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Second-order approximation to the crack trajectory equation
Engineering Fracture MechwicsVol. Printed in Great Britain.
24, No. I, pp. 121-126.1986 0
0013-7944/86 %3.00+ .I0 1986 Pergamon Press Ltd.
SECOND-ORDER APPROXIMATION TO THE CRACK TRAJECTORY EQUATION E. VIOLA Istituto di Scienza delle Costruzioni, Universita di Bologna, Bologna, Italy and A. PIVA Dipartimento di Fisica, Universita di Bologna, Bologna, Italy Abstract-According to an approach recently proposed[5], this analysis includes the second-order approximation in order to obtain the deviation of the extended crack tip from the original direction. The dependence of the curvature on the crack path upon the inplane stress fields and the derivatives of the stress intensity factors with respect to the original crack length is shown. In particular, the analysis indicates that, for some values of the above-mentioned parameters, an asymptote perpendicular to the original crack direction occurs which specifies the unstable behaviour of the crack path. In addition, the asymptotic crack path behaviour is obtained to the same order of approximation when stability is assumed.
1. INTRODUCTION IN RECENT years a number of papers have been concerned with the incipient stages ofcrack growth[i-
61. Various analytical procedures for calculating characteristic quantities occurring in the fracture criteria predicting the crack tip behaviour have been used. Perturbation methods for crack path prediction have been used in [l] and placed in [2] within the framework of Muskhelishvili’s complex potential method. As far as the paths of crack extension are concerned, results were obtained in [3] by using full field solutions and a step-by-step procedure. A similar technique as in [l] and [2] has been adopted in [4], giving both the first- and second-order perturbation results. In [S] an alternative procedure avoiding integral representations and based on the maximum opening criterion has been proposed to predict crack paths in sheets of brittle materials under biaxial loading. In [6] the method proposed in [S] is extended to a system of two collinear cracks embedded in an infinite medium subjected to biaxial loading. In this paper, the above procedure, based on the maximum tensile stress criterion, is used to obtain the crack trajectory taking into account the second-order terms. The results derived through the foregoing procedure are in agreement, apart from some numerical coefficients, with those obtained in [4] where the local symmetry criterion was employed. Further features concerning the crack tip stability are described and the effects of both lateral stress and higher-order terms are indicated. Finally, the asymptotic expression for the crack trajectory is derived to the same order of approximation when stability occurs.
2. CRACK PATH EQUATION Let us consider an inifinite two-dimensional homogeneous and isotropic elastic body, with a traction-free straight center-crack of length L = 2a, subjected to some external biaxial loading to infinity allowing the crack tip extension through Mode I and Mode II as well. The elastostatic stress field is required to satisfy the well-known equilibrium equations[7] :
ox,+ oyy= ayv-a,,
ww +WI,
+ 2ia,, = 2[(z-z)~(z)-~(z)+~(z)],
(2.1)
where 0._ a,,,,,crXY are the Cartesian stress components, and m(z), n(z) are holomorphic functions in the 121
122
E. VIOLA and A. PIVA
whole cut z plane. The boundary value problem solution gives (see also [7]) ~z) = (2r + P)z -- fi’ J&G? 2’
(2.2)
R(z) = CD(z) + i=‘,
where r and r’ are two constants related to the applied stress field to infinity. Confining attention to the right crack tip, z = L/2, the following relations hold between the fundamental stress combinations when expressed in either set of local polar coordinates (r, 0) and local Cartesian coordinates (x, y) :
or,+ @,I3=
Qxx + cyy’
(2.3)
gee- cr, + 2ia,, = (a,, - (T,, + 2io,,) e”‘.
By expanding (2.2) into power series, eqns (2.1H2.3) lead to the following asymptotic representations of the polar stress components cge and or0 :
crge= A sin2 8+ ___ 4&Y
3cos~+cos~)-*(3sin~+3sin~) ( + O(r), ’
tJ,,g= -
(24) .
