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The theory of non-steady state fracture kinetics analysis—1: General theory of crack propagation
EngineerirgFrmrwe MechonicrVol. 13, pp. ?Sl-758 Pergamon Press Ltd., MO. Printed in Great Britain
THE THEORY OF NON-STEADY STATE FRACTURE KINETICS ANALYSIS-l: GENERAL THEORY OF CRACK PROPAGATION A. S. KRAUSZ Department of Mechanical Engineering, University of Ottawa, Ottawa, Ontario, Canada, KIN 6N5
IL KKAUSZ 300 Driveway, Ottawa, Ontario, Canada, KIN 6N5 Abstract-The kinetics theory of thermally activated time dependent crack propagation is extended to describe the crack size dist~but~on in non-steady state. The dis~but~on is represented by a series of n differential equations, each expressing the rate of crack tip concentration change over the system of n consecutive energy barriers. The general solution for the set of homogeneous linear first order differential equations developed in this report is of the form pi = &c,,~e -“f + Ci, where pi is the crack tip concentration in the ith valley. The theory takes into consideration the discrete character of solids in contrast with the usual continuum models of fracture studies. The analysis is readily applicable to Regions I and II of stress corrosion cracking.
IT ISWELL recognized that thermally activated crack propagation is the result of bond breaking and healing processes [l-3], and that the breaking and healing steps are controlled by a system of consecutive energy barriers [4-71. Consider crack tips in the valley in front of the ith barrier. When a single bond breakmg activation takes place they move over the ith barrier into the next valley. Because bond healing activations also have to be considered, the net flow of crack tips per unit time over the energy barrier can be expressed as (Fig. 1): Net flow per unit time = the rate of crack tip growth by bond breaking, moving from the ith valley over the ith barrier into the (i + 1)th valley-the rate of crack tip shrinkage by bond healing, moving from the (i + l)th valley over the ith barrier into the ith valley that is, Fi = Fbreaking - Kaling = P&b -Pi+l4+*.h i-l
r i-l
I
/
t
iti
Fig. 1. Schematic representation of the consecutive energy barrier system associated with thermally activated crack propagation. A bond breaking activation moves the crack tip from the ith valley into the (it l)th valley. 751
752
4. S. KRAUSZ and K. KRAUSZ
where i is a counter from 1 to n, p is the number of crack tips, and b and h denote the breaking and healing steps, respectively. The rate constants k,, and kh are given by the absolute rate theory as [8,9]
In eqns (1) k is the Boltzmann constant, T is the absolute temperature, h is Planck’s constant, AGi is the activation free energy and W is the work contributed to the activation process by the stress (T. The net flow over each of the n consecutive barriers is expressed by a system of equations as [7,10,11] F, =pI
,k’-p?‘k?
&_, = pi_, i_,k’-’
-p, “k,
Figure 2 shows schematically the i - 1, i, i + I segment of the consecutive barrier system, making clear the meaning of the directional notations of the subscripts and superscripts for the breaking and healing activations. In steady state, by definition, the concentration of the crack tips in the ith valley remains constant with time. Consequently, the flow into the valley is equal to the flow out of the valley, that is, the net flow over each of the barriers is the same: F, =
. = F;-, = 1;1= F],, = . . = F,, = F.
(3)
The descriptions, as given by eqns (2) and (3) can also be represented for the ith valley as flow in-flow out=@,
I,.,k’~‘+p,+,k,+,)-pi(‘~‘ki+ik’)=O
and flow in-flow out=E_,-F,=O. Fl-I
Fig. 2. The i - 1, i, i + 1 and i + 2 segment of the consecutive energy barrier system, showing the sense of direction for the breaking and healing subscripts and superscripts of the rate constants in eqns (2).
The theory of non-steady state fracture kinetics analysis
153
Because F,-1- F;:= (pi_* i_lk’-’ -pi ‘-‘ki) - (pi ik’ - pi+, ‘ki+l), it can be seen that the two statements are equivalent. Fracture kinetics theories are usually carried out for steady-state crack propagation[4-7, 10-201, but in many experimental conditions and engineering applications crack propagation occurs in non-steady state: it is the purpose of this paper to present the kinetics analysis of nonsteady state crack propagation. The development of the general method in Part 1 will be followed by an application to stress corrosion cracking in Part 2. The development and the symbolism of the general theory may cause some difficulties at first, but the parallel reading of the simpler applied analysis of Part 2 will be of advantage. THE DEFINITION OF THE PHYSICAL CONDITIONS Consider the propagation of a crack tip located initially at X1 in a plate of finite length L (Fig. 3a). Because thermally activated crack propagation is the net result of bond breaking and healing steps (forward and backward activations, respectively), the crack tip “pulsates” as it propagates. These temporary fluctuations can be of any size, each with a well-defined probability [21]. The crack, however, cannot shrink below the initial size X1, where the free energy field goes to infinity presenting a non-scalable reflecting boundary (Fig. 3b). Generally, the plate is finite and when the crack tip reaches X = L (that is, i = n), backward activation is not possible. The nth barrier presents an infinitely large backward activation free energy and the barrier system is terminated by an absorbing boundary (Fig. 3b). The analysis can be easily modified to consider an infinitely long plate. In non-steady state, the flux is different over each barrier and pi is a function of time as well as location. The process then has to be described by a system of the rate equations, each expressing the rate of concentration change of the mobile cracks pi at Xi as dpi _- flow per unit time into valley i -flow dt
per unit time out of valley
i + rate of crack nucleation (RI) -rate of crack immobilization (Ry),
and the system of rate equations for N barriers is dp, _ dt---p~
,k’+pz’k2+R;--R;
dpz& -PI rk’-pdkz+zkZ)+ps2k3+R;-R;
dpi_ dt - pi-1 i_lki-‘-pi(‘-‘ki din _
dt
-
pn-1 .-,k”-’
t iki)+pi+l iki+l + R:-R;
- p.(“-‘k,, t .k”).
