Rigid body rotation surrounding the crack

Ensuring

FracNn

Mechanics

Voi. 30, No. 4,451-459,

0013-7944188 $3.00 + .oo @ 1988 Pcrgamon mss pk.

1988

Primed in Great Britain.

RIGID BODY ROTATION SURROUNDING THE, CRACK Department

A. BHATTACHARYA and B. KISHOR of Mechanical Engineering, Institute of Technology, Banaras Hindu University, Varanasi 221005. India

Abstract-An analysis of the rigid body rotation is presented which is based on the conventional Westergaard function and the stress function in complex form. The crack opening displacement is obtained on the basis of rotation at points lying on the crack edge. Using an expression of rotation near the crack tip the rotation at the Dugdale’s crack tip plastic zone is shown to be a material constant. With the help of an analysis of rotation for the elastic region, the distribution of rotation in cracked plates having flnite width are obtained.

NOMENCLATURE Rotation at a point on crack edge; half of the crack angle at Dugdale crack tip. Orientation of a point near the crack tip. Orientation of a point on the constant rotation curve. A polar coordinate. Poisson’s ratio. The applied stress. Yield stress. Stress function. The rotation at Dugdale’s crack tip plastic zone. RB rotation on x-y plane. Elasticity modulus. Modulus of rigidity. Mode I stress intensity factor. Width of SEC and DEC plates. A constant for SEC and DEC plates. An analytic function. The semi crack length of center crack; the length of edge crack. A polar coordinate. The size of plastic zone. The displacement along x direction. The displacement along y direction.

1. ~T~ODUCTION IN THE PRESENCE of a crack in stressed material body the rotational factors become important for explaining the geometry of the deformed region surrounding the crack. If an infinite plate with center crack is subjected to tensile loads, the opening of the crack is associated with the rotation at the points adjacent to the crack. Similarly, it may be interesting to study the rotation at the points in close vicinity of the Dugdale crack tip in the plastic zone. To combine the nature of rotation with the aspects of crack geometry an LEFM based analysis is presented in this work. Using stress function in complex form, and also the Westergaard function[l, 21, the general expression of rigid body rotation at a point is obtained for the plane strain and the plane stress condition. Subsequently the rotations at different points within the space of a crack are obtained as specific cases. Considering rotation at a point on the crack edge an expression of crack opening displacement is derived. Also, the rotation at a point near the crack tip is found in terms of the stress intensity factor and the functions of coordinate r and 8. It is shown that the rotation at the Dugdale’s crack tip plastic zone[3,4] is a material constant, and it is independent of the stress intensity factor. At last, an expression of rotation in elastic region ahead of the crack tip is used to describe the variation of rotation in single edge cracked plate and the double edge cracked plate. 451

A. BHATTA~HARYA

452

and B. KISHOR

2. RIGID BODY ROTATION IN INFINITE PLATE CONTAINING CENTER CRACK The rigid body rotation (RB rotation) at any point in the stress/strain field is a vector which is an angular measure of reorientation of a smah element of sides dx, dy and dz at the point. For a piane element (dx, dy) shown in Fig. 1, the rotation vector is given by w,k and it is measured by the angle through which the diagonal of the element rotates on x-y plane. Its magnitude is given by w, = 1/2(&/~3x - au/ay), where u and 21are respectively the displacement along x and y direction. In the present work the analysis of RB rotation will be confined to the rotation of plane elements only. An infinite plate of finite thickness and containing no crack is shown in Fig. 2(a). When it is subjected to longitudinal stress CT,the rotation w, = 0 at every point because &_Y/&x = 0, and u = 0. If a similar plate containing center crack is loaded with stress u, as shown in Fig. 2(b), the rotational factors become important in explaining the geometry of deformation in a region surrounding the crack. Therefore the displacements u and u at a point, and hence the rotation o, can be determined by assuming a stress function in complex form. Referring to the problem in Fig. 2(bf, the stress function # can be written in terms of the analytic function Z and its derivative Z’, as b, = ReZ + y ImZ’,

(1)

where Z is a function of the variable z = x + iy.

Fig. 1. Rotation of plane element

cr t

t

(a)

t

tb)

Fig. 2. (a) Infinite plate without crack. (b) Infinite plate with a crack of length 2a.

