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Fracture toughness evaluation of 20MnMoNi55 pressure vessel steel in the ductile to brittle transition regime: Experiment & numerical simulations
Journal of Nuclear Materials 465 (2015) 424e432
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Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat
Fracture toughness evaluation of 20MnMoNi55 pressure vessel steel in the ductile to brittle transition regime: Experiment & numerical simulations Avinash Gopalan a, M.K. Samal b, J.K. Chakravartty a, * a b
Mechanical Metallurgy Division (MMD), Bhabha Atomic Research Center (BARC), Mumbai 400085, India Reactor Safety Division, BARC, Mumbai 400085, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 July 2014 Received in revised form 1 June 2015 Accepted 5 June 2015 Available online 9 June 2015
In this work, fracture behaviour of 20MnMoNi55 reactor pressure vessel (RPV) steel in the ductile to brittle transition regime (DBTT) is characterised. Compact tension (CT) and single edged notched bend (SENB) specimens of two different sizes were tested in the DBTT regime. Reference temperature ‘T0’ was evaluated according to the ASTM E1921 standard. The effect of size and geometry on the T0 was studied and T0 was found to be lower for SENB geometry. In order to understand the fracture behaviour numerically, finite element (FE) simulations were performed using Beremin's model for cleavage and Rousselier's model for ductile failure mechanisms. The simulated fracture behaviour was found to be in good agreement with the experiment. © 2015 Elsevier B.V. All rights reserved.
Keywords: DBTT Master curve Damage mechanics FE simulations Pressure vessels
1. Introduction 20MnMoNi55 steel is a type of ferritic steel which is used as reactor pressure vessel (RPV) in the Nuclear power plants. It is important to characterise critical mechanical properties like strength and fracture toughness to ensure safety of the reactor during operation as well as during unexpected accidental conditions. The ductile to brittle transition temperature (DBTT) increases due to the neutron irradiation of the RPV. In order to ensure safety of the reactor, the fracture toughness of the steel used as RPV should be sufficiently high in the DBTT regime [1]. A considerable effort has been made over the last three decades to characterize fracture behaviour of pressure vessel steel with emphasis on the understanding of the fracture behaviour in the DBTT regime. A large volume of literatures is available where the evolutions in the experimental techniques as well as methodologies for fracture control have been discussed [2e4]. The ASME Boiler and Pressure Vessel code section III and XI [5,6] provides a method to characterise the fracture behaviour of the steel by plotting the variation of KIC (plain strain fracture toughness) with temperature using an indexing parameter (RTNDT). A major drawback of using the ASME code is the use of large sample * Corresponding author. E-mail address: [email protected] (J.K. Chakravartty). http://dx.doi.org/10.1016/j.jnucmat.2015.06.009 0022-3115/© 2015 Elsevier B.V. All rights reserved.
size for estimating KIC values as the plastic zone size at the crack tip must be very small compared to the specimen dimensions [1]. Secondly, this approach does not deal with the inherent scatter of the fracture toughness values in the DBTT regime. Further, the indexing parameter (RTNDT) is determined by using results of Charpy V-notch and drop weight tests which are not fracture mechanics based test methods and often leads to an over conservative prediction of fracture toughness in the DBTT regime [7]. The master curve methodology introduced by Wallin [8e11]has overcome the above mentioned drawbacks of the ASME code in evaluating the fracture toughness in the DBTT regime. This method has been later adopted into ASTM E1921 standard. It has been found that grain boundary carbides act as the initiator for cleavage fracture in the ferritic steels [1]. Since these carbides are randomly distributed in the material, the failure of the material at a macroscopical level will be stochastic in nature [12,13]. Hence, the probability of a cleavage failure in ferritic steels is described by a three parameter Weibull distribution [10] in ASTM E1921. The median fracture toughness is plotted as an exponential function of temperature using a single parameter called the reference temperature ‘T0’. The reference temperature T0 is defined as the temperature at which the median fracture toughness value of 1T specimen equals 100 MPa√m. T0 is considered to be unique for a ferritic steel with a given microstructure and is sufficient for
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describing the fracture behaviour of ferritic steels in the DBTT regime. But several studies have shown that the reference temperature ‘T0’ is dependent on the size, geometry and constraint of the specimens used for fracture toughness evaluation [14e16]. In real application like a RPV the cracks are superficial in nature and are of low constraint and subjected to multi axial loading conditions, hence the applicability of T0 is such case have been questioned [12,17,18]. It has been reported that micro-mechanical (damage mechanics model) based approach is best suited for predicting the fracture behaviour of ferritic steel in DBTT regime based on a quantitative framework [17]. The finite element (FE) analysis of fracture behaviour in the DBTT region is quite challenging because in the DBTT regime both cleavage and ductile tearing failure compete with each other [19] and therefore the FE models must be able to predict the possibility of a cleavage failure while also accounting the ductile failure mechanism simultaneously. Traditional methods of fracture analysis using stress intensity factor or Jeintegral cannot be employed here as they do not consider the voids, nucleation of voids and their coalescence. Damage mechanics model are capable of modelling the above mentioned phenomenon and the approach is referred as micro mechanical modelling [20]. The damage models are classified into uncoupled [21] and coupled types [22e26]. The damage in uncoupled models is evaluated from the local stress and strain fields from post processing. Whereas in coupled models, the damage is taken as an internal variable within the yield function. The coupled models are further classified in to local and nonlocal approaches. While local approaches [22e26]evaluate the damage from an element without the consideration of effects of the neighbouring elements on the current element. In non-local model [27,28], damage is evaluated such that it includes the effect of damage from other volumes affecting/influencing the current volume. To model the cleavage aspect of the material Landes [29], Beremin [30], Mudry [31], Wallin et al. [32]) have used the weakest link concept. The most simple and promising model is given by Beremin [30] and the model is derived from Weibull's statistical description. The model is briefly described in a section to follow. It is clear from above that to model the behaviour of materials in the DBTT region it is necessary to couple one of the cleavage models with the damage model to capture the complete phenomenon in the transition region. In the recent past Samal et al. [18]have successfully employed a FE based model to predict the fracture behaviour of two low alloy steels using CT geometry by coupling the non-local Rousselier's model with Beremin's cleavage model. In this study, experiments were performed on 1T-CT, 1/2T-CT, single edge notch bend (SENB) (10 10 55) and SENB (5 5 27) specimens to estimate T0 reference temperature. The effect of geometry and size on T0 was studied for 20MnMoNi55 steel. Subsequently FE simulations were performed on these geometries using the model developed by Samal et al. [18]to predict the fracture toughness behaviour of the 20MnMoNi55 steel in the DBTT regime and the numerical prediction of the fracture in the DBTT regime was compared with the experimental result to demonstrate their validity. 2. Experimental The material tested is a RPV grade 20MnMoNi55 steel and the chemical composition of the steel is given in Table 1. The microstructural studies were carried out on the samples using both optical and Scanning Electron Microscope (SEM). The tensile tests were performed on cylindrical specimens of diameter 4 mm and gauge length of 20 mm. The specimens were made according to ASTM E8/E8M-09 [33] standard, where the gauge length is in the longitudinal direction of the rolled plate. The
425
tensile tests were carried out in a servo-hydraulic machine from a temperature 140 C to room temperature at a strain rate of 103/s. The sub-zero temperatures were achieved by flowing liquid nitrogen from self-pressurized Dewar flask into the environmental chamber. The temperature was controlled within ±2 C. Soaking time of 30 min was given to ensure uniformity of temperature in the specimen. Fracture toughness test were performed on 1T-CT (W ¼ 50 mm, B ¼ 25 mm),1/2T-CT (W ¼ 25 mm,B ¼ 10 mm) specimen, SENB specimens of size 10 mm 10 mm 55 mm and 5mm 5 mm 27 mm according to ASTM Standard E1921 [34] for temperatures ranging from 80 C to 140 C. The specimens were fabricated in LT orientation and pre-cracked to a/W ratio of 0.45e0.55 in a resonance fatigue machine. The fracture toughness tests were carried out in servo-hydraulic machine fitted with an environmental chamber. Clip gauges were attached to CT specimens for measuring the crack opening displacement to estimate load line displacement (LLD). In case of SENB, the clip gauges were not used and as suggested by the ASTM E1921 standard, LLD from the machine was used for KJC calculations. The J-integral at crack instability Jc is determined and converted to KJC through the relation given in ASTM standard E1921. Reference temperature T0 was evaluated from both single temperature and multi temperature methods which are explained in detail in the ASTM standard E1921 [34].