Equations (4) are the expressions of the crack conservation law, similar to other transport processes. Often, two or more of the crack tip locations have identical free energy surroundings, that is, the rate constants for these locations are identical. The corresponding differential equations then can be combined into a single one. For example, if
!g= p/-l
,_,k’-’ - pPkr
dprn_ yg- pm-,m-,km-’
+ rk’) + pi+, ‘k,+,;
- pnSm-‘km + mk”) + pm+, “k,,,+,;
A. S. KRAUSZ
754
and K. KRAUSZ
4
(a)
ib)
Fig. 3. A pre-cracked plate of length S = L with the initial crack tip at X, (a); the energy barrier system associated with thermally activated crack propagation. indicating the reflecting and the absorbing boundaries fhl.
with
[k’ = ,k” = .k”; and ‘klil = mkm+,= “kp+,, the combination gives
and
dpv_ P*-~ ._-,k”-” -pt,(“-fkv+,.k’)+Pv;I dt-
Ykg,+l.
In these expressions the nucleation and immobilization rates were considered to be negligiblea realistic and convenient condition for the illustration of the combination concept. The symbol Y signifies that the combined equation provides only the combined 1 and m, but will not determine them as separate entities. THE DEVELOPMENT OF CRACK-SIZE DISTRIBUTION RELATIONS
To solve the system of simultaneous differential eqns (4), two conditions are imposed: (1) the rate constants do not change in time, that is, at constant stress and temperature k# f(t); (2) cracks are neither nucleated nor immobilized, that is, R: = 0 and R:! = 0. It can be shown that the general solution of a system of homogeneous linear first order differential equations with constant coefficients, giving the number of cracks pi of size Xi, is
pi2
Ci,j
exp ( - A$)+ Ci9
The theory of non-steady state fracture kinetics analysis
755
where C, is an integration constant. Substitution of eqn (5) into eqns (4) leads to a system of algebraic equations of the following type for i# 1 and if n: -
T
eXP ( -
Ci,jAj
Af)
=
[T
Ci-1.j exP
-
+
[T
AjOli_,ki-l
(-
Ci,j exp ( - Ajt)](i-lki
C C i
Ci+l,j exp(-A+)
+ Ci_l i_lk’-’ -
+ ik’)
1ki+l
(6)
i
Ci(‘-‘ki+ i/C’)+ Ci+l iki+l.
In the initial position at i = 1, the number of cracks changes at the rate of
-2 c,,jAjexp(-Air)=
-[C
c,,jexp(-A+)],k, J
i
+
Ci C2.jexp ( -
-
c, ,k’ + c, ‘kg
A$) lk2 I
(7)
and at i = n, that is, in front of the last barrier before the plate fractures, the number of crack tips changes per unit time as -2
i
- [T
c,jAjexP(-A#)=
[x c”-,,jexp(-Ajl)]“_,k”-, i
Cn,jexp ( - Ajt)](‘-lk,
+ C”_, ._Jc-
+ A”)
(8)
- c,(“-‘k” + .k”).
Each of eqns (6)-(8) must be separately satisfied by both the time dependent and time independent terms. For the time dependent terms it follows that
T[cl,j(Aj- ok’)+ C2,j1k21 = 0, T
[Ci,j(Aj-
'-'ki t ik') t Ci_~,ji_,k'-' t Ci+l,j ‘ki+l] = 0,
Z$[Cn,j(Aj - R-lk~-
(9)
n/C”)+ C,_*,j“-l/C-‘1 = 0;
and for the time independent terms the following expressions apply - c, ,k’ t c, ‘k, = 0, Ci_, i-*/C'-' - Ci('-'ki C”_, ,_#-*
t i/C’)+ Ci+l ‘ki+l= 0,
(10)
- c”(“-‘k” + “k”) = 0.
The set of algebraic equations given by eqns (9) has non-trivial solutions if the determinant of the coefficients, that are formed by the rate constants and A, is zero. Then, the algebraic
7%
A.S.KRAUSZandK.KRAUSZ
equations are expressed in explicit form as (hj - ]k’)Cl,,
+ ‘k>C*.j =
lk’C1.j + (hj -
0
‘k* - zk2fC2,j + *k3Cx,j =
0
(II.4
zk2C2,, +(hi-"kl-,k')c,,, +'k,cd,, =O ,,:,kn-'~i l,, +(Aj'"'k,-,k")c,j=iO.
