NASA Contractor Report 177925 

A CLAMPED RECTANGULAR PLATE 

CONTAINING A CPACK 

Renji Tang and F. Erdogan 

LEHIGH UNIVERSITY 

Bethlehem, Pennsylvania 

Grant NGR 39-007-011 

March 1985 

NI\S/\ 

National AeronautiCS and 

Sp~ce Administration 

Langley Research Center 

Hampton, V:rglnla 23665 

NASA-CR-177925 

19850023239 

)-

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Ll\r~CL[Y f, r ::":ri'-ICH CEI'Jr [~ 

L/8f'ARY, l'J'\Si\ 

H~' 'pro,'I. VIR:"II:.\ 

1111111111111 1111 111111111111111111111111 1111 

NF00689 
A CLAMPED RECTANGULAR PLATE CONTAINING A CRACK 

by 

Renji Tang * and F. Erdogan 

Department of Mechanical Engineering and Mechanics 

Lehigh University, Bethlehem, PA 18015 

Abstract 

In this paper the general problem of a rectangular plate clamped along 

two parallel sides and containing a crack parallel to the clamps is considered. The problem is formulated in terms of a system of singular integral equations and the asymptotic behavior of the stress state near the corners 

is investigated. Numerical examples are considered for a clamped plate without a crack and with a centrally located cracK, and the stress intensity factors and the stresses along the clamps are calculated. 

1. Introduction 

A long strip containing a central or an edge crack and a rectangular 

block with an edge crack are two of the more widely studied geometries in 

fracture mechanics. The former ~enera11y simulates a single-edge-notched 

or a center-notched specimen and the latter the compact tension specimen. 

These specimens are often used for fatigue crack growth and fracture characterization of engineering materials. Typical studies using a variety of 

methods such as finite elements, Wiener-Hopf, weight function, Laurent 

series, conformal mapping, boundary collocation, and integral equations may 

be found in references [1]-[13]. In all these and similar studies the 

external loads are usually assumed to be such that the boundary conditions 

may be prescribed in terms of tractions only. However, in some cases the 

loads are applied to the specimen through IIgripsll. In these problems it 

would be more appropriate to prescribe the boundary conditions along the 

grips in terms of displacements rather than tractions. The problem of a 

* Permanent address: Associate Professor, Department of Mathematics and 

Mechanics, Lanzhou University, China 

-1-
rectangular strip without a crack loaded through two rigid grips was considered in [14J, where it was shown that the contact stresses along the 

grips are singular at the corners. In this paper the problem of a strip 

containing a crack is considered by using a technique which is somewhat 

different than that of [14J and which may be used to solve the clamped 

plate problem for an arbitrarily oriented crack. 

2. Formulation of the Problem 

We will first formulate the problem described in Fig. 1 where the 

clamped strip contains three cracks as shown. The "external load" in the 

problem is the relative clamp displacement uo• One may express the stress 

and displacement components in the strip as follows: 

bk 

°ij(x,y) = °sij(x,y) + k!l J Gijk(x,y,Sk,t)fk(t)dt, 

ak 

(i,j=x,y; k=l, •.• ,S), 

bk 

ui(x,y) = usi(x,y) + ~ J Vik{x,y,Sk,t)fk(t)dt. k=l ak 

(i=(x,y) , k=l, .•. ,S) 

(1) 

(2) 

where 0sij and usi are associated with an infinite strip without any cracks 

and the functions Gijk and Vik represent the stresses and displacements 

in an infinite plate at the point (x,y) due to dislocations fk distributed 

along the cuts (ak<t<bk). In terms of the crack surface displacements the 

functions fk are defined as follows (Fig. 1): 

f;(x) = d~ [uy(x,s;+O) - uy(x,s;-O)] , (;=1,2) 

f3(Y) = d~ [ux(s3+0,y) - ux(s3-0,y)] 

fj(x) = :x [ux(x,Sj+O) - ux(x,Sj-O)] , (j=4,5) 

fS(Y) = ;y [uy{s6+0,y) - uy{sS-O,y)] , (3a-d) 

-2-
where 

SO+3=So, ao+3=a o, bo+3 111111111 =b o, ao<s.<b., (;=1,2,3) 

and the Green1s functions Gijk and Vij are given in Appendix A. 

