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Fracture mechanical analysis of tungsten armor failure of a water-cooled divertor target
Fusion Engineering and Design 89 (2014) 2716–2725
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Fracture mechanical analysis of tungsten armor failure of a water-cooled divertor target Muyuan Li a , Ewald Werner a , Jeong-Ha You b,∗ a b
Lehrstuhl für Werkstoffkunde und Werkstoffmechanik, Technische Universität München, Boltzmannstr. 15, 85748 Garching, Germany Max-Planck-Institut für Plasmaphysik, Boltzmannstr. 2, 85748 Garching, Germany
h i g h l i g h t s • • • •
The FEM-based VCE method and XFEM were employed for computing KI (or J-integral) and predicting progressive cracking, respectively. The most probable pattern of crack formation is radial cracking in the tungsten armor block. The most probable site of cracking is the upper interfacial region of the tungsten armor block adjacent to the top position of the copper interlayer. The initiation of a major crack becomes likely, only when the strength of tungsten armor block is significantly reduced from its original strength.
a r t i c l e
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Article history: Received 3 April 2014 Received in revised form 17 July 2014 Accepted 18 July 2014 Available online 8 August 2014 Keywords: Divertor Tungsten armor High heat flux loads Fracture mechanics Crack Stress intensity factor
a b s t r a c t The inherent brittleness of tungsten at low temperature and the embrittlement by neutron irradiation are its most critical weaknesses for fusion applications. In the current design of the ITER and DEMO divertor, the high heat flux loads during the operation impose a strong constraint on the structure–mechanical performance of the divertor. Thus, the combination of brittleness and the thermally induced stress fields due to the high heat flux loads raises a serious reliability issue in terms of the structural integrity of tungsten armor. In this study, quantitative estimates of the vulnerability of the tungsten monoblock armor cracking under stationary high heat flux loads are presented. A comparative fracture mechanical investigation has been carried out by means of two different types of computational approaches, namely, the extended finite element method (XFEM) and the finite element method (FEM)-based virtual crack tip extension (VCE) method. The fracture analysis indicates that the most probable pattern of crack formation is radial cracking in the tungsten armor starting from the interface to tube and the most probable site of cracking is the upper interfacial region of the tungsten armor adjacent to the top position of the copper interlayer. The strength threshold for crack initiation and the high heat flux load threshold for crack propagation are evaluated based on XFEM simulations and computations of stress intensity factors and J-integrals. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The divertor is an important in-vessel plasma-facing component (PFC) of a fusion reactor. The essential function of the divertor is to exhaust the edge plasma in the scrape-off layer in order for helium ash and other impurities to be continuously removed from the burning plasma core. The plasma exhaust is achieved by intense particle bombardment onto the divertor target plate generating a high heat flux load into the target surface [1]. The maximum stationary heat flux load for the ITER divertor target reaches up to
∗ Corresponding author. Tel.: +49 089 3299 1373; fax:+49 089 3299 1212. E-mail address: [email protected] (J.-H. You). http://dx.doi.org/10.1016/j.fusengdes.2014.07.011 0920-3796/© 2014 Elsevier B.V. All rights reserved.
10 MW/m2 , and slow thermal transient loads up to 20 MW/m2 are expected [2]. In the case of the DEMO divertor, the range of possible heat flux load may be even larger. Under such high heat flux loads, the divertor target component (a bi-material joint structure) is subjected to high thermal stresses. Thus, the high heat flux loads impose a strong constraint on the structure–mechanical performance of the divertor. In the current design concept for the DEMO as well as the ITER divertor, the material for the plasma-facing target armor is tungsten. This choice is owing to the advantageous properties of tungsten such as an extremely low sputtering yield, the highest melting point of metallic materials, an extremely low tritium solubility and a good thermal conductivity [3,4]. On the other hand, the inherent brittleness of tungsten at low temperature and the
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Fig. 1. Picture of representative mock-ups with 13 tungsten blocks [12].
