Computer simulation of effect of grain size distribution on Weibull parameters

Theoretical and Applied Fracture Mechanics 35 (2001) 255±260

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Computer simulation of e€ect of grain size distribution on Weibull parameters S.A. Kotrechko a,*, Yu. Meshkov a, I. Dlouhy b a

G.V. Kurdyumov Institute for Metal Physics, National Academy of Sciences of the Ukraine, 36 Vernadsky Blvd., Kyiv-142, 01680, Ukraine b Institute of Physics of Materials ASCR, Zizkova 22, CZ-61662 Brno, Czech Republic

Abstract Computer simulation of crack nucleation associated with unstable equilibrium in polycrystal consisting of grains of random sizes and orientations are presented. Micro-stress and micro-strain ¯uctuations were taken into account. Distribution functions for probability of fracture for di€erent parameters of grain structure were obtained. It is shown that these functions may be approximated by three-parameter Weibull distribution with high accuracy. Ascertained are the dependencies of Weibull parameters on both grain sizes and their variance. Ó 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction Weibull distribution has been widely used for description of quasi-brittle fracture in metals and alloys. Such a distribution has been used as a ``local approach'' to cleavage fracture [1±3]. Material was assumed to consist of a number of statistically independent elements (cells). Probability of fracture of separate cell can be described by Weibull distribution. One of the problems concerned with the grain size distribution on Weibull parameters is of great interest. Statistical modelling usually considers micro-crack formation as a result of carbide particle cracking [4±7]. In this case, Weibull parameters are determined according to the data on particle size distribution from quantitative measurements of the particle size.

*

Corresponding author. E-mail address: [email protected] (S.A. Kotrechko).

Grith's treatment for a crack has been applied where the micro-crack length distribution was assumed to be equivalent to one of the carbide particle diameters. Such an approach cannot be used to obtain the statistics of micro-cracks forming according to dislocation mechanism, i.e. at the pile-up tip of micro-crack nucleation. Polycrystalline metals without brittle second-phase particles as well as high-strength steels with disperse ®ne particles, however, have been focused to fracture by such a micro-mechanism. A recent attempt was made to determine the relations between Weibull parameters and metal micro-structure characteristics [8]. However, it is rather dicult to change the value of the most probable grain size at constant value of its variance, and vice versa. This work aims to determine the general regularities of in¯uence of grain distribution characteristics on Weibull parameters by computer simulation for the micro- and macrofracture of polycrystals.

0167-8442/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 0 1 ) 0 0 0 4 9 - 0

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2. Proposed model The model in [9] is used to solve the proposed problem. The main regularities of crack nucleation and catastrophic propagation in polycrystals with random size and orientation will be described. 2.1. Crack nucleation Cleavage fracture of metals and alloys occurs by micro-crack nucleation assisted by the plastic deformation. According to classical model of fracture, the crack nucleus forms at the tip of dislocation pile-up where the corresponding stress must reach its critical level sCR : N sef P sCR ;

…1†

where N is the number of dislocations in pile-up and sef is the e€ective stress. For small plastic strains ranging from 0.002 to 0.1, the magnitude of sef depends on the micro-strain of the grain q …2† sef ˆ b e=dg ; where b is a constant, dg the grain size and e is the equivalent micro-strain in the grain. Using Eqs. (1) and (2), a relation among the crack nucleus size a, grain size dg and micro-strain e can be obtained:  2 p dsCR dg ; …3† aˆ e c b where p is the constant (for iron p ˆ 6  10 10 H) dependent on the lattice parameters only while c is the speci®c energy of fracture of lattice at the nucleus crack tip, d  0:3 [9]. Note that in Eq. (3), e P eC ;

…4†

where eC is the minimum micro-strain that is required for crack nucleation. In a polycrystal, the grain size dg and the plastic strain e are random quantities. Accordingly, a is also a random quantity.

stress ®eld have been considered. The crack was modelled in two dimensions. The critical principal micro-stress nC at the onset of crack propagation is given by k  …5† nC ˆ u…h; g† n; a where n is the e€ective micro-stress induced by dislocations, k the coecient that characterizes resistance of the crystal to the crack propagation, u…h; g† the function that describes in¯uence of micro-stress state g (note that g ˆ n22 =n11 , 1 n11 and n22 are the principal tensile micro-stresses) and orientation of the crack h. As a ®rst approximation, there results 1 u…h; g† ˆ q : sin2 h ‡ g cos2 h

