A thickness criterion for fracture toughness testing based on a plane stress compatible solution

Engineering Printed

Frac~un Mechanics Vol. 29, No. 1, pp. 41-47,

1988

0013-7944/88 @ 1988 Pergamon

in Great Britain.

$3.00 + .tXI Journals Ltd.

A THICKNESS CRITERION FOR FRACTURE TOUGHNESS TESTING BASED ON A PLANE STRESS COMPATIBLE SOLUTION Department

of Mechanics,

S. A. ZHOU and R. K. T. HSIEH Royal Institute of Technology, S-100 44 Stockholm,

Sweden

Abstract-In this paper, an elastic compatible plane stress solution for a thin plate with an elliptic hole or a crack is obtained by a complex variable method. The result is used to study the stress field near tensile crack-tips. It is shown that the effect of incompatibility of conventional plane stress solutions on the stress distribution near the crack tips may not be ignored. A model for the study of the thickness effect of fracture toughness in plane stress states is then proposed. Comparisons of this model with available experimental data and other models are also presented.

1. INTRODUCTION

AT PRESENTthere seems to be no universally accepted methods for the determination of fracture toughness under plane stress states. This is due to difficulties in understanding the observed phenomena and the plane stress behavior. However, many structures, especially in aircraft, are built up of sheets and consequently the plane stress problem is of great technical importance. Experiments show that fracture toughness under a plane stress state displays a dependence on the specimen thickness and material constants. This phenomenon can not be explained by means of conventional plane stress solutions which, as we know, do not satisfy all compatible equations (see Timoshenko and Goodier[l]). Several semi-empirical models have been proposed to describe this phenomenon, e.g. by Bluhm[2], Anderson, Borek and Vlieger (a comprehensive discussion on this problem is given in Broek[3]). In this paper we shall study the influence of the incompatibility of the usual plane stress solution on the fracture toughness. A new plane stress compatible solution will be found and a quantitative formula for the description of the thickness effect on fracture toughness in plane stress states will be given. Comparisons of the new model with experimental data and with other models will also be performed.

2. AN ELASTIC COMPATIBLE PLANE STRESS SOLUTION WITH AN ELLIPTIC HOLE

OF A THIN PLATE

It is known that conventional plane stress solution does not satisfy all the compatibility equations which for vanishing body forces read 1 V2Uki + 1+u

%wn.kl

=

0,

(1)

where V2 is the Laplace’s operator in three-dimensional space. A compatible solution under plane stress assumptions, i.e. CT*, =

a,, = uy = 0

can be given with the use of the Airy’s stress function[l],

a2u v a2 uxx=----(v:U)t2, a? 2(1+ 0) ay2 41

(2) which has the form

(34

S. A. ZHOU and R. K. T. HSIEH

42

a=u

UYY

GY

a=

v

=-------(v:U)r=, 8X2 2( 1 -I-u) ax= =__++-

a2 u

ax

ay

V

a*

2(1+ v) ax ay

(Vf W2,

where VT = a2/ax2 -ta2/ay2. U is the Airy’s stress function, v the Poisson’s ratio and c the measure of the thickness. Explicit expressions of the stresses are thus reduced to the finding of a biharmonic function U under given boundary conditions. A complex method of solving boundary-value problems governed by a biharmonic equation has been developed principally by ~uskheiishviii~4~. The general solution for the biharmonic function U may be written as

where x(z) = j*(z) dz. (P(Z) and W(z) are analytical functions of the complex z = x + iy. By ey. (4), the stress components in eq. (3) may be expressed as a,, = R,

2@‘(z)-W’(z)-t’(z)++&

variable

(Sa)

PC) The solution to an individual problem is then reduced to the finding of the functions@(z) and *IF(z) satisfying its corresponding boundary conditions. For the first boundary-value problem, the boundary condition may he expressed in the following manner: __

___

w(z)+Z@yZ)+~(t)-~ f*w( 2) = f,

(6)

where f = i j (X, 4 iY,) ds. Since we are dealing with a thin plate, it may be reasonable to consider the thickness averaged ~undary condition approximately by Saint-Venant’s principle. After the thickness averaging, eq. (6) becomes --

W(z)+zW(z)--cYw(Z)=f*,

(7)

where LY= z&*/6(1 + v), and f* = j.!Y&(f/h) dt. The comparison of this boundary condition with the one from conventional formuiati~~n of incompatible plane stress problems shows that an additional term --C&“(Z) is presented here. The difference between the solution satisfying the boundary eq. (7) and the conventional incompatible plane stress solution is that our solution satisfies all the compatible eq. (1) at every point inside the plate. Now, let us consider the case of an infinite elastic plate with a surface force-free elliptic hole or a crack (see Fig. 1). The averaged boundary condition of the problem reads -cD’(2) + zW( 2) + 9(z) - a@“( 2) = 0 (8) and the components

of the stress at infinity write rrTx = (N,( 1 - cos (2@)) + TW(1+ cos (2/3)))/2,

t9a)

oTy = (NJ 1 + cos (2P)) + TcQ(1- cos (2P)))/2,

(9b)

r;i = ((T, - AL) sin (28))/2.

