THE EFFECT OF COUPLE-STRESSES ON THE STRESS CONCENTRATION

THE EFFECT OF COUPLE-STRESSES ON THE STRESS CONCENTRATION

AROUND A MOVING CRACK

S. ITOU

Department of Mechanical Engineering Hachinohe Institute of Technology

Hachlnohe 031 Japan (Received March 20, 1980)

ABSTRACT. The problem of a uniformly propagating finite crack in an infinite medium is solved within the llnearized couple-stress theory. The self-equill- brated system of pressure is applied to the crack surfaces. The problem is reduced to dual integral equations and solved by a serles-expanslon method. ^The dynamic stress-lntenslty factor ^is computed numerically.

KEY WORDS AND PHRASES. Couple-Stress, A moving 6rack.

1980 MATHEMATICS SUBJECT CLASSIFICATION CODES. 73C.

i. INTRODUCTION.

The classical theory of elasticity is based on the assumption that matter is continuously distributed in any elastic body. Therefore, it is an approximation for polycrystalline metals, granular materials, porous materials, discrete mate-

rials, steel-fiber reinforced

concrete

and so

on,

^which are generallyknown as materials with microstructures.

An attempt to

drop the continuity of

matter as-

sumption is bound

to

make the analysis for the modified theory extremely diffi- cult.

Therefore,

for materials with

microstructures,

^some models of continua

.are

constructed in such a

way

as

to represent

a better approximation.

Among

the sev- eral theories available, the linearized couple-stress theory, which was developed by Mindlin and Tiersten

(1),

^is the simplest.

In

the

context

of this

theory,

a large number of analytical solutions have been published

as

shown in Ref.

(2).

On the other

hand,

^some research has been carried

out

regarding the experimen- tal evaluations of the new material

constant

which was introduced in the couple-

stress

theory.

In

Refs.

()

and

(4),

bending

tests

were performed on aluminium alloy plate and low-carbon steel and the upper limits of the material

constant

are presented.

Later,

^Savin and his coauthors accurately determined the

constant

bymeasuring the velocity of the

transverse

ultrasonic

wave (5).

The results showed that the value for falls within the limits from I0^-2

to

I0^-S mm for

brass, bronz,

duralmin and aluminium.

In

this

case, ^was

approximately one order of magnitude less than the

mean

grain size. This

means

that couple-stresses do

not

significantly affect the

stress

concentrations caused by the existense of circular holes or inclusions.

However,

the effect of couple-stresses ^{is serious} in fracture mechanics.

According

to

the linear theory of elasticity, the

stresses

in the vicinity of the ends of the cracks

are

inversely proportional

to

the

square root

of

r,

^the

distance from the end of the crack. Stress-intensity factor

Ki

is defined from the coefficient. If the plastic zone is very small relative

to

the size of the

crack,

we can assume that

K

is proportional

to

the applied tensile load and is a function of the crack and specimen sizes.

In

linear fracture

mechanics,

it is considered that when the stress-intensity

K K& ,

value and is characteristic of the

material,

^{an unstable} crack propagation

occurs.

This idea was established by Irwim

(6)

and is equivalent

to

Griffith’s original

concept (7),

that a crack will begin

to propagate

if the elastic energy released by its growth is

greater

than the

energy

required

to

produce the fractured sur- face

s.

The stress-intensity factor calculated according

to

the couple-stress theory is always larger than the classical

solution,

and

furthermore,

it becomes larger as the new material

constant

decreases

(8)(9).

For this

reason,

crack problems in the theory of couple-stresses

are very

important physically

(10). Nevertheless,

few studies have been carried

out to

reveal the effects of couple-stresses on the stress-intensity factor

(2),

^{because of the severe} mathematical complexity en- countered in finding the solutions fit

to

the geometries.

In

the

present paper, Yoff’s

model

(11)

is solved by the two-dimensional linearized couple-stress theory. The crack

propagates

only

to

the right, main- taining its

constant

length. Such an idealization will affect the magnitude of the local

stress

field

to

some

extent,

^{but does}

not

alter the qualitative features of the

stress

solution

(12).

