A mathematical model of Panin’s prefracture zones and stability of subcritical cracks Текст научной статьи по специальности «Физика»

A mathematical model of Panin’s prefracture zones and stability of subcritical cracks

M.P. Wnuk, M. Alavi1, A. Rouzbehani

University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA 1 Finele Consulting Engineers, Inc., Hercules, CA 94547, USA

Basic concept underlying Griffith’s theory of fracture of solids was that, similar to liquids, solids possess surface energy and, in order to propagate a crack by increasing its surface area, the corresponding surface energy must be compensated through the externally added or internally released energy. This assumption works well for brittle solids, but is not sufficient for quasi-brittle and ductile solids.

Some new forms of energy components must be incorporated into the energy balance equation, from which the input of energy needed to propagate the crack and subsequently the stress at the onset of fracture can be determined. The additional energy that significantly dominates over the surface energy is the irreversible energy dissipated by the way of the plastic strains that precede the leading edge of a moving crack. For stationary cracks the additional terms within the energy balance equation were introduced by Irwin and Orowan. An extension of these concepts is found in the experimental work of V. Panin, who has shown that the irreversible deformation is primarily confined to the prefracture zones associated with a stationary or a slowly growing crack.

The present study is based on the structured cohesive crack model equipped with the “unit step growth” or “fracture quantum”. This model is capable to encompass all the essential issues such as stability of subcritical cracks, quantization of the fracture process and fractal geometry of crack surfaces, and incorporate them into one consistent theoretical representation.

Keywords: brittle, quasi-brittle and ductile fracture, stationary and propagating crack, subcritical crack, Panin’s prefracture zones, unit step growth, fracture quantum, structured cohesive crack model, instabilities in ductile fracture, fractal fracture mechanics

1. Introduction

Inspiration for writing this paper was provided by the experimental work of V. Panin and his group [1] relevant to the better understanding of the phenomenon of prefracture strain accumulation, concentration and redistribution, which occurs within the small pre-fracture zone adjacent to the leading edge of crack and being of paramount importance in determining the early stages of fracture, point of fracture onset — followed at first by the stable crack growth — and then by a terminal instability, which when the positive stress intensity factor K-gradient is maintained, leads to a catastrophic propagation.

In order to be able to construct a mathematical model of these non-linear deformation and fracture processes it is necessary to introduce the “quantized model” of fracture, or “quantized fracture mechanics”. We shall be working here with a structured cohesive model of crack equipped with the “unit growth step” — or, equivalently — the Neu-ber’s particle [2] or Novozhilov’s “fracture quantum” [3]. Such an approach represents a substantial departure from the classic theory of Griffith who predicts no subcritical

cracks. Griffith crack is either stationary or catastrophic under the positive K-gradient loading configuration. What visibly is missing in the classic theory is the transition period from a stationary state to a moving crack, which is accomplished by the insertion of the period of slow stable crack growth (SCG) and made possible by accounting for the highly non-linear deformation processes preceding fracture.

To this end similar works have been done in recent past [4, 5], but none of these investigations have succeeded in presenting a mathematically complete theory departing from the continuum based approximations and consistent with the latest trends in the computational fracture mechanics, cf. [6]. It is noteworthy that the mathematical model of “structured cohesive crack” has been successfully applied to the studies of the effects of specimen geometry and loading configuration on occurrence of instabilities in ductile fracture [7], an in modeling the fatigue phenomenon at nano scale levels [8].

To follow this line of approach the quantization of the fracture process is needed, and it was implemented via the

© Wnuk M.P., Alavi M., Rouzbehani A., 2013

8 (crack opening displacement) criterion of Wnuk. The notions of Neuber’s particle and Novozhilov’s fracture quantum are invoked in order to accomplish the quantization procedure. A prior knowledge of the strain distribution within the Panin zones is required. The governing differential equations of slow stable crack growth based on the Panin’s study and on the theoretical model proposed [5], have been refined. It has been demonstrated that the nature provides certain mechanisms of enhancing or reducing the material resistance to fracture. The first one is related to material ductility and energy dissipation that precedes the final act of decohesion, while the other factor is purely geometrical as it derives from the roughness of the crack surfaces (not accounted for by the Euclidean geometry of a smooth classic crack). The conclusion is that while ductility significantly improves the fracture toughness, the increased roughness of crack surface suppresses the subcriti-cal crack growth and it tends to induce a more brittle-like fracture. This feature is described in our model by the fractal fracture mechanics. The theory developed here is based on certain key equations involving the fractal representation of stationary and growing cracks due to the fundamental research [4, 9].

2. Displacements and strains associated with a structured cohesive crack model

For many years one of the primary subjects of Panin’s research [1] were the experimental studies and recording of the strains at micro- and mesolevels as well as observation of their subsequent build up and redistribution occurring within a certain small process zone adjacent to the leading edge of the crack. Here we set out to construct a simple mathematical model describing such a phenomenon of prefracture strain accumulation and concentration within the regions close to the crack front — in what follows re-

Fig. 1. Examples of quasi-brittle and ductile material behavior. Two materials with identical yield strain eY and similar yield points gy, but with widely different ductility are compared: Material 1 (quasi-brittle) shows the ratio ef /8Y close to one, while for the material 2 (ductile) the ratio 82 /8Y >> 1. In the structured cohesive crack model the ductility index Rini/A is defined as 8f/8Y. The length of the cohesive zone at the onset of fracture Rini equals n/8(Kc/gy)2, while A is the size of the Neuber’s particle or “fracture quantum”

ferred to as “prefracture” or “Panin zones”. We intend to study both the stationary and slowly moving cracks. For this purpose we shall employ the structured cohesive crack model equipped with a Neuber’s particle or fracture quantum A. An assumption regarding existence of such particle embedded within the cohesive zone is necessary if the quantization of the fracture process is anticipated and necessary to provide a complete mathematical representation of the fracture process for a ductile (or quasi-brittle) solid. It is noted that the terms “pre-fracture zone”, “Panin zone” or “cohesive zone” are to be understood as synonyms.

