Crack propagation in an orthotropic medium under general loading

Engineering Fracture Mechanics Vol. 34, No. 516, pp. Printed in Great Britain.

1155-I174, 1989

0013-7944/89 $3.00 + 0.00 0 1989 Pergamon Press plc.

CRACK PROPAGATION IN AN ORTHOTROPIC MEDIUM UNDER GENERAL LOADING E. VIOLA, A. PIVA and E. RADI Istituto

di Scienza delle Costruxioni, University of Bologna, Italy

Abatrati-The elastodynamic response of a finite crack steadily propagating in an orthotropic medium, acted upon at infinity by uniform biaxial and shear loads, is studied. In particular, the effects of varying crack velocity as well as the ratio of shear load to tensile load are described The action of material orthotropy on various quantities describing the crack propagation characteristics is pointed out.

1. INTRODUCTION TBBORE-DCAL studies of crack propagation in orthotropic media have mostly dealt with problems where a steady-state is considered. In particular, for finite length cracks, the trailing tip of the crack has been considered to be self-sealing so that the crack length remains constant. This fact should not affect the local fields in the neighbourhood of the advancing crack tip, what has been postulated firstly by Yoffe[l] and also by Bilby and Bullough[2] in solving elastodynamic crack problems in isotropic materials. Problems with moving cracks in orthotropic media were considered by Kassir and Tse[3], and Arcisz and Sih[4], among others. These investigators solved the Griffith crack problem by using integral transform techniques. Danyluk and Singh[S] obtained closed form solutions to antiplane problems of a crack propagating in an orthotropic strip. The analysis of [5] has been extended to a strip made up of an anisotropic material with one plane of symmetry, through a complex variable approach[6]. Of concern in the present paper is a study of the steady-state propagating features of a crack in an orthotropic medium. The analysis is carried out according to the results recently obtained in [7] where a complex variable formulation, alternative to that one developed by Lekhnitski@], is reported. A uniform state of biaxial and shear loads applied at infinity and unperturbed by the propagating crack, is assumed. In view of understanding the process and nature of fracture of orthotropic materials the asymptotic stress and displacement fields near the propagating crack tip have been obtained for two types of materials. Next, the dynamic energy release rates are determined in terms of the crack velocity, using the asymptotic stress and displacement fields, and the concept of crack closure energy. Finally, the above quantities are represented for various cases and widely discussed.

2. BASIC PRELIMINARIES For an orthotropic medium the Cartesian coordinate axes X, y and z are assumed to be co-incident with the axes of elastic symmetry and the displacement component along the z-axis is supposed to vanish, with all its derivatives, with respect to z. Under plane strain or plane stress conditions the stress components are related to the displacement components u and u along the x and y-axes, respectively, by the equations:

(2.1)

1156

E. VIOLA et al.

in which cii are parameters related to the elastic constants by the relations: Cl1= EI /[l -

w4)V:,I~

czz= 6%/4 h I9 Cl2 =

(2.2)

VI2 Cl2 =v21c11,

for generalized plane stress, and by CII= (WA)(l

- w&,

~22 =

(E2lW

-

~12 =

W~(YII

A

=

1 -

vnvw),

+$,,v,,),

v12v21

-

v23v32

-

hI%3

-

~12v23v31

-

v13v21

v32,

(2.3)

for plane strain. As usual, Ej, pii and viistand for the Young’s moduli, the shear moduli and the Poisson ratios, respectively. The system of equations of motion governing elastodynamic problems in the (x, y) plane reduce to d2U Cl1Q +

d2U A2

-ay2

a2u

+

(h2

+

Cl21

d2u

axay

=

?F’

d2V

+ (IL12 + Cl21 -&=p$, (2.4) dY where t is time and p the mass density of the material. Since steady-state problems will be discussed, it is convenient to introduce the transformation P*2$+

c22 2

C=x-ct,

Y=y,

t=t,

where c is a constant velocity. Assuming u = u(X, Y) and v = v(X9 Y), eq. (2.4) becomes

2

$+28,-

a2V

(2.5)