+$&(sing-sinT)++&(5cosT-cos$+O(r), where K, and K, are the stress intensity factors at the crack tip, b, and b, are the coefficients of the terms proportional to the square root of the distance from the crack tip, and A is a constant depending on the applied lateral stress. From now on it is assumed that crack extension occurs accordingly with the maximum tensile stress criterion and that the deviation of the crack tip from its original direction is described by the function I(x) (see Fig. l), which is supposed to be small with respect to its argument. Assuming a slight curvature for the crack extension, the following approximations may be considered up to second-order terms : sin 6 N 19‘v X(x), cos
8N
i - 12(~)/2,
(2.5)
r N x[l +X2(x)/2].
The maximum tensile stress criterion requires that, for some distance r from the crack tip, the angle 8
6
Fig. 1. Geometry of the cracked body and external loading.
Second-order approx~ation
to the crack trajectory equation
123
of extension is such as to satisfy the following conditions of positive maximum:
(2.6)
Hence, from (Z-4), and (2.5), the conditions (2.6) lead respectively to (4A~-~K,~~b,x)~2-(6K,+10b,x)R’+4(K,+b,x)
(1-~J2;;;-2$ 1
I’_
1
>
> 0,
(z!!L+gz, >jp+~+gz,=o, 1
1
(27K,-145b,x-128AJj;;;;-)Y2+(84K,-t620b,~~~+(64A~-32K,f40b,x) By noting from (2.4) that, in the neighbourhood approximations are valid:
(2.7)
1
< 0.
of the preexisting crack tip, the following
,/t%-qqB = K,+b,x+0(;1/x), ,,hh,+ = K,
1
+ b,x + O(~/x),
P-8)
defining the stress intensity factors at the extended crack tip as
(2.9)
one obtains
(2.10)
In addition, in view of (2.9),, the following approximate expression may he obtained from (2.4), :
(2.11)
which leads to A’(0)= - 2K,/K, because K,, = 0 when x = 0. It follows from (2.10), that b2 is of the first order in A’.Hence, in the required approximation, eqn (2.7), gives
X(x) =
X(0) -( lOb,,‘3K,)x 1 -(8A/3K,)~-(5~~/3K,)xy
(2.12)
where its range of validity is bounded by constraints (2.7), and (2.7),. By expanding (2.12) for 81Aj,/%/3K, << 1 [n’(x) a: A/x], the near-tip crack path behaviour is obtained : L(x) =
(11x +
/!h J12+ yx2+ o(x5’2),
(2.13)
124
E. VIOLA
and
A. PIVA
where a = -2K,/K,, (2.14)
B = ~a(AIKJJ2n,
It may be noticed that, apart from some numerical coefficients, the behaviour (2.13) is the same as that predicted in [4]. It should also be pointed out from (2.12) that depending on the lateral stress as well as on b,, an asymptote perpendicular to the x-axis may occur when 3K, - 8Ae5b,x = 0. Henceforth the crack path stability depends not only on the lateral load applied but also upon the derivatives of both Mode I and Mode II stress intensity factors, as pointed out also in [4]. In addition, when large values of x are in accord with restrictions (2.7), and (2.7),, eqn (2.12) leads to nb, = 2bz/b, = -l’(O), which is physically admissible when 3K, -8A,,f?&-5b,x # 0. When stability occurs, the approximation L’(x) N n/x is allowed also for large values of x so that the following “asymptotic” behaviour may be obtained from (2.12) : n(x) = -i’(O)[x-
~~+2?gy+o(x-‘:‘)].
3. CRACK PATH
(2.15)
PICTURES
Some graphical representations of the results obtained in Section 2 are reported in the following. In Fig. 2 the dimensionless crack path I(x)/a is represented against the normalized distance x/a from the crack tip for various values of the biaxial load parameter, when the crack inclination angle with respect to the lateral stress ko is 6 = 2”. Dashed lines refer to second-order approximation as predicted by eqn (2.12), whereas solid lines refer to crack path equation in which only first-order terms are retained. It may be noted that, in both cases, the crack path is unstable for k > 1 and stable for k c 1. In the last case, it tends to revert to its original straight path. In the particular case of k = 1, the crack will extend in a self-similar manner. Different order approximations on the crack path stability and instability are illustrated in Figs. 3 and 4, respectively, for lateral compressive and tensile load. It should be noted that the second-
Fig. 2. Comparison
-
First order *ppmximafion
----
Second order approximation
between first- and second-order approximations angle is 6 = 2”.