Rigid body rotation

453

Using the stress function given by eq. (l), the stresses are obtained and subsequently displacements u and u for the plane strain condition can be written as 2Gu=(l-2v)ReZ’-yImZ”

the

(2)

2Gu=2(1-v)ImZ’-yReZ” where G is modulus of rigidity, and Y is the Poisson’s ratio. After getting the partial derivatives of the above equations with respect to y and x respectively, and subtracting the first equation from the second, the following is obtained. =2(1-v)iImZ’-y;ReZ’ -(l-2v)-$ReZ’+ImZ”+y&ImZ”. By rearranging

the terms in eq. (3) the expression of rotation is written as

fl&

=

v2)

al-

Im

z

(4)

E

where E is elasticity modulus of the plate material. Equation (4) provides a simple expression of rotation W, in terms of the harmonic function ImZ”. For rotation under the plane stress condition, the factor (1 - v*) should be dropped from eq. (4). It implies that for similar loading condition and geometry the RB rotation in case of plane stress will be higher than that under the plane strain condition. 2.1. Crack opening displacement based on rotation It is known that when a tensile stress acts normally over a sharp crack (Fig. 2b), the crack opens to assume an elliptic shape[2,5]. The crack opening displacement (COD) is given by COD = 2u, where u is displacement at the crack edge along y direction. Similar results on crack opening are obtained by considering rotation of an element (dx, dy) at the edge of the crack. RB rotation of an element at a point P on the crack edge is shown in Fig. 3. Since the normal stresses, a, and aY, and the shear stress T~,,are absent at P, there is no shear deformation of the element and the pure rotation is obtained from LCPT = (Y.Hence the following equation is written for small values of Q.

C T

Er, a

dY

P

a

dx

‘t

Fig. 3. RB rotation at the crack edge.

A. BHATTA~HARYA

454

and B. KISHOR

where dyfdx is the slope at P. The Westergaard function[l, 2] for this problem is taken as 2” = (rz/m, where s is the applied stress and a is the semi crack length. Next, applying the condition, -a s x < + a and y = 0 the rotation u, is obtained from eq. (4), as w, = -

2(1-

V2)

E

(TX (6)

w’

After equating the terms on right side of eqs (5) and (6) and integrating,

it can be shown that

COD=2y=(I+)$JaZ-. Thus, the equation of COD is conveniently derived on the basis of rotation at the crack edge. Generally an expression of COD similar to eq. (7) is obtained after determining the displacement v on the edge of the crack[2]. 2.2. RB Rotation near crack tip; nature of rotation at the Dugdale’s crack tip plastic zone For an ignite plate (Fig. 2b) the RB rotation near crack tip is found from the general the expression of rotation given in eq. (4). Using Westergaard function as Z” = a~/-, rotation w, in the vicinity of crack tip is obtained. It can be shown that, as z --+ a, KI

ImZ”=-,‘Z;;;sinf

(8)

where Ki = a& is the mode I stress intensity factor. Next, using eq. (4) for the plane stress condition the rotation is obtained in the form w, =

2 KI ---zsm-2.

.

e

(9)

In the above expression the rotation at a point near crack tip is obtained in terms of the stress intensity factor and the functions of coordinates r and 8 of the point. In eq. (9), the negative sign on the right hand side indicates the sense of rotation according to the right hand vector rule. The rotation is negative for 8 > O”, and it is positive for 8 < 0”. This is made clear in Fig. 4. Yb wl=--

2 KI EJ2a7

Fig. 4. RB rotation near the crack tip.

SinB

45.5

Rigid body rotation

A particular case of interest will be to find o, at 8 = 7cand r = ry, where r,, is size of the crack tip plastic zone. For this case o, can be written with the help of eq. (9) as

w=---

2

KI

(10)

EJ27TTy’

As long as the applied stress u is small compared to the yield stress u,,, and the plastic zone size ry is small compared to the semi-crack length a, size of Dugdale plastic zone can be written as ry = (rr/8)(K:/a$[2,6]. Using this value of r,, in eq. (lo), the following is obtained.

(JJz=w=---.

4 UY

(11)

TE

In eq. (1 l), w, is expressed in terms of the material parameters oy and E only. Therefore, the rotation w, is constant for a given material. It can be verified geometrically that eq. (11) gives the value of constant rotation at Dugdale’s crack tip. Also, it can be shown by comparing with the rotation at various points near crack tip, that the rotation w, provides the maximum value. The Dugdale model is known to give reasonable estimates for the extent of plasticity at the crack tip[4], especially for non-work-hardening materials under the plane stress condition. Figure 5 illustrates the Dugdale crack with a sharp crack tip in plastic zone. When the applied stress a-+ aY, opening at the tip of elastic portion of the crack (MN), i.e. the crack tip opening displacement (CTOD) is written as CTOD = KfIEa,. The plastic zone size, ry = (7r/8)(K:/(+$. Referring to Fig. 5(b) the rotation of an element at the point M is found from the following equations. Wz=-ff,