The characteristic equation D = 0 of this system of algebraic equations is Aj- ,k’ ,k’ 0 0 :.....
0
‘kl hi- 'k-->k=
'k, Aj-*kj-3k3
2k2 0
3k' ,z_lk"-i
0
0
0
0
0 0 4k,
'k4 Ai-3k,-cjk4 Aj-*-'k~-nk"
Lzz
0
O
The expansion of the determinant results in an algebraic equation of the nth degree in A and defines n roots. It can be shown [22] that one root must be zero and all others positive to have physical meaning. Substitution of the Aj values into the system of algebraic equations (11) defines the ci.j ratios only. To obtain unique values, first the integration constants Cj of eqn (5) have to be determined from the boundary conditions [23]. In the initial state when t = 0, the number of crack tips pi”in the ith valley is expressed as
and hence the integration constant is
By using the ith terms of eqns (9) and (lo), unique values of the Ci,j constants can be determined and the boundary conditions are also satisfied. When all the roots are distinct the crack size distribution is expressed by eqn (5). When the expansion of the determinant results in multiple roots, then the crack-size distribution expression [24] is described by a polynomial form of time. For example, if Al= A2 = A?,then the number of crack tips at the ith valley is [23] pi = (Ci.1 + Ci,lt f $
2
Ci,j
Ci,3t2) exp (I Art)
exp ( - hit) + Ci*
j=4
It was discussed in a previous study [21] that crack propagation is a random walk prticess, controlled essentially by energy transport. The phenomenon of crack propagation is an integral part of the transport processes: conductive, convective, and radiation heat transfer; mass transfer by diffusion or flow; the flow of electric current; chemical reactions. The unity of heat and mass transfer is widely recognized [25-281. The phenomena of electric current, chemical reactions, heat and mass transfer as well as other processes were developed into a single system by Bosworth [29]. The interrelation among the above processes has led to the recognition that characteristic values (such as viscosity, diffusion coefficients, thermal diffusivity, etc.) are also interrelated [4,25-28,301, resulting in the welI-known dimensionless numbers of Prandtl, Smith, etc. The first indication that fracture, and plastic flow as well, are integral parts
Thetheoryof non-steadystatefracturekineticsanalysis
757
of the unified system came from Eyring [31] and Tobolsky and Eyring [ll through their realization that both are thermally activated processes. Rate equations for steady and nonsteady states were developed for plastic flow by Krausz and Faucher [32-341, and for fracture by Krausz [35,36]. Their analyses led to differential equations identical with Fick’s laws of diffusion, and to Fourier’s equations of heat conduction. The identity is not coincidental, but the consequence of the fact that the same principles control heat and mass flow, plastic deformation and crack propagation. These principles are rooted in the molecular theory of statistical mechanics for non-equilibrium systems. The kinetics analysis studies, referred to in the foregoing, follow from the statistical mechanics approach to fracture and plastic deformations processes. The interrelations were also discussed by Krausz and Eyring[9]. Fracture studies, similarly to the usual transport process analyses, are carried out for continuum models that result in the partial differential equations of transport phenomena. An alternative approach is to consider the medium as a discrete system (as all materials are, being composed of particles), and to apply random walk theory in the investigations [37]. The corresponding mathematical development is more involved than the differential equation formulation of the continuum model. It was shown, however, in a recent study by Krausz [21] that the analysis of crack propagation through random walk theory illuminates the operation of the actual physical processes. The present investigation aims to introduce a method of crack propagation study that also considers the discrete character of solids and may be thought of as a variation of the random walk analysis. Mathematically it is expected to be of similar convenience as the differential equation formulation, while at the same time preserving the physical reality of discreteness and the clarity that results from representing the actual physical process. The method is closely related to the kinetics analysis of consecutive chemical reactions [22,23]-one of the transport processes. The theory developed in Part 1 will be illustrated in Part 2 through an application to a relatively simple system of two barriers in series that describes the rate controlling processes of stress corrosion cracking. For an alternative method of crack propagation study the use of Laplace transform is currently under development.
Acknowledgement-The financialassistanceprovidedto one of the authors(A. S. K.) by the Engineering Research Council Canada is gratefully acknowledged.