embrittlement by neutron irradiation are the most critical weaknesses, in particular, when tungsten is considered as a structural material of a pressurized component [5]. The brittleness of tungsten should be a critical issue even for the functional application as an armor, if the operation temperature is below the ductile-to-brittle transition temperature (DBTT). This is the case in a water-cooled tungsten monoblock divertor. The DBTT of a commercial tungsten material ranges between 400 ◦ C and 700 ◦ C depending on the loading modes [6]. This means that most part of the tungsten armor in the water-cooled monoblock target will remain below the DBTT during typical high heat flux loadings. Furthermore, one has to consider the irradiation embrittlement effect in addition. Thus, the combination of brittleness and the thermally induced stress fields due to the high heat flux loads raises a serious reliability issue in terms of the structural integrity of tungsten armor. In the literature one finds few previous works dealing with this issue. One relevant paper is the finite element method (FEM)based probabilistic failure risk analysis of the tungsten armor in a water-cooled divertor target published by You and Komarova [7]. They used the weakest-link failure theory expressed by the Weibull statistics and calculated the impact of embrittlement on the failure risk probability of tungsten armor considering four different cracking criteria based on linear elastic fracture mechanics. Another related work [8] is the crack loading analysis of the bond interface between a tungsten flat tile and a copper heat sink in a water-cooled target model. Blanchard and Martin studied the fracture and creep behavior of an all-tungsten divertor for ARIES [9]. However, there is no previous report to be found in the literature dedicated to a rigorous fracture analysis of tungsten armor apart from these. The aim of this study is to deliver quantitative estimates of the vulnerability of a tungsten monoblock armor to cracking under stationary high heat flux loads. To this end, a comparative fracture mechanical investigation has been carried out by means of two different types of computational approaches, namely, the extended finite element method (XFEM) and the FEM-based virtual crack tip extension (VCE) method. In the VCE method, the crack tip loading is described in terms of the stress intensity factor (SIF) or the Jintegral. The results of a comprehensive parametric study obtained from both simulation methods are presented. In total, nine different load cases are considered as a combination of three different heat flux loads and three different coolant temperatures. The impact of the temperature level and the temperature gradient resulting from the different loading cases on crack initiation is discussed. It is noted that the present study is not necessarily limited to specific target geometry (say, the ITER divertor), but rather devoted to investigation of generic failure features of water-cooled tungsten monoblock target under DEMO-relevant operation conditions. The geometry and dimension of the present monoblock model was
taken from the optimized divertor target design of the ITER-like target concept obtained in a previous EFDA power plant Design Assessment Studies carried out in 2013 [10]. As we have no specified boundary conditions (e.g. constraint) yet for DEMO divertor, we assumed rather generic boundary conditions. The thermal loading conditions and thermohydraulic cooling conditions considered here are our best estimates for DEMO divertor operation conditions at the current stage. 2. FE model 2.1. Geometry, FE mesh and materials The monoblock type divertor target model has already been applied to the water-cooled divertor target of ITER. Furthermore, it was also considered for the water-cooled divertor target of a fusion power plant in the framework of the Power Plant Conceptual Study (model A: WCLL) [11]. The monoblock target consists of a number of small rectangular tungsten blocks which are connected by a long cooling tube of high conductivity metal (e.g. copper alloy) running through the central region of each block, see Fig. 1. Two neighboring blocks are separated by a thin gap (∼0.3 mm). The deposited heat is transported from the surface to the cooling
Fig. 2. The FE mesh of the monoblock divertor model. Due to symmetry only one half of the structure was considered.
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Fig. 3. Schematic drawing of the thermal excursion of the PFC. 1: Cooling from the stress-free temperature to room temperature, 2: preheating to the coolant temperature, 3: high heat flux loading, 4: cooling to the coolant temperature.
tube through the tungsten block. The block functions as sacrificing armor whereas the tube acts as heat sink. The typical dimension of a block is roughly 20 mm (3–5 mm thick), and the tube has an inner diameter of 10 mm (1 mm thick). The distance (i.e. thickness) from the loading surface to the tube is determined by the predicted erosion rate and envisaged erosion lifetime. At the brazed bond interface between the tungsten block and the copper alloy tube, a thin (∼0.5 mm) interlayer of soft copper is placed in order to reduce the residual or thermal stress. The model PFC considered for the FEM study is a water-cooled tungsten monoblock duplex structure consisting of a tungsten armor block and a CuCrZr alloy coolant tube (heat sink). The geometry, the finite element (FE) mesh and constituent materials of the considered model PFC are illustrated in Fig. 2. The tungsten armor block has dimensions of 23 mm × 22 mm × 4 mm. The heat sink tube has a thickness of 1.0 mm and an inner diameter of 12 mm. The thickness of the copper interlayer is 0.5 mm. The commercial FEM code ABAQUS was employed for the numerical studies using quadratic brick elements of 20 nodes. In total 8496 elements were used. The mesh in the critical region of the component was refined. For the thermo-mechanical elasto-plastic simulations, data of several materials in the PFC model were used. Cross-rolled and stress-relieved tungsten was applied for the tungsten armor block. For simulation, tungsten was assumed to be linear elastic and ideal-plastic. A precipitation-hardened CuCrZr alloy was
Heat transfer coefficient (KW/(m2K))
400
350
considered for the heat sink tube and soft-annealed copper constituted the interlayer. The Frederick–Armstrong constitutive model applied for copper and the CuCrZr alloy is based on the combination of non-linear isotropic and kinematic hardening laws [13–15]. Temperature-dependent material properties are listed in Table 1 at selected temperatures, which correspond to the operation temperatures for considered materials.
2.2. Loads and boundary conditions coolant temperature: 150ϒC
The thermal excursion (as shown in Fig. 3) of the PFC consists of 4 steps. The first step is cooling the PFC from the stress-free temperature1 to room temperature. Subsequently, a pre-heating of the PFC to the coolant temperature is applied. Then, the HHF load is applied on the top surface of the PFC for 30 s. After HHF loading, the PFC is cooled to the coolant temperature within 10 s. The heat transfer coefficient between the inner wall of the heat sink tube and the coolant water is plotted in Fig. 4. The pressure of the coolant water was 5 MPa. The displacement of the end crosssections of the tube is fully fixed in the tube axis direction, as the tube is assumed to have infinite length. The bottom surface of the PFC is fixed in vertical direction. We took this full constraint boundary condition as an extreme case to produce conservative prediction (i.e. higher stress in tungsten block due to constraint).
coolant temperature: 200ϒC coolant temperature: 250ϒC
300
250
200
150
100 0
Fig. 5. Temperature field of the tungsten armor block for an HHF load of 15 MW/m2 and a coolant temperature of 200 ◦ C at the end of HHF loading. Two positions defined for characterizing temperatures due to variations of HHF loads and coolant temperatures. The heat sink tube and the interlayer are not shown.
50
100
150 200 250 Wall temperature (ϒC)
300
350
Fig. 4. Heat transfer coefficient between the inner wall of the heat sink tube and the coolant water.
