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Crack propagation in cylindrical shells
Engineering Printed
Fracrure
Mechanics
Vol. 44, No. 6, pp. 931-947,
1993
0013-7944/93
in Great Britain.
CRACK
$6.00
+ 0.00
0 1993Pergamon Press Ltd.
PROPAGATION M. FARSHAD
IN CYLINDRICAL
SHELLS
and P. FLOELER
Swiss Federal Laboratories for Materials Testing and Research (EMPA), Uberlandstrasse CH-8600 Dubendorf. Switzerland
129.
Ahstrati-Results of a theoretical study of dynamic steady crack propagation in a circular cylindrical shell are presented, Equations of thin circular cylindrical shells are utilized for theoretical investigation. The semi-analytic crack tip stress solutions to shell equations are obtained by the perturbation method as well as an iterative method. Plots of internal stresses in the shell with a stationary as well as a steadily running crack are presented. These results are compared with those related to flat plates. It is concluded that the shell curvature has a p~noun~ effect on increasing the values of crack tip stresses. It would also qu~itatively affect the variation of stress field around the crack tip. In some cases, the curvature action is shown to be even more pronounced than the inertia effects.
1. INTRODUCTION OF PARTICULAR importance in the dynamic response of polymer pipes are the phenomena of crack initiation and crack propagation, and in particular the phenomenon of rapid crack propagation could affect several meters of a pipe’s length. For example, in polyethylene (FE) gas pipes, operating at pressures up to 5 bars, rapid crack propagation (RCP) having speeds of several hundreds of me&es per second (amounting to about 350 m/set for PE pipes) could be considered a global failure. In fact, a number of brittle failures of this type have been reported in gas dist~bution systems. Such global failures are not only troublesome, from the economic point of view, but also, gas and water distribution systems being significant “lifelines”, could cause some social hazards. Hence, safe designs, controlled production, and proper application of polymeric pipes appear to be important and certainly deserve due professional and research consideration. So far, a number of investigations have been carried out on certain aspects of the rapid crack propagation and dynamic feature phenomena. As a result, a set of recommendations, taking such phenomena into consideration in the design of polymer pipes, have, to some extent, been deduced and, in certain countries, have also been incorporated in related codes of practice. For example, the IS0 Standard sets a provisional operating pressure restriction in order to insure the prevention of long propagating fractures, It appears, however, that many aspects of this subject still remain unexplored and, therefore, deserve further investigation. References (l-71 represent some related literature. Some results on the theoretical modeling of crack propagation in flat plates are also available. However, theoretical studies on the RCP in polymer pipes that would highlight the influence of the pipe geometry, i.e. the curvature effects, seem to be lacking. Due to the fact that the curvature generally plays a very important role in the behavior of curved bodies, it is expected that it would also influence the static and dynamic crack behavior of pipes. The aim of the present study is to perform a theoretical investigation of the influence of the pipe geometry on the static and dynamic crack tip stress and displacement fields. The pipe geometric parameters include the pipe wall thickness and the pipe radius. In this work, an analytical solution to the problem will be sought. To this end, the governing equations of thin circular cylindrical shells will be utilized. An attempt will be made to find analytical solutions to these equations. 2. PERTURBATION
SOLUTION TO CRACK PROPAGATION CYLINDRICAL SHELLS
IN
2.1. Governing relations Consider an isotropic elastic (E, v) circular cylindrical thin shell with radius, R, and thickness, h. In Fig. 1 the shell geometry and a hypothesized longitudinal crack in the shell are demonstrated. 937
M. FARSHAD and P. FLUELER
938
Direction
of crack notion
\ ~~Rg~~~d~~~l crack Fig. 1. Shell geometry, loqitudinal
crack, and defined coordinate systems and the state of stress at the crack tip.
Also in this figure, two sets of coordinate systems are defined. One set of coordinates, consisting of x and # (or, equivalently, w) variables, are chosen to denote the longitudinal and hoop directions of the shell. The other set, consisting of r and 8, are the polar coordinates around the crack tip_ Both coordinate systems have their origin at the crack tip. Due to s~rnet~* only one half of the angle 8 variation, i.e. the range 0 < 6 < ‘ic,will be considered, Thus, the value B = 0 designates the axis of the crack v&Se the value 8 = a~signifies one of the stress-free crack surfaces. The s~rnp~~~~ governing partial hernial equations of a cylindricaf sheff could be presenred in terms of two coupled relations expressed in terms of radial displacement, W(X,cfr), and a stress function, Y&C, 4), as follows (see, for example, ref. [8]).
(1) D(1 - 9) R WI*.
v4y,-
The symbols introduced
in these eq~tio~s
m
have the following de~n~~ons.
v4=v2v29 V2=(
)“+(
p*,
13
where
$-+( >*,d$(
)‘,
(4)
The symbols K and D denote the bending and membrane stiffnesses, respectively, They have the following definitions:
The resultant of internal forces (per unit length)), consisting of the hoop forces, NgS axial force> Iv,, and membrane shear force, N,,, can be expressed in terms of stress functions by the following relations. A$ = Y”, Moroever,
the constitutive
Iv, = !Po,
A?&= - !P’O.
(6)
relation for the hoop stress, needed for the present analysis, is N+=D(w
+vo.v,‘))
(7)
where u, u, and w are the displacement components in the axial, hoop, and radial (normal to the shell) directions, respectively.
Crack propagation in cylindricalshells
939
There exist other relations related to the present theory of cylindrical shells which are not needed for the present analysis and thus are not given here. 2.2. Static solution to crack tip stresses Let us define the shell thickness-radius
ratio as h &=-* R
(8)
For relatively thin shells and pipes, this parameter is small and is of the order of magnitude of l/10 or less. In the special case of a flat plate, L = 0. With this definition, the governing differential equations (1) and (2) can be written in the following forms. 63’c14w
=2 _12(1
-V2)
lp”
E V4 Y = cEw”.
SO)
Now, assuming that the parameter t: is small, we consider the following perturbation to these equations.
series solution
Y(x,(cb)= !P(J+tY,+&J~+
***
(11)
w(x,#)=w$)+EW,+E2W*+
.+. .
