Showing papers on "Linear elasticity published in 1972"

A minimum principle for frictionless elastic contact with application to non-Hertzian half-space contact problems

Abstract: A variational principle governing the frictionless contact between two elastic bodies is established, which is valid both for linear and for non-linear elasticity. In the case of linear elasticity it appears to lead to an infinite dimensional convex quadratic programming problem. It is applied to the half-space geometry in linear elasticity and it is established that non-Hertzian normal half-space contact problems are physically meaningful. A Hertzian and a non-Hertzian normal contact problem are investigated numerically, to which end the principle is discretised on a triangular network. In the case of the Hertz problem it is found that the exact relationships between penetration, maximum pressure, and total normal force are well satisfied. The form of the contact area is given only crudely, unless the discretisation network is considerably refined. It appeared that such a refinement is only necessary close to the edge, in which case passable results will be obtained.

The fracture mechanics of slit-like cracks in anisotropic elastic media

Abstract: U sing the method of continuously distributed dislocations, the problem of a slit-like crack in an arbitrarily-anisotropic linear elastic medium stressed uniformly at infinity is formulated and solved. The crack faces may be either freely-slipping or loaded by arbitrary equal and opposite tractions. If there is no net dislocation content in the crack, then the tractions and stress concentrations on the plane of the crack are independent of the elastic constants and the anisotropy; the same is true of the elastic stress intensity factors. The crack extension force depends on anisotropy only through the inverse matrix elements Kmg−1, where [K] is the pre-logarithmic energy factor matrix for a single dislocation parallel to the crack front. Numerical results for crack extension forces are presented for three media of cubic symmetry.

A new representation of the strain field associated with the cube‐edge dislocation in a model of a α‐iron

Abstract: The displacement field around an edge dislocation core is examined in an attempt to quantitatively modify the linear elastic Volterra solution. The approach derives from a computer simulation to obtain the equilibrium atomic positions in a dislocated bcc crystallite using Johnson's potential to model α‐iron. Unlike many previous calculations of this nature which employ rigid boundary conditions throughout the relaxation, the elastic media bounding the crystallite is here allowed to adjust in response to atomic readjustments within the crystallite. The procedure used to achieve flexible boundaries and its application to this problem is discussed in detail. This scheme permits linear elastic displacements at distances relatively far from the dislocation center. The difference between the computed strain field and the Volterra prescription results from nonlinear effects in the core, and is found to correspond to a net expansion of 0.25b2 per unit dislocation length. This is in accord with findings of experim...

An atomistic study of cracks in diamond-structure crystals

Abstract: The structuxe of atomically sharp equilibrium cracks in diamond, silicon and germanium is calculated. The treatment considers a long plane crack formed by bond rupture across the (111) cleavage plane, critically loaded in tension. Within a small 'core' region immediately surrounding the crack tip the interatomic interactions are represented by a potential function specially constructed to match macroscopic fracture parameters. Anisotropic linear elastic theory is invoked to provide boundary conditions for the core region, and a first approximation for lattice-point displacements within. The core atoms are then relaxed to a configuration of minimum potential energy by computer. The results indicate that continuum theory is capable of giving remarkably accurate predictions of the crack-tip displacement field, except within about three atom spacings from the tip, despite marked nonlinearity in the interatomic force function. These results are discussed in terms of existing continuum models of crack-tip structure: in particular, Barenblatt's model of a cusp-shaped tip region is found to be inapplicable to diamond-structure crystals. The crack-tip geometry is better pictured as a narrow slit terminated by a single line of bonds close to the rupture point. Brief reference is made to the possible extension of the treatment to other classes of highly brittle solid, especially glassy materials, and to the relevance of the results to some fracture problems of practical importance. The fracture of an ideally brittle solid is essentially an atomic process, in which cohesive bonds are ruptured at the tip of the growing crack. Yet traditionally the rnathematical treatment of the mechanics of fracture propagation has been developed almost exclusively from continuum concepts. The chief reason for this lies in the interest of simplicity, a proper description of the atomic configuration at a crack tip requiring seemingly formidable analysis in terms of a suitable structural model for the given solid. The continuum approach, based on linear elasticity theory, has in fact proved adequate in many fracture-mechanics problems: in particular, the growth of a semi-brittle crack in most 'engineering materials' can be described in terms of a macroscopic 'plastic zone' encasing the tip. Many mechanical properties, on the other hand, are highly sensitive to events occurring over distances no greater than a few interatomic spacings. For instance, the energetics of dislocations in plastic crystals, particularly covalently-bonded crystals, may depend largely on the atomic structure of the dislocation core. The ideally brittle crack provides a similar case, the crack front advancing one atomic