on crack path behaviour
when the crack
Second-order approx~ation
to the crack trajectory equation
-
First
----
Semnd cndei approxi~tion
ox&r
125
sppmximation
Fig. 3. Different order approximation effect on the crack path stability for k = -0.5.
order-approximation effect on the crack path behaviour increases when the crack angle tends to vanish and the biaxial load parameter k is negative. An opposite effect may be observed when lateral tensile stresses are considered, with k > 1. From Figs. 2 and 4 it may be noted that the crack path deviation when the second-order terms are retained is always smaller with respect to the first-order approximation effect. Note that the figures refer to eqn (2.12) where the first-order approximation is obtained by setting b, = b2 = 0. Figure 5 shows a comparison between crack path predictions obtained according to the present theory [see eqn (2.13) and eqn (24) of Ref. 41. It should be noted that the comparison is valid only up to small distances from the original crack tip because of the required approximations. Figure 6 shows the variation of the dimensionless distance xJa of the asymptote perpendicular to the crack-axis against the biaxial load parameter k. When increasing values of k are considered, the limit distance from the original crack tip for the validity of the aforesaid approach decreases very rapidly. It may be noted that when second-order terms are retained (dashed line), the validity range is reduced with respect to the first-order approximation (solid line). In addition, note that the distance x,/a is an upper bound as a function of k and is independent of the derivatives of the stress intensity factors with respect to the initial crack length. In reality, the validity of the crack path equation &x)/a is limited to a smaller range according to the required approximations. 4. REMARKS At this stage some remarks are reported to point out the differences between the procedure and the results reported in this paper and the results in [4] and, more recently, also in [8]. It should be noted that both investigations refer to mathematical modeIs concerning the virtual extension of the crack tip up to second-order effects. Both models give the same formal representation for crack paths which coincide, with the same inclination, when the original crack tip is approached. However, except
-XIX) a 0.10
-
First order approximation
----
Second order ,qqroximation
t
0 L!L
0.05
0.10
0.16
Fig. 4. Different order approximation effect on the crack path instability r for k = 1.5.
126
E. VIOLA UXI
T O.W8
I
and A. PIVA
-Present
theory [equation (2.131]
-‘-.-
Reference ]4] [[equation
(24)]
~,
- 0.006
Fig. 5. Comparison
0.10
between asymptotic
crack path predictions.
-
First order apprwimafion
----
Semnd order approximation
t
Fig. 6. Second-order-terms
effect on the upper bound
for path validity
range.
for the linear term, the numerical coefficients do not coincide which, in the authors’ opinion, depends on the different fracture criteria employed. In fact, in [4] the local symmetry criterion (K,, = 0) is used and the family of crack trajectories must be assumed in order to evaluate the numerical coefficients of the path equation. In this paper the maximum hoop stress criterion is used and the crack path equation is obtained without preliminary assumption. Acknowledgement-This
work was supported
by the National
(Italian)
Research
Council
(C.N.R.).
REFERENCES [l] R. V. Goldstein and R. L. Salganik, Brittle fracture of solids with arbitrary cracks. Int. J. Fracture 10, 5077523 (1974). [2] B. Cotterel and J. R. Rice, Slightly curved or kinked cracks. Int. J. Fracture 18, 155-169 (1980). [3] H. Liebowitz, J. D. Lee and N. Subramonian, Criteria for predicting crack extension angle and path in plane crack problems. Proc. Int. Conf. on Analytical and Exp. Fracture Mech., Roma (1980), pp. 239-250. [4] B. L. Karihaloo, L. M. Keer, S. Nemat-Nasser and A. Cranratmachai, Approximate description of crack kinking and curving. J. Appl. Mech. 48,515-519 (1981). [S] E. Viola and A. Piva, Crack paths in sheets of brittle material. Engng Fracture Mech. 19,1069-1084(1984). [6] A Piva and E. Viola, On paths of two equal collinear cracks. Engng Fracture Me&, 20,735-741(1984). [7] N. I. Muskhelishvili, Some basic problems of the Mathematical Theory of Elasticity, 3rd edn. Noordhoof, Groningen (1963). [S] Y. Sumi, S. Nemat-Nasser and L. M. Keer, On crack branching and curving in a finite body. Int. .r. Fracture 21,67-79 (1983).
