Crack Paths 2009

Further for the spherical cavity under tri-axial loading, 5, with internal pressure, p, the

result in spherical coordinates is:

— 3 — 3 M a n d O ' , I O T — M f O r F Z R (6)

o 6 : o ' ¢ : o T +

2r

2r

Both of these formulae give the stresses on a prospective crack plane extending outward

from the cavity. For the latter case in Eq. (6) the radial displacement from the surface of

the cavity outward is given Eq. (7) where E is the elastic modulus. _ 3 _ rimvity : (1+ v)(o' + p) R_2 + (1 2v)o'r

(7)

M

2 E r

E

For the spherical inclusion with external pressure, p, the radial displacement of the

surface is:

1— 2 '

R

urIinclusion :_

%

where E’ and v’ are the elastic constants of the inclusion.

If the spherical inclusion is larger than the spherical cavity it occupies, then there will

be a contact pressure, p, which will also depend on the external hydrostatic tension, 5.

The mismatch shall be that of an inclusion which is larger by a radial amount, A. Then

the compatibility of the radial displacements between the cavity and inclusion can be

expressed as:

ur—inclusion+ A : ur—cavity (at r : CombiningEqs. (7, 8 and 9) leads to an additional stress outside the cavity as:

A l + v 1—2v' e@(r-R)-[2R)/[2E+ E, )

(10>

which should be added to the previous stress result Eq. (6) with r I R and [2:0 Under

such circumstances the pressure, p, generated between the inclusion and the matrix is

given Eq. (11). The contact between inclusion and matrix is lost if

A / R§3(1—v)5/(2E).

A _ 2 (l—v)fi]/i:l+v +

(compression)

(11)

p211? 2 E

2E E’

N o w ,if there is no mismatchand the inclusion is bondedto the external body, then the

stress concentration is: KIl1—l*V1/l*”'+“1 2

2 E E’

2 E

Notice that for E ’ : 0that K, :3/2; for E’:E,v’: v that K, : 1;

and for E ’ :00 that KZ :i which is as expected. With this in mind the stress in the

1+ v

body next to the inclusion is:

1

_ A 1+ v 1— 2v’

U 6 i m a XR( r) :K I U

E , j

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