Quantification of brittle-ductile failure behavior of ferritic reactor pressure vessel steels using the Small-Punch-Test and micromechanical damage models

Materials Science & Engineering A 614 (2014) 136–147

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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea

Quantification of brittle-ductile failure behavior of ferritic reactor pressure vessel steels using the Small-Punch-Test and micromechanical damage models T. Linse a,n, M. Kuna b, H.-W. Viehrig c a

Technische Universität Dresden, Institute for Solid Mechanics, 01062 Dresden, Germany Technische Universität Bergakademie Freiberg, Institute of Mechanics and Fluid Dynamics, Lampadiusstrasse 4, 09596 Freiberg, Germany c Helmholtz-Zentrum Dresden-Rossendorf, P.O. Box 510119, 01328 Dresden, Germany b

art ic l e i nf o

a b s t r a c t

Article history: Received 8 January 2014 Received in revised form 22 May 2014 Accepted 28 May 2014 Available online 19 July 2014

Two German ferritic pressure vessel steels are examined in the brittle to ductile transition regime as a function of temperature and irradiation. The experiments are done by a miniaturized Small-Punch-Test in hot cells within the temperature range of
185 1C up to 70 1C. From the load–displacement curve of the SPT, the yield curves and parameters of both a non-local GURSON-TVERGAARD-NEEDLEMAN ductile damage model and a modified BEREMIN model are identified. The influence of temperature and irradiation on the model parameters is analyzed. All parameters are verified by comparison with results from standard test methods. The parameters, identified from SPT, are used to simulate the failure behavior in standard fracture mechanics specimens. In the upper shelf, the non-local GTN-model is applied to simulate crack resistance curves, from where the fracture toughness data could be successfully predicted. In the lower shelf, the WEIBULL-stress of the specimens was computed to find out the statistics of fracture toughness values. Finally, the modified BEREMIN model and the non-local ductile damage model were combined to evaluate the failure of fracture specimens in the brittle-ductile transition region. This way, an acceptable agreement with Master-curve data for non-irradiated steels could be achieved in the whole temperature range. & 2014 Elsevier B.V. All rights reserved.

Keywords: Non-local ductile damage model Beremin-Model Small-Punch-Test Brittle-ductile transition

1. Introduction To ensure the safety of light water reactors, both exact calculation of the mechanical stresses and realistic modeling of the deformation and failure behavior of the used ferritic steels are required. Due to micro-defects and inhomogeneities in the microstructure of ferritic steels, deformations can cause the formation of microcracks [1], for example by breaking of carbide particles [2]. In the brittle and brittle-ductile transition region trans- or intergranular cleavage fracture initiates at these microcracks, leading to macroscopic brittle failure at sufficiently high stress levels. In the upper shelf microvoids nucleate as a result of plastic deformation. Further loading leads to the growth and coalescence of voids. The associated irreversible changes in the microstructure are referred to as ductile damage. Damage mechanics can be used to assess the integrity of a mechanical structure with micro-defects by evaluating local criteria only. The concept of LOCAL APPROACH [3] is, in contrast to the

n

Corresponding author. E-mail address: [email protected] (T. Linse).

http://dx.doi.org/10.1016/j.msea.2014.05.095 0921-5093/& 2014 Elsevier B.V. All rights reserved.

methods of fracture mechanics, applicable to any structure. However, the application of Local Approach for the determination of fracture characteristics requires precise knowledge of the materials hardening and softening behavior. Because of the neutron irradiation embrittlement [4] during the operation of nuclear power plants and the associated scatter of fracture properties, statistical methods must be used for the characterization of material properties. This requires a large number of experiments. Therefore, testing of miniaturized specimens is predestinated for determining the current state of the material. The Small-Punch-Test (SPT) is one of the most important miniaturized test procedures used in the reactor safety research. It has been developed to monitor the shift of the brittle-ductile transition temperature due to neutron irradiation during the operation of nuclear power plants [5–9]. The SPT offers excellent possibilities for monitoring safety-related components in power plants, since the small specimens can easily be sampled. The Small-Punch-Testing derive is simply installed in standard testing machines. To adjust a desired temperature requires heating or loading of a minimal volume only. The load–displacement curve (LDC) measured in the SPT can be divided into distinctive parts as depicted in Fig. 3, which depend on the elastic, plastic and damage