(4) 

The solution which corresponds to an infinite strip may be expressed 

as follows: ex> 

usx(x,y) = 2!i J {[Al+(~ + x)A2]e -aX + [-A3+(~ - x)A4]eax e -iay da, 

ex> 

USy(X,y) = in J [(Al+A2

x)e-ax + (A3+A4x)eaxJe-iaYda , (Sa,b) 

(Sa-c) 

The unknown functions fl, ••• ,fS and Al (a), ••• ,A4(a) are determined from the 

boundary conditions prescribed on the crack surfaces and the plate boundaries x=O and y=H. In the actual problem the crack surfaces are free from 

tractions and along the boundaries of the plate we have (Fig. 1) 

The superposition technique may be used to solve the problem by first 

obtaining the stress state in a clamped plate without any cracks under the 

boundary conditions (7). The equal and opposite of the stresses found in 

this solution may then be used as crack surface tractions in the perturbation problem. The ten unknowns fl, .•• ,fS' Al .•. A4 would then have to be 

determined from 

-3-
(8) 

3-1<: llUo 

= - ""-1 -H ' (y.=s.,s=x,a.<s<b.,j=1,2) " J J J J 

(9) 

By substituting from equations (2), (5) and (A13)-(A20) (of Appendix 

A) into (8), by inverting the Fourier integrals and by evaluating the re1ated infinite integrals, the functions Al , .•• ,A4 may be expressed 

of integrals of the following form: 

6 

Ai(a) = E 

j=l 

b. 

J J Bjj(a.t)fj(t)dt • (;=1 ••••• 4) • 

a. 

J 

in terms 

(10) 

where the functions Bij depend also on the constants sl' s2' s3' H, and 1<:. 

From (10) and (6) the stress components asij' (i,j=x,y) are then obtained 

as follows: 

, (i ,j=x,y) , (11) 

where the kernels Cijk are given in Appendix B. 

The integral equations for the unknown functions f 1, ••• ,f6 are 

obtained by substituting from (1) and (11) into the crack surface boundary 

conditions (9) and may be expressed as 

-4-
(12) 

where the kernels hnk (n,k=1, ••• ,6) are known in terms of Gijk and Cijk (i,j=x,y; k=1, ••• ,6) (see equations (1), (11), (9), Appendix A and Appendix B). For the crack problem shown in Fig. 1 (i.e., for O<ak<bk<H, 

k=1,2 and s2<a3<b3<sl) from appendices A and B it can be shown that the main 

diagonal elements of the kernels hnk which are contributed by Gijk are 

Cauchy kernels and consequently hnk can be expressed as follows: 

2 °nk hnk(s,t) = ~(l~K) t-s + knk(s,t) , (n,k=1, •.• ,6) (13) 

where 0nk=l for n=k, 0nk=O for n~k, and k

nk , (n,k=1, ••• ,6) are known bounded 

functions in the closed intervals, ak~(t,s)<bk' (k=1, ••• ,6). If one wants 

to solve the problem shown in Fig. 1 for any crack geometry provided 

O<ak<bk<H, k=1,2 and s2<a3<b3<sl' one could let 

(14) 

and use a Gaussian integration formula to determine the unknown bounded 

functions Fk(t), (k=1, ••• ,6)[15]. From the definition of the unknown functions fk(t) as given by (3) and from Fig. 1 it is clear that the integral 

equations (12) must be solved under the following singlevaluedness conditions: 

bk J fk(t)dt = 0 , (k=1, •.• ,6). (15) 

ak 

-5-
3. The Clamped Plate 

From Fig. 1 it may be seen that the problem would reduce to that of a 

clamped plate if one lets ai=O and bi=H, (i=1,2). In this case technically the integral equations (12) and the singlevaluedness conditions (15) 

are still valid and the input functions Pn(s} are still given by (9). 

