3.1. Probabilistic modelling of safe crack growth and estimation of the durability of structures

Crack growth up to a critical size under cyclic and long-term static loading is a rather complex process, which can be described by various crack growth equations. Methods for estimation of the lifetime of structures containing defects can be developed on the basis these equations. However, there are insufficient studies of the probabilistic aspects of crack growth, which greatly limit the opportunity for practical applications of these methods. To overcome this restriction, probabilistic models of the crack growth have been developed. This part presents

the results, in a generalized manner, of these studies involving the probabilistic modelling of safe crack growth and the estimation of the durability of a structure [18, 19].

Probabilistic factors of crack growth are present both at the micro- and macro-levels of deforming materials. At a micro-level, these factors are the structural heterogeneity of materials and the heterogeneity of the stress-deformed conditions of local zones at the level of grain size. The important factors at the macro-level include the heterogeneity of intensely deformed zones of structural elements, the uncertainty of form, size, and orientation of cracks, and the dispersion in the evaluation of the cyclic crack growth resistance of materials. It is an extremely complex problem to develop probabilistic models of crack growth that reflect all levels of the process. Therefore, our main attention is directed to probabilistic models that handle macro-level factors.

Three models can represent crack growth: a discrete model with casual moments of time; a continuous model with casual increments at fixed time intervals; and discrete continuous model with casual increments of both types. In all cases, the conditions of irreversibility δl<sup>τ</sup> ≥ 0 and kinetic conditions apply

$$\frac{dl}{d\tau} = \varphi(\Delta\sigma, l\_{\tau})\tag{18}$$

understood, then the sense of component w remains unclear. Nevertheless, as will be shown

Using a Monte Carlo method can be considered as another effective approach. The advantage of this method is the possibility to use determined forms of the equations of the crack growth with casual parameters. Let us consider an example of the kinetic equation. As is known, it has

If one accepts that parameters C, m, Δσ, l are casual variables with given probability distribution functions, use of the Monte Carlo method allows one to obtain casual realizations of the

ð Þ <sup>φ</sup>ð Þ<sup>l</sup> <sup>m</sup>�<sup>1</sup>

Distribution densities f(l|τ) or f(τ|l) in the case are a result of frequent realizations of the model expressed by Eq. (23). Obviously, the determination of specified probability distributions is a complex and labour-consuming task, and representative statistics on the parameters of cyclic

The most productive approach is a creation of special probabilistic models, the parameters of which can be determined by means of fracture mechanics. Assuming the fundamental basis of the mechanics of the crack growth, a probabilistic kinetic model can be formulated in the

Increments can be calculated by means of fracture mechanics. As a first approximation, it is possible to suppose that these values are the parts of the plastic zone at the crack front, where deformations reach the fracture state ef. Using an energy notation of crack growth, the proba-

Obviously, the values of WN and W<sup>ε</sup> are widely variable, while WfN and Wf<sup>ε</sup> only slowly change. It is possible to estimate values of WN and W<sup>ε</sup> by means of computational fracture

The presented models contain initial defect sizes l(0). Therefore, probabilistic models of the distributions of the defect sizes must be included into the main models. Statistical studies of defects in welded joints show that it is possible to use a two-parameter Weibull distribution as

3 7 5

�1

8 < :

C X N

j¼1 σm j

9 = ;

Probabilistic Modelling in Solving Analytical Problems of System Engineering

http://dx.doi.org/10.5772/intechopen.75686

lð Þ¼ τ εNPN þ ετp<sup>τ</sup> (24)

� �<sup>β</sup> n o (25)

, p<sup>ε</sup> ¼ 1 � exp � Wε=Wf<sup>ε</sup>

<sup>π</sup><sup>l</sup> <sup>p</sup> <sup>φ</sup>ð Þ<sup>l</sup> n o<sup>m</sup>

(22)

15

(23)

dN <sup>¼</sup> <sup>C</sup>ð Þ <sup>Δ</sup><sup>K</sup> <sup>m</sup> <sup>¼</sup> <sup>C</sup> <sup>Δ</sup><sup>σ</sup> ffiffiffiffiffi

later, the practical use of this approach gives rather efficient results.

dl

process trajectory l(N) for a given number N of loading cycles:

l Nð Þ¼ <sup>ð</sup>

pN <sup>¼</sup> <sup>1</sup> � exp � WN

a basic probabilistic model for distribution of defect sizes:

WfN � � � �<sup>α</sup>

crack resistance are not available at present.

bilities can be written as follows:

lf

2 6 4

l0 l �m=2

the form:

manner of:

mechanics.

