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Non-recoverable elastic energy and crack propagation in brittle materials
Mat. Res. Bull. Vol. 7, pp. 769-77Z, in the United States.
197Z.
Pergamon
Press, Inc. Printed
NON-RECOVERABLE ELASTIC ENERGY AND CRACK PROPAGATION IN BRITTLE MATERIALS
D. P. H. Hasselman~ J. A. Coppola and D. A. Krohn Physical Ceramics Laboratory Materials Research Center Lehigh University Bethlehem, Pennsylvania 18015 and R. C. Bradt Department of Materials Science Ceramic Science Section Pennsylvania State University University Park, pennsylvania 16802 (Received June Z0, 197Z; Refereed)
A B S T R A C T
A discussion of the Griffith criteria for brittle fracture is presented. It is suggested that catastrophic crack propagation can only be achieved with the expenditure of unrecoverable elastic strain energy in addition to the surface free energy of the new surfaces. The discrepency then between the Griffith surface energy and measured fracture energies is concluded to be partly attributed to non-recoverable elastic energy dissipated during fracture. The low tensile strength of brittle materials has been attributed by Griffith (I) to the presence of microcracks.
Assuming fracture to
be a thermodynamically reversible process, Griffith hypothesized that crack instability will occur when on crack extension the change in the elastic energy of the stress field of the crack equals or exceeds the surface free energy of the resulting new fracture surfaces, expressed by:
dW/d~ >- 2Ys
(1)
where W is the energy of the stress field of the crack, % is the crack length and Ys is the surface free energy. 769
Results of cleavage
770
BRITTLE
MATERIALS
Vol. 7, No. 8
experiments on single crystals established the validity of the Griffith hypothesis (2-4). More recent studies (5-9) showed that in many brittle solids such as polycrystalline ceramics, the actual energy (hereafter referred to as the fracture energy, yf) greatly exceeds the surface free energy (ys), in apparent contradiction with eq. i.
This discrepancy has been
attributed to surface roughness and to energy dissipation at the crack tip by non-linear deformation such as plastic and viscous flow (9-13). The purpose of this communication is to suggest an additional mechanism for the fracture energy to exceed the surface free energy even with the complete absence of viscous and/or plastic flow.
It
should be realized that the Griffith criterion (eq. i) strictly represents the condition of crack instability on thermodynamic grounds. Crack propagation can only occur when, on crack extension, the total change in elastic energy from all regions of the stress field of the crack is transferred to the crack tip simultaneously, at an efficiency of one hundred percent.
Such an efficiency exists in few, if any,
chemical or mechanical processes which require the transfer of energy from one form into another, and it is unlikely that the fracture process is also one hundred percent efficient.
Hence, the fracture
energy expended in propagating a crack, in addition to the surface free energy, is expected to contain, in part, non-recoverable elastic strain energy which on crack propagation is transformed into lattice vibrations (heat) and/or acoustic energy. Although the elastic energy distribution near cracks is complex, the concept of non-recoverable elastic strain energy can be illustrated in a simple manner by noting that during crack propagation, fracture at the crack tip occurs at the theoretical strength of the material.
Under these conditions, sufficient energy is available at
the crack tip to provide the surface free energy of the new crack surfaces.
However, applying a stress directly at a crack tip requires
the expenditure of energy in stressing the material adjacent to the plane of crack propagation as well.
Now, suppose that fracture at
the plane of crack propagation were to occur instantaneously, this will cause the material adjacent to the plane of crack propagation to relax, but the elastic energy released no longer can be turned into surface energy since the crack surfaces were already created.
The
Vol. 7, No. 8
BRITTLE
MATERIALS
771
relaxed energy is simply transformed into heat or acoustic energy, as stated.