12 (d) 9 (d) 17 (t)

5 (d) 20 (d) 20 (t) 19 (b)

5 (d)

70 1C
30 1C

22 1C

4 5 5 5 10 (d) 20 (t) 19 (t) 5 (t) 14 (t)


105 1C


80 1C

22 1C

mechanical properties of the material [10–13]. Therefore it contains all information for the identification of these material properties. However, as with all miniaturized test methods, the determination of transferable material properties from experimental results is a challenging task. In this paper, two ferritic reactor pressure vessel (RPV) steels are examined in the entire toughness region, covering brittle, brittle-ductile and ductile region. The parameters of mechanical models describing the hardening and softening behavior of the materials are identified from the SPT with the help of Neural Networks and parallel optimization algorithms. Fracture mechanical parameters are calculated following the concept of LOCAL APPROACH.

137

(d) (d) (d) (d)

T. Linse et al. / Materials Science & Engineering A 614 (2014) 136–147

Designation

JRQ

Low Medium High

7.28 54.85 98.18

RH6 RH7 far (cf) RH7 close (cn)

JFL

Low Medium High

6.52 51.21 86.69

RH6 RH7 far (cf) RH7 close (cn)

5 (d) 3 (t) 2 (t) 21 (b) 2 (b) 2 (b)


100 1C
120 1C
140 1C Test temperature T


150 1C

11 (b) 13 (b) 6 (b)


150 1C
175 1C Test temperature T


185 1C

5 (d) 19 (t) 19 (b)


70 1C

8 (t)


125 1C
135 1C

4 (t) 20 (b) 20 (b) 19 (b)

Non-irradiated RH6 RH7 far RH7 close

Neutron fluence F ne

JRQ

Irradiation level

Non-irradiated RH6 RH7 far RH7 close

RPV steel

JFL

Table 1 Irradiation levels of the tested RPV steels (F ne in 1018 neutrons/cm2 ðE 4 1 MeVÞ).

Table 2 SPT test program for the RPV steels JFL and JRQ (numbers of tests; (b)-brittle, (t)-transition, (d)-ductile).

This section provides a brief overview over the experimental data that provides the basis for the parameter identification method presented in this paper, the experimental results have partly been published in [12,13]. The experiments were carried out at the Forschungszentrum Dresden-Rossendorf (FZD). Remnants of CHARPY specimens of two reactor vessel steels, A533B Cl.1 (IAEA JRQ) and A508 Cl.3 (IAEA JFL), were used for manufacturing and preparing Small Punch Test (SPT, see sketch in Fig. 3) specimens. Both non-irradiated and irradiated (irradiation levels see Table 1) specimens of the steels were tested extensively using the SPT to investigate the influence of temperature and irradiation on the material properties. The complete test program is shown in Table 2. Two major influences were observed from the measured load– displacement curves: First, for both irradiated and non-irradiated materials higher forces are measured as the test temperature is decreased, while the observed displacements at failure diminish. Second, irradiation leads to embrittlement of the material, reflected by the expected shift of the measured curves. In [12,13] the type of failure was determined visually from remains of the tested specimen and with the help of SEM fractography of the fracture surface. It was shown that the test program covered the complete range of brittle, ductile and brittle-ductile transition region. Furthermore, shape and size of the fracture surface of tested SPT specimens depends on the failure mode: there is one single crack with a circular shape in the ductile region while there are several straight cracks in the brittle region. The specific fracture energy, i.e. the work done up to specimens failure related to the size of the resulting fracture surfaces, is shown in Fig. 1. A clear shift of the ductile-brittle transition temperature T SPT is observed for the material JRQ, while the 0 specific fracture energy of the material JFL is less sensitive to neutron irradiation. Here the temperature at which the specific fracture energy reaches a mean value between the ductile and brittle region is understood as ductile-brittle transition temperature T SPT 0 . In the non-irradiated state both materials JFL and JRQ show little differences regarding the observed failure mechanism.