1 The stress-free temperature is assumed to be 450 ◦ C, at which it is thought that there is zero stress in the PFC.
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Table 1 Properties of the considered materials at selected temperatures [16,21]. Tungstena ◦
Young’s modulus (GPa) Yield stress (MPa) Qd (MPa) bd Cd (MPa) d Heat conductivity (W/m K) Coefficient of thermal expansion (10−6 /K) a b c d
CuCrZrb 400 C
1200 C
20 C
400 C
20 ◦ C
400 ◦ C
398 1385
393 1100
356 346
115 273 −43 6 148,575 930 318 16.7
106 238 −68 10 117,500 1023 347 17.8
115 3 76 8 64,257 888 379 17.8
95 3 36 25 31,461 952 352 18.1
140 4.6
◦
105 5.3
◦
Copperc
20 C
175 4.5
◦
◦
Rolled and stress-relieved state. Precipitation-hardened state, the reference alloy: Elmedur-X (code: CuCr1Zr, Cr: 0.8%, Zr: 0.08%). Softened by annealing at 700 ◦ C for 1 h. Material parameters entering the Frederick–Armstrong constitutive model.
Fig. 6. Stresses of the tungsten armor block for an HHF load of 15 MW/m2 and a coolant temperature of 200 ◦ C at the end of HHF loading (left column) and after final cooling (right column). The heat sink tube and the interlayer are not shown. Cylindrical coordinates with their origin at the center of the tube on the plane of the right end cross-section of the tube are applied for the plots.
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M. Li et al. / Fusion Engineering and Design 89 (2014) 2716–2725 Table 2 Peak temperatures (◦ C) for different HHF loads and coolant temperatures at the two positions, as shown in Fig. 5. Coolant temperature
Fig. 7. Precracks in the radial direction through the tungsten armor block. The heat sink tube and the interlayer are not shown.
3. Thermo-mechanical simulations In this work, the heat transfer problem was solved first, then its solution was read into the corresponding mechanical simulation as a predefined temperature field.
3.1. Temperature fields In order to cover loading conditions in DEMO and ITER, thermal simulations were performed for nine loading combinations of HHF loads of 10 MW/m2 , 15 MW/m2 and 18 MW/m2 and coolant temperatures of 150 ◦ C, 200 ◦ C and 250 ◦ C. The temperature field of the tungsten armor block during steady state is illustrated for a HHF load of 15 MW/m2 and a coolant temperature of 200 ◦ C in Fig. 5. Two positions in the tungsten armor block are selected to characterize temperatures due to variations of HHF loads and coolant temperatures. The peak temperature in the tungsten armor block occurs at position 1. The temperature
150 ◦ C
200 ◦ C
250 ◦ C
Position 1 10 MW/m2 15 MW/m2 18 MW/m2
820 1221 1478
879 1279 1531
924 1315 1562
Position 2 10 MW/m2 15 MW/m2 18 MW/m2
294 362 404
339 398 434
368 415 447
at position 2 is interesting, since this region is considered to be vulnerable, as shown by the later mechanical simulation. The peak temperatures at two positions for different HHF loads and coolant temperatures are listed in Table 2. The peak temperature in the tungsten armor block ranges from 820 ◦ C to 1562 ◦ C, while the peak temperature at position 2 lies between 294 ◦ C and 447 ◦ C, indicating that this region of the PFC is mostly below the DBTT during the HHF loading. A higher HHF load enlarges the peak temperature as well as the temperature difference between the peak temperature and the coolant temperature, while a higher coolant temperature only enlarges the peak temperature. The difference between the peak temperature and the coolant temperature remains almost the same, irrespective of the coolant temperature. 3.2. Stress fields To illustrate the stress distribution, the stress field is plotted for an HHF load of 15 MW/m2 and a coolant temperature of 200 ◦ C in Fig. 6. The stress field in the tungsten armor block results from the temperature distribution in the tungsten armor block as well as the thermal mismatch between the tungsten armor block and the interlayer. Stresses are, therefore, most critical, when the temperature and the temperature gradient are highest, namely, at the end of the HHF loading. At the end of the HHF loading, compressive hoop stresses concentrate at the top surface close to the plane of symmetry and the back surface in the narrow strip-shaped domains of the tungsten armor block. Tensile hoop stresses are concentrated in the region, which is close to the interlayer in the upper part of
Fig. 8. Stress intensity factor KI for precracks of different angles and lengths at the end of the HHF loading for the HHF load of 15 MW/m2 and the coolant temperature of 200 ◦ C, see Fig. 7.
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Fig. 9. Stress intensity factor KI for precracks of different angles and lengths at the end of cooling for an HHF load of 15 MW/m2 and a coolant temperature of 200 ◦ C, see Fig. 7.
