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Showing papers on "Displacement field published in 1970"


Journal ArticleDOI
TL;DR: In this paper, the problem of determining the mean (i.e., statistical average) field quantities in a statistical sample of heterogeneous linearly elastic solids is discussed, and a formulation is obtained that determines {ui(x)}, where ui denotes the displacement field and the braces indicate an ensemble average.

123 citations


DOI
01 Jan 1970
TL;DR: In this article, an original software for the displacement field measurement is presented, which uses the digital picture correlation principle, and the accuracy of the measure could reach 1/60 th of a pixel and, it is possible to measure strains between 5.10-5 to 0.8.
Abstract: Despite the powerful of current calculus codes by finite element or analytical, their reliability and validity must be justified by experimental tests. An original software for the displacement field measurement is presented in this paper. This software use the digital picture correlation principle. Due to this technique, the accuracy of the measure could reach 1/60 th of a pixel and, it is possible to measure strains between 5.10-5 to 0.8. Initially, this software was developed for measuring strains on a sheet testing in metal forming. But, it is revealed that this application field is more important: biomechanical field, geotechnical field, metal characterization, control tests This software could be used, nowadays, for all application which need the knowledge of displacement and strain field for a plane surface. Of course, the experimental conditions must be made in such a way that any deterioration of the random aspect, deposed on the piece, will occur. Lost, alteration or little modification of the speckle aspect will not guarantee the success of the measure. Finally, the purpose of this paper will be, first, to give clearly the mathematical principle of the displacement field determination by a correlation method. Then, various experimental tests using the correlation method, are presented. The powerful of this technique is show by their variety. This method of measure without contact can be an another tool for validating, in a short time, a numerical or analytical result.

57 citations


Journal ArticleDOI
TL;DR: In this article, the displacement field, stress field, and energy of a hexagonal dislocation of arbitrary orientation and arbitrary Burgers vector were derived analytically for hexagonal materials.

34 citations


Book ChapterDOI
01 Jan 1970
TL;DR: In this article, it is shown that only large earthquakes contribute significantly to the displacement fields at large distances from a fault zone, and that these earthquakes are very inefficient in relieving the regional strain indicated by displacement of triangulation stations.
Abstract: The standard model for the calculation of the displacement fields of earthquakes is briefly reviewed. The source dimensions of a set of 27 strike-slip faults are used to obtain empirical relationships between fault dimensions and earthquake magnitude. It is shown that only large earthquakes contribute significantly to the displacement fields at large distances from a fault zone. The mechanics of fault slippage, and some of the problems of the elastic rebound theory are discussed. It is shown that earthquakes on some portions of the San Andreas system appear to account for only a small part of the total movement on the zone indicated by geodetic measurements. Furthermore, these earthquakes are very inefficient in relieving the regional strain indicated by displacement of triangulation stations. It is concluded that aseismic creep in the upper portions of the crust must be an important contribution to the fault slippage in some regions. Also, the regional strain can only be relieved by movements in the lower crust and upper mantle. It is suggested that these movements almost certainly precede surface breakage. They could be detected by geodetic observations at 30-50 km from the fault, and might give valuable premonitory indications of destructive earthquakes.

17 citations


Journal ArticleDOI
TL;DR: In this paper, the stiffness equation for curved elements of orthotropic axi-symmetric thin shells is derived, and equivalent applied loads are found for shells subjected to initial strains, applied surface loads and body forces.
Abstract: The stiffness equation is derived for curved elements of orthotropic axi-symmetric thin shells, and equivalent applied loads are found for shells subjected to initial strains, applied surface loads and body forces. The Lure approximation of thin shells and displacement field approximation by polynomials of arbitrary degree are included in the formulae derived.

17 citations


Journal ArticleDOI
TL;DR: In this paper, an exact solution for the thermal displacements and stresses in a finite, right circular cylinder of homogeneous, isotropic, elastric material having a known radial temperature distribution is given.

16 citations


Journal ArticleDOI
TL;DR: In this article, the propagation of infinite trains of symmetric harmonic waves traveling in infinitely long, right circular cylindrical shells is investigated on the basis of the three-dimensional theory of elasticity.
Abstract: The propagation of infinite trains of symmetric harmonic waves traveling in infinitely long, right circular cylindrical shells is investigated on the basis of the three-dimensional theory of elasticity. The shells are assumed made of three concentric, transversely isotropic cylinders, each of different materials, bonded perfectly at their interfaces. The frequency equation is established by representing the displacement field in each cylinder in terms of potential functions and satisfying the Navier equations of motion and the boundary and interface conditions of the cylinder. The frequency equation has been programmed for numerical evaluation on an IBM 7044/7094 DCS computer, and the influence of the mechanical properties of the layers on the frequencies of the first few modes is investigated.