50 1C

2. Experimental data

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GRIFFITH criterion, the BEREMIN model was developed to describe cleavage of ferritic steels [1,15]: The large scatter of fracture toughness values in the brittle and transition region, which results from the statistical distribution of microcracks, is taken into account by using the WEIBULL theory. Consequently, using the BEREMIN-model, failure by cleavage is not strictly modeled in a micromechanical sense by treating individual microcracks but rather by quantifying the cleavage fracture probability of an ensemble of microcracks. For this purpose, a constant power law probability density function is assumed for microcracks created during plastic deformations. Applying the GRIFFITH criterion for each microcrack and the weakest-link hypothesis, the probability of failure for a specific load level L is given by 
  σ W ðLÞ m P f ðLÞ ¼ 1
exp
: ð1Þ

σu

Fig. 1. Specific fracture energy of the SPT.

Fig. 2. Difference of transition temperatures identified from SPT and impact tests (Charpy-V specimens, [14]).

Cleavage fracture occurs at about
150 1C and
140 1C for JFL and JRQ, respectively. Significant differences between the two materials can be observed with respect to their irradiation sensitivity. While for JFL brittle fracture occurs at approximately the same temperatures for all irradiation levels, a strong influence of neutron irradiation on the transition temperature is observed for JRQ. Due to lower stress triaxiality in the SPT specimen, significantly lower transition temperatures are observed compared to T 41J values from impact tests using Charpy-V specimens [14] (Fig. 2). These temperature differences also dependent on the irradiation condition. Here again the material JRQ proves more irradiation-sensitive.

3. Damage mechanics Since the entire toughness region of ferritic steels is to be analyzed, a consistent and independent description of both micromechanical processes—cleavage fracture and ductile damage—is required.

In (1) the scaling parameter σ u (WEIBULL reference stress) and the WEIBULL-modulus m are used. The WEIBULL-stress characterizes the loading situation due to an inhomogeneous stress field in the structure and is calculated as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 m σ W ðLÞ ¼ ðσ I ðLÞÞm dV : ð2Þ V 0 V pl Here σ I is the maximal principal stress, V pl the whole plastic part of the structure. The reference volume V 0 must be large enough to represent the microstructure of the steel as well as small enough to fulfill the requirements of the Griffith criterion, i.e. stress gradients being small. Thus, if the reference volume is chosen constant, the probability of failure is therefore determined by the two model parameters σ u and m. To account for ductile crack growth preceding cleavage in the transition region, the combined use of the BEREMIN-model for the calculation of the probability of cleavage fracture together with a ductile damage model is conceivable in principle [16,17]. However, the problems coming along with the original BEREMIN model are getting worse when ductile damage is involved, i.e. dependence of the WEIBULL parameters with regard to sample shape, temperature and strain rate, difficulties in the iterative determination of the WEIBULL parameters, and large variations in the calculated probabilities of cleavage fracture. Bordet et al. [18] discussed that the above problems are a result of a oversimplified description of local cleavage in the BEREMIN-model. Already in [15], a proposal has been made, how to consider the influence of plastic strain. Further modifications of the BEREMIN-model, mainly concerning the calculation of the WEIBULL-stress can be found in [19–24], among others. In this paper, we apply a modification proposed by Bernauer et al. [2]. This modification takes into account that the nucleation of voids is promoted by the presence of carbide particles, either by the debonding of the surrounding matrix or the breaking of the particles. Both effects lead to the formation of voids, whereby the initiation of cleavage fracture is no longer possible at this point. Therefore, the number of active micro-cracks is variable in the modification proposed by Bernauer. The modified WEIBULL-stress is calculated numerically as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N pl   m V B σ W ðtÞ ¼ ∑ i  max σ B1 þ σ B2 ð3Þ τrt i ¼ 1 V0 where the terms

σ B1 ¼ ðσ iI ðτÞÞm  1
3.1. Cleavage fracture


Subsequent to classical linear-elastic fracture mechanics models based on stress criteria or energetic concepts such as the

cn i ∑ f ð Δt j Þ f n tj r τ N c fn

σ B2 ¼ ∑ ðσ iI ðt j
1 ÞÞm  n f iN ðΔt j Þ tj r τ

! ð41 Þ

 ð42 Þ

T. Linse et al. / Materials Science & Engineering A 614 (2014) 136–147

denote the volume fraction of voids nucleated during the last time increment. As in the original model, the probability of cleavage fracture is determined according to (1), where σ W is now replaced by σ BW . Thus, by means of (3) the nucleation of voids ðf N Þ and the associated reduction of cleavage initiation points can be considered, if a ductile damage model is used supplementary.

