Yielding fracture mechanics

In practice, the inevitable yield zone ahead of the crack leads to a   blunt crack tip. Thus, LEFM predicts a sharp tip to the crack of length   a, as indicated in figure ( k) here. Further considerations require a yield zone, of size   r_y as shown in the previous section. In actuality ( l) the tip is stretched by an amount   δ - the   crack tip opening displacement or COD - which may be confirmed and measured experimentally.

Plastic effects are sometimes allowed for approximately by applying LEFM with an   'equivalent crack size',   a'   figure ( m), equal to the actual crack size plus some proportion of the plastic zone size. This   'small scale yielding correction' becomes increasingly inaccurate when the background stress reaches an appreciable fraction of yield.

δ, being strain-based, is a more suitable crack characterising parameter in the post -yield region, than is   K which is stress -based. It may be shown that, for a centrally cracked infinite plate, the crack opening displacement and the plastic zone size are

( ii)	δ     =   a 8/π   Sy /E'   ln sec ( π/2 σ/Sy )   ;	where   E' = E     plane stress
	ry   =   a ( sec ( π/2 σ/Sy ) - 1 )	or   E' = E/( 1 -ν2)     plane strain

Crack opening displacement is similar to stress intensity in that it is a function of crack length   a and load ( eg. background stress   σ ), however it also takes material properties - particularly S_y - into account. The ratio of background stress to yield stress is particularly relevant as the following results from ( ii) illustrate.

σ --> Sy		Tending towards plastic collapse; both   δ and   ry --> infinity
σ = 2/3 Sy		The plastic zone and crack sizes are equal
σ --> 0		Tending towards LEFM ( negligible plasticity ) and enabling binomial expansion of ( ii)

( iii)       δ   -->   π a σ² / E' S_y     and since this refers to LEFM, then from ( 1) :
                      -->   K_I² / E' S_y

The advantages of COD as a crack characterising parameter should now be apparent - it is applicable over the whole range of loading, from linear elastic fracture to plastic collapse. One may expand on the COD theme as was done for stress intensity - configuration factors for non -infinite shapes, a critical maximum COD ( δ_c ) being a material property, and so on - however we shall restrict ourselves to the adaptation of the COD equation to provide an interaction formula for correlating the two extreme mechanisms noted above.

We take   σ_E as the failure load when the mechanism is perfectly elastic ( LEFM ), and   σ_P as the failure load when the mechanism is perfectly plastic ( plastic collapse ) - both of these can be computed by the methods of the previous sections. To find   σ_F, the failure load when both elastic fracture and plastic collapse occur, we proceed as follows.
            δ_c   =   a ⁸/_π ( S_y /E' ) ln sec ( ^π/₂ σ_F /S_y )       is the critical COD from ( ii) under failure load σ_F
                  =   K_Ic² / E' S_y                       on equating critical properties from ( iii).
But for an infinite plate   K_Ic = σ_E √( π a ) and   σ_P = S_y , so the interaction is

( 3 )         ^π/₂ ( σ_F /σ_P )   =   arccos ( exp [ - ¹/₂ { ^π/₂ ( σ_E /σ_P ) }² ] )

Predicting the failure load in this way via the COD equation, is known as the   'two criteria' or   'CEGB R6' approach to failure assessment - the failure locus from ( 3) plots as shown. Although ( 3) was deduced on the basis of a centrally cracked infinite plate, it is applicable to any configuration provided the axes are interpreted as load ratios rather than merely stress ratios.
 
This approach is just one of many possible interaction models. Recalling the Column Analogy and adapting the nomenclature used with imperfect columns, it is apparent that the R6 locus is a particular case of the general interaction model shown here, which requires   θ --> 1 as ψ --> 0 and   θ --> 0 as ψ --> 1.
Various empirical correlations such as   ( 1/θ - 1) ∗ ( 1/ψ - 1) = constant   have been proposed - some of which describe particular experimental results better than others. One model which finds wide acceptance is the 'circular' failure locus   θ² + ψ² = 1,   that is

( 4)         1/σ_F²   =   1/σ_E² + 1/σ_P²

For an infinite plate   K_Ic = σ_E √( π a ) and   σ_P = S_y as above, so this may be written

( 4a)         σ_F   =   S_y / √( 1 + a / a^* )

                        in which   a^* = ¹/_π ( K_Ic / S_y )²   is a material property, like   b_o ( 2a)

The implication of critical crack size and limitations of crack size detection during inspection must be appreciated. If a crack of the critical size remains undetected by the inspection process then no amount of inspection will avert potential catastrophe.