24, No. I, pp. 121-126.1986 0
0013-7944/86 %3.00+ .I0 1986 Pergamon Press Ltd.
SECOND-ORDER APPROXIMATION TO THE CRACK TRAJECTORY EQUATION E. VIOLA Istituto di Scienza delle Costruzioni, Universita di Bologna, Bologna, Italy and A. PIVA Dipartimento di Fisica, Universita di Bologna, Bologna, Italy Abstract-According to an approach recently proposed[5], this analysis includes the second-order approximation in order to obtain the deviation of the extended crack tip from the original direction. The dependence of the curvature on the crack path upon the inplane stress fields and the derivatives of the stress intensity factors with respect to the original crack length is shown. In particular, the analysis indicates that, for some values of the above-mentioned parameters, an asymptote perpendicular to the original crack direction occurs which specifies the unstable behaviour of the crack path. In addition, the asymptotic crack path behaviour is obtained to the same order of approximation when stability is assumed.
1. INTRODUCTION IN RECENT years a number of papers have been concerned with the incipient stages ofcrack growth[i-
61. Various analytical procedures for calculating characteristic quantities occurring in the fracture criteria predicting the crack tip behaviour have been used. Perturbation methods for crack path prediction have been used in [l] and placed in [2] within the framework of Muskhelishvili’s complex potential method. As far as the paths of crack extension are concerned, results were obtained in [3] by using full field solutions and a step-by-step procedure. A similar technique as in [l] and [2] has been adopted in [4], giving both the first- and second-order perturbation results. In [S] an alternative procedure avoiding integral representations and based on the maximum opening criterion has been proposed to predict crack paths in sheets of brittle materials under biaxial loading. In [6] the method proposed in [S] is extended to a system of two collinear cracks embedded in an infinite medium subjected to biaxial loading. In this paper, the above procedure, based on the maximum tensile stress criterion, is used to obtain the crack trajectory taking into account the second-order terms. The results derived through the foregoing procedure are in agreement, apart from some numerical coefficients, with those obtained in [4] where the local symmetry criterion was employed. Further features concerning the crack tip stability are described and the effects of both lateral stress and higher-order terms are indicated. Finally, the asymptotic expression for the crack trajectory is derived to the same order of approximation when stability occurs.
2. CRACK PATH EQUATION Let us consider an inifinite two-dimensional homogeneous and isotropic elastic body, with a traction-free straight center-crack of length L = 2a, subjected to some external biaxial loading to infinity allowing the crack tip extension through Mode I and Mode II as well. The elastostatic stress field is required to satisfy the well-known equilibrium equations[7] :
ox,+ oyy= ayv-a,,
ww +WI,
+ 2ia,, = 2[(z-z)~(z)-~(z)+~(z)],
(2.1)
where 0._ a,,,,,crXY are the Cartesian stress components, and m(z), n(z) are holomorphic functions in the 121
122
E. VIOLA and A. PIVA
whole cut z plane. The boundary value problem solution gives (see also [7]) ~z) = (2r + P)z -- fi’ J&G? 2’
(2.2)
R(z) = CD(z) + i=‘,
where r and r’ are two constants related to the applied stress field to infinity. Confining attention to the right crack tip, z = L/2, the following relations hold between the fundamental stress combinations when expressed in either set of local polar coordinates (r, 0) and local Cartesian coordinates (x, y) :
or,+ @,I3=
Qxx + cyy’
(2.3)
gee- cr, + 2ia,, = (a,, - (T,, + 2io,,) e”‘.
By expanding (2.2) into power series, eqns (2.1H2.3) lead to the following asymptotic representations of the polar stress components cge and or0 :
crge= A sin2 8+ ___ 4&Y
3cos~+cos~)-*(3sin~+3sin~) ( + O(r), ’
tJ,,g= -
(24) .
+$&(sing-sinT)++&(5cosT-cos$+O(r), where K, and K, are the stress intensity factors at the crack tip, b, and b, are the coefficients of the terms proportional to the square root of the distance from the crack tip, and A is a constant depending on the applied lateral stress. From now on it is assumed that crack extension occurs accordingly with the maximum tensile stress criterion and that the deviation of the crack tip from its original direction is described by the function I(x) (see Fig. l), which is supposed to be small with respect to its argument. Assuming a slight curvature for the crack extension, the following approximations may be considered up to second-order terms : sin 6 N 19‘v X(x), cos
8N
i - 12(~)/2,
(2.5)
r N x[l +X2(x)/2].