Material ductility will be one of the parameters of primary concern. For the structured cohesive crack model, see Figs. 1 and 2, the following quantity will be used as a measure of the ductility

8 Y + 8

pl

(1)

A 8y 8 Y

Here A denotes the fracture quantum, while the length of the cohesive zone measured at the onset of fracture (usually occurring in form of slow stable crack growth that in ductile media precedes the catastrophic propagation) is denoted by Rini and this quantity is related to the yield stress gy, Young modulus E and the fracture toughness measured by Kc or Rice’s integral Jc in a familiar fashion

R _n Rni _ —

Kc ] 2 n f EJc ]

Ig y j _ 8 Ig y j

(2)

This quantity is often identified with the material characteristic length, say Lch, cf. [10]. In order to estimate the size of the other important length parameter, the fracture quantum A, it suffices at this point to say that in brittle and quasi-brittle materials A and Lch are of the same order of

Fig. 2. Wnuk-Yavari’s model of an embedded fractal crack and the associated cohesive zones of length R. Note that the crack has a certain fractal dimension D and that prior to addition of the end-zones R the order of singularity of the near tip stress field is r_a, where the fractal exponent a = (2 - D)/2

magnitude, while for the ductile materials A is much smaller than the characteristic length.

Roughness of the crack surfaces represented via fractal geometry will be treated as a secondary variable that influences the early stages of fracture, i.e., the stable crack extension and the onset of the unstable propagation. For comparison, both the Euclidean and the fractal geometries of a crack will be considered. An approximate model proposed in [9] known as the “embedded fractal crack”, whose fractal dimension D may differ from one will be employed, see Fig. 2. Prior to the addition of the cohesive zones an embedded fractal crack exhibits a singular near tip stress field proportional to r_a, where r is the distance measured from the crack tip and the so-called fractal exponent is related to the dimension D and the roughness measure H as follows

2-d 2h-1 (3)

a- -

a - -

2 2H

Using the fractal crack model Wnuk and Yavari [9] and also Khezrzadeh et al. [4] have estimated the stress intensity factor as

Kjf - x(a)Vna2aG, (4)

where the function %(a) is defined by the integral

d rn-

X(a)- na I

n 0

1 1(1 + w)2 a + (1 -w)2a

(1 - s 2)a

- na-1r(a) r(a +1/2) 1 }

Here r is the Euler gamma function. We note that for 1 < D < 2 the fractal exponent a varies within the range [0.5, 0]. According to the principle of correspondence all quantities describing a fractal crack reduce to the classic expressions valid for a smooth crack when a ^1/2. As shown in [4] the Wnuk-Yavari model of a fractal crack holds only for cracks with relatively small roughness, and thus in the considerations that follow the range of a will be limited to [0.5, 0.4].

First we shall consider the case of a smooth Euclidean crack represented by a structured cohesive crack model as shown in Fig. 3. Two sets of coordinates are used, the dimensional coordinates are shown in Fig. 3, a, while the nondimensional are explained in Fig. 3, b. The distance measured from the origin of coordinates is denoted by x (or 5 = x/ a), while the distance measured from the tip of the physical crack is denoted by x1 (or X = xjR), and the ratio a/a1 - k. The crack length is a, the length of the extended crack is a1 = a + R, and the profile of the entire crack is described, cf. [4, 11], by the following expression involving the inverse hyperbolic functions

uy -y nE

4Gya Re] coth

-1 /1 - k2 s 2 1 - k2

- s coth

-1

k 2 s 2 1 - k2

(6)

Fig. 3. Dimensional coordinates associated with an extended structured cohesive crack (a). Note the location of the “fracture quantum” A, which is adjacent to the crack leading edge and is embedded within the cohesive zone. According to our model, the brittle behavior is observed when A and R are of approximately same size, while for the ductile behavior A is deeply embedded within the cohesive zone and therefore it is much smaller than R. P denotes the control point for measuring the increment of the crack opening displacement (COD) for a slowly moving crack during the early stages of fracture. Symbols CCOD and CTOD designate the “crack center opening displacement” and the “crack tip opening displacement”, respectively. Nondimensional coordinates 5 = x/a and X = xjR (b). Note that when these coordinates are used, the tip of the extended crack falls at 5 = 1/ k, in where k is related to the nondimensional loading parameter Q = n2(g/gy) by this formula k = cos Q

At 5 = 0 we obtain the expression for the crack center opening displacement, namely

r 1 V/2

u center - 40^ coth-1

y %E

4Gy a coth-1

1

1 - k2

nE

sin Q

(7)

The nondimensional loading parameter Q - tcg/2gy enters the last equation due to the known Dugdale formula valid for our model

k- cos Q. (8)

At 5 = 1 we obtain the crack tip opening displacement, namely

utip -4GYa ln( 1 A

nE

k

4gya.

-—^ln nE

1

cos Q

(9)

Fig. 4. Shows the opening displacements at the center of a cohesive crack and at the tip of the physical crack as functions of the applied load Q. Shows the ratio of CTOD (crack tip opening displacement) to CCOD (crack center opening displacement). Despite the nonlinear nature of the problem these results show that CTOD is roughly one half of CCOD through the entire range of the loading parameter Q. The ratio depicted in the figure can be very closely approximated by the simple equation CTOD = = 0.504 CCOD

Figure 4, a shows the dependence of the tip displacement and the center displacement on the applied load Q, while Figure 4, b illustrates the fact that the ratio of these two quantities CTOD/CCOD remains constant almost throughout the entire range of loading. The constant is 0.504, which provides a good rule of thumb: the tip displacement of the cohesive crack is roughly one half of the mouth displacement measured at the crack center. This observation provides helpful information for an experimentalist, who utilizing various clip gages can access the center of the crack much easier than the tip of the physical crack. Thus, once CCOD is measured, the tip displacement, CTOD, can be estimated with good accuracy.

