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Crack propagation directions in anisotropic media
CRACK
PROPAGATION
DIRECTIONS MEDIA
IN
ANISOTROPIC
B. W. CHERRY and N. L. HARRISON
Materials Engineering Department, Monash University, Clayton, Victoria 3168 (Australia) (Received : 12 December, 1969)
INTRODUCTION
Several authors 1-3 have discussed the development of a criterion to determine the direction in which a crack will propagate in an anisotropic medium. Gurney and Hunt 1 propose that, under quasi-static conditions, the crack will propagate in the direction for which dX/du is least (where X is the applied force and u the spacing between the grips), and have shown that this condition makes the entropy increase of an isothermal system greatest. The object of this work is to develop, from Gurney and Hunt's criterion, a general criterion which is independent of the conditions of loading, and which is compatible with criteria suggested by other authors.
DISCUSSION
Cook and Gordon 2 approach the problem from the point of view of tensile and shear stresses near the tip of a crack and have suggested that the crack will propagate in the direction for which the local stress corresponding to a given mode of failure exceeds the strength of the material at the smallest applied stress. Cooper and Kelly 3 interpreted their results on crack propagation in a series of fibre-reinforced composites for which the crack geometry was constant as showing that a transition took place from crack propagation parallel to the reinforcing fibres to crack propagation transverse to the fibres when the ratios of the strengths of the material in the different directions reached a specific value. It is implicit in the approaches of both these sets of authors that unstable crack propagation under conditions of continuously increasing load occurs in the direction requiring the least applied load. 299 Fibre Science and Technology (2) (1970)--C) Elsevier Publishing Company Ltd, England--Printed in Great Britain
300
B. W. CHERRY, N. L. HARRISON
Gurney and Hunt t suggest that under conditions of quasi-static crack propagation the crack will propagate in the direction for which dX/du is least. If" a crack has propagated quasi-statically until u has reached a value Uo and is about to propagate further to u = u 1, then in the absence of discontinuities in the material, X and d X / d u are already determined at u = Uo. Now in the interval Uo - u~, Xo - X t = (dX/du) ( U 0 - - Ul) -~- terms of higher order in (u o - u~). Hence for small values of (u o - ul), neglecting the terms of higher order in (Uo - u~), the path which makes d X / d u least at u = u~ also makes X least at this point. Thus Gurney and Hunt's criterion for quasi-static crack propagation is also a least load criterion. If R is the energy required to produce unit area of crack and if U is the strain energy of the body, then during quasi-static crack propagation . . . .
,
Xdu = RdA + dU
(1)
Now if the linear elastic compliance of the material is C and if the energy associated with self-equilibrating strains in the body is negligible: U = ½uX = ½CX 2
(2)
F r o m eqn. (2) it can be seen that dU/du = ½[X + u(dX/du)]
(3)
From eqn. (3) it can be seen that the path for which dX/du and hence X are least is the path which makes the strain energy of the system after crack propagation least. Similarly, from eqn. (2), d U = ½X 2 d C + X C d X
in eqn. (1) yields 2 / X 2 = (1/R)(dC/dA)
(4)
From eqn. (4) it can be seen that the path for which X is least is the path for which (1/R)(dC/dA) is greatest. Hence from eqns. (3) and (4) additional criteria for the direction of crack propagation under quasi-static conditions are deduced that the crack will propagate in the direction which leads to the smallest value of the strain energy after an interval of crack propagation, and that this is the direction for which (1/R)(dC/dA) is greatest. If crack propagation is not quasi-static, there is no definite relationship between X and u after propagation has commenced and hence the criterion that propagation is in the direction for which dX/du is least can not be applied in such instances.
C R A C K P R O P A G A T I O N DIRECTIONS IN ANISOTROPIC MEDIA
However, if propagation propagation is the strain then
301
the load on the b o d y has increased steadily from a low value, crack will occur in the direction for which the load required to cause is least. It is a necessary condition for crack propagation that, if G energy release rate during crack propagation in the given direction, G>~R
N o w since G = ½X2(dC/dA), then if it is assumed that crack propagation is initiated as soon as G = R, the direction for which this condition is satisfied at the lowest value of load is also the direction for which (1/R)(dC/dA) is greatest. I f the criterion G = R is universally true for crack propagation, then a crack will always propagate in the direction for which ( I / R ) ( d C / d A ) is greatest. A special case arises when the failure of a uniaxially oriented fibre-reinforced composite is considered. F o r such a material a crack may propagate in a direction transverse to the fibres or in a direction parallel to the fibres. I f these two modes o f failure are termed cracking and splitting respectively and denoted by a subscript 'c" or 's' then the transition from cracking to splitting occurs when [(1/R)(dC/dA)]¢ = [(I/R)(dC/dA)]~ or
R~/R~ = (dC/dA~)/(dC/dA)~
(5)
ACKNOWLEDGEMENT
One of us (N.L.H.) gratefully acknowledges receipt of a M o n a s h Graduate Scholarship during the period in which this work was carried out.
REFERENCES
1. C. GURNEYand J. HUNT, Quasi-static crack propagation, Proc. Roy. Soc., A 299 (1967) 508. 2. J. COOKand J. E. GORDON,A mechanism for the control of crack propagation in all brittle systems, Proc. Roy. Soc., A282 (1964) 508. 3. G. A. COOPERand A. KELLY,Tensile properties of fibre-reinforced metals: Fracture mechanics. J. Mech. Phys. Solids, 15 (1967) 279. 4. G. R. IRWIN,Structural aspects of brittle failure, Appl. Mater. Res., 3 (1964) 65.