the tungsten armor block. Radial and axial stresses are negligible compared to the hoop stress. At the end of cooling, the hoop stress is significantly smaller than that occurring at the end of the HHF loading. Radial and axial stresses are similar to those at the end of HHF loading. 4. Fracture simulation 4.1. Fracture mechanics parameters To capture the crack features in a quantitative way, stress intensity factors (SIFs) and J-integrals for precracks are calculated by the FEM-based VCE method. As shown in Fig. 6, the tensile hoop stress is concentrated in the upper part the tungsten armor block at the end of HHF loading, while the stresses in other directions are negligible during the whole thermal excursion. Based on this result, radial precracks for computing SIFs and J-integrals are inserted in the tungsten armor block, as shown in Fig. 7. The direction of the virtual crack tip extension is along the positive radial direction. All precracks are through-thickness cracks along the z-axis. For
the purpose of a parametric study, the length of precracks in the radial direction varies from 0.5 mm to 1.5 mm. In each simulation the length of the precrack is fixed and only one precrack is set up avoiding the interaction with other precracks. 4.1.1. Stress intensity factors Cracking is governed by the stresses in the vicinity of the crack tip and the SIF is the magnitude of the stress singularity at the tip of a mathematically sharp crack in a linear elastic material. The elasto-plastic simulations show that no plastic yielding occurred in the domain of the PFC, where precracks were inserted in the tungsten armor block. This result ensures the prerequisites for applying failure analysis based on linear elastic fracture mechanics to precracks inserted in the tungsten armor block. Modes I, II and III of SIFs are used to describe the cracking modes of normal opening, inplane shear and out-of-plane shear, respectively. Based on the stress analysis, mode I is considered to be the critical crack mode in this work. A fracture criterion for crack propagation is defined as follows: As the stress intensity factor of mode I reaches the critical value, unstable fracture occurs and cracks can propagate. The 25
20
15
10 10 MW/m2 2
5
15 MW/m
Stress intensity factor KI (MPa√m)
Stress intensity factor KI (MPa√m)
25
20
15
10 Tsf: 20°C 5
Tsf: 450°C
18 MW/m2 KIC 0
150
200 Coolant temperature (°C)
250
Fig. 10. KI for the precrack at the end of HHF loading for different HHF loads and coolant temperatures. The angle of the precrack is 4.5◦ , and the length of the precrack is 1 mm.
Tsf: 900°C 0
150
200 Coolant temperature (°C)
250
Fig. 11. KI for a precrack subjected to different stress-free temperatures for an HHF load of 10 MW/m2 and different coolant temperatures. The angle of the precrack is 4.5◦ , and the length of the precrack is 1 mm.
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J−integral (mJ/mm2)
1.5
1
0.5
direct converted from SIFs 0
150
200 Coolant temperature (ϒC)
250
Fig. 12. J-integrals for a precrack for an HHF load of 15 MW/m2 and different coolant temperatures. The angle of the precrack is 4.5◦ , the length of the precrack is 1 mm.
√ critical value of KI (KIC : 10 MPa m) refers to the data gained from ◦ fracture toughness tests at 400 C [17]. As stresses are not constant along the z-axis, KI is calculated at each plane perpendicular to the z-axis in the tungsten armor block. Figs. 8 and 9 depict KI at the end of HHF loading and cooling. KI calculated for the end plane is slightly smaller than the one calculated for the center plane, as the stresses are smaller at the free surface, as shown in Fig. 6. However, the tungsten armor block is more vulnerable at the free surface due to the possible defects during fabrication, which means that a crack is more likely to be initiated at the free surface. Therefore, in the following work, if it is not particularly mentioned, KI is evaluated at the free surface (end plane). In Fig. 8, KI decreases, as the angle of the precrack increases. While KI for the precrack in the upper part (ϕ ≤ 9◦ ) of the tungsten armor block is larger than the critical value, KI for the precrack in the lower part (ϕ > 90◦ ) is much smaller, because here the temperature is not massively influenced by the HHF loads. When the angle is 157◦ , KI even attains a small negative value, which indicates that the precrack tends to close. For precracks with small angles in the upper part KI increases, if the precrack length is increased from 0.5 mm to 1.0 mm. The difference of KI for precracks of 1 mm and 1.5 mm is not significant.
Fig. 13. XFEM simulations for different HHF loads and coolant temperatures with an MPS of 400 MPa. The STATUSXFEM of an enriched element is 1.0, if the element is completely cracked and 0.0, if the element contains no crack. If the element is partially cracked (the cohesive stiffness is degraded), the value of STATUSXFEM lies between 1.0 and 0.0. The heat sink tube and the interlayer are not shown in the plots.
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Fig. 14. XFEM simulations for different HHF loads and coolant temperatures with an MPS of 600 MPa. The heat sink tube and the interlayer are not shown in the plots.
