Stress intensity factor for an infinite transversely isotropic solid containing a penny-shaped crack

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Stress intensity factor for an infinite transversely

isotropic solid containing a penny-shaped crack

M. Dahan

To cite this version:

M. Dahan.

Stress intensity factor for an infinite transversely isotropic solid containing a

penny-shaped crack.

Journal de Physique Lettres, Edp sciences, 1980, 41 (9), pp.213-215.

�10.1051/jphyslet:01980004109021300�. �jpa-00231762�

L-213

Stress

_intensity

factor for

an

infinite

transversely

isotropic

solid

containing

a

_penny-shaped

crack

(*)

M. Dahan

Laboratoire de Mécanique des Solides, Ecole Polytechnique (**), 91128 Palaiseau, France

(Re~u le 24 _{janvier 1980, accepte} le 13 mars ₁₉₈₀₎

Résumé. 2014 On donne la solution

explicite du _{problème axi-symétrique quand} des forces intérieures concentrées

sur une _surface, sont

_appliquées

à une distance finie de la fissure. Pour des charges appliquées sur les lèvres de la _fissure, les résultats donnés sont

_{indépendants}

des coefficients

_élastiques

et ont les mêmes valeurs que pour un

milieu _isotrope.

Abstract. 2014 A closed form solution of _an

axisymmetric

problem

is obtained when an interior surface concentrated

loading is

_applied

at a finite distance from the crack. For a _{general loading}

_applied

to the crack _surface, the results

are

_independent

of the elastic constants and hence, they have the same values as in an

_isotropic

case.

J. _Physique - LETTRES 41 (1980) L-213-L-215 1 ~’’ MAI _1980, Classification Physics Abstracts 03.40D - 46.30N 1. Introduction. - Let us consider a

_homogeneous

elastic medium

_occupying

the three-dimensional eucli-dian space, referred to a fixed

rectangular

cartesian

coordinate _system

_(Oxyz).

We shall assume that the

medium is characterized

_by

transverse

_isotropy,

having

z-axis as elastic _symmetry axis.

_By

_using

the

cylindrical

coordinates

_(r,

0,

z),

we shall note

_(u,

_0,

_w)

the

_components

of the

_displacement

_field,

(o~.,

7~, ~zz~

O’rz)

the non-zero _components of the

stress tensor and

_(err,

_{G66, ~,}

_8~)

the _components of the strain tensor.

Then,

the strain-stress relations

can be defined

_{by :}

where all, a12, a13, a33, a44 are the five

_independent

elastic coefficients of the medium.

(*) La version française de cet article a etc acceptee aux _Comptes

Rendus de 1’Academie des Sciences et est insérée dans le n~ du

14 janvier 1980.

(**) ERA C.N.R.S. no 455.

The solid contains a

_penny-shaped

crack defined

_by

0 r ro, z = 0 and it is

subjected

to

body

forces

field

_parallel

to the

z-axis,

Fz(r,

z),

assumed to be odd functions of z. In this case, we shall solve the

equi-valent

_problem

for the

_half-space

z >

_0,

actionned

internally by

Fz(r,

z)

and

_subjected

to the mixed

boundary

conditions :

and

_regularity

conditions at

_infinity.

Let us assume that the

body

forces

_{F z}

are

concentrat-ed on a

_plane

interior

_surface,

round the

z-axis,

i.e. of the form :

where p is an

arbitrary

function defined for r ~ 0

such as the Hankel transform of order zero

pH

_exists,

6 is the Dirac distribution and h is a

_positive

constant.

In a

_previous

_paper

_[1],

it has been shown that a

potential

function of the Love _type exists for this

problem

and is defined

_{by :}

L-214 JOURNAL DE _{PHYSIQUE -} LETTRES

where the functions A and B are determined

by

the

boundary

conditions

(2).

In the last

expression,

we have

used the

_following

notations :

and :

Using

the

_potential

function,

the

_boundary

conditions

₍₂₎

are

_given

_{by :}

2. Resolution. -

First,

we introduce the constants :

Then,

the condition

₍₇₎

_implies

the relation :

If we introduce a

_{supplementary}

function _X such that :

with

_XeD)

=

0,

the condition

(9),

transformed

by

(11),

is

_identically

verified. The condition

₍₈₎

can be reduced

to an Abel

integral equation

for which the

following

solution exists :

Then,

the stress

_intensity

factor defined

_by

the limit :

is

_equal

to :

When the loads are

applied

on the crack

surface,

we have h = 0

and,

from

(15),

_we deduce :

3. Discussion. - It is

important

to remark that the last result

_depends

_only

on the

_{loading p(r) applied}

on the crack

_through

its

Hankel transform

pH(m).

Thus,

the elastic coefficients of the material do not

appear in formula

(16),

and,

therefore,

every result

existing previously

for the stress

_intensity

factors in an

isotropic

medium is

_{generalizable}

to a

_transversely

isotropic

medium when the loads are

applied

on the

crack surfaces. This is

_surely

not true when h is dif-ferent from zero.

The

_{expressions (15)}

and

₍₁₆₎

_give

closed form results to most of the

_{loadings. Specially,}

we obtain

for a uniform

_{load po}

over a circular area of radius _rl,

below ro, and at a non-zero distance h from the

crack ;

L-215 STRESS INTENSITY FACTOR IN ANISOTROPIC SOLID

with : If the

_loading

is actionned over the crack

surfaces,

we can tend h to zero and we deduce :

This last

_expression

_generalizes

Sneddon’s result

_[2]

given

for an

isotropic

material.

References

[1] DAHAN, M., PREDELEANU, M., C. R. Hebd. Séan. Acad. Sci., Paris 289 (1979) 147.

[2] KASSIR, M. K., SIH, G. C., Three dimensional crack _problems (Noordhoff Int. Publishing, Leyden) 1975.