However, the form of the solution is no longer given by (14). To determine the fundamental functions of the system of integral equations (12) 

the singular nature of the kernels knk(s,t), (n,k=1, ••• ,6) for ak=O, 

bk=H, (k=1,2,4,5) must be investigated. This may be done through the asymptotic analysis of the infinite integrals giving the appropriate functions 

Cijk in Appendix B which are related to knk (see, for example, [14] and 

[15]). First we observe that from (l), (11), (9), (12), (13) and Appendices A and B, the kernels knk(s,t) may be expressed as follows: 

( 16) 

Q) 

k~k(s,t) = f K~.k(s't,a)da, .(n,k=l , ••• ,6) (17) 

o 

where k~E are the nonsingular parts of Gijk (see equation (1) and Appendix 

A) and knk are the kernels contributed by Cijk . If we now denote the asymptotic values of K~k for a~ by K~kQ)(s,t,a), the singular and Fredholm parts 

of the kernels k~k may be separated as follows: 

CD 

k~k(s,t) = f K~kCD(s,t,a)da, (n,k=1, ••• ,6) , (19) 

o CD 

k~k(s,t) = I [K~k(s,t,a)-K~kCD(s,t,a)]da , (nbk=1, .•• ,6) (20) 

o 

where the kernels k~k(s,t) (n,k=1, •.• ,6) are bounded in their respective 

closed domains. For s2<a3<b3

<sl' after a relatively straightforward analysis 

it may be shown that 

k~k(S,t) = k~n(S,t)onk ' (n,k=1, ..• ,6) , (21) 

-6-
(22) 

s ( _ 1 2 d 2 d2 J 1 knn s,t) - 2K [(3-K )+12s ds + 4s W s+t 

1 d 2 d2 1 - 2K [(3-K2)-12{H-s) ds + 4(H-s) ds Z] 2H-s-t ' 

(n=1,2,4,5) (23) 

Because of (21) and (22) the dominant part of the integral equations 

(12) for n=3 and n=6 has only Cauchy type singularity and the unknown 

functions f3 and f6 have the form 

F .(t) 

f. (t) = J , (j=3,6) 

J l(t-a3)(b3-t) (24) 

Expressing now the remaining unknowns as 

(25) 

through a function-theoretic analysis it may be shown that (see, for example [15]) 

B = y, 2KcosnB-(1+K2 )+4(S-1)2 = 0 • (26) 

The characteristic equation (26) is identical to that found in [14] and 

in, for example, [16] for a quarter elastic plane fixed along one of its 

straight boundaries. 

4. The Stress Intensity Factors 

In the crack problem shown in Fig. 1, once the functions f

1, ••• ,f6 

giving the crack surface displacement derivatives or the bounded functions 

F1, •.. ,F6 defined in (14) are determined, the Modes I and II stress 

intensity factors at the crack tips may be obtained from the following 

standard relations: 

-7-
2 2 F.(a.) k

l

(a 1.) = -ll-lim 12{t-a.) f.(t) =...9:!.... 1 1 , (i=1,2,3) l+K t4a i 1 1 1+K l{bi

-aiJ/2 

(a.=a. 3' b.=b. 3' j=4,5,6) , J J- J J2 2 Fi(bi ) kl(b;) = - f.tL lim 12(b;-t) fi(t) = - 1: ' (i=1,2,3), 

K t~. 111 K l(b.-a.)/2 

2 2 F.(b.) 

= - 2.. lim 12(b.-t) f.(t) = - -r- J J , 

l+K t~. J J +K l(b.-a.)/2 J J J 

(27a-d) 

These relations are based on the definition of the stress intensity factors 

in terms of normal and shear cleavage stresses,at the crack tips of the 

form (Fig. 1) 

, ... 

(28a,b) 

In the "clamped plate" problem it-was shown that at the corners y=si' 

x=O, x=H, (i=1,2) the displacement derivatives have a singularity of the 

order r- S, where r is a small distance (in x direction) from the corners 

(see eqs. (25) and (26». This implies that at the corners the stresses 

may also have a singularity of the order r-S, r now being the radial distance from the corner. In practical applications one may be particularly 

interested in the behavior of interface stresses crxx and crxy along the 

clamps. We first observe that the stress state at any point in the medium 

is given in terms of the density functions fl , .•. ,f6 and the kernels Gijk 

-8-
and Cijk (see (1) and (11». Next, we note that the singular behavior of 

the stresses is determined by the dominant parts of the related kernels 

only and these dominant kernels can be separated through a relatively simple asymptotic analysis of Gijk and Cijk given in the Appendices A and 

