Elastic monopole modelling of defects in thermoelastic media

ELASTIC

Department

MONOPOLE MODELLING OF DEFECTS IN THERMOELASTIC MEDIA

of Mechanics.

S. A. ZHOLJi_ Royal Institute

and R. K. T. HSIEH S-100 44 Stockholm of Technolog!.

70. Sweden

Abstract-A unttied theory for the treatment of the quantitative properties of lattices defects thermoelastic media has been developed in this artrcle. It is shown that the defects may represented by a distribution of surface and volume elastic monopoles. It is further shown that arbitraq three-dimensional (planar or curved) crack can also bc modclled h) a distributton surface and volume elastic monopoles. the monopoles of which habe to satisfy a system boundary integral equations.

tn he an ol of

INTRODUCTION LATTICE defects are always present in crystalline solids. They change considerably many properties of the crystal. A quantitative treatment of their properties can be given by the continuum model of a lattice defect [l-3]. However. each type of defect has to be treated as a separate problem. Using the concept of elastic multipoles it is possible to give a unified approach to describe the quantitative physical properties of the defects in various materials, for example. elastic [4], micropolar elastic [5]. or nonlocal elastic [6] and electromagnetic elastic [7]. In this paper we shall treat the case of thermoelastic materials. In addition. in all these previous works, emphasis has been laid on the study of the properties of the lattice defects, such as line defects (dislocation and disclination) and volume defects. Here we shall also be concerned with the study of the properties of cracks. the influence of which on the mechanical behaviors of material is considerable and is one of the main tasks of fracture mechanics and material science. A lot of different methods have been developed to treat crack problems [S-lo]. The approach used here is based on the concept of elastic monopoles and it is expected that both lattice defects and cracks can be treated on a common basis. I.

BASIC

EQUATIONS

OF

THERMOELASTICITY

In thermoelasticity [ 1 I] the anisotropic medium reads

mechanical

t,, =

AND

constitutive

ELASTIC

equation

MULTIPOLES

for a homogeneous

C,,XIUk.I - P$.

(I)

where t,, is the stress, uk is the displacement component and 0 denotes small changes in temperature from the reference temperature, p,, are the thermal moduli, C,,L, are the elastic moduli. The condition of static equilibrium is 4,., +

x, = 0,

(2)

where X, is the body force. In the case of steady temperature of heat conduction is

where k,, are the heat conduction following symmetry properties:

coefficients.

The coefficients

distribution,

C,,L,, P,, and k,, have the

A, = P,, 7

kij t On leave from the Department

=

of Applied

k,,

.

Mechanics. II97

Fudan

University.

the equation

Shanghai.

China

I IYX

Substituting

S

‘L

Lt101

~11x1

eqn ( 1) into the equilibrium

K

h

condition

tlSIt.II

I

we get

C~i,h/k./, ~~~/jiiH., t \,

0.

(-1)

The solution of this equation can be found using the Duhamel-Neumann Analogy theorem [12]. This states that instead of considering our thermal elastic solid with the constitutive eqn (1) and body forces ,Y,. we can consider an equivalent elastic solid with body forces./; and elastic stresses o,,. the displacement II, being unchanged and the thermal changes 0 being identically equal to zero. We thus have

The equilibrium condition:

conditions

are (3) and

(4). thus

reduced

to the elastic

equilibrium

C’,,#~,,, + .j; = 0. Equation

(5)

(5) can be satisfied by the solution u,(x)

Here, G/,(x, x’) is the Green’s

=

function

,/;(x’)G,;(x,

s

tensor

(x, x’) + ~‘,,l,,,G‘/,,m,

x’)dL’.

defined

by the equation

6,,,6(x ~ x’) = 0,

where 6(x - x’) is the Dirac 6 function. Let us consider volume centered at x’ of an arbitrary elastic continuum. can be written as u,(x) =

c

(6)

(7)

N point forces acting in a small The resulting displacement field

,f:Glk(x. x’ + d”),

(8)

a=I

where x’ + d” is the position vector of the force f‘:. Expanding Taylor series about (x, x’) we obtain

where we introduced

In the simplest

the elastic multipoles

case, the elastic monopole

the Green’s

function

in a

of order q

is defined

as

h

Pp = C.f‘Fd:‘> where we have used the equilibrium condition P,= 0.The interaction monopole with other strain fields e$ can be found as C’ = -P,kei.