The distribution density for nC can be written as follows: Z f3 …g† F1 …nC † ˆ (Z 

Lh

Lg

"Z Z

f4 …h†

Ln

a

amax

# )  da dn dh dg; f5 …a; n† 

…7† where f4 …h† and f3 …g† are distribution densities for the nucleus crack orientation angle h and the micro-stress state mode parameter g, respectively.  is the function of joint distribution density f5 …a; n†  Explicit expressions for these functions for a and n. can be found in [9]. Eq. (5) determines the magnitude of micro-stresses, nC , at which micro-crack reaches its unstable equilibrium. At a given macrostress rF , the probability of this event F2 …rF † is described by [9] Z nmax C F2 …rF † ˆ 0:5 f1 …nC † "

nmin C

 1

2.2. Crack propagation When modelling the micro-crack unstable equilibrium, the e€ect of dislocation micro-stress, random crack orientation and multiaxial micro-

…6†

1

erf

n rF pC 2I…n11 †rFc

!# dnC ;

The relation n22 =n11 determines the lower bound of nC .

…8†

S.A. Kotrechko et al. / Theoretical and Applied Fracture Mechanics 35 (2001) 255±260

where f1 …nC † ˆ dF1 …nC †=dnC

…9†

is determined by the distribution density for nC . Here, I…n11 † is the coecient of variation of microstress n11 created by  elastic deformations of the p grains I…n11 † ˆ Dn11 =hn11 i. Note that D…n11 † is the micro-stress variance, hn11 i  r1 . The mean value of micro-stress and r1 is the principal macrostress. The distribution for micro-stress was approximated by the Gauss law. Eq. (8) determines the cumulative probability of cell failure. Commonly accepted models use the Weibull distribution to describe this probability.

3. Computer simulation The procedure (algorithm) used for the calculations follows the ¯ow chart in Fig. 1. The structure of metal is speci®ed by grain size distri-

257

bution that is approximated by a logarithmic normal law. Calculations are made for uniaxial tension (r1 6ˆ 0; r2 ˆ 0; r3 ˆ 0). At the microscopic level, the stress state is triaxial nature in character. It means that at hn22 i ˆ 0 and hn33 i ˆ 0, the variances of these stresses are not equal to zero. For quasi-isotropic (non-textured) polycrystal under uniaxial tension, the values of D…n11 †, D…n22 † and D…n33 † are determined as D…n11 † ˆ DI r21 ;

…10†

D…n22 † ˆ D…n33 † ˆ DII r21 ;

…11†

where DI and DII are coecients dependent on the elastic constants of the lattice. For polycrystalline a-iron, DI ˆ 1:7  10 2 and DII ˆ 0:66  10 2 [9]. According to Eqs. (5) and (6), the relation g ˆ n22 =n11 a€ects the critical micro-stress nC . For quasi-brittle fracture of metals, the residual strain ranges from 0.002 to 0.2. According to low-temperature test data of unnotched tensile specimens,

Fig. 1. Flowchart for modelling of nucleation and unstable equilibrium of cracks in polycrystal.

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the brittle fracture stress for iron and structural steels are non-monotonic depending on the strain. The minimum cleavage fracture stress corresponds to certain critical strain [10]. For structural carbon steels, this strain is of the order of 0.02 which was used in the calculations. Distribution of microplastic strains was approximated by normal law [9]. Micro-strain coecient variation 0.30 was used for the calculations. According to the distribution data for grain size, orientation, and micro-plastic strain, the distribution function for crack nucleus size f1 …a† were evaluated. For quasi-isotropic polycrystal distribution function, the crack nucleus orientation f4 …h† may be described by normal law with good accuracy. The mean value is 0:25p while the coecient of variation is 0:023p . The critical micro-stress for unstable equilibrium of the crack nucleus can be estimated by Eq. (6). The level of e€ective tensile micro-stress n depends on the micro-plastic strain and grain size. For example, with a macro-strain of e ˆ 0:02 and the mean grain size 0.03 mm the value of n is equal to 372 MPa. Based on the distribution f1 …nC † and the value of variance D…n11 †, the cumulative probability for the catastrophic propagation of the crack nucleus F2 …rF † at a macrostresses of rF was estimated. In the Weibull model, this function is interpreted as the strength distribution of cells. The calculations were made for the most probable grain size dmpv that varies from 5 to 120 lm. The variance logarithmic size ranges from 0.05 to 0.45. It corresponds to the range of dmax =dmpv from 1.72 to 5.0, where dmax is the maximum grain size evaluated at a probability of 0.999. Results of computer calculations were approximated with three-parameter Weibull distribution. Standard deviation for F2 …rF † did not exceed 1:5  10 4 .