B. Also, the singular behaviors of the stresses in the medium at all 

four corners have the same form. Thus, after following a procedure similar to that described by the equations (16) through (23), for a small 

distance r from the corner x=O, y=sl the stresses along the clamp x=O 

may be expressed as 

H H 

°xx(O,sl-r) - TI(~¥K) [J K~l(y,t)fl(t)dt + J K~4(y,t)f4(t)dtJ o 0 

H H 

°xy(O,sl-r) - TI(~¥K) [J K~l(y,t)fl(t)dt + J K~(y,t)f4(t)dt] o 0 

(y=sl-r) , (29a,b) 

where 

(30a-d) 

The kernels given in (30) are "singular" in the sense that they become 

unbounded as t and r approach the end point (t=O, r=O) simultaneously and 

have the order t-l or r-l . To evaluate the stresses given in (29) we 

first note that by substituting fl(t) and f4(t) from (25), the leading 

terms of the typical integrals in (29) may be expressed as follows (see, 

for example [15]): 

-9-
(31a-d) 

Thus, it is seen that the leading terms in the expressions of the 

interface stresses 0xx and 0xy around the corner x=O, y=sl have a singularity of the form r-S• Similar to (28), defining now the "stress intensity 

factors" by 

and substituting from (30) and (31) into (29), we obtain 

k,(O) = 12 ~ [(K+1-2s)cos ~2S Fl (0)-(K+3-2s)sin ~2S F

4(0)] KHSsin~s 

(32a,b) 

~ ~ ITK-1+2s)sin ~: F1(0)+(K-3+2S)cos ~: F4(0)]. 

K Sln~s 

(33a,b) 

The stresses elsewhere along the interfaces x=O and x=H, of course, may be 

evaluated from eqs. (1) and (11) by using the appropriate kernels from 

Appendices A and B. The stress intensity factors in the other corners may 

be obtained by following a procedure similar to that described in (29)-

(33) . 

-10-
5. The Results 

The first example considered is the problem of a clamped plate without a crack in order to compare the results with that given in [14J where 

a different technique was used to soive the problem. In [14J the measure 

of the external loads was the resultant force rather than the displacement U

o used in this paper. Hence, the easiest direct comparison may be 

made for the ratio of the stress intensity factors, k2(O)/kl (O) (see (32) 

for the definitions). For the length to width ratio H/2s1 = 2 and 

K=3-4v this comparison is shown in Table 1. 

v 

Table 1. Comparison of k2(0)/kl(0) obtained in this paper (second row) ana in [14J (third row), H/2s1=2. 

0 0.1 0.2 0.3 0.4 0.5 

- k

2/kl 0 0.118610 0.212352 0.302276 0.396751 0.504213 

(- k

2

/k1) 

(from [14]) 0 0.118614 0.212355 0.302280 0'.396751 0.504209 

For various relative plate dimensions and for K=1.8, the normalized 

stress intensity factors defined by (32) are shown in Table 2. In this 

case (26) gives the power of stress singularity as 8=0.28882. 

Table 2. Normalized stress intensity factors at the corner x=O, 

y=sl of a rectangular clamped plate for K=1.8, plate 1ength=H, plate width=2s1 (see insert in Fig. 3) ~u s 8 ~u s 8 

Hlsl k (0)/( 0 1 ) k (O)/( 0 1 ) -k2(0)/k1(O) 1 sl 2 51 

1 

2 

4 

6 

8 

10 

2.030255 

1.024510 

0.478944 

0.318621 

0.240840 

0.194190 

-0.613697 

-0.309685 

-0.144773 

-0.096312 

-0.072800 

-0.058699 

-11-

0.302276 

0.302276 

0.302276 

0.302276 

0.302276 

0.302276 
Figures 2 and 3 show the normalized stresses crxx(O,y) and crxy(O,y) 

along the clamp for the case of plane strain and for K=1.8. The corresponding Mode II stress intensity factor is shown in Fig. 4. 

