Crack propagation in an orthotropic medium

Engineering Fracture Mechanics Printed in Great Britain.

Vol. 29, No. 5, PP. 535-548,

CRACK

0013-7944/88 $3.00+ @ 1988 Pergamon Press

1988

.OO

pk.

PROPAGATION IN AN ORTHOTROPIC MEDIUM A. PIVA

Dipartimento

di Fisica, Universita di Bologna, Bologna, Italy

E. VIOLA Istituto di Scienza delle Construzioni, Universita di Bologna, Bologna, Italy Abstract-In this paper, the partial differential equations related to elastodynamic plane problems in an orthotropic medium are reduced by means of composite transformations to a first order stress

elliptic system of the Cauchy-Riemann type. This result justifies the representation of and displacement fields in terms of holomorphic functions defined into appropriate complex domains. The above formulation has been used to solve the boundary value problem concerned with the steady state propagation of a finite straight crack in an orthotropic medium. Some asymptotic results are also represented and briefly discussed.

1. INTRODUCTION

usually encountered kinds of damage in composite materials is the occurrence and propagation of cracks. Although significant progress has been made in solving elastostatic crack problems in anisotropic materials the literature on elastodynamic problems is somewhat meager. Atkinson[l] studied the steady-state propagation of a semi-infinite crack in an aeolotropic material by means of the Cauchy integral formula. The Griffith crack propagation under steady-state conditions, has been solved by Kassir and Tse[2] through an integral transform technique. Danyluk and Singh[3] obtained closed form solutions to antiplane problems of a crack moving in an orthotropic strip. Arcisz and Sih[4] examined the same problem as in [2] by reducing the related boundary value problem to a system of dual integral equations. A complex variable approach has been used[5] to extend the analysis of [3] to a strip, made up of an anisotropic material, with one plane of symmetry. An aim of this paper is to point out a new method, recently proposed in [6,7], allowing a ONE OF the most

complex variable solution, to formulate and solve steady-state elastodynamic crack problems in an orthotropic medium. The approach deals with the methods of linear algebra to transform the equations of motion into a first order elliptic system of the Cauchy-Riemann type, so justifying the introduction of the complex variable notation. The expressions for the stress and displacement components are then employed to solve the boundary value problem concerned with the steady-state propagation of a Griffith crack under a self-equilibrating load on its sides. In particular, the stress and displacements fields as well as the relation between the energy release rate and the crack speed are obtained. Finally, some asymptotic results are represented with some comments in the topic of fracture mechanics.

2. MATHEMATICAL

PRELIMINARIES

Consider an infinite orthotropic elastic body with the axes of elastic symmetry coinciding with the Cartesian coordinate axes X, y and z. By assuming the displacement component along the z-axis to vanish with all its derivatives with respect to z, the system of equations of motion governing elastodynamic problems in the 535

536

A. PIVA and E. VIOLA

xy-plane reduces to@]

Cl1

$+ c,,$+(c,,+ C66)G=P$y a2u

2

(2-l) 2

c66$+ c**$+(c,,+ c&35= axay pS' in which u = U(X, y, t), u = v(x, y, t) are the displacement components in the x and y directions, t is time, p is the mass density of the material and Crj are the elastic coefficients. The stress-strain equations may be written as follows

(2.2) au au WXY= C66(-+-, ay ax> with a,,, a,, and uxy being the Cartesian stress components. By setting X=x-a,

Y=y,

t= t,

(2.3)

where c is a constant speed and assuming u = u(X, Y), v = u(X, Y),

eqs

(2.1) become

2

a*u +cu%=o, $+2p-axay au2 (2.4)

with

2p=

cl*+ G6 Gl(1 -MT)' C

a=C11(1 k:y

2&=

cl*+ G6 C66U - M3 C

a1= C6b(l

(2.5)

L43

The Mach numbers A4j= c/Uj (j = 1,2), in which z11= ( CII/p)“* and 712= (C&p)“*, assumed less than one (subsonic propagation). The system (2.4) may be rewritten as

will be

Crack

propagation

in an orthotropic

medium

in which I is the 4 X 4 identity matrix and

It is assumed that A has no real eigenvalues characteristic equation

so that the system (2.6) is elliptic. Thence, the

m4+2aim2+a*=0,

(2.8)

where 2a1=cu+cw1-4/3/31,

a2=aa1,

(2.9)

provides four distinct either complex or purely imaginary roots, which occur in conjugate 3. THE CASE OF COMPLEX When the elastic properties

of the orthotropic l@
EIGENVALUES

material are such that (3.1)

a;?>%

the eq. (2.8) has complex roots. Without loss of generality, ml = y1+

iY2,

where a bar denotes complex conjugation,

Yl =

The corresponding

it is always possible to have

m2=-ml,

(3.2)

Y2>0,

and

[;(J;II+ a1,]1’2.[$i& q2. y2

eigenvectors

pairs.