13 citations


Journal ArticleDOI
TL;DR: In this article, the question of including rigid body modes in the prescribed displacement field is discussed and it is found that the constant strain condition, other than the trivial one, cannot be satisfied in the case of curved plates and shells.

12 citations


Journal ArticleDOI
TL;DR: In this article, the displacement field for body waves, surface waves, static deformations and free oscillations is examined in a unified algebraic treatment relating it to the parameters of various source mechanisms.
Abstract: Summary The displacement field for body waves, surface waves, static deformations and free oscillations are examined in a unified algebraic treatment relating it to the parameters of various source mechanisms. By using a convenient notation of the source parameters θ, δ, γ (representing the strike angle, dip angle and slip angle, respectively), handling of the algebraic expressions becomes simplified. It is shown that the ambiguity between the fault plane and the auxiliary plane exists for all the above-mentioned fields. The various conventions for expressing the parameters and by different authors are also related so that the method may be useful to all.

12 citations


Journal ArticleDOI
TL;DR: In this paper, the displacement and the stress field around a prismatic infinitesimal dislocation loop lying near the welded boundary and for the displacement field near the slipping boundary are given.
Abstract: The expressions for the displacement and the stress field around a prismatic infinitesimal dislocation loop lying near the welded boundary and for the displacement field near the slipping boundary are given. The expression for the total force by which the loop is either attracted or repelled from the grain boundary (welded or slipping) was calculated. The forces are graphically illustrated for different combinations of material constants. The calculations were carried out within the isotropic elasticity.

11 citations


Journal ArticleDOI
TL;DR: In this article, the system of equations of the linear theory of isotropic elastic shells of revolution for stress and strain sinusoidal in the polar angle of the base plane with period 2 π is reduced to two simultaneous second order ordinary differential equations which are remarkably similar to the corresponding equations for symmetric bending.

Journal ArticleDOI
TL;DR: In this article, the Laplace transforms of the displacement components of an axisymmetrical poloidal pulse were derived for a semi-infinite, homogeneous, isotropic elastic solid by applying a uniform time-dependent normal pressure over a circular portion of the surface of the half-space.

Journal ArticleDOI
TL;DR: In this article, the authors derived the convergence criteria for the trial displacement functions employed in the finite element method for plate bending by using a variational principle, and proposed a method of improving the convergence of approximate finite element solutions.

Book ChapterDOI
01 Jan 1970
TL;DR: In this paper, the principal parts of displacement fields of reflected and head waves propagating in elastic media near the point of origin of the head wave were derived for a simple case.
Abstract: In the present paper we obtain computational formulas for the principal parts of displacement fields of multiple reflected and head waves propagating in elastic media near the point of origin of the head wave. Similar formulas for simpler cases have been considered in [1, 2].

DOI
01 Jan 1970
TL;DR: In this article, it is shown that the second derivative of the potential energy and of the dissipated energy play a fundamental role in the rate of propagation of a plane crack in an elastic solid.
Abstract: The extension of a plane crack of arbitrary shape in an elastic solid is discussed for G-based laws of propagation. It is shown that the rate of extension is governed by a variational inequality in which the second derivative of the potential energy and of the dissipated energy play a fundamental role. Crack surface is the principal unknown, the differentiation of energy must be performed with respect to a variable domain with moving boundary. Bifurcation and stability of the crack front curve can be discussed as in plasticity. The obtained results are illustrated by some simple analytical examples. INTRODUCTION The propagation of plane cracks of arbitrary shape is an interesting problem in fatigue or in fracture analysis. For example, the study of a plane crack of delamination propagating in laminated composites, or of interface cracks in thin films or in surface coatings of different kind, has been the subject of many discussions in the recent literature [1,3,9,10,14,15]. On the other hand, some models of damage mechanics also lead to study the extension of a damage zone in a solid [2,4,6,7,12,14]. The objective of this paper is to present some general results on the subject concerning the rate problem and stability or bifurcation analysis. GENERAL EQUATIONS An elastic solid with a propagating plane crack is a mechanical system undergoing irreversible transformation. The associated irreversible parameter is the crack surface, a plane domain Q of boundary S as shown in Fig. 1. Its evolution is associated with a total potential energy W : Transactions on Modelling and Simulation vol 6, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 410 Free and Moving Boundary Problems W = w(e(u)) dV F(X).u ds (1) In this expression, w(e) denotes the volumic density of elastic deformation and F is the applied forces, assumed to depend on a force or displacement control parameter X. If only quasi-static evolutions are considered, it is well known that the displacement field at equilibrium u can be implicitly defined as a function of the given state of crack 0 and of control X :

