Local and global instabilities in ductile failure Текст научной статьи по специальности «Физика»

Local and global instabilities in ductile failure

M.P. Wnuk

University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA

Catastrophic fracture in ductile solids is usually preceded by a certain amount of quasistatic crack growth that occurs as a result of void expansion and coalescence process associated with large deformations localized in the narrow zone adjacent to the crack leading edge. This zone is subject to a tri-axial state of stress, and its local properties may vary from those of the bulk material. To describe this condition a modified cohesive crack model is suggested based on the mesomechanical law of the 5-stress distribution and equipped with the “fine structure” feature that is lacking in the standard model.

Subcritical crack growth may be likened to the phenomenon of “preliminary displacements” known in the studies encountered in the physics of tribology. Microscopic sliding of a solid block placed on an elastic-plastic substrate located on the inclined plane is observed to begin at angles somewhat smaller than the critical angle 0 = tg -1 (a), where ¡a denotes the coefficient of friction. With careful observational techniques these displacements can indeed be measured. Likewise, in the course of the early stages of ductile fracture, quasistatic crack growth is detected between the lower bound Ki = Kinj, tantamount to the onset of stable growth, and the upper bound Ki = Kf, equivalent to occurrence of the catastrophic failure. While K^ is believed to be a material constant, the other quantity, Kf, is determined not only by the material properties, but it also depends on specimen geometry, crack configuration and type of the external loading. The exact shape of the terminal instability locus represented in the plane (load, crack length) must be established by employment of the ft-curve technique, in which the second variations of the energy terms are involved.

When the Liapunov criterion is invoked, then it appears that the propagation of a stable crack should be viewed as a sequence of local instability states, while transition to an unstable propagation becomes equivalent to the loss of global stability, as then the entire component breaks up. A moving quasistatic crack is described on the basis of the Wnuk criterion of final stretch, which leads to the nonlinear differential equations governing the resistance curves for various materials. Both the ductile and brittle limits of material response are discussed.

One of the essential results of this study is the partition of energy available for fracture within the end zone, accomplished by means of considerations of the pre-fracture states developed at the mesolevel. This, in turn, leads to a discovery of the energy screening effect, which manifests itself by a significant enhancement of material fracture toughness prior to the catastrophic failure state. Such phenomena are being confirmed by the brilliant experimental work of the Panin group in Tomsk, and Popov’s team in Berlin.

1. Early stages of crack growth in non-elastic solids

To assess the residual strength and the resistance to fatigue crack propagation in welded structures, it is necessary to define a set of parameters relevant for safe service. Due to a high level of heterogeneity and non-elastic response to fracture encountered in the materials involved in the welded joints, the required definitions fall out of the LEFM range and they must be derived from the nonlinear fracture mechanics. When the residual strength of welded joints is considered, almost all experimental techniques recommended by the British Standards 7448, just as the appropriate ASTM standards, involve determination of the CTOD, usually based on the 85 concept — as defined by the German research group at GKSS, or the /-integral resistance curves, as preferred by the American standards set by the ASTM. These R-curves should be measured independently for the base metal,

the weld metal, the heat-affected zone, and then for the entire specimen that consists of all these components.

If all the various fracture mechanics parameters, used to measure an enhancement of fracture toughness during the early stages of fracture, such as S5 or JR, are denoted by a common symbol R, then the rate of toughness increase associated with growth of the subcritical crack, can be predicted as follows

dR=M -1 - ilogi ±R da 2 2 I A

(1)

This equation was first proposed by Wnuk [1, 2] on the basis of his theory of quasistatic crack and assuming a structured nature ofthe end-zone adjacent to the crack front, and several years later it was derived independently by Rice and Sorensen [3] and Rice et al. [4] from considerations of the Prandtl slip-line field in the near-tip region. Equation (1)

© M.P. Wnuk, 2004

defines material resistance ft-curve for the small scale yielding range. However, studies have shown the equation remains valid and produces correct results for loads a raised to 70 % of the yield stress aY [5, 6]. Symbol M in Eq. (1) denotes the tearing modulus, while A is the characteristic microstructural length parameter identified with the size of the process zone, i.e., the zone of intensive necking occurring just prior to the final act of fracture.

For the range of crack tip plasticity considered here the resistance parameter ft and the JR variable are directly related, namely R = (tcE/8ctY)JR, while the nondimensional tearing modulus M is related to Paris’ tearing modulus TJ = (E aY)(dJ/ da)ini and to Shih’s crack tip opening angle CTOA = S/A, with 8 denoting Wnuk’s constant of final stretch [7] in the following way: M= (rc/8)TJ and M = = (tcE/8cty)CTOA . Here, E denotes the Young modulus and aY is the effective yield stress at the crack front, while a0 is the uni-axial yield stress. For a pressure-vessel steel such as A533B, the approximate values of the pertinent material constants are, a0/E = 3• 10-3, TJ = 50, CTOA = 0.15, A = (1/10)Rc, where the characteristic material length parameter Rc = (n/ 8) KfC/aY.