3. Experimental results The SEM micrographs are shown in Fig. 1. It is seen that the microstructure is bainitic and the ferrite phase (a) is seen within the prior austenite grains. The yield strength (YS) and the ultimate tensile strength (UTS) were evaluated for all the tensile tests over the temperature range tested. Variations of the YS and UTS are depicted in Fig. 2. It is noted that there is an increase in strength with decrease in temperature. The stress-temperature data were fitted to a polynomial of second order as given in Eqs. (1) and (2):
Yield Strength :
sys ¼ 0:0061T 2 0:445T þ 485:13
Ultimate Tensile Strength :
(1)
suts ¼ 0:004T 2 0:86T þ 631:5 (2) or C
The temperature, T is in K and strength in MPa. The J-integral values at crack instability JC, KJC, a/W ratio of all the specimens tested are given in the Table 2. The fractograph (Fig. 3) of a 1T-CT specimen tested at 80 C reveals some stable (ductile) crack growth before failure by cleavage at the fatigued crack tip. This was a common feature of all specimens tested in the DBTT regime indicating that both brittle and ductile fracture mechanisms are competing with each other in the DBTT regime. This also implies that the initial plastic deformation (development of micro cracks) is followed by an unstable propagation of micro cracks. Fig. 4 is the master curve plot for 1T-CT specimens from single temperature evaluation at 80 C. While the solid line indicates the fracture toughness for 50% probability of failure through cleavage mechanism, the dotted lines (confidence curve) represent 5% and 95% cleavage probability fracture toughness values. It is seen the scatter of the fracture toughness values is described well by the master curve plot and the confidence curves. Reference temperature T0 for 1T-CT specimens using single temperature analysis for the test temperatures of 80 C, 110 C & 140 C was 110 C, 133 C & 144 C. Similarly the T0 is obtained for 1/2T-CT specimens tested at 110 C& 140 C were 122 C and 135 C from single temperature methodology.
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Table 1 Chemical composition of 20MnMoNi55 steel. Name of element
C
Si
Mn
P
S
Al
Ni
Mo
Cr
Nb
Percentage composition (weight)
0.20
0.24
1.38
0.011
0.005
0.068
0.52
0.30
0.06
0.032
The T0 values evaluated from a multi temperature analysis for 1T-CT &1/2T-CT are 131 C and 127 C respectively. For SENB specimens only multi temperature analysis is performed as the validity criteria for KJC was not met by many of the samples to perform a single temperature analysis and to obtain valid KJC values the test temperature needs to be lowered than the target test temperatures. The T0 calculated from multi temperature analysis for SENB (10 10) and SENB (5 5) is 135 C and 144 C respectively. It has been observed that irrespective of the geometry and size the scatter and mean fracture toughness values increases with temperature. At the lower shelf regions, cleavage is the dominating mode of failure and this regime is characterised by less scatter in the fracture toughness values. As the temperature is increased stable ductile crack growth increases prior to the initiation of unstable cleavage fracture. This region shows increase in mean fracture toughness values as well as an increase in scatter values. The increase in fracture toughness values is due to the ductility improvement with increase in temperature while the scatter in the fracture toughness values is due to the interplay of the two different mechanisms [19]. The difference in T0 evaluated from SENB (10 10) and SENB (5 5) is 9 C indicating the effect of size on the T0 parameter. The T0 temperature obtained by combining all data are 130 C and 139 C (Fig. 5) for CT and SENB geometries respectively. There is a difference of 10 C, which is primarily attributed to the constrain loss at the crack tip in case of SENB specimens. Similar observation was made by Wallin et al. in an earlier work [16]. The SENB specimens give lower values of T0 when compared to CT geometry specimens in all cases of this investigation. The loading in SENB specimens are predominantly bending type and they have
Fig. 2. Variation of tensile strength with temperature.
smaller ligament length and thickness which promotes plastic deformation ahead of the crack in the specimens. The standard deviation for T0 has been calculated according to the formula given in the ASTM E1921-13 standard [34].
b sðT0 Þ ¼ pffiffiffi r
where b is sample size uncertainty factor and r is the number of valid tests. In this work b is taken 18 C as all the median fracture toughness values are greater than 83 MPa√m] [34]. The T0 values obtained at different test temperature by different sized specimen are compared with the T0 obtained by multi-temperature method. The Fig. 6 compares the variation of reference temperature obtained at different test temperatures for single temperature analysis and multi temperature analysis. The deviation obtained from the Eq. (3) is 13.5 C for CT geometry and this will be considered as the standard because the deviation obtained from the CT geometry is the least and the number of valid test is also higher. The reference temperature obtained from the single temperature analysis for the individual geometry has the highest value of standard deviation. It is because the number of tests done at that temperature are small (usually six), but the values were consistent when the multi temperature analysis was used to determine the reference temperature. Hence the multi temperature analysis appears to be a better method to evaluate the reference temperature in this case. It is seen that T0 obtained varies depending on the type of specimen which results primarily due to different constraints associated with their geometry and sizes. Based on the study by Joyce &Tregoning [35] on RPV steels and Mueller et al. [36] on tempered martensitic steel, an attempt has been made to nullify the effect of constraint by increasing the M values. The following is the formula used for limiting the KJC values for their validity as per the ASTM standard E1921-13:
KJCðlimitÞ Fig. 1. SEM micrograph of 20MnMoNi55 RPV.
(3)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ebo sys ¼ 1 n2 M
(4)
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Table 2 J-Integral experimental results. Geometry
Test temperature ( C)
Sample no
a/w
Jc (KJ/m2)
Kjc (MPa√m) (1T equivalent)
Kjc(limit) (MPa√m) for M ¼ 30
1T-CT
50 80 80 80 80 80 80 110 110 110 110 110 110 110 140 140 140 140 140 140 140 80 80 100 110 110 110 110 110 110 110 140 140 140 140 140 140 140 100 120 120 120 120 120 120 120 140 140 140 140 140 140 140 100 100 120 120 120 120 120 120 120 140 140 140 140 140 140 140
2 9 10 11 12 13 14 1 3 4 6 7 8 5 16 17 18 19 20 15 21 1 2 4 5 12 13 14 16 17 18 6 7 8 9 10 11 3 19 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 8 9 10 11 12 13 14 2 3 4 5 6 7 15
0.51 0.54 0.51 0.54 0.52 0.57 0.52 0.53 0.54 0.52 0.54 0.56 0.50 0.52 0.52 0.51 0.52 0.51 0.51 0.52 0.51 0.48 0.55 0.42 0.49 0.46 0.52 0.50 0.47 0.48 0.43 0.43 0.37 0.46 0.41 0.47 0.48 0.45 0.51 0.46 0.48 0.47 0.50 0.51 0.51 0.51 0.48 0.50 0.48 0.51 0.53 0.53 0.49 0.45 0.45 0.46 0.47 0.47 0.50 0.48 0.48 0.44 0.49 0.41 0.44 0.46 0.45 0.47 0.45
559 59 209 192 85 100 89 93 41 104 8 133 156 73 23 20 111 25 81 32 45 285 122 186 201 67 87 51 71 22 80 115 78 72 65 57 24 39 952 89 119 360 37 175 181 30 33 65 57 13 19 27 46 309 202 271 52 9 235 258 46 335 66 105 24 200 43 54 120
340a 117 208 199 132 144 135 138 93 141 41 165 179 122 68 64 151 72 129 81 97 197 123 160 184 98 111 85 101 58 107 127 105 101 97 90 60 75 444a 113 128 273a 74 190a 194a 67 70 96 91 46 53 63 82 253a 204a 234a 76 36 220a 231a 72 263a 84 148a 54 203a 70 82 157a
319 318 327 319 324 308 326 327 320 330 323 315 334 327 348 348 346 350 350 347 348 231 216 262 239 246 232 236 243 240 251 266 280 259 272 257 255 262 154 164 162 164 159 158 156 157 164 161 164 160 157 157 164 124 124 123 122 122 119 121 121 125 120 129 125 124 125 122 124
1/2 T-CT
SENB (10 10)
SENB (5 5)
a
Validity criteria not satisfied.