=

(3.3)

may be chosen to be 2pm: (a + 6:)

h(l) =

h(2) =

(> Wfil

--(a+r$)

(3.4)

fil

1

In the new basis

(Im h(l), Re h(l), Im hc2), Re hc2’), the matrix A is represented

by the matrix I? = T-‘AT, -2&‘3

T=

wP2

in which the following contractions

with

2pP4

-2pP3

WPl

wP2

-2PP1

-Yl

-y2

1

0

have been used

(3.5)

Yl

1

’ >

det T # 0,

(3.6)

538

A. PIVA and E. VIOLA

ml

@2, a+mT - PI + By considering

the invertible

(3.7)

transformation

where *(X, Y) is a 4 x 1 matrix valued function, and Y, the system (2.6) may be rewritten as

with real entries

of the independent

Ig+B~y=o, where

it may be shown

variables

X

(3.9)

that

(3.10)

By setting

x1=x--=

d+ 7:

Y,

=L

Yl

Y:+ d

Y, (3.11)

x,=x+L the system

r:+

r;

Y,

Y2=

Yl,

(3.9) yields

a*1 _- a*2

ax,-aY*'

a** ---

a-

ax,’

aq3 -= ax, au,' au,

aq3 aq4 --=-_

a*‘2 (3.12)

aq4 ax,'

Hence, assuming continuity of the partial derivatives up to the second order, the CauchyRiemann eqs (3.12) grant that ‘Pj(j = 1,2) and qk(k = 3,4) are pairs of conjugate harmonic functions into the z, = X1 + iY, and z2 = X2 + iY2 plane, respectively. Introducing complex notation by setting

R,(z,) = *, + iT2, (3.13) f12( z2) = T3 + iT4, where aI components

and flz(z2) are holomorphic (2.2) become

functions

and combining

(2.7),,

(3.6), (3.8), the stress

rrxx = Ch6{klImKh(zd + fb(zdl- k2ReKh(zd - fMzdlL

(3.14)

cw = C,,{k3Im[fh(zd + %(.41- kaRGh(zd - fM4lL

(3.15)

axy

= C6,{ ks Re[%(zJ

+ %(zdl+

k6 Im[Kh(zd - ~2(zdll,

(3.16)

Crack

propagation

in an orthotropic

medium

539

in which

(3.17)

k = Wpz

4. THE

CASE

-

k6 = %$I

~2,

OF IMAGINARY

In order for all roots of (2.8) to be purely conditions

which,

the two eigenvalues

the orthotropic

employed

with

(a: - a1)1’2]1’2,

for the complex

eigenvalues,

Ig+cFy=o, in which

c=

( 1 0

-p

p 0

0 0

0 O-q’ 0

0

0

q

0

must satisfy the

(4.1)

q = [al +

- a2)1’2]1’2,

method

material

al>&,

ml = ip and m2 = iq may be chosen

p = [al -(a? positive constants. By following the same becomes

~1.

EIGENVALUES

imaginary

a2>0,

under

-

(4.2)

the system

(2.6)

(4.3)

0

@W, v

= uwx,

Y),

(4.4)

0

with

0

U=

( VP

a - p2

-P 0

2PP2 (Y- p2

wq2 ff - q2

0

WI

0

(Y -

0 1

0

q2

.