The profile of the entire extended crack, normalized by the constant C = 4gy a/(nE), is shown in Fig. 5, a, while Figure 5, b shows the same profile normalized by CTOD. In constructing these figures the following equations were used

u _ — _ Re i coth C I

7 2 2 - k s

1 - k2

and

- s coth

Uy

I —_±.

-1

1 1

s

_kV 1 - k2

u^ C ln(1/k)

_1 11 - k2 s 2

fcoth 1

1 1

s

j 2 2 - k s

1 - k2

(11)

Due to this normalization procedure all v-profiles pass through one at the tip of the physical crack, x = a, or 5 = 1. Figure 6, a shows the graphs resulting from the expression (11) drawn for three values of load Q and plotted within the range of x that corresponds to the “cusp” of the cohesive crack, i.e., for a < x < a1 or 1 < s < 1/k. Finally in Fig. 6, b these ^-graphs representing the cusp are compared to the curves that result from a known, cf. [12, 13], approximate formula valid under the Barenblatt’s restriction of R being much smaller than the crack length

1 + yj 1 - xj R

„cusp _ 4GyR

coh _ nE

(12)

0.8

0.4

0.0

i!

Q = tt/5

Q = n/6 '—

0.0

0.5

1.0

Q = 7i/5^—

Q = tc/4

0.0

0.5

1.0

Fig. 5. Profiles of the cusp region of the cohesive crack: when the crack opening displacements are normalized by the constant C _ 4gy / (nE), see Eq. (10) (a), and when half of CTOD is used as the normalization constant, see Eq. (11) (b). Nondimensional loading parameter, proportional to g/ gy , is denoted by Q

Fig. 6. Profiles of the cusp of the cohesive crack plotted for three values of the loading parameter Q according to Eq. (11) (a); comparison of the profiles shown in (a) with those that result from simplified formula (dashed lines) valid under Barenblatt’s restriction of R being much smaller than the crack length, cf. Eq. (14) (b)

It is not difficult to show that for R << a the expression for the tip displacement (9) reduces to the constant shown in front of the square bracket in (12), namely

tip uj _

nE

4g y a nE

Kkj

4 g y

ln

nE a + R^

ln

a

J-

a1

R << a

nE

R.

(13)

When this constant is used to normalize the displacement (12), one obtains

^coh

„cusp

~coh

tip

x1 xn „I+n/L

_,/1 x1R

R R 1 1 - xj R

_VT-X--ln1 + ^'

■X

‘ I-------- (14)

2 1 -VT-X

Figure 6, b shows that the agreement between the exact and the approximate formulae for the cohesive crack opening displacements within the cusp region is indeed good for all values of the loading parameter Q. Therefore, to simplify all further calculations we shall employ formula (14). Of particular interest will be the strains within the Panin zone, which are defined by the derivative

ecoh _

dChP _

1 d^hp _

dx1 R

1 4gyR d^coh

dX

4gY dycoh

R nE dX nE dX

(15)

Fig. 7. Strains within the pre-fracture zone obtained as gradient of the crack opening displacement, see Eq. (17)

Applying (14) and carrying out the derivative yields the closed form expression for the strains within the prefracture zone. For convenience the strains are expressed in terms of the variable s, related to the variable X as follows

X=

1

m _1, k _ cos Q. k

(16)

m -1

Thus the expression for strains within the prefracture zone associated with the structured cohesive crack model reads

„coh

_ 4gy

nE _ 4gy

11ln^-------------i

[2 1 + 71 - X(s, m) J

nE

X(s, m) _ -

^1 -X (s, m) ^1 - X( s, m)

a/1 - k - V1 - ks

H-

k + *J1~

(s - 1)cos Q

ks

(17)

m

-1 1 - cosQ

The graphs showing the strains as functions of the loading parameter Q and the coordinate s are shown in Fig. 7.

3. Quantization of the Panin strain and the criterion for subcritical crack growth

Closer examination of the expression (17) reveals that the strains within the prefracture zone are infinite at the tip of the physical crack. Therefore, any use of this entity for the purpose of predicting the onset of fracture propagation will fail, unless it is preceded by the quantization procedure, which in essence is tantamount to evaluation of the strain averaged over the Neuber length A, namely

1 A 1 state

(*>a,a+A_ 1c0"^*, _ 1 J

state 2 j cusp

d^dx1 _

state1

dx1

’cusp(state 2)- (state 1)]. (18)

When this quantity is set equal the average critical strain 8”cr _ U/A we obtain the following criterion defining the onset of fracture propagation

(6)a a+A=Fcr _ UlA,

I a ,a+A

ycohp (state 2)

- <uhp(state 1 _ U.

Let us define the two neighboring states: using the time t and the time-like variable x1 (t) = x - a(t) for a slowly progressing crack, we define these states as follows: state 1, (t - 8t, x1 = A),

state 2, (t, x = 0). (20)

This means that at instant “t - St” defining state 1, the front of the advancing crack is a distance A away from the control point P (see Fig. 3), while at the instant t describing state 2, the tip of the physical crack has reached the control point P. This is indicative that the crack has advanced the “unit growth step” or “fracture quantum” A between the two states considered. The constancy of the increment of the crack opening displacement u (the so-called “final stretch”) measured at the control point P constitutes the necessary condition for the stable crack to propagate. In essence this requirement is tantamount to stating Wnuk’s criterion of the final stretch or the S (COD) criterion for subcritical crack extension [13]. It is noteworthy that the physical foundation for the criterion is the same as the one postulated by McClintock [14], which is the critical strain. As equations (18) and (19) demonstrate, the quantization technique and the attributes of the cohesive crack model allow one to by-pass the long expression for pre-fracture strains and to reduce all the essential considerations to the displacements only, namely the function v™hp (X) given by (14). Similar techniques of the quantized fracture mechanics were employed in [15-17].