The problem of probabilistic modelling of the crack growth consists of the assignation of probabilistic features of trajectories l(τ), which adequately describe real processes. The problem of the probabilistic estimation of functions f(l|τ) and f(τ|l) on given probabilistic features of the trajectories. Modelling trajectories can be carried out on the basis of models of the theory of casual processes, empirical models, and probabilistic models of fracture mechanics.

The theory of casual processes offers a wide spectrum of models. Among the analytical models, it is possible to consider diffusive models as being the most respective. The use of diffusive models allows one to write the kinetic function in the manner of:

$$l(\tau) = a(l, \tau)d\tau + b(l, \tau)dw(\tau) \tag{19}$$

Modelling of processes with jumps requires that the kinetic function given by Eq. (19) must contain an additional component, that is:

$$l(\tau) = l(0) + \int\_{\tau} a\{l(\tau), \tau\} d\tau + \int\_{\tau} b\{l(\tau), \tau\} dw(\tau) + \int\_{\tau} \theta(l, \tau) d\tau \tag{20}$$

The models represented by Eqs. (18)–(20) allow one to directly obtain the densities of the distribution of defects f(l|τ) or durability f(τ|l). In particular, Eq. (19) creates a diffusive durability distribution of kind:

$$f\left(\pi|l\_f\right) = \Phi\left\{\frac{a\pi - l\_f}{b\sqrt{a\pi/l\_f}}\right\} + \exp\left\{\frac{2a^2}{b^2}\right\}\Phi\left\{-\frac{a\pi + l\_f}{b\sqrt{a\pi/l\_c}}\right\}\tag{21}$$

It is necessary to point out that such diffusive models present difficulties with respect to a physical interpretation of the parameters. So, if the physical sense of functions a, b, θ are understood, then the sense of component w remains unclear. Nevertheless, as will be shown later, the practical use of this approach gives rather efficient results.

the results, in a generalized manner, of these studies involving the probabilistic modelling of

Probabilistic factors of crack growth are present both at the micro- and macro-levels of deforming materials. At a micro-level, these factors are the structural heterogeneity of materials and the heterogeneity of the stress-deformed conditions of local zones at the level of grain size. The important factors at the macro-level include the heterogeneity of intensely deformed zones of structural elements, the uncertainty of form, size, and orientation of cracks, and the dispersion in the evaluation of the cyclic crack growth resistance of materials. It is an extremely complex problem to develop probabilistic models of crack growth that reflect all levels of the process. Therefore, our main attention is directed to probabilistic models that handle macro-level factors. Three models can represent crack growth: a discrete model with casual moments of time; a continuous model with casual increments at fixed time intervals; and discrete continuous model with casual increments of both types. In all cases, the conditions of irreversibility δl<sup>τ</sup> ≥

safe crack growth and the estimation of the durability of a structure [18, 19].

dl

of casual processes, empirical models, and probabilistic models of fracture mechanics.

diffusive models allows one to write the kinetic function in the manner of:

a l f g ð Þτ ; τ dτ þ

aτ � lf b ffiffiffiffiffiffiffiffiffiffi aτ=lf p ( )

The problem of probabilistic modelling of the crack growth consists of the assignation of probabilistic features of trajectories l(τ), which adequately describe real processes. The problem of the probabilistic estimation of functions f(l|τ) and f(τ|l) on given probabilistic features of the trajectories. Modelling trajectories can be carried out on the basis of models of the theory

The theory of casual processes offers a wide spectrum of models. Among the analytical models, it is possible to consider diffusive models as being the most respective. The use of

