Now, we have the following answer:

The average reduced crack extension force (averaged over the length of the oscillatory segmented crack-front) is = (v0 - v2 / 3) in pure mode I lower than = (v0 + v2 ) in pure mode II lower than = (v0 + 3 v2) in pure mode III; v0 and v2 are positive quantities. =1 for a planar crack under general loading mixed mode I+II+III. may largely exceed 1 for certain non-planar configurations. Hence by adopting a non-planar shape, the propagating crack under general loading improves its motion by increasing the average crack extension force. We may say that the oscillatory crack-front configuration (non-planar crack) corresponds to the natural shape of a propagating crack in brittle homogeneous material. For v0 and v2, please see our contributions: A STUDY OF THE MIXED MODE I+III LOADING OF A NON- PLANAR CRACK USING INFINITESIMAL DISLOCATIONS ; An analysis of a non-planar crack under mixed mode I+III loading using infinitesimal dislocations with edge and screw average characters ; An analysis of a non-planar crack under general loading using continuously distributed sinusoidal edge and screw dislocations (ResearchGate).

It is not about fracture toughness but about higher stresses in case of rough surface which lead to faster crack propagation. You can find some information on this at my book I.I. Kudish and M.J. Covitch, "Modeling and Analytical Methods in Tribology", 2010, CRC.

Before I read your book, please go through the experimental paper quoted above "A study of the brittle fracture characteristics of CoSi2 using laser beam reflections, Acta Metall. et Mater. 43 (1995) 2275 - 2285" and let me know your view about fracture toughness measurement we have performed there. From the experiments, I would say that it is about lower applied stresses which lead  to easier propagation of the non-planar crack with  oscillatory segmented crack-front and, from the calculations, in quasi-static conditions of crack propagation

If you consider that " lower applied stresses which lead  to easier propagation of the non-planar crack" not higher applied stresses lead to easier propagation of cracks then my book will probably not help you.

I confirm that these non-planar cracks propagate under lower applied stresses. I am confident that other experimentalists have observed this fact. This observation encourages me to study, theoretically, the conditions for non-planar crack propagation.

Patrick, you have an experimental observation suggesting you are dealing with real material.  I do not believe you can find an answer to your question by approximating it as homogeneous and seeking classical fracture mechanics explanation.  It is quite probable that the oscillating crack is following a path through existing volume defects, say pores. In this case, the ligaments that need to be broken as the crack evolves may have a total area smaller than the area of a straight crack.  The result would be less dissipated energy in surface separation, hence lower measured toughness.

We have fabricated ourselves and prepared carefully the fracture specimens under the guidance of Prof. Samuel G. Steinemann, analysed the samples through various techniques including XRD, SEM, TEM, Microanalysis (electronic and ionic sondes)... I am confident in the quality of our result. Other experimentalists may confirm that rough cracks propagate under general loading, more easily in brittle materials, as compared to planar cracks(most of the broken surfaces in brittle materials are rough); Please discuss this fact with your colleague experimentalists. When you will trust in this experimental observation, please go through our theoretical analysis.

May be a simple physical argument, to explain why most of the broken surfaces in brittle materials are rough, is as follows.

Assume that the crack is planar in Ox1x3 under mixed mode I+II+III loading: observation shows that propagation is essentially due to the tension applied stress t22 (mode I) in the x2-direction. Now , let the crack to be non-planar with a small oscillatory segmented crack-front in planes perpendicular to the x1-direction. With this new configuration, the crack allows the shear stress t23 (mode III) to become more efficient; observation shows that the direction of crack propagation is now the x1-direction; hence, in addition to mode I, a second mode of applied loading (mode III) comes into play more efficiently. The reasoning can be pursued to include the mode II of loading by allowing the crack to be also oscillatory in planes perpendicular to the x3-direction. In summary, the non-planar oscillatory segmented crack configuration allows the three modes of loading to contribute efficiently to the promotion of crack motion.

Next is: what is the expression of the crack extension force when the crack is non-planar?

If you look at rough surface at microscopic level, it contains number of small/microscopic cracks, which has much prone to propagate cracks through out the body of the component during stressed condition.particularly components, which has tiny cracks(rough surfaces on the surface), when undergo cyclical tensile and compressive stresses will react very fast and propagate the crack through out the body of the component resulting in catastrophic failure of the component. In order to avoid such failures we need to close these crack by some way.Burnish the surface of the component so as to close all small cracks on the surface of the component is one method.Another method to close these tiny cracks responsible for failure can be closed by sand blasting the surface.

It seems to me that you are describing surface (boundary) effects. What is the effect of the roughness character (planar or non-planar ) of the starter crack? We are better concerned with the effect of the non-planar shape of the crack (oscillatory segmented crack-front, when viewing the broken surface along the direction of fracture propagation) on the conditions for crack propagation under stress in quasi static conditions.

In 3-point bending test, when the fractured surface at the centre of the broken specimen is perfectly planar and vertical (i.e. parallel to the loading nose), the fracture process proceeded without shear (pure bending). Our very recent analysis (see below) indicates that applied bending moments required to propagate fracture are larger under pure bending than under bending in presence of shear. As soon as the fractured surface deviates from the central vertical plane (in the non-planar form of "oscillatory segmented facets" when viewed in planes perpendicular to the direction of fracture propagation, for instance), shearing stresses develop during the fracture process; in this latter situation, smaller bending moments are sufficient to promote crack motion and the applied force at the central loading nose may be lowered.

Hence the long standing question from our experiments on CoSi2 (1995) "why striation structures on broken surfaces are observed under lower applied forces" has now a convincing explanation.

Our recent analysis: A theory of the fracture of rectangular bars bent by terminal transverse load and couple.