Statistical fracture modeling of weld joint for nuclear reactor components

Theoretical and Applied Fracture Mechanics 29 (1998) 103±107

Statistical fracture modeling of weld joint for nuclear reactor components A. Lepikhin *, V. Moskvichev, S. Doronin Computing Center, Siberian Branch of Russian Academy of Sciences, 660036 Krasonoyarsk, Russian Federation

Abstract Probabilistic safety and risk assessment of nuclear reactor equipment are referred to as integrated safety assessment which is implemented by the application of probabilistic fracture mechanics (PFM). Such a study is based on statistical simulation of weld damages under dynamic condition. Because of damage accumulation in welded nuclear reactor components under dynamic load, the need for evaluating the risk of fracture is apparent. This work considers the statistical simulation of dynamic damage of weld joint by fracture. Considered in the analysis are weld defects, damage accumulation, crack initiation and crack growth. Ó 1998 Elsevier Science Ltd. All rights reserved. Keywords: Fracture; Risk; Modeling; Weld joint

1. Introduction Risk is a measure of total probable impact failure. There are three basic ways to assess risk. The simplest approach is to extrapolate past failure date and experience while ignoring explicit mechanical and physical details of the system. Such a statistical analysis is called the ``date-base'' approach that has been used in [1,2]. The second approach deals with probabilistic engineering analysis whereby probabilistic fracture mechanics (PFM) is included as a subset [3,4]. In this case the statistical variations of each model parameter are quanti®ed and combined to compute safety or risk. The third approach known as combined analysis utilizes both the data-base and PFM approaches [5]. The second and third approaches

* Corresponding author. Tel.: +7 3912 494804; fax: +7 3912 434657; e-mail: [email protected].

0167-8442/98/$19.00 Ó 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 9 8 ) 0 0 0 2 2 - 6

cannot be determined without the statistical models of damage accumulation and fracture. 2. Basic concept The function of safety S…t† and of risk R…t† can be presented as: S…t† ˆ P fD…t† 2 XS g; R…t† ˆ P fD…t† 2 XR g;

…1† …2†

where D…t† is the vector of damage, XS the safety area, and XR the risk area. The safety and risk areas are presented as XS ˆ fD j g…D† > 0g; and XD ˆ fD j g…D† < 0g; …3† where g…D† is a failure function. The failure function will be de®ned for crack initiation, hermetic breach, and fracture. In the ®rst case, D…t† serves as a scalar measure of

104

A. Lepikhin et al. / Theoretical and Applied Fracture Mechanics 29 (1998) 103±107

accumulated damage: W…t† ˆ ‰0; 1Š. In second and third case, use is made of the vector of basic variable a…t† ˆ fai …t†; i ˆ 1; mg, which is characteristic of crack geometry. For all cases, there are conditions of irreversibility of damage dD P 0 and the dynamic equations dD ˆ u…D; t; Q†; …4† dt where Q is load on the weld. This equation is basic for studying the dynamic behavior of weld damages by using the probability density function f …D; t†, safety S…t† or risk R…t† function. 3. Defect distributions Among all of the parameters in¯uencing the risk of weld joint fracture, the defect distribution plays a dominant role. Functions S…t† and R…t† are very sensitive to the selection of weld defect distribution which will be presented for the type and size of weld defect. Some typical weld defects are shown in Fig. 1 while the associated relative frequency for each of the defects A, B, D and F can be found in Fig. 2(a). The corresponding size distributions are given in Fig. 2(b). They are obtained using statistical analysis of non-destructive inspection data of welds. Diculty arises in selection of defect size distribution because of the lack of statistical data and probability of non-destruction detection data. Moreover, quantitative assessment of defect formation and detectability are not known. In the absence of any accepted rules for selecting ¯aw size, a variety of distribution functions have been employed by di€erent investigators. Fig. 2. Distribution of type and size of weld defects.

Fig. 1. Types of weld defects.

They involve an exponential, Weibull, normal, lognormal, gamma and beta distribution. As theoretical and experimental studies [6] of weld joint defects show that distribution f …a† of defect size a can be approximated in the Weibull form     b  a bÿ1 a b ; …5† exp ÿ f …a† ˆ h h h

A. Lepikhin et al. / Theoretical and Applied Fracture Mechanics 29 (1998) 103±107

where h and b are statistical parameters. The following conclusions can be made: · Weld defect models can be described in terms of mean numbers of weld defects, distribution of type and distribution of size. · Distributions of defect size can be approximated by the Weibull function. 4. Damage accumulation and crack initiation The initial distribution of weld defects in Eq. (5) can be modi®ed for nuclear components in service as the defect grows under the action of fatigue loading. The probability and mean time (number cycle of load) of crack initiation from weld defect can be calculated by using the model of damage accumulation. The damage accumulation model is based on the interaction-free theory such that the damage increment in a stress cycle depends only on the damage accumulated at the beginning of the stress cycle and stress cycle itself DDn ˆ u…Dnÿ1 ; Qn †;

n ˆ 1; 2; 3; . . .

