Torsion of a cylinder with a shallow external crack

International Journal of Solids and Structures 46 (2009) 3061–3067

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Torsion of a cylinder with a shallow external crack P. Malits Center for Applied and Industrial Mathematics, Holon Institute of Technology, 52 Golomb Street, Hanaviim 12/3, 77473 Holon, Ashdod, Israel

a r t i c l e

i n f o

Article history: Received 25 September 2008 Received in revised form 21 March 2009 Available online 17 April 2009 Keywords: Torsion Elastic cylinder Crack Stress intensity factor Dual series equations Asymptotic solution

a b s t r a c t The torsional problem of a finite elastic cylinder with a circumferential edge crack is studied in this paper. An efficient solution to the problem is achieved by using a new form of regularization applied to dual Dini series equations. Unlike the Srivastav approach, this regularization transforms dual equations into a Fredholm integral equation of the second kind given on the crack surface. Hence, exact asymptotic expansions of the Fredholm equation solution, the stress intensity factor and the torque are derived for the case of a shallow crack. The asymptotic expansions are certain power-logarithmic series of the normalized crack depth. Coefficients of these series are found from recurrent relations. Calculations for a shallow crack manifest that the stress intensity factor exhibits the rather weak dependence upon the cylinder length when the torque is fixed and the triple length is larger than the diameter. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The present investigation considers a torsion of an elastic circular cylinder of finite length having a circumferential edge crack. The cylinder is twisted by rigid discs bonded to its bases z ¼ L. The lateral surface of the cylinder r ¼ R and the crack z ¼ 0; a < r 6 R are unloaded. The geometry of the problem is shown in Fig. 1. The torsion of a circular finite cylinder with an external crack was first studied by Freeman and Keer (1967). Employing the long cylinder approximation and the solution to dual Dini series equations suggested by Srivastav (1964), these authors reduced arising dual series equations to a Fredholm integral equation of the second kind given on the interval ½0; a=R. The solution has been found by using the numerical quadrature method to replace the integral equation by a finite system of linear algebraic equation. When a=R < 0:5, the approximate solution solution was determined by neglecting the integral term. An infinite cylinder with an external crack twisted by the torque at infinity was analyzed in the paper by Kudryavtsev and Parton (1973) who have also used Srivastav’s technique. The solution of the Fredholm integral equation was derived in the form of a power series whose coefficients obeys a quasi-regular infinite system of linear algebraic equations solved numerically. When the crack is shallow ða=R ! 1Þ, a numerical solution may be unreliable. In this case a solution of a certain antiplane problem has been taken by authors to find the asymptotic value of the stress intensity factor. The results by Kudryavtsev and Parton were reexamined by Andreikiv in his book (1982) with asymptotic analysis for E-mail address: [email protected] 0020-7683/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2009.04.009

small a=R. An approximate formula for the stress intensity factor, which is suitable throughout the interval 0 6 a=R 6 1, has been established by combining the exact asymptotic solution for small a=R and the Kudryavtsev–Parton leading asymptotic term for a=R ! 1. The torsion of a hollow cylinder with an external crack was treated by Kazuyoshi et al. (1978) and Yuanhan (1990). Mode I and mode II stress intensity factors for elastic cylinders with an external crack were investigated in the papers by Kudryavtsev and Parton (1973), Keer et al. (1977), Atsumi and Shindo (1979), Nied and Erdogan (1983), Singh et al. (1987), Li et al. (1999), Uyaner et al. (2000), Lee (2002), Suat Kadioglu (2005), Mavrothanasis and Pavlou (2008). The exact asymptotics in the small parameter b ¼ 1  a=R is studied for the torsion of a finite cylinder in this paper. In Section 2, the problem is reduced to dual Dini series equations. To achieve an efficient solution for small values of b, a novel technique based on a new discontinuous Dini series is suggested. Our regularization of dual equations is alternative to that of Srivastav and leads to a Fredholm integral equation of the second kind given on the crack surface. Elegant simple formulas for mechanical characteristics are established in Section 3. Hence, when the crack is shallow, exact asymptotic expansions are derived for the solution of the Fredholm equation solution, the stress intensity factor, and the relation connecting the angle of angular twist with the torque (Section 4). We obtain the asymptotic expansions in the form of certain power-logarithmic series. Coefficients of these series can be determined from recurrent relations. Explicit expressions for ten leading terms are found. Results for mechanical characteristics are discussed. In particular, calculations show that the solution obtained can be employed for 0:5 < a=R < 1 if the cylinder is not too short.