1By2=0,

ZY+OL

with 2/I =

Cl2 +

Pi2

c12 + 28,

CllU

-MY

PlZ(l

PI2 a

=c,,(l

-Mf)’

PI2

= -

m’

c22

“=p,2(1

-M;)’

(2.6)

Here, the Mach numbers Mi = c/v, (j = 1,2) with vi = c,,/p and vi = ~,~lp, are assumed less than one. According to a recent paper[q eq. (2.5) may be transformed into a first order elliptic system of the Cauchy-Riemann type, so justifying the formulation of plane problems in terms of complex variables. Setting a, = (a + al - 4/3&)/2,

a2 = aa, B 0,

two types of orthotropic materials need to be considered according as a, > & la, I < & (Type W and correspondingly the following stress and displacement found[7].

(Type I) or fields can be

Crack propagation under general loading

Type I. The stress and displacement components are

and

---&~*)+ol_ [

IA’) =

-

1

&%I2

VP2

u(l) = Im

w2(z2)

a-p

P

Q

(2.8)

Re[po,tzl I+ w2tz2)l.

The Fisons Q,(q) and Q&z,) are holomo~~c fictions of the complex variables z, = X + iY/p and z2= X + iYfq, respectively, and w, (z,), wz(z2) are their primitives. The coefficients apprearing in (2.7) and (2.8) are defined as follows P = [a, - (a: - u#q”2,

q = [a, + (a: - a2)“7’4

M:), i, = -(l, + M:).

I, = -(l, +

Type II. The stress and displa~m~t

~rn~nen~

(2.9) are

a!‘)= ~l,~{k,W~lW+Q2(~2)1 -k2ReInl(z,)-~(z,ll},

@= P~~WW&,) +Q2(z2)l-h W’Mz,) -Q2(z2>lh z%?=P& RdfMz,) +Q2Cdl +kWWzl) -Q2(z2>lI~

(2.10)

and u(fI)= -28 ImE(P3+ 4&t (~1+ (~3 v(lu

=

-W(r,

+

iy2hl(d

-

ip4)02@2)1,

(2.11)

ti2~~2(z2)l.

(19 -

The coefficients appearing in (2.10) and (2.11) are defined as follows Yi= M&+~,ll”*~ p1 +

(Yl +

+ (yl;;;

P3 + ips = (rt a + (rl

k, = cE

-

a, 11”2

+ iY2)(PI

-

ipz)

+ ir212’

WP3C,l CL12

k3scE-

-

iy2)

ip2 =

’

kz = zflP$

WP3 Cl2 k4=Wp4~, PC112

5

72 = tf

= 215~2-

~2,

) k

= WP,

-

Y, .

(2.12)

For this type of orthotropic material, the functions Q,(z,) and Q(z,) are holomo~~c functions of the complex variables

and

EFM 34-W-l

E. VIOLAet al.

1158

In addition, the functions o, (z,) and 02(z2) are the analytic primitives of Q,(z,) and f&(z2), respectively. 3. FORMULATION

AND SOLUTION

OF THE PROBLEM

The problem under consideration is illustrated in Fig. 1. It consists of a line crack of constant length 2a propagating in an orthotropic medium with constant velocity c along the x-axis. Traction-free conditions at the edges of the crack and uniform biaxial and shear loads, acting at infinity, are assumed. The solution to the elastodynamic problem is obtained by applying the superposition to the solution of three problems. The first two problems are those of a symmetrically (mode I) and a skew-symmetrically (mode II) loaded crack, respectively, with zero stress at infinity. The third problem is the deformation of the unbroken medium under known stress at infinity. The solutions to the mode I and mode II problems may be easily obtained by formulating the corresponding boundary value problems as appropriate Dirichlet problems (see for example ref. [7’J). By superposing to the above solutions that one corresponding to the unbroken medium, allows to obtain the following expressions for the stress components (2.7) 1213P

-ImF,(z,)+,.$j) =

T+py

4

ImF,(z,)