The length A represents the constant crack growth step and it can be estimated as follows

- for brittle materials 2yE

A=-

mol

- for quasi-brittle or ductile materials A =

(2)

(3)

Here, the symbol y is used to denote surface tension, amol is the molecular strength (= E/ 30), Gf (equivalent to 2y for the brittle fracture case) is the true work of fracture, while 5max is the maximum stress in the Wnuk-Legat cohesive-stress distribution law proposed for a quasistatic crack [8]:

S (A, n, a) = SQXn exp[a(1 - A)],

(4)

A- Ü. R

This formula implies a certain “separation law”, which within the framework of the continuum mechanics must be appended to the constitutive equations whenever possibility of fracture is considered. In Eq. (4) the distance of any point within the end-zone from the leading edge of crack (xl) is represented by the nondimesional variable A, given as the ratio of x and the length of the end-zone ft. Symbol S0 denotes the local value of the yield point, which is aY corrected for the triaxiality constraint factor prevailing in the vicinity of the crack front. Coefficients n and a reflect the effects of the microstructure on the physical properties of the end zone associated with a quasistatic crack. Wnuk and

Legat [8] showed how these mesomechanical quantities might be connected to macroscopic parameters such as the over-stress ratio Smax/S0 and the ductility index pi, obtainable in laboratory testing.

Note that for a propagating crack ft is a function of the current crack length, ft = ft(a). This function results upon integration of Eq. (1) subject to the initial conditions, at a = a0, ft = Rjni. The modulus Mis a very important parameter, and it may be shown to be proportional to the natural logarithm of the strain ratio efpje0, namely

M - log(ep/60) (5)

where the numerator represents the plastic component of the strain at fracture, while the denominator is the strain measured at the onset of yield.

Setting the derivative dS/dA to zero results in determination of the maximum stress Smax and the location of its occurrence relative to the crack front, Amax = A/R- This means that the point at which one observes the maximum stress coincides with the outer edge of the process zone, X = A, while Smax is given as

Smax S0

exp

a i -

(6)

(7)

If we invoke the Wnuk and Mura rule [9]

A- i

R 1 + eP/eo’

then Smax can be expressed as a function of two microstructural variables, S0 and the ratio e = e0jep, namely

Smax = Snax^ ^ e)- (8)

For ductile steels e is small. Using the power series expansions expression (6) may be reduced to

Smax = So en exp[a(1 -e)].

(9)

Hence, we obtain an assessment of the over-stress factor Smax/S0, valid for small e

• = exp[n log e+ a(1 - e)].

(10)

Substituting in here the values typical for a low-carbon steel

e0 = 2 %, ep = 20 %, e = 0.1,

n = 0.1, a = 1 we obtain an estimate

Smax = 1.95 So.

(11)

(12)

This indicates an almost two-fold increase ofthe S-stress within the microstructural length A, relative to the reference level, S0. It is noteworthy that for the metals of high ductility the ratio Smax/S0 approaches 1, while at the brittle end of the material behavior spectrum this ratio is an order of magnitude greater.

Equation (1) can be readily integrated numerically and it yields a monotonically increasing material resistance cur-

ve, representing such quantities as the CTOD (given by S5) or JR , from the initial threshold of the crack propagation Rjni to the steady-state level, Rss. Of course, the occurrence of the terminal instability is expected long before ft reaches Rss. Yet, these two bounds, Rjni and Rss, determine the shape of the entire ft-curve. The initial slope of an ft-curve follows from Eq. (1) as

— | = M -1 - 1log[ ^Rida L 2 2 &l A

(13)

For the slow stable crack growth to exist, this slope must be positive, which implies

'4Rn

=1 +—lo 22

A

(14)

Using the Wnuk-Mura equality (7) one may estimate the minimum tearing modulus

(„ f A

: log

(15)

which justifies statement (5).

At the steady-state limit, when R ^ Rss, the slope dR/da approaches zero. This leads to a relation

0 = M - — - —log[ 4Rss 2 2 I A

Hence, it follows A

Rss = — exp[2M -1].

(16)

(17)

Introducing the ductility index, pi = Rini/ A and dividing both sides of Eq. (17) by Rini, we obtain a new quantity

m = Rs^ = ^exp[2M -1].

Rini 4pi

(18)

As it turns out all the pertinent entities that describe growth of a quasistatic crack or a fatigue crack in welded joints exhibit a strong dependence on the coefficient m. This coefficient, in turn, is a function of two essential material parameters, the strain ratio e (or ductility index pi) and the fracture enhancement factor k, defined as follows

k = —M—. (19)

M

min

In order for quasistatic crack growth to precede the event of catastrophic failure, k must be greater than one. For example, for pi = 10 and k = 1.2, one obtains M min = 2.3444 and M = 2.8133.