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Fig. 1. Geometry of the problem in the coordinates q ¼ r=R; h; f ¼ z=R: (a) a cracked cylinder twisted by terminal discs, (b) the cross-section f ¼ 0.

2. Reducing the problem to a Fredholm integral equation of the second kind In the dimensionless cylindrical coordinates q ¼ r=R; h; f ¼ z=R whose f-axis coincided with the axis of the cylinder, the only non-vanishing displacement uh ðq; fÞ obeys the partial differential equation

  1 @ @u u @2u q h  h2 þ 2h ¼ 0: q @q @q q @f

ð1Þ

The non-trivial components of the stress tensor are expressed via uh ðr; zÞ

Rshr

  @ uh @uh ¼ Gq ; ; Rshz ¼ G @q q @f

ð2Þ

where G is the shear modulus. Denoting l ¼ L=R and a ¼ a=R; we have the boundary conditions on the surface of the cylinder

shr ðq; fÞ ¼ 0; jfj 6 l; q ¼ 1;

ð3Þ

uh ðq; lÞ ¼ cþ Rq; 0 6 q 6 1; uh ðq; lÞ ¼ c Rq; 0 6 q 6 1;

ð4Þ ð5Þ

and the conditions in the cross-section f ¼ 0

shz ðq; þ0Þ ¼ shz ðq; 0Þ; 0 6 q 6 1; shz ðq; 0Þ ¼ 0; a < q 6 1; uh ðq; þ0Þ ¼ uh ðq; 0Þ; 0 6 q 6 a:

ð6Þ ð7Þ ð8Þ

The additional condition of equilibrium

Z

1

q2 shz ðq; lÞdq ¼

Z

0

1

q2 shz ðq; lÞdq ¼

0

M 2pR3

The remaining conditions (7) and (8) give rise to the following dual series equations

  cþ  c Aþn J 1 ðkn qÞ þ Bþ l þ R q ¼ 0; 0 6 q 6 a; 2 n¼1 1 X k J ðk q Þ n 1 n Aþn ¼ Bþ q; a < q 6 1: tanhðkn lÞ n¼1

1 X

1 X

An

n¼1

sinh kn ðl  jfÞ J 1 ðkn qÞ þ ðB ðl  jfjÞ þ c RÞq; sinh kn l

2

 1 X 0; 0 6 q < 1  t; J 0 ðkn tÞJ 1 ðkn qÞ þ 4q ¼ J ðk Þ q Þ; 1  t < q 6 1: Sðt; n 1 n¼1

2M

pGR2

:

I Lm

2J 0 ðntÞJ 1 ðnqÞ dn: pJ2 ðnÞ

ð16Þ

The contour Lm consists of the arc 0 6 arg n 6 p2 ; jnj ¼ nm with km < nm < kmþ1 , the arcs of small radius e with centers n ¼ 0 and n ¼ kn ; n 6 m; lying within the first quadrant, and the segments of the real and imaginary axes joining the ends of the arcs (see Fig. 2). The integrand is an analytic function within the domain bounded by Lm and possesses poles at n ¼ kn . Hence the integral

ð9Þ

ð10Þ

ð11Þ

The condition (6) is satisfied if

Aþn ¼ An :

ð12Þ

ð15Þ

One might prove by simple asymptotic analysis that the above series converges to a function which is continuous as q–1  t and possesses an integrable singularity at q ¼ 1  t: An explicit expression for the function Sðt; qÞ in the interval 1  t < q 6 1 is omitted because it will be not essential below. In the interval 0 6 q < 1  t, the value of the series (15) is found by inspecting the contour integral

where J m ðxÞ is the Bessel function of the first kind, kn are positive zeros of the Bessel function J2 ðxÞ, and the upper and lower superscripts are taken for positive and negative values of f, respectively. The constants B can be expressed in terms of the torque M by means formulas (2) and (9)

Bþ ¼ B ¼ 

ð14Þ

To solve the dual series equations, we shall make use of the following discontinuous Dini series

allows to relate the torque M rotating the terminal discs and the angles c . The solution satisfying the boundary conditions on the lateral surface q ¼ 1 and the bases f ¼ l of the cylinder is given by

uh ðq; fÞ ¼

ð13Þ

Fig. 2. The contour Lm .

