Crack in uniform tension 7 Griffith energy-balance concept, a formal statement of which is given by the equilibrium requirement du/dc =0. Here then was a criterion for predicting the fracture behaviour of a body, firmly rooted in the laws of energy conservation. a crack would extend or retract reversibly for small displacements from the equilibrium length, according to whether the left-hand side of (1.6) were negative or positive. This criterion remains the building block for all brittle fracture theory 1.3 Crack in uniform tension The griffith concept provided a fundamental starting point for any fracture problem in which the operative forces could be considered to be conservative. Griffith sought to confirm his theory by applying it to a real crack configuration. First he needed an elastic model for a crack, in order to calculate the energy terms in(1.5). For this he took advantage of the Inglis analysis, considering the case of an infinitely narrow elliptical cavity (6-0, fig. 1. 1)of length 2c in a remote, uniform tensile stress field aa Then, for experimental verification, he had to find a well-behaved, 'model material, isotropic and closely obeying Hooke's law at all stresses prior to acture. Glass was selected as the most easily accessible material satisfying these requirements In evaluating the mechanical energy of his model crack system Griffith invoked a result from linear elasticity theory (cf sect. 2.2), namely that for any body under constant applied stress during crack formation, U.=-2Ur, (constant load) (1.7) so that UM=-UE. The negative sign indicates a mechanical energy reduction on crack formation. Then from the Inglis solution of the stress and strain fields the strain energy density is readily computed for each volume element about the crack. Integrating over dimensions large compared with the length of the crack then gives, for unit width along the crack front, UE=TCOA/E (18)
Crack in uniform tension 7 Griffith energy-balance concept, a formal statement of which is given by the equilibrium requirement dU/dc = 0. (1.6) Here then was a criterion for predicting the fracture behaviour of a body, firmly rooted in the laws of energy conservation. A crack would extend or retract reversibly for small displacements from the equilibrium length, according to whether the left-hand side of (1.6) were negative or positive. This criterion remains the building block for all brittle fracture theory. 1.3 Crack in uniform tension The Griffith concept provided a fundamental starting point for any fracture problem in which the operative forces could be considered to be conservative. Griffith sought to confirm his theory by applying it to a real crack configuration. First he needed an elastic model for a crack, in order to calculate the energy terms in (1.5). For this he took advantage of the Inglis analysis, considering the case of an infinitely narrow elliptical cavity (6->0, fig. 1.1) of length 2c in a remote, uniform tensile stress field aA. Then, for experimental verification, he had to find a well-behaved, 'model' material, isotropic and closely obeying Hooke's law at all stresses prior to fracture. Glass was selected as the most easily accessible material satisfying these requirements. In evaluating the mechanical energy of his model crack system Griffith invoked a result from linear elasticity theory (cf. sect. 2.2), namely that for any body under constant applied stress during crack formation, C/ A =-2£/ B , (constant load) (1.7) so that UM= —UK. The negative sign indicates a mechanical energy reduction on crack formation. Then from the Inglis solution of the stress and strain fields the strain energy density is readily computed for each volume element about the crack. Integrating over dimensions large compared with the length of the crack then gives, for unit width along the crack front, UB = nc*al/E' (1.8)
The griffith concept Fig. 1.5. Energetics of Griffith crack in uniform tension, plane stres Data for glass from Griffith: ,=1.75Jm-, E= 62GPa, 0=2.63 MPa (chosen to give equilibrium at co =10 mm) where E'identifies with Youngs modulus E in plane stress( thinplates) and E/(1-v2)in plane strain(thick plates), with v Poissons ratio. The ation of additional loading parallel to the crack plane has negligible n the strain energy terms in(1.8 ). For the surface energy of the crack Griffith wrote, again for unit width of front, (1.9) with y the free surface energy per unit area. The total system energy (1.5) U(c)=-Tc20A/E+4cy Fig. 1.5 shows plots of the mechanical energy UM(c), surface energy Us(c), and total energy U(c) observe that, according to the Inglis treatment, an edge crack of length c(limiting case of surface notch, b0, fig. 1. 2)may be considered to possess very nearly one- half the energy of an internal crack The Griffith equilibrium condition (1.6)may now be applied to(I 10)