NaturalSciences and
REFERENCES [I] A. TobolskyandH. Eyring, Mechanical properties of polymeric materials. J. Chem. Phys. 11, 125-134(1943). [2] S. N. Zhurkov, Kinetic concept of the strength of solids. ht. L Fracture Mech. 1, 311-316(1965). [3] M. L. Williams, Initiation and growth of viscoelastic fracture. Int. J. Fracture Mech. 1,292-310 (1%5). [4] R. Thomson, C. Hsieh and V. Rana, Lattice trapping of fracture cracks. J. Appl. Phys. 42,31563160 (1971). [5] B. R. Lawn and T. R. Wilshaw, Fracture of Brittle Solids. Cambridge University Press (1975). [6] S. D. Brown, The multibarrier kinetics of subcritical brittle crack growth, in Proc. 6th Canadian Congr. Appl. Mech., Vancouver, B.C. (1977). [7] A. S. Krausz, Time dependent crack propagation in lhearelastic solids. 1. Appl. Phys. 49,3774-3778(1978). [8] S. Glasstone, K. J. Laidler and H. Eyring, The Theory of Rate Processes. McGraw-Hill, New York (1941). [9] A. S. Krausz and H. Eyring, Deformation Kinetics. Wiley-Interscience, New York (1975). [IO] A. S. Krausz, The deformation and fracture kinetics of stress corrosion cracking. Int. J. Fracture 14,5-15 (1978). [I I] A. S. Krausz, The theory of thermally activated processes in brittle stress corrosion cracking. Engng Fracture Mech. 11, 33-42 (1979). [12] H. M. Cekiige, W. R. Tyson and A. S. Krausz, Static corrosion and static fatigue of glass. J. Am. Cer. Sot. 58,265-266
(1976). [I31 S. D. Brown, Subcritical crack growth: A treatment based on multibarrier kinetics and an electrical network analog. In Environmental Effects on Engineering Materials (Ed. M. R. Louthan and R. P. McNitt), Virginia Polytechnic Institute, Blacksburg, Va. (1977). [ 141S. D. Brown, A multibarrier rate process approach to subcritical crack growth. In Fracture Mechanics of Ceramics (Ed. R. C. Bradt, D. P. H. Hasselman and F. F. Lange). Plenum Press, New York (1978). [IS] C. Hsieh and R. Thomson, Lattice theory of fracture and crack creep. J. Appl. Phys. 44,2051-2063(1973). [16] B. R. Lawn, An atomistic model of kinetic crack growth in brittle solids. I. Mater. Sci. 10,469-4&l (1975). [17] J.-C. Pallet and S. J. Bums, Thermally activated crack propagation theory. Int. J. Fracture l&667-679 (1977). [I81 J.-C. Pollet and S. J. Bums, An analysis of slow crack propagation data in PMMA and brittle materials. Int. J. Fracture 13,775-786 (1977). 1191S. M. Wiederhorn, Influence of water vapor on crack propagation in soda-lime glass. J. Am. Cer. Sot. 50, 407-414 (1%7).
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A. S. KRAUSZ and K. KRAUSZ
[20] S. M. Wiederhorn, Mechanisms of subcritical crack growth in glass. In Fracture Mechanics of Ceramics (Eds. R. C Brad&D. P. H. Hasselman and F. F. Lange). Plenum Press. New York (1978). [21] A. S. Krausz, The random walk theory of crack propagation. Engng Fracture Mech. 12,499_Ctf4 ( 19791. 12213. J. Zwolinski and H. Eyring, The non-equilibrium theory of absolute rates of reaction. L .Qn. (‘hem. Sue, 69. 2702-2707(1947). [23] S. W. Renson, The Foundations of Chemical Kinetics. McGraw-Hill, New York (1%0). [24] I. S. Sikolnikoff and R. M. Redheffer, Mathematics of Physics and Modem Engineering. McGraw-Hill, New York (1958). 1251W. M. Rcl:senow and H. Y. Choi. Heat and Muss Transfer. Prentice Hall, New York (I%1 i. 1261E. R. G. Eckert and R. M. Drake, Jr., Analysis of Heat and Mass Tru~sfer. McG~~w~Hiil,New York 11972). [27] J. Szekely and N. J. Themelis, Rate P~enome~a in Process ~efulla~y. Wiley-Inters~ience. New York (197t). [28] D. A. Frank-Kamenetskii, Diflusion and Heat Transfer in Chemical Kinetics. Plenum Press. New York (1969). [29] R. C. L. Bosworth, 7’runspoti Processes in Applied Chemistry. Wiley, New York (1956). [30] H. Eyring, D. Henderson, B. J. Stover and E. M. Eyring, Statisfical Mechanics and Dynumics. Wiley. New York (1964). [3l] H. Eyring, Viscosity, plasticity, and diffusion as examples of absolute reaction rates. .I. C’hrm.Phys. 4, 283-290(19633. 1321B. Faucher and A. S. Krausz, Consecutive energy _. barriers and repeated stress relaxation. Scripta Met. 12 913-918 (1978). 1331B. Faucher and A. S. Krausz, Non-steady state mechanism of yield. Scripta Met 12, 175-17911978). 1341A. S. Krausz and B. Faucher, The steady-state kinetics of double-kink spreading. Scripta Met. 13.91-94 ( 1979). 1351A. S. Krausz. The theory of non-steady state fracture propagation rate. Inf. J. Fracture 12. 239-242(1976). i36j A. S. Krausz, Crack-size distribution in homogeneous solids. Int. .I. Fracture 15, 337-342 (1979) [37] S. Chandrasekhar, Stochastic problems in physics and astronomy. In Selected Papers on Noise ~md Stctchastit Professes (Ed, N. Wax), Dover, New York (1954). (Received 12 December 1979;receiued fur pu~ijcaii~n I9 March 1980)
THE THEORY OF NON-STEADY STATE FRACTURE KINETICS ANALYSIS-l: GENERAL THEORY OF CRACK PROPAGATION A. S. KRAUSZ Department of Mechanical Engineering, University of Ottawa, Ottawa, Ontario, Canada, KIN 6N5
IL KKAUSZ 300 Driveway, Ottawa, Ontario, Canada, KIN 6N5 Abstract-The kinetics theory of thermally activated time dependent crack propagation is extended to describe the crack size dist~but~on in non-steady state. The dis~but~on is represented by a series of n differential equations, each expressing the rate of crack tip concentration change over the system of n consecutive energy barriers. The general solution for the set of homogeneous linear first order differential equations developed in this report is of the form pi = &c,,~e -“f + Ci, where pi is the crack tip concentration in the ith valley. The theory takes into consideration the discrete character of solids in contrast with the usual continuum models of fracture studies. The analysis is readily applicable to Regions I and II of stress corrosion cracking.