At the end of cooling, KI of all precracks is far smaller than the critical value (see Fig. 9), which suggests that radial cracks of mode I will not propagate in the tungsten armor block at the end of cooling. This result is in accordance with the stress analysis. Fig. 10 shows KI for the precrack at the end of HHF loading for different HHF loads and coolant temperatures. KI increases as HHF loads and coolant temperatures increase. The increase of KI is more significant as the HHF load changes from 10 MW/m2 to 18 MW/m2 than when the coolant temperature changes from 150 ◦ C to 250 ◦ C. KI for the HHF load of 10 MW/m2 is about the critical value and KI for the HHF loads of 15 MW/m2 and 18 MW/m2 is far larger than the critical value. 4.1.1.1. Effect of stress-free temperature. In this study, the effective stress-free temperature is set equal to the heat treatment temperature of the joined mock-up that is assumed for prime hardening of the CuCrZr alloy (i.e. for precipitation hardening of the tube). This annealing shall take place after joining at higher temperature and subsequent cooling to room temperature. It is assumed that the previous residual stress disappears during annealing, and the final residual stress is solely dictated by the thermal expansion mismatch stress due to cooling from this annealing temperature (450 ◦ C) to room temperature. To study the effect of the stress-free temperature, simulations with different stress-free temperatures (20 ◦ C, 450 ◦ C and 900 ◦ C) were conducted for heat flux load of 10 MW/m2 . KI for the precrack subjected to different stress-free
temperatures is depicted in Fig. 11. KI is much larger for a stressfree temperature of 20 ◦ C than for the other two. When the PFC is cooled to room temperature from the stress-free temperature, elastic deformation, which could be used to counteract the deformation resulting from preheating and HHF loading, is formed in the tungsten block. Therefore, when there is no such elastic deformation in the tungsten armor block (e.g. with a stress-free temperature of 20 ◦ C), the deformation resulting from the preheating and HHF loading is larger and accordingly, KI is larger. However, the stored elastic deformation is limited by the onset of the plastic deformation in the interlayer. When the interlayer behavior is dominated by plasticity, the increase of stored elastic deformation in the tungsten armor block due to an increase of the stress-free temperature is limited. This is the reason why the decrease of KI resulting from the stress-free temperature change from 450 ◦ C to 900 ◦ C is not significant, compared to the decrease of KI resulting from the stress-free temperature change from 20 ◦ C to 450 ◦ C. 4.1.2. J-integrals The J-integral as introduced by Rice [18] is defined in terms of the energy release rate associated with crack advance and it can be evaluated by a path independent contour integral. In the theory of elasticity, SIFs can be converted to J-integrals. Fig. 12 shows Jintegrals for the specified precrack at the end of the HHF loading for an HHF load of 15 MW/m2 and different coolant temperatures. J-integrals show the same tendency as KI : the J-integral increases as
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the coolant temperature increases. The J-integral results calculated from the SIFs and directly with ABAQUS are generally in accordance with each other. The minor difference between J-integrals calculated from SIFs and directly with ABAQUS is expected as well, since the method of calculating J-integrals from the SIF is more sensitive to numerical precision than calculating J-integrals in ABAQUS [19]. Moreover, the J-integral could be used as an elastic-plastic fracture mechanics parameter, with which it is convenient to study the fracture behavior of tungsten, when plasticity occurs due to softening of the material by recrystallization or with long term cyclic loading. 4.2. XFEM simulations2 For computing the fracture mechanics parameters, the lengths and locations of the cracks are fixed and predefined in each simulation. Of course, the driving force for cracking is strongly dependent on the location and orientation of a initial crack. In this regard, a question arises as to how to determine the most probable location of crack initiation and the preferred orientation of initial crack growth. This issue can be treated by means of XFEM technique. With XFEM one is able to study crack growth along an arbitrary, solution-dependent path without needing to remesh the model. For predicting progressive cracking using XFEM, the maximum principal stress criterion (MPS) and the linear traction–separation response are defined as the damage initiation criterion and the damage evolution law, respectively. Once the principal stress exceeds its maximum allowable value, a crack is initiated. At the same time, the cohesive stiffness in the elements in which crack formation occurs, is degraded. The cohesive stiffness degradation is described by the damage evolution law. If the energy dissipation associated with crack extension is larger than the fracture energy, the cohesive stiffness is reduced to zero and the crack is completely opened. The value of the ultimate tensile strength could in principle be used for the MPS. In Table 2, the peak temperature in the region, where the hoop stress is concentrated, is about 400 ◦ C and the ultimate tensile strength of tungsten at 400 ◦ C is about 950 MPa [20]. Strictly speaking, the ultimate tensile strength cannot be reached in the simulation. However, to study a possible crack propagation pattern and to take the possible reduction of the strength of tungsten during the long term operation into consideration, we make a parametric study of the MPS assuming values of 400 MPa, 600 MPa and 800 MPa. The fracture energy is set to be 0.25 mJ/mm2 ,a value which is gained from fracture toughness tests of tungsten at 400 ◦ C [17]. In this work, we focus on the fracture behavior of the tungsten armor block, and cracking is only allowed in the tungsten armor block in the XFEM simulations. Fig. 13 shows the results of XFEM simulations with an MPS of 400 MPa for different HHF loads and coolant temperatures. Under the HHF load of 10 MW/m2 , no crack appears. When the HHF load is above 15 MW/m2 , radial cracks occur in the critical region as predicted by the stress analysis: Cracks are initiated near the interlayer in the upper part of the tungsten armor block and propagate along the radial direction. The influence of multiple cracks is obvious in the simulations. The most critical region for cracking, as predicted by SIF, lies in the region, which is close to the interlayer and the plane of symmetry in the upper part of the tungsten armor block. However, when multiple cracks occur simultaneously in this region, the stress concentration is largely relaxed and cracks cannot grow significantly. At the same time, the radial crack at a larger angle has the possibility to be initiated and propagate, as shown in the HHF loading cases of 15 MW/m2 and 18 MW/m2 in Fig. 13. If
2 In this work, linear brick elements were used for the XFEM simulations, as quadratic brick elements are not supported in the ABAQUS XFEM code.