1 [ +-$

y

ReFz(z,)-5

ReI;,h) 1s

[ImF,(r,)-Im~~(2211-~[dSRe4(z,)-ql~l~ReF,(r,ll, y

[Im FI (z,) - Im F2(z2)],

(3.1)

where D = pl,l, - ql,l, and F,(zJ = 1 - zj(zf - a*)-‘/*, j = 1, 2. Correspondingly (2.8) for the displacement field become

the expressions

~(j,,=S+~[ql.,1,ReF,(z,)-pl,l,ReF,(z,)]+

@=28{[&44

-5j)+&j(B2+5-#

+P[~(A,+~)-g_~~~,2D]y

+

,420

+-

14 Im Gdzd a- P

PllS 1, [--+G2(z2)---_i ~1 -

4

T

ql5 ReG(zd a- 4

Pl6 ---_?ReG(zA-----_i cl12D a -P

--

414s

’

AIPD

1 1 9

(B,+B,)+-

X+ )

I’

[

where Gj(zj) = (zf - a*)‘/*, j = 1, 2 and S

B

_

*-

WI - M*ld /.42(4*~J4- P21213)'

(3.3)

Crack propa~tion

1159

under genhal Ioadiug

sT---______-_____---_____,

-I f

!I-

I

Fig. 1. Crack geometry and loading conditions.

In order to obtain the crack propagation ~haracte~stics it is impo~ant to represent the fieid quantities (3.1) and (3.2) in their asymptotic forms. Introducing polar coordinates (r, 9) measured from the right hand crack tip and following the approach described in [7], the expressions (3.1) and (3.2) render the local fields as

and

E. VIOLA er k

r

-6

-6

- 10

- 12

- 14

-16 60’

39

0”

9b

12o”

1500

Fig, 2. Variation of Q$ vs 9, and various values of s (M2 = 0.4, k = 1).

In (3.4) and (3.5) the following definitions hold sin29 1’2 I 4=tg-‘(F), Pj={;!:;; q(S) = cos219 +P/’ > ( Since the solution for the propagating crack is known, the relation between the energy release rate and the crack velocity can be derived through the expression for the crack-closure energy CI+blI G= [a#, a) Au(X, u + Sa) + tXY(X,a) Au(X, u -I-&Z)]dx. (3.6) 4 I la

jirn&

Substituting the appropriate

expressions given by (3.4) and (3.9, yields

G(I) = ?_

4

T2(la - Z,) + 2j3pqS2 -

2cL12D

14

q2

a-

- a -p2

>I’

(3.7)

4

2

0

-2

-4

III

-6 00

III 300

I lb

I

II w

I

III 1200

I

F

150”

Fig. 3. Variation of uf vs 8 for various values of s (M2 = 0.4, k = 1).

I” 160’

Crack propagation

under general loading

1161

6

4

2

0

I

-2

l

I

I

0’

I

3o"

I

I

II

I

606

I

I

900

I

I

I

1200

I ISOO

I

11 180'

Fig. 4. Variation of r& vs 9, for various values of s (M2 = 0.4, k = 1).

which, making use of (2.9) may be written as

WI2 - P2b

G(l) =

2Pl2mf

-P2>b

[T2

+pq(l

- M:)Sq.

(3.8)

-q2>

In what follows the crack propagation features in an orthotropic material of type I, will be described through some significant representations of the quantities obtained above.

4. CRACK PROPAGATION

CHARACTERISTICS

FOR MATERIALS

OF TYPE I

In Figs 2-17 the graphical representations refer to a steel-aluminium composite whose elastic coefficients, under plane strain conditions, are cl1/p ,2 = 3.952, c~/JA~~= 4.155 and c12/p,2= 1.959 (see for example ref. [4]).

-12 0’

3d

60-

90’

120-

150'

Fig. 5. Variation of 0: vs 9, for various values of s (M, = 0.4, k = 1).

180'

1162

E. VIOLA et

al.

1.5

1

.5

0

- .5

I

00

I

I 30'

I

1

I 600

I

I

I 900

I

I

I

I

1200

I

I

I

150'

14 1600

Fig. 6. Variation of u* vs 9 and various values of s (M2 = 0.4, k = 1).