The values of M were increased from 30 to 50, 80 and then to 150. The effect of M on the number of valid test and the value of T0(multi temperature analysis) obtained for different specimens are
given in the Table 3 and Table 4. As observed from above tables, the T0 values obtained from CT geometry is consistent for all the values of M. The T0 values obtained for SENB specimens are not consistent,
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constraint in this material the suggested value for M is 50. 4. Numerical analysis using continuum damage models The FEM modelling of the fracture toughness in the ductile to brittle temperature region has to account for both ductile fracture and cleavage fracture, as these two phenomena compete with each other in this region. The scatter in fracture toughness and its variation with respect to temperature have been predicted using a coupled model developed by Samal et al. [18]. The model uses nonlocal Rousselier's model for ductile fracture and Beremin's model to predict cleavage failure. The model can predict both the scatter and mean values of fracture toughness variation over the DBTT regime once the stress strain data for the material in the DBTT regime and the Weibull parameters are determined. The following are the brief description of the concepts used in the model. 4.1. Beremin's model for cleavage Fig. 3. Fractograph of the fractured Sample.
The Fracture Toughness had high scatter in the DBTT region, to account for this Beremin's model uses Weibull's Distribution along with weakest link theory [37]. The probability of cleavage fracture is calculated using the following expression:
m sw Pf ¼ 1 exp su
(5)
The relation used in Eq. (5) is popularly known as the two parameter Weibull distribution, where the parameter ‘m’ denotes the Weibull modulus (shape parameter) quantifying the scatter and ‘su’ is the scaling parameter which is the value of Weibull Stress at Pf ¼ 0.632. ‘sw’ is the loading parameter at the crack tip. This parameter is calculated by summing up the maximum principal stress over a small volume at the crack tip using the expression:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n X m Vi u m sw ¼ t si l V0 i¼0 Fig. 4. 1T-CT master curve from single temperature analysis for test at 80 C.
the value of T0 increased with increase in M and this is due to the censored fracture toughness values. The value of M ¼ 50 is taken to be optimal as the T0 values obtained for different specimens were similar. The T0 values obtained for SENB(10 10) and SENB(5 5) are 126 C and 127 C after implementing M ¼ 50 in the multitemperature analysis respectively. Hence to nullify the effect of
(6)
where si l is the maximum principal stress in the material point ‘i’, Viis the volume assigned to the point (due to FEM discretization) and V0 is the characteristic volume. 4.2. Ductile tearing model The load displacement and fracture resistance behaviour of materials are predicted by ductile models like Rousselier's [25] and
Fig. 5. Master curve for (a) CT and (b) SENB geometry.
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429
Fig. 6. Comparison of reference temperature obtained by different test temperature.
Table 3 Number of valid test obtained for different M values. Specimen
1T-CT 1/2T-CT SENB(10 10) SENB(5 5) ALL CT ALL SENB
Total test
21 17 15 16 38 31
d0 ðxÞ ¼
No of valid test M ¼ 30
M ¼ 50
M ¼ 80
M ¼ 150
20 17 11 7 37 18
20 16 10 7 36 17
18 16 9 4 34 13
15 11 5 2 26 7
Gurson-Needleman-Tvergaard [22,23,27]. The constitutive equations of these ductile models include micro-mechanical effects such as void nucleation, growth and coalescence which are taken as the damage in the material. If the damage evolution at a point in a material are dependent on the stress and strain field at that point, then such a model is called the “local damage model”. The disadvantages of using local damage models are the use of constant mesh size at the fracture zone, numerical instabilities and loss of uniqueness of solution [28]. Damage in a characteristic volume depends on the stress, strain and damage of the surrounding regions which in turn depends on the microstructure of the material. Microstructural features such as grain boundaries, second phase particles etc. are known to hinder dislocations there by influencing plastic deformation and void nucleation phenomenon. These features are to be included in the continuum mechanics formulation explicitly. In this model, damage on a characteristic volume U is considered to be Gaussian distribution (J) such that regions closer to the characteristic volume has higher weight values and it decreases exponentially with distance from the characteristic volume. The rate of damage at a point d’, is mathematically defined as the weighted average of rate of change of void volume fraction (f’) in a domain U:
1 jðxÞ
Z
jðy; xÞf 0 ðyÞdUðyÞ
(7)
U
where ‘y’ is the position vector of the small domain dU and J is a Gaussian function. The diffusion equation for damage derived from Eq. (7) [27,28]:
d0 f 0 Clength V2 d0 ¼ 0
(8)
where ‘Clength ‘ is the characteristic length parameter and ‘d’ is the nonlocal variable linked to the void volume fraction ‘f’. In this investigation, simulation is based on nonlocal Rousselier's model for modelling the ductile behaviour of the specimens. The yield function for Rousselier's model is given as:
f¼
shydro seq þ Dsk dexp R εeq ¼ 0 1d ð1 dÞsk
(9)
where, ‘d’ is the non-local material damage (replacing the void volume ‘f’ which is used in the classical Rousselier model), seq is von Mises equivalent stress, shydro is mean hydrostatic stress, R(εeq) is the resistance of the material (stress at the equivalent strain εeq in the stress strain curve), D and sk are Rousselier's constants. Implementing the non-local damage in to the FEM makes the model less dependent on the mesh sizes at the crack tip and also accurate measures of crack growth is obtained from the model [27,28]. This diffusion equation is combined along with mechanical equilibrium equation to get the following equation:
V$s þ fb ¼ 0
(10)
where, fb is the body force. The boundary condition that were associated are given below:
Table 4 Variation of T0 values (multi-temperature analysis) obtained for different M values. Specimen
T0 values ( C) M ¼ 30
M ¼ 50
M ¼ 80
M ¼ 150
1T-CT 1/2T-CT SENB(10 10) SENB(5 5) ALL CT ALL SENB
131 127 135 144 130 139
131 127 126 127 130 127
133 125 120 Condition for multi temperature not met 131 120
136 121 Condition for multi temperature not met Condition for multi temperature not met 133 114
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s$njGf ¼ fm
(11)
njGf ¼ u0
(12)
Vd$njGd ¼ 0
(13)
where fm is the applied mechanical force, the Eq. (11), Eq. (12) and Eq. (13) are the force boundary condition, essential boundary condition and Neumann boundary condition (for damage degree of freedom). njGf , njGd are the normal to the surfaces Gf and Gd. The total strain is decomposed to elastic (_εe ) and plastic (_εp ) parts,
ε_ ¼ ε_ e þ ε_ p
(14)
The yield function is written in terms of mean hydrostatic ‘p’, deviatoric ‘q’ stresses:
fðp; q; Ha ; dÞ ¼ 0
(15)
where Ha is the internal state variable representing hardening. The void volume fraction increases during plastic deformation, this is represented by the following function:
Fðp; q; H a ; dÞ ¼ f_ ¼ f_growth þ f_nucleation ¼ ð1 dÞ_εp : I þ A εeq ε_ eq (16) where ε_ p is the increment in plasticity, I is the Kronecker delta function, ε_ eq is the increment in the equivalent plastic strain. A is the void nucleation constant obeying Gaussian distribution:
A εeq ¼
fN p ffiffiffiffiffiffi exp SN 2p
! 1 εeq εN 2 2 SN
(17)
εN is the mean strain at which void nucleation activity is maximum, SN is the standard deviation for void nucleation strain. fN is the volume fraction of the void nucleating particles corresponding to saturated condition of nucleation(i.e. nucleation reaches near completion). The details of the final governing weak form of the equilibrium equations can be obtained in the works of Samal et al. [18,27,28,38,39]. The above explained equations are used to characterise the ductile fracture behaviour of the material in transition regime.
