The general solutions of an infinite plate with doubly periodic cracks under bending and twisting

Engineering Frocfure Meckunics I’Med in the U.S.A.

Vol. 20. No. 3, pp. 561-567,

oD13-7944/&o

1964

$3.00

+ .OO

Pcwmon Press Ltd.

THE GENERAL SOLUTIONS OF AN INFINITE PLATE WITH DOUBLY PERIODIC CRACKS UNDER BENDING AND TWISTING S. S. CHANG Peking Institute of Civil Engineering, Beijing, China Abstrac-The objective of the present paper is to use the basic theorem of Jacobi elliptic functions and residues to find the general solutions of the stress intensity factor of a plate with doubly periodic cracks subjected to a pair of concentrated couples Mb and M, on the faces of each crack. In this paper the general solutions are expressed in simply closed forms, and they can also be applied to solve practical problems of bending and torsion of a rectangular sheet containing a central crack. The general solutions may be used as a fundamental Green’s fimction for tinding other solutions of a plate with doubly periodic cracks under arbitrary bending and twisting moments. It is easily proved that all problems of bending and torsion of an intinite plate with a central crack or singly periodic cracks, are the special cases of the general solutions in this paper.

FUNDAMENTAL EQUATIONS AND RELATIONS FOR CONVENIENCE, we will only find the complex stress-function? for determining intensity factor of bending and torsion of a cracked-plate because it is the key problem in The maximum values of the stress intensity factors K, and K,,.. occurring at the surface cracked-plate subjected to the bending and twisting moments, can be-calculated by following formulas:

[

the stress this paper. layer of a use of the

& lim J2(z -z*) oymax(z) &I xJmax(z) [z 1' 1=r-4

(1) (1’)

where z stands for the complex coordinate of an arbitrary point in the plate; z* is the complex coordinate of a crack tip. If [ denotes the complex coordinate of an arbitrary point on the crack surface, the expressions (1) and (1’) can be rewritten by

(1”) (1”) because the limit results of (1) and (1’) are same as (1”) and (1”‘). In small deflection of a thin plate, according to KirchofPs theory the maximum stress components of the plate for bending and twisting, may be expressed in terms of the moments:

in which m, and m,,are the bending moments per unit-length; mXyis the twisting moment per unit length, h stands for the thickness of the plate. The moments are related to the normal deflection w as follows:

mx=

-D($$+v$);

my=

--Dr$+v$);

-Y

=-D(l

-v)-.

PW

ibay

tit is the same as the complex stress function in the plane problem[Sl. 561

(3”)

S. S. CHANG

562

The relations between the shearing forces per unit length and w are

I

(Qx, Q,] = -D

;

v2w,$

vlwj,

where V2= (a2/8x2) + (a2/8y2), D = (Eh3/12(1 - v2)), called the flexural rigidity of the plate, E stands for Young’s modulus, v denotes Poisson’s ratio, x and y axes are the same as shown in Fig. 1. Considering the normal load in the plate is vanished, the normal displacement w must be governed by a biharmonic equation $+2-

a% +so. ax2ay2af

(5)

It is well known that the solution of the partial differential equation can be given by means of two complex functions 4(z) and $(z)

(6) Substituting (6) into (3)-(4’) leads to m,(z) = -DRe[2(1

+ v)cp(z) + (1 -v)@‘(z)

+ (1 -VW(Z)];

(7)

m,(z) = -DRe[2(1

+ v)cp(z) - (1 - V&J’(Z) - (1 - v)+(z)];

(7’)

m,(z) =

D(1 - v)Z,[.+‘(z) + W)l,

(7”)

Q,(z) = - 4DRecp ‘(z);

(8)

Q,(z) = 4DZmcp’(z),

(8’)

in which ReF(z) is the real part of the analytic function F(z); ZmF(z) is the imaginary part of the analytic function F(z), z = x + iy; i = x - iy, j? = - 1, q’(z) = (d/dz)cp(z). THE GENERAL SOLUTIONS OF DOUBLY PERIODIC CRACES UNDER BENDING AND TWISTING Let us consider an infinite plate of thickness h, containing an infinite series of the doubly periodic cracks of equal length 2u as shown in Fig. 1. For convenience we will choose the x-axis along the line of the cracks and the y-axis along the direction of the perpendicular bisector of the central crack in the middle plane of the plate. The w-axis is perpendicular to the middle plane,

2K’

t

I

!e

,

I I

4K

i

IFig. 1.

