The size of the fully developed softening zone associated with a crack in a strain-softening material—II. A crack in a double-cantilever-beam specimen

Int. J. Engng Sci. Vol. 21, No. 3, pp. 309-314, Printed in Great Britain. All rights reserved

0020-7225/89 $3.00 + 0.00 Copyright @ 1989 Pergamon Press plc

1989

THE’SIZE OF THE FULLY DEVELOPED SOFTENING ZONE ASSOCIATED WITH A CRACK IN A STRAINSOFTENING MATERIAL-II. A CRACK IN A DOUBLECANTILEVER-BEAM SPECIMEN E. SMITH Joint

Manchester University/UMIST Department of Metallurgy and Materials Science, Grosvenor Street, Manchester Ml 7HS, U.K. (Communicated by B. A. BILBY)

Abstract-Earlier theoretical work has shown that the fully developed softening zone size in a strain softening material, for a semi-infinite crack in a remotely loaded infinite solid, is relatively insensitive to the details of the softening law for a wide range of force laws, but depends primarily on the maximum stress and displacement within the softening zone. This work is extended in the present paper to the case of a long initial crack in a double cantilever beam specimen, for which it is also shown that the fully developed softening zone size, though dependent on the beam height, and the maximum stress and displacement within the softening zone, is again essentially insensitive to the details of the softening law.

1. INTRODUCTION An appropriate model of a crack in a strain softening material such as for example concrete, a soil or a ceramic, is that of a zone of material, coplanar with the crack, in which the material behaviour is nonlinear, the form of the nonlinearity being governed by the strain-softening characteristics of the material under consideration. When an initially cracked solid is progressively loaded, the crack extends and a strain softening zone develops behind the crack tip, i.e. the leading edge of the softening zone. Two important parameters are the value of the crack tip stress intensity, KA, associated with the attainment of a fully developed softened zone, i.e. when cohesion just vanishes at the initial crack tip, and the size, R, of this zone. In Part I [l] it has been shown that, for the case of a semi-infinite crack in a remotely loaded infinite solid (Mode I plane strain deformation), R is relatively insensitive to the details of the softening law for a wide range of force laws, but depends primarily on the maximum stress pc and displacement S, within the softening zone. The case of a long initial crack in a double cantilever beam specimen has also been considered [2]. In this case it has been demonstrated that R, though dependent on the beam height, is approximately the same for a linear softening law and for the case where the stress retains a constant value within the softening zone, with pc and 6, being the same in the two cases. The present paper extends the scope of this latter study to a wider range of softening laws, with particular emphasis being given to the effects of the tail portion of the force law. It is again shown that R, though being very dependent on pc and a,, is relatively independent of the details of the softening law.

2. THEORY

The double-cantilever-beam (DCB) specimen is shown in Fig. 1; the height of each arm of the beam specimen is h, whilst the beam has unit thickness. The specimen contains a crack of initial depth ao, which extends as a result of the application of the loads F to a length a = a0 + R, the length of the softening zone being R. It is assumed that the softening zone is fully developed in the sense that the cohesion at the initial crack tip just vanishes, this occurring when the relative displacement of the two arms of the beam at the initial crack tip attains a critical value 6,; the maximum stress (at the crack tip) in the softening zone is pc. The simplest possible representation for the two components of the specimen is where the two arms behave as Euler-Bernoulli beams and are built-in at the crack tip; in the ensuing theory, 309

310

E. SMITH

Fig. 1. A double-cantilever-beam specimen contains a crack of initial depth a, which extends to a depth a due to the application of the forces F. There is a fully developed softening zone of size R = a -a, behind the tip of the crack.

because of symmetry, it is necessary to consider only one arm and we will therefore concentrate on the lower arm. The forces acting on the lower arm are shown in Fig. 2, there being a restraining stress p(x) acting over the interval which simulates the softening zone. This stress will vary with the distance x as measured backwards from the crack tip, and will have a maximum value pc at the crack tip where the displacement of the arms is zero, and be zero at the trailing edge of the softening zone (i.e. at the initial crack tip position) where the displacement is 6,. The moment and reaction force at the crack tip are respectively M and Q. With the Euler-Bernoulli beam theory, the relative displacement u (i.e. twice the beam deflection) as a function of distance x behind the crack tip is given by the second order differential equation

I

I(X-t)p(t)dt 0

F$=M-Qx-

for 0
Rp(t) dt,

(2)

I0 R

M=Fai0

tp (4 dt,

(3)

while the boundary conditions at the built-in end, i.e. the crack tip, are u = du/& = 0; the boundary conditions at the trailing edge of the softening zone, where the cohesion vanishes, is u=&whenx=R. In the real situation, the net stress intensity factor at the crack tip is numerically equal to the intrinsic fracture toughness Kit of the matrix material; for example with fibre cements Rio is the toughness of the reinforced matrix material and in concretes it is that of the cement paste matrix. In the simulation model Kit is related to the moment M (at the crack tip) by the expression (4) x

p(x)

/-

&p”’

F

M

QI *

R

00

*

Fig. 2. The forces acting on the lower arm of the beam in the simulation model.