(12)
The governing differential equation to the zeroth-order
part is
v‘Yv, = 0 and the governing equation to the first-order perturbation
(13) would be
V4Y, = Ew,“.
(14)
It is to be noted that the zeroth-order biharmonic governing equation (13) corresponds to the case of a flat plate in a state of plane stress. For an isotropic flat plate with a crack, the solution to the mode I cracking of this plane stress problem is available and can be found in texts on fracture mechanics. The first term of its series expression is
yo(r, 0) = Cd-
(15)
where C,, is a constant coefficient proportional to the stress intensity factor at the crack tip. Having the stress function, the stress field in the vicinity of the crack tip can be determined by the well-known stress function solution to the two-dimensional elasticity problem. Specifically, the component of crack tip stress in the y-direction of a cracked plate can be expressed in terms of polar coordinates as (see, for example Ref. [93) .
1 +sin~~in~
cTyO=$CZOs~ (
>
(16)
To utilize the flat plate results as the zeroth-order approximation to a cylindrical shell, we argue as follows. We rely on the point that, for zeroth-order approximation, the hoop stress is zero. In implementing this argument, we use displacement components (U and Y), obtained from a cracked flat plate, as the in-plane displacement components of a cylindrical shell. Hence, we write A& = D(w, + 02 + vu;) = 0.
(17)
From the above relation we obtain, utilizing the results of linear fracture mechanics of cracked plates in the state of plane stress,
940
M. FARSHAD and P. FLOELER
Substituting from (16) into the above relation, and by performing some simple manipulations, obtain i.
we
(191
This expression yields an infinite value for the radial displacement at r = 0, which is obviously not physically meaningful. That is, of course, also the case for the corresponding stress field. However, such infinite values should be treated as theoretical predictions of linear elastic fracture mechanics which are to be subjected to refinements. To find the first-order approximation, we utilize eq. (14), the right-hand side of which is now a known function. In order to derive the second derivative of w(r, #) with respect to X, and also for other necessary conversions of derivatives, we use the well-known coordinate t~nsfo~at~on relations from polar (r, 8) to Cartesian coordinate systems (x, y or, ~uivalently, 4). So, with the help of such transformations, the second x-derivative of (19) is evaluated to be
w~(r,e)=~c*,r( -llcos-+ “2” 15cos- y) 5’2 We now return to the first-order perturbation equation (14) and, having been inspired by the mathematical form of (23), we assume the following solution to that equation. Y,(r, e) = Qr”‘+2”2f,(t3). Substituting this solution into (14) and comparing functions, we determine f(e). Therefore
(21)
the coefficients of similar trigonometric
(22) Having obtained the stress function corresponding to first-order approximation and also utilizing the zeroth-order solution, eq. (15), we can determine the internal force resultant in the vicinity of the crack tip according to relations (6). The combined zeroth- and first-order approximate relations for these quantities are Ntis = Y,N+ GP;
(23)
N XS= !Pu,oO+ ?Pp”
(24)
NXdts = !P;O + c!P ;* s
(25)
Vacations of static crack tip shell stresses, dete~ined by relations (26)-(28) are plotted and are presented in Figs 2-4. These figures are plotted for some selected numerical values of the perturbation parameter E = h/R. The curve identified by t: = 0 signifies the solution to the flat plate problem. Of importance in crack development in pipes is the variation of normal circumferential stress around the crack tip. For a brittle material, and/or in brittle behavior, the fracture would occur
02
0 -0.2
Fig. 2. Static axial stress in a longitudinally cracked cylindrical shell--perturbation
solution.
Crack propagation in cylindrical shells
941
I .o
1: .
0.6
z”
0.6 0.4
I .5
2.0
2.5
3.0
8 U?adl
Fig. 3. Static stress normal to crack axis at the crack tip in a cylindrical shell-perturbation
solution.
due to the principal normal tensile stress in the vicinity of the crack tip. Thus, using the well-known transfo~ation formulas together with the expressions for internal forces in two perpendicular directions, we can find the principal stresses as well as the principal directions. The circ~ferential stress around the crack tip, as a function of the polar angle, is plotted in Fig. 4. As we observe, the location and the value of absolute tensile stress is dependent on the thickness parameter. For higher (h/R) ratios, the location of absolute maxima is shifted more towards the axis of the shell. The values of maxima are also accordingly increased. In all events, the shell curvature seems to have a magnifying effect of increasing the crack tip stresses relative to those in a flat plate. 2.3. Propagating crack in cylindrical shell Next, we consider the problem of a steadily running crack in a shell. A simplified governing equation of motion in the direction normal to the shell surface can be derived to have the following form. $‘?++$pw
= -$ph,,
ah
(26)
in which t is the time parameter and p is mass density of the shell material. We assume that the running crack has a steady speed of propagation, V. Then, at any instant of time, the coordinate of the crack tip would be 5 =x - vt.
(27)
1.0
p 2
‘Cmck
0.8
\
0.6
Fig. 4. Static ~~~fe~t~~
tip
stress around the crack crack-perturbation
tip in the cylindrical shell with a lon~t~~~ solution.
942
M. FARSHAD and P. FLUELER
025 -
8 1 Rad)
Fig. 5. Variation of stress component normal to the crack direction with the angle around the crack tip. (a) Flat plate with stationary crack. (b) Flat plate with propagating crack. (c) Cylindrical shell with stationary crack. (d) Cylindrical shell with propagating crack-perturbation solution.
Utilizing (30), we can easily convert the second time derivative to the spatial derivative. The result is
(28) Now, considering the right-hand side of eq. (26), we define the following parameter.
/3=$
(29)
Physically, this parameter is the ratio of square of crack speed to the square of the so-called “bar velocity”, that is, the velocity of the first mode of longitudinal waves in a slender elastic bar. The latter quantity is defined as
c; =-E
(30)
P’
To determine a general perturbation solution, involving two parameters, c and /I, we consider a typical perturbation expansion as follows. (31) which can be rewritten as N,,=N,,-bw,“.