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P. Malits / International Journal of Solids and Structures 46 (2009) 3061–3067

is equal to zero. In the limit as e ! 0 and m ! 1; Eq. (15) arises for 0 6 q < 1  t upon computing residues and partitioning the imaginary part off. The Dini series expansion (Watson, 1965)

f ðqÞ ¼

1 X

cothðkn lÞ ¼ 1 þ g n ; g n ¼

Z

we arrive at the Fredholm integral equation of the second kind

2 J 21 ðkn Þ Z 1

1

qf ðqÞJ1 ðkn qÞdq;

0

-ðtÞ ¼

qf ðqÞdq;

a0 ¼ 4

Z

ð25Þ

b

x-ðxÞKðx; tÞdx þ 1; 0 6 t < b;

ð26Þ

0

ð17Þ

0

with the continuous kernel

leads from (15) to the relations

SðJ 1 ðkn qÞÞ ¼ J 0 ðkn tÞJ 1 ðkn Þ; SðqÞ ¼ 1; Z

Kðx; tÞ ¼ Mðx; tÞ  p

ð19Þ

Note that the integrand in (24) rapidly decays for any b; 0 6 t; x 6 b; as n tends to infinity, and the kernel (27) remains continuous even in the limiting case b ¼ 1.

1

qSðt; qÞf ðqÞdq:

1 X

ð18Þ

where the operator Sðf ðqÞÞ is given by

Sðf ðqÞÞ ¼

expðkn lÞ ; sinhðkn lÞ

an J 1 ðkn qÞ þ a0 q; 0 < q < 1;

n¼1

an ¼

that h  h0 ¼ 0:4294849652087197 þ 1:2813737976560965i; ð2Þ H2 ðh0 Þ ¼ ð3:8 þ 2:1iÞ  1017 . This value of h is also confirmed to 9 significant digits by Kerimov and Ckorokhodov (1984). Writing

1t

kn g n J 0 ðkn tÞJ 0 ðkn xÞ:

ð27Þ

n¼1

Applying the operator S to Eq. (13) yields 1 X

Aþn

n¼1

3. Mechanical characteristics

kn J 1 ðkn Þ J ðkn tÞ ¼ Bþ ; 0 < t 6 b; tanhðkn lÞ 0

ð20Þ

where b ¼ 1  a is the normalized depth of the crack. The solution of the dual series is sought in the form of the representation

Aþn

¼

Z

pBþ J 1 ðkn Þ

b

x-ðxÞJ 0 ðkn xÞdx;

0

Rðcþ  c Þ ¼ 4pBþ

Z

ð21aÞ

x-ðxÞdx  2Bþ l;

ð21bÞ

with the subsidiary function -ðtÞ to be determined. On substituting (21a)  (21b) into (13) and interchanging the order of summation and integration we obtain

  cþ  c Aþn J 1 ðkn qÞ þ Bþ l þ R q 2 n¼1 ! þ Z b 1 X J ðkn xÞJ ðkn qÞ pB 0 1 þ 4q dx; 0 6 q 6 1  b: ¼ x-ðxÞ 2 J 1 ðkn Þ 2 0 n¼1

1 X

This manifests that the above representation satisfies Eq. (13) by virtue of the identity (15). Substituting the representation (21a) into (20) gives 1 X

kn cothðkn lÞJ0 ðkn tÞ

Z

x-ðxÞJ 0 ðkn xÞdx ¼ 1; 0 < t 6 b:

ð22Þ

Furthermore we invoke the formula (see Appendix and Malits (2004))

p

1 X

kn J 0 ðkn tÞ

Z

¼ -ðtÞ 

b

x-ðxÞJ 0 ðkn xÞdx

Z

x-ðxÞMðx; tÞdx; 0 6 t < b:

ð23Þ

Here Mðx; tÞ ¼ Mðt; xÞ. For t P x

Z

bJ 1 ðkn bÞ-ðbÞ 

Z 0

b

 xJ 1 ðkn xÞd-ðxÞ :

ð28Þ

Employing the above expression, we compute

Z

Bþ G q; q < a; R 0 " # 1 X Bþ Gx J ðkn xÞJ 1 ðkn qÞ ; Xðx; qÞ ¼ X0 ðx; qÞ þ p gn 1 R J 1 ðkn Þ n¼1

shz ðq; 0Þ ¼ Xðb; qÞ-ðbÞ 

b

Xðx; qÞd-ðxÞ 

1 X J 1 ðkn xÞJ 1 ðkn qÞ ; J 1 ðkn Þ n¼1

X0 ðx; qÞ ¼ p

ð29Þ ð30Þ ð31Þ

where the series containing g n is a continuous function for any x and q. The function X0 ðx; qÞ can be represented in the integral form