DOI
01 Jan 1970
TL;DR: In this paper, a method for analyzing an interface crack problem based on Body Force Method is proposed to treat the so-called oscillatory stress and displacement field in the vicinity of the crack tip strictly.
Abstract: A method for analyzing an interface crack problem is proposed based on Body Force Method. In order to treat the so-called oscillatory stress and displacement field in the vicinity of the crack tip strictly, a new basic density function for an interface crack is introduced. In addition to the formulae for an interface crack problem, many problems of the elastic bi-material containing arbitrary cracks are solved numerically by using a personal computer and the results are demonstrated in tables and graphs.

Journal ArticleDOI
TL;DR: In this paper, a perturbation solution of differential character for a layer subject to plane strain conditions and indented by a rigid cylinder, and valid for high contact widths, is extended to the analogous axisymmetric problem of a layer compressed by rigid paraboloidal punch approximating a spherical indenter.
Abstract: The deflections are analytically studied in a flat, deformable layer, firmly anchored to a rigid substrate, and frictionlessly indented by a rigid paraboloidal punch, having in mind the design of hip replacements possessing elastomeric layers. An existing perturbation solution of differential character, developed for a layer subject to plane strain conditions and indented by a rigid cylinder, and valid for high contact widths, is here extended to the analogous axisymmetric problem of a layer compressed by a rigid paraboloidal punch approximating a spherical indenter. Perturbed solutions up to the second order are obtained, and the contact pressure profiles activated by a paraboloidal punch are derived for both compressible and incompressible layers. Numerical examples exploring the sensitivity of the stress field to perturbations of the Poisson's ratio are also included. INTRODUCTION This paper deals with the deflections of a flat, deformable layer, firmly bonded to a rigid substrate and frictionlessly compressed by a rigid paraboloidal indenter, Fig. 1. This axisymmetric study is relevant in the design of hip replacements whose cup is covered with an elastomeric layer, and compressed by a ball replacing the femoral head, [1] . An analytical perturbation solution of differential character, developed in [2] for a layer subject to plane strain conditions and indented by a rigid cylinder, valid for high contact widths, is here extended to the axisymmetric problem of a layer compressed by a rigid paraboloidal punch approximating a spherical indenter. This paper is organized as follows. First, the equations describing the layer deflections as functions of an imposed axisymmetric pressure profile are developed up to the second perturbation order. A subsequent section solves the more relevant situation where the Transactions on Engineering Sciences vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3533 398 Contact Mechanics indenting profile is known, and the contact pressure has to be determined. Numerical examples exploring the sensitivity of the solution to perturbations of the Poisson's ratio end the paper. THE PERTURBED PRESSURE-DEFLECTION SOLUTION Following Armstrong [2], the subsequent steps are developed: a) the axisymmetric equilibrium equations are expressed in terms of the displacement field; b) the equilibrium equations are normalized with respect to proper variables, and a small parameter e is identified, which renders this problem amenable to perturbation solution techniques; c) an approximate solution is achieved for parameter € up to the second order, which connects the pressure (as well as its derivatives) acting upon an elastomeric layer to its local deflection. The axisymmetric equilibrium equations in terms of stresses are [3] : By introducing into (1) the Hooke law linking stresses to strains, expressed in cylindrical coordinates in terms of radial, u , and axial , w , displacements, [3], the equilibrium equations are formulated in terms of the displacement field as :

Journal ArticleDOI
TL;DR: In this article, a miniaturized piezoelectric transducer based on an annular array structure is proposed to compute the radiated acoustic field in a homogeneous isotropic fluid.
Abstract: A design procedure is proposed to develop a miniaturized piezoelectric transducer based on an annular array structure. The first step consists in comparing different structures by using a time domain analysis of the corresponding radiated acoustic fields in an homogenous isotropic fluid, considering a uniform excitation displacement at the radiating surface of the transducer A finite element calculation is then performed for the selected transducer geometries to optimize their harmonic electrical response. The resulting displacement field at the transducer radiating surface is then used to compute the radiated acoustic field in the fluid. The results provided by both time domain and finite element analyses can then be compared to check the validity of the proposed approach.