(32)
The operations related to the perturbation scheme in the dynamic case are similar to what was described for the static problem. Figure 5 shows the plot of circumferential force around the crack tip of the flat plate and the cylindrical shell with a propagating longitudinal crack. In this graph, four plots, each signifying a particular class of problem, are presented. As we note, the curvature parameter, C, and the running crack parameter, /3, have pronounced influence on the distribution and maximal values of the internal hoop forces. Again, we observe that the shell curvature has a magnifying effect on the stress intensities. Moreover, we see that the location of maximum stress has shifted as compared with that of the flat plate problem. 3. ITERATIVE
SOLUTION
TO CRACK
TIP STRESSES
In this section, an iterative solution to the generally nonlinear equation of cylindrical shells having a longitudinal crack will be developed. The methodology of such iterative analysis is summarized in Fig. 6.
The problem of a steadily propagating crack in a flat plate is treated in the literature (see, for example, ref. [6]). To determine the dynamic crack tip solution to the cylindrical shell problem, we will utilize the results of dynamic crack tip stresses in a flat plate. In the following, a summary of the available flat plate solution is presented. Utiliing the so-called Helmholz resolution, the anal~i~al solution to the wave equations in an elastic medium may be obtained (see, for example, ref. [6]). In this connection, and for a steadily propagating crack wave in a plate, having a propagation velocity V, the following parameters can be defined (33) where c is the dimensionless crack speed, c,, is the bar velocity, and c,, c, are the speeds of dilatational and shear waves in the associated infinite medium. For a steadily propagating crack in a flat plate, the dynamic crack tip stress field is [6]
(35)
(36) where
(1-baf>
A=[dadasad =
fl -
(1 + af)]
+Z]“*
(37) (38)
cr,= [l - bfC2]“2
(39)
CL= r2(cos2 t? + a& six? 6)
(40)
tan 8& = #& tan 8.
(41)
and
the nondimensional membrane stress components. The The quantities nour, noYY,noxv represent ’ relations between these quantities and the internal membrane forces in the shell (N,,, NoY,,,NoX,,) are %xX= &XXIW
n0J.J= %J JffE,
noxy= No&W
(42)
where R is the shell radius and E is the elastic modulus of the shell material. The nonlinear equilibrium equations of cylindrical shells have also been developed and are available in the related literature (see, for example, ref. [IO]). If w(x, y) denotes the radial displacement of the shell (positive outwards), then the decoupled nonlinear governing equation of a thin elastic shell is
WE
FV4W
+ny-(n,w,+2n,w,+n,w,)=~,
where v4 W = W,, + 2W,,,, + WYj.JSjl.
(44)
In this equation, p is the radially applied load and K is the plate bending stiffness defined by EP K= 12(1 - v2)’
(45)
944
M. FARSHAD
and P. FLUELER
As mentioned before, the internal stress quantities nlrX,nyY, and nXVhave been nondimensionalized by dividing the physical internal force components by (R E), where R is the shell radius and E is the elastic modulus. The radial displacement, W, is also, with respect to R, already nondimensionalized. The partial derivatives with respect to the axial variable, x, and the hoop variable, y, are denoted by subscripts carrying these variables. Thus, for example, W, represents the second partial derivative of w relative to x, and so on. The constitutive relations of the elastic shell are
As an axisymmetric approximation
n, = (U’ + vu0 + vw)/(l - v2)
(46)
nY= (UO+ vu’ + W)/( 1 - v2)
(47)
n,. = (UO + vv’)/2(1 + v).
(48)
to (43) one may write W = (1 - VZ)nY.
(49)
All of the displacement components are nondimensionalized by dividing the physical components by the shell radius, R. In setting up an iterative scheme, we assume that the shell is thin enough so that the term including bending stiffness can be neglected. This assumption is validated by comparing the first term of eq. (43), having a coefficient proportional to (!I/R)~, with the other terms of this equation. On the basis of such an assumption, and with the help of relation (43), we obtain a first improved expression for the radial displacement. Thus, denoting the improved expression by w,, we have wi = (1 - v *) (4x w.v.x+ 2&y wx),+ nyywyy1.
(50)
Having obtained this improved expression, we now return to relations (46)-(49), from which we may determine an iterative expression for the crack tip stresses in the cylindrical shell. The iterative methodology, outlined herein, is summarized in Fig. 6. Having obtained the improved expressions for the crack tip stress field &, npi, n,,), we can use the following well-known stress transformation formula to find the normal component of the stress vector (circumferential stress) in an arbitrary direction 8 (measured from the shell axis). n,, = nxi sin’
e + nyicos2 e - 2n,,, sin e cos e.
(51)
The governing equilibrium equations can be further developed to include the inertia (flapping) effects. Thus, for a steadily propagating crack, the modified version of the improved expression for the radial displacement with radial inertial effects can be derived. The graphical results of the first iterative solution for the crack tip stresses in the shell are presented in Figs (7-10). In all these figures, the internal force field components versus the half circle around the crack tip (see Fig. 1) are plotted. The force quantities are nondimensionalized with respect to the static circumferential force at 0 = 0 in a flat plate with a stationary crack. In Fig. 7, plots of the static internal force in the direction normal to the crack axis in the flat plate and in the cylindrical shell are presented. Comparison of these two curves shows certain qualitative as well as quantitative differences between the flat plate and shell behavior. Specifically, one can see that the induced crack tip stresses in the shell are, in some locations, more than twice those of the flat plate. Moreover, two distinct regions of high stress in the shell are observed whereas the stress field in the plate remains fairly monotonous. Figure 8 shows the static circumferential stress at the crack tip in the flat plate and in the cylindrical shell. Again, we observe that the stress magnitudes in the shell are higher than those in the flat plate. Figure 9 presents plots of static as well as propagating crack solutions to the flat plate and the cylindrical shell. To determine these variations, the following numerical values for the parameters have been assumed: v = 0.35, b,= 0.8, b, = 1.4, c = 0.5.