For what concerns, the stress and displacement fields (3.1) and (3.2) are represented vs the polar angle 9 for various values of the load parameter s = S/T, the biaxial parameter k and the Mach number M,, for a fixed value of the dimensionless distance r/a = 0.05. Another quantity of physical interest in fracture mechanics will be represented, i.e. the circumferential stress, which is related to cartesian stress components by c9 = bXsin* 9 + oYcos* 9 - tXYsin 29.

(4.1)

Finally, the strain energy release rate (3.8) is represented as a function of the load parameter s, for various values of the crack velocity. In Figs 2-7 the biaxial load factor is taken at k = 1 and the Mach number is taken at M2 = 0.4. In Fig. 2 the dimensionless longitudinal stress component a,,/m/T is shown to be a decreasing function of the load parameter. The influence of the above parameter on the angular

2.5

OD

300

60'

900

1200

160°

Fig. 7. Variation of u* vs 9 for various values of s (M2 = 0.4, k = 1).

Crack propagation under general loading

1163

u @Ia

u*=--

1

T k=

3

10 -

2

k=5 1 k=3

0

-1 k=-3

-2

o-

3P

600

900

1200

1500

1800

Fig. 8. Variation of u* vs 9 for various values of k (M, = 0.8, s = 1).

variation of the dimensionless stress component a&&6&T is represented in Fig. 3. It should be noted that until the polar angle has reached the value 9 ‘Y60” the stress component is an increasing function of the applied shear load. A reverse behaviour is shown for larger values of the polar angle. In Fig. 4 the behaviour of the dimensionless stress component rXYJ@@/T is represented where it is noted the relevant influence of the applied shear load for small values of the polar angle. The dimensionless circumferential stress QgV/(‘ZT’;jSI)T is shown in Fig. 5. The curves show that for higher values of S the rna~rn~ moves from 9 N 30” (corresponding to s = 0) to 9 = 0. The displacement components plzu/aT and plzv/aT are described in Figs 6 and 7.

.26

0 ff

30

6W

12P

Fig. 9. Variation of v* vs 9 for various values of k (M, = 0.8, s = 1).

16oD

E. VIOLA et al.

1164

0’

300

600

900

1200

15P

Fig. 10. Variation of af vs 9 for various values of M2 (k = 1, s = 1).

When the above mentioned quantities are represented for the biaxial load factor taken at k = 0 and the Mach number at M2 = 0 (static case), it appears that a decrease of the crack velocity and the occurrence of a longitudinal stress component equal to the normal stress component, does not alfect significantly the angular variations of stresses and displacements. The effects obtained by combining a high crack velocity with a varying biaxial load parameter on the angular variation of the displacement components are shown in Figs 8 and 9. The influence of varying crack velocity, for k = 1 and s = 1, on the stress and displacement components is shown in Figs 10-15.

o.+ 2r -1 f a 2

0

Fig. 11. Variation of a$ vs 9, for various valuer of M2 (k = 1,s = I).

Crack propagation under general loading

-

I

.5

00

I

I 300

I

I

I

600

I1

I 900

I

I

I 126

1165

I

I

I 15OQ

I

I”1 180°

Fig. 12. Variation of rh vs 9, for various values of M2 (R = 1, s = 1).

It should be noted in Fig. 13 that the first positive maximum of the circumferential stress occurs at 9 = 0 until the value M2 9 0.7. A further increase of the crack velocity affects remarkably the angular position as well as the value of the maximum. In Fig. 16 an effect of increasing the shear load, up to the value s = 3 on the circumferential stress component is shown. From this figure it is evident that the shear load can have a significant influence upon the values of the represented quantity. A comparison between Fig. 16 and Fig. 13 shows that the action of the shear stress is to flatten the maximum of the circumferential stress which remains in the plane of crack propagation (9 = 0) for all values of the crack velocity.

3 1

.3

0

-.9

-1

- I.5 (P

34

w

90.

1200

1500

Fig. 13. Variation of as+ vs 9, for various values of A& (R = 1, s = 1).