the minimum threshold value taken to 20Mpa√m and K0 is the fracture toughness value for a the failure probability of 0.632 for the given test temperature. In an earlier work, Samal et al. [27] have demonstrated that Weibull parameters are not dependent on temperature in non-local damage model for predicting the fracture behaviour in the DBTT regime for a material. This is due to the ability of the model to predict the pre cleavage ductile crack growth of few microns. The crack growth affects the state of stress at the crack tip which affects the sw value in the Beremin's model. The Weibull parameters employed in the analysis are: m ¼ 22 and su ¼ 2125 MPa.The parameters are determined by matching the simulated probability with the experimentally obtained probability using the Eq. (18) for 1T-CT specimens at 140 C.The parameters fixed at 140 C for 1T-CT specimen is used for all the geometries and temperatures in this analysis. Fig. 7 plots the fracture toughness vs. crack growth obtained in the simulation for 1T-CT specimen at 80 C. The pre-cleavage crack growth from experiment for 1T-CT specimen at 80 C is of the order of 50 mm as seen in the fractograph (Fig. 3). Comparing Fig. 7 with Fig. 3, it is observed that the pre-cleavage crack growth obtained from simulation matches reasonably well with the stable crack growth obtained in the fractograph. The simulated results are compared with the experimental failure probability for 1T-CT specimen in Fig. 8. It is observed that for 1T-CT specimen at 80 C the simulated probability of cleavage failure saturates (flat region of the curve), this is due to the pre-cleavage crack growth which relaxes the stresses at the crack tip hence decreasing the ‘sw’.The probability of failure P remains constant till the loading condition where the s increases from its previous maximum value. The above effect was noticed only at 80 C and at lower temperatures there was no noticeable change in the shape of the curves as seen in Fig. 8. This implies that the pre-cleavage crack growth occurs only at the upper shelf of the transition region and as the temperature was decreased towards the lower shelf the pre-cleavage crack growth doesn't occur. Similar observations were noticed in other specimens also but due to the effect of size of the specimen the precleavage crack growth is observed till 120 C (for SENB specimens). For all the specimens, it has been observed that at 140 C there is no pre-cleavage crack growth as this represents the lower shelf region where cleavage failure is the dominant failure mechanism. The above numerical observations are consistent with experimental results. The fracture toughness variation and its scatter with
4.3. Numerical simulation results & discussions Finite Element (FE) analysis was carried out using the in house code developed [27,28]. The element type used in the analysis was 3D-isoparametric 20 noded brick elements. Only one quarter of the specimen was modelled due to the symmetry and therefore symmetrical boundary conditions were applied. The material stress strain data obtained from the tensile test performed were fed as input for material property in the simulation. The stress, strain and damage for all the elements were obtained and these were used for finding out the probability of failure through cleavage using Beremin's model. . The experimental cleavage probability was obtained by using the three parameter Weibull distribution:
"
KJCð1TÞ Kmin Pf ¼ 1 exp K0 Kmin
4 # (18)
where KJC(1T) is the equivalent 1T fracture toughness values, Kmin is
Fig. 7. Simulated crack growth at 80 C for 1T-CT specimens.
A. Gopalan et al. / Journal of Nuclear Materials 465 (2015) 424e432
Fig. 8. Probability of cleavage fracture of 1T-CT Specimen.
temperature have been plotted along with experimental fracture values and master curve obtained by experiments for all the specimen types in Figs. 9e12. It is seen that the simulated results match reasonably well with the experiments, however there are certain deviations at 80 C in 1T-CT and at 120 C in the case of SENB(5 5) samples. As seen from Fig. 9 at 80 C the scatter is within the confidence bound for both numerical and master curve predictions, but the simulation predicts lower median fracture toughness when compared to master curve predictions. This is expected because the master curve over predicts the fracture toughness for temperature regions close to T0þ50 C, due to the assumption of exponential variation of fracture toughness with temperature. The variation for SENB specimen (Fig. 12) at 120 C is due to small ligament length (~2.5 mm) which induces large plastic deformation at the crack tip there by censoring fracture toughness values to the limiting values given by the Eq. (4). This results in narrow scatter band prediction. The following are the observations made when comparing the master curve results with the numerically simulated results. T0 values must be known for plotting the master curve, but form experiments it is observed that T0 values are dependent on the crack
Fig. 9. Variation of fracture toughness in DBTT for 1T-CT (experiment & FE simulation).
431
Fig. 10. Variation of fracture toughness in DBTT for 1/2T-CT (experiment & FE simulation).
configurations, size, geometry and loading conditions. For a component, which may have varying and complicated geometry it is difficult to estimate T0 value a priori. Hence a conservative T0 values from the standard geometry is used for the safety analysis which may not be necessarily be the optimum value. It can be concluded that the nonlocal damage model which considers both physical and microscopic phenomenon of material behaviour can predict the fracture behaviour of the ferritic steel without any dependence on the geometry, loading configuration and size of the specimen. The Weibull parameters ‘su’ and ‘m’ are material properties and they are not dependent on the geometry and size of the material, as well as on temperature in the DBTT regime. 5. Conclusion The reference temperature T0 is obtained for 20MnMoNi55 steel for CT and bend geometries using both single temperature and multi temperature methods. The FEM modelling of the above is also done and compared. On the basis of the results obtained the following conclusions can be drawn:
Fig. 11. Variation of fracture toughness in DBTT for SENB (10 10) (experiment & FE simulation).
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corrected master curve was compared with simulated results which were in close agreement. This proves that the FEM model is constraint independent. It is seen that Weibull parameter (‘su'and ‘m’) is constant throughout the DBTT regime. Same parameters were used for all the specimen type; hence these parameters not dependent on size and geometry of the samples tested.
References
Fig. 12. Variation of fracture toughness in DBTT for SENB (5 5) (experiment & FE simulation).
The value of T0 was obtained for different crack geometries with varying dimensions. T0 for CT specimens was found to be greater compared to that of SENB specimens. SENB specimens were of smaller ligament length and thickness and the loading is predominantly bending. This promotes plastic deformation and hence lowers T0 values. T0 values obtained from the single temperature analysis has shown considerable variations compared to that obtained from the multi temperature method. Further, the error due to extrapolation is prevalent in the single temperature analysis. The test temperature has minimum effects on the T0 values for CT geometry when tested within the range of T0±50 C. In case of bend type specimen it is safe to test below the T0 temperature so that the validity of KJC is met, this due to loss of constrain (plastic deformation) in the bend specimen. The nonlocal damage mechanics model has been used to predict the fracture behaviour of 20MnMoNi55 steel in the DBTT regime numerically. The model's ability to predict fracture toughness in DBTT without any dependence on the specimen geometry, size and loading configuration has been demonstrated. It was observed from both experiment and damage model that the failure probability predicted by master curve method is not valid for the temperature beyond T0þ50 C. The constraint correction were performed by increasing M from 30 to 50 for master curve in this material. The constrained
[1] S. Kim, B. Hwang, S. Lee, S. Lee, Metallurgical Mater. Trans. A 34 (2003) 1275e1281. [2] K. Suzuki, J. Nucl. Mater. 108e109 (1982) 443e450. [3] M. Brumovsky, Int. J. Press. Vessels Pip. 79 (2002) 715e721. [4] H.W. Viehrig, J. Boehmert, J. Dzugan, Nucl. Eng. Des. 212 (2002) 115e124. [5] A. Boiler, P. V. Code, section III, in: Division, 1998. [6] A. Boiler, P. V. Code, Subsection, IWL, 2001. [7] S.T. Rosinski, W.L. Server, Int. J. Press. Vessels Pip. 77 (2000) 591e598. [8] K. Wallin, Eng. Fract. Mech. 19 (1984) 1085e1093. [9] K. Wallin, Eng. Fract. Mech. 22 (1985) 149e163. [10] K. Wallin, Eng. Fract. Mech. 69 (2002) 451e481. [11] K. Wallin, Eng. Fract. Mech. 77 (2010) 285e292. [12] J. Hohe, J. Hebel, V. Friedmann, D. Siegele, Eng. Fract. Mech. 74 (2007) 1274e1292. [13] K. Wallin, A. Laukkanen, Eng. Fract. Mech. 75 (2008) 3367e3377. [14] X. Gao, R.H. Dodds Jr., Eng. Fract. Mech. 68 (2001) 263e283. [15] K. Wallin, Eng. Fract. Mech. 68 (2001) 303e328. [16] K. Wallin, T. Planman, M. Valo, R. Rintamaa, Eng. Fract. Mech. 68 (2001) 1265e1296. [17] J.P. Petti, R.H. Dodds Jr., Eng. Fract. Mech. 72 (2005) 91e120. [18] M.K. Samal, J.K. Chakravartty, M. Seidenfuss, E. Roos, Eng. Fail. Anal. 18 (2011) 172e185. [19] M. Kroon, J. Faleskog, H. Oberg, Eng. Fract. Mech. 75 (2008) 2398e2417. [20] R.H. Dodds Jr., M. Tang, T.L. Anderson, Eng. Fract. Mech. 46 (1993) 253e264. [21] J.R. Rice, J. Appl. Mech. 35 (1968) 379e386. [22] A.L. Gurson, J. Eng. Mater. Technol. 99 (1977) 2e15. [23] A. Needleman, V. Tvergaard, J. Mech. Phys. Solids 32 (1984) 461e490. [24] J.R. Rice, D.M. Tracey, J. Mech. Phys. Solids 17 (1969) 201e217. [25] G. Rousselier, Nucl. Eng. Des. 105 (1987) 97e111. [26] V. Tvergaard, A. Needleman, Acta metall. 32 (1984) 157e169. [27] M.K. Samal, M. Seidenfuss, E. Roos, B.K. Dutta, H.S. Kushwaha, Mater. Sci. Eng. A 496 (2008) 25e35. [28] M.K. Samal, M. Seidenfuss, E. Roos, B.K. Dutta, H.S. Kushwaha, Finite Elem. Analysis Des. 44 (2008) 358e371. [29] J.D. Landes, D.H. Shaffer, ASTM STP 700 (1980) 368e382. [30] F.M. Beremin, A. Pineau, F. Mudry, J.-C. Devaux, Y. D'Escatha, P. Ledermann, Metall. Trans. A 14 (1983) 2277e2287. [31] F. Mudry, Nucl. Eng. Des. 105 (1987) 65e76. [32] K. Wallin, T. Saario, K. Torronen, Theoretical scatter in brittle fracture toughness results described by the Weibull distribution, in: Application of Fracture Mechanics to Materials and Structures, Springer, 1984, pp. 511e518. [33] A. Standard, 2009, ASTM International, West Conhohochen, PA, USA. [34] A. Standard, 2013, ASTM International, West Conhohochen, PA, USA. [35] J.A. Joyce, R.L. Tregoning, Eng. Fract. Mech. 72 (2005) 1559e1579. [36] P. Mueller, P. Spatig, R. Bonade, G.R. Odette, D. Gragg, J. Nucl. Mater. 386e388 (2009) 323e327. [37] A. Khalili, K. Kromp, J. Mater. Sci. 26 (1991) 6741e6752. [38] M. Seidenfuss, M.K. Samal, E. Roos, Int. J. Solids Struct. 48 (2011) 3365e3381. [39] M.K. Samal, M. Siedenfuss, E. Roos, Int. J. Mech. Sci. 51 (2009) 619e630.