The second example considered is a clamped plate containing a central crack (see the insert in Fig. 5). The normalized Mode I stress 

intensity factor calculated at the crack tips y=+c, x=H/2 is shown in 

Table 3 for K=1.8 and K=2.2. For comparison the results obtained in 

[6] by using the technique of conformal mapping for K=2.2 are also 

included in the table. The agreement seems to be fairly good for small 

cracks. However, for larger cracks there is some discrepency. For K=2.2 

and H/2s1=1 the normal stress along the clamp is shown in Fig. 5. 

Table 3. Crack tip stress intensity factor kl(a3) in clamped plate containing a symmetrically located central 

crack, H/2s1=1, c=(b3-a3)/2 (Fig. 5). 

2c 

If 

O. 1 

0.2 

0.4 

0.5 

0.667 

0.8 

REFERENCES 

3. 15135 

3.00550 

2.78474 

2.34322 

1.98095 

1. 71592 

llUo k1(a3)/(T Ie) 

2.60029 

2.49602 

2.17958 

2.00829 

1.73227 

1.52883 

K=2.2 (Ref. 6) 

2.60000 

2.50000 

2.17391 

2.01724 

1.78663 

1.64815 

1. W.K. Wilson, in Methods of Analysis and Solutions of Crack Problems, 

ed. G.C. Sih, Noordhoff, Leyden (1973) 484-515. 

2. W.T. Koiter, Journal of Applied Mechanics Transactions 32, ASME 

(1965) 237-242. 

3. H.F. Bueckner, Zeitschrift fur angewandte Mathematik und Mechanik 51 

(1971) 97-109. 

4. M. Isida, in Method of Analysis and Solutions of Crack Problems, 

G.C. Sih, ed., Noordhoff, Leyden (1973) 56-130. 

-12-
5. H. Terada and M. Isida, "Analysis of Stress Intensity Factors for 

Eccentric Cracks in Plate and Strip", NAL TR-436, National Aerospace Lab., Tokyo, Japan (Dec. 1975). 

6. O.L. Bowie, in Methods of Analysis and Solutions of Crack Problems, 

ed. G.C. Sih, Noordhoff, Leyden (1973) 1-55. 

7. B. Gross and J.E. Srawley, "Stress Intensity Factors for Three 

Point Bend Specimens by Boundary Collocation", NASA-TN 03092 

(Dec. 1965). 

8. J.E. Srawley, International Journal of Fracture 12 (1976) RCR 475-6. 

9. J.C. Newman, Jr., "An Improved Method of Collocation for the Stress 

Analysis of Cracked Plates with Various Shaped Boundaries", 

NASA-TN-D6376 (1971). 

10. H.F. Nied and F. Erdogan, International Journal of Solids and Structures, 15 (1979) 951-965. 

11. A.C. Kaya and F. Erdogan, International Journal of Fracture 16 

(1980) 171-190. 

12. F. Erdogan and H. Terada, "Wedge Loading of a Semi-Infinite Strip with an Edge Crack", Int. Journal of Fracture, 14 (1978}, 399-415. 

13. M.B. Cive1ek and F. Erdogan, "Crack Problems for a Rectangular Plate 

and an Infinite Strip", Int. Journal of Fracture, 19 (1982) 139-159. 

14. G.D. Gupta, "The Problem of a Finite Strip Compressed Between Two 

Rough Rigid Stamps", J. of Appl. Mech., 42 (1975) 81-87. 

15. F. Erdogan, "Mixed Boundary Value Problems in Mechanics", Mechanics 

Today, Vol. 4, S. Nemat-Nasser, ed., pp. 1-86, 1978 Pergamon Press 

Inc. 

16. V.L. Hein and F. Erdogan, "Stress Singularities in a Two-Material 

Wedge", Int. J. of Fracture Mechanics, 7 (l971) 317-330. 

ACKNOWLEDGEMENT 

This work was supported by NASA-Langley under the Grant NGR 39-007-011 

and by NSF under the Grant MEA-8414477 and was completed when the second 

author was an Alexander von Humboldt Senior U.S. Scientist Awardee in 

Freiburg (i. Br.) Germany. 

-13-
APPENDIX A 

The Green's Functions Gijk which appear in equations (1) and (2). 