X0 ðx; qÞ ¼ 

Z

1

0

I1 ðkxÞI1 ðkqÞ dk; 0 6 q < 1  x; I2 ðkÞ

I Lm

ð32Þ

J 1 ðnxÞJ 1 ðnqÞ dn ¼ 0: pJ2 ðnÞ

in a manner which is analogous to that for the integral (16). Then Xðx; qÞ is seen to be continuous for q < 1  x. As q!1x0

pffiffiffi Z Bþ G x

b 0

Mðx; tÞ ¼

kn J 1 ðkn Þ

0

n¼1



pBþ

by inspecting the contour integral

b

0

n¼1

Aþn ¼ 

b

0

p

It is readily seen from the Fredholm integral Eq. (26) that -ðxÞ is a differentiable function. Then, integrating by parts by means of the relation kn xJ 0 ðkn xÞdx ¼ dðxJ1 ðkn xÞÞ, one can rewrite the representation (21a) as

Xðx; qÞ ¼  pffiffiffiffiffiffiffiffiffiffi

2pqR

0

1

expðkðq þ x  1ÞÞ pffiffiffi dk þ Oð1Þ k

pffiffiffi Bþ G x ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Oð1Þ: R 2qð1  x  qÞ

ð33Þ

1

K 2 ðnÞI0 ðntÞ  I2 ðnÞK 0 ðntÞ h i I0 ðnxÞdn 0 p2 I22 ðnÞ þ K 22 ðnÞ I2 ðnÞ h i ð2Þ  pRe hJ 0 ðhxÞH0 ðhtÞ ;

Thus Eqs. (29) and (11) involve the stress intensity factor for the cylinder under the action of the applied torque M

ð24Þ ð2Þ

Im ðzÞ and K m ðzÞ are modified Bessel functions, Hm ðzÞ is the Bessel ð2Þ function of the third kind, and h is the only zero of H2 ðzÞ lying within the first quadrant (Watson, 1965). Our calculations show

pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2M b 2pRða  qÞshz ðq; 0Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5=2 -ðbÞ: q!a pð1  bÞR

K III ¼ lim

ð34Þ

The equation related the angle of the angular twist cþ  c and the torque M follows from (21b) and (11)

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P. Malits / International Journal of Solids and Structures 46 (2009) 3061–3067

GR3 þ l ðc  c Þ  ¼ 2 4M p

Z

b

x-ðxÞdx:

ð35Þ

Finally, the stress intensity factor induced by the angular twist of angle cþ  c is given by

K III ¼

w11 ðsÞ ¼ 

0

pffiffiffiffiffiffiffiffiffi pRbGðcþ  c Þ-ðbÞ i : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih Rb 2l ð1  bÞ 1 þ 2lp 0 x-ðxÞdx

ð36Þ

15 225 ; w22 ðsÞ ¼ ; w33 ðsÞ ¼ 0:8239746; 16 256

w10 ðsÞ ¼ 2:033134 

15 2 d0 s  ; 32 2

w20 ðsÞ ¼ 0:0384521s4 þ 1:2355904s2 þ 4:8557992 þ ð2s2 þ 1Þ þ ð0:234375s2  1:9159465Þd0 þ

4. Asymptotic analysis for a shallow crack

w21 ðsÞ ¼ 0:3076172s2  1:0726016 þ

In the case of a shallow crack b  1 the solution of the integral Eq. (26) can be found with asymptotic analysis. Upon making changes t ¼ bs, x ¼ bu, (26) becomes

-ðsbÞ ¼ b2

Z

1

u-ðubÞKðub; sbÞdu þ 1; 0 6 s < 1:

ð37Þ

X

b2 Kðub; sbÞ ¼

b2nþ2 ½K n ðu; sÞ þ Q n ðu; sÞ ln b;

n¼0 n X

Q n ðu; sÞ ¼

(

bm;n

m¼0

u2m s2n2m ; u P s s2m u2n2m ; u 6 s

n X

K n ðu; sÞ ¼

ð38Þ

 ð0:0192261s4 þ 0:196211s þ 6:9996645Þd0   d1 d0 15s4  ð2s2 þ 3Þ   0:195548s2  0:34703294 d1 ; 32 1024