4K

J

The general solutions of an infinite plate with doubly periodic cracks

563

but it is not drawn in Fig. 1. The centers of a pair of neighbouring cracks are a distance 4K apart, and the parallel cracks are spaced apart periodically by a distance 2K’. Assuming the crack faces are subjected to the concentrated bending moment Mb and twisting moment M, at the point z = xi + 4mK + 2nKi, forming a self-balanced system. According to the right-hand rule, the sense of rotation of M6 or M, is represented by use of double-arrows in Fig, 1. For saving the length of a piece of writing, we may only discuss the general solution of bending of the plate because of the general solution of torsion of the plate can be easily obtained by us of the similar method.? In this case the boundary conditions for bending of the plate with doubly periodic cracks may be given as follows: m,(z) = m(z),

as

-a+4mK+2nK’i
P,(z) = 0,

as

-a + 4mK + 2nK’i c z ca + 4mK + 2nK’i;

(10)

p;(z) = 0,

as

IzI > a + 4mK + 2nK’i,

(11)

(9

in which m(z) = Mbd (x, + 4mK + 2nK’i - z); VJz) = Q,,(z) + (a/c%x)m,(z) and m, n = 0, &-1, +2,.... If [ stands for the complex coordinate of an arbitrary point on the crack surface, from the boundary conditions (10) and (11) we can get &(5) = 0.

(12)

Replacing z by < in the equations (7’), (7”) and (S’), my, mxy and Q,, may be rewritten as

m,(C)= -DWzU + v)cpUJ- (1- WP’(O - (1- WUX m,(0

= Nl

- v)Zm [&‘K) +

WI;

(13) (14)

Q,
(19

Inserting (14) and (15) into (12), and using (a/&~)F(c) = (d/d[)F(c)

we have

(16) Then, we find that Zm &‘P(S)+~~‘(l)+Jl(i) [ With the aid of the Cauchy-Riemann

1

=co.

(17)

equation leads to

in which cl is a real number. Substituting (18) into (13) and (14), the moment components mYand mxv on the crack surfaces can be expressed by use of only one function, i.e. m,(c) = -20(3

+ v)Recp(c) + D(1 - v)c,;

m&X = - 4DZmcp([). Letting

CPUJ = - 2D(;+v)‘Pa(c)+&jc19 tPlease see Appendix.

(1% (20)

S. S. CHANG

564

the expressions (19) and (20) will become

(21) (22) where z = [, Zmc, = 0. the components of the moment are doubly periodic with 4K and 2K’i, the. complex function (P*(Z) must be the same periods 4K and 2K’i. Proof Considering the formulas’ (21) and (22) exist everywhere on the crack surfaces, putting z = z + 4mK + 2nK’i and substituting it into the representations (21) and (22), we have LEMMA. If

m,(z + 4mK + 2nK’i) = Rerp,(z + 4mK + 2nK’i); m,(z

+ 4mK + 2nK’i) = &

Zm(z + 4mK + 2nK’i).

(23) (230

According to the known condition, we get m,,(z) = m,(z + 4mK + 2nK’i); m,(z)

Making a comparison

= m,(z

(24)

+ 4mK + 2nK’i).

(240

between the expressions (23), (23’) and (24), (247, we obtain Rep,(z)

= Req,(z

-t 4mK + 2nK’i);

(25)

Zmq,(z) = Zmq,(z + 4mK + 2nK’i).

(25’)

Then this Lemma is proved, i.e. q+,(z) = q+,(z + 4mK + 2nK’i).

(26)

With the help of the boundary condition (9) we can easily find that the bending moment issingularities at z = x, + 4mK + 2nK’i with the double periods 4K and 2K’i. In this case the complex function (pb(z) is also singularities at z = xl + 4mK + 2nK’i with the same periods 4K and 2K’i. According to theorem of pole point, the function (pb(z) must contain the factor of (Snz - &xl)-‘. Considering the singularities at the crack tips z = -t-a + 4mK + 2nK’i with the double periods 4K and 2K’i, it is well known that the function (P*(Z) should contain the factor of (Sri***- &*a)-I’*. To sum up the above analysis, the complex function (pb(z) may be represented by the following basic form[5]:

%(Z)=

c (Snz - Snx,)J(Sn*z

- Sn*a)’

z=[

(27)

where c is a constant, and it can be determined by the boundary equilibrium conditions on the faces of each crack. From any one of the cracks we get

s

o+4mK+hri

-a+4mK+2nZi

c

dz

(Snz - Snx,)J(Sn’z

- %*a)

= Mb,

(28)

in which it has been employed the expression (21) and the doubly periodic equation[S], so the symbol “Re” is automatically vanished.

565

The general solutions of an infinite plate with doubly periodic cracks

Making use of the theorem of the residue, the constant c can be dete~ned C = i ~~Cnx,Dnx, J(Sn’a

by eqn (28).

- Sn*x,).