(t-X)[(t-X)2-(Y-Yk)2] 

Gxxk(x'Y'Yk,t) = n(~~K) [(t-x)2+(Y-Yk)zJ 2 , (k=1,2; Yk=sk) , 

(Al) 

2 (t-x)[3(Y-Yk)2+{t-x)2] 

Gyyk(x'Y'Yk,t) = n(l~K) [(t-x)z+(Y-Yk)Z)Z , (k=1,2; Yk=sk), 

(A2) 

2 (Y-Yk)[(Y-Yk)2_(t-X)2] 

Gxyk(x'Y'Yk,t) = n(l~K) [(t-x)z+(Y-Yk)Z]Z , (k=1,2; Yk=sk) , 

(A3) 

(A7) 

-14-
(AB) 

(A9) 

_ 1 K+1 x-x3 (t-y)(x-x3) 

Vx3 (x,Y,X3 ,t) - - 'IT(1+K) {-2- Arctan t-y - (t-y)2+(X-x3)2}, 

(x3=s3) , (A15) 

-15-
1 1+K Y-Yj (x-t)(Y-Yj) 

Vxj(x'Y'Yj,t) = - 1T(l+K) {-2-Arctan x-t + (x-t)2+(y-y.)2} , J 

(j-4,5; Yj=Sj_3)' (Al?) 

( ) 1 K-l [( )2 ( )2J (X-t)2 Vyj X'Y'Yj,t = 1T(l+K) {~log x-t + Y-Yj + (x-t)Z+(y-y.)Z}, 

J 

(A19) 

(A20) 

-16-
APPENDIX B 

The kernels C;jk which appear in equation (11) 

ex> 

Cxxk = ~(l~K) f [rnxxk(x,t,a)-rnxxk(H-x,H-t,a)]cosa(Y-Yk)da, 

o 

ex> 

Cxx3 = ~(l+K) J [rnxx3(x,x3,a)+mxx3(H-x,H-x3,a)]sina(t-y)da, 

o 

(B2) 

ex> 

Cxxj = ~(l+K) J [rnxxj(x,t,a)+mxxj(H-x,H-t,a)]sina{Y-Yj)da , 

o 

(j=4,5; y.=s. 3) (B3) J Jex> • 

Cxx6 = ~(l+K) J [rnxx6(x,x3,a)-mxx6(H-x,H-x3,a)]cosa(t-y)da, 

o 

(B4) 

ex> 

Cyyk = ~(l~K) J [rnYYk(x,t,a)-rnyyk(H-x,H-t,a)]cosa(Y-Yk)da, 

o 

ex> 

Cyy3 = ~(l+K) J [myy3(x,x3,a)+rnyy3(H-x,H-x3,a)]s;na(t-y)da , 

o 

(B6) 

ex> 

Cyyj = ~(l~K) J [rnyyj(x,t,a)+myyj(H-x,H-t,a)]sina(y-Yj)da 

o 

-17-
<X> 

Cyy6 = rr(l+K) I [myy6(x,x3,a)-myy6(H-X,H-x3,a)]cosa(t-y)da , 

o 

(B8) 

<X> 

Cxyk = rr(1+K) J [mxyk(x,t,a)+mxyk(H-x,H-t,a)]sina(y-Yk)da , 

o 

00 

Cxy3 = rr(l~K) J [mXy3(x,x3,a)-mxY3(H-x,H-X3,a)]cosa(t-y)da , 

o 

(810) 

<X> 

Cxyj = rr(l~K) I [mxyj(x,t,a)-mXyj(H-X,H-t,a)]cosa(y-Yj)da , 

o 

<X> 

Cxy6 = rr(l+K) J [mXy6(x'X3,a)+mXy6(H-x,H-X3,a)]sina(t-y)da , 

o 

(812) 

(k=1,2) , (813) 

-18-
(B14) 

(j=4,5) (B15) 

rnXX6=O-1(a){[K(e2aH_1)(1-K2+2aX-2aX3-4a2xx3)+4KaH(-ax-aX3

+aH)]e-a(X+X3) 

+[2K2(e-2aH_1)(-ax+ax3)+2aH(-1+K2-2aX+2aH-2ax3+4a2XH-4a2xx3)]e-a(x-x3)}, 

(B16) 

(k=1,2) , (B17) 

(B18) 

-19-
(j=4,5), (819) 

(820) 