E. VIOLA ef al.

1166

30’

0'

90’

60’

150’

120’

Fig. 14. Variation of u* vs 9, for various values of M2 (k = 1, s = 1).

In Fig. 17 the strain energy release rate is represented as a function of the crack velocity for various values of the load parameter s. It is seen again the significant influence of the shear load upon the values of the above quantity. In order to point out the effects of material orthotropy on crack propagation characteristics, some results are graphed for a graphittipoxy composite which is distinctly orthotropic. The material parameters are (seen again ref. [4]) cIl/~,z = 3.504, c~J~,~ = 29.822 and c,~/~,~ = 1.723. In Fig. 18, the dimensionless circumferential stress is shown for s = 0, k = 0 and various values of the crack velocity.

t

I

0

00

I

I 300

I

I

I 60.

I

I

I 900

I

I

I 1200

I

I 1500

Fig. 15. Variation U* vs 9, for various values of M2 (k = 1, s = 1).

I

I"1 160a

o;=;f 2 a

a

-1

-2

-3

-4

-5

I

I

I

I

I

I

30’

I

I

I

I

I

I

I

120°

900

600

I

I*

I

150’

180’

Fig. 16. Variation of ai vs 9, for various values of M2 (k = I, s = 3).

I

0

I

I .2

’

I

I.

.4

I .6

I

1

Ma

.8

1

Fig. 17. Variation of G$) vs M,, for various values of s.

ct2 = 29.822 IhI WI = 1.723 IrIl

Fig. 18. Variation of 88 vs 8, for various values of M2 (k= 0,s = 0). 1167

E. VIOLA et al.

1168

0

- .5

30’

0’

600

900

1200

1500

1800

Fig. 19. Variation of a: vs 9, for various values of M2 (k = 1, s = 1).

It is worthwhile noting that the maximum becomes more pronounced as the velocity increases. The above behaviour changes severely when the parameters take the values k = s = 1 as shown in Fig. 19. The effect of orthotropy is given by comparison with Fig. 13. In Figs 20 and 21 the dimensionless circumferential stress is represented for k = 3, s = 1 and k = 3, s = 3, respectively. It should be noted the remarkable effect due to an increase of the load parameter which is also depicted in Fig. 22. In Fig. 23 the strain energy release rate is represented versus the crack velocity for various values of the load parameter.

OD

300

600

w

120°

150’

Fig. 20. Variation of as vs 9, for various values of s (M2 5 0.1, k = 1).

Crack propagation under general loading

.25

00

3P

600

900

1200

1500

Fig. 21. Variation of tug+vs 9, for various values of M2 (/c = 3, s = 1).

5. LOCAL FIELDS FOR MATERIALS OF TYPE II When an orthotropic material of type II is considered the solution to the elastodynamic crack problem may be obtained by the same approach mentioned in Section 4 and reported in ref. [7].

I 00

llIrlIlIIlIIllIIl'[ 304

60.

900

1204

1500

Fig. 22. Variation of a8 vs 9, for various values of A& (k = 3, s = 3).

160

1170

E. VIOLA et al. f

I

G(” -CI~

G!" = -

T’

a

J

150 ['-s/r] t

t=0 M2

I

I

0

0

I

I

0.2

,,

I

0.4

I Od

I

I 0.8

I 1

Fig. 23. Variation of G!j vs I&, for various values of s.

Limiting

attention to the asymptotic expressions, the stress component (2.10) become

sin 9, /2

-(k,k,-k,k,)

---

sin 9,/2

( ~~

+ S(k: + k:)

cos a,/2

>I

cos a,/2

201

cos92’2

Jai

1

+~~~~k~~k~k~)~~-~)+D,~~+~)],

(5.1)

in which D, = ksk, - k4k5 and the following definitions hold (see ref. [I) c,(S) = (cos29 + i2sin29 + 6+*sin 29)‘“, I2 i (y: + y$)-‘, Sj=

tg-’

- 1,j = 1, I,j-2.