The Griffith concept 100 0 100 >00 1 Equilibrium s - U" A' N N N \ - \ 1 1 ^ ^ \ " \ \ \ 1 \ 0 10 20 Crack length, c (mm) 30 Fig. 1.5. Energetics of Griffith crack in uniform tension, plane stress. Data for glass from Griffith: y = 1.75Jnr2 , E = 62GPa, aA = 2.63MPa (chosen to give equilibrium at c0 = 10 mm). where Ef identifies with Young's modulus ^i n plane stress ('thin' plates) and E/{\ — v 2 ) in plane strain ('thick' plates), with v Poisson's ratio. The application of additional loading parallel to the crack plane has negligible effect on the strain energy terms in (1.8). For the surface energy of the crack system Griffith wrote, again for unit width of front, = 4cy (1.9) with y the free surface energy per unit area. The total system energy (1.5) becomes U(c)= - (1.10) Fig. 1.5 shows plots of the mechanical energy UM(c), surface energy Us(c), and total energy U(c). Observe that, according to the Inglis treatment, an edge crack of length c (limiting case of surface notch, b -> 0, fig. 1.2) may be considered to possess very nearly one-half the energy of an internal crack of length 2c. The Griffith equilibrium condition (1.6) may now be applied to (1.10)
Obreimoft's experiment We thereby calculate the critical conditions at which failure'occurs dF=(2Ey/Ico) As we see from fig. 1.5, or from the negative value of dU/dc the system energy is a maximum at equilibrium, so the configuration is unstable. That is, at oa O the crack remains stationary at its original size co; atoa>OF it propagates spontaneously without limit. Equation(1. 11)is the famous Griffith strength relation For experimental confirmation, Griffith prepared glass fracture speci mens from thin round tubes and spherical bulbs. Cracks of le 4-23 mm were introduced with a glass cutter and the specimens annealed prior to testing. The hollow tubes and bulbs were then burst by pumping in a fluid. and the critical stresses determined from the internal fluid pressure. As predicted, only the stress component normal to the cracl plane was found to be important; the application of end loads to tube containing longitudinal cracks had no detectable effect on the critical conditions. The results could be represented by the relation Fp c0/ 2=0.26 MPa m" with a scatter x 5%o, thus verifying the essential form of a(co)in(1. 11) If we now take this result, along with Griffith's measured value of oung's modulus, E= 62 GPa, and insert into(1. lI)at plane stress, we obtain ,=1.75Jm-2 as an estimate of the surface energy of glass. Griffith attempted to substantiate his model by obtaining an independent estimate f ,. He measured the surface tension within the temperature range 1020-1383 K, where the glass flows easily, and extrapolated linearly back to room temperature to find y=0.54 J m. Considering that even present day techniques are barely capable of measuring surface energies of solids to very much better tha an a factor of two this 'agreement measured values is an impressive vindication of the griffith theory. 1. 4 Obreimoff's experiment Plane cracks in uniform tension represent just one application of the energy-balance equation(1.6). To emphasise the generality of the griffith concept we digress briefly to discuss an important experiment carried out
ObreimofTs experiment 9 We thereby calculate the critical conditions at which 'failure' occurs, dA = (TF, c = c0, say : aF = (2E'y/ncor\ (1.11) As we see from fig. 1.5, or from the negative value of d2U/dc2 , the system energy is a maximum at equilibrium, so the configuration is unstable. That is, at aA < G¥ the crack remains stationary at its original size c0; at aA > o¥ it propagates spontaneously without limit. Equation (1.11) is the famous Griffith strength relation. For experimental confirmation, Griffith prepared glass fracture specimens from thin round tubes and spherical bulbs. Cracks of length 4-23 mm were introduced with a glass cutter and the specimens annealed prior to testing. The hollow tubes and bulbs were then burst by pumping in a fluid, and the critical stresses determined from the internal fluid pressure. As predicted, only the stress component normal to the crack plane was found to be important; the application of end loads to tubes containing longitudinal cracks had no detectable effect on the critical conditions. The results could be represented by the relation with a scatter « 5%, thus verifying the essential form of aF(c0) in (1.11). If we now take this result, along with Griffith's measured value of Young's modulus, E = 62 GPa, and insert into (1.11) at plane stress, we obtain y = 1.75 J m"2 as an estimate of the surface energy of glass. Griffith attempted to substantiate his model by obtaining an independent estimate of y. He measured the surface tension within the temperature range 1020-1383 K, where the glass flows easily, and extrapolated linearly back to room temperature to find y = 0.54 J m~2 . Considering that even presentday techniques are barely capable of measuring surface energies of solids to very much better than a factor of two, this 'agreement' between measured values is an impressive vindication of the Griffith theory. 1.4 Obreimoff's experiment Plane cracks in uniform tension represent just one application of the energy-balance equation (1.6). To emphasise the generality of the Griffith concept we digress briefly to discuss an important experiment carried out
The Griffith concept Fig. 1.6. Obreimoff's experiment on mica. Wedge of thickness h inserted to peel off cleavage flake of thickness d and width unity. In this configu both crack origin O and tip C translate with wedge Equilibrium Crack length, c(mm) Fig. 1.7. Energetics of Obreimoff crack. Data for mica from Obreimoff: y=0.38 J m"(air), E= 200 GPa, h=0.48 mm, d=75 um (chosen to give equilibrium at co= 10 mm) by Obreimoff (1930)on the cleavage of mica. This second example provides an interesting contrast to the one treated by griffith, in that the equilibrium configuration is stable The basic arrangement used by Obreimoff is shown in fig. 1.6. a glass wedge of thickness h is inserted beneath a thin flake of mica attached to a parent block, and is made to drive a crack along the cleavage plane. In this case we may determine the energy of the crack system by treating the cleavage lamina as a freely loaded cantilever, of thickness d and width ity, built-in at the crack front distant c from the point of application of the wedge. We note that on allowing the crack to form under constant