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Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat
Fracture toughness evaluation of 20MnMoNi55 pressure vessel steel in the ductile to brittle transition regime: Experiment & numerical simulations Avinash Gopalan a, M.K. Samal b, J.K. Chakravartty a, * a b
Mechanical Metallurgy Division (MMD), Bhabha Atomic Research Center (BARC), Mumbai 400085, India Reactor Safety Division, BARC, Mumbai 400085, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 July 2014 Received in revised form 1 June 2015 Accepted 5 June 2015 Available online 9 June 2015
In this work, fracture behaviour of 20MnMoNi55 reactor pressure vessel (RPV) steel in the ductile to brittle transition regime (DBTT) is characterised. Compact tension (CT) and single edged notched bend (SENB) specimens of two different sizes were tested in the DBTT regime. Reference temperature ‘T0’ was evaluated according to the ASTM E1921 standard. The effect of size and geometry on the T0 was studied and T0 was found to be lower for SENB geometry. In order to understand the fracture behaviour numerically, finite element (FE) simulations were performed using Beremin's model for cleavage and Rousselier's model for ductile failure mechanisms. The simulated fracture behaviour was found to be in good agreement with the experiment. © 2015 Elsevier B.V. All rights reserved.
Keywords: DBTT Master curve Damage mechanics FE simulations Pressure vessels
1. Introduction 20MnMoNi55 steel is a type of ferritic steel which is used as reactor pressure vessel (RPV) in the Nuclear power plants. It is important to characterise critical mechanical properties like strength and fracture toughness to ensure safety of the reactor during operation as well as during unexpected accidental conditions. The ductile to brittle transition temperature (DBTT) increases due to the neutron irradiation of the RPV. In order to ensure safety of the reactor, the fracture toughness of the steel used as RPV should be sufficiently high in the DBTT regime [1]. A considerable effort has been made over the last three decades to characterize fracture behaviour of pressure vessel steel with emphasis on the understanding of the fracture behaviour in the DBTT regime. A large volume of literatures is available where the evolutions in the experimental techniques as well as methodologies for fracture control have been discussed [2e4]. The ASME Boiler and Pressure Vessel code section III and XI [5,6] provides a method to characterise the fracture behaviour of the steel by plotting the variation of KIC (plain strain fracture toughness) with temperature using an indexing parameter (RTNDT). A major drawback of using the ASME code is the use of large sample * Corresponding author. E-mail address: [email protected] (J.K. Chakravartty). http://dx.doi.org/10.1016/j.jnucmat.2015.06.009 0022-3115/© 2015 Elsevier B.V. All rights reserved.
size for estimating KIC values as the plastic zone size at the crack tip must be very small compared to the specimen dimensions [1]. Secondly, this approach does not deal with the inherent scatter of the fracture toughness values in the DBTT regime. Further, the indexing parameter (RTNDT) is determined by using results of Charpy V-notch and drop weight tests which are not fracture mechanics based test methods and often leads to an over conservative prediction of fracture toughness in the DBTT regime [7]. The master curve methodology introduced by Wallin [8e11]has overcome the above mentioned drawbacks of the ASME code in evaluating the fracture toughness in the DBTT regime. This method has been later adopted into ASTM E1921 standard. It has been found that grain boundary carbides act as the initiator for cleavage fracture in the ferritic steels [1]. Since these carbides are randomly distributed in the material, the failure of the material at a macroscopical level will be stochastic in nature [12,13]. Hence, the probability of a cleavage failure in ferritic steels is described by a three parameter Weibull distribution [10] in ASTM E1921. The median fracture toughness is plotted as an exponential function of temperature using a single parameter called the reference temperature ‘T0’. The reference temperature T0 is defined as the temperature at which the median fracture toughness value of 1T specimen equals 100 MPa√m. T0 is considered to be unique for a ferritic steel with a given microstructure and is sufficient for
A. Gopalan et al. / Journal of Nuclear Materials 465 (2015) 424e432
describing the fracture behaviour of ferritic steels in the DBTT regime. But several studies have shown that the reference temperature ‘T0’ is dependent on the size, geometry and constraint of the specimens used for fracture toughness evaluation [14e16]. In real application like a RPV the cracks are superficial in nature and are of low constraint and subjected to multi axial loading conditions, hence the applicability of T0 is such case have been questioned [12,17,18]. It has been reported that micro-mechanical (damage mechanics model) based approach is best suited for predicting the fracture behaviour of ferritic steel in DBTT regime based on a quantitative framework [17]. The finite element (FE) analysis of fracture behaviour in the DBTT region is quite challenging because in the DBTT regime both cleavage and ductile tearing failure compete with each other [19] and therefore the FE models must be able to predict the possibility of a cleavage failure while also accounting the ductile failure mechanism simultaneously. Traditional methods of fracture analysis using stress intensity factor or Jeintegral cannot be employed here as they do not consider the voids, nucleation of voids and their coalescence. Damage mechanics model are capable of modelling the above mentioned phenomenon and the approach is referred as micro mechanical modelling [20]. The damage models are classified into uncoupled [21] and coupled types [22e26]. The damage in uncoupled models is evaluated from the local stress and strain fields from post processing. Whereas in coupled models, the damage is taken as an internal variable within the yield function. The coupled models are further classified in to local and nonlocal approaches. While local approaches [22e26]evaluate the damage from an element without the consideration of effects of the neighbouring elements on the current element. In non-local model [27,28], damage is evaluated such that it includes the effect of damage from other volumes affecting/influencing the current volume. To model the cleavage aspect of the material Landes [29], Beremin [30], Mudry [31], Wallin et al. [32]) have used the weakest link concept. The most simple and promising model is given by Beremin [30] and the model is derived from Weibull's statistical description. The model is briefly described in a section to follow. It is clear from above that to model the behaviour of materials in the DBTT region it is necessary to couple one of the cleavage models with the damage model to capture the complete phenomenon in the transition region. In the recent past Samal et al. [18]have successfully employed a FE based model to predict the fracture behaviour of two low alloy steels using CT geometry by coupling the non-local Rousselier's model with Beremin's cleavage model. In this study, experiments were performed on 1T-CT, 1/2T-CT, single edge notch bend (SENB) (10 10 55) and SENB (5 5 27) specimens to estimate T0 reference temperature. The effect of geometry and size on T0 was studied for 20MnMoNi55 steel. Subsequently FE simulations were performed on these geometries using the model developed by Samal et al. [18]to predict the fracture toughness behaviour of the 20MnMoNi55 steel in the DBTT regime and the numerical prediction of the fracture in the DBTT regime was compared with the experimental result to demonstrate their validity. 2. Experimental The material tested is a RPV grade 20MnMoNi55 steel and the chemical composition of the steel is given in Table 1. The microstructural studies were carried out on the samples using both optical and Scanning Electron Microscope (SEM). The tensile tests were performed on cylindrical specimens of diameter 4 mm and gauge length of 20 mm. The specimens were made according to ASTM E8/E8M-09 [33] standard, where the gauge length is in the longitudinal direction of the rolled plate. The
425
tensile tests were carried out in a servo-hydraulic machine from a temperature 140 C to room temperature at a strain rate of 103/s. The sub-zero temperatures were achieved by flowing liquid nitrogen from self-pressurized Dewar flask into the environmental chamber. The temperature was controlled within ±2 C. Soaking time of 30 min was given to ensure uniformity of temperature in the specimen. Fracture toughness test were performed on 1T-CT (W ¼ 50 mm, B ¼ 25 mm),1/2T-CT (W ¼ 25 mm,B ¼ 10 mm) specimen, SENB specimens of size 10 mm 10 mm 55 mm and 5mm 5 mm 27 mm according to ASTM Standard E1921 [34] for temperatures ranging from 80 C to 140 C. The specimens were fabricated in LT orientation and pre-cracked to a/W ratio of 0.45e0.55 in a resonance fatigue machine. The fracture toughness tests were carried out in servo-hydraulic machine fitted with an environmental chamber. Clip gauges were attached to CT specimens for measuring the crack opening displacement to estimate load line displacement (LLD). In case of SENB, the clip gauges were not used and as suggested by the ASTM E1921 standard, LLD from the machine was used for KJC calculations. The J-integral at crack instability Jc is determined and converted to KJC through the relation given in ASTM standard E1921. Reference temperature T0 was evaluated from both single temperature and multi temperature methods which are explained in detail in the ASTM standard E1921 [34].
