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Analysis of the crack motion with varying velocity according to the dugdale-panasyuk model
Engineering Fractwe mechanics Vol. 39, No. 2, pp. 329-338.
0013-7944/91 $3.00 + 0.00 0 1991 Pergamon Press pk.
1991
Printed in Great Britain.
ANALYSIS OF THE CRACK MOTION WITH VARYING VELOCITY ACCORDING TO THE DUGDALE-PANASYUK MODEL A. NEIMITZ Kielce University of Technology, Kielce, Poland Abstract-Kostrov’s and Achenbach’s approach is adopted to the analysis of the fast unstable Dugdale-Panasyuk mode III crack motion. The results obtained suggest a new formulation of the crack motion equation.
ONEOF THE most important problems in the crack propagation analysis is a proper formulation of the crack-tip equation of motion in order to select an actual motion from the class of all dynamically admissible motions. The most useful form of the crack motion equation would be the one that in limit, when the crack tip speed u-+0, leads to a criterion of crack motion initiation. For linear-elastic materials the best candidates to be applied to const~ct an equation of crack motion are either the stress intensity factor (SF) or energy release rate (ERR). By analogy to the criterion predicting the onset of the crack motion one can postulate the following equation of the crack-tip propagation: ki,, (a,, a, u, geometry) = ki,,c(a, temperature, environment),
(1)
where kItI is dynamic stress intensity factor that shouid be calculated from the boundary-value problem and klllc represents the resistance of material to the crack motion and is considered as a material parameter. Unfortunately, the above simple equation cannot be utilized to the analysis of the moving cracks in the non-elastic materials. The asymptotic analyses presented by Achenbach and Dunayevsky[ l] and Gao and Nemat-Nasser[2] for elastic-perf~tly-plastic materials, or by Achenbath et a1.[3] for plastic-strain-hardening materials, or by Lo[4] for elastic-visco-plastic materials do not provide us with an appropriate amplitude factor that might be adopted to propose an equation of motion, equivalent to eq. (1). Thus, for elastic-plastic materials another crack motion equation must be proposed. For example, the concept of the critical strains at a certain point ahead of the crack tip, as proposed by Freund and Douglas[5] for the steady-state motion of a mode III crack may be utilized. The above approach is conceptually similar to the equation of crack motion based on a crack tip opening displacement concept that is particularly useful in the Dugdale[6]-Panasyuk[7] crack propagation analysis. Both equations can be written in the following form: e&(cr, ,a, 0, spec. geometry)/,, =,~~i = ezr(v, temp., environment),
(2)
S&(6,, a, t’, spec. geometry) = S&(u, temp., environment),
(3)
where e& is the strain component (in mode III) ahead of the moving crack tip, S$ denotes the crack tip opening displacement, the subscript i = I, II, or III depending on mode of loading, the superscript d indicates that particular quantity follows from the dynamic analysis and ed,, ;S& are the material parameters. One of the most often used equations of motion is based on a concept of the energy release rate Gf, that for a moving crack can be defined as follows[S, 91:
329
330
A. NEIMITZ
where t’ is a crack tip speed, L is an arbitrary contour that begins on one traction-free face of the moving crack, surrounds the tip and ends on the opposite traction-free face, cij and ui are the stress tensor and displacement vector components respectively, ni is a component of the unit vector normal to the contour L and dot denotes derivative with respect to time. The integral over contour L in eq. (4) represents the rate of energy flow F through L and can be calculated from the power balance equation according to procedure proposed by Freund[8], first proposed by Atkinson and Eshelby[lO]. Thus, the integral in eq. (4) must be path independent. In general it is true for the steady-state crack motion only as was shown, e.g. by Sih[l I] and Nilsson[9]. However, it turns out that for relatively wide class of materials, for which the displacement of the crack faces satisfies the relation u, - I-*
(5)
where 0 < CI< 1 and r is a distance from the crack tip, an arbitrary motion of the crack can be considered as an asymptotically steady+tate[8,9]. Thus for the linear and nonlinear elastic materials the equation of motion in a form GQ(cr,,a, u) = G$(u, temp., environment)
(6)
can be adopted to discuss certain crack propagation problems. However, in any case, path independence of the definition of Gf should be checked before eq. (6) is used for a particular problem. It is the three-fold purpose of the presented paper (a) to recast the Achenbach and Neimitz[l2] analysis of the fast, mode III, Dugdale-Panasyuk (D-P) crack motion in the framework of an energy balance including varying crack-tip speed, (b) to reconsider the validity of an eq. (6) for an arbitrary D-P fast crack motion and to propose a new equation of motion and (c) to compare eqs (3) and (6) for an arbitrary motion. FAST CRACK MOTION
ACCORDING
TO THE DUGDALE-PANASYUK
MODEL
The Dugdale-Panasyuk model of the crack plays a very specific role in fracture mechanics. Thanks to its basic postulates[6] the complex problem of the crack in elastic-plastic materials is replaced by a much easier one of a crack in linear-elastic material with an additional boundary condition on a crack face. It was shown in several experimental studies, e.g. refs [6] and [13] that despite strong simplifying assumptions the D-P model provides very good results, at least for stable cracks. The weak point of the model is its rather limited applicability-only for the plane stress situation. Nevertheless, the simplicity of the model encourages its utilization in crack dynamics analyses in anticipation of certain results that are impossible or very difficult to obtain using more realistic models. The fast motion of the D-P cracks has already been discussed by several authors. Goodier and Field[ 141, Kanninen[ 151, Achenbach and Neimitz[ 121 and Neimitz[ 16, 171, utilizing various techniques, adopted this model to analyse crack motion with a constant, high velocity. Glennie and Willis [I 81 were the first who included the effect of a varying crack tip speed on certain features of the fast crack motion phenomenon. They utilized Friedlander’s method[l9] to obtain the stress and displacement fields within the crack tip region, moving with an arbitrary velocity, and adopted it to the D-P model. Kostrov[20] and Achenbach[21] solved the same problem adopting Evvard’s method[22]. This method was later adopted by Achenbach and Neimitz[l2] to the analysis of the D-P crack motion. Both methods lead to similar results, but Glennie and Willis’ method is more general since it provides a displacement field at an arbitrary point around the crack tip, not only in the plane of the crack. Besides, in the Glennie and Willis method the effect of body forces can also be considered. However, the advantage of the Kostrov and Achenbach method lies, in the author’s opinion, in a clear, convincing, physical picture of this complex phenomenon and in an easy computational procedure. There is, however, one serious disadvantage in both methods (if one accepts the D-P model) that is particularly in computing displacement within the D-P zone or crack tip opening displacement. Namely, one should know both leading and trailing edge trajectories prior to computing displacements. However, in turn, a particular motion can be determined only with the help of the crack opening displacement. Thus the analysis may only be
Crack model with varying velocity
331
