the relation for the polarization of the displacement vector associated with any plane wave. It is shown that the polarization of plane quasi‐P and quasi‐SV waves depends strongly on the sign of a particular elastic modulus, call it A, whereas the shape of the slowness surface is independent of the sign of A. When A is positive, which is the usual case, the particle displacement vector rotates in the same sense as the slowness vector. When A is negative, which is the “anomalous” case, the sense of rotation of the particle displacement vector is opposite to that of the slowness vector. Thus there is a direction in the medium for which the displacement vector associated with the...

Through‐the‐thickness stress predictions for laminated plates of advanced composite materials

Abstract: A finite element formulation is developed with emphasis primarily focused on providing stress predictions for thin to moderately thick plate (shell) type structures. Plate element behaviour is specified by prescribing independently the neutral surface displacements and rotations, thus relaxing the Kirchhoff hypothesis. Numerical efficiency is achieved due to the simplicity of the element formulation, i.e. the approach yields a displacement dependent multi-layer model. In-plane layer stresses are determined via the constitutive equations, while the transverse shear and short-transverse normal stresses are determined via the equilibrium equations. Accurate transverse stress variations are obtained by appropriately selecting the displacement field for the element. A selective reduced integration technique is utilized in computing element stiffness matrices. Static and spectral (eigenvalue) tests are performed to demonstrate the element modelling capability.

Determining K and related stress-field parameters from displacement fields

Abstract: A general algorithm is presented that computes the stress-field parameters for opening-mode crack problems in a least-squares sense from full-field moire or speckle-displacement fringe patterns. The algorithm can be used in the presence of rigid-body rotation and does not require absolute fringe numbering. Extensive numerical experiments were conducted with the algorithm to determine the sensitivity of the method to experimental errors. Small random position errors in locating the fringe maxima were found to have a negligible influence on the stress-intensity-factor calculation when the number of data points was about ten times greater than the number of unknown stress-field parameters. It was also found that systematic position errors due to an incorrectly specified crack-tip location could be minimized by assuming various crack-tip locations in the vicinity of the actual crack tip and selecting the best fit results. Bothu andv fields were found to be equally suitable for determination of the stress-intensity factor.

On topology optimization of trusses

Abstract: The selection of a minimum weight truss, out of a large set of candidate trusses, is treated. An initial configuration is generated by connecting all the n nodal points with n(n − 1)\2 truss members. A feasible displacement field, satisfying displacement and stress constraints, is obtained from a static analysis of this truss, followed by a uniform scaling of all truss dimensions. The finite element formulation for the initial configuration is then reformulated to yield a linear programming problem. The solution to this leads to a new configuration which is further optimized by solving a small nonlinear programming problem. With the method proposed, trusses subject to one loading condition and subject to stress and displacement constraints can be selected and optimized using a modest computational effort. Three examples are given to demonstrate the usefulness of the proposed method.

Nonlinear finite element analysis of thick composite plates using cubic spline functions

Abstract: A nonlinear, thick, composite plate element is developed in which the usual Kirchhoff hypothesis of plane sections remaining plane and undeformed after loading is abandoned The displacement field is characterized by the sum of displacements with respect to a reference surface and displacements through the thickness The through-the-thickness deformations are modeled by imposing a cubic spline function and allowing the rotations at interlaminar boundaries to be degrees of freedom in the element The theory is developed by considering the Lagrangian strains in conjunction with the second Piola-Kirchhoff stress This formulation leads to a quasi-threedimensional element that encompasses large displacements with moderately large rotations but is restricted to small strains Comparisons of linear and nonlinear thick orthotropic plate solutions with those of previously published analytical and numerical results show the validity of the method

The displacement field of a triangular dislocation loop

Abstract: A formula is developed, in coordinate-free form, for the displacement field of a triangular dislocation loop in an infinite isotropic linear elastic solid. For purposes of numerical calculations, the present prescription may be more useful than one given previously by Hirth and Lothe (1982), since no special care need be taken in dealing with the branches of inverse trigonometric functions which appear in the ‘solid angle’ contribution to the displacement field. The method developed for computing the solid angle is also valid for a triangular loop in an an isotropic elastic solid.

On the effect of the local overall interaction on the postbuckling of uniformly compressed channels

Abstract: On the basis of the general theory of elastic stability due to Koiter, postbuckling analysis of simply supported channels under uniform compression is performed. Attention is essentially focused on local/eularian and local/flexural-torsional simultaneous bucklin modes interaction. The column is treated as a plate assemblage. Linearized expressions for the displacement field are employed while assuming strain-displacements relationships that are linear for the curvatures and up to second order terms for the in-plane strains. The total potential energy is hence written up to third order terms in order to investigate asymmetric buckling phenomena. A discrete model is developed through an automatic procedure of algebraic manipulation, and an extensive parametric analysis is performed. After determining the range of geometric parameters which characterize different types of interaction, it is found that in the postbuckling range the local/eulerian interaction is more dangerous than the local/flexural-torsional one, due to the column higher imperfection sensitivity.