Finite Element Methods in Continuum Mechanics

Abstract: Publisher Summary The chapter presents a brief introduction to the different finite element formulations for linear elastic solids and discusses similar formulations for several other field problems. The chapter presents detailed illustrations for several typical finite element formulations. In the finite element formulation, displacement and stress fields are assumed to be continuous within each discrete element. This formulation calls for modified variational principles for which the continuity or equilibrium conditions along the interelement boundaries are introduced as conditions of constraint and appropriate boundary variables are used as the corresponding Lagrangian multipliers. The chapter presents the several variational principles and the corresponding models used in the finite element formulation. The large majority of the existing finite element formulations are based on the assumed displacement approach. The chapter discusses equilibrium problems of linear elastic solids. There are several other problems in solid mechanics, which can be formulated by means of variational principles and hence can be solved by finite element methods. The finite element methods have also been extended to nonlinear problems resulting from elastic-plastic material properties or from large deflections or finite strains.

Computer simulation studies of the a/2 〈111〉 edge dislocation in α‐iron

Abstract: Core characteristics of the stable a/2 〈111〉 edge dislocation in α‐iron are presented for three two‐body interatomic potentials, and the importance of properly defining boundary conditions is demonstrated. The boundary requirements adopted for this analysis are as follows: (i) The coordinate of the dislocation line along the axis parallel to the burgers vector is such that, above and below the slip plane, the lattice to the right and left of the extra half‐planes of atoms forming the dislocation is symmetric. (ii) The coordinate of the dislocation line in the direction perpendicular to the slip plane is such that the slope of the strain energy curve in the linear region of the lattice agrees with the slope in the continuum. (iii) The positions of atoms at the lattice‐continuum interface are made to agree with the predictions of linear elasticity theory. Four interatomic potentials are examined, one of which is shown to be unsatisfactory. Results with the other three potentials are quite reasonable. Core radii of 4.6, 4.8, and 5.4 A and corresponding core energies of 0.479, 0.539, and 0.701 eV per (112) plane are calculated. The stable core configuration in each case is consistent with the particular character of the interatomic potential. The results suggest that the core characteristics are not sensitive to the value of the cohesive energy associated with each potential.

Dispersion of flexural waves in circular cylindrical shells

Abstract: The imaginary and complex branches of the dispersion spectra corresponding to flexural waves in circular cylindrical shells of various wall thicknesses including the solid cylinder have been constructed by utilizing exact three-dimensional equations of linear elasticity. The effects of wall thickness and Poisson ratio on the cut-off frequencies have been studied. Complex branches emanate from the points of frequency extrema on the purely imaginary or purely real branches and intersect the zero frequency plane, either as purely imaginary or as complex branches. The waves associated with complex branches emerging from points on the real plane are less decaying at higher frequencies.

Theoretical Formulation of Finite-Element Methods in Linear-Elastic Analysis of General Shells∗

Abstract: A systematic classification of the variational functional whose stationarity conditions (Euler equations) can be used alternately to solve for the various unknowns in a boundary-value problem in linear-shell theory is made. The application of these alternate variational principles to a finite-element assembly of a shell and thus, the development of the properties of an individual discrete element are studied in detail. A classification of the finite-element methods, formulated from the variational principles by systematically relaxing the continuity requirements at the interelement boundaries of adjoining discrete elements is made.

Factors affecting the elastic deformations of concrete

Abstract: The shape of a load-deformation diagram of concrete is influenced by the properties of the material as well as by the testing conditions. Several of the numerical approximations presented for the stress-strain diagram can take some of these effects into consideration. The modulus of elasticity of a concrete is influenced by the same factors as the stress-strain diagrams. Formulae are offered for the estimation of the effects of concrete composition, including the porosity, on the modulus of elasticity. A discussion of the other elastic constants, poisson's ratio, etc., concludes the paper. A part of this paper presents new information, the other part is an organized summary of the state of the art.