IT ISWELL recognized that thermally activated crack propagation is the result of bond breaking and healing processes [l-3], and that the breaking and healing steps are controlled by a system of consecutive energy barriers [4-71. Consider crack tips in the valley in front of the ith barrier. When a single bond breakmg activation takes place they move over the ith barrier into the next valley. Because bond healing activations also have to be considered, the net flow of crack tips per unit time over the energy barrier can be expressed as (Fig. 1): Net flow per unit time = the rate of crack tip growth by bond breaking, moving from the ith valley over the ith barrier into the (i + 1)th valley-the rate of crack tip shrinkage by bond healing, moving from the (i + l)th valley over the ith barrier into the ith valley that is, Fi = Fbreaking - Kaling = P&b -Pi+l4+*.h i-l
r i-l
I
/
t
iti
Fig. 1. Schematic representation of the consecutive energy barrier system associated with thermally activated crack propagation. A bond breaking activation moves the crack tip from the ith valley into the (it l)th valley. 751
752
4. S. KRAUSZ and K. KRAUSZ
where i is a counter from 1 to n, p is the number of crack tips, and b and h denote the breaking and healing steps, respectively. The rate constants k,, and kh are given by the absolute rate theory as [8,9]
In eqns (1) k is the Boltzmann constant, T is the absolute temperature, h is Planck’s constant, AGi is the activation free energy and W is the work contributed to the activation process by the stress (T. The net flow over each of the n consecutive barriers is expressed by a system of equations as [7,10,11] F, =pI
,k’-p?‘k?
&_, = pi_, i_,k’-’
-p, “k,
Figure 2 shows schematically the i - 1, i, i + I segment of the consecutive barrier system, making clear the meaning of the directional notations of the subscripts and superscripts for the breaking and healing activations. In steady state, by definition, the concentration of the crack tips in the ith valley remains constant with time. Consequently, the flow into the valley is equal to the flow out of the valley, that is, the net flow over each of the barriers is the same: F, =
. = F;-, = 1;1= F],, = . . = F,, = F.
(3)
The descriptions, as given by eqns (2) and (3) can also be represented for the ith valley as flow in-flow out=@,
I,.,k’~‘+p,+,k,+,)-pi(‘~‘ki+ik’)=O
and flow in-flow out=E_,-F,=O. Fl-I
Fig. 2. The i - 1, i, i + 1 and i + 2 segment of the consecutive energy barrier system, showing the sense of direction for the breaking and healing subscripts and superscripts of the rate constants in eqns (2).
The theory of non-steady state fracture kinetics analysis
153
Because F,-1- F;:= (pi_* i_lk’-’ -pi ‘-‘ki) - (pi ik’ - pi+, ‘ki+l), it can be seen that the two statements are equivalent. Fracture kinetics theories are usually carried out for steady-state crack propagation[4-7, 10-201, but in many experimental conditions and engineering applications crack propagation occurs in non-steady state: it is the purpose of this paper to present the kinetics analysis of nonsteady state crack propagation. The development of the general method in Part 1 will be followed by an application to stress corrosion cracking in Part 2. The development and the symbolism of the general theory may cause some difficulties at first, but the parallel reading of the simpler applied analysis of Part 2 will be of advantage. THE DEFINITION OF THE PHYSICAL CONDITIONS Consider the propagation of a crack tip located initially at X1 in a plate of finite length L (Fig. 3a). Because thermally activated crack propagation is the net result of bond breaking and healing steps (forward and backward activations, respectively), the crack tip “pulsates” as it propagates. These temporary fluctuations can be of any size, each with a well-defined probability [21]. The crack, however, cannot shrink below the initial size X1, where the free energy field goes to infinity presenting a non-scalable reflecting boundary (Fig. 3b). Generally, the plate is finite and when the crack tip reaches X = L (that is, i = n), backward activation is not possible. The nth barrier presents an infinitely large backward activation free energy and the barrier system is terminated by an absorbing boundary (Fig. 3b). The analysis can be easily modified to consider an infinitely long plate. In non-steady state, the flux is different over each barrier and pi is a function of time as well as location. The process then has to be described by a system of the rate equations, each expressing the rate of concentration change of the mobile cracks pi at Xi as dpi _- flow per unit time into valley i -flow dt
per unit time out of valley
i + rate of crack nucleation (RI) -rate of crack immobilization (Ry),
and the system of rate equations for N barriers is dp, _ dt---p~
,k’+pz’k2+R;--R;
dpz& -PI rk’-pdkz+zkZ)+ps2k3+R;-R;
dpi_ dt - pi-1 i_lki-‘-pi(‘-‘ki din _
dt
-
pn-1 .-,k”-’
t iki)+pi+l iki+l + R:-R;
- p.(“-‘k,, t .k”).