(11)

energy of an elastic

(12)

Elastic monopole

The interaction

2.

force .f’::’ acting

LATTICE

DEkt(‘TS VOLlJME

modelllng

on a monopole

IN

THERMOLLASTIC‘

DISTRIBIJTION

OF

I IYY

oE defect<

reads

MEDIA Et-AS-TI(‘

AS

SURFACE

.AND

MONOPOLES

It is known that real materials always contain defects. These defects arc then sources of internal elastic stresses. In the case of cut surface defects. they can be defined as the following. We tirst make a cut in the body along the surface S,. bounded by the curve C’ and slip the two sides of the cut relative to each other by a displacement trk(x’) (see Fig.

I)

We then weld the two sides (perhaps b,, adding or removing material) and create in such a way a defect along the curve C’. To find the displacement field ~r,jx) in the medium due to the so created defects. let us multiply the Green’s equation (7) by 2(,(x’) and integrate over the volume of the body, excluding the cut surfaces. by a surface S’. Then for an infinite medium when C;,,Jx. x’) = G,,,,(x - x’) one obtains

u,,(x) = -

Applying

Id,(X) =

the Gauss

theorem

s

s, [W)G/,,(x

we let S’ tend

(15)

C;,,& ,,,.,,,,(x - x’)z!,(x’)d I “.

we get

- x’) + C~,/,JLn,(x- x’b,(x’)W; -

Now

sI

s

to the cut

I, [G,,,(x - x’)t,m.m,(x’) + P,mfW)G,n.n,(x- x'lldt".

surface

S,.. In equilibrium.

Fig. 1.

the

stresses

have

( 16)

to be

S. A. ZHOll

I 200

continuous,

and R. K. T

HSIEH

and if the body force is absent (.I-, =- 0) we obtain the displacement

held due

to the defect eqn (14) as

u,(x) = ~

where

I-‘” denotes

C’r,,,nG,,w,~ 6’ W:

the whole space. Noting

.-

J II

\i

~/,rr~/,w(x

x’)%( s’)d I .‘.

(17)

eqns (9) and (10) we can define

dp,,, = C’,,/,rru,dS; + P/,,,%db”.

(18)

Equations ( 17) and ( 18) show that cut surface defects in a thermoelastic medium can be modelled by a continuous distribution of surface and volume elastic monopoles. The point thermal contributions are given by the second term on the right sides of eqns (13) and (I). In particular. eqn (18) shows that the thermal effect is equivalent to that of a distribution of volume elastic monopoles. The interaction energy between the defect and the other strain field ei can be easily calculated by using eqn (I 2). This leads to

(19)

By eqn (I 3) the total force .I’:’ acting

on the defect writes

(20) Equations (17) and (18) are equally applicable to any type of cut surface defects in a thermoelastic medium. For a certain type of defect it is necessary to give the surface SC and its transformation uk(x’). In what follows we give these definitions for some wellknown basic defects. In the case of line defects (dislocation and disclination), according to the Volterra model, they can be created by a rigid body motion defined by

adx’) = bk + Q,~Q~(x; -

x:,,

(21)

where hk is the Burgers vector. R, is the Frank vector. It can be seen that in the case of a dislocation (hk = const, Q,, = 0) the displacement field is

(22)

while in the case of a disclination

u,(x)

= 4,

(b, = 0, C& = const) the displacement

s se

s

Q&I - x~)C,,,mGn,md~; -

V=-S,

field becomes

P~mG/n,m% d v’.

(23)

The interaction energy and force of the defect with the other strain field e$ can be calculated by eqns (19) and (20). This method is also applicable to volume defects (close surface) by applying Gauss’s theorem to eqn (17). 3. CRACKS

IN

THERMOELASTIC DISTRIBUTION

MEDIA OF

AS

ELASTIC

THE

SURFACE

AND

VOLUME

MONOPOLES

Let us consider an infinite homogeneous anisotropic thermoelastic medium, in which there is a slit on a smooth bounded surface SC (a three-dimensional crack). Without loss of generality, we consider the crack subjected to the external traction ty on the crack

Elastic monopole

modelling

of defects

IX1

surfaces .S = SC?U S; . where .S’t are parts of Ljapunov surfaces differing only in their unit normal vectors II,!/,~;= -nkls; (see Fig. I ). The stress boundary condition on the crack surfaces can be written as r,,fx)n,(x)

=

IT(X)

x E

s.