(

2m 2n2m

u s

; uPs

s2m u2n2m ; u 6 s

w32 ðsÞ ¼ 0:28839111s2 þ 2:7101007  ð40Þ

with the coefficients bm;n and am;n being combinations of the series

k2kþ1 g n ; lim dk ¼ 0; n l!1

n¼1

w31 ðsÞ ¼ 0:0360489s4  1:6728956s2  8:2143923 

ð39Þ

;

½am;n þ bm;n lnðmaxðu; sÞÞ

1 X

ð41Þ

and certain integrals whose explicit expressions are omitted. In particular, upon evaluating the integrals we have

Q 2 ðu; sÞ ¼ 0:11329651ðs4 þ 4s2 u2 þ u4 Þ; K 0 ðu; sÞ ¼ 3:128768  d0  1:875lnðmaxðu;sÞÞ; d1 K 1 ðu; sÞ ¼ ð4:2454079 þ 4  0:26367188lnðmaxðu;sÞÞÞðs2 þ u2 Þ

xðbsÞ ¼

d11 ¼ 

ð42Þ

k1 Z X l¼0

ð43Þ

1

ð47Þ

m¼0

d30 ¼ 

d0 d2 d1 ; d20 ¼ 0  1:798759d0 þ þ 5:4864118; 2 4 8

d30 d1 d0 5 d2 þ 1:3490693d20  þ 0:43992413d1  384 8 8

7 165 d0 ; d32 ¼ 2:5659051  d0 ; 32 256 11 2 15 ¼ 9:0388238 þ 2:6051296d0  d  d1 ; 32 0 64

d21 ¼ 0:918793 þ d31

ð48Þ

and

u½wkl1;m ðsÞK l ðu; sÞ þ wkl1;m1 ðsÞQ l ðu; sÞdu;

0

k P 1; m ¼ 0; 1; . . . ; k; w0;0 ðsÞ ¼ 1; wk;1 ðsÞ ¼ 0; wk;n ðsÞ ¼ 0 for k < n:

ð46Þ

 7:1368806d0 þ 0:6294092

Inserting (38) and (43) into (37), we obtain the recurrent equations for wk;m ðsÞ by equating coefficients of like asymptotic terms

wk;m ðsÞ ¼

ð45Þ

15 225 ; d22 ¼ ; d33 ¼ 0:82397461; 16 256

d10 ¼ 1:798759 

m¼0

k¼0

165 d0 : 256

with the coefficients dkm given by

Now, write xðbsÞ in the form of the asymptotic expansion k X m b2k wk;m ðsÞln b:

15 d1 256

In the case of a long cylinder l  1 one might neglect the terms containing dk . Below it will be shown by numerical calculations that the long cylinder approximation for coefficients of asymptotic expansions can be taken even when a cylinder is rather short. Asymptotic expansions of the mechanical characteristics become

k¼1

 0:2636718maxðu2 ;s2 Þ;   3 K 2 ðu; sÞ ¼ 0:11329651 Q 2 ðu; sÞ lnðmaxðu; sÞÞ  maxðu2 ;s2 Þ  4s2 u2 2  0:22488794ðs4 þ s2 u2 þ u4 Þ d2  ðs4 þ 4s2 u2 þ u4 Þ: 64

11 2 d ; 32 0

GR3 þ l 4 ðc  c Þ  ¼ WðbÞ þ Oðb10 ln bÞ; 4M p 3 k X X m WðbÞ ¼ b2 þ b2kþ2 dkm ln b;

Q 0 ðu; sÞ ¼ 1:875; Q 1 ðu;sÞ ¼ 0:26367188ðs2 þ u2 Þ;

X

7 d0 ; 32

þ ð2:6234401  0:0366211s2 Þd0  ð2s2 þ 3Þ

m¼0

dk ¼ p

d20 ; 4

w30 ðsÞ ¼ 0:058651s6  0:3615551s4 þ 1:4705296s2 þ 12:892385   1 15 2 2 d2  d30 þ 1:407663  s d0  ð3s4 þ 6s2 þ 1Þ 8 128 384

0

Expand the kernel Kðub; sbÞ into the asymptotic series by inserting the standard series representations of Bessel functions (Watson, 1965, Abramowitz and Stegun, 1964) and integrating term-by-term,

d1 16

ð44Þ

Elementary but cumbersome computations implemented with Maple yield the polynomial expressions

pffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffi  pR5 4 b17 ln b ; K III ¼ XðbÞ þ O 2M sffiffiffiffiffiffiffiffiffiffiffiffi" # 3 k X X b m XðbÞ ¼ b2k ckm ln b ; 1þ 1b m¼0 k¼1