Crack propagation
in cylindrical shells
945
Shell properbes v Determine the static solution for StreSw and displacements at the crock tip N,, N, ,Nxy ,a,~ as functions of polar variable
1 Use transformation relations (from x,ytor,S) to evaluate second derivatives I Use the shett equation (461 to find an improved hoop force
Employ relation (SO) to find the improved expression for radial dtsptacement
I Prescribe
crock speed
Ublize expressions I34)-(36) to determine dynomlc stresses at the crack ttp in o flat plate
1 Use the shell eauotion 150) to find on improved hoop stress in shell and hence the radiot displacement
-i
Fig. 6. The methodology for iterative analysis of shells for dynamic cracking.
The plots labeled C, D, and E in Fig. 9 are the dynamic contributions to the crack tip stresses. Therefore, the total crack tip stress would be obtained by the superposition of these dynamic values and the corresponding static quantities. Comparison of dynamic crack tip stresses with the static field of Fig. 9 shows that the dynamic stresses in the flat plate are relatively higher than the dynamic stresses in the shell. However, the total crack tip stress in the shell is always higher than that in the flat plate. In Fig. 10, plots of circumferential stress variation around the crack tip in the flat plate and the shell are presented. The same numerical values as in Fig. 8 have been assumed. From the plots of Fig. 10, the relative influence of the dynamic effects and the curvature effects can be compared.
Cylindrical
Angle
around the cmck tip,
Fig. 7. Static stress component direction. EFM44/6-H
normal
8 (~1
to the crack
shell
~
Angle around the crack tip, 8 (t)
Fig. 8. Static circumferential stress around the crack tip.
946
M. FARSHAD and P. FLUELER A: Shell with stationary crack B: Ftat plate with stationary
crack 0 Flat plate with DroDaaatina crack
g .x x ; 0) 5 5 P ?! 5 i 2
(c=O.5) D: Flat &ate with propa$tini crack (c=O.3) E: Shell with propagating crack (c-O.51
Static
case
(A) Flat plate (B) Cylindrical shell Dynamic case (c=O.5) (C) Flat plate
2.0
1.5
6 I .o
0.5
0
Angle around the crack tip, 8
( s)
Polar angle around the cmck tip, 8
(R 1
Fig. 9. Static and dynamic stress components normal to the crack direction.
Fig. 10. Static and dynamic circumferential stress.
As we observe, the curvature has a more pronounced the dynamic effect.
influence on the crack tip stresses than does
4. CONCLUSIONS Based on the theoretical drawn.
study carried out in this work, the following conclusions may be
(1) Both the perturbation
solution and the iterative solution to cylindrical shells show a pronounced influence of curvature on the crack tip stress field. The presence of curvature generally seems to increase the stress intensities in the vicinity of the crack tip. This result could be expected since the curvature is known to have a remarkable stiffening effect. In fact, the high stiffness of shell structures, as compared with flat plates, is due to the curved geometry of the shells. (2) In cylindrical shells, as in flat plates, the dynamic nature of a moving crack influences the stress field at the crack tip. The dynamic field is obviously dependent on the speed of crack propagation as well as on the material and geometric parameters of the shell. The iterative solution to the nonlinear shell equation shows that the influence of curvature is, in general, greater than the dynamic effects. The static and dynamic crack tip stress solutions in the cylindrical shell predict two distinct (3) regions of pronounced stress build-up. One of these regions is around the pipe axis (0 = 0), while the second region of high stress occurs around 8 = 120”. According to this result, one might expect that the curving of the crack path would potentially occur in the directions falling in one of these two zones. The stress magnitudes in the first zone are, however, higher, and so crack curving in the first region is more probable. The results obtained herein, for the pipe, have been based on the assumptions of the thin shell theories. Real polymer pipes, however, have a wall thickness to diameter ratio ranging from 0.10 to 0.20. For the higher ratios, some of these assumptions need certain refinements. The theoretical conclusions arrived at in this study should be correlated with the appropriate experimental findings. No experimental data could be found, however, relating to the crack tip stresses in cylindrical shells. One of the aims of the ongoing work in this direction is the performance of crack propagation experiments on polymer pipes, the results of which could be utilized for such correlations.
Crack propagation
in cylindrical shells
941
REFERENCES (11 C. G. Bragaw, Rapid crack propagation in MDPE. Proceedings 7th American Gas Association Plastic Fuel Pipe Symposium, New Orleans (1980). [2] J. M. Greig and L. Ewing, Fracture propagation in polyethylene (PE) gas pipes. Proc. 5th International Conf. Plastic Pipes, The Plastics and Rubber Institute, London, pp. 13.1-13.11 (1982). [3] J. M. Greig, Rapid crack propagation in hydrostatically pressurized 250 mm polyethylene pipe. Proc. 7th International Conf. Plastic Pipes, The Plastics and Rubber Institute, London, pp. 12.1-12.9 (1988). [4] R. Vancrombrugge, Fracture propagation in plastic pipes. Proc. 5th International Conf. Plastic Pipes, The Plastics and Rubber Institute, London, pp. 11.1-I 1.8 (1982). [5] M. Wolters, Rapid-crack propagation in PE pipes studied by modified Robertson tests. Proc. 6th International Co@ Plastic Pipes, The Plastic and Rubber Institute, London, pp. 22.1-22.6 (1985). [6] J. G. Williams, Fracture Mechanics of Polymers, p. 292. Ellis Hotwood, Chichester (1984). [7] P. Yayla, Rapid crack propagation in polyethylene gas pipes. Ph.D. thesis, Dept. Mech. Engng, Imperial College, London (1991). [8] W. Fliigge, Shells, in Handbook of Engineering Mechanics (Edited by W. Flilgge), pp. 40/17-40/18 (1962). [9] K. Hellan, introduction to Fracture Mechanics, pp. 236-237. McGraw-Hill, New York (1984). [IO] D. 0. Brush and B. 0. Almroth, Buckling of Bars, Plates, und Shells, pp. 142-163. McGraw-Hill, New York (1975). (Received 8 July 1992)
Fracrure
Mechanics
Vol. 44, No. 6, pp. 931-947,
1993
0013-7944/93
in Great Britain.