3. Experimental results The SEM micrographs are shown in Fig. 1. It is seen that the microstructure is bainitic and the ferrite phase (a) is seen within the prior austenite grains. The yield strength (YS) and the ultimate tensile strength (UTS) were evaluated for all the tensile tests over the temperature range tested. Variations of the YS and UTS are depicted in Fig. 2. It is noted that there is an increase in strength with decrease in temperature. The stress-temperature data were fitted to a polynomial of second order as given in Eqs. (1) and (2):
Yield Strength :
sys ¼ 0:0061T 2 0:445T þ 485:13
Ultimate Tensile Strength :
(1)
suts ¼ 0:004T 2 0:86T þ 631:5 (2) or C
The temperature, T is in K and strength in MPa. The J-integral values at crack instability JC, KJC, a/W ratio of all the specimens tested are given in the Table 2. The fractograph (Fig. 3) of a 1T-CT specimen tested at 80 C reveals some stable (ductile) crack growth before failure by cleavage at the fatigued crack tip. This was a common feature of all specimens tested in the DBTT regime indicating that both brittle and ductile fracture mechanisms are competing with each other in the DBTT regime. This also implies that the initial plastic deformation (development of micro cracks) is followed by an unstable propagation of micro cracks. Fig. 4 is the master curve plot for 1T-CT specimens from single temperature evaluation at 80 C. While the solid line indicates the fracture toughness for 50% probability of failure through cleavage mechanism, the dotted lines (confidence curve) represent 5% and 95% cleavage probability fracture toughness values. It is seen the scatter of the fracture toughness values is described well by the master curve plot and the confidence curves. Reference temperature T0 for 1T-CT specimens using single temperature analysis for the test temperatures of 80 C, 110 C & 140 C was 110 C, 133 C & 144 C. Similarly the T0 is obtained for 1/2T-CT specimens tested at 110 C& 140 C were 122 C and 135 C from single temperature methodology.
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Table 1 Chemical composition of 20MnMoNi55 steel. Name of element
C
Si
Mn
P
S
Al
Ni
Mo
Cr
Nb
Percentage composition (weight)
0.20
0.24
1.38
0.011
0.005
0.068
0.52
0.30
0.06
0.032
The T0 values evaluated from a multi temperature analysis for 1T-CT &1/2T-CT are 131 C and 127 C respectively. For SENB specimens only multi temperature analysis is performed as the validity criteria for KJC was not met by many of the samples to perform a single temperature analysis and to obtain valid KJC values the test temperature needs to be lowered than the target test temperatures. The T0 calculated from multi temperature analysis for SENB (10 10) and SENB (5 5) is 135 C and 144 C respectively. It has been observed that irrespective of the geometry and size the scatter and mean fracture toughness values increases with temperature. At the lower shelf regions, cleavage is the dominating mode of failure and this regime is characterised by less scatter in the fracture toughness values. As the temperature is increased stable ductile crack growth increases prior to the initiation of unstable cleavage fracture. This region shows increase in mean fracture toughness values as well as an increase in scatter values. The increase in fracture toughness values is due to the ductility improvement with increase in temperature while the scatter in the fracture toughness values is due to the interplay of the two different mechanisms [19]. The difference in T0 evaluated from SENB (10 10) and SENB (5 5) is 9 C indicating the effect of size on the T0 parameter. The T0 temperature obtained by combining all data are 130 C and 139 C (Fig. 5) for CT and SENB geometries respectively. There is a difference of 10 C, which is primarily attributed to the constrain loss at the crack tip in case of SENB specimens. Similar observation was made by Wallin et al. in an earlier work [16]. The SENB specimens give lower values of T0 when compared to CT geometry specimens in all cases of this investigation. The loading in SENB specimens are predominantly bending type and they have
Fig. 2. Variation of tensile strength with temperature.
smaller ligament length and thickness which promotes plastic deformation ahead of the crack in the specimens. The standard deviation for T0 has been calculated according to the formula given in the ASTM E1921-13 standard [34].
b sðT0 Þ ¼ pffiffiffi r
where b is sample size uncertainty factor and r is the number of valid tests. In this work b is taken 18 C as all the median fracture toughness values are greater than 83 MPa√m] [34]. The T0 values obtained at different test temperature by different sized specimen are compared with the T0 obtained by multi-temperature method. The Fig. 6 compares the variation of reference temperature obtained at different test temperatures for single temperature analysis and multi temperature analysis. The deviation obtained from the Eq. (3) is 13.5 C for CT geometry and this will be considered as the standard because the deviation obtained from the CT geometry is the least and the number of valid test is also higher. The reference temperature obtained from the single temperature analysis for the individual geometry has the highest value of standard deviation. It is because the number of tests done at that temperature are small (usually six), but the values were consistent when the multi temperature analysis was used to determine the reference temperature. Hence the multi temperature analysis appears to be a better method to evaluate the reference temperature in this case. It is seen that T0 obtained varies depending on the type of specimen which results primarily due to different constraints associated with their geometry and sizes. Based on the study by Joyce &Tregoning [35] on RPV steels and Mueller et al. [36] on tempered martensitic steel, an attempt has been made to nullify the effect of constraint by increasing the M values. The following is the formula used for limiting the KJC values for their validity as per the ASTM standard E1921-13:
KJCðlimitÞ Fig. 1. SEM micrograph of 20MnMoNi55 RPV.