an approximate one and such was proposed by Glennie and Willis[l8]. They assumed that the critical crack tip opening displacement was constant, independent of the crack tip speed, the length of the plastic zone was constant as for steady state motion, and the equation of energy flow out of the body consisted only of one term, neglecting the influence of the variation in the length of the D-P zone. The real crack tip speed was approximated by a piecewise-linear u(t) function. With all the above assumptions they solved given equations on a computer. It turned out that the energy released during crack tip acceleration increased over its steady-state value and decreased during crack tip deceleration. In the present paper the analysis will be carried out in another direction. It is assumed that both crack tip (trailing edge) and plastic zone tip (leading edge) move with constant velocities and flL = constant # & = constant, where
With this assumption, the length of the plastic zone either increases (/& > &) or decreases (BT> /JL) in time. The length of the plastic zone rr, follows from the stress analysis and Barenblatt’s postulate that stresses should be finite ahead of the crack tip[l2]:
From the above relation one can conclude that when crack tip speed increases the length of the plastic zone decreases to zero at PT = 1. For PT = 0 one obtains the static value of yp. Thus the assumption that BT= constant # /?r = constant simulates the process of acceleration or deceleration of the crack while the real shapes of the crack trajectories are approximated by a piecewise-linear functions. We hope that such an approach will provide us with certain additional information not available from the elastic or elastic-plastic steady state analysis and will be a good starting point to more advanced computer simulation procedures. It should be noted, however, that while discussing the whole history of the crack tip propagation the approximation procedure of the crack trajectories by a piecewise-linear functions cannot be arbitrary. This problem will be discussed in more detail later on. As a result of a computational procedure proposed by Achenbach[23] the stress-ahead of the crack tip, moving with a high speed in a mode III loading system, can be found from the relation: 1
r(5,r/)=L or in the x,, s coordinate
t(x,
N(r)f(4, rl)[N(<) -
71[‘t- wt1)1”*s 5
V-U
UP2du 3
(9)
system
s)
=
1 71 [x
-
X(s)]“2
’
(10)
where most of the symbols are defined in Fig. 1; s = cTt, cT = (~/p)“~,f(<, ‘I) is given for the D-P crack by the relation (12). The opening of the crack faces can be calculated from the relation:
Equations (16)<18) together with Fig. l(a) can be considered as a “receipt” to solve various problems for mode III fast moving cracks. The “key” to the solution of the D-P crack is a proper definition of f(<, q) or f(x, , s) functions. In the present article it is assumed that the specimen is loaded in infinity. (The solution for the crack loaded on a crack faces by concentrated forces was presented by Neimitz[l7].) Thus the f(x,, s) function may be defined as follows: f(X, 3s) = -%(X, )faW)
-
x,lff(s) + z,wx, - ~(~w[u~) - x, W(s).
(12)
332
A. NEIMITZ ;(
*
/
/
E \ (a)
(b) Fig. 1. Scheme of the crack trajectories.
where T0 = (27~)-“~K,,i(L) represents the instantaneous static stress that is removed by a moving crack, K,,,(L) is an instantaneous, static SIF, H [- -1 is a Heaviside function, ~~is the shearing stress within D-P zone. It is here assumed that rf is constant, independent of the crack lip speed. We simply neglect (as a first approximation) the viscous effect. The viscous effect within the plastic zone was discussed by Glennie[25]. It is here assumed that the strain rate, or more precisely the crack tip speed, will influence the parameters characterizing fracture toughness of the material only. It should also be recalled here that the presented analysis is strictly valid for a semi-infinite crack within an infinite body only. The formulae defining the length of the plastic zone rp (eq. 8 in this paper) and the crack tip opening displa~ment (CTOD) were the main results of the paper[ 121 that is the starting point to the present analysis. The CTOD can be expressed by the relation (13) that for steady-state motion reduces to: (14) In ref. [12] the motion of the D-P crack was also analysed but limiting discussion to the steady-state situation only. Equation (6) was used as an equation of motion. In the present paper the analysis is extended for an arbitrary motion with varying velocity. The integral in eq. (4) denotes the energy flow through an arbitrary contour L, surrounding the crack tip and moving along with it with a speed U. For mode III it will be denoted by F,,,. Analysing the D-P model, it is convenient to select contour L in this way that it begins at the lower crack face at the trailing edge, terminates at the upper face of the crack at the trailing edge and it is drawn along the lower and upper faces, surrounding the leading edge. For such a contour the integral in eq. (4) reduces to the form
F,,,=
s$ -0)
dx, ,
(15)
since n, = 0 for all points along L. Locating the coordinate system at the moving leading edge in this way that [ = x1 - vr, the derivative &5(.x,, r)/& may be replaced by -v &5(& t)/ac + &5([, t)f& and one may rewrite eq. (15) in the form:
Crack model with varying velocity
333
The first integral in eq. (16) can be evaluated yielding to the following relation (17) To evaluate the second integral in eq. (16) we must first calculate the function S(c, t) for a moving crack. It may be done utilizing eqs (18) and (19), Fig. l(b) and certain geometrical relations following from the assumed piecewise-linear trajectories of the leading and trailing edges of the crack. qlj,
s) =
2A2qp
- $J“*(rp + blj/)“2 + -& (rp - $) In
II9(18)
where A ==iZ(l
b
+/?r)“2(1 -/IT)-“2,
-- BT- PL _
1+PL
1+ BT
c=iTp e= -i,
rp = rpo + 4PL - BT)r
s=c,t
and rpo is a length of the BP zone, measured at the instant when either fiT or /IL or both changed their values. As a next step in computing the energy release rate we must calculate a6 asas -=_-
at
asat’
(19)
where
and
Inserting eqs (17), (21) and (20) into eq. (16) we finally obtain relatively simple relations for the energy flow F,,, F
=
111
PC
K:,, 1-P Ii2 T2p (->1+p
when /?r = /IL = /I
(22)
A. NEIMITZ
334
F,,,
=
BLcT~(!A)li1{2_ (L#
+y,,) _
(!k$~(~~’
tg(khr},
arc
Before we proceed to discuss the D-P crack motion equation introduced in the form
(24)
the function Sf,, will be
1 sp,, = F,,, = ! F,,, V BLcT
(25)
and plotted in Fig. 2. The function St,, is in some sense similar to the function Gg, but in general it is not the energy release rate. Henceforth, it will be called the “driving force” (DF) on the D-P crack. In Fig. 2 the dependence of the DF, normalized with respect to the static critical ERR-G,,,c on the crack tip speed /IT and the ratio N = /IL//IT is shown. The dot-dash lines represent solution of the first integral in eq. (16) (divided by v)-the “steady-state” term. For all flT this term has maximum for N = 1 and this maximum decreases with increasing /IT. The solution of the second integral in eq. (16) (divided by /ILcT) is represented by a dash line. The seemingly complex relations (23) and (24) have a very simple graphical representation. They are plotted as almost straight lines intersecting the line Q&/G,iic = 0 at a point N = 1. In the domain where the crack accelerates the second integral in eq. (16) is positive and it is negative in the deceleration domain. The actual DF is plotted as a solid line. For the moderate crack tip speeds BT< 0.4 these lines are almost straight. For higher velocities the S& function is not linear in the domain 0 < N < 1. Several interesting conclusions may be drawn from Fig. 2.
