Since the work done is the product of force and distance, the work done seperating unit area is the product of force/unit area (stress) and distance. As stress (s) is a function of distance the work done/unit area is found by taking the integral of s(x) with respect to x (distance).
Approximate the force - deflection curve to a sinewave such that
Now, solving the integral we can determine the work of fracture and equate this to the surface energy (g).
In the elastic region we can assume a linear elastic material where stress and strain are related through Hookes Law
Griffith (A.Griffith Phil.Trans. Roy. Soc. A221, 163 (1920)) proposed that the much lower experimentally determined strengths of brittle solids such as ceramics, where there is little plasticity because of the difficulty of moving dislocations, were the result of the presence within the materials of a population of crack-like defects each of which was capable of concentrating the stress at its crack tip. The magnitude of the stress concentration was dependent on the crack length and the applied stress. Failure would occur when the stress local to the largest crack exceeded the theoretical fracture strength even though the macroscopic stress was relatively low.
In order to determine the magnitude of the Griffith effect we can consider a simple defect - an ellipitical crack of length 2a oriented perpendicular to the maximum principal stress. The concentrated value of the stress at each end of the ellipse would be
where r is the radius of curvature of the crack tip - for an atomically sharp crack, the radius of curvature is similar in magnitude to the burgers vector of a dislocation, and the ratio a/r is subsequently much greater than 1. So...
It should be seen immediately that a defect of about 1mm in length is sufficient to reduce the fracture stress by 2 orders of magnitude.
Griffith's main achievement in providing a basis for fracture strengths of materials containing cracks was his realisation that it was possible to derive a thermodynamic criterion for fracture by considering the change in energy of the material as a crack in it increased in length. Only if the total energy decreased would the crack extend spontaneously under the applied stress. The value of the energetic approach to fracture is that it defocuses attention from the microscopic details of deformation and fracture in the immediate vicinity of the crack tip.
Consider a crack of length 2a is situated in an infinite body and is oriented normal to the applied stress s. Now let us evaluate the changes in energy that occur as the crack is extended by a small distance, da.
Firstly, new crack surfaces are created - absorbs energy - 2 surfaces of area wda multiplied by g, the surface energy per unit area. Secondly, as the crack is assumed to advance only a small amount, the stress and displacement at the crack tip are unchanged. However, these are not the only source of changes in energy. We should consider the macroscopic load displacement curves for a material with a crack of length a and length (a+da).
The material with the larger crack behaves like a weaker spring. Under conditions where a there is a fixed deflection, the extension of the crack is accompanied by a reduction in the load. Thus there is a reduction in the stored elastic strain energy in the body from 1/2P1u1 to 1/2P2u1 because at the same displacement the weaker spring requires less load. Thus at constant deflection the extension of the crack results in a decrease in the elastic strain energy of 1/2(P1-P2)u1 and increase in the surface energy of 2gwda - where w is the thickness of the sample.
If, however, we now consider the conditions of constant load the situation is slightly more complicated but as we shall demonstrate the nett effect is the same. Here the weaker spring will extend more under a constant load there is thus an increase in the elastic strain energy from 1/2P1u1 to 1/2P1u2. However, since there is an extension in the sample, the applied load must fall from u1 to u2 and thus there is a decrease in the potential energy of the load from Pu1 to Pu2. Thus the energy in the material has decreased by an amout P1(u2-u1)-1/2P1(u2-u1).
Thus under condition of constant load there is a reduction in potential energy while under conditions of constant deflection there is a reduction in stored elstic strain energy. Now, the strain energy released = -1/2udP and the potential energy released = -Pdu +1/2Pdu = -1/2pdu
The relationship between deflection u and load P is gibven by
where C is the compliance of the system and
Now substitute for u in the strain energy released and for du in the potential energy released we see that the two are identical with the change in energy being
So where is all this leading?
Well, Griffith recognised that the driving force (thermodynamics again) for crack extension was the difference between the reduction in elastic strain energy/potential energy and that required to create the two new surfaces. Simple! Well almost.
From the figure above, we see
for unit thickness,while S is
for unit thickness
The crack will propogate when any impending increase in length results in a decrease in total energy, i.e. when any crack is longer than acrit, this is simply the value of a at the maximum in the total energy curve i.e. where dW/da=0,
In very brittle solids the term g usally takes the value of the surface energy. However, where there are other energy absorbing processes taking place at the crack tip, such as dislocation motion (plasticity), then g should be replaced with G, the strain energy release rate.
Continue with an exploration of the toughening effect of Fibre Pullout in short fibre composites.