Linear Isotropic Elasticity Research Articles

Analysis and finite element approximation of compressible and incompressible linear isotropic elasticity based upon a variational principle

We propose a variational principle for compressible and incompressible linear isotropic elasticity involving four dependent variables: the strain tensor, the augmented stress tensor, pressure and displacement. The variational formulation derived from the principle is analyzed employing a general result due to Brezzi. Finite element methods using the Galerkin and SBB methods are shown to converge for a wide family of finite element interpolations.

The lamellar inclusion problem in plane elasticity

Within the framework of linear isotropic plane elasticity, a general method giving the elastic field in an internally stressed three-phase material is determined. The internal stress source may be located either in or near the lamellar component. General closed-form formulas solving the limiting case of a bimetallic interface are also obtained. As an example of application, the interaction between a lamellar inclusion and a dilatation centre is analyzed.

Elastic fields about a perturbed spherical inclusion

The elastic fields about an inclusion having the shape of a sphere perturbed by a spherical harmonic are calculated by using isotropic linear elasticity theory. This shape is chosen in order to facilitate coupling of this problem with a morphological stability analysis. The components of the displacement field u( r, θ, φ) about the inclusion are found to be proportional to the components of the gradient of the perturbing harmonic. Two alternative boundary conditions are considered, corresponding to 1. (1) a coherent interface, in which forces and displacements are continuous across the interface 2. (2) a “greased” interface, in which forces tangential to the interface vanish at the interface. For a coherent interface, the sign of the dilatation in the matrix depends on the ratio of the shear moduli of the inclusion and matrix; whereas for a greased interface, the sign of the dilatation in the matrix is independent of this ratio. Also, for a greased interface, the perturbation affects the dilatation in the inclusion; this is not the case for a coherent interface.

Shape effects on edge dislocation climb over an elliptical inhomogeneity

The recently solved linear-isotropic elasticity problem of the interaction between an edge dislocation and an elliptical inhomogeneity is extended to the case in which the matrix is subjected to a shear stress applied at infinity. The resolutive tool is obtained by means of a slight modification of an already available computer program. Numerical results show the existence of atrapping height ytfor a sufficiently elongated inhomogeneity, such that dislocations aty<ytare totally trapped. The consequences on the creep of dispersion-strengthened metals (Holbrook and Nix's model), as well as the dependence of trapping on shape, material constants, applied stress, and orientation, are discussed.

Inhomogeneous plane waves

This paper deals with the propagation of elliptically polarised inhomogeneous, time-harmonic plane waves. Such waves arise in many areas. Examples include Rayleigh, Love and Stoneley waves in classical linear isotropic elasticity theory, gravity waves in ideal fluids, TE and TM waves in electromagnetism, and viscoelastic waves. For the most part even though the applications given here are in the theory of isotropic and anisotropic elastic bodies, it should be apparent that the results have application in other areas, such as electromagnetism. The purpose of the paper is to show how the theory of complex vectors, or “bivectors”as Hamilton and Gibbs called them, may be used to give results on the polarisations of inhomogeneous plane waves. Also, the use of bivectors leads to a simple direct formulation of the eigenvalue problem for these waves.

The classical pressure vessel problems for linear elastic materials with voids

The traditional problems of the thick walled spherical and circular cylindrical shells under internal and external pressure are solved in the context of the theory of linear elastic materials with voids. The solutions are quasi-static. The stress distributions are those predicted by isotropic linear elasticity. The displacement and solid volume fraction charge fields exhibit a volumetric viscoelasticity induced by a rate dependence of the volume fraction change.

Hybrid quadratic isoparametric finite element-boundary element code for underground excavation analysis : Yeung, D; Brady, B H G Proc 23rd Symposium on Rock Mechanics, Berkeley, 25–27 August 1982, P692–703. Publ New York: AIME, 1982

A hybrid finite element-boundary element code with eight noded isoparametric finite elements and three noded boundary elements is described for plane strain and linear isotropic elasticity. In the direct formulation of the boundary element method, quadratic variation of geometry and displacements, and linear variation of tractions with respect to the element intrinsic coordinates are used. The boundary element region is treated as a super-element in the finite element sense. For the covering abstract of the symposium see TRIS 452576. (Author/TRRL)