0.2
I
0.6
I
I
I .o
I .4
I
1.e
7%
2.2
N
Fig. 2. The plot of the 9#, as a function
of N (details
in text).
6
335
Crack model with varying velocity
(1) When the loading is quasi-static and the crack tip is sharp then at the onset of unstable crack motion K,,,(L) = J&C (neglecting the subcritical crack growth). Thus the crack tip speed must not exceed a certain terminal value, /& < 0.267 provided that K,,,(L) does not increase. Above this terminal value the $$, is always smaller than G,,,e. This problem will be discussed in more detail in the next paragraph. (2) It turns out that the DF-g$, becomes negative when the ratio N = /IL/& is greater than a certain value, close to 4, that depends on PT. For example Sf,, < 0 for & = 0.1 and N 2 3.99, for PT = 0.4 and N > 3.93 or for BT= 0.9 and N 2 3.89. It means that sudden arrest of the crack tip while the leading edge is still moving may lead to the situation that $,, < 0. It cannot be accepted from the physical point of view, thus the crack arrest should be considered as a continuous process. If, however, the trailing edge is suddenly arrested the leading edge must move with a speed not greater than quadruple speed of the crack before arrest. It is in contrast to the situation observed for elastic materials. Freund has shown[25] that when the crack in an elastic body is suddenly arrested the stationary field is immediately generated. In Fig. 3 the influence of the crack tip speed and the ratio N = fiLlfiTon the value of the crack tip opening displacement S$ is shown. For the range of crack tip speed 0 < PT < 1 the CTOD assumes the maximum for N = /?Jj& = 1. The slower the crack tip speed & the weaker maximum is observed. For example, for & = 0.1 the difference between the maximum and minimum values of St (in the range 0 < N < 4) is about 0.24%. For PT = 0.95 this difference increases to about 16%. Thus, taking into account the range of the crack tip speed observed in the experiment it will not be a great simplification if one assumes the equation of motion (based on the CTOD concept) for steady-state crack propagation. It also follows from eq. (13) that S$ is practically independent of N and for flT < 0.6 may be approximated by a straight line with a slope equal to -0.833.
CRACK MOTION
EQUATION
FOR D-P CRACK
In the first paragraph three equations of motion based on three different parameters have been presented. For D-P crack the SIF is equal to zero. Therefore we will concentrate on the two remaining quantities. (a) According to the definition (4) the energy flow into the plastic zone, divided by the crack tip speed may be interpreted as an energy release rate for the steady state motion only. For arbitrary motion with varying velocity of the D-P crack it cannot be proved that the motion is asymptotically steady-state since the contour L is precisely defined. The quantity Gt,,e in eq. (4) is usually called the resistance of the material to the crack motion. It may be noticed that for flT smaller than a
0.1-
I I I I I I III 0.2 0.4 0.6 0.6 I.0 1.2 I..41.6 I.6 2.0 :
0
N
I 0
1
I
1
I
I
1
1
1
1
I
I
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 I.0 I.1 BT
Fig. 3. Dependence of the CTOD-ut
on & and N.
336
A. NEIMITZ
certain critical value the quantity S& is greater than G,,, . There is no doubt that for many materials the material parameter Gt,,, is a function of a crack tip speed. The physical intuition and the observation of the experimental results suggest that the Gdillc either increases with /jT (initially increases slowly and from a certain value of /IT rapidly) or initially decreases with & (because for higher values of /IT there is little time for plastic deformation) and then increases with /?r (because of crack branching processes). Thus the equation in the form: s;l,, = %c
(26)
is certainly not correct for the first stage of propagation and probably is also not true for later stages except for a steady-state motion. Taking into account the above arguments one may assume the equation of motion of the D-P crack in the form: @ii - %c = Ma,
(27)
interpreting S& as a “driving force” on D-P crack, Ye,,, as a resistance of the material to the crack motion, M as an “equivalent” mass of the plastic zone and uT as a crack tip acceleration (in the proposed model of the crack tip kinetics ur should be interpreted as a “mean” acceleration of a trailing edge). The equivalent mass M can be assumed to be proportional to the plastic zone mass. When the crack motion is steady-state eq. (27) reduces to eq. (6). In general eq. (27) contains three unknown /IT, pL and aT . This differential equation could be solved for given loading history and material parameter %&c(~T, bL) if is supplemented by an additional relation between Pr and BL. For proposed model of crack kinetics (Fig. 4) one may introduce two additional equations derived from purely geometrical analysis along the crack trajectories. From eq. (8) and relation rf
the following, approximate
P
=
yi-.I
p
+ CTfu%. - 89
(28)
formulae follow:
(30) The computer simulation techniques may now be proposed to predict the crack motion history for given external loading and material properties. It will be a subject of the next paper. (b) Another equation of motion for the D-P crack can be postulated in more straightforward way. The concept of the CTOD is usually extended to the dynamic case and is written in the form of eq. (3). When both sides of eq. (3) are multiplied by rf we obtain an equation of motion in the form of eq. (6) which, as was shown earlier, is strictly true for steady-state motion only. The first
Fig. 4. Hypothetical scheme to determine the crack trajectories according to the u+ criterion.