withpthe hydrostatic stress p ¼
1=3σ : I, the equivalent stress ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q ¼ 3=2σ D : σ D ðσ D ¼ σ
1=3pIÞ, and q1 , q2 being constant model parameters. The yield stress of the matrix material σ m ðε Þ is a function of the equivalent plastic strain ε of the matrix. In this paper, the VOCE-type ansatz ( σ1 0 r ε r εL σ m ðε Þ ¼ σ 1 þ σ 2 ðε
εL Þ þ ðσ 3
σ 2
σ 1 Þð1
e
kðε
εL Þ Þ εL o ε ð6Þ is used for the mathematical description of the yield curves. The evolution equation of equivalent plastic strain ε reads
pε_ p þ qε_ q : ð1
f Þσ m

ð8Þ

Because of mass conservation, void growth is related to the dilatational part of plastic flow f_ G ¼ ð1
f Þε_ p :

ð101 Þ


f 1 ε
εn pnffiffiffiffiffiffi exp
2 sn sn 2π

2 # :

ð102 Þ

Coalescence of voids is modeled in a simplified form by introdun cing the modified damage variable f as proposed by TVERGAARD and NEEDLEMAN 8 f f rf c > < n n f þ ðf
f ÞK f f ¼ f ðf Þ ¼ c c c of rf f > :f f u f of K¼

f u
f c ; f f
f c

fu ¼

1 : q1

ð11Þ

Material failure occurs when f f is reached. Motivated by the approach of generalized continua, the GTN continuum damage model is modified by replacing the dilatational part of the plastic strain rate ε_ p by its non-local spatial average ε_ p in the evolution equation for the growth of existing voids (9) which now reads nl f_ G ¼ ð1
f Þε_ p

ð12Þ

and thus nl f_ ¼ f_ G þ f_ N

ð13Þ

is used instead of (8). All other equations of the GTN model remain untouched. In the considered case of elastic strains being small compared to plastic strains, the volume change of the porous plastic material becomes an additional degree of freedom. Therefore, the non-local dilatational part of the plastic strain ε p is treated as an additional independent field quantity that is governed by the partial differential equation of the HELMHOLTZ type

ε p
c∇2 ε p ¼ εp ;

ð14Þ

where the local dilatational part of the plastic strain εp acts as a pffiffiffi source term. The constant c can be regarded as an internal length parameter that controls the influence of the surrounding material. To complete the implicit gradient formulation, homogeneous NEUMANN boundary conditions ∇ε p  n ¼ 0

ð15Þ

are chosen at all boundaries to solve the averaging equation. Details of the non-local damage model and its numerical implementation into the commercial FEM software ABAQUS can be found in [33,34].

4. Parameter identification

ð7Þ

whereby ε_ p and ε_ q denote the dilatational and deviatoric parts of plastic strain rate tensor, resp. The change of the void volume fraction f is caused by the growth of existing voids and the nucleation of new voids in the matrix f_ ¼ f_ G þ f_ N :

f_ N ¼ Aðε Þε_ Aðε Þ ¼

When standard continuum damage models are implemented numerically using the finite element method (FEM), they exhibit a high sensitivity of the results to the spatial discretization size [25,26]. Material softening due to the evolution of damage localizes in a small portion of the model that is controlled by the finite element mesh, i.e. the finite element size becomes an additional model parameter. As a consequence, it is generally not possible to use the same damage model parameters in dissimilar FE models with very different element sizes. Furthermore, the dissipated energy converges to zero for infinitesimal small elements. Consequently, the combination of a local ductile damage model with a model of the BEREMIN-type, which requires very fine meshes in the presence of high stress gradients, would always overestimate ductile damage and thus predict the wrong failure mode. A common idea of different methods to reduce the mesh sensitivity of damage models is to account for the influence of the surrounding material at a given material point in the constitutive equations. This results in a formulation that introduces a characteristic length scale into the model. Non-local formulations [27,28] of damage models may be divided into three main types: formulations of the integral type, explicit gradient formulations and implicit gradient models. The continuum damage model established by Gurson and modified by Tvergaard and Needleman (GTN, [29–31]) provides the basis for the constitutive equations applied in the present work. It was developed to describe the growth and coalescence of initially present or later nucleating voids in isotropic ductile materials. The GTN model accounts for the growth, nucleation and coalescence of voids. Its yield condition is written as
2
 q 3 p n n F¼ þ 2q1 f cosh
q2 ð5Þ
ð1 þ ðq1 f Þ2 Þ ¼ 0 2 σm σm