Application of the Fourier transformation technique reduces the problem

to

that of solving

two

dual integral equations. To solve these equations, a

very

simple method is

used,

namely, the displacement and

rota-

tion on the crack’ s surfaces

are

expanded bya series of Jacobi polynomials with the Schmidt method being employed.

Numerical calculations

are

carried

out

and compared

to

those given by Yoff

to

clarify the influence of couple-stresses on the dynamic stress-intensity factor.

With

respect to

the fixed rectangular coordinate

system (, , ),

^the

equations of motion in the plane

state

of strain of the linearized couple-stress theory

are

with

(2ol)

(2.2)

where ^c,

((& ^+2#2 )/} ^w

^cv

^{(%z/ )w}

^{are the} dilatational and shear wave

velocities,

respectively. The

Lam constants are

represented by

J4

and

% ,

^is

the density of the

material,

is

time,

^{is the new material’s}

constant,

^u

=

_and

w

are

defined

as

the and

components

of the displacement, respectively, and the indices following a

coua

indicate the partial differentiation with

respect to

the

variable,

e.g.

, */0 .

^Consider the problem of

an

infinite elastic solid which contains a crack with a length represented by 2a along the x axis as shown in Fig.

1. It

is assumed that the crack is

opened at one

end and colsed

at

the other with

constant

speed

U. For

a

constant

velecity

crack,

it is convenient

to

introduce the Galilean transformation

x=-, y=, z=, t=E, (2.)

with

(x, y, z)

being the translating coordinate

system

attached

to

the moving

crack. In

the moving

coordinates,

the equations of motion become independent of the time variable

t,

(2.4)

of the moving crack with

respect to

longitudinal and transversal elastic

waves,

respectively, and is the Poisson ratio.

The displacements,

rotation,

force-stresses and couple-stresses

are

expressed by the following

1/2(/,= ⁺ /,= ^),

(2.8)

The boundary conditions for the problem

to

be studied

are

"7"i/v(+= -p(x),

^{for z}

o, Ixl ^{< a,}

J/= o, o=

^z

o, Ixl< ^a,

@ o,

^for

. O, Ix ^{> a,}

I ^O,

^{for z}

^O, Ix ^>a,

"z’ ^o,

^{_o" z}

^o, Ixl<’.

ANALYSIS

To find the solutions for the

wave

equation

(2.4),

^{we use the Fourier}

trans-

folNns

P( f(x)xp(i a x),

f(x) ( exp(-i x)d .

Using this

theory,

we

can

reduce equation

(2.4)

in x and z

to

the following ordinary equation in z

Due

to

the

symmetry

conditions in equation

(2.9),

it is possible

to

consider only the problem for the

half-space,

^z

_0.

The solutions

to

equation

(.2)

appropriate

to

z

_0

will take the following

forms,

;

A( )expC-gz),

B( )exp(-. z)+C( )exp(-,z),

with

(3.4)

where

A(), B(), C()

are independent of

z.

Then,

^{it can be} shown that the mixed boundaryvalue equation

(2.9)

yields the following four integral equations

(

J-

c|’ ^k.()+iJ k2()} ^{exp(-ix)d =-p(x),}

2 @

i@ks )+(e ^k( )} ^{exp(-i x)d O,}

^for

Ixl< a, (3.5.1)

with

@exp(-i x)d

- ^{oo exp(-i x)d O,}

^for

^{Ixl > a, (.5.2)}

where

o eJ

^{are the} transformed displacement and rotation on z

O,

respectively.

If we

assume

that

p(x)

is

an

even function of

x,

the solutions

can

be repre- sented by the series

.,..(v.)

(x/a)( 1-x/a )V

w

a-

2q-2

11=1

,/2)

(xla)(1_la )

for

Ixl< ^a,

w

eJ ^O,

^for

Ixl > ^a,

where

a, b

^are the coefficients

to

be determined and

p(V,)(x)

is a Jacobi Polynomial. The Fourier transformations of equation

(3.7) are (I 3)

o 2,,/,’ ^a..(-1 )"’-’ r-’(2n..,-) _Jz’n-,

=i

(2n-2)

, ^2-

ⁱ-i

^{b(- )"/’} (2n-1)! J2 (a), (.8)

where

(), J ^are

^{the Gamua and Bessel}

^function,

respectively. Equation

(3.7)

already satisfies equation

(3.5.2).

