## PROBABILISTIC MODELING OF SURFACE CRACK

C a b. Figure 1. A roll of continuous casting machine (CCM): a) loading ... fracture toughness at 375°C. The average value of critical SIF is Кϲ = 78.0 MPa m .

Int J Fract (2011) 172:113–120 DOI 10.1007/s10704-011-9641-7

LETTERS IN FRACTURE AND MICROMECHANICS

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q

d

D A

a

C

L a

b

Figure 1. A roll of continuous casting machine (CCM): a) loading scheme; b) geometrical parameters of crosssection with a semi-elliptical crack

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Probabilistic modeling of surface crack growth in a roll of continuous casting machine

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The roll has the form of a thick-wall hollow cylinder with the outer diameter D = 320 mm and a cooling hole with a diameter d = 80 mm. The distance between supports is equal to L = 2000 mm. A semi-elliptical surface crack is located in the central section of the roll perpendicular to its longitudinal axis. The residual lifetime of CCM roll is assessed under the following assumptions. A semi-elliptical surface crack with an initial depth ɚ0 = 15 mm is considered. We assume for simplicity that the temperature fluctuation during one rotation of the roll is insignificant. The temperature in the median section of the roll is equal to 375°ɋ, and, on the roll surface, it is 600°ɋ. The parameter ɋ of Paris equation has a statistical scatter described by a suitable statistical distribution law. The external loading of the roll is the pressure of the liquid metal. The FCG of the semi-elliptical crack will be modeled under stress ratio R = Kmin/Kmax = 0, where Kmin and Kmax are the minimum and the maximum stress intensity factors (SIF), respectively. Stress range is equal to Δı= ımax- ımin = 257 MPa, with ımax and ımin are the maximum and the minimum stresses, respectively. For this modeling, the SIF at the deepest point A (Fig. 1b) and at the point C on the surface of the semi-elliptical crack in the hollow roll is approximated using the data from Carpinteri (1992). The surfaces of the dimensionless SIF at the points A and C YA(C ) = K A(C ) σ bg πa as a function of the normalized crack depth ζ = a/D and shape of its front a/c are built. They are presented in Fig. 2. K A(C ) = σ bg πaYA(C )

(

YA = K A σ bg πa

(1)

)

(

YC = K C σ bg πa

)

0.6 0.8

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Figure 2. The surface of the dimensionless SIF: a) at the deepest point A; b) at point ɋ on the surface of the semielliptical crack

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The FCG rate da/dN in a steel 25Cr1Mo1V is determined under uniaxial loading of single edge notch tension specimens of 5 mm in thickness and of 25 mm in width. Specimens were tested under tension with stress ratio R = 0 according to the ASTM E647 Standard, 1992. The tests were carried out on an electrohydraulic testing machine of STM 100 type under a computer control. The loading frequency was 0.1 Hz, cycle waveform was triangular. Mechanical properties of the steel at 375°C are the following: the yield stress Sy = 670 MPa and the ultimate tensile stress Su = 690 MPa. The compact tension specimens with a thickness of 19 mm were tested on fracture toughness at 375°C. The average value of critical SIF is Ʉɫ = 78.0 MPa m . The obtained experimental dependencies of da/dN on ΔK (Fig. 3) are approximated by Paris equation da/dN = ɋ(ΔK)m, (2) At 375°ɋ these parameters are: ɋ = 9.26·10-11 mm/cycles/(ΜPɚ m ), m = 4.09 and, at 600°ɋ, they are ɋ = 6.6·10-9 mm/cycles/(MPa m ), m = 3.26. da/dN mm/cycle

10-2

da / dN = 6,6 ⋅ 10 −9 ΔK 3, 26

10-3 da / dN = 9,26 ⋅ 10 −11 ΔK 4,09 -4

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40

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ǻɄ, MPa m

Figure 3. Dependencies of FCG rate in a steel 25Cr1Mo1V on SIF range

Parameter C for the steel 25Cr1Mo1V was treated as a normal random variable. We approximated the experimental data by the least squares method and determined the distribution parameters of ɋ. The assumptions about distribution function were verified by Anderson-Darling (A-D) test according to methods proposed in Varfolomeev and Yasniy, 2008. The statistical value of this criterion is calculated as :M 2i − 1 [ln(F0 [z(i ) ]) + ln(1 − F0 [z(M −i+1) ])] (3) AD = − M − ¦ i =1 M where F0 is the distribution function for which the assumption is checked on, z(i) is the i-th sorted standardized value of the sample, and M is its size. For the normal distribution, standardization is performed in the following way:

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z (i ) =

117

x (i ) − μ

σ

(4)