Shock waves in a mixture of linear elastic materials

Abstract: This work concerns the behavior of shock waves propagating in a mixture of an isotropic elastic solid and an elastic fluid. The displacement of each constituent are presumed to be small and the resulting field equations are linear. Contrary to the usual behavior of shocks predicted by linear theories, we find that the shock amplitudes are influenced by strain gradients behind the waves. Also, we find that the difference of the amplitude of the shock associated with the solid and that associated with the fluid are damped due to the presence of diffusion, but a certain weighted sum of the amplitudes is unaffected.

On the growth and decay of wave fronts in a mixture of linear elastic materials

Abstract: This work concerns the propagation of wave fronts in a mixture of fluids and isotropic solids. Only mechanical effects are included. In the statement of the field equations the importance of buoyancy forces is stressed. These forces arise from a local interaction among the constituents of the mixture and are present even in the absence of diffusion. It is shown that there are as many longitudinal waves possible as there are constituents and that there are as many transverse waves possible as there are solid constitutents. Rules for the evolution of discontinuities on wave fronts are deduced, and it is shown that diffusion causes a decay in the wave strength in addition to the usual geometric decay familiar from classical elasticity.

Viscoelastic Plate on Poroelastic Foundation

Abstract: A solution for the quasistatic response of a linear viscoelastic infinite plate supported on a poroelastic half-space, subjected to a uniform circular loading, is presented. The poroelastic material is defined to be an interacting mixture of materials, consisting of a linear elastic porous solid and an incompressible fluid. Mathematical descriptions of poroelastic medium and viscoelastic plate are presented. The initial-boundary value problem that models the deformation response to load application is solved using iterated Laplace-Hankel integral transformations. Numerical inversions of the transformed solution images that are obtained are considered. An example of numerical solution for the plate deflection and foundation reaction as functions of time is presented.

Viscoelastic plate on poroelastic foundation

Abstract: A SOLUTION FOR THE QUASISTATIC RESPONSE OF A LINEAR VISCOELASTIC INFINITE PLATE SUPPORTED ON A POROELASTIC HALF-SPACE, SUBJECTED TO A UNIFORM CIRCULAR LOADING, IS PRESENTED. THE POROELASTIC MATERIAL IS DEFINED TO BE AN INTERACTING MIXTURE OF MATERIALS, CONSISTING OF A LINEAR ELASTIC POROUS SOLID AND AN INCOMPRESSIBLE FLUID. MATHEMATICAL DESCRIPTIONS OF POROELASTIC MEDIUM AND VISCOELASTIC PLATE ARE PRESENTED. THE INITIAL BOUNDARY VALUE PROBLEM THAT MODELS THE DEFORMATION RESPONSE TO LOAD APPLICATION IS SOLVED BY USING ITERATED LAPLACE-HANKEL INTEGRAL TRANSFORMATIONS. NUMERICAL INVERSIONS OF THE TRANSFORMED SOLUTION IMAGES THAT ARE OBTAINED ARE CONSIDERED. AN EXAMPLE OF A NUMERICAL SOLUTION FOR THE PLATE DEFLECTION AND FOUNDATION REACTION AS FUNCTIONS OF TIME IS PRESENTED. /AUTHOR/

Some partial solutions of finite elasticity

Abstract: Elastic deformations beyond the range of the classical infinitesimal theory of elasticity are governed by highly non-linear partial differential equations and in general there exists no clear statement of these equations which does not involve stress components. Orthodox methods of solution are not usually applicable and consequently only a few exact solutions are known. In this thesis for some special classes of deformations and for some special materials we obtain substantial new reductions of the equilibrium equations which involve the deformation only. From these reduced equilibrium equations a number of new exact partial solutions are derived for the neo-hookean and Mooney materials.

On Wave Propagation in a Mixture of Linear Elastic Materials

Abstract: : The work concerns the propagation of wave fronts in a mixture of fluids and isotropic solids. Only mechanical effects are included. In the derivation of the equations the importance of buoyancy forces is demonstrated. It is shown that these forces arise from a local interaction among the constituents of the mixture and are present even in the absence of diffusion. It is also shown that there are as many longitudinal waves possible as there are constituents and that there are as may transverse waves possible as there are solid constituents. Rules for the evolution of discontinuities on wave fronts are deduced. Diffusion causes a decay in wave strength which is additional to the usual geometric decay familiar from classical elasticity. (Author)

Dynamic Response of Model Pavement Structure

Abstract: Measurements of the stresses, strains and deflections occuring in a three-layer pavement structure subjected to dynamic surface loading are reported. These results are compared with theoretical values determined by linear elastic theory and finite element analysis. The silty clay subgrade and crushed stone subbase exhibited nonlinear stress-strain relationships. With a knowledge of these relationships, elastic theory could be used to calculate critical effects adequately for design purposes. An attempt has been made to characterize nonlinear materials approximately by using bulk and shear moduli rather than the conventional Young’s modulus and Poisson’s ratio.