Equations (4) are the expressions of the crack conservation law, similar to other transport processes. Often, two or more of the crack tip locations have identical free energy surroundings, that is, the rate constants for these locations are identical. The corresponding differential equations then can be combined into a single one. For example, if
!g= p/-l
,_,k’-’ - pPkr
dprn_ yg- pm-,m-,km-’
+ rk’) + pi+, ‘k,+,;
- pnSm-‘km + mk”) + pm+, “k,,,+,;
A. S. KRAUSZ
754
and K. KRAUSZ
4
(a)
ib)
Fig. 3. A pre-cracked plate of length S = L with the initial crack tip at X, (a); the energy barrier system associated with thermally activated crack propagation. indicating the reflecting and the absorbing boundaries fhl.
with
[k’ = ,k” = .k”; and ‘klil = mkm+,= “kp+,, the combination gives
and
dpv_ P*-~ ._-,k”-” -pt,(“-fkv+,.k’)+Pv;I dt-
Ykg,+l.
In these expressions the nucleation and immobilization rates were considered to be negligiblea realistic and convenient condition for the illustration of the combination concept. The symbol Y signifies that the combined equation provides only the combined 1 and m, but will not determine them as separate entities. THE DEVELOPMENT OF CRACK-SIZE DISTRIBUTION RELATIONS
To solve the system of simultaneous differential eqns (4), two conditions are imposed: (1) the rate constants do not change in time, that is, at constant stress and temperature k# f(t); (2) cracks are neither nucleated nor immobilized, that is, R: = 0 and R:! = 0. It can be shown that the general solution of a system of homogeneous linear first order differential equations with constant coefficients, giving the number of cracks pi of size Xi, is
pi2
Ci,j
exp ( - A$)+ Ci9
The theory of non-steady state fracture kinetics analysis
755
where C, is an integration constant. Substitution of eqn (5) into eqns (4) leads to a system of algebraic equations of the following type for i# 1 and if n: -
T
eXP ( -
Ci,jAj
Af)
=
[T
Ci-1.j exP
-
+
[T
AjOli_,ki-l
(-
Ci,j exp ( - Ajt)](i-lki
C C i
Ci+l,j exp(-A+)
+ Ci_l i_lk’-’ -
+ ik’)
1ki+l
(6)
i
Ci(‘-‘ki+ i/C’)+ Ci+l iki+l.
In the initial position at i = 1, the number of cracks changes at the rate of
-2 c,,jAjexp(-Air)=
-[C
c,,jexp(-A+)],k, J
i
+
Ci C2.jexp ( -
-
c, ,k’ + c, ‘kg
A$) lk2 I
(7)
and at i = n, that is, in front of the last barrier before the plate fractures, the number of crack tips changes per unit time as -2
i
- [T
c,jAjexP(-A#)=
[x c”-,,jexp(-Ajl)]“_,k”-, i
Cn,jexp ( - Ajt)](‘-lk,
+ C”_, ._Jc-
+ A”)
(8)
- c,(“-‘k” + .k”).
Each of eqns (6)-(8) must be separately satisfied by both the time dependent and time independent terms. For the time dependent terms it follows that
T[cl,j(Aj- ok’)+ C2,j1k21 = 0, T
[Ci,j(Aj-
'-'ki t ik') t Ci_~,ji_,k'-' t Ci+l,j ‘ki+l] = 0,
Z$[Cn,j(Aj - R-lk~-
(9)
n/C”)+ C,_*,j“-l/C-‘1 = 0;
and for the time independent terms the following expressions apply - c, ,k’ t c, ‘k, = 0, Ci_, i-*/C'-' - Ci('-'ki C”_, ,_#-*
t i/C’)+ Ci+l ‘ki+l= 0,
(10)
- c”(“-‘k” + “k”) = 0.
The set of algebraic equations given by eqns (9) has non-trivial solutions if the determinant of the coefficients, that are formed by the rate constants and A, is zero. Then, the algebraic
7%
A.S.KRAUSZandK.KRAUSZ
equations are expressed in explicit form as (hj - ]k’)Cl,,
+ ‘k>C*.j =
lk’C1.j + (hj -
0
‘k* - zk2fC2,j + *k3Cx,j =
0
(II.4
zk2C2,, +(hi-"kl-,k')c,,, +'k,cd,, =O ,,:,kn-'~i l,, +(Aj'"'k,-,k")c,j=iO.
