Crack-face displacements for an embedded elliptic crack in an infinite solid subjected to arbitrary tractions

Engineering Fracture Mechanics Printed in Great Britain.

Vol. 36, No. 3, pp. 417-428, 1990

0013-7944/90 53.00 + 0.00 Pergamon Press pk.

CRACK-FACE DISPLACEMENTS FOR AN EMBEDDED ELLIPTIC CRACK IN AN INFINITE SOLID SUBJECTED TO ARBITRARY TRACTIONS I. S. PAJUt Analytical Services and Materials, Inc. Hampton, VA 23666, U.S.A. Abstract-Analytical expressions for the crack-face displacements of an embedded elliptic crack in infinite solid subjected to arbitrary tractions are obtained. The tractions on the crack-faces are assumed to be expressed in a polynomial form. These displacement expressions complete the exact solution of Vijayakumar and Atluri, and Nishioka and Atluri. For the special case of an embedded crack in an infinite solid subjected to uniform pressure loading, the present displacements agree with those by Green and S&don. The displacement equations derived in the paper can be directly used with the finite-element or boundary element alternating methods for the analysis of a semi-elliptic surface and quarter-elliptic comer cracks in finite solids.

NOMENCLATURE

P

K2 K'2

is ud

semi-major and semi-minor axes of the embedded elliptic crack magnitudes of the polynomial pressure loading on the crack faces defined by eq. (1) elliptic integral of the second kind potential functions potential function defined by eq. (7) elliptic integral of the first kind degree of truncation of the polynomials displacement along X, direction Cartesian coordinates with the origin at center of the elliptic crack with x, along the major axis and x2 along the minor axis of the ellipse shear modulus of the isotropic material of the solid (0: -&lo: a:@: Poisson’s ratio of the material ellipsoidal coordinates defined by eq. (3) stresses in the solid.

INTRODUCTION DAMAGE tolerant analyses require accurate calculations of stress-intensity factors at crack tips in two-dimensional and around crack fronts in three-dimensional bodies. Experience with several crack configurations have shown that these cracks tend to grow in nearly elliptic, semi-elliptic and quarter-elliptic shapes. Therefore, considerable attention has been devoted to analytical and experimental studies of these crack configurations. While considerable data exists on the stress-intensity factors for these crack configurations[l+, very little information exists on crack-face displacements. Crack-face displacements are directly measured in an experiment, while the stress-intensity factors are deduced from the experimental observations (such as stress freezing). Therefore, a need exists for accurate analytical crack-face displacements for comparison with experimental observations for three-dimensional crack configurations. Also, crack-face displacements are needed to develop three-dimensional weight function methods[5,6]. Recent literature has shown that the three-dimensional finite-element alternating method (FEAM) is a very economical and an accurate method for obtaining stress-intensity factors for elliptical and part-elliptical crack configurations[7-lo]. The FEAM uses the exact solution of an embedded elliptical crack in an infinite solid subjected to arbitrary crack-face tractions obtained by Vijakumar and Atluri[l 11, and Nishioka and Atluri[7,8]. While this solution is comprehensive, tPresent address: Department of Mechanical Engineering, North Carolina A&T State University, Greensboro, NC 27411, U.S.A. 417

418

I. S. RAJU

the crack-face displacements are not available in refs [7], [8] and [ll]. In order to use the FEAM to calculate crack-face displacements, the exact solution of refs[7], [8] and [l I] needs to be extended to derive the crack-face displacements for each of the polynomial pressure loadings on the crack faces. The purpose of this paper is to obtain the analytical crack-face displacements for an embedded elliptic crack in an infinite elastic solid subjected to arbitrary polynomial normal and shear tractions. First, embedded crack solutions in the literature are reviewed. Next the fundamental equations relevant to the embedded elliptic crack problem are summarized. Then the crackface displacements for mode I loading are obtained. Next the crack-face displacements for the mixed-mode loadings are derived. The displacements for the special cases of uniform pressure on the crack faces are compared with those from Green and Sneddon[l2] and Kassir and Sih[l3].