(12)

and cr=y Y

1

If the values of CTOD and ry for the Dugdale crack tip are used in eq. (12) it will lead to an expression of w, given by eq. (11). In fact the rotation w, from eq. (11) is independent of applied stress u. As the stress intensity factor KI increases, the crack tip opening displacement and the plastic zone size ry become higher. However, the sharpness of the Dugdale crack tip in the plastic zone (Z(Y) remains the same since it is not affected by the applied stress. Referring to Fig. 5(b), it is found that the rotation o, given by eq. (11) remains constant in a range 0 s r s ry. Hence there should be a zone of constant rotation at the crack tip. This is found

CTOD

Fig. 5. (a) Dugdale crack. (b) The crack tip geometry.

456

A. BHATTACHARYA

and B. KISHOR

r-ry,2(I-Cos8) /

crack

(al

(b) Fig. 6. Constant-rotation

zone at the crack tip.

from eqs (9) and (11) as a relation between coordinates r = ? (1 - cos 0),

r and 8, which can be written as

(13)

where ry = (~/S)(Kfla$). The relation between r and 0 from eq. (13) is shown graphically in Fig. 6(a). The area enclosed by the curve in Fig. 6(a) represents the zone of constant rotation. The constant-rotation zone on both ends of a crack is shown by Fig. 6(b). The area under the constant rotation zone is found as (3&3)(r$, and this should increase with the stress intensity factor. 3. APPLICATION To compare the values of rotation at different points in the vicinity of the Dugdale crack tip the rotation was calculated using eqs (9) and (11). For fixed ordinates y. (in Fig. 6a) and the values of x ranging from x = 0 to x = r,,, rotation at the points were obtained as fractions of o=, as shown in Fig. 7. In Fig. 7(a), the variation in rotation are shown for four different values of y. corresponding to four values of L y in Fig. 6(a). For example, y. = O.O8r,, 0.22r,, 0.41 y, and 0.5r, against y = 40”, 60”, 80” and 90”, respectively. Similarly, in Fig. 7(b), y. = 0.58r,,, 0.63r, and 0.65r, against y = loo”, 1 lo” and 120”. It may be observed in Fig. 7(a) and (b) that the values of rotation diminish rapidly with the distance from the crack tip. At x = ry, the rotation values are in the range between 4% to 25% of o,(w, = 1.27 a,,/E). At larger distance, i.e. x + r,,, it can be shown that rotation is negligibly small compared to o, at the constant-rotation zone. Also, it may be observed that the rotation decreases more steeply at the ordinates y. which are closer to the crack axis x. 3.1. RB Rotation in plates having finite width In the previous section the rotation was determined for points lying very close to the plastic region at the crack tip. It was shown that RB rotation decreases rapidly with the distance from crack tip. In this section the rotation is obtained at the points lying in the broad elastic region in the single edge cracked (SEC) plate and the double edge cracked (DEC) plate. The problem is shown in Fig. 8. The plates are loaded with tensile stress u along the y direction. It should be noted that only first term of a series was taken into account in eqs (8) and (9). Hence w, at larger distances from the crack tip can be determined by including higher terms in eq. (9). However, the direct application of Westergaard function is an alternative to obtain w, at

Rigid body rotation

457

0.7 2 0.6 s P 2o

0.5

c

0.4 I

0.1

0

I

I 0.1

I 0.2

I 0.3

I

I 0.5

I 0.6

I

I

0.T

0.8

Distance from the crock t&x

fr,l

0.4

I 0.9

t I 1 ’ -0.4

I 1.0

I

I

I

I

1

I

-0.2

0

0.2

0.4

0.6

0.8

1 1.0

Distance frem the cmcktip,x(r,l

Fig. 7. The rotation OJ, vs x near the Dugdale crack tip.

a point (r, 0) in the elastic region. Therefore, 6

0

oz=-y

z

where a denotes the semi-crack I’= 1.99-0.41

o, is obtained as

sin @(a + r-..cos -.8) 2a2$a2+r2+2arcos _-_-_ 8)2

(14)

length and Y is a constant which is written as [7,8] ?Lv)

+18 7 -?_ *-3848 . L>

.

a.

M

3+53 85 % .

M4~

for SEC plate,

a

(b) Fig. 8. Plates of finite width W under tensile load; (a) SEC plate, (b) DEC plate.