Using (12) we express the opening displacements for both considered states as follows

v™!! (state 1) =

4cty R(A)

1 -

nE

A

1 + 71 -A/ R(A) R(A) 2R(A)1,11 -^ 1 -A/ R(A)

ln

(21)

and

Vcdi (state 2) =

4aY R(0) nE

4aY

nE

R(A) + A dR da

(22)

Subtracting (21) from (22) yields the left hand of the second equation in (19), which now reads

d R

R(A) + A — - R(A) da

1 —

R(A)

1+ V1 -A/ R(A) 2R(A) ^ 1 -^ 1 -A/R(A)

-ln

4aY/ (nE)

(23)

For A << R this expression readily reduces to

dR

da

nE

4aY I A

With the notation nE

“ 1-i -

2 2

4(R,m/ A)R

M =-

4a,,

UU | p = Rini A 1 ’ P A

this becomes the ordinary differential equation which governs the motion of a stable crack in the early stages of fracture

dR 11,

— = M-----------ln

da 2 2

( 4pR I

(26)

Two constants which enter the equation above are the tearing modulus M and the material ductility p. For a smooth crack this is the result of Wnuk [13] and Rice et al. [18]. In the next section we shall investigate modifications of this equation extending the range of its validity into the fractal geometry domain. The rate dR/da reflects the rate of material energy demand; and since R and the integral J differ by just a constant, and J = -dn/(2da), thus the left hand side of Eq. (26) also represents the second derivative - d2n/(2da2) where n denotes the potential energy of the loaded body containing a crack. The R-a curve defined by the differential equation (26) is often referred to as the material resistance curve. On the other hand, the rate of the energy supply due to external applied stress field is measured by the quantity R hidden in Eq. (8). For the case of R << a we may expand both sides of this equation in the corresponding power series

a + R i R 1 , Q

-------= 1 + — +..., ---------r = 1 + — +....

a a

Q 2 (27)

cos Q 2

Setting both expression equal to each other, we obtain for R << a

R =

aQ2

(28)

The term R in (28) represents the rate of energy supplied by the external effort. For the terminal instability to occur the second derivatives of the energy terms, or the rates dR^/da and dRappl/da must equal. The rate dR^/da is given by (26), while differentiation of the first equation in (28) leads to

dR

da

Qt

2

R

(29)

- Q=const

This quantity represents the external effort, and thus the conditions for the occurrence of the terminal instability are met when

dRmai

da

or when

dR

appl

da

Q=const

.,11,

M------ln

2 2

( 4pR I

R

(30)

(31)

It is noted that R which appears on the left hand side of (31) represents the material resistance to an extending crack,

(24) so really it should be read as Rmat, while the R shown on the right hand side of (31) symbolizes the driving force applied to the crack and truly it should be denoted by Rappl.

(25) Since at all points of the stable crack growth including the point of terminal instability described by (31) both these

dY/dX, Y/X

Fig. 8. Apparent material resistance to cracking Y = RRini at various levels of material ductility shown as functions of the current crack length during the stable growth process up to the points of terminal instability marked by little circles. All R-curves shown here were obtained from the governing differential equation (26) subject to the initial condition Y = 1 at X0 = 10

quantities remain in equilibrium, Rmat = Rappl, we have skipped the subscripts and wrote the equation defining the critical state as shown in (31). An alternative way to write equation (31) is to define the difference between the energy demand and energy supply. A suitable name for such a difference is “stability index” S, namely

dRma

da

ppl

da

= M-

-ln

4pR

R

Q=const \

R

(32)

To solve for the parameters characterizing the critical state, i.e. the parameter Rc, the critical load Qc, and the critical crack length ac, we need to integrate equation (26) and then inspect the results and eventually solve equations (31) and/or (32). This is best done in two steps: first we separate the variables in (26) obtaining the solution for Y(X) in this implicit form

dz

X (Y) = Xo + |

M -1/2 - 1/2ln(4pz)'

(33)

The value of the tearing modulus M must be chosen to be somewhat above the value of the minimum modulus

Fig. 9. Loading parameter Q shown during the stable crack growth phase for various material ductility indices. The functions shown pass through maxima denoted by little circles. These points define the critical states (Xc, Qc)

0.16

0.14

0.12

0.10

\a_

p = 60

\^d = 20

—= 10

p = lo\J\

p = 20\ \ p = 60\v

12

14

16

X

Fig. 10. Curves representing the rates of the energy demand (nearly straight lines) and the energy supply intersect each other at the points defining the terminal instability (critical states). Stability indices shown as functions of the crack length. Zeros of these functions determine the critical states attained at the end of the stable crack growth. Both (a) and (b) were obtained for the initial crack of length X0 = 10

below which no stable crack growth may occur. In

this case we choose Mto be 20 % above the minimum value, so the modulus M is determined by this expression

M = 1.2

(34)

Next we calculate the loading parameter Q = ^ 2Y/X and plot it against the nondimensional crack length X= a/ Rini. The symbol Ydenotes the nondimensional length of the pre-fracture zone R, namely Y = RRini, and X = = a/Rini denotes the nondimensional length of the crack, while the initial length is X0 = a0/ Rini. Figure 8 shows the R-curves plotted vs. X for p = 10, 20 and 60 and obtained for the initial crack size of 10^^. Figure 9 shows the Q-curves obtained for the same input data. It is noted that the attainment of the maximum on the Q-curve is equivalent to reaching the terminal instability. The point at which the derivative dQ/da approaches zero is best located when the rates of energy demand and energy supply are compared as it is done in Fig. 10, a. The intersection of these curves determines the critical state (Qc, Xc). In addition to these critical parameters the apparent material fracture toughness encountered at the critical point, Yc = Rc/ Rini can readily be evaluated. A convenient way to determine these intersection points numerically is to inspect the sta-

Table 1

Critical parameter Critical values

p = 10 p = 2 O p = 60

Fracture toughness Yc 1.925 2.159 2.581

Crack length Xc 13.605 14.086 14.842

Load Qc 0.532 0.554 0.590

bility indices graphs shown in Fig. 10, b. The critical states for p = 10, 20 and 60, and X0 = 10 were established in Table 1.