Crack model with varying velocity
337
term in eq. (16), evaluated in eq. (17), may be interpreted as the irrecoverable work per unit area which has been done on the material element at the crack tip in bringing it to the point of separation. It does not take into account the energy dissipated within the plastic zone, which changes its length in time. This problem has been discussed in more detail in the review article by Freund [26]. Nevertheless, eq. (3) has often been utilized in crack motion analyses: it can easily be applied in the presented model of crack kinetics. Equations (13) and (3) are sufficient to perform the computer simulation procedure provided the crack tip trajectories are approximated by a piecewise-linear function. The assumed approximation of the real crack tips trajectories have been shown in Fig. 4. It is assumed that the crack starts propagation with a speed #?,.= fl’: at N = & = 0. The info~ation that the trailing edge has already started propagation reaches the leading edge at time S, . At this moment the leading edge starts motion with the speed fiL = fit and sends information about that to the trailing edge. This information reaches the trailing edge at time S = S,, and the latter corrects its speed to a new speed following from the equation of motion. Again the signal is sent to the leading edge which at time S = S, adjust its speed to fit = /?k and so on until the steady-state is reached. It can easily be shown that if S$ = const and dK,i,/dt = 0 (Fig. 4) the steady state is reached after two steps. The situation is more complex for a more general case, but the computational procedure is the same.
CONCLUSIONS The Achenbach and Neimitz[lf] analysis of the fast, mode III, Dugdal~Panasyuk (D-P) crack motion has been recast in the framework of an energy balance including varying crack-tip speed. The more important conclusions of the presented analysis are repeated below. (1) The crack tip opening displacement 6: reaches the maximum value for steady state motion (a = & = 8). This maximum decreases when the crack tip speed increases. (2) The equation of motion based on the CTOD is equivalent to the one based on ERR concept for steady-state motion only. (3) The equation of motion following from the energy rate balance provides a limitation on the crack tip speed without additional assumptions concerning the viscosity effect or velocity dependent resistance to fracture parameter. (4) The crack with a plastic zone ahead of the tip cannot be suddenly arrested. (5) The new equation of motion has been proposed. REFERENCES [I] J. D. Achenbach and V. Dunayevsky, Fields near a rapidly propagating crack tip in an elastic-perfectly plastic material. J. Me&. Phvs. Solids 29. 283-303 (1981). [2J Y. C. Gao and S. NematiNasser, Dynamic fields near a crack tip growing in an elastic-perfectly plastic. Me&. Marer. 2, 4760 (1983). [3] J. D. Achenbach, M. F. Kannien and C. H. Popelar, Crack tip fields for fast fracture of an elastic-plastic material, J. Me&. Phys. Soli& 29, 211-25 (1981). [4] K. K. Lo, Dynamic crack-tip fields in rate sensitive solids. J. Mech. Phys. Solids 31, 287-305 (1983). [5] L. B. Freund and A. S. Douglas, The influence of inertia on elastic-plastic antiplane shear crack growth. J. Mech. Phys. Solids 30, 59-74 (1982). [6] D. S. Dugdale, Yielding of steel sheets containing slits. J. Mech. Php.
Solids 8, 100-104 (1960). 171 V. V. Panasyuk, On the theory of crack extension durinn the deformation of a brittle bodv. DOD. Akad. A&k Ukr. RSR 9, I lSj-1188 (1960) (in Ukrainian). [8] L. B. Freund, Energy flux into the tip of an extending crack in an elastic solid. J. Elasricity 2, 341-349 (1972). 191 F. Nilsson. Dynamic Fracture Theorv. CISM. Udine. Italv (in mess). [I-O]C. Atkinson and J. D. Eshelby, The f”lowof energy into thetip o?a moving crack. Int. J. Fracture Mech. 4,3-8 (1986). [II] G. C. Sih, Dynamic aspects of crack propagation, in Inelastic Behaviour of Solids (Edited by M. F. Kannien), pp. 607-639. McGraw-Hill, New York (1970). [IZ] J. D. Achenbach and A. Neimitz, Fast fracture and crack arrest according to the Dugdale model. Engag Fracture Me&. 14, 385-395 (1981). (131 0. T. Hahn and R. Rosenheld, Local yielding and extension of a crack under plane stress. Acra Metall. 13,293-306
(19651. [14] J. N. Goodier and F. A. Field, Plastic energy dissipation in crack propagation, in Fracture of Solids (Edited by D. C. Drucker and J. J. Gilman), pp. 103-118. Wiley, New York (1963). [15] M. F. Kannien, An estimate of the limiting speed of a propagating ductile crack. J. Mech. Phys. Solids 16, 215-228 (1968).
A. NEIMITZ
338
[16] A. Neimitz, Crack propagation and arrest in plastic medium. Arch Mech. 33, 815-827 (1981). [17] A. Neimitz, 0 Pewnym dynamicznym zagadnieniu szczeliny w osrodku spr@ystym i spr@ysto-plastycznym. Engng Trans. 29, 629-640 (1981). [18] E. B. Glennie and J. R. Willis, An Examination of the effects of some idealized models of fracture on accelerating cracks. J. Mech. Phys. Solids 19, 1 l-30 (1871). [19] F. G. Friedlander, Q. J. Mech. appl. Math. 4, 344 (1951). [20] B. V. Kostrov, Unsteady propagation of longitudinal shear cracks. Prikl. Mat. Mekh. (English Translation), 30, 1241-1248 (1966). [21] J. D. Achenbach, Extension of a crack by a shear wave. ZAMP 21, 8877900 (1970). [22] J. C. Evvard, Distribution of wave drag and lift in the vicinity of wing tips at supersonic speeds. National Advisory Committee for Aeronautics, Technical Note No. 1382 (1947). [23] J. D. Achenbach, Dynamic effects in brittle fracture. Mechanics Today 1, l-54 (1972). [24] E. B. Glennie, A strain-rate dependent crack model. J. Mech. Phys. Solids 19, 255-272 (1971). [25] L. B. Freund, Crack propagation in an elastic solid subjected to general loading-II. Nonuniform rate of extension. J. Mech. Phys. Solids 20, 141-152 (1972). [26] L. B. Freund, Dynamic crack propagation in the mechanics of fracture. ADM 19 (Edited by F. Erdogan), pp. 1055134 ASME (1976).
APPENDIX The idea of the crack tip opening angle is a very popular the D-P zone suggests a geometrical definition of the CTOA at a particular point. Thus one may propose:
It follows from eq. (18) that CTOA(,=,
= 0 and CTOAI,
s
one in the crack motion theories. as an angle between two tangents
=,~ = co. However
The wedge-like shape of to the D-P zone profile
the mean value of the CTOA
has a finite value
‘pWlL, rp)
(CTOA),,,,
=
’
It is independent of external loading characterizing the resistance of material Thus at a high velocities of a crack the does not practically depend (CTGA),,,, decreases in deceleration domain).
aw TP
=
642)
and specimen geometry. One may consider this magnitude as a parameter increases with a crack tip speed at a given N. to crack propagation. (CTOA),,,, material behaves in a more brittle way. For very small crack tip velocities the on N and & (while for higher speeds it increases in acceleration domain and
(Receiued 13 April 1990)
0013-7944/91 $3.00 + 0.00 0 1991 Pergamon Press pk.