ε_ ¼

The nucleation of voids is driven by the equivalent plastic matrix strain until the total nucleated void volume fraction reaches the volume fraction f n of all nuclei initially in the matrix. For this process, a GAUSSian normal distribution with parameters εn and sn was proposed by CHU and NEEDLEMAN [32] "

3.2. Ductile damage

139

ð9Þ

The complete description of the fracture behavior of the studied ferritic reactor steels in the brittle, ductile and brittleductile transition region first requires the determination of the parameters of the yield curves and the two implemented damage models. This is done by analyzing averaged measured load– displacement curves of the small punch test. The inverse problem of identifying material parameters from experimental data is solved in this work by means of numerical simulations of the experiments. An iterative optimization routine finds those parameters that best describe the experiment by comparison of simulation and experimental result.

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parameters are dependent on the temperature and/or the irradiation level, as listed in Table 4. Seebich [39] shows for the nonirradiated steel 22NiMoCr3-7 that all parameters of his modification of the ductile damage model after Rousselier [40] are independent of temperature. Margolin and Kostylev [41] discuss an influence of irradiation on the nucleation of voids caused by the dispersion of phosphorus. However, nucleation of voids is independent of temperature in large temperature ranges [42,43]. A possible influence of neutron irradiation on the nucleation of voids is accounted for in this work by varying the parameter εn . As suggested in [36] for a material of very similar chemical composition, the influence of temperature and irradiation on the coalescence of voids ðf c Þ is neglected. Also Bernauer [44] assumes these parameters as temperature-independent. Table 4 shows the dependencies adopted in this paper.

The implemented non-local damage model uses ten model pffiffiffi parameters ðεn ; f n ; sn ; f c ; f f ; q1 ; q2 ; q3 ; f 0 ; cÞ. Furthermore, parameters describing elastic properties (Youngs modulus E and Poisson's ratio ν), plastic flow and hardening ðσ 1 ; σ 2 ; σ 3 ; k; εL Þ are required. To calculate the probability of cleavage fracture for a fixed reference volume V 0 another two model parameters ðm; σ u Þ are needed. In total, 20 parameters must be known. 4.1. Assumptions As in [10] the elastic properties ðE; νÞ are taken to be constant and known. The initial void volume fraction f 0 and the maximum nucleating porosity ðf n Þ can be estimated from the chemical composition of the investigated materials [35–37]. Bernauer et al. [38] recommend not to vary the factors q1 , q2 and q3 . Coalescence of voids and final material failure are determined by the parameters f c and f f , but there is a dependency between both. As the void volume fraction at complete failure is strongly bonded to the microstructure of the material, fixing f f and varying f c for model adaptation is recommended. This has already been successfully applied by Abendroth and Kuna [10,11]. The range of plastic deformation where nucleation of voids occurs is characterized by the average εn and the standard deviation sn in ð102 Þ. The influence of sn on the load–displacement curve of the SPT is negligibly small [10,11], so this parameter is fixed to be constant. The value of pffiffiffi c ¼ 0:1 mm was chosen such that simulations of the SPT using different fine FE meshes provide equal load–displacement curves. As shown above, some of the 20 model parameters can be held constant for the parameter identification (Table 3). The other

4.2. Yield curve parameters For the small initial void volume fraction f 0 of both investigated materials, ductile damage is visible in the deformation behavior of the specimen only in the regions of III–V (Fig. 3, right), when equivalent plastic strains ε 4 0:2 occur. It is therefore sufficient to identify only the parameter σ 3 of the yield function (6) as sensitive to ductile damage. The remaining yield curve parameters ðσ 1 ; σ 2 ; k; εL Þ essentially describe the initial region of the yield curves and can be identified in the regions I–III of the load– displacement curve of the SPT (Fig. 3, right) without considering ductile damage. Accordingly, the parameter identification in this paper is performed in two stages: First, the yield curve parameters