(3)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ebo sys ¼ 1 n2 M
(4)
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427
Table 2 J-Integral experimental results. Geometry
Test temperature ( C)
Sample no
a/w
Jc (KJ/m2)
Kjc (MPa√m) (1T equivalent)
Kjc(limit) (MPa√m) for M ¼ 30
1T-CT
50 80 80 80 80 80 80 110 110 110 110 110 110 110 140 140 140 140 140 140 140 80 80 100 110 110 110 110 110 110 110 140 140 140 140 140 140 140 100 120 120 120 120 120 120 120 140 140 140 140 140 140 140 100 100 120 120 120 120 120 120 120 140 140 140 140 140 140 140
2 9 10 11 12 13 14 1 3 4 6 7 8 5 16 17 18 19 20 15 21 1 2 4 5 12 13 14 16 17 18 6 7 8 9 10 11 3 19 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 8 9 10 11 12 13 14 2 3 4 5 6 7 15
0.51 0.54 0.51 0.54 0.52 0.57 0.52 0.53 0.54 0.52 0.54 0.56 0.50 0.52 0.52 0.51 0.52 0.51 0.51 0.52 0.51 0.48 0.55 0.42 0.49 0.46 0.52 0.50 0.47 0.48 0.43 0.43 0.37 0.46 0.41 0.47 0.48 0.45 0.51 0.46 0.48 0.47 0.50 0.51 0.51 0.51 0.48 0.50 0.48 0.51 0.53 0.53 0.49 0.45 0.45 0.46 0.47 0.47 0.50 0.48 0.48 0.44 0.49 0.41 0.44 0.46 0.45 0.47 0.45
559 59 209 192 85 100 89 93 41 104 8 133 156 73 23 20 111 25 81 32 45 285 122 186 201 67 87 51 71 22 80 115 78 72 65 57 24 39 952 89 119 360 37 175 181 30 33 65 57 13 19 27 46 309 202 271 52 9 235 258 46 335 66 105 24 200 43 54 120
340a 117 208 199 132 144 135 138 93 141 41 165 179 122 68 64 151 72 129 81 97 197 123 160 184 98 111 85 101 58 107 127 105 101 97 90 60 75 444a 113 128 273a 74 190a 194a 67 70 96 91 46 53 63 82 253a 204a 234a 76 36 220a 231a 72 263a 84 148a 54 203a 70 82 157a
319 318 327 319 324 308 326 327 320 330 323 315 334 327 348 348 346 350 350 347 348 231 216 262 239 246 232 236 243 240 251 266 280 259 272 257 255 262 154 164 162 164 159 158 156 157 164 161 164 160 157 157 164 124 124 123 122 122 119 121 121 125 120 129 125 124 125 122 124
1/2 T-CT
SENB (10 10)
SENB (5 5)
a
Validity criteria not satisfied.
The values of M were increased from 30 to 50, 80 and then to 150. The effect of M on the number of valid test and the value of T0(multi temperature analysis) obtained for different specimens are
given in the Table 3 and Table 4. As observed from above tables, the T0 values obtained from CT geometry is consistent for all the values of M. The T0 values obtained for SENB specimens are not consistent,
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constraint in this material the suggested value for M is 50. 4. Numerical analysis using continuum damage models The FEM modelling of the fracture toughness in the ductile to brittle temperature region has to account for both ductile fracture and cleavage fracture, as these two phenomena compete with each other in this region. The scatter in fracture toughness and its variation with respect to temperature have been predicted using a coupled model developed by Samal et al. [18]. The model uses nonlocal Rousselier's model for ductile fracture and Beremin's model to predict cleavage failure. The model can predict both the scatter and mean values of fracture toughness variation over the DBTT regime once the stress strain data for the material in the DBTT regime and the Weibull parameters are determined. The following are the brief description of the concepts used in the model. 4.1. Beremin's model for cleavage Fig. 3. Fractograph of the fractured Sample.
The Fracture Toughness had high scatter in the DBTT region, to account for this Beremin's model uses Weibull's Distribution along with weakest link theory [37]. The probability of cleavage fracture is calculated using the following expression:
m sw Pf ¼ 1 exp su
(5)
The relation used in Eq. (5) is popularly known as the two parameter Weibull distribution, where the parameter ‘m’ denotes the Weibull modulus (shape parameter) quantifying the scatter and ‘su’ is the scaling parameter which is the value of Weibull Stress at Pf ¼ 0.632. ‘sw’ is the loading parameter at the crack tip. This parameter is calculated by summing up the maximum principal stress over a small volume at the crack tip using the expression:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n X m Vi u m sw ¼ t si l V0 i¼0 Fig. 4. 1T-CT master curve from single temperature analysis for test at 80 C.
the value of T0 increased with increase in M and this is due to the censored fracture toughness values. The value of M ¼ 50 is taken to be optimal as the T0 values obtained for different specimens were similar. The T0 values obtained for SENB(10 10) and SENB(5 5) are 126 C and 127 C after implementing M ¼ 50 in the multitemperature analysis respectively. Hence to nullify the effect of
(6)
where si l is the maximum principal stress in the material point ‘i’, Viis the volume assigned to the point (due to FEM discretization) and V0 is the characteristic volume. 4.2. Ductile tearing model The load displacement and fracture resistance behaviour of materials are predicted by ductile models like Rousselier's [25] and
Fig. 5. Master curve for (a) CT and (b) SENB geometry.
A. Gopalan et al. / Journal of Nuclear Materials 465 (2015) 424e432
429
Fig. 6. Comparison of reference temperature obtained by different test temperature.
Table 3 Number of valid test obtained for different M values. Specimen
1T-CT 1/2T-CT SENB(10 10) SENB(5 5) ALL CT ALL SENB
Total test
21 17 15 16 38 31
d0 ðxÞ ¼
No of valid test M ¼ 30
M ¼ 50
M ¼ 80
M ¼ 150
20 17 11 7 37 18
20 16 10 7 36 17
18 16 9 4 34 13
15 11 5 2 26 7
Gurson-Needleman-Tvergaard [22,23,27]. The constitutive equations of these ductile models include micro-mechanical effects such as void nucleation, growth and coalescence which are taken as the damage in the material. If the damage evolution at a point in a material are dependent on the stress and strain field at that point, then such a model is called the “local damage model”. The disadvantages of using local damage models are the use of constant mesh size at the fracture zone, numerical instabilities and loss of uniqueness of solution [28]. Damage in a characteristic volume depends on the stress, strain and damage of the surrounding regions which in turn depends on the microstructure of the material. Microstructural features such as grain boundaries, second phase particles etc. are known to hinder dislocations there by influencing plastic deformation and void nucleation phenomenon. These features are to be included in the continuum mechanics formulation explicitly. In this model, damage on a characteristic volume U is considered to be Gaussian distribution (J) such that regions closer to the characteristic volume has higher weight values and it decreases exponentially with distance from the characteristic volume. The rate of damage at a point d’, is mathematically defined as the weighted average of rate of change of void volume fraction (f’) in a domain U:
1 jðxÞ
Z
jðy; xÞf 0 ðyÞdUðyÞ
(7)
U
where ‘y’ is the position vector of the small domain dU and J is a Gaussian function. The diffusion equation for damage derived from Eq. (7) [27,28]:
d0 f 0 Clength V2 d0 ¼ 0
(8)
where ‘Clength ‘ is the characteristic length parameter and ‘d’ is the nonlocal variable linked to the void volume fraction ‘f’. In this investigation, simulation is based on nonlocal Rousselier's model for modelling the ductile behaviour of the specimens. The yield function for Rousselier's model is given as:
f¼
shydro seq þ Dsk dexp R εeq ¼ 0 1d ð1 dÞsk
(9)
where, ‘d’ is the non-local material damage (replacing the void volume ‘f’ which is used in the classical Rousselier model), seq is von Mises equivalent stress, shydro is mean hydrostatic stress, R(εeq) is the resistance of the material (stress at the equivalent strain εeq in the stress strain curve), D and sk are Rousselier's constants. Implementing the non-local damage in to the FEM makes the model less dependent on the mesh sizes at the crack tip and also accurate measures of crack growth is obtained from the model [27,28]. This diffusion equation is combined along with mechanical equilibrium equation to get the following equation:
V$s þ fb ¼ 0
(10)
where, fb is the body force. The boundary condition that were associated are given below:
Table 4 Variation of T0 values (multi-temperature analysis) obtained for different M values. Specimen
T0 values ( C) M ¼ 30
M ¼ 50
M ¼ 80
M ¼ 150
1T-CT 1/2T-CT SENB(10 10) SENB(5 5) ALL CT ALL SENB
131 127 135 144 130 139
131 127 126 127 130 127
133 125 120 Condition for multi temperature not met 131 120
136 121 Condition for multi temperature not met Condition for multi temperature not met 133 114
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s$njGf ¼ fm
(11)
njGf ¼ u0
(12)
Vd$njGd ¼ 0
(13)
where fm is the applied mechanical force, the Eq. (11), Eq. (12) and Eq. (13) are the force boundary condition, essential boundary condition and Neumann boundary condition (for damage degree of freedom). njGf , njGd are the normal to the surfaces Gf and Gd. The total strain is decomposed to elastic (_εe ) and plastic (_εp ) parts,
ε_ ¼ ε_ e þ ε_ p
(14)
The yield function is written in terms of mean hydrostatic ‘p’, deviatoric ‘q’ stresses:
fðp; q; Ha ; dÞ ¼ 0
(15)
where Ha is the internal state variable representing hardening. The void volume fraction increases during plastic deformation, this is represented by the following function:
Fðp; q; H a ; dÞ ¼ f_ ¼ f_growth þ f_nucleation ¼ ð1 dÞ_εp : I þ A εeq ε_ eq (16) where ε_ p is the increment in plasticity, I is the Kronecker delta function, ε_ eq is the increment in the equivalent plastic strain. A is the void nucleation constant obeying Gaussian distribution:
A εeq ¼
fN p ffiffiffiffiffiffi exp SN 2p
! 1 εeq εN 2 2 SN
(17)
εN is the mean strain at which void nucleation activity is maximum, SN is the standard deviation for void nucleation strain. fN is the volume fraction of the void nucleating particles corresponding to saturated condition of nucleation(i.e. nucleation reaches near completion). The details of the final governing weak form of the equilibrium equations can be obtained in the works of Samal et al. [18,27,28,38,39]. The above explained equations are used to characterise the ductile fracture behaviour of the material in transition regime.