the MPS is set to be 600 MPa, as shown in Fig. 14, no opened crack appears for the HHF loads of 10 MW/m2 and 15 MW/m2 . When the MPS is set to 800 MPa, no opened crack is predicted for any HHF load studied in this work. 5. Conclusions In this study, an extensive finite element analysis of the fracture mechanical behavior of a tungsten monoblock divertor target was carried out considering ITER- and DEMO-relevant heat flux loading conditions. Two different computational approaches were employed for the fracture simulation, namely, the FEM-based VCE method for computing KI (or J-integral) and XFEM for predicting progressive cracking. A comparative parametric investigation was conducted for three different heat flux loads and three different coolant temperatures (in total, nine load case combinations). The tungsten armor block exhibited several unique fracture mechanical features allowing to draw several conclusions: 1. The most probable pattern of crack formation is radial cracking in the tungsten armor block starting from the interface with the copper interlayer. 2. The most probable site of cracking is the upper interfacial region of the tungsten armor block adjacent to the top position of the copper interlayer. 3. The driving force of cracking turns out to be heavily dependent on the thermal boundary condition. The driving force increases as either the heat flux load or the coolant temperature increase. 4. The initiation of a major crack becomes likely, only when the strength of tungsten armor block is significantly reduced from its original strength (e.g. due to embrittlement by neutron irradiation or recrystallization). 5. The threshold strength for cracking depends on the specific load case. For heat flux loads below 10 MW/m2 , the threshold value of the failure stress is about 400 MPa. Under heat flux loads of 15 MW/m2 and 18 MW/m2 , the threshold values of the failure stress are about 600 MPa and 800 MPa, respectively. 6. A lower stress-free temperature increases the risk of radial crack growth. 7. Both simulation approaches yielded qualitatively identical predictions. The methodology of the current study could be applied to design-by-analysis of a divertor target on the basis of the ITER Structural Design Criteria, even though the tungsten monoblock armor itself does not belong to the structural component. Acknowledgements The authors are grateful to Mr. Simon McIntosh and Mr. Tom Barrett at Culham Centre for Fusion Energy (CCFE), UK, for providing the data of heat transfer coefficients. References [1] A.R. Raffray, R. Nygren, D.G. Whyte, S. Abdel-Khalik, R. Doerner, F. Escourbiac, et al., High heat flux components – readiness to proceed from near term fusion systems to power plants, Fusion Eng. Design 85 (2010) 93–108. [2] J. Linke, High heat flux performance of plasma facing materials and components under service conditions in future fusion reactors, Fusion Sci. Technol. 49 (2006) 455–464. [3] H. Bolt, V. Barabash, W. Krauss, J. Linke, R. Neu, S. Suzuki, et al., Materials for the plasma-facing components of fusion reactors, J. Nucl. Mater. 66 (2004) 329–333. [4] N. Baluc, Assessment Report on W, Final Report on EFDA Task TW1-TTMA-002 Deliverable 5, 2002. [5] Y. Ishijima, H. Kurishita, K. Yubuta, H. Arakawa, M. Hasegawa, Y. Hiraoka, et al., Current status of ductile tungsten alloy development by mechanical alloying, J. Nucl. Mater. 775 (2004) 329–333. [6] D. Rupp, S.M. Weygand, Loading rate dependence of the fracture toughness of polycrystalline tungsten, J. Nucl. Mater. 417 (2011) 477–480.
M. Li et al. / Fusion Engineering and Design 89 (2014) 2716–2725 [7] J.-H. You, I. Komarova, Probabilistic failure analysis of a water-cooled tungsten divertor: impact of embrittlement, J. Nucl. Mater. 375 (2008) 283–289. [8] M. Miskiewicz, J.-H. You, Impact of plastic softening of over-aged CuCrZr alloy heat sink tube on the structural reliability of a plasma-facing component, Fusion Eng. Design 83 (2008) 66–71. [9] J.P. Blanchard, C. Martin, Fracture and creep in an all-tungsten divertor for ARIES, Fusion Sci. Technol. 64 (2013) 435–439. [10] F. Crescenzi, A. Moriani, S. Roccella, E. Visca, M. Richou, Water-cooled Divertor Target Design Study CuCrZr/W Monoblock Concept, EFDA Report: WP13DAS02-T02-D02, 2013. [11] D. Maisonnier, I. Cook, P. Sardain, R. Andreani, L. Di Pace, R. Forrest, et al., A Conceptual Study of Commercial Fusion Power Plants, EFDA(05)-27/4.10, 2005. [12] E. Visca, B. Riccardi, A. Orsini, C. Testani, Manufacturing and testing of monoblock tungsten small-scale mock-ups, Fusion Eng. Design 56-57 (2001) 343–347. [13] J. Lemaitre, J.-L. Chaboche, Mechanics of Solid Materials, 1st ed., Cambridge University Press, 1994.
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[14] P.J. Armstrong, C.O. Frederick, A Mathematical Representation of the Multiaxial Bauschinger Effect, G.E.G.B. Report RD/B/N, 1966, pp. 731. [15] J. Chaboche, Constitutive equations for cyclic plasticity and cyclic viscoplasticity, Int. J. Plast. 5 (1989) 247–302. [16] ITER structural design criteria for in-vessel components (SDC-IC) appendix A: materials design limit data, ITER, 2001. [17] B. Gludovatz, S. Wurster, A. Hoffmann, R. Pippan, Fracture toughness of polycrystalline tungsten alloys, Int. J. Refract. Metals Hard Mater. 28 (2010) 674–678. [18] J.R. Rice, A path independent integral and approximate analysis of strain concentration by notches and cracks, J. Appl. Mech. 35 (1968) 379–386. [19] Abaqus Analysis User’s Manual 6.12, Dassault Systémes Simulia Corp., Providence, RI, USA. 2012. [20] PLANSEE, Tungsten Material Properties and Applications, 2014 http://www. plansee.com/en/Materials-Tungsten-403.htm [21] J.-H. You, M. Miskiewicz, Material parameters of copper and CuCrZr alloy for cyclic plasticity at elevated temperatures, J. Nucl. Mater 373 (2008) 269–274.