PROPAGATION
DIRECTIONS MEDIA
IN
ANISOTROPIC
B. W. CHERRY and N. L. HARRISON
Materials Engineering Department, Monash University, Clayton, Victoria 3168 (Australia) (Received : 12 December, 1969)
INTRODUCTION
Several authors 1-3 have discussed the development of a criterion to determine the direction in which a crack will propagate in an anisotropic medium. Gurney and Hunt 1 propose that, under quasi-static conditions, the crack will propagate in the direction for which dX/du is least (where X is the applied force and u the spacing between the grips), and have shown that this condition makes the entropy increase of an isothermal system greatest. The object of this work is to develop, from Gurney and Hunt's criterion, a general criterion which is independent of the conditions of loading, and which is compatible with criteria suggested by other authors.
DISCUSSION
Cook and Gordon 2 approach the problem from the point of view of tensile and shear stresses near the tip of a crack and have suggested that the crack will propagate in the direction for which the local stress corresponding to a given mode of failure exceeds the strength of the material at the smallest applied stress. Cooper and Kelly 3 interpreted their results on crack propagation in a series of fibre-reinforced composites for which the crack geometry was constant as showing that a transition took place from crack propagation parallel to the reinforcing fibres to crack propagation transverse to the fibres when the ratios of the strengths of the material in the different directions reached a specific value. It is implicit in the approaches of both these sets of authors that unstable crack propagation under conditions of continuously increasing load occurs in the direction requiring the least applied load. 299 Fibre Science and Technology (2) (1970)--C) Elsevier Publishing Company Ltd, England--Printed in Great Britain
300
B. W. CHERRY, N. L. HARRISON
Gurney and Hunt t suggest that under conditions of quasi-static crack propagation the crack will propagate in the direction for which dX/du is least. If" a crack has propagated quasi-statically until u has reached a value Uo and is about to propagate further to u = u 1, then in the absence of discontinuities in the material, X and d X / d u are already determined at u = Uo. Now in the interval Uo - u~, Xo - X t = (dX/du) ( U 0 - - Ul) -~- terms of higher order in (u o - u~). Hence for small values of (u o - ul), neglecting the terms of higher order in (Uo - u~), the path which makes d X / d u least at u = u~ also makes X least at this point. Thus Gurney and Hunt's criterion for quasi-static crack propagation is also a least load criterion. If R is the energy required to produce unit area of crack and if U is the strain energy of the body, then during quasi-static crack propagation . . . .
,
Xdu = RdA + dU
(1)
Now if the linear elastic compliance of the material is C and if the energy associated with self-equilibrating strains in the body is negligible: U = ½uX = ½CX 2
(2)
F r o m eqn. (2) it can be seen that dU/du = ½[X + u(dX/du)]
(3)
From eqn. (3) it can be seen that the path for which dX/du and hence X are least is the path which makes the strain energy of the system after crack propagation least. Similarly, from eqn. (2), d U = ½X 2 d C + X C d X
in eqn. (1) yields 2 / X 2 = (1/R)(dC/dA)
(4)
From eqn. (4) it can be seen that the path for which X is least is the path for which (1/R)(dC/dA) is greatest. Hence from eqns. (3) and (4) additional criteria for the direction of crack propagation under quasi-static conditions are deduced that the crack will propagate in the direction which leads to the smallest value of the strain energy after an interval of crack propagation, and that this is the direction for which (1/R)(dC/dA) is greatest. If crack propagation is not quasi-static, there is no definite relationship between X and u after propagation has commenced and hence the criterion that propagation is in the direction for which dX/du is least can not be applied in such instances.
C R A C K P R O P A G A T I O N DIRECTIONS IN ANISOTROPIC MEDIA
However, if propagation propagation is the strain then
301
the load on the b o d y has increased steadily from a low value, crack will occur in the direction for which the load required to cause is least. It is a necessary condition for crack propagation that, if G energy release rate during crack propagation in the given direction, G>~R
N o w since G = ½X2(dC/dA), then if it is assumed that crack propagation is initiated as soon as G = R, the direction for which this condition is satisfied at the lowest value of load is also the direction for which (1/R)(dC/dA) is greatest. I f the criterion G = R is universally true for crack propagation, then a crack will always propagate in the direction for which ( I / R ) ( d C / d A ) is greatest. A special case arises when the failure of a uniaxially oriented fibre-reinforced composite is considered. F o r such a material a crack may propagate in a direction transverse to the fibres or in a direction parallel to the fibres. I f these two modes o f failure are termed cracking and splitting respectively and denoted by a subscript 'c" or 's' then the transition from cracking to splitting occurs when [(1/R)(dC/dA)]¢ = [(I/R)(dC/dA)]~ or
R~/R~ = (dC/dA~)/(dC/dA)~
(5)
ACKNOWLEDGEMENT
One of us (N.L.H.) gratefully acknowledges receipt of a M o n a s h Graduate Scholarship during the period in which this work was carried out.
REFERENCES
1. C. GURNEYand J. HUNT, Quasi-static crack propagation, Proc. Roy. Soc., A 299 (1967) 508. 2. J. COOKand J. E. GORDON,A mechanism for the control of crack propagation in all brittle systems, Proc. Roy. Soc., A282 (1964) 508. 3. G. A. COOPERand A. KELLY,Tensile properties of fibre-reinforced metals: Fracture mechanics. J. Mech. Phys. Solids, 15 (1967) 279. 4. G. R. IRWIN,Structural aspects of brittle failure, Appl. Mater. Res., 3 (1964) 65.