(k=1,2) , (821) 

mXY3=O-1(a){[K(e2aH_l)(1-K2-2aX+2ax3-4a2x3x)+4KaH(-ax+aH-ax3)]e-a (X+X3) 

\ 

(822) 

(j=1,2) , (823) 

-20-
mXY6=-D-l(a){[K(e2aH_l)(-1-K2+2aX+2aX3-4a2XX3)+4KaH(1-ax-aX3

+aH)]e-a(X+X3) 

+[2K2(e-2ax_l) (1-ax+ax3)+2aH(1+K2-2aH+2ax3-2ax+4a2xH-4a 2xx3)]e-a(X-X3)}, 

(824) 

(825) 

-21-
x ___ .JUo 

y 

Fig. 1 The basic crack geometry considered, 

- ------- _. --- --- ------ -------- ---------- ------
4 

H 

3 

51= 1 

axx (O.y) 

IJ.Uo 

s1 2 

2 

1 

4 

10 

0 

0.0 0.2 Q4 0.6 0.8 1.0 

L 

S1 

Fig. 2 Normal stress along the clamped boundary in a plate without 

a crack (K=1.8). 
1.0 

0.8 

0.6 

dxy (O,y) 

IJ.Uo 

0.4 

0.2 

0.0

0 .0 0.2 0.4 0.6 0.8 1.0 

Fig. 3 Shear stress along the clamped boundary in a plate without 

a crack (K=1.8). 
0.8 I 

I 

I X 1 ! 

Uo 

0.6 H 

k2(a,= 0) 

ex IJ,Uos, 

s, 0.4 

0.2 

2 4 6 8 10 

H 

-

S, 

Fig. 4 Normalized Mode II stress intensity factor at the corners of 

a clamped rectangular plate without a crack (K=1.8). 
4 

--- ~ __ -.1 

Uo 

3 

dxx (D,Y) .~ b3 i Q3 

J..LUo 

s1 2 y 

. 0.0 

1 0.3 

O~~--~--~--~--~~~~--~--~---

0.0 0.2 0.4 0.6 0.8 1.0 

y 

s, 

Fig. 5 Normal stress along the clamped boundary in a plate with a 

central crack (H/2sl

=1), K=2.2. 
1 Report No 

NASA CR-177925 

I 2 Government AccessIon No _ 3 Recipient's Catalog No 

4 TItle and Subtitle 5 Report DatI' 

~ __ ~r1a~r~c~h~1~9~8~5 ________ . ____ 

A Clamped Rectangular Plate Contalning a Crack 6 Performing Organization Cod" 

7 Author(s) 8 Performing OrganiZatiOn Report No 

Renji Tang and F. Erdogan ~ ___________________________ -I 10 Work UOIt No 

9 Perfor rrung Organization Name and Addr~ 

Lehi9h University Department of Mechanical Engineering and Mechanics 

Bethlehem, PA 18015 

11 Contract or Grant No 

NGR 39-007-011 ~ ____________________________ -I 13 Type of Report and Perrod Covered 

12 SponsOring Agency Name iIIld Address 

National Aeronautics and Space Administration 

Washington, DC 20546 

Contractor Reoort 

14 Spon$Orlng Agency Code 

505-33-23-02 

15 Supplementary Notes 

Langley Technical Monitor: W. Steven Johnson 

16 Abstract 

In this paper the general problem of a rectangular plate clamped along two 

parallel sides and containing a crack parallel to the clamps is considered. The 

problem is formulated in terms of a system of singular integral equations and 

the asymptotic behavior of the stress state near the corners is investigated. Numerical examples are considered for a clamped plate ~/ithout a crack and with a 

centrally located crack, and the stress intensity factors and the stresses along 

the clamps are calculated. 

17 Key Words (Suggested by Author(s) I 

stress lntensity factors, 

clampted plate, cracks 

1 9 S"Cur,ly Oassl' (of this report! 

Unclassified 

. 

1 B Distribution Statement 

Unclassified - Unlimited 

20 Security Cluslf (of this ~ge) 

Unc 1 ass ifi ed 

Subject Category 39 

21 No. of Pages 

27 

For sale bV the Nattonal TechnlCsllnformation ServIce, Spnngfleld. Virginia 22151 

A03 
End of Document