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Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Fracture mechanical analysis of tungsten armor failure of a water-cooled divertor target Muyuan Li a , Ewald Werner a , Jeong-Ha You b,∗ a b
Lehrstuhl für Werkstoffkunde und Werkstoffmechanik, Technische Universität München, Boltzmannstr. 15, 85748 Garching, Germany Max-Planck-Institut für Plasmaphysik, Boltzmannstr. 2, 85748 Garching, Germany
h i g h l i g h t s • • • •
The FEM-based VCE method and XFEM were employed for computing KI (or J-integral) and predicting progressive cracking, respectively. The most probable pattern of crack formation is radial cracking in the tungsten armor block. The most probable site of cracking is the upper interfacial region of the tungsten armor block adjacent to the top position of the copper interlayer. The initiation of a major crack becomes likely, only when the strength of tungsten armor block is significantly reduced from its original strength.
a r t i c l e
i n f o
Article history: Received 3 April 2014 Received in revised form 17 July 2014 Accepted 18 July 2014 Available online 8 August 2014 Keywords: Divertor Tungsten armor High heat flux loads Fracture mechanics Crack Stress intensity factor
a b s t r a c t The inherent brittleness of tungsten at low temperature and the embrittlement by neutron irradiation are its most critical weaknesses for fusion applications. In the current design of the ITER and DEMO divertor, the high heat flux loads during the operation impose a strong constraint on the structure–mechanical performance of the divertor. Thus, the combination of brittleness and the thermally induced stress fields due to the high heat flux loads raises a serious reliability issue in terms of the structural integrity of tungsten armor. In this study, quantitative estimates of the vulnerability of the tungsten monoblock armor cracking under stationary high heat flux loads are presented. A comparative fracture mechanical investigation has been carried out by means of two different types of computational approaches, namely, the extended finite element method (XFEM) and the finite element method (FEM)-based virtual crack tip extension (VCE) method. The fracture analysis indicates that the most probable pattern of crack formation is radial cracking in the tungsten armor starting from the interface to tube and the most probable site of cracking is the upper interfacial region of the tungsten armor adjacent to the top position of the copper interlayer. The strength threshold for crack initiation and the high heat flux load threshold for crack propagation are evaluated based on XFEM simulations and computations of stress intensity factors and J-integrals. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The divertor is an important in-vessel plasma-facing component (PFC) of a fusion reactor. The essential function of the divertor is to exhaust the edge plasma in the scrape-off layer in order for helium ash and other impurities to be continuously removed from the burning plasma core. The plasma exhaust is achieved by intense particle bombardment onto the divertor target plate generating a high heat flux load into the target surface [1]. The maximum stationary heat flux load for the ITER divertor target reaches up to
∗ Corresponding author. Tel.: +49 089 3299 1373; fax:+49 089 3299 1212. E-mail address: [email protected] (J.-H. You). http://dx.doi.org/10.1016/j.fusengdes.2014.07.011 0920-3796/© 2014 Elsevier B.V. All rights reserved.
10 MW/m2 , and slow thermal transient loads up to 20 MW/m2 are expected [2]. In the case of the DEMO divertor, the range of possible heat flux load may be even larger. Under such high heat flux loads, the divertor target component (a bi-material joint structure) is subjected to high thermal stresses. Thus, the high heat flux loads impose a strong constraint on the structure–mechanical performance of the divertor. In the current design concept for the DEMO as well as the ITER divertor, the material for the plasma-facing target armor is tungsten. This choice is owing to the advantageous properties of tungsten such as an extremely low sputtering yield, the highest melting point of metallic materials, an extremely low tritium solubility and a good thermal conductivity [3,4]. On the other hand, the inherent brittleness of tungsten at low temperature and the
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Fig. 1. Picture of representative mock-ups with 13 tungsten blocks [12].
embrittlement by neutron irradiation are the most critical weaknesses, in particular, when tungsten is considered as a structural material of a pressurized component [5]. The brittleness of tungsten should be a critical issue even for the functional application as an armor, if the operation temperature is below the ductile-to-brittle transition temperature (DBTT). This is the case in a water-cooled tungsten monoblock divertor. The DBTT of a commercial tungsten material ranges between 400 ◦ C and 700 ◦ C depending on the loading modes [6]. This means that most part of the tungsten armor in the water-cooled monoblock target will remain below the DBTT during typical high heat flux loadings. Furthermore, one has to consider the irradiation embrittlement effect in addition. Thus, the combination of brittleness and the thermally induced stress fields due to the high heat flux loads raises a serious reliability issue in terms of the structural integrity of tungsten armor. In the literature one finds few previous works dealing with this issue. One relevant paper is the finite element method (FEM)based probabilistic failure risk analysis of the tungsten armor in a water-cooled divertor target published by You and Komarova [7]. They used the weakest-link failure theory expressed by the Weibull statistics and calculated the impact of embrittlement on the failure risk probability of tungsten armor considering four different cracking criteria based on linear elastic fracture mechanics. Another related work [8] is the crack loading analysis of the bond interface between a tungsten flat tile and a copper heat sink in a water-cooled target model. Blanchard and Martin studied the fracture and creep behavior of an all-tungsten divertor for ARIES [9]. However, there is no previous report to be found in the literature dedicated to a rigorous fracture analysis of tungsten armor apart from these. The aim of this study is to deliver quantitative estimates of the vulnerability of a tungsten monoblock armor to cracking under stationary high heat flux loads. To this end, a comparative fracture mechanical investigation has been carried out by means of two different types of computational approaches, namely, the extended finite element method (XFEM) and the FEM-based virtual crack tip extension (VCE) method. In the VCE method, the crack tip loading is described in terms of the stress intensity factor (SIF) or the Jintegral. The results of a comprehensive parametric study obtained from both simulation methods are presented. In total, nine different load cases are considered as a combination of three different heat flux loads and three different coolant temperatures. The impact of the temperature level and the temperature gradient resulting from the different loading cases on crack initiation is discussed. It is noted that the present study is not necessarily limited to specific target geometry (say, the ITER divertor), but rather devoted to investigation of generic failure features of water-cooled tungsten monoblock target under DEMO-relevant operation conditions. The geometry and dimension of the present monoblock model was
taken from the optimized divertor target design of the ITER-like target concept obtained in a previous EFDA power plant Design Assessment Studies carried out in 2013 [10]. As we have no specified boundary conditions (e.g. constraint) yet for DEMO divertor, we assumed rather generic boundary conditions. The thermal loading conditions and thermohydraulic cooling conditions considered here are our best estimates for DEMO divertor operation conditions at the current stage. 2. FE model 2.1. Geometry, FE mesh and materials The monoblock type divertor target model has already been applied to the water-cooled divertor target of ITER. Furthermore, it was also considered for the water-cooled divertor target of a fusion power plant in the framework of the Power Plant Conceptual Study (model A: WCLL) [11]. The monoblock target consists of a number of small rectangular tungsten blocks which are connected by a long cooling tube of high conductivity metal (e.g. copper alloy) running through the central region of each block, see Fig. 1. Two neighboring blocks are separated by a thin gap (∼0.3 mm). The deposited heat is transported from the surface to the cooling
Fig. 2. The FE mesh of the monoblock divertor model. Due to symmetry only one half of the structure was considered.