The maximum tensile stress criterion requires that, for some distance r from the crack tip, the angle 8
6
Fig. 1. Geometry of the cracked body and external loading.
Second-order approx~ation
to the crack trajectory equation
123
of extension is such as to satisfy the following conditions of positive maximum:
(2.6)
Hence, from (Z-4), and (2.5), the conditions (2.6) lead respectively to (4A~-~K,~~b,x)~2-(6K,+10b,x)R’+4(K,+b,x)
(1-~J2;;;-2$ 1
I’_
1
>
> 0,
(z!!L+gz, >jp+~+gz,=o, 1
1
(27K,-145b,x-128AJj;;;;-)Y2+(84K,-t620b,~~~+(64A~-32K,f40b,x) By noting from (2.4) that, in the neighbourhood approximations are valid:
(2.7)
1
< 0.
of the preexisting crack tip, the following
,/t%-qqB = K,+b,x+0(;1/x), ,,hh,+ = K,
1
+ b,x + O(~/x),
P-8)
defining the stress intensity factors at the extended crack tip as
(2.9)
one obtains
(2.10)
In addition, in view of (2.9),, the following approximate expression may he obtained from (2.4), :
(2.11)
which leads to A’(0)= - 2K,/K, because K,, = 0 when x = 0. It follows from (2.10), that b2 is of the first order in A’.Hence, in the required approximation, eqn (2.7), gives
X(x) =
X(0) -( lOb,,‘3K,)x 1 -(8A/3K,)~-(5~~/3K,)xy
(2.12)
where its range of validity is bounded by constraints (2.7), and (2.7),. By expanding (2.12) for 81Aj,/%/3K, << 1 [n’(x) a: A/x], the near-tip crack path behaviour is obtained : L(x) =
(11x +
/!h J12+ yx2+ o(x5’2),
(2.13)
124
E. VIOLA
and
A. PIVA
where a = -2K,/K,, (2.14)
B = ~a(AIKJJ2n,
It may be noticed that, apart from some numerical coefficients, the behaviour (2.13) is the same as that predicted in [4]. It should also be pointed out from (2.12) that depending on the lateral stress as well as on b,, an asymptote perpendicular to the x-axis may occur when 3K, - 8Ae5b,x = 0. Henceforth the crack path stability depends not only on the lateral load applied but also upon the derivatives of both Mode I and Mode II stress intensity factors, as pointed out also in [4]. In addition, when large values of x are in accord with restrictions (2.7), and (2.7),, eqn (2.12) leads to nb, = 2bz/b, = -l’(O), which is physically admissible when 3K, -8A,,f?&-5b,x # 0. When stability occurs, the approximation L’(x) N n/x is allowed also for large values of x so that the following “asymptotic” behaviour may be obtained from (2.12) : n(x) = -i’(O)[x-
~~+2?gy+o(x-‘:‘)].
3. CRACK PATH
(2.15)
PICTURES
Some graphical representations of the results obtained in Section 2 are reported in the following. In Fig. 2 the dimensionless crack path I(x)/a is represented against the normalized distance x/a from the crack tip for various values of the biaxial load parameter, when the crack inclination angle with respect to the lateral stress ko is 6 = 2”. Dashed lines refer to second-order approximation as predicted by eqn (2.12), whereas solid lines refer to crack path equation in which only first-order terms are retained. It may be noted that, in both cases, the crack path is unstable for k > 1 and stable for k c 1. In the last case, it tends to revert to its original straight path. In the particular case of k = 1, the crack will extend in a self-similar manner. Different order approximations on the crack path stability and instability are illustrated in Figs. 3 and 4, respectively, for lateral compressive and tensile load. It should be noted that the second-
Fig. 2. Comparison
-
First order *ppmximafion
----
Second order approximation
between first- and second-order approximations angle is 6 = 2”.