Table 3 Constant model parameters. E ðGPaÞ

ν

f0

fn

ff

sn

q1

q2

q3

pffiffiffi c ðmmÞ

V 0 ðmm3 Þ

210

0.3

2.3  10
4

2.3  10
2

0.2

0.1

1.5

1.0

2.25

0.1

1.0  10
3

Table 4 Model parameters to be identified.

Influence of temperature Influence of irradiation

σ1

σ2

σ3

k

εL

εn

fc

m

σu

þ þ

þ þ

þ þ

þ þ

þ þ

– þ

– –

þ þ

þ þ

Fig. 3. Parameter identification in two subsequent phases (left) and the difference between calculated (dashed line) and measured (full line) load–displacement curve of the SPT (right).

T. Linse et al. / Materials Science & Engineering A 614 (2014) 136–147

ðσ 1 ; σ 2 ; σ 3 ; k; εL Þ are determined for all irradiation levels and test temperatures (Table 2) thereby neglecting any ductile damage. In this first phase the material is modeled as elastic–plastic (phase 1, Fig. 3, left) using standard J 2-flow theory with isotropic hardening. Neural Networks are used for the determination of the yield curve parameters from measured load–displacement curves of the SPT in

Table 5 Parameter identification scheme.

Non-irradiated RH6 RH7 far (cf) RH7 close (cn)

Brittle (b)

Transition (t)

Ductile (d)

σ 3 ðTÞ; σ 3 ðTÞ; σ 3 ðTÞ; σ 3 ðTÞ;

σ 3 ðTÞ; σ 3 ðTÞ; σ 3 ðTÞ; σ 3 ðTÞ;

σ 3 ðTÞ; σ 3 ðTÞ; σ 3 ðTÞ; σ 3 ðTÞ;

m; σ u m; σ u m; σ u m; σ u

m; m; m; m;

σu σu σu σu

εn ðF ne Þ; f c εn ðF ne Þ εn ðF ne Þ εn ðF ne Þ

141

this paper. This avoids the need to carry out FEM simulations during the parameter identification, instead previously trained Neural Networks are used. A brief introduction to the structure and behavior of Neural Networks, the main training algorithms together with a detailed description of the use of Neural Networks for parameter identification from load–displacement curves of the SPT can be found in [10–13]. In order to determine the yield curve parameters the difference between the measured load–displacement curves and the curve approximated by the Neural Networks is minimized. Since ductile damage is not modeled and thus the softening region cannot be represented exactly, parameter identification corresponds to minimizing the areas 1 and 2 in Fig. 3 (right). The optimization algorithm SIMULATED ANNEALING [45] was used. With this gradient free optimization procedure, a global minimum of an objective function is found without the need to calculate derivatives with respect to the parameters.

Fig. 4. Algorithm to identify the WEIBULL-parameters.

Fig. 5. FE model of the CT-specimen with collapsed quadrilateral elements at the crack tip. Box: Distribution of damage f T ¼ 22 1C; Δa  0:2 mm).

n

in front of the crack tip (JRQ non-irradiated,

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T. Linse et al. / Materials Science & Engineering A 614 (2014) 136–147

4.3. Damage parameters The identification of damage parameters f c , εn according to Table 5 is carried out subsequently to the determination of the yield curve parameters described in the previous section. Now the ductile

damage is taken into account (phase 2, Fig. 3, left). The parameters

σ 1 , σ 2 , k and εL are left unchanged for each irradiation and test temperatures. However, next to the parameters f c , εn , the stress σ 3 is redetermined to match the softening part of the load–displacement curves. For test temperatures, in which the load–displacement curves have been fully developed and the specimens have failed due to ductile damage, the strong decrease in region V of the curves is taken into account (minimization of areas 1, 2 and 3, Fig. 3, right). This leads to the exact determination of the parameters of the ductile damage

Fig. 6. Prediction of fracture toughness values corresponding to failure probabilities P f of 5%, 50% and 95%.