1991
Printed in Great Britain.
ANALYSIS OF THE CRACK MOTION WITH VARYING VELOCITY ACCORDING TO THE DUGDALE-PANASYUK MODEL A. NEIMITZ Kielce University of Technology, Kielce, Poland Abstract-Kostrov’s and Achenbach’s approach is adopted to the analysis of the fast unstable Dugdale-Panasyuk mode III crack motion. The results obtained suggest a new formulation of the crack motion equation.
ONEOF THE most important problems in the crack propagation analysis is a proper formulation of the crack-tip equation of motion in order to select an actual motion from the class of all dynamically admissible motions. The most useful form of the crack motion equation would be the one that in limit, when the crack tip speed u-+0, leads to a criterion of crack motion initiation. For linear-elastic materials the best candidates to be applied to const~ct an equation of crack motion are either the stress intensity factor (SF) or energy release rate (ERR). By analogy to the criterion predicting the onset of the crack motion one can postulate the following equation of the crack-tip propagation: ki,, (a,, a, u, geometry) = ki,,c(a, temperature, environment),
(1)
where kItI is dynamic stress intensity factor that shouid be calculated from the boundary-value problem and klllc represents the resistance of material to the crack motion and is considered as a material parameter. Unfortunately, the above simple equation cannot be utilized to the analysis of the moving cracks in the non-elastic materials. The asymptotic analyses presented by Achenbach and Dunayevsky[ l] and Gao and Nemat-Nasser[2] for elastic-perf~tly-plastic materials, or by Achenbath et a1.[3] for plastic-strain-hardening materials, or by Lo[4] for elastic-visco-plastic materials do not provide us with an appropriate amplitude factor that might be adopted to propose an equation of motion, equivalent to eq. (1). Thus, for elastic-plastic materials another crack motion equation must be proposed. For example, the concept of the critical strains at a certain point ahead of the crack tip, as proposed by Freund and Douglas[5] for the steady-state motion of a mode III crack may be utilized. The above approach is conceptually similar to the equation of crack motion based on a crack tip opening displacement concept that is particularly useful in the Dugdale[6]-Panasyuk[7] crack propagation analysis. Both equations can be written in the following form: e&(cr, ,a, 0, spec. geometry)/,, =,~~i = ezr(v, temp., environment),
(2)
S&(6,, a, t’, spec. geometry) = S&(u, temp., environment),
(3)
where e& is the strain component (in mode III) ahead of the moving crack tip, S$ denotes the crack tip opening displacement, the subscript i = I, II, or III depending on mode of loading, the superscript d indicates that particular quantity follows from the dynamic analysis and ed,, ;S& are the material parameters. One of the most often used equations of motion is based on a concept of the energy release rate Gf, that for a moving crack can be defined as follows[S, 91:
329
330
A. NEIMITZ
where t’ is a crack tip speed, L is an arbitrary contour that begins on one traction-free face of the moving crack, surrounds the tip and ends on the opposite traction-free face, cij and ui are the stress tensor and displacement vector components respectively, ni is a component of the unit vector normal to the contour L and dot denotes derivative with respect to time. The integral over contour L in eq. (4) represents the rate of energy flow F through L and can be calculated from the power balance equation according to procedure proposed by Freund[8], first proposed by Atkinson and Eshelby[lO]. Thus, the integral in eq. (4) must be path independent. In general it is true for the steady-state crack motion only as was shown, e.g. by Sih[l I] and Nilsson[9]. However, it turns out that for relatively wide class of materials, for which the displacement of the crack faces satisfies the relation u, - I-*
(5)
where 0 < CI< 1 and r is a distance from the crack tip, an arbitrary motion of the crack can be considered as an asymptotically steady+tate[8,9]. Thus for the linear and nonlinear elastic materials the equation of motion in a form GQ(cr,,a, u) = G$(u, temp., environment)
(6)
can be adopted to discuss certain crack propagation problems. However, in any case, path independence of the definition of Gf should be checked before eq. (6) is used for a particular problem. It is the three-fold purpose of the presented paper (a) to recast the Achenbach and Neimitz[l2] analysis of the fast, mode III, Dugdale-Panasyuk (D-P) crack motion in the framework of an energy balance including varying crack-tip speed, (b) to reconsider the validity of an eq. (6) for an arbitrary D-P fast crack motion and to propose a new equation of motion and (c) to compare eqs (3) and (6) for an arbitrary motion. FAST CRACK MOTION
ACCORDING
TO THE DUGDALE-PANASYUK
MODEL
The Dugdale-Panasyuk model of the crack plays a very specific role in fracture mechanics. Thanks to its basic postulates[6] the complex problem of the crack in elastic-plastic materials is replaced by a much easier one of a crack in linear-elastic material with an additional boundary condition on a crack face. It was shown in several experimental studies, e.g. refs [6] and [13] that despite strong simplifying assumptions the D-P model provides very good results, at least for stable cracks. The weak point of the model is its rather limited applicability-only for the plane stress situation. Nevertheless, the simplicity of the model encourages its utilization in crack dynamics analyses in anticipation of certain results that are impossible or very difficult to obtain using more realistic models. The fast motion of the D-P cracks has already been discussed by several authors. Goodier and Field[ 141, Kanninen[ 151, Achenbach and Neimitz[ 121 and Neimitz[ 16, 171, utilizing various techniques, adopted this model to analyse crack motion with a constant, high velocity. Glennie and Willis [I 81 were the first who included the effect of a varying crack tip speed on certain features of the fast crack motion phenomenon. They utilized Friedlander’s method[l9] to obtain the stress and displacement fields within the crack tip region, moving with an arbitrary velocity, and adopted it to the D-P model. Kostrov[20] and Achenbach[21] solved the same problem adopting Evvard’s method[22]. This method was later adopted by Achenbach and Neimitz[l2] to the analysis of the D-P crack motion. Both methods lead to similar results, but Glennie and Willis’ method is more general since it provides a displacement field at an arbitrary point around the crack tip, not only in the plane of the crack. Besides, in the Glennie and Willis method the effect of body forces can also be considered. However, the advantage of the Kostrov and Achenbach method lies, in the author’s opinion, in a clear, convincing, physical picture of this complex phenomenon and in an easy computational procedure. There is, however, one serious disadvantage in both methods (if one accepts the D-P model) that is particularly in computing displacement within the D-P zone or crack tip opening displacement. Namely, one should know both leading and trailing edge trajectories prior to computing displacements. However, in turn, a particular motion can be determined only with the help of the crack opening displacement. Thus the analysis may only be
Crack model with varying velocity
331