It is readily seen that when the Qc values are compared with the load prevailing at the onset of fracture, Qini = =^/^xo, one comes to a conclusion that for each case represented in Table 1 shown above the loading parameter is enhanced during the process of slow crack growth and the percentage increases of the load are as follows: 19% for p = 10, 24 % for p = 20 and 32 % for p = 60. These are significant numbers.

4. Stability of fractal cracks

In this section attention will be focused on the cusp region of the cohesive crack. Following [4] the crack opening displacements at the center of the crack and at the crack tip will be redefined to accommodate the fractal geometry. First let us define four auxiliary functions

p(a) = 4n

= 4nV (2a )-2

ar(a)

1/ a

and

K(a) =

r(1/2 + a) 1 + (a-1)sin(na)

2a(1 - a)

(35)

(36)

and

N (a, X) = p(a) —

= p(a)

-(1-2a)/a

_(1-2a)/a

Yf =

A

Ru

= N (a, X) p (a)

2 2Y

(1-2a)/a

(37)

(38)

The subscript “f” designates the entities pertinent to the fractal geometry of the crack. We shall limit the considerations to the R << a case and consider rough cracks described by the fractality parameters such as fractal dimension D, fractal exponent a and the roughness measure H. These are related as defined by equations (3).

Since the limitations of the Wnuk-Yavari “embedded crack” representation of a fractal crack need to be accounted for, only the limited range of the fractal exponent will be considered, namely a will be contained within the interval

[0.5, 0.4] and it will not fall below 0.4. When this notation is applied, we can cast the results [4] in the following form

tip

4a

= K(a)—Y R = K(a)u *ip,

nE

Rf = N (a, X) R = 4tc1/(2“)-2 x

ar(a)

1/ a

(1-2a)/a

, ,, R. (39)

r(1/2 +a) -v ”

Upon inspection of the latter expression it is seen that before the length Rf can be determined (and before the profiles of a fractal crack can be sketched), a prior knowledge of the resistance curve Yf (X) is necessary. Therefore, we must at first establish the differential equation that defines Yf as a function of the nondimensional crack length X. Let us return to equation (23), which in view of the first expression in (39) has to be re-written as follows

K(a)4^ v

nE

dR

Rf (A) + A ^ - Rf (A) > da

1-

x ln

Rf (A) 2Rf (A)

1+ V1 -A Rf( A)

1 -^ 1 -A/ Rf(A)

• = u.

(40)

This expression reduces to the ordinary differential equation of the kind similar to (24). When the ductile behavior of the material (A << R) is considered the equation (40) reads

dRf

1 nE

u-1-1 - 1ln A I 2 2

da K(a) 4aY

77 = Mf - 2 - -2.ln(4pYf),

dX 2 2

4( R m/A)

Rini

(41)

Mf =

1 nE

M

K(a) 4aY 1 A I K(a)

The second expression in (37) defines the function R — or its nondimensional equivalent Yf — namely

(o rr^l(1-2“)/a

Rf = N (a, X) R = p(a) - —

Yf = N (a, X )Y = p(a)

R,

\(1-2a )/ a

(42)

y .

Substituting this into (40) yields

( 2 f2V 1(1-2 a )/a

d

dX

P(a)

Y

1 1

= Mf - — -—ln f 2 2

\(1-2a )/ a

4pp(a) -J—

Y

(43)

Carrying out the differentiation in the left hand side of this equation gives

_d_

dX

P(a) -J-tr

+

= p(a) Y— -J—

2 tiY T~2a)/a„,

, v - (44)

n\ X

V J

When this expression is substituted back into (43) and after some simple algebraic manipulations (see the Appendix) one obtains the desired governing differential equation

dY

dX

2a

Mf

—ln

2

(1-2 a)/a

4pp(a) -J—

p( a)

\(1-2 a )/a

-1

+ (1 - 2a)

X

(45)

The fractal tearing modulus Mf will be assumed to be somewhat higher (say by 20 %) than the minimum value of the modulus, at which the stable growth is still possible. The minimum value of the fractal tearing modulus Mfmn is readily established by setting the rate d Y/ dX equal zero at the point of fracture onset, X = X0 and Y = 1, and then evaluating the corresponding modulus. The result is

Mlin = 2 + |ln(4pN,(a, X0)) -1 -2a N0(a, X0)

2a

X„

(46)

N)(a, X0) = p(a)

\(1-2 a)/a

n\ X,

With the modulus M

ff-

min

solu-

assumed to be 1.2Mm tions of Eq. (45) are generated in form Y = Y(X, a), and they are shown in Fig. 11. Three curves shown were drawn for a equal 0.5 (smooth crack) and 0.45 and 0.40, which correspond to the rough cracks of increasing degree of surface roughness. Inspection of Fig. 11 leads to a conclusion that an increased roughness of the crack surface reduces the apparent material fracture toughness attained during the subcritical crack growth. Little circles on the Y-curves in Fig. 11 show the terminal instability points. The location of these points was evaluated by seeking maxima on the Q-curves shown in Fig. 12 or evaluating zeros in the graph representing the stability index (32) — this has been demonstrated in Fig. 13, b. Figure 13, a also shows an alterna-

Fig. 11. The resistance curves for stable cracks with the fractal geometry accounted for. The top curve corresponds to the case of smooth crack, while the lower curves pertain to the rough cracks with the fractal exponent a designated in the figure. Little circles denote the critical states (Yc, Xc)

tive way of determining the terminal instability points by comparing the rate of energy demand with the rate of energy supplied to the system.

Once the function Y(X, a) has been determined from (45), we can proceed to evaluate the profiles of the fractal crack within the pre-fracture zone. The equations used for these evaluations read

cusp _

4aYR

nE

N K(a) x

1 -i.

N

2N

-ln

N = N (a, X) =

1 + 71 -V N 1 -y/1 -V N

1 a

ar(a) r(1/2 + a)

(1-2a)/a

(47)

,tip

4aY R

nE

N (a, X )k (a).