DOI
01 Jan 1970
TL;DR: In this article, the analysis of the interaction of a main crack with an array of microcracks placed near the tip of the main crack has been performed considering 2-D polycarbonate flat specimens.
Abstract: The analysis of the interaction of a main crack with an array of microcracks, placed near the tip of the main crack, has been performed considering 2-D polycarbonate flat specimens. Analytical results, based on the elastic potential theory for the stress intensity factor K, have been well compared with experimental results obtained through the application of the caustic method. Two different specimen configurations have been analysed involving one and two microcracks. INTRODUCTION The problem of the interaction of a main crack with an array of microcracks placed near its tip, of strong relevance in the prediction of the reliability of structural materials, can be experimentally analyzed by applying the caustic method on 2-D specimens. In fact, the method of caustics can be applied to the determination of the stress intensity factors at crack tips approaching other crack tips or boundaries of the specimen. Several problems of interaction of cracks with other cracks or boundaries have been reported [1,2]. In these problems two special cases can be distinguished: in the first case, the crack tip at which the stress intensity factor is determined does not lie very near another crack or boundary [3]. In the second case the crack tip lies very near another crack tip (or boundary) and the shape of the caustic around this crack tip is influenced by the other crack (or boundary). On the other hand, the elastic interaction between both types of cracks can be analytically studied by applying a method based on the combination of the double layer potential technique and the Willis polynomial conservation theorem stating that the COD of a crack embedded into a polynomial stress field of degree N has the form (ellipse)x(polynomial) [4]. Following this approach and Transactions on Modelling and Simulation vol 10, © 1995 WIT Press, www.witpress.com, ISSN 1743-355X 620 Computational Methods and Experimental Measurements expressing the displacement field as well as the stress field by an integral of the double layer potential type, the problem can be reduced to the one of finding vectorial functions-microcrack CODs b(x) and the stress intensity factor K, at the macrocrack tip from the system obtained by solving the system of integral equations which express the boundary conditions on the crack faces. The microcrack CODs are represented in the form (ellipse)x(polynomial) where the first multiplier corresponds to the crack embedded into a uniform stress field and the second multiplier accounts for crack interactions. Thus, the system of singular integral equations is reduced to a system of linear algebraic equations which must be solved to obtain the polynomial coefficients. POTENTIAL REPRESENTATION THEORY The schematic representation of a main crack (-L, i) which can interact with one microcrack is shown in Fig. La and with two microcracks in Fig.l.b. In order to describe the elastic interaction between both cracks, plane stress conditions and Mode I are assumed. Moreover, the case where the microcrack is embedded into the stress field of the main crack tip is considered. Using the constant approximation the stress field within the microcrack line (c-l, c+l) may be approximated by a constant equal to the value of the field at the microcrack centre (x-c) and, following symmetry, a* = â fcj is the only stress component acting along the microcrack line. Fig.l .a: Main crack collinear with one microcrack. Transactions on Modelling and Simulation vol 10, © 1995 WIT Press, www.witpress.com, ISSN 1743-355X Computational Methods and Experimental Measurements 621 Thus, the overall stress field in the vicinity of the main crack tip is given by the superposition:

DOI
01 Jan 1970
TL;DR: In this paper, a numerical approach is presented for the evaluation of the power spectrum of ground vibrations in the vicinity of a roadway, which accounts for actual road surface roughness, vehicle mechanic characteristics, vehicle speed and soil parameters.
Abstract: A numerical approach is presented for the evaluation of the power spectrum of ground vibrations in the vicinity of a roadway. The method accounts for actual road surface roughness, vehicle mechanic characteristics, vehicle speed and soil parameters. Dynamic models for generation and propagation of vibrations are developed, from the source (the vehicle) through the ground, until the building foundations. An application is presented, in which the road roughness power spectrum has been obtained directly from road displacement field measurements in Florence. The reliability of the computational method has then been verified comparing numerical results with experimental measurements in the same conditions.