Vacancy-dislocation interaction energies in MgO

Abstract Calculations based on an atomistic model have been carried out to determine interaction energies of cation and anion vacancies in a region approximately 26 by 46 (where 6 is the magnitude of the dislocation's Burgers vector) surrounding an α/2 [110] edge dislocation in a model MgO crystal. The atomistic results are compared to predictions of a model based on linear isotropic elasticity theory. Throughout most of the core region investigated, the atomistic results differ both qualitatively and quantitatively from the predictions of the linear elastic model. At the furthest defect separation distances investigated, the predictions of the two theories show qualitative—though generally not quantitative—agreement. The variation of the interaction energy as a function of local equilibrium vacancy position from the centre of the dislocation is complex, depending on both angle, θ, and defect separation, R. Ignoring the θ-dependence, the steepest part of the spatial variation of the binding energy has an approximately R −2 dependence. This is in disagreement with an R −9 dependence inferred from an internal-friction study of neutron-irradiated MgO. Possible reasons for this disagreement are discussed.

Boundary discretization using the edge-function method in three dimensional elasticity

The edge function asymptotic boundary discretization method, which generates a partition of the boundary into a finite set of boundary elements, and its application to three-dimensional problems of linear isotropic elasticity are presented. The present solution representation, which is global, is constructed from asymptotic solutions of the Navier equations of elasticity. The unknown parameters are then determined by imposing the boundary conditions separately over each of the boundary elements. A numerical example is included and the results are discussed.

Elastic energies of dissociated Frank loops in f.c.c. metals and alloys

Abstract Elastic energies of dissociated non-regular hexagonal Frank loops have been calculated in the approximation of isotropic linear elasticity theory. A model has been used in which attractive Shockley dislocations could eventually recombine. It has been shown that small loops are strongly dissociated provided that the stacking fault energy is not too high. This is a possible explanation for black spot defects. An evolution scheme for loop growth is suggested. In particular, Frank loops and stacking fault tetrahedra could have a similar origin to that considered in early studies of vacancy clustering.

Volume changes in copper due to point defects

Isotropic linear elasticity theory is used to calculate the volume change due to point defects and the pressure of atomistic helium bubbles. Atomistic defect calculations for the specific case of helium in copper are used to illustrate the technique. It is found that helium atoms introduced into vacancies in copper produce a volume increase of \ensuremath{\sim}0.62 atomic volumes per helium atom. However, this linear increase is found only after the number of helium atoms is greater than the number of vacancies. The pressure of helium as a function of density is also calculated. Extremely high pressures are found for the high densities considered; it is found that the pressure increases nearly linearly with density.

Elastic Interaction between Crowdions and Extended Atomic Defects in Isotropic Media

AbstractThe interactions between the partners of crowdion‐vacancy, crowdion‐crowdion, and crowdion‐dumbbell pairs in face‐centred cubic crystal sare calculated within the framework of the isotropic linear elasticity theory, taking into account the finite extensions of these defects. It is found that among all types of pairs there are configurations which can form bound states at small defect separations. For a number of crowdion‐vacancy and particularly of crowdion‐crowdion configurations, pairs of neighbouring bound states exist which are capable of giving rise to relaxation phenomena.

An Analysis of the Entropy of Thermally Activated Dislocation Motion Based on the Theory of Thermoelasticity

AbstractAn analysis is made of the entropy of thermally activated dislocation motion using the continuum theory of thermoelasticity. A relation for the activation entropy, which is valid for non‐linear, anisotropic elastic solids is derived. The activation entropy is expressed in terms of the general stress state in the solid and is found to be linear in the first temperature derivative of the elastic compliances, a relation often applied to the special case of linear, isotropic elasticity, but not previously derived for the more general case treated here. It is shown that, under the special conditions where pure shear or pure hydrostatic stress states are associated with both the initial and activated states, the present result can be reduced to the one often found in the past for the case of linear isotropic elasticity. However, these special conditions are never satisfied either in the thermally activated motion of dislocations or in diffusion. This suggests that the previous result is not applicable to these situations.

Theoretical seismology

An outstanding achievement of the last 4 years in U.S. research into theoretical seismology is the identification of how much information (about earth structure) is contained in a given set of seismic data. This work, an extension of earler successes in inverse theory, should find wide application in other branches of physical science and in the biological sciences. Advances have also been made in studies of the effect of lateral heterogeneities on short‐period waves, the theoretical effect (on seismic waves) of departures from the usual laws of isotropic linear elasticity, the spectral content of short‐period waves by using maximum entropy methods and homomorphic deconvolution, and the use of finite difference and finite element codes. A beginning has been made in the study of wave scattering in random media. These research fields are all fairly new in relation to the traditional occupation of generating theoretical seismograms in ever more complex earth models, assumed to be spherically symmetric; this classical endeavor too has seen breakthroughs in the last 4 years. Seismic source theory, now taking probably more than half the research effort of theoreticians, is reviewed in a separate report.

