Scaling laws and structural inhomogeneities in solids

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Scaling laws and structural inhomogeneities in solids

F.R.N. Nabarro

To cite this version:

F.R.N. Nabarro. Scaling laws and structural inhomogeneities in solids. Journal de Physique, 1989, 50

(18), pp.2519-2523. �10.1051/jphys:0198900500180251900�. �jpa-00211078�

Scaling ^{laws and} structural inhomogeneities ^{in solids}

F.R.N. Nabarro

Condensed Matter Physics ^Research Unit, University ^{of the} Witwatersrand, Johannesburg, ^and

Division of Materials Science and Technology, CSIR, Pretoria, South Africa

(Reçu ^{le 13}

^1989, accepté le 25 avril 1989)

Résumé. 2014 On montre que la démonstration de Cottrell (que le module de cisaillement doit être nul dans

milieu continu

absence de forces discontinues)

réduit à l’affirmation que, si la contrainte limite de cisaillement dans

cristal est très inférieure

module de cisaillement, ^cette contrainte limite est atteinte pour

^très petite déformation de cisaillement. Une solution

approchée ^est obtenue pour le champ de déformation d’une dislocation coin dans

milieu

incompressible.

Abstract.

Cottrell’s demonstration that the shear modulus must be

in

continuum without discontinuous forces is shown to reduce to the statement that if the limiting ^{shear stress in}

crystal

is much less than the shear modulus, then this limiting ^{stress is achieved at}

very small shear strain. An approximate solution is obtained for the strain field of

edge dislocation in

incompressible ^medium of this kind.

Classification

Physics ^Abstracts

61.70G - 76.30C

1. Introduction.

In a paper with the above title, ^Cottrell [1] discusses the relation between the shear modulus il and limiting ^{shear stress} o-s in ^a continuum, regarded ^{as a} crystal having ^a vanishingly ^small

lattice parameter. The conclusion is « that > - ^{0 in a} continuum without discontinuous forces. The medium behaves as a fluid. Thus it follows from the assumption ^{of neutral} equilibrium ^{that the} concept ^{of a solid is} incompatible with that of a continuum

Such a conclusion is disturbing ^to those accustomed to studying the elastic properties ^of

continuous solids. It therefore may be useful ^to examine the underlying assumption, ^{which is :}

«

It is clearly reasonable to assume that in a continuum the faces of a plane ^{are in neutral} equilibrium ^with respect ^{to shear} along ^the plane. ^The limiting ^{shear stress} (Ts is then zero

If this statement were to hold for infinitesimally ^small strains, ^{the shear} modulus 1£ ^would automatically ^{be zero.} The situation which is actually considered is one in which the shear stresse depends ^{on the} angle of shear strain 0 according ^{to a} law of the form

as long ^as

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500180251900

2520

and remains constant when 10 1 -- 00. The force law is discontinuous at 0

e o, but the force remains continuous.

The limiting ^{shear stress is then}

Cottrell then considers the situation in which 00 «.,c ^{1, and} points ^{out that as} 00 - ^{0 so the}

value of 0 at which the law of force becomes discontinuous also tends to zero. This is

analogous ^to the unrealistic

hard-sphere

model in atomic cohesion theory.

2. Analysis of the basic assumption.

A more realistic basic assumption ^{is not}

^{that in a} continuum the faces of a plane ^{are in}

neutral equilibrium ^with respect ^{to shear} along ^the plane

It is rather a variant of what is stated as a consequence ^{in the next} sentence, namely ^{that we are} considering ^a model in which the limiting ^{shear stress} uo is much less than 1£. It is this assumption which leads to the

discontinuity of the force law at very small strains. This is in fact ^an obvious consequence of the assumption ^that

Suppose ^the limiting ^{shear stress} au is achieved ^{at a} limiting shear strain 6 S. For small strains 0 we have

and, provided that the law of force remains smooth, ^{we must have}

with

It follows from (3) ^and (5) ^that

and thus from (6) ^that

The limiting ^{shear stress} is achieved at very small strains, which is Cottrell’s result. The medium is of the kind considered by Barsch and Krumhansl [2], ^{in which we} may neglect

«

geometric non-linearity » ^while retaining

physical non-linearity ».