The critical value (CV) of the Anderson – Darling criterion depends on the chosen distribution and the significance level Į. The hypothesis on a distribution with given parameters is rejected at a certain significance level Į (usually taken Į = 0.05) for a sample of size M if the statistical value AD of the Anderson – Darling criterion is greater than the critical one (CV), i.e. when AD > CV. Recall that lognormal probability distribution function has a form: ª 1 § x − x0 ·2 º f ( x) = exp « − ln » ( x − x0 ) s 2π «¬ 2s 2 ©¨ n ¹¸ »¼ , 1

(5)

where x0 , s and n are parameters of log-normal distribution. Normal probability density function is: ª 1 § x − α ·2 º 1 f ( x) = exp « − ¨ ¸ » β 2π «¬ 2 © β ¹ ¼»

(6)

where α and β are mean and standard deviation of normal distribution, respectively. Since the parameters of cyclic crack resistance C and m are mutually dependent, the scatter of data at 375Ƞɋ and at 600Ƞɋ was described for constant m and variable C. By forming samples that contain at least five points from Paris region of FCG curve, the statistical distributions of ɋ were constructed. The normal distributions were found acceptable (solid curves on Fig. 4). P

P

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0.6 Norm al di stribution

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Normal di stri bution Empi ri cal dat a

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Empirical dat a

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7·10 11

8·10 11

9· 1011

1·1 010

1.1·1010

C

6·10-9

7·10-9

8· 10-9

9· 10-9

C

Figure 4. Distribution function of the coefficient C with R=0 and f = 0,1 Hz: a) at 375°ɋ and b) at 600°ɋ

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The FCG rate of the surface semi-elliptical crack in a CCM roll at radial direction (point Ⱥ) and circumferential direction (point ɋ) was calculated from a system of two Paris-type equations ­ da m °° dN = C (ΔK A ) , ® ° dc = C (ΔK )m ; C °¯ dN

(7)

where KA, KC are the SIF of mode I in points Ⱥ and ɋ of the crack front, respectively. To model FCG, normal distributions of parameter C (Fig. 4) at temperatures 375°ɋ and 600°ɋ are employed. The statistical values of A-D test for 375 °ɋ and 600 °ɋ are 0.399 and 0.175, respectively. In both cases, the critical value was equal to 0.754. An initial crack with a depth a0 =15 mm and a front aspect ratio of a0/c0=0.5 was considered. The system of differential equations (7) was solved numerically by the fifth order Runge-Kutta method. The FCG rate was modeled in two directions. First, the values of final lifetime were preset. They were: N = 34900, 40480, 42750 cycles for the temperature 375°ɋ and N = 10980, 13250, 14300 cycles for the temperature 600°ɋ. As a result of simulations, the distributions of final crack depth were obtained. Second, we preset that the critical depth of the crack as ɚ = 45, 60, or 70 mm, and then constructed the lifetime distributions. The crack depth a = 70 mm is the critical crack depth for the crack growth resistance criterion - fracture toughness Kɫ = 78.0 MPa m . In each of the two modeling directions (preset lifetime and preset critical depth of the crack), 100 simulations were carried out. P

P

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70 a, mm

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42000N,cycle

b

Fig. 5. a) Distribution of surface crack depth at temperature 375°ɋ for N = 34900 cycles (curve 1), 40480 cycles (curve 2) and 42750 cycles (curve 3) ; b) lifetime at temperature 375°ɋ for ɚ = 45 mm (curve 1), ɚ = 60 mm (curve 2), and ɚ = 70 mm (curve 3)

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The results of modeling were described by the normal distribution function at 375°ɋ for crack depth a (Fig 5ɚ) and lifetime N (Fig 5b). The normal distributions of a and N at 600°ɋ are shown in Figs. 6a and 6b, respectively. P

P

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70 a, mm

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14000 N,cycle

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Fig. 6. a) Distribution of surface crack depth at temperature 600°ɋ for N = 10980 cycles (curve 1), 13250 cycles (curve 2) and 14300 cycles (curve 3) ; b) lifetime at temperature 600°ɋ for ɚ = 45 mm (curve 1), ɚ = 60 mm (curve 2), and ɚ = 70 mm (curve 3)

For Nf = 34900, 40480, 42750 cycles, the median of random variable of crack depth is equal to af = 45, 60, 70 mm, respectively. The average values of random variable of lifetime for final crack depth 45, 60, 70 mm are approximately equal to 35000 cycles (1), 40500 cycles (2), 43000 cycles. To illustrate the crack propagation, we present the fronts of surface cracks (Fig. 7a) and the dependence of the crack shape parameter a/c on the normalized crack depth a/D (Fig. 7b). a/c 0,8

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a

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b Fig. 7. a) Propagation of the crack front as the depth increases; b) crack shape parameter vs. normalized depth

Crack shape factor a/c increases from 0.5 to about 0.8 with an increase of its relative depth of 0.05 to 0.14 (Fig. 7b). With further increase of the crack depth up to 0.22, the crack shape parameter decreases.