Fig. 9. Identified initial yield stresses JRQ.

Table 6 Identified ductile damage parameters. fc

εn

JFL Non-irradiated RH6 RH7 (cf) RH7 (cn)

0.0473 0.0473 0.0473 0.0473

0.4 0.34 0.2 0.2

JRQ Non-irradiated RH6 RH7 (cf) RH7 (cn)

0.0375 0.0375 0.0375 0.0375

0.41 0.16 0.12 0.16

Fig. 7. Comparison with results from tensile tests.

Fig. 8. Identified initial yield stresses JFL.

Fig. 10. Identified WEIBULL reference stresses JFL.

T. Linse et al. / Materials Science & Engineering A 614 (2014) 136–147

Fig. 11. Identified WEIBULL reference stresses JRQ.

143

model ðf c ; εn Þ. In the brittle and transition region, the experimental curves are evaluated until the maximum force (minimization of area 1 or areas 1 and 2, Fig. 3, right). Here, minimization is done by using the asynchronous parallel optimization algorithms APPSPACK or HOPSPACK [46–48]. Both algorithms are based on so-called Generating Set Search Methods [49,50], a class of parallelizable derivative-free optimization procedures. Compared to the algorithm SIMULATED ANNEALING these algorithms are better suited for phase 2, since a relatively small number of material parameters have to be identified, whereas computationally expensive simulations need to be performed. Both algorithms start multiple FE simulations on different processors. WEIBULL parameters σ u , m are identified from the load–displacement curves for experiments in which the specimens have failed due to cleavage fracture. The SPTs are simulated by FEM and the stress state at the time of failure is analyzed. The adjustment of the WEIBULL parameters is carried out using the MAXIMUM LIKELIHOOD METHOD [51–53], thereby fitting the calculated to the experimental distribution of the failure probability (Fig. 4). Specimen failure is identified with the sudden drop in the load–displacement curve.

Fig. 12. Comparison of predicted (blue) and measured (red) fracture toughness values JFL: (a) non-irradiated t and (b) RH6. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

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4.4. Prediction of fracture mechanical properties Fracture toughness values are predicted by simulating fracture mechanics tests using the yield curves and damage parameters identified previously from the SPT. This method relies on the assumption that the crack initiation in both the SPT specimen and the fracture mechanics specimen can be predicted using the same damage models. In this paper, the prediction of fracture toughness values is done by simulating an experiment with a CT specimen with a 20% side groove (dimensions W ¼50 mm , crack length a0 ¼ 0.57 W and H ¼1.2 W). A two-dimensional plane strain FE model of the specimen is set up, see Fig. 5. Due to the non-local ductile damage formulation, there is no worry about mesh size dependency of the solution and fracture characteristics. At the crack tip, collapsed quadrilateral elements with independent nodal degrees of freedom and reduced integration are used as suggested in [54,55]. They have an element edge length of 0.0125 mm ð o5  10
4 a0 Þ. In the brittle and transition region, the relationship between crack loading (quantified by stress intensity factor KI) and WEIBULL-stress

σ W is provided by evaluating the stress state in the fracture mechanics specimen for failure probabilities P f of 5%, 50% and 95% (Fig. 6). According to linear-elastic fracture mechanics, these values are equalized to the respective fracture toughness K Jc . In the ductile region, fracture toughness values are determined from simulated crack resistance curves J
Δa applying the evaluation of the test standard [56,57]. In a corresponding way, critical J c;Δa ¼ 0:2 values are found from the simulations at a crack growth of Δa ¼ 0:2 mm.

5. Application and results 5.1. Identified yield curves and damage parameters The comparison of initial yield stresses Rp02 and tensile strengths Rm identified from the SPT with the values determined by conventional tensile tests [58] gives a good agreement (Fig. 7). Here, initial yield stresses are defined as Rp02 ¼ σ m ðε ¼ 0; 002Þ, the

Fig. 13. Comparison of predicted (blue) and measured (red) fracture toughness values JFL: (a) RH7 far and (b) RH7 close. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