Modelling of processes with jumps requires that the kinetic function given by Eq. (19) must

ð

τ

The models represented by Eqs. (18)–(20) allow one to directly obtain the densities of the distribution of defects f(l|τ) or durability f(τ|l). In particular, Eq. (19) creates a diffusive durability

þ exp

It is necessary to point out that such diffusive models present difficulties with respect to a physical interpretation of the parameters. So, if the physical sense of functions a, b, θ are

2a<sup>2</sup> b2 � �

<sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>φ</sup>ð Þ <sup>Δ</sup>σ; <sup>l</sup><sup>τ</sup> (18)

lð Þ¼ τ a lð Þ ; τ dτ þ b lð Þ ; τ dwð Þτ (19)

ð

θð Þ l; τ dτ (20)

(21)

τ

<sup>Φ</sup> � <sup>a</sup><sup>τ</sup> <sup>þ</sup> lf b ffiffiffiffiffiffiffiffiffiffi aτ=lc p ( )

b l f g ð Þτ ; τ dwð Þþ τ

0 and kinetic conditions apply

14 Probabilistic Modeling in System Engineering

contain an additional component, that is:

distribution of kind:

lð Þ¼ τ lð Þþ 0

f τjlf � � <sup>¼</sup> <sup>Φ</sup>

ð

τ

Using a Monte Carlo method can be considered as another effective approach. The advantage of this method is the possibility to use determined forms of the equations of the crack growth with casual parameters. Let us consider an example of the kinetic equation. As is known, it has the form:

$$\frac{d\mathcal{l}}{d\mathcal{N}} = \mathcal{C}(\Delta \mathcal{K})^m = \mathcal{C}\left\{\Delta \sigma \sqrt{\pi l} \rho(l)\right\}^m \tag{22}$$

If one accepts that parameters C, m, Δσ, l are casual variables with given probability distribution functions, use of the Monte Carlo method allows one to obtain casual realizations of the process trajectory l(N) for a given number N of loading cycles:

$$I(\mathbf{N}) = \left[ \int\_{l\_0}^{l\_f} l^{-m/2} (\boldsymbol{\varrho}(\boldsymbol{I}))^{m-1} \right]^{-1} \left\{ \mathbb{C} \sum\_{j=1}^{N} \sigma\_j^m \right\} \tag{23}$$

Distribution densities f(l|τ) or f(τ|l) in the case are a result of frequent realizations of the model expressed by Eq. (23). Obviously, the determination of specified probability distributions is a complex and labour-consuming task, and representative statistics on the parameters of cyclic crack resistance are not available at present.

The most productive approach is a creation of special probabilistic models, the parameters of which can be determined by means of fracture mechanics. Assuming the fundamental basis of the mechanics of the crack growth, a probabilistic kinetic model can be formulated in the manner of:

$$l(\tau) = \varepsilon\_N P\_N + \varepsilon\_\tau p\_\tau \tag{24}$$

Increments can be calculated by means of fracture mechanics. As a first approximation, it is possible to suppose that these values are the parts of the plastic zone at the crack front, where deformations reach the fracture state ef. Using an energy notation of crack growth, the probabilities can be written as follows:

$$p\_N = 1 - \exp\left\{-\left(\frac{\mathcal{W}\_N}{\mathcal{W}\_{\notin}}\right)^a\right\}, p\_\varepsilon = 1 - \exp\left\{-\left(\mathcal{W}\_\varepsilon/\mathcal{W}\_{\notin}\right)^\beta\right\} \tag{25}$$

Obviously, the values of WN and W<sup>ε</sup> are widely variable, while WfN and Wf<sup>ε</sup> only slowly change. It is possible to estimate values of WN and W<sup>ε</sup> by means of computational fracture mechanics.