ð49Þ ð50Þ

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P. Malits / International Journal of Solids and Structures 46 (2009) 3061–3067

with the coefficients ckm given by

15 225 1 ; c22 ¼ ; c33 ¼ 0:82397461; c10 ¼ 1:564384  d0 16 256 2 1 2 3 ¼ 6:1298417  1:6815715d0 þ d0 þ d1 ; 4 16 7 ¼ 0:76498441 þ d0 ; 32 3 d 5d1 d0 ¼ 1:1676255  0 þ 1:2904755d20  7:2151016d0  8 32 d2 þ 0:5279325d1  ; 192 75 11 2 ¼ 9:851239 þ 2:586819d0  d1  d ; 256 32 0 165 ¼ 2:4217096  ð51Þ d0 : 256

c11 ¼  c20 c21 c30

c31 c32

Results for the normalized stress intensity factor and angle of the angular twist, which are evaluated by the asymptotic formulas (49) and (46), are summarized in Tables 1 and 2, respectively. It is seen that the shallow crack asymptotic expansions of the mechanical characteristics give close results for the finite and infinitely long cylinders if the value of the applied torque M is fixed and the cylinder is sufficiently long, l P 1=3. When b 6 0:3 or l P 0:4, the relative discrepancies are within 1 per cent. For l P 0:5 the relative discrepancies are within 0.25 per cent. The Table 3 gives comparative analysis of the normalized stress pffiffiffiffiffiffiffiffiffi pffiffiffiffiffi R5 K III =2M is intensity factor K ¼ pR5 ð1  bÞ3 K III =2M, where evaluated by the asymptotic formulas for infinitely long cylinder, e obtained by with the approximate values of the same quantity K Kudryavtsev and Parton (1973). When the angle of the angular twist cþ  c is fixed, the asymptotic formula for the stress intensity factor can be readily derived from (36) by combining (49) and (46),

pffiffiffiffi

K III pXðbÞ pffiffiffi ¼  : RGðcþ  c Þ 2l 1 þ pl WðbÞ

ð52Þ

pffiffiffi Fig. 3. Stress intensity factors K III = RGðcþ  c Þ evaluated by formula (52).

In this case the strong dependence of the stress intensity factor upon the length of the cylinder is observed (Fig. 3). When l P p, the asymptotic expansion

K III

sffiffiffiffiffiffiffiffiffiffiffiffi" # 3 k X X pb m 2k b pkm ln b 1þ 1b m¼0 k¼1 qffiffiffiffiffiffiffi 4 þ Oð b17 ln bÞ

1 pffiffiffi ¼ þ  2l RGðc  c Þ

with the coefficients

p 15 p p2 p10 ¼ 1:564384  ; p11 ¼  ; p20 ¼ 6:1298417  3:363143 þ 2 ; 16 l l l p p2 225 p21 ¼ 0:7649844 þ 1:875 þ 2 ; p22 ¼ ; 256 l l p

p

p2

p31 ¼ 9:8761113 þ 4:836724  2:8125 2 ; l l

p

lnb

0.05

0.1

0.2

0.25

0.3

0.35

0.4

1 0.75 0.5 0.4 1/3

0.261725 0.261723 0.261704 0.261663 0.261592

0.39062 0.39061 0.39051 0.39028 0.38990

0.64321 0.64317 0.64266 0.64161 0.63987

0.79413 0.79406 0.79328 0.79164 0.78875

0.97724 0.97715 0.97605 0.97357 0.96837

1.20772 1.20761 1.20609 1.20193 1.19091

1.50560 1.50546 1.50313 1.49501 1.46935

Table 2 h 3 i ðcþ  c Þ  pLR evaluated by formula (46). Relations 100 GR 4M lnb

0.05

0.1

0.2

0.25

0.3

0.35

0.4

1 0.75 0.5 0.4 1/3

0.25291 0.25290 0.25289 0.25285 0.25278

1.0409 1.0408 1.0405 1.0399 1.0388

4.5948 4.5945 4.5903 4.5815 4.5669

7.698 7.697 7.687 7.667 7.635

12.054 12.053 12.034 11.996 11.930

18.103 18.101 18.069 18.001 17.870

26.465 26.461 26.409 26.288 26.014

Table p ffiffiffiffiffiffi 3 pR5 ð1  bÞ3 K III for infinitely long cylinder. 2M b