During the process of slow stable crack extension the energy criterion of fracture, dictated by the First Law of Thermodynamics, requires that the rate of energy demand equals the rate of energy supplied to the crack tip (same as it is in the Griffith criterion), namely

R(a) = Rappl- (20)

It is noted that this equation involves the first variations of the appropriate energy terms.

However, at the point of terminal instability, another condition must be satisfied. Slopes of the material ft-curve and the curve representing the dependence of the rate of energy supply on the current crack length, (dRAPP^ da)x, must equal each other. Thus, at the point of catastrophic failure we have

da

rappl

da

(21)

Note that this equation contains second variations of the energy terms, as it would be expected for an analysis of loss of stability for the system. Subscript x added at the foot of the symbol of partial derivative indicates that the derivative should be evaluated at a certain specific condition of loading. In particular, two limiting cases are considered; when loading is controlled by the stress (then x = Q) and when loading is displacement controlled (then x = u). Details of such considerations were discussed by Wnuk and Omid-var [10].

With the symbol Q denoting nondimensional loading parameter

q=<22) 2S0

we can define the applied energy rate in the way consistent with the cohesive zone model of a crack, namely

R

APPL

=1 aQ 2. 2

(23)

as a normalizing constant for R

If we choose to use R and a, i.e.,

R = Y _a

Rini Rii„

then the applied energy rate (23) assumes the form Yappl = i XQ2 and the derivative in Eq. (21) reads

r dY ^

W—APPL

dX

= 1Q2 = YAPPL

2Q X

(24)

(25)

(26)

Now, applying both equations (20) and (21) to describe the terminal instability, we may drop the subscript “APPL” (since Y and YAPPL equal each other), and rewrite the condition (21) as

Y

(27)

dY

dX

X

or

T = —— = — = 0 at failure.

dX X

(28)

Of course, the derivative dY/ dX that appears in both equations above has to be replaced by the right-hand-side of the Wnuk-Rice-Sorensen equation (1), rewritten in a normalized form as follows

e

Fig. 1. Three ft-curves shown as plots of Y vs. the increment of crack extension, AX=X- X0, obtained for the fracture toughness enhancement parameter m assuming values 1.5 (low curve), 2 (middle curve) and 3 (upper curve)

= M -1 -ilog(4piY), or dX 2 2

dY 1. Y

----= — log-^.

dX 2 Y

(29)

The variable T, known as stability index, is very helpful in describing the slow stable crack growth process starting with the onset of crack propagation, when T is a positive number, and ending with the transition to a catastrophic growth, when T attains zero. For the unstable phase of propagation the index T assumes negative values. The steady-state limit of the nondimensional measure of material resistance, Yss, is identical with our parameter of fracture toughness enhancement, m.

Summarizing, we note that in order to study quasistatic crack extension, sometimes referred to as “static fatigue” case, we have at our disposal various mathematical tools. Equations (29) generate material ft-curves for a given set of input data, such as M and p i (note that the shape of the ft-curve does not depend on the initial crack length X 0). This curve is described by the function Y = Y(X - X0) that results from integration of either of equations (29). Examples of such solutions are shown in Fig. 1. An alternative way is to study the so-called Q-curve, which describes the dependence of the loading parameter Q on the current crack length X during the stable phase of crack growth, when Q remains in equilibrium with a crack of a given length. Function Q = Q(X) can be obtained as an integral of this nonlinear differential equation (which accounts only for the simplest geometry of a center crack, but it can readily be generalized for other crack and loading configurations):

dQ = log(2m/ Q2 X) - Q2 dX 2QX '

When this equation is subject to the initial conditions, at X = X0, Q = Qini =yj2/X0, and integrated for various

Fig. 2. Three Q-curves illustrating the dependence of the loading parameter Q on the current crack length X, obtained for three different material parameters m (1.5, 2 and 3). Maxima on these curves are indicative of the occurrence of the terminal instability. The critical crack lengths for the three curves shown are 11.445 for the lowest curve, 12.712 for the middle one, and 14.163 for the upper one

parameters m, see definition (18), the resulting Q-curves emerge — as shown in Fig. 2. It is readily seen that the maxima on these curves coincide with the points of transition from stable to unstable growth. Finally, when all the data are combined, one can generate the graphs of the corresponding stability indices, the so-called T-curves, as shown in Fig. 3. The initial crack length X0 = 10 was used in all runs presented in the figures, while the m-coefficient was allowed to vary, assuming three values, 1.5 then 2 and 3. With the ductility index p i = 10, this would correspond to three values of the tearing modulus, namely

Fig. 3. Three T-curves shown represent the transition from stable to unstable crack propagation. Each point where T-curve intersects the axis T= 0 determines the onset of the catastrophic fracture. The lowest curve corresponds to m = 1.5, the middle one results for m = 2, while m = 3 for the upper curve

M =

2.5472, 2.6910, k = 2.8937,

1.09,

1.15,

1.23.