458

A. BHATTACHARYA

and B. KISHOR

0 Distance

from

the

crack

-0.1

tip,x(W),log

0.2

0.3

Distance

0.4 from

0.5 the

0.6

crack

0.7

0.6

0.9

tip,x(W)

Fig. 9. The rotation o, vs x in SEC plate.

and Y=

1.99+0.76(+)-8.48 (;)2+27.36 (;)3 for DEC plate,

where W is the width of the plates. Equation (14) is used to determine w, at different points of SEC plate (Fig. 8a). The ratio a/ W is taken as 0.1, so that the constant Y is found to be 2.103. The distribution of rotation in SEC plate is shown in Fig. 9(a) and Fig. 9(b). The rotation o, at the points are expressed as

0.3

oec ptats a-O.lW

y -II

0.2

0

0

0.2 Distance

0.4 from

the

0.6 cmck

tip,

x (WI

0.6

0

0.2 Distance

Fig. 10. The rotation o, vs x in DEC plate.

.x?L 0.6

0.4

from

the

crock tip,x

(WI

0.6

Rigid body rotation

4.59

fractions of (a/E) and these are plotted against distance x from the crack tip. The vaiues of x are shown as fractions of plate width W. In Fig. 9(a), the relation between o, and x: is shown for four different values of y, namely y = 0.025 W, 0.05 W, 0.075 W and 0.1 W. Due to large variation of rotation from x = 0 to x = 0.9 W, the graph is plotted on log-log scale for convenience. It may be observed in Fig. 9(a) that w, decreases consistently for all values of y from y = 0.025 W to y = 0.1 W. The crack length is taken as 0.1 W. The rotation at a point in the elastic region is found from superposition of rotation calculated with respect to each of the crack tips. Values of o, which are expressed as fractions of (a/E) are shown in Fig. 10(a) and (b). It should be noted that the rotation is of opposite nature on the two sides of the plane of symmetry. The plane of symmetry is shown by dotted lines which intersect the crack axis at x = 0.4 W in Fig. 10(a) and (b). It is observed in Fig. 10(a) that the maximum rotation occurs over the crack tips up to y = 0.1 W. The rotation decreases rapidly and it vanishes on the plane of symmetry at x = 0.4 W. In Fig. 10(b), the relation between o, and x is shown against y = 0.2 W to y = 0.5 W. It is observed that for higher ordinates the maximum rotation occurs at some distance from the crack tips. For example with y = 0.4 W, o, = O.Ob(cr/E) is the maximum value which occurs at x=O.l2Wand x=O.68W. 4. CONCLUSION The present study deals with the analysis of rigid body rotation based on the conventional Westergaard function and stress function in complex form. The rotation of plane elements on the x-y plane are analysed. RB rotation at a point near the crack tip is expressed in terms of the stress intensity factor and functions of coordinates r and 8 of the point. Major conclusions obtained are as follows: (1) The crack opening displacement is obtained by analysing the rotation of a small element on the edge of the crack. (2) For non-work-hardening materials the rotation at Dugdale crack tip plastic zone is shown to be independent of the applied stress. A constant rotation zone surrounding the crack tip is obtained, in which the rotation w, = - (4/~)(~~/E). Since the rotation depends on material parameters such as a,, and E, the sharpness of Dugdale crack tip is independent of stress intensity factor. (3) It is shown that the magnitude of rotation decreases rapidly with the distance from crack tip. (4) RB rotation is determined at the points lying in the elastic region of single edge cracked plate and the double edge cracked plate. The distribution of rotation on SEC plate and DEC plate are described. REEERENCES [l] H. M. Westergaard,Bearing pressureand cracks. 1. appt. Meek. 61, A49-A53 (1939). [2] D. Broek, Solution to crack problems. Elementary EngineeringFractureMechanics, pp. 69-72. Martinus Nijhoff, The Hague (1982). [3] D. S. Dugdale, Yielding of steel sheets containing slits. J. Meek. Pkys. Solids, 8, 100-108 (1960). [4] T. C. Lindley and L. N. McCartney, Mechanics and mechanisms of fatigue crack growth. Development in Fracture Mechanics-2, p. 264. Applied Science, U.K. (1981). [S] A. A. Wells, Unstable crack propagation in metals; cleavage and fast fracture. The Crack ~o~g~~io~ Symp., Cranfield, pp. 210-230 (1961). [6] J. F. Knott, F~du~nf~Zs of Fracture Mechanics, Butterworths, U.K. pp. 68-69. (1973). [7] M. Isida, On the tension of a strip with a central elliptical hole. Truns. Japan Sot. Meek. Engrs 21 (1955). [S] P. C. Paris and G. C. Sih, Stress analysis of cracks. ASTM STP 391, 30-81 (1965). (Received 3 August 1987)