3. The edge dislocation in an incompressible ^médium of this kind.

In a medium of this latter kind, Hooke’s Law is replaced by ^a non-linear constitutive relation at small strains where the geometry ^{of linear} elasticity is still valid. This means that we can

study the effects of the non-linear stress-strain relation without introducing the mathematical

complications associated with finite strains.

We further simplify ^this preliminary investigation by taking the medium to be isotropic ^and

incompressible, ^{and consider} only ^{the case of} plane ^strain.

Let the stress tensor be pij. ^{Then the} deviatoric stress, which alone determines the strain in

an incompressible medium, ^is

The only ^{invariant of} pÚ ^is

In an isotropic incompressible medium, the strain tensor eij ^must be similar and similarly

orientated to the deviatoric stress tensor. It must therefore be the form

where _IL is the shear modulus for infinitesimal strains and

We shall consider the simplest possible ^case

Here 00 ^{is the} angle of shear strain at which deviations from Hooke’s Law become

appreciable. ^{In a} conventional linear medium, 00

^{oo. The form} (13) ^{ensures that the non-} linearity is of the type already considered, in which the shear strain increases monotonically

and more rapidly ^than linearly with the shear stress.

In cylindrical coordinates we have

giving

Since the strains are assumed to be small, the medium will be in equilibrium ^without body

forces if the stresses are derived from a stress function X by the relations

and

For the case of a dislocation at the origin, the strains must be derivable from a displacement

field which is single-valued around any circuit which does ^not enclose the origin. ^The

condition of compatibility, converted from Jaeger [3] ^{into tensor} components, ^is

2522

We obtain the first two terms of ^a solution in ascending powers of b 2/r2 CP6 ^{which represents}

an edge dislocation at the origin. ^{The first} term, which alone remains as (Jo --+ 00, ^{is the} ordinary linear elastic solution, which is valid when r > b. In the present type ^of medium, 00 ^is small, and the second term is a valid correction to the linear solution provided r > bloo.

The proposed solution has the stress function

The first term is the standard stress function for an edge dislocation in ^an incompressible medium, ^{free from} body ^{forces at all} points including ^the origin. ^The danger that there may be

a force at the origin ^{when the rest} of the medium is free from body forces arises only ^{when the}

stresses are proportional ^to 1/r, ^{and is not present} for the second term in (24).

Inserting (24) ^into (21) ^and (22) ^{we obtain}

and

If we substitute (25) ^and (26) ⁱⁿ (10), and retain only ^{the term} of lowest order in

b2/ cP5 ^{r2, we obtain}

Substituting (25), (26) ^and (27) ^into (11) ^and (13), ^we obtain, ^to first order in

b2/ ~20 r2,

The last term in (29) represents the non-linear strain arising directly from the zero-order shear stress.

At large distances, ^the non-linearity introduces additional strains

and

which are larger than the direct non-linear strain.

We have to verify that the strains (28) ^and (29) satisfy the condition of compatibility (23).

We first note that, ^{because we consider} only ^the region where the strains are small, ^this

condition is linear in the strains. The strains

are those of an edge dislocation in ^a linear medium. They satisfy the condition of

compatibility, and may be neglected. ^An elementary ^but tedious calculation confirms that the choice of the coefficient of the last term in (24) ^causes (28) ^and (29) ^to satisfy (23).

The solenoidal displacements ^must be derivable from a potential tp by the relations

The appropriate function is

from which we derive the displacements

and the strains given by (28) ^and (29).

References

[1] ^{COTTRELL A. H., S.} Afr. ^J. Phys. ⁹ (1986) ^44.

[2] BARSCH G. R. and KRUMHANSL J. A., Phys. Rev. Lett. 53 (1984) ^1069.

[3] ^{JAEGER J.} C., Elasticity, Fracture and Flow, Methuen, ^London (1956) p. 45.