CRACK
$6.00
+ 0.00
0 1993Pergamon Press Ltd.
PROPAGATION M. FARSHAD
IN CYLINDRICAL
SHELLS
and P. FLOELER
Swiss Federal Laboratories for Materials Testing and Research (EMPA), Uberlandstrasse CH-8600 Dubendorf. Switzerland
129.
Ahstrati-Results of a theoretical study of dynamic steady crack propagation in a circular cylindrical shell are presented, Equations of thin circular cylindrical shells are utilized for theoretical investigation. The semi-analytic crack tip stress solutions to shell equations are obtained by the perturbation method as well as an iterative method. Plots of internal stresses in the shell with a stationary as well as a steadily running crack are presented. These results are compared with those related to flat plates. It is concluded that the shell curvature has a p~noun~ effect on increasing the values of crack tip stresses. It would also qu~itatively affect the variation of stress field around the crack tip. In some cases, the curvature action is shown to be even more pronounced than the inertia effects.
1. INTRODUCTION OF PARTICULAR importance in the dynamic response of polymer pipes are the phenomena of crack initiation and crack propagation, and in particular the phenomenon of rapid crack propagation could affect several meters of a pipe’s length. For example, in polyethylene (FE) gas pipes, operating at pressures up to 5 bars, rapid crack propagation (RCP) having speeds of several hundreds of me&es per second (amounting to about 350 m/set for PE pipes) could be considered a global failure. In fact, a number of brittle failures of this type have been reported in gas dist~bution systems. Such global failures are not only troublesome, from the economic point of view, but also, gas and water distribution systems being significant “lifelines”, could cause some social hazards. Hence, safe designs, controlled production, and proper application of polymeric pipes appear to be important and certainly deserve due professional and research consideration. So far, a number of investigations have been carried out on certain aspects of the rapid crack propagation and dynamic feature phenomena. As a result, a set of recommendations, taking such phenomena into consideration in the design of polymer pipes, have, to some extent, been deduced and, in certain countries, have also been incorporated in related codes of practice. For example, the IS0 Standard sets a provisional operating pressure restriction in order to insure the prevention of long propagating fractures, It appears, however, that many aspects of this subject still remain unexplored and, therefore, deserve further investigation. References (l-71 represent some related literature. Some results on the theoretical modeling of crack propagation in flat plates are also available. However, theoretical studies on the RCP in polymer pipes that would highlight the influence of the pipe geometry, i.e. the curvature effects, seem to be lacking. Due to the fact that the curvature generally plays a very important role in the behavior of curved bodies, it is expected that it would also influence the static and dynamic crack behavior of pipes. The aim of the present study is to perform a theoretical investigation of the influence of the pipe geometry on the static and dynamic crack tip stress and displacement fields. The pipe geometric parameters include the pipe wall thickness and the pipe radius. In this work, an analytical solution to the problem will be sought. To this end, the governing equations of thin circular cylindrical shells will be utilized. An attempt will be made to find analytical solutions to these equations. 2. PERTURBATION
SOLUTION TO CRACK PROPAGATION CYLINDRICAL SHELLS
IN
2.1. Governing relations Consider an isotropic elastic (E, v) circular cylindrical thin shell with radius, R, and thickness, h. In Fig. 1 the shell geometry and a hypothesized longitudinal crack in the shell are demonstrated. 937
M. FARSHAD and P. FLUELER
938
Direction
of crack notion
\ ~~Rg~~~d~~~l crack Fig. 1. Shell geometry, loqitudinal
crack, and defined coordinate systems and the state of stress at the crack tip.
Also in this figure, two sets of coordinate systems are defined. One set of coordinates, consisting of x and # (or, equivalently, w) variables, are chosen to denote the longitudinal and hoop directions of the shell. The other set, consisting of r and 8, are the polar coordinates around the crack tip_ Both coordinate systems have their origin at the crack tip. Due to s~rnet~* only one half of the angle 8 variation, i.e. the range 0 < 6 < ‘ic,will be considered, Thus, the value B = 0 designates the axis of the crack v&Se the value 8 = a~signifies one of the stress-free crack surfaces. The s~rnp~~~~ governing partial hernial equations of a cylindricaf sheff could be presenred in terms of two coupled relations expressed in terms of radial displacement, W(X,cfr), and a stress function, Y&C, 4), as follows (see, for example, ref. [8]).
(1) D(1 - 9) R WI*.
v4y,-
The symbols introduced
in these eq~tio~s
m
have the following de~n~~ons.
v4=v2v29 V2=(
)“+(
p*,
13
where
$-+( >*,d$(
)‘,
(4)
The symbols K and D denote the bending and membrane stiffnesses, respectively, They have the following definitions:
The resultant of internal forces (per unit length)), consisting of the hoop forces, NgS axial force> Iv,, and membrane shear force, N,,, can be expressed in terms of stress functions by the following relations. A$ = Y”, Moroever,
the constitutive
Iv, = !Po,
A?&= - !P’O.
(6)
relation for the hoop stress, needed for the present analysis, is N+=D(w
+vo.v,‘))
(7)
where u, u, and w are the displacement components in the axial, hoop, and radial (normal to the shell) directions, respectively.
Crack propagation in cylindricalshells
939
There exist other relations related to the present theory of cylindrical shells which are not needed for the present analysis and thus are not given here. 2.2. Static solution to crack tip stresses Let us define the shell thickness-radius
ratio as h &=-* R
(8)
For relatively thin shells and pipes, this parameter is small and is of the order of magnitude of l/10 or less. In the special case of a flat plate, L = 0. With this definition, the governing differential equations (1) and (2) can be written in the following forms. 63’c14w
=2 _12(1
-V2)
lp”
E V4 Y = cEw”.
SO)
Now, assuming that the parameter t: is small, we consider the following perturbation to these equations.
series solution
Y(x,(cb)= !P(J+tY,+&J~+
***
(11)
w(x,#)=w$)+EW,+E2W*+
.+. .