The characteristic equation D = 0 of this system of algebraic equations is Aj- ,k’ ,k’ 0 0 :.....
0
‘kl hi- 'k-->k=
'k, Aj-*kj-3k3
2k2 0
3k' ,z_lk"-i
0
0
0
0
0 0 4k,
'k4 Ai-3k,-cjk4 Aj-*-'k~-nk"
Lzz
0
O
The expansion of the determinant results in an algebraic equation of the nth degree in A and defines n roots. It can be shown [22] that one root must be zero and all others positive to have physical meaning. Substitution of the Aj values into the system of algebraic equations (11) defines the ci.j ratios only. To obtain unique values, first the integration constants Cj of eqn (5) have to be determined from the boundary conditions [23]. In the initial state when t = 0, the number of crack tips pi”in the ith valley is expressed as
and hence the integration constant is
By using the ith terms of eqns (9) and (lo), unique values of the Ci,j constants can be determined and the boundary conditions are also satisfied. When all the roots are distinct the crack size distribution is expressed by eqn (5). When the expansion of the determinant results in multiple roots, then the crack-size distribution expression [24] is described by a polynomial form of time. For example, if Al= A2 = A?,then the number of crack tips at the ith valley is [23] pi = (Ci.1 + Ci,lt f $
2
Ci,j
Ci,3t2) exp (I Art)
exp ( - hit) + Ci*
j=4
It was discussed in a previous study [21] that crack propagation is a random walk prticess, controlled essentially by energy transport. The phenomenon of crack propagation is an integral part of the transport processes: conductive, convective, and radiation heat transfer; mass transfer by diffusion or flow; the flow of electric current; chemical reactions. The unity of heat and mass transfer is widely recognized [25-281. The phenomena of electric current, chemical reactions, heat and mass transfer as well as other processes were developed into a single system by Bosworth [29]. The interrelation among the above processes has led to the recognition that characteristic values (such as viscosity, diffusion coefficients, thermal diffusivity, etc.) are also interrelated [4,25-28,301, resulting in the welI-known dimensionless numbers of Prandtl, Smith, etc. The first indication that fracture, and plastic flow as well, are integral parts
Thetheoryof non-steadystatefracturekineticsanalysis
757
of the unified system came from Eyring [31] and Tobolsky and Eyring [ll through their realization that both are thermally activated processes. Rate equations for steady and nonsteady states were developed for plastic flow by Krausz and Faucher [32-341, and for fracture by Krausz [35,36]. Their analyses led to differential equations identical with Fick’s laws of diffusion, and to Fourier’s equations of heat conduction. The identity is not coincidental, but the consequence of the fact that the same principles control heat and mass flow, plastic deformation and crack propagation. These principles are rooted in the molecular theory of statistical mechanics for non-equilibrium systems. The kinetics analysis studies, referred to in the foregoing, follow from the statistical mechanics approach to fracture and plastic deformations processes. The interrelations were also discussed by Krausz and Eyring[9]. Fracture studies, similarly to the usual transport process analyses, are carried out for continuum models that result in the partial differential equations of transport phenomena. An alternative approach is to consider the medium as a discrete system (as all materials are, being composed of particles), and to apply random walk theory in the investigations [37]. The corresponding mathematical development is more involved than the differential equation formulation of the continuum model. It was shown, however, in a recent study by Krausz [21] that the analysis of crack propagation through random walk theory illuminates the operation of the actual physical processes. The present investigation aims to introduce a method of crack propagation study that also considers the discrete character of solids and may be thought of as a variation of the random walk analysis. Mathematically it is expected to be of similar convenience as the differential equation formulation, while at the same time preserving the physical reality of discreteness and the clarity that results from representing the actual physical process. The method is closely related to the kinetics analysis of consecutive chemical reactions [22,23]-one of the transport processes. The theory developed in Part 1 will be illustrated in Part 2 through an application to a relatively simple system of two barriers in series that describes the rate controlling processes of stress corrosion cracking. For an alternative method of crack propagation study the use of Laplace transform is currently under development.
Acknowledgement-The financialassistanceprovidedto one of the authors(A. S. K.) by the Engineering Research Council Canada is gratefully acknowledged.