(4.5)

0 1 J

-4 0

The above eq. (4.3) confirms that the functions ‘l’j(i = 1,2) and qk(k = 3,4) are pairs of conjugate harmonic functions into the z1 = X + iY, and z2 = X + iY2 plane respectively, with Yl = Y/p and Y2 = Y/q. Introducing again the complex notation, the stress components (2.2) become

oitx = 2

Im[hfh(zd

CYY

Im[p2Mh(zA

~XY

=

=

+ ~2fl2(~2)1,

(4.6)

q2W2(z2)1,

(4.7)

ce.6Re[p15Ql(Zd+ +-6fi2(22)1,

(4.8)

C66

+

540

A. PIVA and E. VIOLA

where a3P2 l’=(a_p2)(1_~)+(2P-00,

w2 [Z=(~_q2)(1_M:)+(2p-a)

I,=(l-i%@-2P

5. THE DISPLACEMENT

FIELD

When complex eigenvalues occur, the displacement (3.8) by detailing the following expressions

% = g

= -2P{p4

Re[fh(zd

% = Fy = Idfh(zd

where the analytic functions relations

components

fM.41 + p3 Im[fh(zd

-

(4.9)

a-q2’

may be obtained

+ fMzdb,

from

(5.1)

w-9

+ fi2(~2)1,

Qj(zj) admit analytic primitives

tij(zj) satisfying the following

(5.3)

w2(z2) =

Hence, integrating

I

C&(z2)dX =

‘(‘,:=i,‘:; j-f&(.4 2

d Y.

(5.4)

1

(5.1) and (5.2), gives respectively u(X, Y) = -2P Id@3

+ @dw(~J

4X, V = -Im[(y, + ~~2Mz1) In the case of imaginary eigenvalues

-

(p3 - ip4b2(z2)1,

(71- iy2)02(22)1.

m2 2PP2 2 %(z1) + --_?fi2(z2)

[ ‘y-p

ax

(5.5) (5.6)

eq. (4.4)2 lead to

au

-=Im

g

+

= ImWdzd

“-9

1

+ fi2(z2)1,

9

(5.7)

(5.8)

where the following realtions hold w1(z1)=

02(z2)

=

I

I

fl,(zl)

dX =;

&(z2)

dX = 6

‘I

‘I

Ri(zi) d Y,

(5.9)

fi2(z2) d Y.

(5.10)

Crack propagation

Hence, integrating

in an orthotropic

medium

541

Y(Z24

(5.11)

(5.7) and (5.8) yields u(X, Y)=Im

-

2PP2

%(Zi) + -3

[ a-p2

u(x, y) = -RdpwW

+ qdzz)].

6. THE PROPAGATING

(5.12)

CRACK

PROBLEM

Consider a straight crack of constant length 2a, with edges, propagating along the x-axis with constant speed c, in infinity. Taking into account the symmetry and referring (X, Y) attached to the mid-point of the crack, the boundary

1X1-c a, 1x1cm, (Xl> a,

UYYW,0) = -Po, UXYW,0) = 0, u(X, 0) = 0,

uxx,

UYY,

When purely imaginary eigenvalues as follows

uyy

UXY

=

=

0,

\ZjI+m.

occur, the stress components

fr =

@XY

uniform tractions po applied to its an orthotropic medium unloaded at to the moving coordinate system value problem may be written as

- ph&A1(zdl,

Re[qLl&(zz)

F

(4.6-4.8) may be rewritten

Im[Al(zJ

(6.2)

- A2(z2)],

in which

(6.3) A.Z(z.Z)= - +z(zz), 5 and A = $3 16- 41415. In view of (6.2), the conditions (6.l)z and (6.1)4 give A,(X)

=

AZ(X),

1x1

(6.4)


so that eq. (6.2)2 by virtue of (6.1), leads to Re A(X) = B0 J

7

lX(
The Dirichlet problems (6.5) with the symmetry properties value problem (6.1).

j=l,2. Ai

(6.5) = Aj(Zj) solve the boundary

542

A. PIVA and E. VIOLA

Following the approach of [9] the solutions are A.(z.) I

I

=

-J!L

I;‘(z.)

Ct%

I

I

7

(6.6)

j = 1,2,

with (6.7) By utilizing (6.6) in (6.2) the expressions for the stress components

may be obtained

(6.8)

It can also be shown from (5.11) and (5.12) that

(6.9)

4x

Y) =

&

Im[&G(zd - fd%(zdl,

66

with Gj(Zj)=(~,2_a’)~‘~-Zi, j= 1,2. When complex eigenvalues occur the boundary value problem (6.1) may be solved by the same approach as for the imaginary eigenvalues. The stress components in eqs (3.16) can thus be written as

WY =

flxu

=

E

POW: + kg) 2A1

ReCFdzd- Mdl,

in which Ai = k3k6- k4k5 and the corresponding 24(X,

(6.10)

14 RetF;(zd + f3.dl-t (bk + k.+hJWFdzd - WZZ)]),

I?=-$

((p3k + p&J Re[G,(zd

66

+

displacements

(5.5) and (5.6) become

G(zJl f (p&s - p&d ImCGdzl) - G(zdl~

1

(6.11) v(X, Y)=-*

_

66 1

((y1k6-t- y2ks) Re[Gdzt)-

G2(4+

tY,ks-

7. THE LOCAL STRESS AND DISPLACEMENT

nkh) MG&d+

Ghd.