Figure 14 shows the profiles of the cusp (47) normalized by the tip displacement rfip. It is now seen that for the enhanced roughness of crack surfaces (diminishing

Fig. 12. Loading parameter Q plotted for the rough cracks (two lower curves) and for a smooth crack (top curve) as a function of the current crack length. It is noted that an increasing roughness of the crack surface reduces the effects of stable crack extension

+

Fig. 13. (a) Rates of energy demand (dY/dX) and energy supply (Y/X) for rough cracks, a = 0.45 and a = 0.40; the curve drawn for a = 0.5 corresponds to a smooth crack. Initial crack length is set as X0 = 100, and the material ductility index is p = 10. (b) Stability index S sketched for the same input data as in (a). Little circles on both graphs indicate the terminal instability points

fractal exponent a) the prefracture zones diminish and the entire pre-fracture zones shrink. This phenomenon reflects on the earlier attainment of the critical state at the end of stable crack growth phase. To document this fact all three

Fig. 14. Profiles of the cusp of the cohesive fractal crack drawn according to (47). All curves have been normalized by the tip displacement vfip = N (a, X )K(a)(4aY/(nE)). An increased crack surface roughness (smaller values of the fractal exponent a) causes the cohesive zone to shrink, which is indicative of a more brittle material behavior

parameters: apparent fracture toughness Yc established from the available R-curves, critical nondimensional crack length Xc and the loading parameter at the terminal instability Qc, characterizing the critical state (terminal instability) have been grouped in sets (Yc,Xc,Qc) and collected in Table 2. The numbers shown in Table 2 were obtained for different initial inputs of the pertinent parameters characterizing a cracked body such as material ductility index p, the initial crack length X0 and the fractal exponent a.

5. Conclusions

It has been established that the unstable (catastrophic) fracture propagation in the Griffith sense is almost always preceded by slow stable crack extension that is associated with accumulation and redistribution of strains within the prefracture zone adjacent to the front of the propagating crack [1]. Solutions for advancing cracks significantly differ from those for stationary cracks. Exact solutions address only few loading configurations such as anti-plane mode of loading considered in [19-21]. By analogy with anti-plane case the crack advance under a tensile loading has been researched by Krafft et al. [22], who reformulated the problem and restated it in terms of a universal resistance curve. This view is supported by the studies at the microstructural level of ductile fracture occurring in metals and metallic alloys, where it was found that certain mechanisms exist that facilitate slow crack growth by a sequence of debonding of hard inclusions followed by the formation of voids and their growth and coalescence [12]. It is noteworthy that due to high strain levels and the redistribution associated with crack motion, the deformation theory of plasticity is not sufficient as a mathematical tool. Perhaps the path-dependent relations between stresses and strains, as those described by the incremental theory of plasticity of Prandtl and Reuss would be more appropriate to construct a theoretical model based on continuum mechanics.

With exception of Prandtl’s slip lines field suggested by Rice et al. [18], no theory has been proposed that would provide exact mathematical treatment of the problem at hand. Therefore, in this research we have employed an approximation based on the cohesive crack model equipped with the “unit step growth” or “fracture quantum” combined with an “embedded crack” model of Wnuk and Yavari [9, 17], which accounts for a non-Euclidean geometry of a crack represented by a certain fractal. The fractal dimension D for such a crack can vary between 1 (straight line) and 2 (a two-dimensional object). It has been shown that the present “structured cohesive crack model” yields the same essential result as that of Rice et al. [18], namely the governing equation which defines the universal R-curve. This statement is true for a smooth crack only. For fractal geometry there have been two papers published that address the slow crack advancement, namely [4, 5]. When

the stability of the fractal cracks is reconsidered in the con- Perhaps the best way to explain the essential conclu-

text of the present model, the pertinent results somewhat sions of this paper is to take a look at Figs. 15 and 16, and

diverge from the previous findings. Specifically, the rela- also — to examine the summary of the results collected in

tion between the extent of the slow cracking and the fractal Table 2. From all the pertinent parameters used in the theo-

exponent a is opposite to what was suggested earlier. retical considerations we choose just one: the increase in

Table 2

Characteristic parameters of the critical states resulting for various input data (p, a, X0)