(12)
The governing differential equation to the zeroth-order
part is
v‘Yv, = 0 and the governing equation to the first-order perturbation
(13) would be
V4Y, = Ew,“.
(14)
It is to be noted that the zeroth-order biharmonic governing equation (13) corresponds to the case of a flat plate in a state of plane stress. For an isotropic flat plate with a crack, the solution to the mode I cracking of this plane stress problem is available and can be found in texts on fracture mechanics. The first term of its series expression is
yo(r, 0) = Cd-
(15)
where C,, is a constant coefficient proportional to the stress intensity factor at the crack tip. Having the stress function, the stress field in the vicinity of the crack tip can be determined by the well-known stress function solution to the two-dimensional elasticity problem. Specifically, the component of crack tip stress in the y-direction of a cracked plate can be expressed in terms of polar coordinates as (see, for example Ref. [93) .
1 +sin~~in~
cTyO=$CZOs~ (
>
(16)
To utilize the flat plate results as the zeroth-order approximation to a cylindrical shell, we argue as follows. We rely on the point that, for zeroth-order approximation, the hoop stress is zero. In implementing this argument, we use displacement components (U and Y), obtained from a cracked flat plate, as the in-plane displacement components of a cylindrical shell. Hence, we write A& = D(w, + 02 + vu;) = 0.
(17)
From the above relation we obtain, utilizing the results of linear fracture mechanics of cracked plates in the state of plane stress,
940
M. FARSHAD and P. FLOELER
Substituting from (16) into the above relation, and by performing some simple manipulations, obtain i.
we
(191
This expression yields an infinite value for the radial displacement at r = 0, which is obviously not physically meaningful. That is, of course, also the case for the corresponding stress field. However, such infinite values should be treated as theoretical predictions of linear elastic fracture mechanics which are to be subjected to refinements. To find the first-order approximation, we utilize eq. (14), the right-hand side of which is now a known function. In order to derive the second derivative of w(r, #) with respect to X, and also for other necessary conversions of derivatives, we use the well-known coordinate t~nsfo~at~on relations from polar (r, 8) to Cartesian coordinate systems (x, y or, ~uivalently, 4). So, with the help of such transformations, the second x-derivative of (19) is evaluated to be
w~(r,e)=~c*,r( -llcos-+ “2” 15cos- y) 5’2 We now return to the first-order perturbation equation (14) and, having been inspired by the mathematical form of (23), we assume the following solution to that equation. Y,(r, e) = Qr”‘+2”2f,(t3). Substituting this solution into (14) and comparing functions, we determine f(e). Therefore
(21)
the coefficients of similar trigonometric
(22) Having obtained the stress function corresponding to first-order approximation and also utilizing the zeroth-order solution, eq. (15), we can determine the internal force resultant in the vicinity of the crack tip according to relations (6). The combined zeroth- and first-order approximate relations for these quantities are Ntis = Y,N+ GP;
(23)
N XS= !Pu,oO+ ?Pp”
(24)
NXdts = !P;O + c!P ;* s
(25)
Vacations of static crack tip shell stresses, dete~ined by relations (26)-(28) are plotted and are presented in Figs 2-4. These figures are plotted for some selected numerical values of the perturbation parameter E = h/R. The curve identified by t: = 0 signifies the solution to the flat plate problem. Of importance in crack development in pipes is the variation of normal circumferential stress around the crack tip. For a brittle material, and/or in brittle behavior, the fracture would occur
02
0 -0.2
Fig. 2. Static axial stress in a longitudinally cracked cylindrical shell--perturbation
solution.
Crack propagation in cylindrical shells
941
I .o
1: .
0.6
z”
0.6 0.4
I .5
2.0
2.5
3.0
8 U?adl
Fig. 3. Static stress normal to crack axis at the crack tip in a cylindrical shell-perturbation
solution.
due to the principal normal tensile stress in the vicinity of the crack tip. Thus, using the well-known transfo~ation formulas together with the expressions for internal forces in two perpendicular directions, we can find the principal stresses as well as the principal directions. The circ~ferential stress around the crack tip, as a function of the polar angle, is plotted in Fig. 4. As we observe, the location and the value of absolute tensile stress is dependent on the thickness parameter. For higher (h/R) ratios, the location of absolute maxima is shifted more towards the axis of the shell. The values of maxima are also accordingly increased. In all events, the shell curvature seems to have a magnifying effect of increasing the crack tip stresses relative to those in a flat plate. 2.3. Propagating crack in cylindrical shell Next, we consider the problem of a steadily running crack in a shell. A simplified governing equation of motion in the direction normal to the shell surface can be derived to have the following form. $‘?++$pw
= -$ph,,
ah
(26)
in which t is the time parameter and p is mass density of the shell material. We assume that the running crack has a steady speed of propagation, V. Then, at any instant of time, the coordinate of the crack tip would be 5 =x - vt.
(27)
1.0
p 2
‘Cmck
0.8
\
0.6
Fig. 4. Static ~~~fe~t~~
tip
stress around the crack crack-perturbation
tip in the cylindrical shell with a lon~t~~~ solution.
942
M. FARSHAD and P. FLUELER
025 -
8 1 Rad)
Fig. 5. Variation of stress component normal to the crack direction with the angle around the crack tip. (a) Flat plate with stationary crack. (b) Flat plate with propagating crack. (c) Cylindrical shell with stationary crack. (d) Cylindrical shell with propagating crack-perturbation solution.
Utilizing (30), we can easily convert the second time derivative to the spatial derivative. The result is
(28) Now, considering the right-hand side of eq. (26), we define the following parameter.
/3=$
(29)
Physically, this parameter is the ratio of square of crack speed to the square of the so-called “bar velocity”, that is, the velocity of the first mode of longitudinal waves in a slender elastic bar. The latter quantity is defined as
c; =-E
(30)
P’
To determine a general perturbation solution, involving two parameters, c and /I, we consider a typical perturbation expansion as follows. (31) which can be rewritten as N,,=N,,-bw,“.