Solitons and electroacoustic interactions in ferroelectric crystals. II. Interactions of solitons and radiations.

Abstract: Multiple-soliton solutions representative of the motion and interaction of walls in ferroelectric crystals of the same type as ${\mathrm{NaNO}}_{2}$ are studied, both analytically and numerically, on the basis of a set of coupled nonlinear electroacoustic equations deduced in paper I [Phys. Rev. B 30, 5036 (1984)] of the present series. Electromechanical couplings are duly taken into account and, in fact, are responsible for the nonlinear coupling between a d'Alembert wave equation for the transverse elastic displacement and a sine-Gordon equation that governs the orientation of the dipole-carrying molecular group in each lattice cell. To solve this rather complex problem, a singular perturbation technique associated with the methods used for solving nonlinear wave equations (Baaumlcklund transformations, etc.) is developed at the first order in the coupling parameter. This allows one to exhibit the first-order corrective radiative terms which superimpose on the nonlinear zeroth-order solution (multiple soliton), the free parameters of which are modulated in order to satisfy the secularity condition at the first order. Simultaneously, a numerical solution using a Lax-Wendroff finite-difference scheme is obtained which illustrates the analytical considerations and models the soliton-antisoliton collision, the soliton-soliton collision, the ``oscillatory soliton,'' and the one-soliton solution (already exhibited in paper I on the basis of a double sine-Gordon equation) along with the accompanying elastic displacement field and the radiative contributions.

The elastodynamic near field

Abstract: Summary. The elastodynamic fields of point forces and shear dislocations of finite source duration are analysed with the aim of establishing the frequency and time-domain characteristics of the field in the near-source region. Criteria are obtained for amplitude dominance in regions where the source -sensor distance is much smaller than the wavelength. It is shown that in the frequency domain, the Green’s tensor (and hence the displacement field of a single point force) attenuates like R-’ in the nearsource region and there exists no region in which the ‘near-field’ term becomes dominant such that the ‘far-field’ term can be neglected. Hence, there is no real ‘near-field’ term for the elastodynamic Green’s tensor. The near-field terms of the displacements, velocities and accelerations excited by a shear dislocation attenuate like R-’, since the R-3 and R-4 terms tend to be eliminated due to mutual cancellation of P and S motions in the near-source region. In the time domain, the corresponding near field of the displacement field is defined for the steady amplitude interval (away from transients) RIP < t < R/a + T by the condition R < PT where is the shear velocity and T is the source’s duration. The relative strengths of all other arrivals will depend on the particular time window under consideration. The particle motion patterns due to a single force in the near-source region are shown to be similar to rotating hyperbolas with an axis along the force direction, which are quite different from the ‘smoke ring’ motion patterns of the so-called ‘near-field’ term itself.

Calculation of approximate weight functions in fracture mechanics by FEM

Abstract: A method is presented for the calculation of weight functions used in fracture mechanics to determine stress intensity factors of cracks loaded by stress gradients. The reference solution for the stress intensity factor and for the reference crack opening displacement field is computed numerically by use of finite elements. The accuracy of the method is checked by comparison with well-known solutions from the literature.

Material Forces and Configurational Forces in the Interaction of Elastic Singularities

Abstract: It is generally held that the forces acting on and between elastic defects are ‘configurational forces’ analogous to, but entirely different in kind from, the Newtonian mechanical forces that may act on the material in which the elastic defects are present. However, in the process of Cottrell locking, solute atoms migrate towards a dislocation under the influence of the interaction between the dislocation and each atom. This transport of mass must be the result of a mechanical force, and the rate of migration is correctly predicted by equating this mechanical force to the calculated configurational force. In the diffusion of a substitutional solute, the detailed process is not simply the motion of the solute atom, but includes the back flow of the solvent atom that replaces it. In a continuum model, one will expect to find displacements corresponding to a quadrupole of forces, a central force acting on the substitutional atom, flanked by opposite forces producing the back flow of the matrix. If real mechanical forces of the kind suggested actually exist, they are proportional to the product of the strength of the dislocation and of the centre of dilatation that represents the solute atom. Both the dislocation and the centre of dilatation may be represented by prescribed displacements of surfaces in the elastic medium. The elastic field in linear elasticity is the sum of the fields corresponding to these two displacements, and cannot contain product terms. Such product terms could appear in second-order elastic theory. To investigate this possibility, we apply second-order elastic theory in the plane-strain formalism of Green and Adkins to the case of an edge dislocation interacting with a parallel dilated cylinder with its axis lying in the inserted half-plane. The medium is assumed to be incompressible. Only the interaction terms between the two defects are considered. The displacement field is analysed in the neighbourhood of the boundary between the dilated cylinder and the matrix. The dominant term in the displacement is found to coincide with the dominant term in the displacement that would be produced by a mechanical force per unit length equal to the configurational force, which is conventionally calculated. There are additional terms corresponding to the quadrupole of forces that produces the back flow.