the minimum threshold value taken to 20Mpa√m and K0 is the fracture toughness value for a the failure probability of 0.632 for the given test temperature. In an earlier work, Samal et al. [27] have demonstrated that Weibull parameters are not dependent on temperature in non-local damage model for predicting the fracture behaviour in the DBTT regime for a material. This is due to the ability of the model to predict the pre cleavage ductile crack growth of few microns. The crack growth affects the state of stress at the crack tip which affects the sw value in the Beremin's model. The Weibull parameters employed in the analysis are: m ¼ 22 and su ¼ 2125 MPa.The parameters are determined by matching the simulated probability with the experimentally obtained probability using the Eq. (18) for 1T-CT specimens at 140 C.The parameters fixed at 140 C for 1T-CT specimen is used for all the geometries and temperatures in this analysis. Fig. 7 plots the fracture toughness vs. crack growth obtained in the simulation for 1T-CT specimen at 80 C. The pre-cleavage crack growth from experiment for 1T-CT specimen at 80 C is of the order of 50 mm as seen in the fractograph (Fig. 3). Comparing Fig. 7 with Fig. 3, it is observed that the pre-cleavage crack growth obtained from simulation matches reasonably well with the stable crack growth obtained in the fractograph. The simulated results are compared with the experimental failure probability for 1T-CT specimen in Fig. 8. It is observed that for 1T-CT specimen at 80 C the simulated probability of cleavage failure saturates (flat region of the curve), this is due to the pre-cleavage crack growth which relaxes the stresses at the crack tip hence decreasing the ‘sw’.The probability of failure P remains constant till the loading condition where the s increases from its previous maximum value. The above effect was noticed only at 80 C and at lower temperatures there was no noticeable change in the shape of the curves as seen in Fig. 8. This implies that the pre-cleavage crack growth occurs only at the upper shelf of the transition region and as the temperature was decreased towards the lower shelf the pre-cleavage crack growth doesn't occur. Similar observations were noticed in other specimens also but due to the effect of size of the specimen the precleavage crack growth is observed till 120 C (for SENB specimens). For all the specimens, it has been observed that at 140 C there is no pre-cleavage crack growth as this represents the lower shelf region where cleavage failure is the dominant failure mechanism. The above numerical observations are consistent with experimental results. The fracture toughness variation and its scatter with
4.3. Numerical simulation results & discussions Finite Element (FE) analysis was carried out using the in house code developed [27,28]. The element type used in the analysis was 3D-isoparametric 20 noded brick elements. Only one quarter of the specimen was modelled due to the symmetry and therefore symmetrical boundary conditions were applied. The material stress strain data obtained from the tensile test performed were fed as input for material property in the simulation. The stress, strain and damage for all the elements were obtained and these were used for finding out the probability of failure through cleavage using Beremin's model. . The experimental cleavage probability was obtained by using the three parameter Weibull distribution:
"
KJCð1TÞ Kmin Pf ¼ 1 exp K0 Kmin
4 # (18)
where KJC(1T) is the equivalent 1T fracture toughness values, Kmin is
Fig. 7. Simulated crack growth at 80 C for 1T-CT specimens.
A. Gopalan et al. / Journal of Nuclear Materials 465 (2015) 424e432
Fig. 8. Probability of cleavage fracture of 1T-CT Specimen.
temperature have been plotted along with experimental fracture values and master curve obtained by experiments for all the specimen types in Figs. 9e12. It is seen that the simulated results match reasonably well with the experiments, however there are certain deviations at 80 C in 1T-CT and at 120 C in the case of SENB(5 5) samples. As seen from Fig. 9 at 80 C the scatter is within the confidence bound for both numerical and master curve predictions, but the simulation predicts lower median fracture toughness when compared to master curve predictions. This is expected because the master curve over predicts the fracture toughness for temperature regions close to T0þ50 C, due to the assumption of exponential variation of fracture toughness with temperature. The variation for SENB specimen (Fig. 12) at 120 C is due to small ligament length (~2.5 mm) which induces large plastic deformation at the crack tip there by censoring fracture toughness values to the limiting values given by the Eq. (4). This results in narrow scatter band prediction. The following are the observations made when comparing the master curve results with the numerically simulated results. T0 values must be known for plotting the master curve, but form experiments it is observed that T0 values are dependent on the crack
Fig. 9. Variation of fracture toughness in DBTT for 1T-CT (experiment & FE simulation).
431
Fig. 10. Variation of fracture toughness in DBTT for 1/2T-CT (experiment & FE simulation).
configurations, size, geometry and loading conditions. For a component, which may have varying and complicated geometry it is difficult to estimate T0 value a priori. Hence a conservative T0 values from the standard geometry is used for the safety analysis which may not be necessarily be the optimum value. It can be concluded that the nonlocal damage model which considers both physical and microscopic phenomenon of material behaviour can predict the fracture behaviour of the ferritic steel without any dependence on the geometry, loading configuration and size of the specimen. The Weibull parameters ‘su’ and ‘m’ are material properties and they are not dependent on the geometry and size of the material, as well as on temperature in the DBTT regime. 5. Conclusion The reference temperature T0 is obtained for 20MnMoNi55 steel for CT and bend geometries using both single temperature and multi temperature methods. The FEM modelling of the above is also done and compared. On the basis of the results obtained the following conclusions can be drawn:
Fig. 11. Variation of fracture toughness in DBTT for SENB (10 10) (experiment & FE simulation).
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A. Gopalan et al. / Journal of Nuclear Materials 465 (2015) 424e432
corrected master curve was compared with simulated results which were in close agreement. This proves that the FEM model is constraint independent. It is seen that Weibull parameter (‘su'and ‘m’) is constant throughout the DBTT regime. Same parameters were used for all the specimen type; hence these parameters not dependent on size and geometry of the samples tested.
References
Fig. 12. Variation of fracture toughness in DBTT for SENB (5 5) (experiment & FE simulation).
The value of T0 was obtained for different crack geometries with varying dimensions. T0 for CT specimens was found to be greater compared to that of SENB specimens. SENB specimens were of smaller ligament length and thickness and the loading is predominantly bending. This promotes plastic deformation and hence lowers T0 values. T0 values obtained from the single temperature analysis has shown considerable variations compared to that obtained from the multi temperature method. Further, the error due to extrapolation is prevalent in the single temperature analysis. The test temperature has minimum effects on the T0 values for CT geometry when tested within the range of T0±50 C. In case of bend type specimen it is safe to test below the T0 temperature so that the validity of KJC is met, this due to loss of constrain (plastic deformation) in the bend specimen. The nonlocal damage mechanics model has been used to predict the fracture behaviour of 20MnMoNi55 steel in the DBTT regime numerically. The model's ability to predict fracture toughness in DBTT without any dependence on the specimen geometry, size and loading configuration has been demonstrated. It was observed from both experiment and damage model that the failure probability predicted by master curve method is not valid for the temperature beyond T0þ50 C. The constraint correction were performed by increasing M from 30 to 50 for master curve in this material. The constrained
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