Substituting equation

(3.8)

into equation

(3.5.1),

^we obtain for

with

.(x)+ , ^{a.F (x) p(x)x,}

=, ^{bG(x)+ -,E s.,H(x) O,}

ECx) 2(-I) - ^{C2n+) [ {k, C )-2}/’J C a)sin(} x)d+2Cl-x’/a ) xsin{2n ^sin-’ Cx/a)}/C4n:-1 )-(x/a)cos{2n ^sin-’ (x/a)}/{n(4-I ^)I ) _’

() 2(-’)’-’ F (2n-) ^,.

(2n-2).v o{ ^{k’ ()/’}

^-k,

^{}/ J,_, ( a)sin(, x)d}

+k/(2a-1) sin{(2n-1 )sin-’(x/a)}),

e(x) _(2n-)

sin( x)d-2sin{2n ^sin-’ (x/a)}-(4-M%)/(4 z)E ^{(1-xZ/a ){} sinI2n

x

sin-’ (X/a)}/C4n:-1)-x/a consin-’ (x/a)}/(2n(4n’-1 )I)

(2n-2)

-1/

^zsin

{(2n-I ^)sin-’ Cx/a)}/C2n-I )), ^(3.so)

and

where the first equation in

(3.9)

has been integrated with

respect to x.

The functions

flC) kC)-2} ^J2Ca)/ ,

(..2)

behave as

t, ()-- o(-’),

t, )-- o( -" ^),

t, )-- o( -" ^),

t’, (a)--- o(-"),

^{for large}

, ^3.13

so that the semi-infinite integrals in equation

(3.10) ^can

easily be evaluated numerically by Filon’s

method.

Equation

(3.9)

^can be solved for the coefficients

a, b

^{by a} modified version of Schmit’s method

(14). Once

the displacement and rotation

at

the boundary

are found,

this analysis is considered

to

be complete.

4. STRESS-INTENSITY FACTOR

The coefficients

a,

b,

are known,

so that the entire

stress

field is obtain-

able. In

the fracture

theory, however,

^the significant quantity

to

be calculated is the direct

stress

acting across the radius from the tip of the crack.

It can

be written as

x

a+r

cos(e)

z r

sin(O (4.1)

and the

stresses

for the small value of

r can

be considered. The required

stress

’

^{is given by}

Te= Tx ^{sin (e)+z} cs ()-(Tz + zx)sin( ^)cos() ). (4.2)

For

the small value of

r,

it is shown

J_,( a)exp{-(1-m ^)V z } ^{cos( x)d}

(-1) (cos ^{e +(l-re’ sin" e )v} }{ ^/ { ^{2(1-m’ sin’ +O(r ),}

/" ^J,n_, a)exp{-(1-m" ){ _z } ^{sin( x)d}

(-1)"*’ {-COS ^{e +(1- sin’@} )v-} ^/ { ^{27 (l-re’ sin’ )} } ^{+O(r ).}

(4.)

Using equations

(4.2)

and

(4.),

we obtain the stress-intensity factor

K

_i

K ^{(e r/a/ r-}

⁰

(2n-2)! (M-2) (2+M-2Mt)

_s

_O

_+q

"M ⁽ 1-MI )*

^{co sin}

e )/q

with

cos(e )+qcos(e )/+ M4#+cos(e ^{(s+/-n’ e-cos’e}

+

8

41-cose ^{sin(e )cos(}

4(M-2) _Jq-cos( e ^sin( e )cos(e )/q

M

(4.4)

5. NUMERICAL EXAMPLES AND RESULTS

Numerical calculations arecarried

out

for

# 0.25

^and

p(x) P (constant).