an approximate one and such was proposed by Glennie and Willis[l8]. They assumed that the critical crack tip opening displacement was constant, independent of the crack tip speed, the length of the plastic zone was constant as for steady state motion, and the equation of energy flow out of the body consisted only of one term, neglecting the influence of the variation in the length of the D-P zone. The real crack tip speed was approximated by a piecewise-linear u(t) function. With all the above assumptions they solved given equations on a computer. It turned out that the energy released during crack tip acceleration increased over its steady-state value and decreased during crack tip deceleration. In the present paper the analysis will be carried out in another direction. It is assumed that both crack tip (trailing edge) and plastic zone tip (leading edge) move with constant velocities and flL = constant # & = constant, where
With this assumption, the length of the plastic zone either increases (/& > &) or decreases (BT> /JL) in time. The length of the plastic zone rr, follows from the stress analysis and Barenblatt’s postulate that stresses should be finite ahead of the crack tip[l2]:
From the above relation one can conclude that when crack tip speed increases the length of the plastic zone decreases to zero at PT = 1. For PT = 0 one obtains the static value of yp. Thus the assumption that BT= constant # /?r = constant simulates the process of acceleration or deceleration of the crack while the real shapes of the crack trajectories are approximated by a piecewise-linear functions. We hope that such an approach will provide us with certain additional information not available from the elastic or elastic-plastic steady state analysis and will be a good starting point to more advanced computer simulation procedures. It should be noted, however, that while discussing the whole history of the crack tip propagation the approximation procedure of the crack trajectories by a piecewise-linear functions cannot be arbitrary. This problem will be discussed in more detail later on. As a result of a computational procedure proposed by Achenbach[23] the stress-ahead of the crack tip, moving with a high speed in a mode III loading system, can be found from the relation: 1
r(5,r/)=L or in the x,, s coordinate
t(x,
N(r)f(4, rl)[N(<) -
71[‘t- wt1)1”*s 5
V-U
UP2du 3
(9)
system
s)
=
1 71 [x
-
X(s)]“2
’
(10)
where most of the symbols are defined in Fig. 1; s = cTt, cT = (~/p)“~,f(<, ‘I) is given for the D-P crack by the relation (12). The opening of the crack faces can be calculated from the relation:
Equations (16)<18) together with Fig. l(a) can be considered as a “receipt” to solve various problems for mode III fast moving cracks. The “key” to the solution of the D-P crack is a proper definition of f(<, q) or f(x, , s) functions. In the present article it is assumed that the specimen is loaded in infinity. (The solution for the crack loaded on a crack faces by concentrated forces was presented by Neimitz[l7].) Thus the f(x,, s) function may be defined as follows: f(X, 3s) = -%(X, )faW)
-
x,lff(s) + z,wx, - ~(~w[u~) - x, W(s).
(12)
332
A. NEIMITZ ;(
*
/
/
E \ (a)
(b) Fig. 1. Scheme of the crack trajectories.
where T0 = (27~)-“~K,,i(L) represents the instantaneous static stress that is removed by a moving crack, K,,,(L) is an instantaneous, static SIF, H [- -1 is a Heaviside function, ~~is the shearing stress within D-P zone. It is here assumed that rf is constant, independent of the crack lip speed. We simply neglect (as a first approximation) the viscous effect. The viscous effect within the plastic zone was discussed by Glennie[25]. It is here assumed that the strain rate, or more precisely the crack tip speed, will influence the parameters characterizing fracture toughness of the material only. It should also be recalled here that the presented analysis is strictly valid for a semi-infinite crack within an infinite body only. The formulae defining the length of the plastic zone rp (eq. 8 in this paper) and the crack tip opening displa~ment (CTOD) were the main results of the paper[ 121 that is the starting point to the present analysis. The CTOD can be expressed by the relation (13) that for steady-state motion reduces to: (14) In ref. [12] the motion of the D-P crack was also analysed but limiting discussion to the steady-state situation only. Equation (6) was used as an equation of motion. In the present paper the analysis is extended for an arbitrary motion with varying velocity. The integral in eq. (4) denotes the energy flow through an arbitrary contour L, surrounding the crack tip and moving along with it with a speed U. For mode III it will be denoted by F,,,. Analysing the D-P model, it is convenient to select contour L in this way that it begins at the lower crack face at the trailing edge, terminates at the upper face of the crack at the trailing edge and it is drawn along the lower and upper faces, surrounding the leading edge. For such a contour the integral in eq. (4) reduces to the form
F,,,=
s$ -0)
dx, ,
(15)
since n, = 0 for all points along L. Locating the coordinate system at the moving leading edge in this way that [ = x1 - vr, the derivative &5(.x,, r)/& may be replaced by -v &5(& t)/ac + &5([, t)f& and one may rewrite eq. (15) in the form:
Crack model with varying velocity
333
The first integral in eq. (16) can be evaluated yielding to the following relation (17) To evaluate the second integral in eq. (16) we must first calculate the function S(c, t) for a moving crack. It may be done utilizing eqs (18) and (19), Fig. l(b) and certain geometrical relations following from the assumed piecewise-linear trajectories of the leading and trailing edges of the crack. qlj,
s) =
2A2qp
- $J“*(rp + blj/)“2 + -& (rp - $) In
II9(18)
where A ==iZ(l
b
+/?r)“2(1 -/IT)-“2,
-- BT- PL _
1+PL
1+ BT
c=iTp e= -i,
rp = rpo + 4PL - BT)r
s=c,t
and rpo is a length of the BP zone, measured at the instant when either fiT or /IL or both changed their values. As a next step in computing the energy release rate we must calculate a6 asas -=_-
at
asat’
(19)
where
and
Inserting eqs (17), (21) and (20) into eq. (16) we finally obtain relatively simple relations for the energy flow F,,, F
=
111
PC
K:,, 1-P Ii2 T2p (->1+p
when /?r = /IL = /I
(22)
A. NEIMITZ
334
F,,,
=
BLcT~(!A)li1{2_ (L#
+y,,) _
(!k$~(~~’
tg(khr},
arc
Before we proceed to discuss the D-P crack motion equation introduced in the form
(24)
the function Sf,, will be
1 sp,, = F,,, = ! F,,, V BLcT
(25)
and plotted in Fig. 2. The function St,, is in some sense similar to the function Gg, but in general it is not the energy release rate. Henceforth, it will be called the “driving force” (DF) on the D-P crack. In Fig. 2 the dependence of the DF, normalized with respect to the static critical ERR-G,,,c on the crack tip speed /IT and the ratio N = /IL//IT is shown. The dot-dash lines represent solution of the first integral in eq. (16) (divided by v)-the “steady-state” term. For all flT this term has maximum for N = 1 and this maximum decreases with increasing /IT. The solution of the second integral in eq. (16) (divided by /ILcT) is represented by a dash line. The seemingly complex relations (23) and (24) have a very simple graphical representation. They are plotted as almost straight lines intersecting the line Q&/G,iic = 0 at a point N = 1. In the domain where the crack accelerates the second integral in eq. (16) is positive and it is negative in the deceleration domain. The actual DF is plotted as a solid line. For the moderate crack tip speeds BT< 0.4 these lines are almost straight. For higher velocities the S& function is not linear in the domain 0 < N < 1. Several interesting conclusions may be drawn from Fig. 2.