Fatigue crack growth

A typical crack history under a cyclic load of constant amplitude,   Δσ, is sketched. A crack of size   a₁ exists initially, and grows in a stable, controlled manner until the critical crack size is approached - when crack growth rates increase out of hand and disaster strikes.
For a given material, the instantaneous rate of crack growth, the slope   da/dN, is found to depend mainly upon the stress intensity range,   ΔK, as expected - since

-	it is the near tip field ( characterised by K ) which affects crack advance, and
-	fatigue is known to be greatly dependent upon the range of stress and of load ( the   'S' of the S-N diagram ) which is proportional to the range of stress intensity, as suggested here :

Growth rate   da/dN is affected also by the mean component of intensity - typically characterised by the load ratio   R = K_min/K_max - but this is largely a reflection of   K_max approaching the critical   K_c , ie. of impending fracture.

Closure of the crack, if and when the load becomes compressive, is usually allowed for by simply ignoring any sub -zero compressive excursions. Further evidence suggests that as the load changes from tensile to compressive, the crack closes while the load is still tensile - a result of residual stresses; however we shall neglect such complications here. So, accepting that intensity range is the main contributor to growth rate, the relationship is found to be as follows - this being a unique sigmoidal curve for each particular material. There are three stages

I	Initiation.     Relates to cleavage along grain boundaries at rates of the order of one lattice spacing per cycle ( 4 E-7 mm/cycle ); growth requires the stress intensity range to exceed some threshold value; influenced greatly by environment. This stage is byepassed if a crack exists prior to loading ( eg. the point 1 sketched may be the initial state ).
II	Stable Propagation.     This is the most important stage, dealing with identifiable cracks (say > 0.1 mm ) growing in a stable manner. The direction of propagation is less random than in stage I and the material behaves more homogeneously. The material characteristic crack growth rate vs intensity range is approximately log-log linear throughout stage II and so we may write
( 5 )	da/dN   =   C ΔKn       - the so-called   Paris equation in which   C and the index   n are constant material properties. Selected values appear in the literature, it being customary to cite   C indirectly by the stress intensity range,   ΔKo , which corresponds to a certain crack growth rate,   (da/dN)o - often 1 mm/Mc   thus   C = (da/dN)o / ΔKon
III	Instability.     Although important, this stage exists only for a very small fraction of the component's life, since the instability is catastrophic. The onset of stage III is dictated by the critical crack size being approached, that is by   Kmax tending to   Kc   - 2 in the sketch. One empirical modification of ( 5) which caters for both stages II and III, is :
( 5a)	da/dN   =   C ΔKn / { 1 - ( Kmax /K c )n } Thus if   Kmax << Kc, then the RHS denominator --> 1 corresponding to stage II; alternatively if   Kmax --> Kc then   da/dN tends to infinity ( stage III ).

Integration of ( 5a) yields the component life ( ΔN₁₂ cycles ) which elapses during crack growth from   a₁ to a₂, thus, in normalised form :

( 5b )         { (da/dN)_o ΔN₁₂ /w } { Δσ √( πw) / ΔK_o }ⁿ   =   ∫₁²( Y √α ) ^-n dα   - ( Y_c √α_c ) ^-n ( α₂ - α₁ )
                          in which   α_c is the normalised critical crack size corresponding to   K_c, and   α₂ ≤ α_c.

The Pascal program Crack Growth enables integration along these lines; it assumes that the configuration factors, both elastic and plastic, are single functions of crack size.

When plane strain conditions are assured, then   K_c ≡ K_Ic in the foregoing; otherwise it is suggested that the   K_c predicted by the R6 ( or other elastic -plastic interaction relationship ) should be used.

 
It is usually the initial, rather than the final state of affairs which has largest bearing on component life, as is demonstrated here.