(31)

These numbers were calculated from a relation M = (1/2)[1 + log(4pim)], which is the inverse relation with respect to Eq. (18). Since for pi = 10 the minimum tearing modulus defined by Eq. (14), Mmin = 0.5log(4epi) = = 2.3444, each case results in a different K-coefficient, as shown in (31).

Finally, the data obtained from this analysis lead to the predicted values of fracture enhancement and increase in crack length and the loading parameter prior to terminal instability. With the definitions

Y - Y del(Y) = Y-------^,

del( X) = Xf X0

del(Q) =

X0 Qf - Qin

(32)

Qin

we have these results for the three cases considered m = 1.5 m = 2 m = 3

del(Y) = 21.3 % del(Y) = 56.4 % del(Y) = 119 %

del(X) = 14.5 % del(X) = 27.1 % del(X) = 41.6 % (33)

del(Q) = 2.96 % del(Q) = 10.9 % del(Q) = 24.6 %.

It is seen that an increase in the fracture toughness en-

hancement coefficient substantially affects the increments del(Y), del(X) and del(Q).

Theory presented here may be further developed to include the effects of specimen size and geometry as well as the loading configuration. The guidelines for such an investigation were provided by Budiansky [11], Xia, Shih and Hutchinson [12] and Wnuk and Omidvar [10]. One should also be able incorporate here the insight provided by the earlier experimental and theoretical work on direct measurements of /-integral in welded structures by Read and Petrovski [13], Read et al. [14], Petrovski and Kocak [15] and Petrovski [16] will be incorporated in the present research plan.

Numerical approach suggested by Wnuk (1992) may prove useful in predicting fatigue life of welded joints. The basic concept underlying this approach consists in associating each load cycle with an incremental crack extension governed by the equation developed for “static fatigue”, such as Eq. (30). Thus, the integral

Qmax

dX

dN

= J

2QX0 dQ

(34)

Qmin log(2m/QX) - Q2(<02 + 2X00')

represents the rate of fatigue crack growth. Integration process is completed under an assumption that the length X remains almost constant during a single cycle. The shape function O = O(X) is a geometry dependent entity that enters into the general definition of the stress intensity factor

KI = aJna0(a/W). (35)

Here, W denotes width of the specimen, while symbol O' is used to denote the derivative dO/dX. For O = 1 the integrand in (1.34) reduces to the reciprocal of the function on the right-hand-side of Eq. (30).

2. Brittle and ductile limits of material behavior. Energy screening effect

If the profile of the crack within the structured cohesive zone, 0 < Xj < R, or 0 < A <1, with A denoting the ratio Xj/ R, is determined as a certain function of A, say A(A), then the Wnuk criterion of final stretch reduces to the following nonlinear differential equation defining the time-dependent material resistance to crack propagation [2, 5]

dR R R_ (R — = M- — + — F\ — da A A I A

(36)

Here, the current crack length a is treated as a time-like variable, M is a material constant known as the tearing modulus, while the function F (R/ A) is identical with A(1/ A). This function is essential in all the considerations that follow, and it has to be evaluated by solving an appropriate mixed boundary value problem associated with an extended crack, i.e., the crack plus the cohesive zones added at both ends. Some considerations related to such a treatment are shown in Appendix A.

To determine the key function A(A), one has to assume a certain distribution of the S-stresses, which are the cohesive stresses imposed at the mesomechanical level through material microstructure and the properties near the leading edge of the crack. For the Wnuk-Legat cohesive stress distribution law involving two mesomechanical parameters a and n, namely

S(A, a, n) = S0An exp [a(1 - A)], (37)

the function A has been established as follows

A(A, a, n) = n+a[A1(A)](a+n) -

a +1

n -1 a +1

[A0(A)](a+1).

(38)

Here, the functions A0 and A1 are defined as

A

A0 (A) = V1 - A - — ln

1 + v 1 -A 1 -V1 -A

A1(A) = V 1 -A| 1 -Al-Apln

1 + v 1 -A 1 -yl 1 -A

(39)

Note that A = 1 for A = 0, and A equals zero for A = 1, which designates the outer edge of the ft-zone. The normalization constant for the displacement v = vtipA(A, a, n) is the opening displacement at the tip of the crack, vtip = = 4S0R/nE. The function A = A(A, a, n) with a = 1, and n = 0.2, is depicted in Fig. 4. The interpolation formula

Fig. 4. Profiles of the cohesive crack within the end-zone. The first two curves on the top show the solution resulting from the Knauss and Dugdale models, respectively, while the other two curves correspond an a certain choice of the mesomechanical parameters a and n shown as the arguments of the function V in the figure

(38) has been tested numerically for the range of the parameters a and n belonging to the interval (0, 1).