(32)
The operations related to the perturbation scheme in the dynamic case are similar to what was described for the static problem. Figure 5 shows the plot of circumferential force around the crack tip of the flat plate and the cylindrical shell with a propagating longitudinal crack. In this graph, four plots, each signifying a particular class of problem, are presented. As we note, the curvature parameter, C, and the running crack parameter, /3, have pronounced influence on the distribution and maximal values of the internal hoop forces. Again, we observe that the shell curvature has a magnifying effect on the stress intensities. Moreover, we see that the location of maximum stress has shifted as compared with that of the flat plate problem. 3. ITERATIVE
SOLUTION
TO CRACK
TIP STRESSES
In this section, an iterative solution to the generally nonlinear equation of cylindrical shells having a longitudinal crack will be developed. The methodology of such iterative analysis is summarized in Fig. 6.
The problem of a steadily propagating crack in a flat plate is treated in the literature (see, for example, ref. [6]). To determine the dynamic crack tip solution to the cylindrical shell problem, we will utilize the results of dynamic crack tip stresses in a flat plate. In the following, a summary of the available flat plate solution is presented. Utiliing the so-called Helmholz resolution, the anal~i~al solution to the wave equations in an elastic medium may be obtained (see, for example, ref. [6]). In this connection, and for a steadily propagating crack wave in a plate, having a propagation velocity V, the following parameters can be defined (33) where c is the dimensionless crack speed, c,, is the bar velocity, and c,, c, are the speeds of dilatational and shear waves in the associated infinite medium. For a steadily propagating crack in a flat plate, the dynamic crack tip stress field is [6]
(35)
(36) where
(1-baf>
A=[dadasad =
fl -
(1 + af)]
+Z]“*
(37) (38)
cr,= [l - bfC2]“2
(39)
CL= r2(cos2 t? + a& six? 6)
(40)
tan 8& = #& tan 8.
(41)
and
the nondimensional membrane stress components. The The quantities nour, noYY,noxv represent ’ relations between these quantities and the internal membrane forces in the shell (N,,, NoY,,,NoX,,) are %xX= &XXIW
n0J.J= %J JffE,
noxy= No&W
(42)
where R is the shell radius and E is the elastic modulus of the shell material. The nonlinear equilibrium equations of cylindrical shells have also been developed and are available in the related literature (see, for example, ref. [IO]). If w(x, y) denotes the radial displacement of the shell (positive outwards), then the decoupled nonlinear governing equation of a thin elastic shell is
WE
FV4W
+ny-(n,w,+2n,w,+n,w,)=~,
where v4 W = W,, + 2W,,,, + WYj.JSjl.
(44)
In this equation, p is the radially applied load and K is the plate bending stiffness defined by EP K= 12(1 - v2)’
(45)
944
M. FARSHAD
and P. FLUELER
As mentioned before, the internal stress quantities nlrX,nyY, and nXVhave been nondimensionalized by dividing the physical internal force components by (R E), where R is the shell radius and E is the elastic modulus. The radial displacement, W, is also, with respect to R, already nondimensionalized. The partial derivatives with respect to the axial variable, x, and the hoop variable, y, are denoted by subscripts carrying these variables. Thus, for example, W, represents the second partial derivative of w relative to x, and so on. The constitutive relations of the elastic shell are
As an axisymmetric approximation
n, = (U’ + vu0 + vw)/(l - v2)
(46)
nY= (UO+ vu’ + W)/( 1 - v2)
(47)
n,. = (UO + vv’)/2(1 + v).
(48)
to (43) one may write W = (1 - VZ)nY.
(49)
All of the displacement components are nondimensionalized by dividing the physical components by the shell radius, R. In setting up an iterative scheme, we assume that the shell is thin enough so that the term including bending stiffness can be neglected. This assumption is validated by comparing the first term of eq. (43), having a coefficient proportional to (!I/R)~, with the other terms of this equation. On the basis of such an assumption, and with the help of relation (43), we obtain a first improved expression for the radial displacement. Thus, denoting the improved expression by w,, we have wi = (1 - v *) (4x w.v.x+ 2&y wx),+ nyywyy1.
(50)
Having obtained this improved expression, we now return to relations (46)-(49), from which we may determine an iterative expression for the crack tip stresses in the cylindrical shell. The iterative methodology, outlined herein, is summarized in Fig. 6. Having obtained the improved expressions for the crack tip stress field &, npi, n,,), we can use the following well-known stress transformation formula to find the normal component of the stress vector (circumferential stress) in an arbitrary direction 8 (measured from the shell axis). n,, = nxi sin’
e + nyicos2 e - 2n,,, sin e cos e.
(51)
The governing equilibrium equations can be further developed to include the inertia (flapping) effects. Thus, for a steadily propagating crack, the modified version of the improved expression for the radial displacement with radial inertial effects can be derived. The graphical results of the first iterative solution for the crack tip stresses in the shell are presented in Figs (7-10). In all these figures, the internal force field components versus the half circle around the crack tip (see Fig. 1) are plotted. The force quantities are nondimensionalized with respect to the static circumferential force at 0 = 0 in a flat plate with a stationary crack. In Fig. 7, plots of the static internal force in the direction normal to the crack axis in the flat plate and in the cylindrical shell are presented. Comparison of these two curves shows certain qualitative as well as quantitative differences between the flat plate and shell behavior. Specifically, one can see that the induced crack tip stresses in the shell are, in some locations, more than twice those of the flat plate. Moreover, two distinct regions of high stress in the shell are observed whereas the stress field in the plate remains fairly monotonous. Figure 8 shows the static circumferential stress at the crack tip in the flat plate and in the cylindrical shell. Again, we observe that the stress magnitudes in the shell are higher than those in the flat plate. Figure 9 presents plots of static as well as propagating crack solutions to the flat plate and the cylindrical shell. To determine these variations, the following numerical values for the parameters have been assumed: v = 0.35, b,= 0.8, b, = 1.4, c = 0.5.