p a X0

3 10 20 60 100 200

2 0.50 Yc Unstable 1.462 1.619 1.757 1.791 1.820

Xc 12.407 24.153 67.358 108.998 211.334

Qc 0.486 0.366 0.228 0.181 0.131

AQ 8.569 15.785 25.109 28.200 31.228

0.45 Yc 1.410 1.515 1.569 1.570 1.559

Xc 12.096 23.328 65.092 105.801 206.626

Qc 0.483 0.360 0.220 0.172 0.123

AQ 7.958 13.972 20.255 21.797 22.851

0.40 Yc 1.348 1.399 1.383 1.360 1.326

Xc 11.734 22.453 63.090 103.207 203.228

Qc 0.479 0.353 0.209 0.162 0.114

AQ 7.912 11.637 14.676 14.805 14.223

4 0.50 Yc 1.088 1.648 1.835 2.006 2.049 2.085

Xc 3.249 12.945 24.851 68.420 110.269 212.919

Qc 0.818 0.505 0.384 0.242 0.193 0.140

AQ 0.205 12.848 21.513 32.616 36.315 39.957

0.45 Yc 1.059 1.580 1.704 1.773 1.775 1.765

Xc 3.170 12.575 23.911 65.880 106.684 207.629

Qc 0.817 0.501 0.378 0.232 0.182 0.130

AQ 0.051 12.102 19.401 27.066 29.006 30.401

0.40 Yc 1.037 1.501 1.560 1.545 1.52 1.481

Xc 3.110 12.143 22.912 63.619 103.754 203.782

Qc 0.817 0.497 0.369 0.220 0.171 0.121

AQ 0.0 11.184 16.705 20.697 21.034 20.579

8 0.50 Yc 1.123 1.854 2.076 2.287 2.342 2.389

Xc 3.591 13.447 25.530 69.503 111.587 214.590

Qc 0.828 0.525 0.403 0.257 0.205 0.149

AQ 1.403 17.422 27.531 40.516 44.885 49.218

0.45 Yc 1.197 1.768 1.915 2.002 2.008 1.998

Xc 3.514 13.020 24.478 66.682 107.597 208.681

Qc 0.825 0.521 0.396 0.245 0.193 0.138

AQ 1.095 16.517 25.095 34.227 36.607 38.391

0.40 Yc 1.172 1.668 1.739 1.725 1.698 1.655

Xc 3.454 12.521 23.357 64.153 104.312 204.352

Qc 0.824 0.516 0.386 0.232 0.180 0.127

AQ 0.903 15.419 22.012 27.020 27.587 27.280

Table 2 continuation

p a X0

3 10 20 60 100 200

10 0.50 Yc 1.278 1.925 2.160 2.386 2.445 2.496

Xc 3.692 13.604 25.748 69.859 112.023 215.150

Qc 0.832 0.532 0.410 0.261 0.209 0.152

AQ 1.908 18.946 29.524 43.142 47.741 52.313

0.45 Yc 1.243 1.832 1.988 2.082 2.089 2.080

Xc 3.615 13.159 24.660 66.946 107.900 209.036

Qc 0.829 0.528 0.402 0.249 0.197 0.141

AQ 1.549 17.984 26.978 36.606 39.139 41.059

0.40 Yc 1.216 1.725 1.800 1.787 1.760 1.715

Xc 3.554 12.640 23.499 64.328 104.497 204.542

Qc 0.827 0.522 0.391 0.236 0.184 0.130

AQ 1.325 16.823 23.766 29.119 29.766 29.513

100 0.50 Yc 1.827 2.800 3.218 3.666 3.794 3.908

Xc 4.590 15.194 28.060 73.871 117.052 221.752

Qc 0.892 0.607 0.479 0.315 0.2 5 0.188

AQ 9.929 35.755 51.460 72.553 80.040 87.743

0.45 Yc 1.762 2.620 2.901 3.107 3.135 3.135

Xc 4.495 14.575 26.610 69.955 111.430 213.241

Qc 0.886 0.600 0.467 0.298 0.237 0.171

AQ 8.452 34.085 47.671 63.243 67.736 71.487

0.40 Yc 1.708 2.419 2.561 2.575 2.540 2.479

Xc 4.402 13.852 25.044 66.349 106.665 206.819

Qc 0.881 0.591 0.452 0.279 0.218 0.155

AQ 7.887 32.162 43.016 52.600 54.326 54.841

200 0.50 Yc 2.017 3.122 3.617 4.163 4.324 4.468

Xc 4.832 15.678 28.796 75.224 118.784 224.082

Qc 0.914 0.631 0.501 0.333 0.270 0.200

AQ 11.921 41.123 58.501 82.223 90.788 99.705

0.45 Yc 1.940 2.909 3.243 3.500 3.540 3.546

Xc 4.728 15.010 27.240 70.984 112.665 214.748

Qc 0.906 0.623 0.488 0.314 0.251 0.182

AQ 10.964 39.204 54.304 72.008 77.253 81.739

0.40 Yc 1.874 2.671 2.843 2.873 2.837 2.770

Xc 4.623 14.229 25.551 67.051 107.435 207.646

Qc 0.900 0.613 0.472 0.293 0.230 0.163

AQ 10.274 37.019 42.186 60.329 62.488 63.334

the nondimensional loading parameter AQ due to the slow cracking process that precedes the critical point of terminal instability. This quantity is represented on the vertical axes in Figs. 15 and 16. It is seen the material ductility p significantly enhances the slow stable crack growth, leading to an increase in the applied load measured as a difference between the load at the point of catastrophic fracture

Qc and the load at the onset of stable crack growth Qini: AQ = , Qini =. E. (48)

Qini V X 0

This observation is entirely in agreement with the previous researches [4, 5]. However, accounting for the fractal geometry leads to an opposite conclusion: higher rough-

Fig. 15. Enhancement of the load measured at the terminal instability, see (48), shown as a function of material ductility. Case denoted by a = 0.5 corresponds to a smooth crack, while the other curves describe the rough cracks represented by the fractal geometry. The initial crack length for all three curves is X0 = 100

ness of crack surface (a less than 0.5) reduces the slow crack growth and results in a decrease of the observed load increase AQ, see Fig. 16. In other words, the roughness of the crack surface is conducive to a more brittle material response. Closer examination of the data gathered in Table 2 reveals an interesting phenomenon. It indicates that for a very small crack, X0 = 3, and low material ductility, p = 2, the stable growth does not exist at all. We note that in this particular case the initial crack is of the size on the order of magnitude of the characteristic length Rni. For such a small crack a new effect of “over-stressing” comes to light, similar to the phenomenon known in physics of fluids as “super-cooled” liquid. The effect can be explained as follows: despite the sufficient energy accumulated within the immediate surrounding of the small crack, the crack does not begin to propagate until a certain hypercritical load level is reached. What happens then is a sudden transition from a stationary to a dynamically propagating crack, compare [23, 24].

Whether or not the phenomenon of stable crack extension may exist strongly depends on material ductility and tearing modulus of Paris, proportional to the initial slope

Fig. 16. Load increase attained during the subcritical crack extension at X0 = 100 and various levels of the ductility index shown as a function of the increasing crack surface roughness. It is seen that an increase in roughness causes a reduction of AQ

of the R-curve; and to a much lesser degree on the level of crack surface roughness. If the tearing modulus in the governing equation (26) for a smooth crack and (45) for a rough crack does not meet the condition of being greater than the minimum modulus calculated in Sects. 3 and 4, i.e.,

nE

4a.