(9c)

43

Thickness criterion for fracture toughness

Fig. 1. An infinite large thin plate with an elliptic hole.

If N, = T, and p = 0, one has cry1 = N,, u& = N,, T& = 0. If N, = -T, and p = rrl4, one has -_ m = 0, and r:,, = T,. After some manipulations, the solution of the boundary value (TX,- @yy problem can be obtained as

in which

(5+y,)

z=o([)=c

and 4I = N, + T,, and 2I’ = (N, - T,) exp (-2ip). respective!y by

@(I)=-

*y(l)=

c (ry+m5-

P

(if

(12)

(c>O,O
#‘([)

and e(l)

c(mr+T’) 5

(13)

,

d)(T'+2mr)l

MC2- ml

are defined

2@+

)

2mr)p

c(12- m)3

’

(14)

The solution of crack problems can now be easily obtained by letting m = 1 and 2c = L where L is the semi-length of the crack. For crack problems of type I, i.e. Nm=

P =o,

Tm,

m=

1,

2c= L

(15)

we can get

(16)

ql({)

=

_

LN-4-_ 4aN-13

p=i-

L(12- 1)3’

(17)

which shows that for (r = 0, the well-known incompatible solution of plane stress is recovered. Similarly, solutions for crack problems of type II can also be derived from eqs (lo), (1 l), (13) and (14) by letting N, = - 7’,, /3 = T+, and m = 1. The elastic compatible plane stress field near the

S. A. ZHOU and R. K. T. HSIEH

44

crack-tip at 8 = 0 (or y = 0 and x > L) can be found for the problem of type I as

(184

’

K1 UYY

=

J2?rr

7xy

=

0,

I _ v(12rZ- h2)

(18b)

16(1+ u)r2 ’

(18~)

in which one lets z = L + re’ and omits higher order byusingeqs(5),(10),(11),(12),(16)and(17), terms in r with the condition L P r. Here, K1 = IV,= is the stress intensity factor. It can be shown that the stress field from eq. (18) after thickness average is just the one given by conventional incompatible plane stress solution. It is shown also, from eq. (18) that the effect of incompatibility of plane stress field may become important when we consider the stress field closed to the crack-tips due to the elastic stress singularity at the crack-tips. 3. THICRNESS

EFFECT

OF FRACTURE

TOUGHNESS

IN PLANE

STRESS STATE

By using eq. (18), the strain energy stored in a unit volume element at 6 = 0 can be obtained as dW -= dV

K:(l-v) 2 ?rEr

(1%

The thickness average of the strain energy per unit volume then reads I+ ’

(20)

which shows that the thickness average of the strain energy density does not recover its corresponding one from conventional incompatible plane stress solution though the stress field does. An additional term accounting for the effect of the plate thickness appears on the right-hand-side of eq. (20). In what follows, an approach based on an energy criterion will be used to study the well-known thickness effect of fracture toughness in plane stress states. It is known that Sih’s strain energy density criterion[5] is based on the hypothesis that fracture occurs when a small element of the material near the crack-tip has absorbed a critical amount of energy, and then releases it to cause material separation. This material element is always kept at a finite distance, say r*, from the crack-tip (see Fig. 2). The concept of a local core region in Fig. 2 stems from a concern that in the immediate vicinity of the crack, inhomogeneity of the material due to grain boundaries, microcracks and dislocations precludes an accurate analytical solution, whereas an analysis presuming a valid continuum mechanics solution external to the core region may provide a sufficiently accurate measure of the failure behaviour of the material. The continuum description requires the size of the core region to be limited to the smallest macroscopic element. Here, considering the limitation of the effective region of the elastic solution, we assume that the crack starts to propagate when the average strain energy density (d W/d V) at the joint (r = rc) separating the elastic region and the plastic region reaches

Fig. 2. A local core region near crack-tips.

Thickness criterion for fracture toughness

45

its critical value (d W/d V),, which is supposed to be a material parameter. The use of the critical value of strain energy density (d W/d V), as the material parameter has been demonstrated by Gillemot[6]. In particular, Gillemot and his co-workers have performed many experiments to measure the value of (d W/d V), for numerous engineering materials. It has also been pointed out (see Sih[Sj) that (d W/d V), is equivalent to Sih’s critical strain energy density factor SC. By means of this average strain energy density criterion, two ways of determining the thickness effect on fracture toughness Ki, can now be proposed as follows. The first one is that by knowing the maximum value of the fracture toughness, KY,, analytical formula of accounting for the thickness effect on fracture toughness may thus be obtained from eq. (20). By noting that h* 6 1, one can get approximately

(21)