EMBEDDED

CRACK SOLUTIONS

The exact solutions for an embedded crack in an infinite solid are the fundamental solutions used in alternating methods. Therefore, this section briefly reviews the classical works in the literature on embedded crack solutions. Some of the results from these references are used to build the present solution for the crack-face displacements. Exact analytical solution for a penny-shaped crack subjected to uniform pressure loading on the crack surfaces was obtained by Sneddon[l4]. Later, Green and Sneddon[l2] obtained the exact solution for an embedded elliptic crack in an infinite solid subjected to uniform pressure loading. Kassir and Sih[l3] obtained an exact solution for an elliptic crack subjected to shear loading on the crack faces that can be expressed by a simple trignometric functions. Shah and Kobayashi[lS] derived close form solutions for an embedded elliptic crack subjected normal loading that can be described by polynomial functions up to the third degree in terms of the two Cartesian coordinates that describe the ellipse. Smith et a/.[161 obtained the solution for a penny-shaped crack subjected to a polynomial pressure distribution on the crack faces. Smith and Sorensen[l7] obtained the solution to an embedded elliptical crack in an infinite solid subjected to shear tractions that can be described by polynomial functions up to the third degree in terms of the two Cartesian coordinates that describe the ellipse. Later, Vijayakumar and Atluri[ 1I] and Nishioka and Atluri[7,8] obtained the exact solution for an embedded elliptic crack subjected to arbitrary normal and shear loadings on the crack surfaces. Note that the solutions in refs[12-171 are special cases of this general solution. As mentioned previously, the purpose of this paper is to extend the solution in refs [7], [8] and [l l] and derive the crack face displacements.

FUNDAMENTAL

EQUATIONS

FOR MODE I LOADINGS

For completeness this section presents the equations that are relevant to an embedded elliptical crack in an infinite solid subjected to arbitrary tractions on the crack faces. The notation of ref. [7] is followed. Some of these equations can also be found in refs[l l-171.

Fig. 1. Embedded elliptic crack in an infinite solid.

Embedded elliptical cracks

419

Consider an embedded elliptic crack in an infinite solid as shown in Fig. 1. The crack faces are subjected to prescribed normal tractions 033 which can be expressed in a polynomial form as

where i and j specify the symmetries of the load with respect to the axes of the ellipse, A$,,_,,, are constants that give the magnitude of the load, and A4 is the degree of truncation of the polynomial. The stress dist~bution everywhere in the solid and the stress-intensity factors all along the crack border due to the prescribed pressure distribution of eq. (1) has been obtained in [7], [S] and [ 1I]. Here the crack opening displacements corresponding to the arbitrary pressure distribution of eq. (1) will be obtained. E~li~~o~l coordinates The elliptic crack perimeter can be described by

(xIh)*+(x2/a2)*=1,

(2)

q>a2.

The coordinates xi, (i = 1,2,3) of any point in the solid may be expressed in terms of the ellipsoidal coordinates, <, , (CL= 1,2,3) in the form 44

- 4)~: = (4 + &)(4 + 52)(4 +

t3)

a%4

-

53)

4)x:

=

+:x:

(4

+

t,)(4

+

t2)Q4

+

(3)

= 5,<2 <3

where

Note that the ellipsoidal coordinates

e, are defined by the roots of the cubic equation

x: o(s) = 1 - u:+s

x: - - = ~(~)/Q(~) S

4 - ~ u:+s

(4)

where es>

=

6

-t,)@

-

r2M

-

531,

and Q(s) = s(s +

u:>.

(5)

In the x3 = 0 plane (the plane of the crack), the inside of the ellipse is given by r3 = 0, outside of the ellipse is given by &, = 0, and the elliptic crack front is described by (f2= t3 = 0. Pote~t~l functions, stresses, and dis~~u~e~e~ts

Using Trefftz’s formulation[7,8,

1l] for mode I problems, a potential function

f3is defined such

that

where (7)

I. S. RAJU

420

In eq. (6), C,,, are undetermined constants and o(s) and Q(S) in eq. (7) are defined in eqs (4) and (5), respectively. The partial derivatives of the potential functionf, are needed to obtain the stresses and displacements. They are denoted as follows.

h,fl = 1 k

c C3,k,lFMS

(8)

