such as planar crack, LEFM, equivalent stress intensity factor for multi-modal frac- ture, and crack retardation model are not individually quantified. However, in the model validation step, the difference between the model prediction and the experi- mental observation obviously includes the contribution of these sources.
The rest of the chapter is organized as follows. Section 5.2 discusses the crack growth modeling procedure. Section 5.3 discusses several sources of uncertainty and proposes methods to handle them.Section 5.4 discusses global sensitivity analysis to screen the model parameters for calibration, and outlines the proposed Bayesian inference technique for calibrating these parameters.Section 5.5 extends Bayesian hy- pothesis testing to time-dependent problems, in order to assess the validity and the confidence in the crack growth model prediction.Section 5.6 illustrates the proposed methods using a numerical example, surface cracking in a cylindrical component.
model [184] as:
da
dN =φrC(∆K)n
1−∆Kth
∆K m
(5.1) In Eq. 5.1, ∆Kth refers to the threshold stress intensity factor range andφr refers to the retardation parameter which can be defined as:
φr =
r
p,i
aOL+rp,OL−ai
λ
if ai +rp,i < aOL+rp,OL
1 if ai +rp,i ≥aOL+rp,OL
(5.2)
In Eq. 5.2, aOL is the crack length at which the overload is applied, ai is the current crack length,rp,OLis the size of the plastic zone produced by the overload ataOL,rp,iis the size of the plastic zone produced at the current crack lengthai, and λis the curve fitting parameter for the original Wheeler model termed the shape exponent [185].
Sheu et al. [186] and Song. et al. [187] observed that crack growth retardation actually takes place within an effective plastic zone. Hence the size of the plastic zone can be calculated in terms of the applied stress intensity factor (K) and yield strength (σ) as:
rp =αK σ
2
(5.3) where α is known as the effective plastic zone size constant which is obtained exper- imentally [186]. Eq. 5.2 and Eq. 5.3 can be used in combination with Eq. 5.1 under the assumption of small-scale plasticity, where the plastic zone size is estimated using linear-elastic fracture mechanics (LEFM).
The expressions in Eq. 5.2 and Eq. 5.3 can be combined with Eq. 5.1 and used to calculate the crack growth as a function of number of cycles. In each cycle, the stress intensity factor can be expressed as a function of the crack size (a), loading (L) and
angle of orientation (χ). Hence, the crack growth law in Eq. 5.1 can be rewritten as da
dN =g(a, L, χ) (5.4)
Note that the long crack growth model is not applicable to the short crack growth regime and the concept of an equivalent initial flaw size (EIFS) was proposed to bypass short crack growth analysis and make direct use of a long crack growth law for fatigue life prediction. The equivalent initial flaw sizeθ, i.e., the initial condition of the differential equation in Eq. 5.1, can be calculated from material properties (∆Kth, the threshold stress intensity factor and ∆σf, the fatigue limit) and geometric shape factor (Y) as derived by Liu and Mahadevan [188]:
θ = 1 π
∆Kth
Y∆σf
2
(5.5)
By integrating the expression in Eq. 5.1 starting fromθ, the number of cycles (N) to reach a particular crack size aN can be calculated as:
N = Z
dN = Z aN
θ
1
φrC(∆K)n 1− ∆K∆Kthmda (5.6) The stress intensity factor range ∆K in Eq. 5.6 can be expressed as a closed form function of the crack size for specimens with simple geometry subjected to constant amplitude loading. However, this is not the case in many mechanical components, where ∆K depends on the loading conditions, geometry and the crack size. Further, if the loading is multi-axial (for example, simultaneous tension, torsion and bending), then the stress intensity factors corresponding to three modes need to be taken into account. This can be accomplished using an equivalent stress intensity factor. If KI, KII, KIII represent the mode-I, mode-II and mode-III stress intensity factors respectively, then the equivalent stress intensity factor Keqv can be calculated using
a characteristic plane approach proposed by Liu and Mahadevan [189] as:
Keqv = 1 B
r
(KI)2+ (KII
s )2+ (KIII
s )2+A(KH
s )2 (5.7)
In Eq. 5.7, KH is related to hydrostatic stress, and s is the ratio of KII and KI
evaluated at a specific crack growth rate (da/dN). Aand B are material parameters.
The characteristic plane approach is applicable only when the crack surface can be approximated to be planar. The use of the characteristic plane approach for crack growth prediction under multi-axial variable amplitude loading has been applied to cracks in railroad wheels [189] and validated earlier with several data sets [190, 191].
Each cycle in the integration of Eq. 5.6 involves the computation of ∆Keqv us- ing a finite element analysis owing to (1) complicated geometry, and (2) variable amplitude, multi-axial loading. Repeated evaluation of this finite element analysis renders this integration extremely expensive. Hence, it is computationally more af- fordable to substitute the finite element model with an inexpensive surrogate model (also known as response surface model). Different kinds of surrogate models (polyno- mial chaos [45], support vector regression [46], relevance vector regression [47], and Gaussian Process interpolation [51, 52] have been explored and the Gaussian process (GP) modeling technique has been found to have the least error for the purpose of predicting ∆Keqv [181] . A few runs of the finite element analysis are used to train the GP surrogate model and then, this GP model is used to predict the stress intensity factor for other crack sizes and loading cases (for which finite element analysis has not been carried out).
The details of the construction of the Gaussian process were discussed earlier in Section 2.8. Further, the construction of the Gaussian process surrogate model in order to predict the equivalent stress intensity factor has been documented [179, 181].
This equivalent stress intensity factor is then used in cycle-by-cycle integration of the
crack growth law, thereby calculating the crack size (A) as a function of number of load cycles (N). The entire procedure for the adopted crack growth analysis is summarized in Fig. 5.1.
Finite element analysis (Generate training points)
Characteristic plane model
Surrogate model
Loading
∆Keqv
EIFS
Crack growth law (Modified Paris Law
+ Wheeler model)
Predict final crack size (A) as a function of number of load cycles (N)
Material properties
∆Kth, ∆σf
Figure 5.1: Deterministic Crack Propagation Analysis
In Fig. 5.1, note that the finite element analysis and the construction of surrogate model are performed offline, i.e. before the start of crack growth analysis. Crack propagation analysis is done only with the surrogate model.
The algorithm shown in Fig. 5.1 for crack propagation analysis is deterministic and does not account for errors and sources of uncertainty. The next section discusses different sources of uncertainty associated with each of the blocks in Fig. 5.1.