10 The Griffith concept 1 h T Fig. 1.6. Obreimoff's experiment on mica. Wedge of thickness h inserted to peel off cleavage flake of thickness d and width unity. In this configuration both crack origin O and tip C translate with wedge. Crack length, c (mm) Fig. 1.7. Energetics of Obreimoff crack. Data for mica from Obreimoff: y = 0.38 J m 2 (air), E = 200 GPa, h = 0.48 mm, d = 75 um (chosen to give equilibrium at c0 = 10 mm). by Obreimoff (1930) on the cleavage of mica. This second example provides an interesting contrast to the one treated by Griffith, in that the equilibrium configuration is stable. The basic arrangement used by Obreimoff is shown in fig. 1.6. A glass wedge of thickness h is inserted beneath a thin flake of mica attached to a parent block, and is made to drive a crack along the cleavage plane. In this case we may determine the energy of the crack system by treating the cleavage lamina as a freely loaded cantilever, of thickness d and width unity, built-in at the crack front distant c from the point of application of the wedge. We note that on allowing the crack to form under constant
lI dging conditions the bending(line)force Fsuffers no displacement, so the net work done by this force is zero,i.e U.=O. (1.12) At the same time we have, from simple beam theory, the elastic strain energy in the cantilever arm UE=Ed°h2/8c3 (1.13) The surface energy U=2 (1.14) The total system energy U(c)in(1.5)now follows, and application of the Griffith condition(1.6)leads finally to the equilibrium crack length Co=(3Ed°h2/16) The energy terms UM(c), Us(c), and U(c) are plotted in fig. 1. 7. It is evident from the minimum at U(co) that (1. 15)corresponds to a stable con figuration. In this instance the fracture is 'controlled': the crack advances into the material at the same rate as that of the wedge Equation (1. 15)indicates that, as in Griffith's uniform tension example, knowledge of equilibrium crack geometry uniquely determines the surface energy. Obreimoff proceeded thus to evaluate the surface energy of mica under different test conditions, and found a dramatic increase from y=0.38J m-at normal atmosphere(100 kPa pressure)to y= 5.0J in a vacuum(100 HPa). The test environment was clearly an important factor to be considered in evaluating material strength. Moreover, Obreimoff noticed that on insertion of the glass wedge the crack did not grow immediately to its equilibrium length: in air equilibrium was reached within seconds, whereas in a vacuum the crack continued to creep for several days. Thus the time element was another complicating factor to be considered. These observations provided the first indication of the role of chemical kinetics in fracture processes Obreimoff also observed phenomena that raised the question of reversi bility in crack growth. Propagation of the crack was often erratic, with an accompanying visible electrostatic discharge (triboluminescence), especially in a vacuum. On partial withdrawal of the glass wedge the
ObreimofPs experiment 11 wedging conditions the bending (line) force F suffers no displacement, so the net work done by this force is zero, i.e. UA = 0. (1.12) At the same time we have, from simple beam theory, the elastic strain energy in the cantilever arm, UB = Ed3 h2 /Sc3 . (1.13) The surface energy is Us = 2cy. (1.14) The total system energy U(c) in (1.5) now follows, and application of the Griffith condition (1.6) leads finally to the equilibrium crack length co = (3£tf3 /*2 /16y)1/4. (1.15) The energy terms UM(c), Us(c), and U(c) are plotted in fig. 1.7. It is evident from the minimum at U(c0) that (1.15) corresponds to a stable configuration. In this instance the fracture is 'controlled': the crack advances into the material at the same rate as that of the wedge. Equation (1.15) indicates that, as in Griffith's uniform tension example, a knowledge of equilibrium crack geometry uniquely determines the surface energy. Obreimoff proceeded thus to evaluate the surface energy of mica under different test conditions, and found a dramatic increase from y = 0.38 J m~2 at normal atmosphere (100 kPa pressure) to y = 5.0 J m~2 in a vacuum (100 uPa). The test environment was clearly an important factor to be considered in evaluating material strength. Moreover, Obreimoff noticed that on insertion of the glass wedge the crack did not grow immediately to its equilibrium length: in air equilibrium was reached within seconds, whereas in a vacuum the crack continued to creep for several days. Thus the time element was another complicating factor to be considered. These observations provided the first indication of the role of chemical kinetics in fracture processes. Obreimoff also observed phenomena that raised the question of reversibility in crack growth. Propagation of the crack was often erratic, with an accompanying visible electrostatic discharge (' triboluminescence'), especially in a vacuum. On partial withdrawal of the glass wedge the