The semi-infinite integrals which occur in equation

(3.10)

can easily be evaluated numerically, because the values for

f ), _f2 ), fs ^), f4

decay rapidly

when-

^becomes

large,

as shown in Table

1.

Adopting the first five

terms

of the infinite series of equation

(3.9),

we utilize the Schmidt procedure.

In

order

to

given in Table 2 for

/a 0.1, 0.5

^and

MT ^0.6. From this,

^{it is clear} that the

accuracy

of Schmidt’s method is satisfactory.

In

Table

3,

the

stress

intensity factor

KI/P

^for

=

⁰

^{@, M= 0.01, 0.6, 0.8, 0.9}

^is

^shown,

^{in which the values}

^put

in the circular-type brackets

are

those obtained from the diagram in Ref.

(8)

and the results given by Atkinson and Leppington

(9)

are also written in the

square- type brackets.

The values for My

0.01

coincide well with those corresponding

to

the static

solutions. In

Figs. 2 and

3,

^the

stress-intensity

factor

K2

^is

plotted against

MT

for

/a 0.1, 0.5

and

9= 0", 27 , 54 ^In

Fig.4,

Ki at

e

0 is plotted against

/a. It

is difficult

to carry out

numerical calcula- tions in the

/a<0.1 range,

because we

cannot expect

that the integrands in equation

(3. 0) to

rapidly decay when becomes large for a small value of

/a.

In

Fig.

4,

the broken lines

are

likely curves drawn suitably. Figure

5

^shows stress-intensity factor

Ki

^{versus for}

/a 0.2

and

My= 0.01, 0.8, 0.9.

As a result of the above

calculations,

we are able

to

deduce the following information.

i) In contrast to

the classical

solution,

the stress-intensity factor

K ^at

⁼⁰

is dependent on the propagating speed and becomes larger

as

the speed increased.

i)

The maximum value of the stress-intensity factor

K

^occurs

^at

^{0 even}

though the crack speed is increased.

Therefore,

the crack branching does

not

occur for materials which take a

nonzero

value for

iii)

The moving velocity of the crack has less of an effect on stress-intensity factor

K%

^{when the value for increases.}

iv)

Variations of the

stress-intensity

factor

KI

^{with the angle is much}

different from that of the classical solution.

Table

0.1

Values of f,

(), f2(), fs(), f4()

for n

I, MT= ^0.6

^and

/a _0.I, 0.5

^against

a.

0.01 0.21

40.0

-0.24950x.10 0.11099x10’ 0.11.350x10 0. 49899x102

-0.2880x10 O. 1067x-10 ^O.

^10821:x10:

0.4767210:

-0..047610-’ 0.96084x10-’* 0.1259x10

^-6

0.3944210

^-2

80.01 0.01

0.21

40.)1

80.01

-0.6740010 - -0.55057x10 - ^0.27258xI0 - -0.22575x10 -

-0.24751x10 0o11012x10’ 0.45002x10 ’ O. 19801x10’

-0.2011610 _{0.90954x10 0.35930210} 0o160610 -0.12456x10 - ^0.9156x10 - ^{0.95798’10 -’ 0.649910 -}

-0.2715410 -" -0.21915x10 - 0.84226x10-" -0.,56254x10 -

Table 2

0.1

Values of

., bE(x)+,, ^{aF (x) /P}

^and

h.G(x)+.= ^{(x) /}

for

My 0.6

d

/a 0.1 0.5.

a { ^{(x)+n F (x)}/P} { b(x)+,a(X)}/p

0.001 0.1

0.9 0.999 0.001 0.1

0.9 0.999

-0.9996835xI

-0.9997904xl 0-’

-0.499997010

-0.899998x10

-o. 998999

5,1

o -0.999972 5

^x

0-’

-0.999988 xl 0-’

-0.49999981 o"

-0.899998?xl

⁰

-0.998999210 =

O. 149504310 -’

0.113874x10 -

-0.5581461,,10

^-4

0.3729001x10

^’’4

-0.434080’10

^-:

-0.4316153x10 -

-0.69 69440xl

⁰

^-e -0.5124155x10 -

-0. 884164

^xl0-8

-0.1314380

^10-e

Table

3

Stress-intensity factor

Ki/P ^{at @} 0for

MT= 0.01, 0.6, 0.8, 0.9

and

/a 0.1, 0.2, 0.5, 1.0.