Lattice statics applied to vacancy-screw dislocation interactions in cubic metals

Abstract: The interaction of point defects in a model harmonic cubic crystal has been analysed from a lattice statics point of view by Hardy and Bullough (1967 a, 1967 b), and their results are in serious disagreement with the corresponding results of elasticity theory. Here we extend their work to include the interaction of point defects with line defects by considering specifically the interaction of a vacancy with a screw dislocation in certain cubic metals. Results are presented for potassium and α-iron and three model b.c.c. crystals which are expected to be falid except in the region of close interdefect separations. The model crystals were designed to show the effect of elastic isotropy, and the results for these crystals show that while elastic isotropy considerably reduces the range of the vacancy-screw dislocation interaction, this interaction does not vanish as predicted by linear elasticity theory.

Characterization of asphalt concrete and cement-treated granular base course

Abstract: IN THE CONTEXT OF THE AVAILABLE METHODS OF STRUCTURAL DESIGN, LINEAR ELASTICITY AND VISCOELASTICITY CAN ADEQUATELY CHARACTERIZE ASPHALT CONCRETE PROVIDED THAT CHARACTERISTICS NOT MODELLED BY THE CHOSEN CONSTITUTIVE LAW ARE RECOGNIZED. STRESS NONLINEARITIES CAN BE PARTIALLY ACCOUNTED FOR IF THE AVERAGE STRESS STATE EXPECTED IS USED WHEN SELECTING THE APPROPRIATE LINEAR ELASTIC AND VISCOELASTIC PARAMETERS. FOR AN ELASTIC CHARACTERIZATION, IF THE EFFECTS OF LOAD DURATION, TEMPERATURE AND PERMANENT DEFORMATION ARE INCLUDED, THE STRESS-STRAIN PREDICTIONS SHOULD BE COMPARABLE TO THOSE OBTAINED USING LINEAR VISCOELASTICITY AND THERMORHEOLOGIC MODELS. HOWEVER, IT IS PREFERABLE TO DIRECTLY MODEL AS MANY OF THE RESPONSE CHARACTERISTICS AS POSSIBLE, AND THEREFORE, FUTURE WORK SHOULD BE DIRECTED TOWARDS UTILIZING LINEAR VISCOELASTICITY. DISTRESS CRITERIA WHICH CONSIDER ASPHALT CONCRETE AS A VISCOELASTIC MATERIAL NEED TO BE DEVELOPED. /AUTHOR/

Vibrations of Laminated Beams

Abstract: This paper is a continuation of Ref. 1 and its extension to dynamic problems. As in Ref. 1 the underlying assumptions about the kinematics of the beam deformation are: a) deformations are elastic and small such that the linear elasticity theory is applicable; b) perfect bond is realized between adjacent laminae; c) Bernoulli's hypothesis of planar cross sections is valid for each lamina independently such that after the deformation each cross section consists of a number of planes interconnected at interface lines; d) deformation is symmetric with respect to vertical plane (no torsion). As shown in Ref. 1 on the basis of these assumptions it is possible to develop a simple and consistent theory without the introduction of additional simplifications (such as neglecting the bending rigidity of the core, etc.). Consider a laminated beam with x axis being the loci of all cross-sectional centroids and denote axial displacement by u(x9 z) and transverse displacement by w(x, z). As a consequence of assumptions about the deformation kinematics, set forth above, both displacement components u(x,z) and w(x, z) are defined as a polygonal graph with vertices at extreme fibers of all three laminae. It is, for example, obvious that to determine the displacement component u(x, z) we ought to know the values of the axial displacement at four vertices at each cross section. Hence, we write

A photoelastic material with variable modulus of elasticity

Abstract: Instructions are given in this paper to manufacture a photoelastic material (polyurethane rubber) exhibiting a linearly variable modulus of elasticity. In the examples given, the range ofE is of the order of 1 to 2.5. Illustrations of applications in stress analysis are included. The new material should be particularly useful in the solution of soil-mechanics problems and it can be used for its photoelastic properties, or as base for grid and moire analysis.