Now we shall consider in some detail certain limiting cases, corresponding to ductile and to brittle material response associated with the process of fracture, and all resulting from the Eq. (36) as certain specific values of the mesomechanical parameters are assumed.

Case 1. When a = n = 0, the S-stress reduces to a constant, S0, over the length of the cohesive zone. This corresponds to the Dugdale model, modified by Wnuk [1] to describe the quasistatic crack extension. As we proceed to show it represents the ductile limit of the material behavior. As is readily seen, equation (38) yields then A = A0(A), and thus

F (p) =A (

1?

p-1______1

P 2p

ln

Vp+Vp~1

VP-VfP-î

(40)

p=

R A.

Since the ductile limit of the material behavior implies R >> A, or equivalently p >> 1, the function defined by (40) reduces to

Fp>>1 =1 ln(4p).

2p 2p

(41)

Therefore, the governing equation for the ft-curve pertinent to this case is obtained by combining Eqs. (36) and (41), yielding

da

-M

yp>>1

1 1

22

ln(4p)

(42)

d—

dX

-M

1 1

22

ln(4pi— ).

(43)

Here X = a/Rini, Y = R/Rini and pi = Rini/A. The propagation threshold Rini represents the measure of the resistance to crack propagation at the onset of crack growth. The outcome of the integration of Eq. (43) is expected to strongly depend on the material ductility represented by the variable pi, and this effect has been illustrated in Fig. 1. According to the Wnuk-Mura relation, the ductility index is related to the plastic component of the strain at fracture epj and the strain at the yield point, e0, as follows

pi = RnL=1+i.

A e0

We recall here that equation (41) is identical with the Wnuk-Rice-Sorensen equation discussed in the previous section. The range of its validity is limited to the small scale yielding condition (R << a) and the large value of the ductility measure, p >> 1. Examples of R-curves resulting from Eq. (43) were discussed in the previous section.

Case 2. It appears to be possible to use the same model, as discussed for the case 1, but in the other limit imposed on the inner structure of the end zone. When the ratio of the length of this zone to the size of the process zone, R/ A, approaches unity, then we are dealing with the brittle limit of material behavior. Let us verify whether or not this statement applies to the extended Dugdale model discussed above. Using a Taylor expansion for p ^ 1, the function F, defined in Eq. (40), can be reduced to a simple form

Fp^1 =

1 —

A

R

3/2

(44)

Therefore, the governing equation of the R-curve reads

dR

da

M

yp^1

R 2 R

■ — +----

A 3 A

A

R

or

—1 = M1 -p— + - p— dX 3 1

3/2

3/2

1 —

p—1

(45)

(46)

or, in a non-dimensional form,

The initial condition for the differential equations defining an ft-curve, for all the cases discussed here, is Y = Y1 = = 1 at X = X0, where the initial crack length X0 = a0/Rini. We shall compare the outcome of the numerical integration of Eq. (46) with the corresponding curves resulting from the next two limiting cases to be discussed.

Case 3. When n = 1 and a = 0, the S-stress distribution assumes the linear form, S = S0A, identical to the one assumed by Knauss [17-19]. Formula (38) reduces then to

A (A, a, n) = A1( A). (47)

We shall further simplify this form by assuming that Knauss’ model is appropriate for representing the brittle end of the material response spectrum, for which one expects p ^ 1. Using a Taylor expansion we have

Fig. 5. Three ft-curves obtained for the extended Knauss model (upper curve), the extended Dugdale model (intermediate curve) and the “most brittle” case (lowest curve)

[A1(À)]^1 = -(1 -À)32

or

[ F (p)]

p^1

1 -1 p

3/2

(48)

(49)

Thus, the governing equation for the R-curve becomes

or

dR

da

d—2

R 4 R

■ — +---

A 3 A

A

R

3/2

= M2 -p— +-p—2 dX 2 3 2

3/2

1 —

pi—2

(50)

(51)

Graphs representing a resistance curve, the stability index and the energy transmission ratios resulting from this equation are shown in Figs. 5, 6 and 7.

Case 4. Here we assume the microstructural parameters in such a way that the ratio R/A equals exactly unity. This occurs when the parameters are chosen as a = n =1, as then the R/A ratio, given by Wnuk and Legat [8] as a/n equals unity. In this way we expect to satisfy all conditions incorporated in the present model to describe the fracture process of brittle nature.

The interpolation formula (38) reduces now to

Aa=n=1 =A2(À).