Softening zone size-11

v

c

311

c

Displacement Fig. 3. The idealized softening law examined in the paper.

while the crack tip stress intensity KA due to the applied loadings is given by the expression tP(9 dt,

(5)

using eqns (3) and (4). In the ensuing analysis, where the objective is to explore the effect of softening law on R, it will be assumed that K Ic = 0; this assumption simplifies the analysis but is unlikely to significantly affect the conclusions. With KIc, and thereby M, being equal to zero [see eqn (4)], the governing differential equation (1) becomes via use of eqns (2) and (3),

for 0
+ (1-

q)w] -P+

for

O
(8)

while E,Z d2u ~~=PJ]@

+ (I-

qbl -pcW

- ql+ PcW2U 2 - 4) -- 4PcX2 2

for

w
(9)

Since u = du/dx = 0 at x = 0, eqns (8) and (9) integrate to give

E;z~=p+[qR+(l_q)w]_p$ for

O
(10)

EoZ 2

O
(11)

--

-u=p~[qR+(l-q)w]-p~

for

--E~z~-p~[qR+(l-q)w]-p~[l-q]+p~(l-q)-~+A

for

w
EoZ -u=p~[qR+(l-q~w]-p~~~-ql+p~~l-q) 2 4Pd4 --+Ax+B

24

for

w
(13)

312

E. SMITH Table 1. The fully developed softening zone size for q = 0 and for different x values X

114

0.2

0.4

0.6

0.8

1.0

0.964

0.862

0.813

0.782

0.760

where A and B are constants. Since u and duldx are continuous at x = w, it follows from relations (lo)-(13) that the constants A and B are given by the expressions

Since u = ~6, when x = w and u = 6, when x = R, it follows from relation (13) that E,,Z6, qR4 + (1 - q)w2R2 _ (1 - q)w3R + (1 - q)w4 -4 6 24 ’ 2p, -8 &CA

_ qw3R : (3 - 4q)w4 24 . 6 2PC

06) (17)

With @ = w/R, these equations can be rewritten as RoZ& -2p,R4 - 84+(l-!?)~2_(l-q)@3+(l-q)@4 4 6 WX~C

--2p,R4 -6

e3 +

24

’

(3 - 4qM4 24 ’

For the special case where q = 1 when the stress is constant within the softening follows from relations (18) and (19), remembering that Z = h3/12, that R=

(

3c;~;pj1’4

= 0.760( (ly;jp,)l’4

(18)

(19) zone, it

(20)

in accord with the result obtained elsewhere [2]. For the special case where q = 0, which simulates the case where there is a long tail in the force law, it follows from relations (18) and (19) that R=

(21)

(

Table 1 gives R for values of x from 0 to 1. Inspection of the results shows that R is relatively insensitive to the magnitude of x for a wide range of x values; thus for x lying between 0.5 and 1, R differs from the x = 1 value by less than 10%.

3. DISCUSSION

The preceding section’s analysis for the case of a long initial crack in a double cantilever beam specimen has shown that the fully developed softening zone size R for the softening law indicated in Fig. 3 can be expressed in the form l/4 R

=

dq,

x,(

cl:;jpc

. >

(22)

Furthermore, and most importantly from the present paper’s perspective, LYfor .the case q = 0 and for x lying between 0.5 and 1 is only slightly different from the value of (Y(i.e. 0.760) for

Softeningzone size-11

313

the case where x = 1, i.e. where the stress is constant within the softening zone. It is also clear that the same conclusion will be valid for other q values, for the same range of x values. Since it has also been shown [2] that LY= 0.84 for a linear softening law, it may be concluded for a wide range of softening laws, that though R is dependent on the beam height h, and the m~imum stress pC and displacement S, within the softening zone, it is relatively insensitive to the details of the softening law. This conclusion is therefore similar to that reached in Part I [l] for the case of a semi-infinite crack in a remotely loaded infinite solid, though of course in this latter case there is no dependency on the solid dimensions with R taking the form

BE&

R=(1 -

(23)

vZ)pC

where j3 is a constant (in fact it is approximately equal to 0.40). The paper has so far focussed on the size R of the fully developed softened zone. The value of KA, the crack tip stress intensity associated with the attainment of this state, for the case of a long initial crack and again for the case where K ic is equal to zero, is given by relations (5), (18) and (19) as

this being a special case of the more general relationship (25) which is applicable for a long initial crack and for any force law. The question arises as to how large the initial crack size must be for the KA value appropriate to an infinitely long crack, i.e. KAm [given by relation (25)J to be approximately applicable. Intuition suggests that the initial crack depth should exceed the size R, of the fully developed softening zone associated with an infinitely deep crack. This view is easily underscored by appealing to the results for the special case where the restraining stress is constant within the softening zone. In this case it is easily shown from the governing differential equation (6), that the size R of the fully developed softening zone associated with a crack of initial depth a0 is given by the equation

while relation (S), with K,, equal to zero, gives the corresponding

KA value as (27)

The values R, expressions

and KAm appropriate

to an infinitely

long initial crack are given by the

Eh3d,

=R4, 3u - V21Pc

(28)

and

KAm=

(29)

It follows from relations (26)-(29) that in order for KA not to differ from K_,+,by more than 20%, then a0 should approximately exceed R,. This result should be fairly general, and consequently the overall conclusion is that the simple formula (25) for the KA value associated with the attainment of the fully developed softening zone should be approximately applicable for cracks whose depths are greater than the critical value R,, irrespective of the details of the force law, provided the pc and 6, values are identical.

314

E. SMITH

REFERENCES [l] E. SMITH, Znt. J. Engng Sci. 27, 301 (1989). [2] E. SMITH, The failure of a strain-softening specimen. Theor. Appl. Fract. Mech. Accepted.

material:

I. Analytical

(Received 9 August 1988)

approach

for a double-cantilever-beam