0.2
I
0.6
I
I
I .o
I .4
I
1.e
7%
2.2
N
Fig. 2. The plot of the 9#, as a function
of N (details
in text).
6
335
Crack model with varying velocity
(1) When the loading is quasi-static and the crack tip is sharp then at the onset of unstable crack motion K,,,(L) = J&C (neglecting the subcritical crack growth). Thus the crack tip speed must not exceed a certain terminal value, /& < 0.267 provided that K,,,(L) does not increase. Above this terminal value the $$, is always smaller than G,,,e. This problem will be discussed in more detail in the next paragraph. (2) It turns out that the DF-g$, becomes negative when the ratio N = /IL/& is greater than a certain value, close to 4, that depends on PT. For example Sf,, < 0 for & = 0.1 and N 2 3.99, for PT = 0.4 and N > 3.93 or for BT= 0.9 and N 2 3.89. It means that sudden arrest of the crack tip while the leading edge is still moving may lead to the situation that $,, < 0. It cannot be accepted from the physical point of view, thus the crack arrest should be considered as a continuous process. If, however, the trailing edge is suddenly arrested the leading edge must move with a speed not greater than quadruple speed of the crack before arrest. It is in contrast to the situation observed for elastic materials. Freund has shown[25] that when the crack in an elastic body is suddenly arrested the stationary field is immediately generated. In Fig. 3 the influence of the crack tip speed and the ratio N = fiLlfiTon the value of the crack tip opening displacement S$ is shown. For the range of crack tip speed 0 < PT < 1 the CTOD assumes the maximum for N = /?Jj& = 1. The slower the crack tip speed & the weaker maximum is observed. For example, for & = 0.1 the difference between the maximum and minimum values of St (in the range 0 < N < 4) is about 0.24%. For PT = 0.95 this difference increases to about 16%. Thus, taking into account the range of the crack tip speed observed in the experiment it will not be a great simplification if one assumes the equation of motion (based on the CTOD concept) for steady-state crack propagation. It also follows from eq. (13) that S$ is practically independent of N and for flT < 0.6 may be approximated by a straight line with a slope equal to -0.833.
CRACK MOTION
EQUATION
FOR D-P CRACK
In the first paragraph three equations of motion based on three different parameters have been presented. For D-P crack the SIF is equal to zero. Therefore we will concentrate on the two remaining quantities. (a) According to the definition (4) the energy flow into the plastic zone, divided by the crack tip speed may be interpreted as an energy release rate for the steady state motion only. For arbitrary motion with varying velocity of the D-P crack it cannot be proved that the motion is asymptotically steady-state since the contour L is precisely defined. The quantity Gt,,e in eq. (4) is usually called the resistance of the material to the crack motion. It may be noticed that for flT smaller than a
0.1-
I I I I I I III 0.2 0.4 0.6 0.6 I.0 1.2 I..41.6 I.6 2.0 :
0
N
I 0
1
I
1
I
I
1
1
1
1
I
I
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 I.0 I.1 BT
Fig. 3. Dependence of the CTOD-ut
on & and N.
336
A. NEIMITZ
certain critical value the quantity S& is greater than G,,, . There is no doubt that for many materials the material parameter Gt,,, is a function of a crack tip speed. The physical intuition and the observation of the experimental results suggest that the Gdillc either increases with /jT (initially increases slowly and from a certain value of /IT rapidly) or initially decreases with & (because for higher values of /IT there is little time for plastic deformation) and then increases with /?r (because of crack branching processes). Thus the equation in the form: s;l,, = %c
(26)
is certainly not correct for the first stage of propagation and probably is also not true for later stages except for a steady-state motion. Taking into account the above arguments one may assume the equation of motion of the D-P crack in the form: @ii - %c = Ma,
(27)
interpreting S& as a “driving force” on D-P crack, Ye,,, as a resistance of the material to the crack motion, M as an “equivalent” mass of the plastic zone and uT as a crack tip acceleration (in the proposed model of the crack tip kinetics ur should be interpreted as a “mean” acceleration of a trailing edge). The equivalent mass M can be assumed to be proportional to the plastic zone mass. When the crack motion is steady-state eq. (27) reduces to eq. (6). In general eq. (27) contains three unknown /IT, pL and aT . This differential equation could be solved for given loading history and material parameter %&c(~T, bL) if is supplemented by an additional relation between Pr and BL. For proposed model of crack kinetics (Fig. 4) one may introduce two additional equations derived from purely geometrical analysis along the crack trajectories. From eq. (8) and relation rf
the following, approximate
P
=
yi-.I
p
+ CTfu%. - 89
(28)
formulae follow:
(30) The computer simulation techniques may now be proposed to predict the crack motion history for given external loading and material properties. It will be a subject of the next paper. (b) Another equation of motion for the D-P crack can be postulated in more straightforward way. The concept of the CTOD is usually extended to the dynamic case and is written in the form of eq. (3). When both sides of eq. (3) are multiplied by rf we obtain an equation of motion in the form of eq. (6) which, as was shown earlier, is strictly true for steady-state motion only. The first
Fig. 4. Hypothetical scheme to determine the crack trajectories according to the u+ criterion.