Only if fracture were to occur very slowly, so that the total
energy released from all regions of the stress field of the crack could be transmitted to the crack tip by thermal or other means, would fracture occur at the energy condition of eq. i. In order to express the concept of non-recoverable elastic energy quantitatively,
it is suggested to extend the definition of fracture
energy (12) to: Yf : KYs + Yp + Y a n
+ Yel
(2)
K is the surface roughness factor, yp and Yan are the energies expended by plastic and viscous flow, respectively, recoverable elastic strain energy.
and Yel represents the non-
Consequently, there is no need to
resort to the presence of plastic or viscous flow to completely explain the discrepancies between fracture and surface free energy. Acknowledgments The thoughts expressed in the present communication were the result of interaction between two research programs supported by the Pennsylvania Science and Engineering Foundation under contract ME-344151 and by the U.S. Army Research Office - Durham, North Carolina, under grant DA-ARO-D-31-124-Gl106,
respectively.
References i.
A.A.
Griffith, Proc. Roy. Soc. (London)
2.
J.J.
Gilman, J. Appl. Phys.
3.
A.R.C. (1963).
4
Y.P.
5
J. Nakayama, J. Am. Ceram. Soc.
6
R.W.
221A, 163 (1920).
31, 2208 (1960).
Westwood and D. L. Goldheim, J. Appl. Phys.
Gupta and A. T. Santhanam, Acta Met.
34, 3335
17, 419 (1969).
48, 583 (1965).
Davidge and G. Tappin, J. Mat. Sci.
3 165 (1968). m
7
A.G.
Evans and R. W. Davidge, Phil. Mag.
20, 373 (1969).
8
A.G.
Evans and R. W. Davidge, J. Mat. Sci.
9
J. Congleton and N. J. Perch, Acta Met. 1-4, 1179 (1966).
i0.
F. J. P. Clarke, H. G. Tattersall and G. Tappin, Proc. Brit. Ceram. Soc. ~, 16] (1966).
~, Z14 (1970).
77Z
BRITTLE
NLATERIALS
ii.
J. Congleton, (1969).
12.
A. G. Evans, Phil. Mag.
13.
S. M. Wiederhorn, Fracture of Ceramics, Publ. 303 (1969).
Vol. 7, No. 8
N. J. Petch and J. A. Shiels, Phil. Mag.
19, 79
22, 84 (1970). p. 217. N.B.S. Spec.
197Z.
Pergamon
Press, Inc. Printed
NON-RECOVERABLE ELASTIC ENERGY AND CRACK PROPAGATION IN BRITTLE MATERIALS
D. P. H. Hasselman~ J. A. Coppola and D. A. Krohn Physical Ceramics Laboratory Materials Research Center Lehigh University Bethlehem, Pennsylvania 18015 and R. C. Bradt Department of Materials Science Ceramic Science Section Pennsylvania State University University Park, pennsylvania 16802 (Received June Z0, 197Z; Refereed)
A B S T R A C T
A discussion of the Griffith criteria for brittle fracture is presented. It is suggested that catastrophic crack propagation can only be achieved with the expenditure of unrecoverable elastic strain energy in addition to the surface free energy of the new surfaces. The discrepency then between the Griffith surface energy and measured fracture energies is concluded to be partly attributed to non-recoverable elastic energy dissipated during fracture. The low tensile strength of brittle materials has been attributed by Griffith (I) to the presence of microcracks.
Assuming fracture to
be a thermodynamically reversible process, Griffith hypothesized that crack instability will occur when on crack extension the change in the elastic energy of the stress field of the crack equals or exceeds the surface free energy of the resulting new fracture surfaces, expressed by:
dW/d~ >- 2Ys
(1)
where W is the energy of the stress field of the crack, % is the crack length and Ys is the surface free energy. 769
Results of cleavage
770
BRITTLE
MATERIALS
Vol. 7, No. 8
experiments on single crystals established the validity of the Griffith hypothesis (2-4). More recent studies (5-9) showed that in many brittle solids such as polycrystalline ceramics, the actual energy (hereafter referred to as the fracture energy, yf) greatly exceeds the surface free energy (ys), in apparent contradiction with eq. i.
This discrepancy has been
attributed to surface roughness and to energy dissipation at the crack tip by non-linear deformation such as plastic and viscous flow (9-13). The purpose of this communication is to suggest an additional mechanism for the fracture energy to exceed the surface free energy even with the complete absence of viscous and/or plastic flow.