FIELDS

It is well known that the crack propagation features are controlled by the local stress field. Limiting attention to the right hand crack tip and introducing polar coordinates (rj, ~j) the

Crack

following

relations

hold when

Zj =

propagation

imaginary

in an orthotropic

eigenvalues

occur

Sj =t

Cj(lY),

U( 1 + Sj e’4),

medium

j=

543

1,2,

(7.1)

where

C,(l9) =

(co,2i3+y2,

?Yj=tg-l(~)~

Pi’

(7.2)

pj=[~:~_~

and (r, 6) are polar coordinates in the physical plane, measured from the tip of the moving crack. Considering the series expansion of the function (6.7) in the neighborhood of the crack tip, the following asymptotic stress field may be obtained from (6.8)

(Tyy

uxyJa A

z.r

&

+ g’g,

(7.3)

2-

7

sin a212 _ ~-~

sin 6,/2

2rXfz$q

&z&q’

J(

)

where constant terms c$& and (T(% have been retained, also in order to satisfy the boundary conditions. It should be noted that the non singular stress a$? depends on the kind of material as well as on the crack speed. This fact reflects also upon the circumferential stress distribution which is given by

where (T$A = &

sin2 6 + a(& cos2 19- cxy sin 2 6,

(7.4)

and &A=

Similarly,

uyy=u

the asymptotic

Yy+u(g,=-

-po[cos2

a+&

stress field associated

PO $[(k,k,+ 281 2r

(F-F)

sin2 $31.

(7.5)

with (6.10) becomes

k,k,,(%-y)

•t A

cos 6,/2 + cos &I2 ( IJc,

JE2-p”

>I

(7.6)

A. PIVA and E. VIOLA

544

in which the following contractions

have been made

Cj(S) = (COS’6 + 1*sin* 8 +

Ej1*

sin 26)“*,

1*= ($ + &‘, (7.7)

-y212 IYj =

tg-’ ( COS

6

+

-l,j=

sin 6

Ejy11*

=

sin 6 > ’

Ej

1

1 1, j=2’

Limiting the attention to the case of imaginary eigenvalues, the following asymtotic expressions for the displacement field may be also obtained from (6.9)

Ii__tY!L_ 1~cos$L(LIL)

2%zr

u(X, Y)=---

"_q*

a-p*

C66A

x(a+Clrcos

&)-

--!&( ff -q*

(a +

a+-~~(-$$

C2rcos $13~),

>

(7.8)

I

r( cl

16

sin a1 - c2 15sin 82).

To complete the analysis it may be of interest to obtain the relation between the energy release rate and the crack speed, which can be derived through the well known formula a+6a

/jrr,& JD

G=

Substituting into (7.9) the corresponding to the following expressions G

=

~

1

ayy(X,

a)Av(X, CL+ sa) dX.

asymptotic expressions of stress and displacement,

-

15) ,

79&~2k6-

Ylks)

x66&

leads

(7.10)

G2=

2 C66A

(7.9)

’

according to the occurrence of pure imaginary or complex eigenvalues respectively. In the limiting case of an isotropic medium the expression (7.10)* becomes meaningless whereas (7.10)1 reduces to G=_

7qGi

Pdl - PI)

1

2cL 4/%~2-(1+~~)*’ in agreement

P:=l-M:,

(7.11)

with the result obtained in previous works (see for example [lo]).

8. RESULTS

AND DISCUSSION

In what follows the results will refer to the steel-aluminium I composite whose elastic coefficients, obtained from [4], lead to imaginary roots for (2.8), which is the occurrence in many cases of practical interest. In Figs l-3 the variations of the dimensionless singular terms of the asymptotic stress components (7.3), with crack speed and positions near the crack tip, are shown. & although nearly independent of the The curves in Fig. 4 show that the ratio a$&/ma crack speed may reach high values, what may play an important role in predicting branch angles. For the same reason, in Figs 5-6 the ratios u&/oxx and ~a(:?/aYY are represented respectively, for the dimensionless distance r/a = 0.05.