’

where KY, might be called the ideal plane stress fracture toughness since it is the value of K,, at the thickness h* = K,,/(37&) where the plane stress state can fully develop (see [3]). The second one is that by knowing the plane strain fracture toughness KI, and the minimum specimen thickness ho which gives the valid KI, (see [7]), another form of the analytical formula for describing the thickness effect of the fracture toughness can then be derived from eq. (20). This reads

KI, = KI,

320(1320(1-

v’)r:+ u2h: v2)r:+ u2h4’

(22)

It is noted that by using formula (21), one can also derive the minimum specimen thickness h,,, which gives a valid plane strain fracture toughness, from the following equation

(23) Finally the thickness effect of the plastic-zone yield condition (f71- a#

size r, can also be determined

by the von Mises’s

+ (02 - OS)2+ (a3 - (Ti)2= 2( a#,

(24)

where a, is the simple tensile yield stress and ul, u2, o3 are the principal stresses. The equation of determining the plastic-zone size r, for 8 = 0 reads h~_A4_3u2(12t2-h2)2=0

256( 1 + ~)~r:

(25)

in which A = rS/ro where r. = K:/(2&) is the conventional plastic-zone size given by an incompatible solution. Equation (25) is an algebraic equation of the fifth degree. There is no general method to obtain the roots of this equation. However, according to Descartes’ rule of signs, there is only one real positive root for this equation. It can be further shown that rS(t) > r. in -h/2 < t < h/2 since the eq. F(h) = 0 has the characteristics F(1) < 0 and F(+m) > 0. The form of the plastic-zone on 0 = 0 is shown in Fig. 3. In order to use eq. (21) or eq. (23), one may choose r, = is, which is defined as the plastic-zone size averaged over the thickness and now is satisfying the following equation

in which A, = is, /rot and rot = K:,/(2ruf). As an illustration, the results shown by eq. (21) in comparison with experimental

data from

S. A. ZHOU

46

Fig. 3. Plastic-zone

and K. K. T. HSIEH

sizes around

a crack-tip

on the surface

y= 0

7075 - T6 A---.-“----

Test data [81 Formula (3-3) Anderson Broek and Vlieger ASTM’s plane strain condition vro.3

K,, = 104 kg/mm3”

IO

5

0

20

15

25

h (mm) (a) TI, 6AI, 6V, 2s” 200

l

!

l

\\ 8

c

Test data

[71

Formula (3-3) Anderson

l

\

l50-

\

\E

i

i - ;__\.L

K,, = IIO kg /mm3”

B

?L’

---

-.-

loo

50 0

-.

.

:

.-

-1

.

I IO

I 5

I 15

h (mm) (b)

Fig. 4. (a) Effect of plate thickness h on fracture toughness K,, in plane stress state with (1) o, = 5 1 kg/mm2 or (2) v, = 45 kg/mm*. (b) Effect of plate thickness h on fracture toughness K, cin plane stress state (a, = 124 kg/mm*, Y = 0.33).

Thickness criterion for fracture toughness

47

[7,8] and other models are presented in Fig. 4 for two kinds of materials. It is shown that the proposed model describes reasonably the thickness effect of the fracture toughness K,, in plane stress states as well as the dependence of the fracture toughness K,, on the yield stress a, and the Poisson’s ratio 2). 4. CONCLUSIONS An elastic plane stress compatible solution for an infinite large thin plate with an elliptical hole or a crack has been obtained by using a complex variable method. A model for studying the thickness effect of fracture toughness in plane stress states is then proposed with the aid of this solution. Analytical formulae for describing the functional relation between the fracture toughess Ki, and the thickness of the plate as well as approaches for determining the size of plastic-zone near the crack-tips and the minimum specimen thickness h, for a valid plane strain fracture toughness Kr, are also presented. The proposed model seems to provide quite reasonable information on the thickness effect of fracture toughness in plane stress states shown by the comparisons with available experimental data and other models. REFERENCES [l] S. Timoshenko and N. Goodier, Theory of EIasticify (2nd edn), p. 241. McGraw-Hill, New York (195 1). [2] J. I. Bluhm, A model for the effect of thickness on fracture toughness. ASTM Pm. 61, 1324 (1961). [3] [4] [5] [6] [7] [8]

D. Broek, Elemenrary Engineering Frucrure Mechanics (3rd revised edn). Martinus Nijhoff, Hague (1982). N. I. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity. Noordhoff, Groningen (1953). G. C. Sih (ed.), Mechanics of Fracture, Vol. 5, Noordhoff, Leyden (1978). L. F. Gillemot, Criterion of crack initiation and spreading. Engng Fracture Mech. 8, 239 (1976). T. C. Ritter, A modified thickness criterion for fracture toughness testing. Engng Fracture Mech. 9, 529 (1977). F. C. Allen, Effect of thickness on the fracture toughness of 7075 aluminium in the T6 and T73 conditions. ASTM STP 486, 16 (1971). (Received

24 February 1987)