(52)

Using the p ^ 1 limit, the pertinent function F(p) can be obtained from Eqs. (48) and (49) as follows

[ F (p)]a=n=1 =16

1 -1

while the equation of the ft-curve assumes the form

-|3

dR

da

ya=n=1

1 --

A

R

(53)

(54)

When converted to a non-dimensional form, this equation reads

Fig. 6. Stability indices T plotted as functions of the current crack lengths for the three cases considered in Section 2: brittle limit of the Knauss model (upper curve), brittle limit of the Dugdale model (intermediate curve) and the “most brittle” case represented by the lowest curve

d—

16

—3 = M3 -pi—3 +—p—3 dX 3 i 3 9 i 3

1 --

pi—3

(55)

Comparison of the three cases pertinent to the brittle material behavior in the process of non-elastic fracture is best accomplished by a careful observation of the three ft-curves shown in Fig. 5. The graphs depict curves Y1, Y2 and Y3 as functions of the crack length. Each of these curves was obtained by the numerical integration of the governing equations, namely (46), (51) and (55), respectively. The top curve represents a curve obtained from the extended Knauss model, intermediate curve resulted from the extended Dugdale model, while the lower curve illustrates the variable material resistance during the course of a quasistatic

Fig. 7. Energy transmission ratios for the three cases considered in Section 2: the “most brittle” case shown by the upper curve, the intermediate curve corresponds to the brittle limit of the Dugdale model, while the lowest one was obtained from the extended Knauss model modified by Wnuk for a quasistatic crack problem

p

crack extension for the most brittle case. Circles indicate the end of the stable crack growth, and they are indicative of the transition from local to the terminal, or global, instability. The initial crack length for all curves shown was assumed to be X0 = 7, while the ductility index pi was set at 1.1. The enhancement coefficient K = M/M^ was assumed as 1.15 for all three cases. This means that the tearing modulus exceeded the minimum value of the modulus, Mmin , by 15 %. The minimum values of each modulus, M1, M2 and M3 have been calculated for a given pi in such a way that there would be no stable crack growth at all, as the problem reverts then to the Griffith case (indicated by a zero slope of the ft-curve at the initial crack length).

The quickest way to determine the points of the transition from local to global instability associated with a quasistatic crack is to plot the “stability indices” as shown in Fig. 6. For the crack configuration considered here, a stability index T is defined as the difference between the rate of energy demand and the rate of energy provided by the external forces and the elastic strain energy stored in the cracked body. The corresponding definition of T is provided by the Eq. (28). When T is positive, stable crack extension is possible; when it becomes negative, propagation changes to catastrophic. The transition point is clearly visible when the curve depicting T vs. a passes through zero.

The numerical data support the cursory observation of the ft-curves depicted in Fig. 5. It is readily seen that the third case, for which a = n = 1, exhibits the least developed ft-curve, and thus the most brittle material response is to be expected. Indeed, the calculations of the increments in the crack length due to stable crack extension are as follows: 2.3 % for the extended Dugdale model (case 2), 3.71 % for the extended Knauss model (case 3), and 1.95 % for the most brittle case (case 4).

For the sake of comparison, let us consider yet another example. If the ductility index was assumed pi = 10, and Eq. (43) was used with the same coefficient k of 1.15 and the same initial crack length of X0 = 7, then the increment in the crack length attained prior to transition to unstable propagation, would have been predicted as 27.05 %.

It is noteworthy that for a case involving a microcrack (when X0 is about 2 or 3), the analysis gets a bit more complex. If we insist on the same value of the coefficient k,

i.e., k = 1.15, it turns out that models considered predict no stable growth at all. This should be interpreted as a manifestation of the fact that the tearing moduli chosen are too low to trigger the dissipative processes, which are associated with the early stages of fracture and which delay the catastrophic propagation of the crack. Here, two very different scenarios are possible. Either the microcrack becomes supercritical, analogous to a drop of supercooled liquid that does not yet solidify, or the crack relaxes the high stresses generated in its vicinity by a certain degree of stable propagation that precedes the onset of the terminal instability.

These results are further corroborated by the studies of the energy transmission ratio (ETR), which reflects the phenomenon of energy screening within the end zone. As it turns out, only for ideally brittle solids the total energy available in the system is delivered to the crack tip. For the range of non-elastic responses, most of the energy is dissipated in the course of irreversible deformation, leaving a small fraction of the total energy available entering the process zone adjacent to the crack tip. To put it briefly, the ETR for a brittle solid approaches unity, while for very ductile materials it is quite small [20].

Figure 7 illustrates the variations of the energy transmission ratios, predicted by the models developed for the cases

2, 3 and 4, as the current crack length increases from its initial value of 7Rini to the value marked by a small circle indicative of the point of transition into the unstable crack propagation, for each case respectively. Equations that were used for the purpose of evaluation of the ETR values shown in Fig. 7, are quoted in Appendix A.