on crack path behaviour
when the crack
Second-order approx~ation
to the crack trajectory equation
-
First
----
Semnd cndei approxi~tion
ox&r
125
sppmximation
Fig. 3. Different order approximation effect on the crack path stability for k = -0.5.
order-approximation effect on the crack path behaviour increases when the crack angle tends to vanish and the biaxial load parameter k is negative. An opposite effect may be observed when lateral tensile stresses are considered, with k > 1. From Figs. 2 and 4 it may be noted that the crack path deviation when the second-order terms are retained is always smaller with respect to the first-order approximation effect. Note that the figures refer to eqn (2.12) where the first-order approximation is obtained by setting b, = b2 = 0. Figure 5 shows a comparison between crack path predictions obtained according to the present theory [see eqn (2.13) and eqn (24) of Ref. 41. It should be noted that the comparison is valid only up to small distances from the original crack tip because of the required approximations. Figure 6 shows the variation of the dimensionless distance xJa of the asymptote perpendicular to the crack-axis against the biaxial load parameter k. When increasing values of k are considered, the limit distance from the original crack tip for the validity of the aforesaid approach decreases very rapidly. It may be noted that when second-order terms are retained (dashed line), the validity range is reduced with respect to the first-order approximation (solid line). In addition, note that the distance x,/a is an upper bound as a function of k and is independent of the derivatives of the stress intensity factors with respect to the initial crack length. In reality, the validity of the crack path equation &x)/a is limited to a smaller range according to the required approximations. 4. REMARKS At this stage some remarks are reported to point out the differences between the procedure and the results reported in this paper and the results in [4] and, more recently, also in [8]. It should be noted that both investigations refer to mathematical modeIs concerning the virtual extension of the crack tip up to second-order effects. Both models give the same formal representation for crack paths which coincide, with the same inclination, when the original crack tip is approached. However, except
-XIX) a 0.10
-
First order approximation
----
Second order ,qqroximation
t
0 L!L
0.05
0.10
0.16
Fig. 4. Different order approximation effect on the crack path instability r for k = 1.5.
126
E. VIOLA UXI
T O.W8
I
and A. PIVA
-Present
theory [equation (2.131]
-‘-.-
Reference ]4] [[equation
(24)]
~,
- 0.006
Fig. 5. Comparison
0.10
between asymptotic
crack path predictions.
-
First order apprwimafion
----
Semnd order approximation
t
Fig. 6. Second-order-terms
effect on the upper bound
for path validity
range.
for the linear term, the numerical coefficients do not coincide which, in the authors’ opinion, depends on the different fracture criteria employed. In fact, in [4] the local symmetry criterion (K,, = 0) is used and the family of crack trajectories must be assumed in order to evaluate the numerical coefficients of the path equation. In this paper the maximum hoop stress criterion is used and the crack path equation is obtained without preliminary assumption. Acknowledgement-This
work was supported
by the National
(Italian)
Research
Council
(C.N.R.).
REFERENCES [l] R. V. Goldstein and R. L. Salganik, Brittle fracture of solids with arbitrary cracks. Int. J. Fracture 10, 5077523 (1974). [2] B. Cotterel and J. R. Rice, Slightly curved or kinked cracks. Int. J. Fracture 18, 155-169 (1980). [3] H. Liebowitz, J. D. Lee and N. Subramonian, Criteria for predicting crack extension angle and path in plane crack problems. Proc. Int. Conf. on Analytical and Exp. Fracture Mech., Roma (1980), pp. 239-250. [4] B. L. Karihaloo, L. M. Keer, S. Nemat-Nasser and A. Cranratmachai, Approximate description of crack kinking and curving. J. Appl. Mech. 48,515-519 (1981). [S] E. Viola and A. Piva, Crack paths in sheets of brittle material. Engng Fracture Mech. 19,1069-1084(1984). [6] A Piva and E. Viola, On paths of two equal collinear cracks. Engng Fracture Me&, 20,735-741(1984). [7] N. I. Muskhelishvili, Some basic problems of the Mathematical Theory of Elasticity, 3rd edn. Noordhoof, Groningen (1963). [S] Y. Sumi, S. Nemat-Nasser and L. M. Keer, On crack branching and curving in a finite body. Int. .r. Fracture 21,67-79 (1983).