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3. Conclusions. A methodology to determine CCM roll residual lifetime is developed. It is based on an analysis of a thick-walled cylinder with a semi-elliptical crack that takes into account the statistical variation of the characteristics of cyclic crack resistance under operating conditions. The growth of semi-elliptical crack in a CCM roll is modeled. The coefficient C of the Paris equation is considered as a normally distributed random variable. The procedure enables to determine the parameters of acceptable distribution of crack depth for a given number of stress cycles and the distribution of durability for a given allowable depth of crack. As a result of simulation, distributions of the final crack depth af for given Nf and fatigue life Nf for af = 45, 60, 70 mm are presented for temperatures 375°C and 600°ɋ. References Mercado-Solis, R. D., Beynon, J. H. (2005). Simulation of thermal fatigue in hot strip mill work rolls. Scandinavian Journal of Metallurgy 34, 175–191. Yasniy, P., Maruschak, P., Lapusta, Y., Hlado, V., Baran, D. (2008). Thermal fatigue material degradation of caster rolls' surface layers. Mechanics of Advanced Materials and Structures 15, 499–507. Colas, R., Ramyrez, J., Sandoval, I., Morales, J. C., and Leduc, L.A. (1999). Damage in hot rolling work rolls. Wear 230, 56–60. Yasniy, P., Maruschak, P., Lapusta, Y. (2006). Experimental study of crack growth in a bimetal under fatigue and fatigue–creep conditions. International Journal of Fracture 139, 545–552. Gorbatikh, L., Kachanov, M., (2000). A simple technique for constructing the full stress and displacement fields in elastic plates with multiple cracks. Engineering Fracture Mechanics 66, 51-63. Lapusta, Y. Henaff-Gardin, C. (2000). An analytical model for periodic Į°-layer cracking in composite laminates. International Journal of Fracture, 102, L73-76 (2000). Loboda V., Lapusta Y. Sheveleva A. (2007). Electro-mechanical pre-fracture zones for an electrically permeable interface crack in a piezoelectric biomaterial. International Journal of Solids and Structures, 44, 5538-5553. Piat R., Lapusta Y., Böhlke T., Guellali M., Reznik B., Gerthsen D., Tengfei Chen, Oberacker R., Hoffmann M.J. (2007). Microstructure-induced thermal stresses in pyrolytic carbon matrices at temperatures up to 2900°C. Journal of the European Ceramic Society, 27, 4813-4820. Lapusta, Y, Wagner, W. (2001). On various material and fibre-matrix interface models in the near-surface instability problems for fibrous composites. Composites Part A: Applied Science and Manufacturing, 32, 413-423. Revel, P., Kircher, D., and Bogard, V. (2000). Experimental and numerical simulation of a stainless steel coating subjected to thermal fatigue. Materials Science and Engineering A 290, 25–32. Srivastava, A., Joshi, V., and Shivpuri, R. (2004). Computer modeling and prediction of thermal fatigue cracking in die-casting tooling. Wear 256, 38–43. Grooteman, F. (2008). A stochastic approach to determine lifetimes and inspection schemes for aircraft components. International Journal of Fatigue 30, 138–149. Castillo, E., Fernández-Canteli, A., Pinto, H., Ruiz-Ripoll, M. L. (2008). A statistical model for crack growth based on tension and compression Wöhler fields. Engineering Fracture Mechanics 75, 4439–4449. Riahi, H., Bressolette, Ph., Chateauneuf, A. (2010). Random fatigue crack growth in mixed mode by stochastic collocation method. Engineering Fracture Mechanics 77, 3292–3309. Beretta, S., Carboni, M. (2006). Experiments and stochastic model for propagation lifetime of railway axles. Engineering Fracture Mechanics 73, 2627–2641. Ichikawa, M., Hamaguchi, M., Nakamura, T. (1983). Statistical characteristics of m and fatigue crack propagation law da/dN=C(ǻK)m (2024-T3 Al alloy). Journal of the Society of Materials Science 33, 8–13. Carpinteri, A. (1992). Elliptical-arc surface cracks in round bars, Fatigue and Fracture of Engineering Materials and Structures 15, 1141-1153. ASTM Standard test method for measurement of fatigue crack growth rates (1992). Annual Book of ASTM Standards, E647-00, Vol 03.01, W. Conshohocken, PA, 674–701. Varfolomeev, I. V., Yasniy, O. P. (2008). Modeling of fracture of cracked structural elements with the use of probabilistic methods. Materials Science 44, 87–96.

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