References

[1] M.P. Wnuk, Accelerating Crack in a Viscoelastic Solid Subject to Subcritical Stress Intensity, in Proceedings of the International Conference on Dynamic Crack Propagation, Ed. by G.C. Sih, Lehigh University, Noordhoff, Leiden (1972) 273.

[2] M.P. Wnuk, Quasi-static extension of a tensile crack contained in a viscoelastic-plastic solid, J. Appl. Mechanics, 41, No. 1 (1974) 234.

[3] J.R. Rice and E.P. Sorensen, Continuing crack-tip deformation and fracture for plane strain crack growth in elastic-plastic solids, J. Mech. Phys. Solids, 26 (1978) 263.

[4] J.R. Rice, W.J. Drugan, and T.L. Sham, Elastic-Plastic Analysis of Growing Cracks, ASTM STP 700, ASTM, Philadelphia (1980) 189.

[5] M.P. Wnuk, Mathematical Modeling of Nonlinear Phenomena in Fracture Mechanics, in Nonlinear Fracture Mechanics, Ed. by M.P. Wnuk, Springer-Verlag, Wien-New York, CISM Course and Lecture No. 314, International Centre for Mechanical Sciences, Udine, Italy, 1990.

[6] M.P. Wnuk, Bridging the Gap between Micro- and Macro-Fracture Mechanics, in Abstr. of Int. Workshop “Mesomechanics: Foundations and Applications”, 26-28 March, 2001, Tomsk (2001) 27.

[7] M.P. Wnuk, Criterion of Final Stretch for a Quasistatic Crack in NonElastic Medium, in Proceedings of ICM3 Conference, Cambridge, England, 3 (1979) 549.

[8] M.P. Wnuk and J. Legat, Work of fracture and cohesive stress distributions resulting from triaxiality dependent cohesive zone model, Int. J. Fracture, 114 ((2002) 29.

[9] M.P. Wnuk and T. Mura, Effect of microstructure on the upper and lower limit of material toughness in elastic-plastic fracture, J. Mech. of Materials, 2, No. 1 (1983) 33.

[10] M.P. Wnuk and B. Omidvar, Local and global instabilities associated with continuing crack extension in dissipative solids, Int. J. Fract., 84 (1997) 237.

[11] B. Budiansky, Resistance Curves for Finite Specimen Geometries, Seminar at Harvard University, 1996.

[12] L. Xia, C.F. Shih, and J.W. Hutchinson, A computational approach to ductile crack growth under large scale yielding, J. Mech. Phys. Solids, 42 (1995) 21.

[13] D.T. Read and B. Petrovski, Elastic-Plastic Fracture at Surface Flaws in HSLA Weldments, in Proc. of the 9th Int. Conf. on Offshore Mechanics and Arctic Engineering, Houston, TX, Vol. III, Materials Engineering, Part B (1990) 461; publ. in Transactions of ASME, J. of Offshore Mechanics and Arctic Engineering, 114, No. 4 (1990) 264.

[14] D. Read, H.I. McHenry, and B. Petrovski, Elastic-plastic models of surface cracks in tensile panels, J. Experimental Mechanics, June (1989) 226.

[15] B. Petrovski and M. Kocak, Fracture of Surface Cracked Undermatched Weld Joint in High Strength Steel, in IIWDocument X-1284-93 presented at the 46th Annual Assembly of International Institute of Welding, August 28 - September 4, 1993, Glasgow, UK, 1993.

[16] B. Petrovski, Evaluation of Fracture Behaviour of Mismatched Steel Weld Joints in Cracked Tensile Panels”, Invited Lecture at First Slo-venian-Japanese Seminar on “Fracture of Welded Structures under Static and Dynamic Loading”, Maribor, July 2001.

[17] T. Ungsuwarungsri and W.G. Knauss, The role of delayed-softened material behavior in the fracture of composites and adhesives, Int. J. of Solids and Structures, 35 (1987) 221.

[18] T. Ungsuwarungsri and W.G. Knauss, A nonlinear analysis of equilibrium craze. Part I. Problem formulation and solution, J Appl. Mechanics, 110 (1987) 44.

[19] T. Ungsuwarungsri and W.G. Knauss, Part II. Simulation of craze and crack growth, J. Appl. Mechanics, (1987) 52.

[20] M.P. Wnuk, Enhancement of fracture toughness due to energy screening effect in the early stages of non-elastic failure, Fatigue and Fracture of Engineering Materials Structures Journal, UK, 26 (2003) 741.

Appendix A

To evaluate the opening displacement within the cohesive zone, a < x < a + ft, a mixed boundary value problem must be solved. The problem is stated in terms of

1) displacement normal to the crack plane, v = 0 for |x| > a + ft, and

2) tractions applied to the crack surface, t(x) = -a for 0 < x < m, and t(x) = -a + S, for m < x < 1, where S is defined as follows

1 - x

S(x, a, n, m) = I ——m I exp 1 - m

a

1-m

, (A1)

m < x < 1.