I

(9)

where f3,s is the partial derivative of f3with respect to xB, (j? = 1,2,3) and so on. The stresses and displacements in the solid are given by 011 =

2PUJl

+

2v!,22

+

d3,3lll

a22 =

2ph22

+

2v!,ll

+

x3.h,3221

012

=

2/4(1

+

x3.h,3121

a33

=

M-h,33

-

%!h2

+.h,3331

+

6 = 2P&,313 032 = 2k%fb3

(11)

and 4

=

(1 -

2V%,,

+X3.&3,

u2 =

(1 -

2"zf,,2

+

u3 =

-w

-

vs.3

x3&32

+

(12)

xfj.33

where p is the shear modulus and v is the Poisson’s ratio of the material. Crack boundary conditions

The boundary conditions on the crack plane (x3 = 0) are in two regions: inside and outside the elliptic crack. Inside the elliptic crack (& = 0), the boundary conditions are

(13)

for (~llal)~ +

(x2/a2j2

<

1,

x3

=

0.

Note that the right hand side of eq. (13) are the prescribed tractions on the crack faces. Outside the elliptic crack (r2 = 0), because of symmetry in this mode I problem u3 = 0,

a#’ = ag = 0.

(14)

for (xl/al)2 + (x2/a2)2 2 1,

x3 = 0.

From the stresses in eq. (1 l), the shear stress boundary conditions are identically satisfied and the normal stress boundary conditions require that -2j.&

= i

i

i-Oj=O

;

i

m-0

n-0

A$,_,,n~f”‘-2zn+i~f’+j.

(15)

Embedded elliptical cracks

421

Because the potential function f, is expressed in terms of the unknown constants C3&,,and an unknown function _Fkl (see eq. 6), eq. (15) provides the relationship between C,,, and A $,, _ ,,*. Such a relationship was obtained from eq. (42a) of [I. Equations (12) define the displacements everywhere in the solid. As the displacements at any point on the crack faces are sought, the displa~ments inside the ellipse (& = 0) on the x3 = 0 plane need to be evaluated. Because the derivatives Fkl.3,,(and hence&) are finite when x3, & = 0 (see Appendix I of ref. [18] for details), the crack-face displacements can be written as Ul = (1 - 2vzj;,, +=(l-2vlfj.z Uj = -2(1 -

vlf,,.

WI

Therefore, to obtain the crack-face displacements one needs the derivatives of the potential function f3 with respect to x,+ MODE I CRACK FACE DISPLACE~~S Fklb must be evaluated to determine the partial derivatives of h with respect to xa. The derivatives of Fkl are obtained by differentiating eq. (7) as (17)

k,=k+d,,,

l,=l+d,,,

and S,,, a,, and 6,, are the well known Kronecker Because o(&) = 0, eq. (17) can be written as

ml=S3#

(18)

deltas.

Fic,,s =

(1%

Equation (19) needs to be evaluated in the interior of the ellipse (& = 0). Substituting w(s) from eq. (4) into eq. (19) and performing the integrations one obtains

where r _

It =

z

t _

(3-3-%I! (p-q)!

G%- 2rN (q-r)!

(2rY

3 r!

$--Zq-4

f4=(2p

”

=

-2q -k,)! 29- 2~ -I,

(2qxi 2r - I,)!

(21)

422

I. s. RAJLJ

Equation (21) is an elliptical integral and can be written in terms of the Jacobian elliptic functions[7, 191 as dt

(22) where (23) The integral Lp.q_l,l was integrated by parts and a recurrence relation is available in ref. [19] (eq. 357.01) and ref. [7] (eq. 30). A slightly different recurrence relationship that is useful for the crack-face displacement evaluation was obtained using integration by parts as shown in Appendix II in ref. [18]. The recurrence relation is 1 JL? - r,r= (2#, _ l)K r2

[WJ

2,-lu)(nc2’-‘u)(nd24-2’-‘u)

-(2P - wp-,,q-r-,,r-1 - K*(2q - 2)~,,~ - ,,I- l 3. (24) Equations

(19-24), provide all the information

necessary to evaluate the partial derivative f3,s.