On some two-dimensional cracks in linear isotropic elasticity

In this paper the method of conformal transformation is used to consider a class of crack problem in linear isotropic elasticity. The conformal transformation used is the Joukowski transformation so that in general the crack profile is an aerofoil. This is a useful profile to consider since as well as providing information about the general aerofoil crack it includes as special cases the circular arc crack, the straight line crack and, as will be shown, can also be used to obtain similar results to those obtained by Bowie (1956) for the crack originating at the boundary of a circular hole. Thus by varying the parameters in the one elastic solution it is possible to obtain the elastic solution for a number of crack profiles. An examination of the stresses at the crack tip is made since this gives useful information about the direction of crack extension and also gives information necessary for the application of the fracture criterion which is derived in section 6. Some particular crack profiles are considered in detail in section 7.

On the Application of a Baecklund Transformation to Linear Isotropic Elasticity

On the Application of a Baecklund Transformation to Linear Isotropic Elasticity

Linear dispersive shear waves in two-layer elastic medium

In linear isotropic elasticity, cases exist where pure shear waves are possible even in bounded media; these have a nondispersive mode with propagation speed equal to that in an infinite medium. For these cases, consideration of a two-layer medium shows the existence of dispersion which vanishes only with equality of the propagation speeds. The present study uses the method of asymptotic expansions; a uniformly valid approximation is obtained to describe the speed and dispersive nature of these waves. The elegance of this approach is brought out by derivation of the basic result of this study viz., Jeffrey equation to describe the farfield structure of these waves.

Stress functions for cosserat elasticity

In this paper a complete solution is obtained to the displacement equations of equilibrium for linear infinitesimal isotropic Cosserat elasticity. This solution is in terms of stress functions analogous to the Papkovitch functions of classical elasticity and those found by Mindlin and Tiersten for couple stress elasticity. Previous complete solutions to these equations given by Mindlin and by Neuber contain three more scalar potentials than are employed in the present solution.

On the formation and migration entropies of vacancies in metals

An approach has been developed for calculating defect entropies from an entirely discrete vibrational point of view. Calculations have been performed within a discrete finite spherical array of atoms which includes all contributions to the defect entropy predicted by linear isotropic continuum elasticity. The method used, however, does not demand an explicit knowledge of the strength of the defect and thus avoids certain conceptual difficulties that arise in both the elastic and the combined discrete-elastic approaches. The vibrational spectra for the various defect configurations have been obtained using a quasiharmonic Einstein model in conjunction with a modified Morse function as force law. Values of the vacancy formation and migration entropies were computed for the cases of copper, nickel and aluminum. Where experimental values were available for comparison, fair agreement was found with the calculated entropies. In addition, Arrhenius D 0's calculated on the assumption that diffusion is controlled by the migration of single vacancies were found to be lower than experimental D 0's, thus lending support to previous suggestions in the literature that divacancies make a significant contribution to the diffusion flux in f.c.c. metals at high temperatures.

The Determination of Stacking Fault Energy by the Tetrahedron Method

At present there are three methods for obtaining values of the stacking fault energy y of face-centred cubic (f.c.c.) materials by direct observation of dislocationstacking fault configurations in the electron microscope. These are based on measurements of extended three-fold dislocation nodes (e.g. Whelan 1958; Brown and ThOlen 1964), faulted dipole configurations (e.g. Haussermann and Wilkens 1966; Steeds 1967), and triangular Frank dislocation loops and stacking fault tetrahedral (e.g. Silcox and Hirsch 1959; Loretto, Clarebrough, and Segall 1965). The main advantages of the third method over the other two are that it is applicable to materials of a very wide range of stacking fault energy and involves only simple length measurements of defects that are easily recognized. However, it has suffered from the disadvantage that the values of y deduced from these measurements relied on an incomplete theory. The present authors have reconsidered this problem and, subject to the limitations of isotropic linear elasticity, have taken into account the major variables that may affect the values of y. It is the purpose of this note to present the results of this theory in a form in which values of y may easily be obtained from measurements of Frank dislocation loops and stacking fault tetrahedral without the resources of a large digital computer.