NaturalSciences and
REFERENCES [I] A. TobolskyandH. Eyring, Mechanical properties of polymeric materials. J. Chem. Phys. 11, 125-134(1943). [2] S. N. Zhurkov, Kinetic concept of the strength of solids. ht. L Fracture Mech. 1, 311-316(1965). [3] M. L. Williams, Initiation and growth of viscoelastic fracture. Int. J. Fracture Mech. 1,292-310 (1%5). [4] R. Thomson, C. Hsieh and V. Rana, Lattice trapping of fracture cracks. J. Appl. Phys. 42,31563160 (1971). [5] B. R. Lawn and T. R. Wilshaw, Fracture of Brittle Solids. Cambridge University Press (1975). [6] S. D. Brown, The multibarrier kinetics of subcritical brittle crack growth, in Proc. 6th Canadian Congr. Appl. Mech., Vancouver, B.C. (1977). [7] A. S. Krausz, Time dependent crack propagation in lhearelastic solids. 1. Appl. Phys. 49,3774-3778(1978). [8] S. Glasstone, K. J. Laidler and H. Eyring, The Theory of Rate Processes. McGraw-Hill, New York (1941). [9] A. S. Krausz and H. Eyring, Deformation Kinetics. Wiley-Interscience, New York (1975). [IO] A. S. Krausz, The deformation and fracture kinetics of stress corrosion cracking. Int. J. Fracture 14,5-15 (1978). [I I] A. S. Krausz, The theory of thermally activated processes in brittle stress corrosion cracking. Engng Fracture Mech. 11, 33-42 (1979). [12] H. M. Cekiige, W. R. Tyson and A. S. Krausz, Static corrosion and static fatigue of glass. J. Am. Cer. Sot. 58,265-266
(1976). [I31 S. D. Brown, Subcritical crack growth: A treatment based on multibarrier kinetics and an electrical network analog. In Environmental Effects on Engineering Materials (Ed. M. R. Louthan and R. P. McNitt), Virginia Polytechnic Institute, Blacksburg, Va. (1977). [ 141S. D. Brown, A multibarrier rate process approach to subcritical crack growth. In Fracture Mechanics of Ceramics (Ed. R. C. Bradt, D. P. H. Hasselman and F. F. Lange). Plenum Press, New York (1978). [IS] C. Hsieh and R. Thomson, Lattice theory of fracture and crack creep. J. Appl. Phys. 44,2051-2063(1973). [16] B. R. Lawn, An atomistic model of kinetic crack growth in brittle solids. I. Mater. Sci. 10,469-4&l (1975). [17] J.-C. Pallet and S. J. Bums, Thermally activated crack propagation theory. Int. J. Fracture l&667-679 (1977). [I81 J.-C. Pollet and S. J. Bums, An analysis of slow crack propagation data in PMMA and brittle materials. Int. J. Fracture 13,775-786 (1977). 1191S. M. Wiederhorn, Influence of water vapor on crack propagation in soda-lime glass. J. Am. Cer. Sot. 50, 407-414 (1%7).
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[20] S. M. Wiederhorn, Mechanisms of subcritical crack growth in glass. In Fracture Mechanics of Ceramics (Eds. R. C Brad&D. P. H. Hasselman and F. F. Lange). Plenum Press. New York (1978). [21] A. S. Krausz, The random walk theory of crack propagation. Engng Fracture Mech. 12,499_Ctf4 ( 19791. 12213. J. Zwolinski and H. Eyring, The non-equilibrium theory of absolute rates of reaction. L .Qn. (‘hem. Sue, 69. 2702-2707(1947). [23] S. W. Renson, The Foundations of Chemical Kinetics. McGraw-Hill, New York (1%0). [24] I. S. Sikolnikoff and R. M. Redheffer, Mathematics of Physics and Modem Engineering. McGraw-Hill, New York (1958). 1251W. M. Rcl:senow and H. Y. Choi. Heat and Muss Transfer. Prentice Hall, New York (I%1 i. 1261E. R. G. Eckert and R. M. Drake, Jr., Analysis of Heat and Mass Tru~sfer. McG~~w~Hiil,New York 11972). [27] J. Szekely and N. J. Themelis, Rate P~enome~a in Process ~efulla~y. Wiley-Inters~ience. New York (197t). [28] D. A. Frank-Kamenetskii, Diflusion and Heat Transfer in Chemical Kinetics. Plenum Press. New York (1969). [29] R. C. L. Bosworth, 7’runspoti Processes in Applied Chemistry. Wiley, New York (1956). [30] H. Eyring, D. Henderson, B. J. Stover and E. M. Eyring, Statisfical Mechanics and Dynumics. Wiley. New York (1964). [3l] H. Eyring, Viscosity, plasticity, and diffusion as examples of absolute reaction rates. .I. C’hrm.Phys. 4, 283-290(19633. 1321B. Faucher and A. S. Krausz, Consecutive energy _. barriers and repeated stress relaxation. Scripta Met. 12 913-918 (1978). 1331B. Faucher and A. S. Krausz, Non-steady state mechanism of yield. Scripta Met 12, 175-17911978). 1341A. S. Krausz and B. Faucher, The steady-state kinetics of double-kink spreading. Scripta Met. 13.91-94 ( 1979). 1351A. S. Krausz. The theory of non-steady state fracture propagation rate. Inf. J. Fracture 12. 239-242(1976). i36j A. S. Krausz, Crack-size distribution in homogeneous solids. Int. .I. Fracture 15, 337-342 (1979) [37] S. Chandrasekhar, Stochastic problems in physics and astronomy. In Selected Papers on Noise ~md Stctchastit Professes (Ed, N. Wax), Dover, New York (1954). (Received 12 December 1979;receiued fur pu~ijcaii~n I9 March 1980)