0.05

0.1

0.2

0.3

0.4

0.5

K e K

0.199 0.210 5.6

0.252 0.231 8.5

0.292 0.274 6.1

0.297 0.286 3.7

0.2882 0.2880 0.07

0.268 0.264 1.5

e j KK K j  100%

p2 p3

p30 ¼ 1:1676255  14:430203 þ 5:161902 2  3 ; l l l

p32 ¼ 2:4217096  2:6367188 ; p33 ¼ 0:8239746; l

Table 1 pffiffiffiffi 5 Stress intensity factors MR K III evaluated by formula (49).

ð53Þ

ð54Þ

can be employed instead of (52). Note that the quantities dk rapidly increase for l 6 1=3. Then the range of applicability of the asymptotic expansions obtained in this section rapidly shrinks as l decreases. 5. Conclusions In this paper we give a new solution to the problem of an elastic finite cylinder with a circumferential edge crack twisted by terminal rigid discs. The dual series of the problem are solved with a new technique which permits an efficient solution in the case of a shallow crack. This basing on a new discontinuous integral technique regularizes the ‘‘edge” singularity of equations and leads to the Fredholm integral equation of the second kind on the crack surface. It enables us to establish asymptotic expansions of the mechanical characteristics when the crack depth is relatively small. Calculations manifest that for 0:5R < a < R the stress intensity factor exhibits the rather weak dependence upon the cylinder length when the torque is fixed and the triple length is larger than the diameter. The strong dependence of the stress intensity factor upon the length is observed if the angular twist of the terminal discs is given.

Appendix A The formula (23) can be derived as follows. We write for t6b<1

3066

p

P. Malits / International Journal of Solids and Structures 46 (2009) 3061–3067

1 X

kn J 0 ðkn tÞ

Z

Fðt; x; gÞ ¼ p

x-ðxÞJ 0 ðkn xÞdx ¼ lim

g!þ0

0

n¼1

1 X

Z

b

b

ð1Þ

Yðx;t; gÞ: ¼ phJ0 ðhxÞH0 ðhtÞexpðhgÞ Z 1 I0 ðnxÞK 0 ðntÞ cosðngÞdn þ 2 2 2 0 p I2 ðnÞ þ K 2 ðnÞ " # Z 2 1 p2 I2 ðnÞ  n sinðngÞdn: I ðnxÞK 0 ðntÞ þ p 0 0 K m ðnÞ½p2 I22 ðnÞ þ K 22 ðnÞ

x-ðxÞFðt; x; gÞdx;

0

kn expðgkn ÞJ 0 ðkn xÞJ 0 ðkn tÞ:

n¼1

Inspecting the contour integral

I

J 0 ðntÞJ0 ðnxÞ ð2Þ

J2 ðnÞH2 ðnÞ

Lm



2

p

e ! 0 in

results as m ! 1 and 1 p2 X

Taking into account the above formula, we obtain by inserting (56) into (55)

expðngÞdn ¼ pihJ 0 ðhtÞJ 0 ðhxÞ expðhgÞ

n¼1

kn expðgkn ÞJ 0 ðkn xÞJ 0 ðkn tÞ þ

n¼1

 expðngÞdn þ

Z

0

1

1 X

Z

1

0



J 0 ðntÞJ 0 ðnxÞ

Z

ð2Þ

involving

b

x-ðxÞJ 0 ðnxÞdx þ lim

g!þ0

 sinðngÞdn;

have been exploited. Partitioning the real parts off, we find that for x þ t < 2

Fðt; x; gÞ ¼ 2phJ 0 ðhtÞJ 0 ðhxÞ expðhgÞ Z 2 1 J 0 ðntÞJ0 ðnxÞ expðngÞ þ dn ð2Þ ð1Þ p 0 H2 ðnÞH2 ðnÞ Z 1 i I0 ðnxÞI0 ðntÞK 2 ðnÞ cosðngÞ h 2 2  ðnÞ p I2 ðnÞ þ K 22 ðnÞ dn I2 0 Z 1 I0 ðnxÞI0 ðntÞ sinðngÞ dn: p p2 I22 ðnÞ þ K 22 ðnÞ 0