Crack model with varying velocity
337
term in eq. (16), evaluated in eq. (17), may be interpreted as the irrecoverable work per unit area which has been done on the material element at the crack tip in bringing it to the point of separation. It does not take into account the energy dissipated within the plastic zone, which changes its length in time. This problem has been discussed in more detail in the review article by Freund [26]. Nevertheless, eq. (3) has often been utilized in crack motion analyses: it can easily be applied in the presented model of crack kinetics. Equations (13) and (3) are sufficient to perform the computer simulation procedure provided the crack tip trajectories are approximated by a piecewise-linear function. The assumed approximation of the real crack tips trajectories have been shown in Fig. 4. It is assumed that the crack starts propagation with a speed #?,.= fl’: at N = & = 0. The info~ation that the trailing edge has already started propagation reaches the leading edge at time S, . At this moment the leading edge starts motion with the speed fiL = fit and sends information about that to the trailing edge. This information reaches the trailing edge at time S = S,, and the latter corrects its speed to a new speed following from the equation of motion. Again the signal is sent to the leading edge which at time S = S, adjust its speed to fit = /?k and so on until the steady-state is reached. It can easily be shown that if S$ = const and dK,i,/dt = 0 (Fig. 4) the steady state is reached after two steps. The situation is more complex for a more general case, but the computational procedure is the same.
CONCLUSIONS The Achenbach and Neimitz[lf] analysis of the fast, mode III, Dugdal~Panasyuk (D-P) crack motion has been recast in the framework of an energy balance including varying crack-tip speed. The more important conclusions of the presented analysis are repeated below. (1) The crack tip opening displacement 6: reaches the maximum value for steady state motion (a = & = 8). This maximum decreases when the crack tip speed increases. (2) The equation of motion based on the CTOD is equivalent to the one based on ERR concept for steady-state motion only. (3) The equation of motion following from the energy rate balance provides a limitation on the crack tip speed without additional assumptions concerning the viscosity effect or velocity dependent resistance to fracture parameter. (4) The crack with a plastic zone ahead of the tip cannot be suddenly arrested. (5) The new equation of motion has been proposed. REFERENCES [I] J. D. Achenbach and V. Dunayevsky, Fields near a rapidly propagating crack tip in an elastic-perfectly plastic material. J. Me&. Phvs. Solids 29. 283-303 (1981). [2J Y. C. Gao and S. NematiNasser, Dynamic fields near a crack tip growing in an elastic-perfectly plastic. Me&. Marer. 2, 4760 (1983). [3] J. D. Achenbach, M. F. Kannien and C. H. Popelar, Crack tip fields for fast fracture of an elastic-plastic material, J. Me&. Phys. Soli& 29, 211-25 (1981). [4] K. K. Lo, Dynamic crack-tip fields in rate sensitive solids. J. Mech. Phys. Solids 31, 287-305 (1983). [5] L. B. Freund and A. S. Douglas, The influence of inertia on elastic-plastic antiplane shear crack growth. J. Mech. Phys. Solids 30, 59-74 (1982). [6] D. S. Dugdale, Yielding of steel sheets containing slits. J. Mech. Php.
Solids 8, 100-104 (1960). 171 V. V. Panasyuk, On the theory of crack extension durinn the deformation of a brittle bodv. DOD. Akad. A&k Ukr. RSR 9, I lSj-1188 (1960) (in Ukrainian). [8] L. B. Freund, Energy flux into the tip of an extending crack in an elastic solid. J. Elasricity 2, 341-349 (1972). 191 F. Nilsson. Dynamic Fracture Theorv. CISM. Udine. Italv (in mess). [I-O]C. Atkinson and J. D. Eshelby, The f”lowof energy into thetip o?a moving crack. Int. J. Fracture Mech. 4,3-8 (1986). [II] G. C. Sih, Dynamic aspects of crack propagation, in Inelastic Behaviour of Solids (Edited by M. F. Kannien), pp. 607-639. McGraw-Hill, New York (1970). [IZ] J. D. Achenbach and A. Neimitz, Fast fracture and crack arrest according to the Dugdale model. Engag Fracture Me&. 14, 385-395 (1981). (131 0. T. Hahn and R. Rosenheld, Local yielding and extension of a crack under plane stress. Acra Metall. 13,293-306
(19651. [14] J. N. Goodier and F. A. Field, Plastic energy dissipation in crack propagation, in Fracture of Solids (Edited by D. C. Drucker and J. J. Gilman), pp. 103-118. Wiley, New York (1963). [15] M. F. Kannien, An estimate of the limiting speed of a propagating ductile crack. J. Mech. Phys. Solids 16, 215-228 (1968).
A. NEIMITZ
338
[16] A. Neimitz, Crack propagation and arrest in plastic medium. Arch Mech. 33, 815-827 (1981). [17] A. Neimitz, 0 Pewnym dynamicznym zagadnieniu szczeliny w osrodku spr@ystym i spr@ysto-plastycznym. Engng Trans. 29, 629-640 (1981). [18] E. B. Glennie and J. R. Willis, An Examination of the effects of some idealized models of fracture on accelerating cracks. J. Mech. Phys. Solids 19, 1 l-30 (1871). [19] F. G. Friedlander, Q. J. Mech. appl. Math. 4, 344 (1951). [20] B. V. Kostrov, Unsteady propagation of longitudinal shear cracks. Prikl. Mat. Mekh. (English Translation), 30, 1241-1248 (1966). [21] J. D. Achenbach, Extension of a crack by a shear wave. ZAMP 21, 8877900 (1970). [22] J. C. Evvard, Distribution of wave drag and lift in the vicinity of wing tips at supersonic speeds. National Advisory Committee for Aeronautics, Technical Note No. 1382 (1947). [23] J. D. Achenbach, Dynamic effects in brittle fracture. Mechanics Today 1, l-54 (1972). [24] E. B. Glennie, A strain-rate dependent crack model. J. Mech. Phys. Solids 19, 255-272 (1971). [25] L. B. Freund, Crack propagation in an elastic solid subjected to general loading-II. Nonuniform rate of extension. J. Mech. Phys. Solids 20, 141-152 (1972). [26] L. B. Freund, Dynamic crack propagation in the mechanics of fracture. ADM 19 (Edited by F. Erdogan), pp. 1055134 ASME (1976).
APPENDIX The idea of the crack tip opening angle is a very popular the D-P zone suggests a geometrical definition of the CTOA at a particular point. Thus one may propose:
It follows from eq. (18) that CTOA(,=,
= 0 and CTOAI,
s
one in the crack motion theories. as an angle between two tangents
=,~ = co. However
The wedge-like shape of to the D-P zone profile
the mean value of the CTOA
has a finite value
‘pWlL, rp)
(CTOA),,,,
=
’
It is independent of external loading characterizing the resistance of material Thus at a high velocities of a crack the does not practically depend (CTGA),,,, decreases in deceleration domain).
aw TP
=
642)
and specimen geometry. One may consider this magnitude as a parameter increases with a crack tip speed at a given N. to crack propagation. (CTOA),,,, material behaves in a more brittle way. For very small crack tip velocities the on N and & (while for higher speeds it increases in acceleration domain and
(Receiued 13 April 1990)