Partial derivatives with respect to x3 Because the crack opening displacements are sought the discussion is first focused on the derivative Fk1,3.To evaluate Fk1,3,k, = k, 1, = 1, and ml = 1 need to be substituted in eq. (20). On the x, = 0 plane and outside the ellipse (c2 = 0), all terms in eqs (23) and (24) are non-zero and the term x?- ’ in eq. (20) is identically zero because r 2 1 and x3 = 0. Hence, Fk,,3,and therefore u3 are identically equal to zero outside the ellipse, as required by the boundary condition of the problem. On the x3 = 0 plane and inside the ellipse (t3 = 0), because cn2u = 0 (see eq. 23), the integral Lp.q_l,r diverges. However, on this plane, XT- ’ vanishes. The term

takes an indefinite form and therefore needs to be examined when x3 = & = 0. The term is written as

- (2p - 1)x:‘-’ =p-l.y-r-l,r-l 1 =

2r-

(sn 28_in)(nd*~-*-‘u)

(2r - 1)X” -(2p

- K2c% -2)~:‘-‘Lp,y-~r,r-,IlxjJ:~j=0

1

- 1)X:‘-‘Lp-,,y_,_,,r_*-K2(2q

-2)x:‘-‘Lp,q-,,,-,

1x3=&=0 (25)

As will be obvious later, the last two terms in the square brackets of eq. (25) are zero when xj = & = 0. Now consider the term in braces in eq. (25). With the help of eq. (23) one obtains

[z]:,I:,_, =[pq::I:,Eo [ 2r-

=a:‘-l

I

x3

Jr,

1 x3=c3=0

*

(26)

Embedded elliptical cracks

423

Noting that o(&) = 0 and utilizing eq. (4), the term in the square brackets in eq. (26) can be written as -X3 [ cnu

1~~~3~o_:-‘[1-$-LJ@-‘~‘*.

(27)

Substituting eq. (27) in eq. (25) and using eq. (23) gives

x:l-~Lp.q-,*rI.r3=,,=o = &-

K

jh[q{l-p$y’.

(28)

Now consider the term x:‘- 'Lp_ ,.9_r_ ,,r_, in eq. (25). As before, considering the product of x3 and ncu terms, one has

1

Zr-3

_x3

x:l-'nc *+I)-IU=X~

[ cnu

.

(29)

The term in eq. (29) is identically zero when x3 = 5, = 0. A similar argument holds for the term x:l-‘Lp,g-,,l-l * The partial derivative Fk,,3when x3 = & = 0 can now be written as k+l+l

p

Fk,,,3Iq=~~=o=(~ +I+ l)! p;.

4

q;or;,+j;li_p), tlf2f3f4f5 (k

1 x (2r - l)! Non-dimensionalizing 413

Ixj = c, = o =

(y-“k

eq. (30) one obtains

a, k+~~:,,(k+‘+1)!k~~‘~~o,~,~~+~~1~_p)ll”i13

x

(x$-*r-‘x

1 * (2p - 2q - k)!(2q - 2r - 1)!(2r - l)! x1x (2r - 1)

iI +?Jo+*.

(31)

Note that in the above equation the valid terms in the triple summation are those whose factorial arguments are either zero or positive integers. The opening displacements u3 are obtained as u3(xI~x2~o)=

-2(1

(32)

-V)CCC3,k,,Fk,,31x,-t)-o. k I

The unknown constants C3,k,,are obtained in terms of the prescribed tractions on the crack faces by eq. (42a) of ref. [7]. The non-dimensionalized opening displacements at the center of the crack (x, = x2 = 0) can be obtained from eq. (32), where

41.3Ix,=

x2

=

x3

=

o=

a:+fa:i,(k+l+ I)!%? 2.

2.

x “2+2+’

r=l

(_l)k2+‘2+r

(k2+/2+1-r)!Xr?X2r

1

2r

(33)

for even values of k and I, with 2k2 = k, 21, = 1, and fil.3 IX,=x2=x3=o = 0

(34)

for odd values of k and 1. Note that for odd values of k and 1 the antisymmetry of deformation requires that u3 be equal to zero along the x, and the x2 axes, respectively. Equations (33) and (34) were evaluated for various values of k and 1 and Fk,,3 is presented in Table 1. Note that eqs (30-34) are equally valid for the penny-shaped crack (a, = a2 = a) because these equations do not contain any indefinite forms.