> M„

(49)

for a smooth crack, and

1 nE

f 1

>M ■ =- +

— mm 2

k(a) 4aY

+ jln (4pN0(a, X0))-^ Nq( Xq) ,

2 2^X 0

(50)

N0(a, Xq) = p (a)

(1-2 a )/ a

2 _2_

W Xo

V /

for a rough crack, the stable crack growth will vanish.

It is seen that while the tearing modulus for a smooth crack (49) depends only on the material property such as the ductility index p, the tearing modulus for a fractal crack, as given by (50) depends also on the purely geometrical parameters such as the measure of the crack surface roughness a and the initial crack length X0. It is noteworthy that in the case when the conditions stated by the inequalities (49) and (50) are not met; slow stable crack growth does not exist. Indeed, for a certain combination of the input parameters such as the material ductility, initial crack length and the fractal exponent, it can be shown that the transition period of slow stable crack growth is missing and one returns to the rules valid for ideally brittle fracture. This may be compared with the experimental data [25]. Further research of this type is needed to fully understand the physics behind the present model.

Appendix A

Let us recall equation (43)

d

dX

p(a)

2 i2Y 'ii_2“ ,,a

1 1 1

= Mf----------------ln

f 2 2

4pp (a)

\(l -2 a )/ a

(A1)

When the product of (2/n)(1-2a^a and p(a) is denoted by f (a), then the left hand side (LHS) of (A1) can be written as

d

f (a)

dX

X

Now we do the differentiation

LHS = f (a) 1

2Y

N|I-2 a )/|2 a)

X

+

2a

1

2Y

X

v j

V(2«)-2

dY

------+

dX

X dY/ dX - Y

X2

(A2)

hence

LHS = f (a)

2Y

X

,1/(2a)-1

i+-1 - il dL

2a j dX

- '< - 2IX ft* - “

(A3)

With G denoting the right hand side of (A1), we have

2Y f2“-2 ( 1 dY | „ 4

f(a)l * 1 [^dX|-f(a)x

X

I - 2

dY

a X This reduces to 2aG

Y I2 ( 2Y Y/(2a)-2 G( )

* I = G(X, Y, a).

(A4)

*=*+2aia- 2T-XI #■ (A5)

dX f (a) (2 Y/X) , ...

With (2/n)(1-2a^ap(a) = f (a) this equation becomes identical with (45).

Appendix B

Fracture in an ideally brittle solid (and for the fractal exponent a = 1/2) occurs when the ductility index p = = R/ A ^ 1. It would be worthwhile to prove that in this case the differential equation governing motion of the sub-critical crack (23) predicts no stable crack growth, and that the 8 (COD) criterion reduces then to the classic case of Griffith. In order to prove this point let us write the governing equation derived from the 8 (COD) criterion, equation (23), in this form

dR R R ( A ^ nE (u ^

— = M---+—Fl — I, M =----------------1 — 1, (B1)

da A A [ R / 4aY [a/

where M is the tearing modulus and the function F is defined as follows

1 + ,/1 -A/ R

f (Д/ R) = v 1 -Д/ R --

-ln-

^ ---------(B2)

2R 1 1 -A/ R

For ductile solids A is much smaller than R, and thus p>> 1. Under this condition the function F reduces as follows

F

, Д Д , ( Д

= 1------------1-------lnl —

R jp>>i 2R 2R 14R

(B3)

This form leads to the differential equation (24) considered in the last section. To obtain the ideally brittle limit we need to expand the function F into a power series for p approaching one. The results is

яД1,=- I (i - R

3/2

(B4)

When this is substituted into (B1) one obtains the differential equation governing an R-curve for quasi-brittle solids, namely

dR = _ R - 2 R (1-Д

da Д 3 ДI R

3j2

(B5)

For the ideally brittle solid two things happen, first, we have R = A and, second, the slope of the R-curve defined by (B5) equals zero (the R-curve reduces now to a horizontal line drawn at the level R = R;ni). Therefore, equation (B5) reduces to

dR = 0 or M = 1.

da

We also know that for an ideally brittle solid the size of the Neuber particle A can be identified with the length of the cohesive zone

nE CTOD (B7)

(B6)

Д = R =-

-CTOD.

The final stretch u is now equal half of CTOD, namely

й = ^CTOD. 2

(B8)

When (B7) and (B8) are substituted into the definition of the tearing modulus shown in (B1), one gets

M =

nE

4ctv

nE

(

(1 2)CTOD

|

, = 1. (B9)

(nE/ (8aY))CTOD

Thus we have confirmed that the requirement of zero slope of the R-curve in the limiting case of an ideally brittle solid, expressed by (B6), is satisfied when U = (1/2)CTOD and A = R. In other words, the 8 (COD) criterion for the onset of fracture reduces to the CTOD criterion of Wells, or, equivalently, to the /-integral criterion of Rice. The latter is in full accord with the Irwin driving force criterion G = Gc, and this yields the result identical to the ubiquitous Griffith expression for the critical stress

2Ey

G„ E Kc

-I— (B10)

na \ na -y na

Similar conclusion may drawn directly from the fact that the R-curve is given as a horizontal line (of zero slope) drawn at the level of Rjni. Setting the equilibrium length of the cohesive zone Rini equal to its critical value Rc leads to (B10), as expected. To complete this consideration we are reminded that the quantities R and Kl are related as follows

R =П

= Rc 4

(к I2

or,

(B11)

KI = Kc or> acr = aG

In this way we have demonstrated that the nonlinear theory described in the preceding sections encompasses the classic theory of fracture, which becomes now a special case of a more general mathematical representation.

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Поступила в редакцию 13.03.2013 г.

CeedeHun 06 aem0pax

Wnuk Michael Peter, Prof., University of Wisconsin-Milwaukee, [email protected]

Alavi Manouchehr, Ph.D., P.E., Dr., Technical Director, Finele Consulting Engineers, Inc., [email protected] Rouzbehani Anousheh, Ph.D. Candidate, Int. P.E., CEng, University of Wisconsin-Milwaukee, [email protected]