Crack propagation

I

0

in an orthotropic

medium

545

IM2=LqI

M_ = 0.90

I

Fig. 1. Variation of a$,

with the crack speed.

In Fig. 7 the dimensionless singular term (7.4) of the circumferential stress is pictured as a function of the angular position from the crack tip, for different values of the crack speed. It is noticed, also at very low speeds, the presence of an absolute maximum which becomes more pronounced as the speed increases. In addition, it may be seen that the angular position of the maximum and consequently the presumable angle of crack deviation, is slightly affected by the crack speed. In Figs 8-9 the behaviour of the local displacement components (7.8) as functions of the angular position, for different vaIues of the crack speed and dimensionless distance r/a = 0.05, is shown. In particular, it should be noticed in Fig. 8 that the crack opening displacement is an increasing function of the crack speed.

M,

=

0.90

Fig. 2. Variation of a!& with the crack speed.

546

A. PIVA

and E. VIOLA

M,

= 0.85

n 0.80

M, = 0.85 M2 = 0.80 . M, = 0.70

Fig. 3. Variation

I

of the shear stress with the crack

speed.

O(O)

4_

-xx

= 0.8

3-

79 I 60’

I 300

O

Fig. 4. Comparison

between

I 900

non singular

I 1200

and singular

I 1500

I180°

terms of the stress component

A 12 -

Oxx I

10 -

-

M

a____

MZ = 0.8

2

= 0.0

8 -

6

-

I I

0 Fig. 5. Comparison

between

the non singular

I*

I

60’

1200

term I$&

1800

and the stress component

oxx.

axx_

I

547

: I' a

0.6 -

YY ____----

0.4 -

0.2 _I

I

60'

300

900

M, = 0.8

- 0.2 -

I -0.4

_--___-M,=O.O

-

Fig. 6. Comparison

between

the singular

term u’&

and the stress component

cyy,

M, = 0.70

M_

=0

/

9 I

300

I

I

60"

900

Fig. 7. Variation

I

I

1200

of a(,‘; with the crack

/

Fig. 8. Angular Ben 29:5-c

variation

of the TVdisplacement

component

w

-

1500

1800

speed.

M2 = 0.90

M, =0.85

with the crack

speed.

548

A. PIVA I 2.0

-

u -7

cffi p,

I’ 300

Fig. 9. Angular

Acknowledgement-This

and E. VIOLA

II

variation

work was supported

9

“I’

60°

0.00 900

of the u displacement

1 s 1200

component

n

1 ‘I 1500

’ 180’

with the crack

speed.

by M.P.I.

REFERENCES [l] C. Atkinson, The propagation of fracture in aelotropic materials. Int. J. Fracture Mech. 1, 47-55 (1965). [2] M. K. Kassir and S. Tse, Moving Griffith crack in an orthotropic material. Inr. J. Engng Sci. 21, 315-325 (1983). [3] H. T. Danyluk and B. M. Singh, Closed form solutions for a finite length crack moving in an orthotropic layer of finite thickness. Len. appl. Engng Sci. 22, 637-644 (1984). [4] M. Arcisz and G. C. Sih, Effect of orthotropy on crack propagation. Theor. appl. Fracture Mech. 1,225-238 (1984). [5] A. Piva, Elastodynamic crack problems in an anisotropic medium through a complex variable approach. Q. Appl. Math. 44, 441-445 (1986). [6] A. Piva, An alternative approach to elastodynamic crack problems in an orthotropic medium. Q. appl. Mada. (In Press). [7] E. Viola and A. Piva, Effect of orthotropy on elastodynamic crack behaviour. Proc. VIIIAIMETA, Torino, Vol. 1, pp. 155-158 (1986). [8] S. G. Lekhnitskii, Theory of Ehsticity of an Anisotropic Ehstic Body. Holden-Day, [9] F. D. Gakhov, Boundary Vafue Problems. Pergamon Press, Oxford (1966). [lo] F. Erdogan, Crack propagation theories. In Fracture II (Edited by H. Liebowitz), New York (1968). (Received

20 July

1987)

San Francisco pp. 498-592.

(1963). Academic

Press,