Crack propagation
in cylindrical shells
945
Shell properbes v Determine the static solution for StreSw and displacements at the crock tip N,, N, ,Nxy ,a,~ as functions of polar variable
1 Use transformation relations (from x,ytor,S) to evaluate second derivatives I Use the shett equation (461 to find an improved hoop force
Employ relation (SO) to find the improved expression for radial dtsptacement
I Prescribe
crock speed
Ublize expressions I34)-(36) to determine dynomlc stresses at the crack ttp in o flat plate
1 Use the shell eauotion 150) to find on improved hoop stress in shell and hence the radiot displacement
-i
Fig. 6. The methodology for iterative analysis of shells for dynamic cracking.
The plots labeled C, D, and E in Fig. 9 are the dynamic contributions to the crack tip stresses. Therefore, the total crack tip stress would be obtained by the superposition of these dynamic values and the corresponding static quantities. Comparison of dynamic crack tip stresses with the static field of Fig. 9 shows that the dynamic stresses in the flat plate are relatively higher than the dynamic stresses in the shell. However, the total crack tip stress in the shell is always higher than that in the flat plate. In Fig. 10, plots of circumferential stress variation around the crack tip in the flat plate and the shell are presented. The same numerical values as in Fig. 8 have been assumed. From the plots of Fig. 10, the relative influence of the dynamic effects and the curvature effects can be compared.
Cylindrical
Angle
around the cmck tip,
Fig. 7. Static stress component direction. EFM44/6-H
normal
8 (~1
to the crack
shell
~
Angle around the crack tip, 8 (t)
Fig. 8. Static circumferential stress around the crack tip.
946
M. FARSHAD and P. FLUELER A: Shell with stationary crack B: Ftat plate with stationary
crack 0 Flat plate with DroDaaatina crack
g .x x ; 0) 5 5 P ?! 5 i 2
(c=O.5) D: Flat &ate with propa$tini crack (c=O.3) E: Shell with propagating crack (c-O.51
Static
case
(A) Flat plate (B) Cylindrical shell Dynamic case (c=O.5) (C) Flat plate
2.0
1.5
6 I .o
0.5
0
Angle around the crack tip, 8
( s)
Polar angle around the cmck tip, 8
(R 1
Fig. 9. Static and dynamic stress components normal to the crack direction.
Fig. 10. Static and dynamic circumferential stress.
As we observe, the curvature has a more pronounced the dynamic effect.
influence on the crack tip stresses than does
4. CONCLUSIONS Based on the theoretical drawn.
study carried out in this work, the following conclusions may be
(1) Both the perturbation
solution and the iterative solution to cylindrical shells show a pronounced influence of curvature on the crack tip stress field. The presence of curvature generally seems to increase the stress intensities in the vicinity of the crack tip. This result could be expected since the curvature is known to have a remarkable stiffening effect. In fact, the high stiffness of shell structures, as compared with flat plates, is due to the curved geometry of the shells. (2) In cylindrical shells, as in flat plates, the dynamic nature of a moving crack influences the stress field at the crack tip. The dynamic field is obviously dependent on the speed of crack propagation as well as on the material and geometric parameters of the shell. The iterative solution to the nonlinear shell equation shows that the influence of curvature is, in general, greater than the dynamic effects. The static and dynamic crack tip stress solutions in the cylindrical shell predict two distinct (3) regions of pronounced stress build-up. One of these regions is around the pipe axis (0 = 0), while the second region of high stress occurs around 8 = 120”. According to this result, one might expect that the curving of the crack path would potentially occur in the directions falling in one of these two zones. The stress magnitudes in the first zone are, however, higher, and so crack curving in the first region is more probable. The results obtained herein, for the pipe, have been based on the assumptions of the thin shell theories. Real polymer pipes, however, have a wall thickness to diameter ratio ranging from 0.10 to 0.20. For the higher ratios, some of these assumptions need certain refinements. The theoretical conclusions arrived at in this study should be correlated with the appropriate experimental findings. No experimental data could be found, however, relating to the crack tip stresses in cylindrical shells. One of the aims of the ongoing work in this direction is the performance of crack propagation experiments on polymer pipes, the results of which could be utilized for such correlations.
Crack propagation
in cylindrical shells
941
REFERENCES (11 C. G. Bragaw, Rapid crack propagation in MDPE. Proceedings 7th American Gas Association Plastic Fuel Pipe Symposium, New Orleans (1980). [2] J. M. Greig and L. Ewing, Fracture propagation in polyethylene (PE) gas pipes. Proc. 5th International Conf. Plastic Pipes, The Plastics and Rubber Institute, London, pp. 13.1-13.11 (1982). [3] J. M. Greig, Rapid crack propagation in hydrostatically pressurized 250 mm polyethylene pipe. Proc. 7th International Conf. Plastic Pipes, The Plastics and Rubber Institute, London, pp. 12.1-12.9 (1988). [4] R. Vancrombrugge, Fracture propagation in plastic pipes. Proc. 5th International Conf. Plastic Pipes, The Plastics and Rubber Institute, London, pp. 11.1-I 1.8 (1982). [5] M. Wolters, Rapid-crack propagation in PE pipes studied by modified Robertson tests. Proc. 6th International Co@ Plastic Pipes, The Plastic and Rubber Institute, London, pp. 22.1-22.6 (1985). [6] J. G. Williams, Fracture Mechanics of Polymers, p. 292. Ellis Hotwood, Chichester (1984). [7] P. Yayla, Rapid crack propagation in polyethylene gas pipes. Ph.D. thesis, Dept. Mech. Engng, Imperial College, London (1991). [8] W. Fliigge, Shells, in Handbook of Engineering Mechanics (Edited by W. Flilgge), pp. 40/17-40/18 (1962). [9] K. Hellan, introduction to Fracture Mechanics, pp. 236-237. McGraw-Hill, New York (1984). [IO] D. 0. Brush and B. 0. Almroth, Buckling of Bars, Plates, und Shells, pp. 142-163. McGraw-Hill, New York (1975). (Received 8 July 1992)