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tensile strength Rm is determined from the maximum of the technical stress–strain curve. The influences of test temperature and neutron irradiation visible in the measured load–displacement curves of the SPT are also found in the identified yield curve parameters. Lower temperatures and high irradiation levels lead to significant increases in the yield stresses in the entire strain range. Figs. 8 and 9 show the identified initial yield stresses. Again, the material JRQ shows a much higher sensitivity to irradiation than the steel JFL, which also applies to the identified ductile damage parameters (Table 6). Using the maximum likelihood method, unreasonable high Weibull modules were obtained for some test series. Therefore, it was decided to set the Weibull modulus to a constant value of m ¼ 30. The parameters of the original BEREMIN model were identified for reference purpose. The parameters of the original and the modified BEREMIN model clearly show dependencies of both irradiation level and test temperature (Figs. 10 and 11, m ¼ 30). The influence of the test temperature on the identified WEIBULL reference stresses is best seen at the non-irradiated materials: Note

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that within one irradiation level, the identified WEIBULL reference stresses σ u decrease at higher test temperatures. 5.2. Predicted fracture mechanical properties Subsequent to the identification of yield curves, ductile damage parameters and the parameters of the original and the modified BEREMIN model for the brittle and the transition region, critical fracture toughness values were predicted indirectly by numerical simulation of fracture mechanics experiments. For this purpose, the developed non-local ductile damage model was used in combination with the original BEREMIN model or the modification of Bernauer for the calculation of the cleavage fracture probability. The fracture toughness values obtained from FEM simulations of fracture mechanics experiments using the identified yield curve and damage parameters are plotted in Figs. 12–15. In each figure, predicted fracture toughness values (blue symbols) are compared with experimental results (red color, [58]). In the brittle region the calculated values agree well with Master Curves from

Fig. 14. Comparison of predicted (blue) and measured (red) fracture toughness values JRQ: (a) non-irradiated t and (b) RH6. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

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Fig. 15. Comparison of predicted (blue) and measured (red) fracture toughness values JRQ: (a) RH7 far and (b) RH7 close. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

experimental results [58]. For the transition region, the calculated fracture toughness values and their scatter are clearly too small. The values predicted using either the original or the modified BEREMIN model differ slightly, in correlation to the specific WEIBULL reference stresses (Figs. 10 and 11) fracture toughness values calculated using the modified model are smaller. In the ductile region fracture toughness values are determined via J-integral at a crack growth of Δa ¼ 0:2 mm. The predicted values are in good agreement with experimental reference data [58].

6. Discussion and conclusion

Fig. 16. Relation between identified yield stresses and WEIBULL reference stresses.

The parameters of both BEREMIN models clearly show dependencies on irradiation level and test temperature. In the relevant range of the load–displacement curve of the SPT, large parts of the specimen are subjected to a biaxial stress state which is characterized by approximately equal first and second principal stresses. Accordingly, the maximum principal stresses at the onset of

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void coalescence and consequently the determined WEIBULL reference stresses are mainly determined by the current yield stress of the material. Fig. 16 shows the almost linear relationship between calculated WEIBULL reference stresses and identified yield stresses. The strong influence of the test temperature and irradiation level on the hardening properties of the matrix material is thus transferred to the particular WEIBULL reference stresses: for higher temperatures, the WEIBULL reference stresses decrease. This result is consistent with the experimental observations on notched tensile specimens [59] that principal stresses at cleavage fracture do depend on temperature. Note that there are several temperature-dependent modifications of the BEREMIN model, e.g. [39,60,61]. However, in these modifications the WEIBULL reference stress has to be increased with higher temperatures to get reasonable results, which lacks a micromechanical motivation [37] and contradicts with our results. The predicted fracture toughness values agree well with experimental results in the brittle region and in the ductile region. In the brittle-ductile transition region, the values determined from the SPT cannot be transferred to the fracture mechanics specimen. The calculated fracture toughness values are much smaller than the experimentally determined values. It is therefore questionable whether the assumptions made in the BEREMIN model are sufficient to capture the size effect and the influence of stress triaxiality. In particular in the transition region, the assumption of the failure of the whole structure by unstable crack growth initiated at a micro-crack (weakest link assumption) seems to be not justified for the fracture mechanics specimen. Acknowledgments The financial support of the projects 1501298 and 1501343 by the German Federal Ministry of Economics and Technology (BMWI) is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

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