Appendix: mathematical relations

Griffith's criterion

For the simple case of a attenuate ellipsoidal bowl with a able erect to the amount Griffith’s access becomes:

G = \frac{\pi \sigma^2 a}{E}\, (1.1)

where G is the ache activity absolution rate, σ is the activated stress, a is bisected the able length, and E is the Young’s modulus. The ache activity absolution amount can contrarily be accepted as: the amount at which activity is captivated by advance of the crack.

However, we aswell accept that:

G_c = \frac{\pi \sigma_f^2 a}{E}\, (1.2)

If G ≥ Gc, this is the archetype for which the able will activate to propagate.

edit Irwin's modifications

Eventually a modification of Griffith’s debris access emerged from this work; a appellation alleged accent acuteness replaced ache activity absolution amount and a appellation alleged breach courage replaced apparent weakness energy. Both of these agreement are artlessly accompanying to the activity agreement that Griffith used:

K_I = \sigma \sqrt{\pi a}\, (2.1)

and

K_c = \sqrt{E G_c}\, (for even stress) (2.2)

K_c = \sqrt{\frac{E G_c}{1 - \nu^2}}\, (for even strain) (2.3)

where KI is the accent intensity, Kc the breach toughness, and ν is Poisson’s ratio. It is important to admit the actuality that breach constant Kc has altered ethics if abstinent beneath even accent and even strain

Fracture occurs if K_I \geq K_c. For the appropriate case of even ache deformation, Kc becomes KIc and is advised a actual property. The subscript I arises because of the altered means of loading a actual to accredit a able to propagate. It refers to alleged "mode I" loading as against to access II or III:

The three breach modes.

There are three means of applying a force to accredit a able to propagate:

Access I able – Opening access (a compactness accent acclimatized to the even of the crack)

Access II able – Sliding access (a microburst accent acting alongside to the even of the able and erect to the able front)

Access III able – Tearing access (a microburst accent acting alongside to the even of the able and alongside to the able front)

We have to agenda that the announcement for KI in blueprint 2.1 will be altered for geometries added than the center-cracked absolute plate, as discussed in the commodity on the accent acuteness factor. Consequently, it is all-important to acquaint a dimensionless alteration factor, Y, in adjustment to characterize the geometry. We appropriately have:

K_I = Y \sigma \sqrt{\pi a}\, (2.4)

where Y is a action of the able breadth and amplitude of area acclimatized by:

Y \left ( \frac{a}{W} \right ) = \sqrt{\sec\left ( \frac{\pi a}{W} \right )}\, (2.5)

for a area of bound amplitude W absolute a through-thickness able of breadth 2a, or

Y \left ( \frac{a}{W} \right ) = 1.12 - \frac{0.41}{\sqrt \pi} \frac{a}{W} + \frac{18.7}{\sqrt \pi} \left ( \frac{a}{W} \right )^2 - \cdots\, (2.6)

for a area of bound amplitude W absolute a through-thickness bend able of breadth a

edit Elasticity and plasticity

Since engineers became acclimatized to application KIc to characterise breach toughness, a affiliation has been acclimated to abate JIc to it:

K_{Ic} = \sqrt{E^* J_{Ic}}\, area E * = E for even accent and E^* = \frac{E}{1 - \nu^2} for even ache (3.1)

The butt of the mathematics active in this access is interesting, but is apparently bigger summarised in alien pages due to its circuitous nature.