The parameter m is defined as the ratio a/(a + R). When the method of Sneddon’s integral transforms is employed, then the opening displacement within the cohesive zone may be evaluated in terms of the integral

i \ n f F(t, a, n, m)

v(x, a, n, m) = C0 I----------, —

0 Vt2 - u 2

dt.

(A2)

Here, the constant C0 = 4S0a/nE, while the auxiliary function F is calculated as follows

F(t, a, n, m) = } 0», a, "№ - m)dA +

0 \ -1 - [(1 - m)X + m]2

I

S(u, a, n, m) du

sit2 -2

•, m < t < 1.

(A3)

If the integrand G is defined as

G(—, a, n) = — exp[a(1 - A)], (A4)

then the first integral in Eq. (A3) can be interpreted as the nondimensional loading parameter Q(a, n, m), determined from the finiteness condition, and then Eq. (A2) may be rewritten in the form

v(x, a, n, m) =

= Co

VT-—2Q(a, n, m) - IFl(t,,a, n, m)tdt

x Vt2 - —2

. (A5)

The second auxiliary function, which appears in (A5), Fj, is defined [8] by

F (t, a, n, m) = |

1 S(u, a, n, m)dt

Vt2-i

(A6)

For the two specific cases, S = S0 as postulated by the Dugdale model (case I), and S = S0A (case II) as assumed by Knauss, the integrals shown above reduce to the fairly simple formulae if m ^ 1, which implies an assumption of the small scale yielding condition. The final results are

(1+V1 -a^

VT-X-—in 2

-—2in

4

i-VT-X (i+7!-— ^

i-41-—

(A7)

v(—) = C

for the case I, and

v(—) = C Vi - —

for the case II.

We readily recognize the two basic interpolating functions A 0(—) and Aj(—) used in Eqs. (38) and (39), while the constant C = 4S0R/nE. Variable — denotes a nondimensional distance from the crack tip, i.e., — = —1 / R, and xf = = x - a.

The energy transmission ratio (ETR) is defined as

Wue =

ETR = -

Wo

A |S( —j, a, n) 0 dv( —j, a, n) d—j d—1

R |S( —j, a, n) 0 dv( —1, a, n) d—1 d—1

(A8)

For the case 2 (the extended Dugdale model), case 3 (the extended Knauss model), and case 4 (the most brittle case) discussed in Section 2, this expression assumes the following forms

(

ETR = 1----

1 -1 P

n3/2

V P

ETR3 =

ETR4 =

I -(1 -—)3/2d—--- (

J 3V ' p 3

1 -1

f/2

| 3(1 -—)32d—

03

1P

| — (1 - —)4 exp (1 - —) d—------------------—

P—

1— P

exp

(A9)

1,6

| — (1 - —)4exp (1 - —) d—

0—

P

Fig. 8. Energy transmission ratios shown as functions of the ductility index p for four different models of material behavior are represented: 1) a “very brittle” material described by the Wnuk-Legat S-distribution cohesive stress, for which a and n are equal 1 (top curve); 2) Dugdale S-distribution cohesive stress with no restrictions on the index p; 3) Dugdale S-distribution cohesive stress valid for the limiting case of p ^ 1; 4) the Knauss linear S-distribution case for the limiting case of brittle behavior, P ^ 1

These equations, combined with the governing equation of the R-curve pertinent to each case considered, have been used in graphing the curves shown in Fig. 7. Eqs. (A9) suggest that all ETR’s are functions of p, a nondimensional measure of the material resistance to cracking. Since during the early stages of non-elastic fracture process, p depends on the crack length X, it is possible to express the energy transmission ratios explicitly as functions of the crack length.

Indeed, when the appropriate equation of the R-curve is employed, p = p(X), the variable p can be eliminated from equations (A9) and replaced by X. For case 2, equation (46) has been used, then Eq. (51) for case 3, and Eq. (55) for case 4. The results of such a transformation are shown in Fig. 7.

It is noteworthy that the equations (A9) resulted from the following general formula valid for any linear S-distribu-

tion case

ETRr

1

v

P

i/ p

+ J v( À)dÀ

0

Jv ( À)dÀ

(A10)

To compare this formula with the “classic” solution for the ETR predicted by the Dugdale model, let us examine the equation

ETRd = 1 -v

T

Vpy

(A11)

For the usual range of p >> 1 and for the limiting case of p ^ 1, the equation above predicts

ETRd =

_L + J_ ln(4p), p>> 1, 2p 2p

1-

1-

3/2

(A12)

, p-1.

Note that the last equation given is identical to the relation ETR2 vs. p as shown in Eq. (A9). The graphs reflecting those two relationships along with the ETR2 and ETR3 results are plotted in Fig. 8.

p