105

5. Probability fracture mechanics The risk of weld fracture of nuclear components as estimated by PFM is governed by the random load, random material properties, and the number and size distribution of crack. The simplest cases of a PFM model involves only two probability distributions f …a† and f …ac †. Hence, the risk function (Fig. 4) can be written as Z1 Za R…a† ˆ 1 ÿ P …a < ac † ˆ f …a† f …ac † dac da 0

0

…9† The critical crack size distribution f …ac ; t† is related to the fracture resistance distribution of the weld.

…6†

Damage accumulation for random loading is based on the model in Ref. [6]. The distribution of crack initiation time F …tc † is given by F …tc † ˆ 1 ÿ P fD…t† < D …t†; t 2 ‰0; tc Šg:

…7†

The damage increment for the statistical loading process is Z Z f …ra ; rm ; t† dra drm ; …8† D…t† ˆ N …ra ; rm ; t† where f …r; t† is the probability density stress function and N …r; t† the number of cycles for crack initiation. The amplitude of stress ra is calculated with use of the stress concentration coecient of weld defect. The value of D is taken as zero in the initial state and as one for failure or crack initiation (Fig. 3). Statistical modeling of damage accumulation is accomplished by using the Monte Carlo technique simulation. Eqs. (4), (5) and (8) give estimate of the mean value Et , variation Vt and probability function f …a; t† of crack size distribution as function of time.

Fig. 3. Diagram of damage accumulation and crack initiation.

Fig. 4. Risk function without stable crack growth.

106

A. Lepikhin et al. / Theoretical and Applied Fracture Mechanics 29 (1998) 103±107

For a Poisson PFM model, the risk function can be estimated as 8 9 < Zt = …10† R…a; t† ˆ 1 ÿ exp ÿ l…ac ; s† ds ; : ; 0

where l…ac ; s† is the mean number of critical crack size at time s which may be evaluated from Z1 …11† l…ac ; s† ˆ l ‰1 ÿ P …a; sja; t†Šf …a† da; ac

where l is mean of number of all cracks sized. The general risk function of the weld Rw …a; t† is Rw …a; t† ˆ R…a† ‡ R…a; t†:

…12†

Fig. 5. Probability density function of crack size.

The transitional probability P …a; sja; t† gives the dynamic probabilistic structure of a given stochastic process of crack growth in the weld. A two parameter relation can be used: da m …13† ˆ C…DK† ; dt where da=dt is the crack growth rate, and DK the change of stress intensity factor. The empirical parameters are C and m. The critical crack size ac can be calculated from f …ac † for each zone in the weld joint [7]. 6. Numerical example The present model can be used to determination the risk of failure function R…t† or safety function S…t† ˆ 1 ÿ R…t† throughout the service life of nuclear components. Here, risk is de®ned as the crack penetrates through the wall thickness. The calculations are based on the following assumptions [8]: · The weld contains initial defects. · The parameters of distributions size of defects are known. · The weld defect corresponds to initiation of crack. · The two parameter crack growth rate relation can be used. · The fracture resistance distribution are known for the HAZ and weld metal. A typical example of modi®ed defect size distribution are shown in Fig. 5. Fig. 6 shows the de-

Fig. 6. Risk functions of weld.

pendence of risk function on operation time for two types of nuclear component welds used in nuclear reactor VVER-1000. 7. Conclusion Discussed in this work are the probabilistic aspects of weld fracture problem. The analytical technique facilitates the risk assessment using fracture mechanics. The statistical treatment of weld fracture problem of nuclear components was initiated several decades ago. But it was not really developed until 1960s. More advanced techniques are under development.

A. Lepikhin et al. / Theoretical and Applied Fracture Mechanics 29 (1998) 103±107

Acknowledgements This study has been ®nancially supported by the Russia Fund of Fundamental Research (RFFR). References [1] A.M. Freudenthal, M. Shinozuka, Structural safety under conditions of ultimate load failure and fatigue, WADD TR, 1961, pp. 61±77. [2] A.M. Freudenthal, New aspects of fatigue and fracture mechanics, Eng. Fract. Mech. 6 (1974) 775±793. [3] G.O. Johnston, A review of probabilistic fracture mechanics literature, Reliability Engineering 3 (1982) 423±448.

107

[4] V.V. Bolotin, Stochastic models of cumulative damage in composite materials, in: Progress in Fatigue and Fracture, 1976, pp. 103±111. [5] C.A. Rau, P.M. Besuner, Statistical aspects of design: risk assessment and structural safety, Philos. Trans. R. Soc. London Ser. A 299 (1981) 111±130. [6] V.V. Bolotin, The Operation Time of Machines and Constructions, Mashinostroenie, Moskow, 1991. [7] A.M. Lepikhin, V.V. Moskvichev, Bases of information about presence of defects and characteristics of crack resistance, Problems of Engineering and Automation 5 (1991) 85±89. [8] A.M. Lepikhin, V.V. Moskvichev, Computer simulation of fracture in problems of welding joints reliability, in: Trans. Sec. Conf. Mater. Sc. Probl., Prod. and Oper. NPP Fac., 1991, pp. 291±297.