A study of dynamic near-crack-tip fracture parameters by digital image analysis

Abstract: The feasibility of quantifying fracture parameters for cracked body problems under impact loading through the use of high speed photographic techniques coupled with computer based digital image processing methods is discussed. The procedure for gathering data and the method of analysis which is applied to the experimentally determined displacement field in the crack tip region is presented.

The trapping of hydrogen at nitrogen in niobium investigated by diffuse X-ray scattering

Abstract: The diffuse X-ray scattering from the displacement field around N and H in single crystals of Nb was studied at various temperatures. The observed increase of the scattering intensity with decreasing temperature can be explained by the trapping of H by N. A trap model can describe the observed changes. The displacement fields of H and N do not add linearly. On the assumption that the force dipole tensor P=(ABB) describing the displacement fields does not change for N, the trace (A+2B) of the tensor for H is lowered by 10-20% and a tetragonality mod A-B mod /(A+2B)=0.08+or-0.02 is observed for the trapped state.

Implementation of Local Effects into Conventional and Non Conventional Finite Element Formulations

Abstract: The first part of this contribution is concerned with the problem of including load dependent stress singularities (such as e.g. those due to concentrated loads) into conventional assumed displacement elements. It is shown that such singularities may be conveniently handled in the form of appropriate initial strain and stresses, each of which may be accounted for following standard finite element technology. A numerical example illustrates the efficiency of the method which does not require any mesh refinement in the vicinity of the singularities. The second part of this contribution presents the concept of the so-called large finite elements (LFEs). This concept, which may be viewed as a finite element form of the Trefftz's method (ref.l), is based on using parametric displacement fields satisfying, a priori, the governing differential problem equations. Any local solution representing a stress singularity or stress concentration may be used as the LFE expansion basis. Boundary conditions and interelernent continuity are implicitly imposed by making use of a simple stationary principle and an auxiliary compatible interelernent displacement field. The excellent efficiency of the approach is demonstrated by several examples.

A New Numerical Technique for Computing Surface Elastic Deformation Caused by a Given Normal Pressure Distribution

Abstract: This paper presents a new numerical method for computing the elastic normal surface displacement field caused by a given normal pressure distribution. The pressure function is approximated by a piecewise biquadratic polynomial on the whole domain analyzed, and the deformation of every node is expressed as a linear combination of the nodal pressures whose coefficients can be combined into a deformation matrix. Consequently, the iterative calculation of elastic deformation is simplified and the amount of work is greatly reduced. It has been proved, in addition, that the numerical accuracy of the new method is higher than that of some others.

Surface analysis of an actively controlled telescope primary mirror under static loads.

Abstract: The performance of an actively controlled mirror with an aspect ratio of 63:1 is modeled by a large scale software system. The model allows generalized static disturbances to be applied to the mirror, generates a corrective force field, and tests the result through a finite element simulation. For convenience in input, disturbances may be in the form of a force field or a displacement field. Use of a displacement field is easier, for example, if one wishes to distort the mirror with a particular aberration term for which the disturbing forces are unknown. The user must have established a particular actuator set to be used in counteracting the disturbance, whereupon the actuator forces for this set are determined by a modal truncation method. These computed actuator forces are then reapplied to the mirror model along with the disturbance to evaluate the corrected behavior of the model. Comparisons are made of the rms surface errors and energy concentrations as predicted by modal truncation and computed via the finite element method. Numerical examples, including some bordering on the pathological, are employed to show that the corrected mirror performance under the action of computed actuator forces and external disturbances agrees generally with the mirror surface as predicted by modal truncation.

Geometrically nonlinear theory of naturally curved and twisted rods with finite rotations

Abstract: The object of this paper is to develop a finite displacement theory of naturally curved and twisted rods undergoing finite rotations. Particular attention is paid to investigate the coupling of finite rotations in space under the Bernoulli-Euler hypothesis. A finite rotation vector is employed to derive the displacement field available for finite rotations. A new variable is introduced as a fourth parameter associated with rotations of cross sections. Then the twist and curvatures after the deformation are expressed in terms of four parameters without using small-strain assumptions. The equilibrium equations and the associated boundary conditions, in which second order terms with respect to displacement components are fully taken into account, are derived from the principle of virtual work. The accuracy of the present equilibrium equations are confirmed through comparisons with those obtained by the equilibrium method.

Elastostatics of Structures with Unilateral Conditions on Stress and Displacement Fields

Abstract: A general analysis of the elastostatic problem for structures with unilateral conditions on the stress distributions and on the displacement fields is developed.

An analytic solution for interlaminar stresses in a fiber- reinforced double-layer cylindrical shell

Abstract: The present paper investigates the basic characteristics of the interlaminar stresses in a double-layer cylindrical shell with both ends simply-supported under uniform external or internal pressure. The double layer shell is composed of a 0° fiber-reinforced composite layer and an isotropic layer. In this paper, this axisymmetric problem is solved exactly with the three-dimensional theory of elasticity. Both the displacement field and the stress field of each layer of the shell are expressed in Fourier series and Fourier-Bessel series. Then we illustrate the effects of the various parameters, such as geometry, material constants, loading conditions and stacking sequence, on the interlaminar stresses.