The presented models contain initial defect sizes l(0). Therefore, probabilistic models of the distributions of the defect sizes must be included into the main models. Statistical studies of defects in welded joints show that it is possible to use a two-parameter Weibull distribution as a basic probabilistic model for distribution of defect sizes:

$$f(l) = \frac{\gamma}{\theta} \left(\frac{l}{\theta}\right)^{\gamma - 1} \exp\left\{-\left(\frac{l}{\theta}\right)^{\gamma}\right\} \tag{26}$$

Producing complete probabilistic diagrams of integrity is a major prospect. These diagrams are based on information gained from the same model. The structure of complete probabilistic diagrams of integrity is shown in Figure 7. A complete probabilistic diagram of integrity presents itself as a number of sections of a surface connecting three parameters: probability P, safety factor n, and durability N. They allow one to estimate the durability and probability of its attainment. Additionally, there is a possibility for a decision of an inverse task—the definition of probability that for a chosen safety factor, a certain durability will be achieved. Thus, a complete probabilistic diagram of integrity presents itself as a number of sections of a surface

Probabilistic Modelling in Solving Analytical Problems of System Engineering

http://dx.doi.org/10.5772/intechopen.75686

17

connecting three parameters: probability P, safety factor, and durability.

vessels

functions of equal safety factors.

3.2. Reliability and risk assessment of metal-liner composite overwrapped pressure

Metal-composite pressure vessels (MCOPVs) have found a wide application in aerospace and aeronautical industries. Such vessels should combine the impermeability and high weight efficiency with enhanced long-term safety and durability. To meet these requirements, theoretical

Figure 7. A complete probabilistic integrity diagrams: (a) probability of fatigue, (b) durability distributions, and (c)

An estimation of the parameters of Eq. (26) for structures of different types shows that parameter of form γ changes from 0.4 to 4.0. The second parameters θ changes over rather wide limits (from 1.5 mm to 25 mm and more).

These probabilistic models were used to study crack kinetics in the welded joints for highpressure vessels and estimation of durability and reliability functions. The results of calculation reflect the character of the model trajectories over a range of dispersion in the function l(τ), which corresponds to those observed in laboratory experiments.

The empirical Paris-Erdogan model, in combination with the statistical simulation method, was used for modelling kinetic of crack growth in the weld joint of a pipeline in the nuclear reactor VVER-1000. Processing of the results of the probabilistic modelling of the crack kinetics allowed one to estimate reliability functions for a weld joint fracture as is shown in Figure 6.

Figure 6. Reliability functions for hermetic breach criterion (1) and fracture criterion (2).

Producing complete probabilistic diagrams of integrity is a major prospect. These diagrams are based on information gained from the same model. The structure of complete probabilistic diagrams of integrity is shown in Figure 7. A complete probabilistic diagram of integrity presents itself as a number of sections of a surface connecting three parameters: probability P, safety factor n, and durability N. They allow one to estimate the durability and probability of its attainment. Additionally, there is a possibility for a decision of an inverse task—the definition of probability that for a chosen safety factor, a certain durability will be achieved. Thus, a complete probabilistic diagram of integrity presents itself as a number of sections of a surface connecting three parameters: probability P, safety factor, and durability.

f lðÞ¼ <sup>γ</sup> θ l θ <sup>γ</sup>�<sup>1</sup>

which corresponds to those observed in laboratory experiments.

Figure 6. Reliability functions for hermetic breach criterion (1) and fracture criterion (2).

limits (from 1.5 mm to 25 mm and more).

16 Probabilistic Modeling in System Engineering

exp � <sup>l</sup>

An estimation of the parameters of Eq. (26) for structures of different types shows that parameter of form γ changes from 0.4 to 4.0. The second parameters θ changes over rather wide

These probabilistic models were used to study crack kinetics in the welded joints for highpressure vessels and estimation of durability and reliability functions. The results of calculation reflect the character of the model trajectories over a range of dispersion in the function l(τ),

The empirical Paris-Erdogan model, in combination with the statistical simulation method, was used for modelling kinetic of crack growth in the weld joint of a pipeline in the nuclear reactor VVER-1000. Processing of the results of the probabilistic modelling of the crack kinetics allowed one to estimate reliability functions for a weld joint fracture as is shown in Figure 6.

θ <sup>γ</sup>

(26)