424

I. S. RAJU Table 1. Values of F,,,, at the center of the elliptic crack x, = x, = x3 = 0 for various values of k and I (0 s k, I s 5) k

I

0

0

-4

1 0

0 1

0 0

2 1 0

0 I 2

32/a: 0

2 3 1 0

01 2 3

8

4

0 1

- 1536/a:

4

a,&,,

32/a:

0 0 0 -512/(aia:) 0

2 3 4

-1536/a:

0

1 2 3 4 5

Partial deriuatiws with respect to x1 and x2

The displacements al and uz (see eq. 12) require the evaluation of& andf3,2, and, hence, Fk,,, and F:k1,,2 with x3 = & = 0. The partial derivative FkJ,, and FkL2are obtained by substituting k,=k+l, k,=k,

l,=l, &=l+l,

m,=O ml=0

(35)

in eq. (20). This substitution yields terms like (36)

&&-W

Evaluation of these terms can be performed in two parts, when r = 0 and when r 2 1. First consider the latter case. When r 1~1, theterm in eq. (36) yields (371

Because (x&u) at x3, CJ= 0 is finite (see eq. 27), the term in eq, (37) vanishes when x3 = 0. Therefore contributions to the displacements do not occur when r 2 1. Now consider the term in eq. (36) when r = 0.

1’sn2pt nd% dt.

[x$‘L~~-,,I,,~ = Lp.q,o= --&

0

Instead of using the recurrence relation of eq. (24), it is convenient to use[l9]

s ”

sn*t nd29 dt =

0

where

Zh =

” nd”t dt s0

(39a)

Embeddedellipticalcracks

425

for 2m 2 0 and Z- %=Gzm=

dn? dt

WW

0

for 2m < 0. The integrals in eq. (39a) and (39b) can be evaluated recursively by using Zan+2

2m(2-Kz)Zzm+(l-2m)Z2,_2-u2snucnunP+’u =

(2m + l)rc’*

(Ma)

for 2m 2 0 and G 2m+2

2m(2-K2)G,+(1

-2m)lc’2G2,_2+~2Sn#~t)I(~2m-‘~

=

(2m + 1)

(Mb)

for 2m < 0. To evaluate the integrals in eq. (39) and (40) one needs the starting values Z,, Go, Z_2, and G2. These are obtained by setting m = 0 and m = 1 in eqs (39a) and (39b). This yields Z, = Go= u, Z_2 = Gt = E(u).

(41)

As the displacements on the crack faces are sought, eqs (39) and (40) need to be evaluated when x3 = r, = 0. Thus, instead of incomplete elliptic integrals, one has complete elliptic integrals [with u = K(K)] and therefore eqs (40a) and (40b) reduce to Z2m+2

=

2m(2 - rc2)Z2,+ (1 - 2m)Z,_, (2m + 1)K’2

Wa)

for 2m 2 0, G2m+2=

2m(2 - K~)G~ + (1 - 2m)~‘~G~_~

(2m + 1)

(42b)

for 2m < 0, and with Z,,= Go = K(K),

Z_, = G2 = E(K)

(42~)

where K(K) and E(K) are complete elliptic integrals of the iirst and second kind, respectively. Therefore, the displacements I(] and 24,at an arbitrary point on the crack faces are given by (43)

where

where y = 1,2. Thus eqs (43) and (44) completely define the crack-face displacements u1 and 24,. This completes the analysis of crack-face displacements for mode I type of deformations. ~K~MODE

CRACK FACE DISP~~~~

For mixed-mode deformation, instead of a single potential functionf, of eq. (6) for the mode I case, there are three potential functions f,, (0:= 1,2,3) f, = ;

where Fkl is the function defined in eq. (7).