It
should be realized that the Griffith criterion (eq. i) strictly represents the condition of crack instability on thermodynamic grounds. Crack propagation can only occur when, on crack extension, the total change in elastic energy from all regions of the stress field of the crack is transferred to the crack tip simultaneously, at an efficiency of one hundred percent.
Such an efficiency exists in few, if any,
chemical or mechanical processes which require the transfer of energy from one form into another, and it is unlikely that the fracture process is also one hundred percent efficient.
Hence, the fracture
energy expended in propagating a crack, in addition to the surface free energy, is expected to contain, in part, non-recoverable elastic strain energy which on crack propagation is transformed into lattice vibrations (heat) and/or acoustic energy. Although the elastic energy distribution near cracks is complex, the concept of non-recoverable elastic strain energy can be illustrated in a simple manner by noting that during crack propagation, fracture at the crack tip occurs at the theoretical strength of the material.
Under these conditions, sufficient energy is available at
the crack tip to provide the surface free energy of the new crack surfaces.
However, applying a stress directly at a crack tip requires
the expenditure of energy in stressing the material adjacent to the plane of crack propagation as well.
Now, suppose that fracture at
the plane of crack propagation were to occur instantaneously, this will cause the material adjacent to the plane of crack propagation to relax, but the elastic energy released no longer can be turned into surface energy since the crack surfaces were already created.
The
Vol. 7, No. 8
BRITTLE
MATERIALS
771
relaxed energy is simply transformed into heat or acoustic energy, as stated.
Only if fracture were to occur very slowly, so that the total
energy released from all regions of the stress field of the crack could be transmitted to the crack tip by thermal or other means, would fracture occur at the energy condition of eq. i. In order to express the concept of non-recoverable elastic energy quantitatively,
it is suggested to extend the definition of fracture
energy (12) to: Yf : KYs + Yp + Y a n
+ Yel
(2)
K is the surface roughness factor, yp and Yan are the energies expended by plastic and viscous flow, respectively, recoverable elastic strain energy.
and Yel represents the non-
Consequently, there is no need to
resort to the presence of plastic or viscous flow to completely explain the discrepancies between fracture and surface free energy. Acknowledgments The thoughts expressed in the present communication were the result of interaction between two research programs supported by the Pennsylvania Science and Engineering Foundation under contract ME-344151 and by the U.S. Army Research Office - Durham, North Carolina, under grant DA-ARO-D-31-124-Gl106,
respectively.
References i.
A.A.
Griffith, Proc. Roy. Soc. (London)
2.
J.J.
Gilman, J. Appl. Phys.
3.
A.R.C. (1963).
4
Y.P.
5
J. Nakayama, J. Am. Ceram. Soc.
6
R.W.
221A, 163 (1920).
31, 2208 (1960).
Westwood and D. L. Goldheim, J. Appl. Phys.
Gupta and A. T. Santhanam, Acta Met.
34, 3335
17, 419 (1969).
48, 583 (1965).
Davidge and G. Tappin, J. Mat. Sci.
3 165 (1968). m
7
A.G.
Evans and R. W. Davidge, Phil. Mag.
20, 373 (1969).
8
A.G.
Evans and R. W. Davidge, J. Mat. Sci.
9
J. Congleton and N. J. Perch, Acta Met. 1-4, 1179 (1966).
i0.
F. J. P. Clarke, H. G. Tattersall and G. Tappin, Proc. Brit. Ceram. Soc. ~, 16] (1966).
~, Z14 (1970).
77Z
BRITTLE
NLATERIALS
ii.
J. Congleton, (1969).
12.
A. G. Evans, Phil. Mag.
13.
S. M. Wiederhorn, Fracture of Ceramics, Publ. 303 (1969).
Vol. 7, No. 8
N. J. Petch and J. A. Shiels, Phil. Mag.
19, 79
22, 84 (1970). p. 217. N.B.S. Spec.