(29)

Inserting (29) into (27), we obtain the complex function of the cracked plate under bending: M&,,x,Dnx, ,/(Sn2a - Sn’x,) %I@) =

a(Snz - snxl)J(sn2z

- S?Az)

(30)

We can easily prove that the boundary conditions (9Hll) and the differential equation (5) are all satisfied by use of the solution (30).‘It is omitted here. Using the formula (l), (2), (21) and (30), and considering the doubly periodic equations of the Jacobi elliptic functions, and with the help of the limit result[5] Lim Snz - Sna

T-B

z-a

= CnaDna,

(31)

the general solution of the plate with doubly periodic cracks under bending is given as K-

6M,Cnx,Dnx,

(32)

’ - zh *~(S~~CnaD~~)

Let us take the solution of torsion of the plate containing the doubly periodic cracks together with the bending cracked-plate by use of the following form. The complex functions are Cnx, Dnx, J(Sn 2a - Sn 2x,) [~~,,]=[~]n(snr-Snx,)~(S~2r-S~2~)’

(33)

and the stress intensity factorsare (34)

SPECIAL CASES 1. The solutions of an infinite sheet containing an inflnite-series of the doubly periodic cracks

subjected to the continuously distributed bending moment mb(xI) and twisting moment m,(x,)-along the surfaces of each crack, can be found by substituting Mb = mb(xl) dx, and M,=m,(x,)dx, into the expressions (33) and (34), and integrating with respect to x1 from --a + 4?nK + 2nK’i to a + 4z~zK+ 2nKi. The complex functions are OfhK+trRI

I 1 q,(x,) m,(x,)

J(Sn*a - Sn2x,) - d(Snx,) n(Snz - Snx,) J(sn’z - Sn’a)’

(35)

and the stress intensity factors are

in which d(Snx,) = Cnx,Dnx, dx,. 2. When an infinite series -of the doubly periodic cracks are loaded by the uniform couples m, and 111,on the faces of each crack, the simple results of this problem may be obtained by use of

566

S.S.CHANG

integrating the equations (35) and (36). The complex functions are

(37) and the stress intensity factors are

(38)

3. The solutions of an infinite series of collinear cracks placed along the x-axis, subjected to the concentrated bending moment Mb and twisting moment M, at the point z = xl + 2mrz of each crack, can be derived from the general solutions (33) and (34). Letting K’ = co, the Jacobi elliptic functions shall become[ I]: (Snz, Cnz, Dnz) = {sin z, cos z, l}. Substituting the results into (33) and (34), we get the complex functions

and the stress intensity factors (40)

4. When an infinite series of collinear cracks are subjected to uniform couples mb and m, on the crack faces, the solutions may be similarly obtained by use of the formulas (37) and (38). The complex functions are (41)

and the stress intensity factors are

[z]=[p$?

(42)

The solutions (39)-(42) are only. corresponded to the period 21~of an infinite series of collinear cracks because K = (x/2) as K’ = co [l]. 5. Let K = co and K’ = co, the solutions found by Sih[3] in 1962, may also be obtained by means of the general solutions (33) and (34). Considering lim Snu = u and lim Gnu = lim Dnu = 1, Id-0 u-0 u-0 the complex functions are

(43) and the stress intensity factors are

The general solutions of an iutinite pfate with doubly periodic cracks

567

REFERENCES [l] W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theoremsfor the Special Functions of Math. Phys., pp. 377394 (NM). {2] H. Tada, P. C. Paris and G. R. Irwin, The Stress Analysisof Cracks Handbook, p. 76 (1973). [3] G. C. Sih, J. Appl. Mech., pp. 306-312 (1962). [4] G. C. Sih, handbook of Stress interim Factors, 6.1-l - 6*2+1-g(1973). [51 S. S. Chang, The general solutions of the doubly periodic cracks. Engng Fracture Mech. 4,887-893 (1983). (Received 19 September 1983)

APPENDIX The complex function of a cracked-plate under twisting. Similarly, we will only find the complex function for expressing the twisting moment on the crack surfaces. If we take the solution of the partial differential equation (5) W(r)=Zm[iSrp(z)dz+SJ~~(z)~dr],

(Al)

from the formulas (3X3”), m, m,, and mxy can be found as follows: m,(z)= -DZm[Z(i +v)ca(z)+(l

-v).@‘(z)+(l

-v)Jl(z)];

m,(z) = -DZm[2(1 + v)9(z) - (1 - v)?(p’(z) - (1 - v)@(z)]; m,,(z) = -D(l - v)Re[i;y,‘(z) + e(z)].

WI (A31 (A41

Putting z = I, we get m,fO

= --~fWW

+ v)cpfO

m,(F)

= -DW2(1+

v)9(0

m,(i

I=

-W

- vYWf9rK)

+ (1 - v)I;p*(I) -

+ (1 - v)tj(c)];

(A%

v)@(c)];

646)

(1 - v)c9’(I) - (1 + 11(1)1.

(A7)

In this case, one of the boundary conditions is

m,(l) = 0. Considering (A8) and employing the Cauchy-Riemann

w-9

condition may lead to

Setting (AS) into (A7) we have mXv= -DRe[Z(l

+ v)rp(l)] + C.

(A9’)

Because of 9(C) is an arbitrary function, we may write (AY) as

m,(z) = Reso,(z) (z = If.

WO)