5:c,,&

(45)

I. S. RAJU

426

The displacements U, on the xj = 0 plane are given by [7,8, 1l-171

-w

u,(x,,x290)=

vYl.3+ (1

u2(x,,x2,0)= -31 - vlv;,,+ (1-

Fkrs are already obtained and are given by eqs (44) and (30). Thus for a general mixed-mode problem of an embedded elliptic crack in an infinite solid subjected to arbitrary tractions of the form tsjcr(x,,x2,0) = i i i=lJ j=O

f

i

m=O

A~m_,,,x;m-2n+ix:n+~

(47)

n=fJ

where a = 1,2,3, the displacements on the crack surfaces are given by eq. (46). The relationships between the magnitudes of A$,,_.,, and the constants Cak,, are already available from eq. (42b) of ref. [7] for mode II and mode III, and eq. (42a) of ref. [7] for mode I deformations. SPECIAL

CASES

Elliptic crack Consider an elliptic crack subjected to a uniform pressure of magnitude S on the crack faces. For this case k = I = 0 and A yi, , , = S. Therefore, the opening displacements are computed using eq. (32) as u3(x,~

x2~

O)

=

-2(1

-

v)C3,0,,&,3

=

-w

-

v)C3,c4,

[z-4{,-$-$Y].

(48)

The relationship between A’&, and C3,,,0is[l2, 151 (49) or (50) Therefore, the opening displacements for an elliptic crack subjected to a uniform pressure S are obtained by substituting eq. (50) into eq. (48) and are u,(x,,x,,O)=~-$$[l which agrees identically displacement is

-(x:/a:)-(x:/a:)]“’

with those due to Green and Sneddon[l2]. (1 -v)

Sa,

u3(0,0,0)=--. P

The maximum

opening

(52)

E(K)

Note that the total opening displacements between the crack faces is the COD and is equal to twice the value of the opening displacement u3(x,, x2, 0). Now consider the displacements u, and u2. The displacements are n, = (1 - 2”)‘%,o&.y

(53)

Embedded elliptical cracks

427

with y = 1,2. Using eq. (44) to evaluate Foo,yone obtains F WI =~W-E(r)]

(54)

and (55) Equations (54) and (55) agree with those obtained displacements ui and u2 at any point on the crack face are

by Kassir and Sih[l3]. Thus the

u,(x,,x*,O)= -(l-2+$-+~$-1] u,(x,,x,,O)=

-(1-2v)

(56)

$?J[$-#

Because of symmetry about the x, and x2 axes, y(x,, 0,O) = u,(O, x2, 0) = 0. Equations (56) identically satisfies this requirement. The maximum u, and u2 displacements occur at the end of major and minor axes, respectively, and is given by eq. (56) with x, = a, and x2 = a,. Penny -shaped crack

For a penny-shaped

crack, a, = a2 = a, E(K) = ~12, and eq. (51) reduces to 24(x,, x2, 0) = i!3c1

=

[1 - (x:/a2) - (x:/a2)]“2

(57)

and the maximum opening displacement is

Similarly, the displacements u, (x, , x2, 0) and u2(x, , x2, 0) can be obtained from eq. (56), using the limit as rc+O (see Appendix II 04181) as u,(x,,xz,O)=

-(I

-2v)%

z+(x,,x2,0)=

-(l-2@.

4P 4P

(59)

CONCLUDING REMARKS The evaluation of the crack-face displacements of an embedded elliptic crack in an infinite solid subjected to arbitrary tractions is addressed. The arbitrary tractions are assumed to be applied to the crack faces and are assumed to be expressed in a polynomial form in terms of the Cartesian coordinates that describe the ellipse. The exact solution obtained by Vijayakumar and Atluri, and Nishioka and Atluri was extended to obtain closed form expressions for the crack-face displacements. The crack-face displacement for special cases of uniform normal tractions agree identically with those obtained by Green and Sneddon, and Kassir and Sih. The evaluation of the crack-face displacements for embedded, surface, and corner cracked solids in finite bodies require the use of numerical methods such as finite element, finite-element alternating and boundary-element alternating methods. The displacement expressions obtained in this paper can be directly used with these numerical methods. Acknowledgements-This work was performed as a part of contract NASI-18599 at the NASA Langley Research Center, Hampton, Virginia U.S.A.

I. S. RAJU

428

REFERENCES [l] H. Tada, P. C. Paris and G. R. Irwin, The Stress Analysis of Cracks Handbook. Del Research Corp., St Louis (1973). [2] [3] [4] [S]

D. G. Y. H.

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