1

J 0 ðntÞJ0 ðnxÞexpðngÞ ð2Þ

0

Yðx; t; gÞ ¼

ð1Þ

H2 ðnÞH2 ðnÞ Z 1 2

dn ¼

pH2ð2Þ ðnÞHð1Þ 2 ðnÞ

0

b

x-ðxÞMðx; t; gÞdx

ð57Þ

0

x 6 t 6 b:

The well-known Hankel integral theorem (Watson, 1965) manifests that the first term at the right side of (57) is equal to -ðtÞ if -ðtÞ is a continuous function. It follows from the asymptotic expansions of modified Bessel functions that the integrals in Mðx; t; gÞ converge absolutely and uniformly with respect to the parameter g. Then we are permitted to look for the limit g ! þ0 inside the integration signs. Finally,

p

1 X

kn J 0 ðkn tÞ

Z

n¼1

b

0

x-ðxÞJ 0 ðkn xÞdx ¼ -ðtÞ

Z

b

x-ðxÞMðx;tÞdx; 0 6 t < b;

0

g!þ0

1 0

n expðngÞJ 0 ðntÞdn

with Mðx; tÞ ¼ lim Mðx; t; gÞ; Mðx; tÞ ¼ Mðt; xÞ as t 6 x, and as t P x

The second term at the right can be decomposed as 2

Z

0

Mðx;t; gÞ ¼ pRe½hJ0 ðhxÞH0 ðhtÞ expðhgÞ Z 1 I0 ðnxÞI0 ðntÞ p sinðngÞdn p2 I22 ðnÞ þ K 22 ðnÞ 0 Z 1 I2 ðnÞK 0 ðntÞ  I0 ðntÞK 2 ðnÞ þ I0 ðnxÞcosðngÞdn I2 ðnÞ½p2 I22 ðnÞ þ K 22 ðnÞ 0 " # Z 2 1 p2 I2 ðnÞ n I ðnxÞK 0 ðntÞ þ p 0 0 K 2 ðnÞ½p2 I22 ðnÞ þ K 22 ðnÞ

dJ2 ðuÞ ð2Þ 2i ; H2 ðuÞju¼kn ¼ du pk n

Z

g!þ0

1

ð2Þ

ð55Þ

dJ2 ðuÞ ð2Þ dH ðuÞ 2i H2 ðuÞ  J 2 ðuÞ 2 ¼ ; du du pu

p

0

x-ðxÞJ 0 ðkn xÞdx ¼ lim

with Mðx; t; gÞ ¼ Mðt; x; gÞ as t 6 x 6 b,

iI0 ðnxÞI0 ðntÞ expðingÞ dn 2Im ðnÞ½I2 ðnÞ þ iK 2 ðnÞ=p

In the foregoing, the Wronskian (Watson, 1965)

Z

Z

b

0

ð2Þ

J 2 ðnÞH2 ðnÞ

¼ p2 hJ 0 ðhtÞJ0 ðhxÞ expðhgÞ:

Z

kn J 0 ðkn tÞ

n expðngÞJl ðntÞJl ðnxÞdn þ Yðx; t; gÞ; !

Mðx; tÞ ¼

Z

1

K 2 ðnÞI0 ðntÞ  I2 ðnÞK 0 ðntÞ

½p2 I22 ðnÞ þ K 22 ðnÞI2 ðnÞ h i ð2Þ  pRe hJ 0 ðhxÞH0 ðhtÞ :

I0 ðnxÞdn

0

 n J0 ðntÞJ 0 ðnxÞ expðngÞdn; ð56Þ

References

where for x 6 t

Yðx; t; gÞ ¼ Re

Z

1

0

2

p

ð2Þ ð1Þ H2 ðnÞH2 ðnÞ

! n

ð1Þ

J 0 ðnxÞH0 ðntÞ expðngÞdn; Yðx; t; gÞ ¼ Yðt; x; gÞ as x P t: The integral Yðx; t; gÞ can be transformed with the integration along the contour consisting of the arc of small radius jnj ¼ e; 0 6 arg n 6 p2 and arc of large radius jnj ¼ f; 0 6 arg n 6 p2 , which are joined by the segments of the real and imaginary axes. As e ! 0 and f ! 1, we arrive by using the Wronskian (Watson, 1965) ð1Þ

ð2Þ

dHm ðuÞ ð2Þ dHm ðuÞ 4i Hm ðuÞ  Hð1Þ ¼ m ðuÞ du du pu at the expression

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