(74)

We shall consider the symmetric crack loading

The displacement vector v~(x), which is a continuous function in all space. with the exception of the surface S,., corresponds to the solution of this problem. On passing through S,.. the displacement [Q(X)varies by a jump since the crack surfaces are displaced by the external loads. Therefore a crack in thermoelastic media may be generally defined as a kind of general motion relative to the crack surfaces by

By eqn ( 18) it can be seen that the mechanical part of such a crack may be modelled by a distribution of surface elastic monopoles and the thermal eIEect may be represented by a distributio~l of voiume elastic monopoles. Obviously. the geometry of a crack is determined by the shape of the slit S,.. but the behavior of the crack under various loading on the crack surface is related to the function uk(x’) which is an unknown beforehand in the case of a crack and which should be selected to satisfy the given boundary condition on the crack surfaces. From the thermoelastic constitutive eqn f 1) and by eqn (17) the stress field created by the crack subjected to some loading may be expressed in terms of the distribution of elastic monopoles as

The stress vector ti(s) on an arbitrary surface S with the unit normal vector n(x) may be expressed as

In order to determine the unknown function a@‘), we can perform the limit process x - r E S,, in eqn (27) and use the given boundary condition (24). The result of the limit process is the following boundary integral equations:

(28) where

r,*(f) = $l) + P,,WhJ1”) + n&n

s

C,,qkn&,&,,mk(i-- x’)@x’)dt”.

V=-S<,

The integrals in eqn (28) exist in the sense of the Cauchy principle value [I 31. The interaction energy between the crack and the other strain field (2%can be easily obtained by using eqn ( 19).

1707

S. 4. ZHOI

and

K. k.

I. IiSIFH

This analysis also shows [see eqn (1X)] that the thermal effect is replaced b> the distribution of volume elastic monopoles. The results for the case of defects in elastic media are therefore obtained by modelling the cracks as distributions of surface elastic monopoles in these media.

It has been shown that lattice defects in thermoelastic media can be modelled as distributions of elastic monopoles. It was also shown that an arbitrary three-dimensional crack, too, can be described as a distribution of surface and volume elastic monopoles. This theory thus gives a unified approach and expressions for the quantitative physical and material properties of defects in both solid state physics and fracture mechanics. The crack problems have been solved by introducing integral equations, the analytical solutions of which may be found in a few cases only. However accurate numerical solutions can be obtained relatively easily which makes this present approach superior to other stress analysis method in many cases. This analysis also shows that by neglecting temperature effects. the results apply for defects in elastic media. For lattice defects. they are in accordance with previous results [4]. while for cracks. they show that these latter can be modelled as distributions of surface elastic monopoles.

RF.FEREN(‘E:S

[‘I PI [31 [41 [51 [61 [71 [81 [91 1101 [I 11 ;rs;

A. H. COTTRELL and B. A. BILBY. /%x K Phjx Sot,. A62, 49 ( 1949). J. D. ESHELBY. Propzss ,n Solid Mwhunic~s, Vol. I I North-Holland. Amsterdam ( I96 I ). I. KOVACS and L. ZSOLDOS. Di.c/ocutron\ und P/u\tic~ Deformation. Pergamon Press, London (1973) I. KOV&S. Ph,wica 94B, 177 (1978). R. K. T. HSIEH. G. VoRiiS and I. KOVACS. f’/?~,,rc,u lOlB, 201 (1980). I. KOVi\CS and G. VeRijS. Phv.srcu 96B, I 1I (I 979). R. K. T. HSIEH. Mulfipolur Lut;icr De/&~s rn kktromugnefic Elust~ C’onrrnua. The Mechunrcul Behavior of‘E/L~ctmrna~nelic Solid C’ontinw. (Edited by G. A. Maugin) p. 3 IS. North-Holland. Amsterdam (1984). B. A. BILBY and J. D. ESHELBY, Di.~locu/ion.c und (he Theor), o/ Frucrwe. Fracture. (Edited by H. Liebowitz). Vol. I. p. 99. Academic Press. New York (1968). D. T. BARR and M. P. CLEARY. Thermoelastic Fracture Solutions using Distributions of Singular Influecce Functions-l, II. Iwt J. Solids Srrrrcrwcs 19, 73 ( 1983). V. SLADEK and J. SLADEK. Three-Dimensional Curved Crack in an Elastic Body. Inr J. Solid.s Strwtw~~~ 19, 325 (1983). Y. C. FUNG. Fowzdutions of Solidr 9fechunrc.t Prentice-Hall, Englewood Cliffs. New Jersey (1965). A. J. A. MORGAN. A proof of Duhamel’s analogy for thermal stresses. J Arro. Scr 25, 466 (1958). V. D. KUPRADZE. Potential Merhods in the Theoq of’E/astiri~y. D. Davey. Jerusalem (1965).

(Received

28 JunwrJs

1984)