where m is a shape factor, rth the threshold stress (the lower bound strength) and ru is the scale parameter. The commonly used threshold stress is rth ˆ 0. For metal, this parameter characterizes the critical stress corresponding to a crack with maximum length and orientation angle of h ˆ p=2. That is the most vulnerable condition for crack propagation. The condition rth ˆ 0 implies that micro-cracks may have in®nite length. It is known that the maximum length of crack nucleus is approximately equal to 0.01d with d being the ferrite grain size or martensite (bainite) packet [9,11]. Hence, for a description of fracture in metals and structural steels, the three-parameter distribution should be used. 4.1. Dependency of Weibull distribution on grain size and its variance Dependency of parameters m, rth and ru on the most probable grain size for four di€erent size variances are given in Fig. 2. Note that the shape factor m is nearly independent of the most

4. Results and discussion The three-parameter Weibull distribution may be written as   m  r rth F …r† ˆ 1 exp ; …12† rth

Fig. 2. E€ect of the most probable grain size on Weibull distribution parameters.

S.A. Kotrechko et al. / Theoretical and Applied Fracture Mechanics 35 (2001) 255±260

Fig. 3. E€ect of the grain size expressed by logarithmic variance Dln d on shape parameter m.

probable grain size dmpv , while rth and ru increase with decreasing dmpv . A linear ®t describes the 1=2 dependence rth and ru on dmpv . The grain size variance a€ects the shape factor m. For polycrystalline metals, the variance of grain size expressed as logarithm of Dlnpd  and the shape factor m are linear functions of Dln d , Fig. 3. As mentioned earlier, the threshold stress rth corresponds to the Grith stress for the largest crack nucleus in metal. Such cracks form in grains of maximum sizes, Eq. (3). The maximum grain size depends on the most probable grain size and grain size variance. This relation explains the regularities of change in rth , Fig. 2. The present results show that for the wide range of grain sizes of 5±120 lm, the normalized scaling stress ru =rth does not depend on the most probable grain size and is a linear function of Dln d as shown in Fig. 4, i.e., ru =rth ˆ a ‡ b Dln d ;

259

Fig. 4. Dependence of normalized scaling stress ru =rth on logarithmic grain size variance Dln d .

fect for quasi-brittle fracture of metals appears to be active for regions ahead of small size notches. For specimens of volume greater than 1000 mm3 , the strength decreases slightly with increasing specimen size [9]. Therefore, a relationship prevails between rth and the minimum cleavage fracture stress, RMC , for unnotched cylindrical specimen in the ductile±brittle transition temperature range [12]. For specimens with diameter of about 5 mm and length about 10 mm, there results rth  0:7RMC :

…14†

This approach enables to estimate the magnitude of rth by testing unnotched specimens. Simple experimental dependence between the grain size and the value of RMC (RMC ˆ Kf d 1=2 , p where Kf ˆ 180 Mpa mm) has been determined in [9] for iron. This is in good agreement with the theoretical value of Dln d ˆ 0:19.

…13†

where for iron a ˆ 2:81  0:07 and b ˆ 11:97 0:22. 4.2. Threshold stress Threshold stress is the fracture stress for an in®nite specimen. As it was shown in [9], scale ef-

5. Conclusions Based on the results of the present study, the following conclusions can be made: · For polycrystalline metals, the valuepof shape parameter m is a linear function of Dln d . Increase in the grain size expressed by logarithmic

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variance Dln d leads to a decrease in the shape parameter. 1=2 · The threshold stress rth is proportional to dmpv and depends on grain size variance. This stress is approximately equal to 0:7RMC . · The normalized scaling stress ru =rth is a linear function of grain size expressed by logarithmic variance Dln d .

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