/a 0.1

0.2

0.5 ^1.0

MT ^0.01 0.6

1.22 1.199

( ^.2,1) ([ ^.202

1.3}1

1.279 1.553 1.418 2.097 1.626

1.121 1.056 ) (1.120) (1.06])

1.15 .o7

1.193 .o85 1.2.5 1.096

’Z

Fig.

Geometry

and coordinate

system.

0

0o0 0.5 MT

Fig 2 Stress-intensity factor

K

^for

^{2/a 0.1,} e=

⁰

2? 54

^versus

MT.

, la=0.5

27

0

0.0 0.5 MT 1.0

Fig

5

Stress-intensity factor

Ki

^for

/a 0.5,

27 , ^54"

^versus

^M-r.

Ki/P

MT-0.9

F/g.

4

Stress-intensity factor

0.9

^versus

/a.

j/a 1.0

at

g=

0

for _MT= 0.01, 0.8,

K/P MT=0.9

,/a=0.2

0.01

Fig.

5

Stress-intensity factor

K

0.8, 0.9

^versus

.

^for

^{/a 0.2, M= 0.01, 90"}

ACKNOWLEDGEMENT: The author wishes to express his grateful thanks ^to Professor A. Atsumi of Tohoku University for his constant help during the preparation of the paper. This work was partially funded by the Ministry of Education.

REFERENCES

[i] Mindlin, R.D. and H.F. Tiersten, "Effects of Couple-Stresses in Linear Elasticity"

Arch. Rat. Mech. Anal. ii (1962) 415-448.

[2]

Hlavaek,

M., "Vliv Momentovych

Napti

v Problmech Koncentrace

Napti",

Strojnicky

asopi.s

24(1973) 373-381.

[3] Schijve, J.,

"Note

on Couple-Stresses", J. Mech. Phys. ^Solids 14(1966) 113-120.

[4] Ellis, R.W. and C.W. Smith,

"A

Thin-plate Analysis and Experimental Evaluation of Couple-Stress Effects",

Exp.

Mech.(1967) 372-380.

[5] Savln, G.N., A.A. Lukasev, E.M. Lysko, S.V. Veremejenko and G.G. Agasjev,

"Elastic Wave Propagation in a Cosserat Continuum with Constrained Particle Rotation", Prikl. Mekh. 6(1970) 37-41.

[6]

Irwin, G.R., "Analysis of Stresses and Strains Near the End of a Crack Tra- versing a Plate", ASME J. Appl. Mech.24(1957) 361-364.

[7]

Griffith, A.A., "The Phenomena of Rupture and Flow in Solids", Phil. Trans.

Roy.

Soc.

London

^Series A, 221(1920) 163-198.

[8] Sternberg, E. and R. Muki, "The Effect of Couple-Stresses on the Stress Con- centration Around a

Crack",

Int. J. Solids Structures 3 (1967) 69-95.

[9] Atklnson, C. and

F.G.

Leppington, "The Effect of Couple-Stresses on the Tip of a

Crack",

Int. J. Solids Structures 13(1977) 1103-1122.

[i0] Tiersten, H.F. and J.L. Bleustein, Generalized Elastic Continua, R.D. Mindlin and Applied Mechanics (Ed. by G. Herrmann), Pergamon Press, 1973, 67-103.

[Ii] Yoffe, E.H, The Moving Griffith Crack, Phil. Mag. 42(1951) 739-750.

[12]

Sih, G.C., Some Elastodynamic Problems of Cracks, Int. J. Fracture Mech.

4(1968) 51-68.

[13] Erdelyi, A. (Editor), Tables of Integral Transforms, Vol. i, McGraw-Hill, 1954.

[14] Yau, W.F., Axisymmetric Slipless Indentation of an Infinite Elastic Cyli.nder, SIAM J. Appl. Math. 15(1967) 219-227.

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