1171

Crack propagation under generalloading

Correspondingly

the asymptotic expressions for the displacement components (2.11) may be

obtained as #It = -W](PJ,

-p44

+ -!!c (p3k6 C”t2a I -

(p3k,

+P4wy,

+P4A,K

+PsB2X21

+p‘,k5)[2(u + r cos 9) - @(&@jcos

-p,k,),,&$,&@sin

- (p3k4 +p4k3)[2Y,

+&

+tP3&

9,/2 - msin

((7, k - Y2k4)W

+ r ~0s

f (Y2k3 + y,k4),/%,,6@b

W

$1-

9,/2)]

9,/2)

9112+&&in

,h(&@b

-

8,/2 f &@jcos

,h&&%cos

&%sin

h/2)1 ,

9, /2 + ,/‘&@i~~s

92/U

92/W,

where the following relations hold X,=(a+rcos9)-y,Z2rsin8, ~2=(~+~cos~)+~,~2rsin~, Y, = y212rsin 9, and

s

B -’ -

-

T

%,zkf, + 2p,zb

B2=:--

s

3” %,2&e + &,2k

[W&e -k&J f (klkst&k4 - k2k3)

k2Wl ’

[k(k4k6+k3kS)-(k,kJ+kzks)l (kk4

- k2k3)

’

To complete the analysis, the energy release rate may be obtained by using again the expression (3.6) for the crack-closure energy. Substituting the appropriate expressions given by (5.1) and (5.2) leads to G,,,)-- &

[(y&6- Y,k,)Y’+ 8@&, +U,)S’].

(5.3)

In the next section graphical representations will be reported to display the characteristics of crack propagation through an orthotropic material of type II. 6. CRACK PROPAGATION CHARACTERISTICS FOR MATERIALS OF TYPE II As a result of the difficulty in finding experimental results concerning materials of type II, fictitious elastic parameters c,,/~,~ = c22/p,2= 3 and c,,/~,~ = 2, which allow the condition /a, 1-c & to be satisfied, have been taken as in ref. [9].

E. VIOLA et al.

1172

I

- .5 00

I

I 30’

I

I

I 6OO

I

I

I

I

90’

1\kJ,

,61

lZ(P

150’

166

Fig. 24. Variation of ot vs 9, for various values of M2 (k = 0, s = 0).

Hence, the aim of the graphical representations which will follow is to give an idea of the action of material orthotropy on the various quantities describing the crack propagation characteristics. In Fig. 24 the angular variation of the dimensionless circumferential stress is represented for s = 0, k = 0 and various values of the crack velocity. A comparison with Fig. 18 shows that the effect of varying crack velocity on the angular position as well as on the value of the maximum of the above quantity is more pronounced for materials of type II. In Fig. 25 where the circumferential stress is represented for s = 1 and k = 1, an unusual bchaviour is shown (see Fig. 13), i.e. the angular displacement of the maximum towards smaller values, as the crack velocity increases.

00

30@

6P

90’

12P

1500

Fig. 25. Variation of at vs 9, for various values of A42(k = 1, s = I).

Crack propagation under general loading

3-y

% f

1173

2r -a 1s

10

6

0

-6

-10

Fig. 26. Variation of b$ vs 9, for various values of s (Mz = 0.4, k = 1).

Analogously, a comparison of Fig. 26 with Fig. 5, where the ~rcu~e~ntial stress is 0.4, k = 1 and various values of s, shows the different effects of increasing represented for M2 = the load parameter according as steel-aluminium or the material of type II is considered. In Fig. 27 the strain energy release rate is represented which exhibits a decreasing behaviour with respect to an increasing load parameter as opposed to what is shown in Figs 17 and 23. It may be noted that all curves are discontinuous at Mz - 0.9 which corresponds to the limiting value for the material of type II taken into consideration. ~c~o~i~e~~~-~s

work was supported by M.P.I.

Material of Tyre

f

0

.2

A

.6

Fig. 27. Variation of G!$) VI M2, for various values of s. EM

34.5X-K

.I)

Ii

1174

E. VIOLA ei al.

REFERENCES [I] E. H. Yoffe, The moving Griffith crack. Phil. Mug. 42, 739-750 (1951).

[2] [3] [4] [SJ

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