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Fig. 3. Schematic drawing of the thermal excursion of the PFC. 1: Cooling from the stress-free temperature to room temperature, 2: preheating to the coolant temperature, 3: high heat flux loading, 4: cooling to the coolant temperature.
tube through the tungsten block. The block functions as sacrificing armor whereas the tube acts as heat sink. The typical dimension of a block is roughly 20 mm (3–5 mm thick), and the tube has an inner diameter of 10 mm (1 mm thick). The distance (i.e. thickness) from the loading surface to the tube is determined by the predicted erosion rate and envisaged erosion lifetime. At the brazed bond interface between the tungsten block and the copper alloy tube, a thin (∼0.5 mm) interlayer of soft copper is placed in order to reduce the residual or thermal stress. The model PFC considered for the FEM study is a water-cooled tungsten monoblock duplex structure consisting of a tungsten armor block and a CuCrZr alloy coolant tube (heat sink). The geometry, the finite element (FE) mesh and constituent materials of the considered model PFC are illustrated in Fig. 2. The tungsten armor block has dimensions of 23 mm × 22 mm × 4 mm. The heat sink tube has a thickness of 1.0 mm and an inner diameter of 12 mm. The thickness of the copper interlayer is 0.5 mm. The commercial FEM code ABAQUS was employed for the numerical studies using quadratic brick elements of 20 nodes. In total 8496 elements were used. The mesh in the critical region of the component was refined. For the thermo-mechanical elasto-plastic simulations, data of several materials in the PFC model were used. Cross-rolled and stress-relieved tungsten was applied for the tungsten armor block. For simulation, tungsten was assumed to be linear elastic and ideal-plastic. A precipitation-hardened CuCrZr alloy was
Heat transfer coefficient (KW/(m2K))
400
350
considered for the heat sink tube and soft-annealed copper constituted the interlayer. The Frederick–Armstrong constitutive model applied for copper and the CuCrZr alloy is based on the combination of non-linear isotropic and kinematic hardening laws [13–15]. Temperature-dependent material properties are listed in Table 1 at selected temperatures, which correspond to the operation temperatures for considered materials.
2.2. Loads and boundary conditions coolant temperature: 150ϒC
The thermal excursion (as shown in Fig. 3) of the PFC consists of 4 steps. The first step is cooling the PFC from the stress-free temperature1 to room temperature. Subsequently, a pre-heating of the PFC to the coolant temperature is applied. Then, the HHF load is applied on the top surface of the PFC for 30 s. After HHF loading, the PFC is cooled to the coolant temperature within 10 s. The heat transfer coefficient between the inner wall of the heat sink tube and the coolant water is plotted in Fig. 4. The pressure of the coolant water was 5 MPa. The displacement of the end crosssections of the tube is fully fixed in the tube axis direction, as the tube is assumed to have infinite length. The bottom surface of the PFC is fixed in vertical direction. We took this full constraint boundary condition as an extreme case to produce conservative prediction (i.e. higher stress in tungsten block due to constraint).
coolant temperature: 200ϒC coolant temperature: 250ϒC
300
250
200
150
100 0
Fig. 5. Temperature field of the tungsten armor block for an HHF load of 15 MW/m2 and a coolant temperature of 200 ◦ C at the end of HHF loading. Two positions defined for characterizing temperatures due to variations of HHF loads and coolant temperatures. The heat sink tube and the interlayer are not shown.
50
100
150 200 250 Wall temperature (ϒC)
300
350
Fig. 4. Heat transfer coefficient between the inner wall of the heat sink tube and the coolant water.
1 The stress-free temperature is assumed to be 450 ◦ C, at which it is thought that there is zero stress in the PFC.
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Table 1 Properties of the considered materials at selected temperatures [16,21]. Tungstena ◦
Young’s modulus (GPa) Yield stress (MPa) Qd (MPa) bd Cd (MPa) d Heat conductivity (W/m K) Coefficient of thermal expansion (10−6 /K) a b c d
CuCrZrb 400 C
1200 C
20 C
400 C
20 ◦ C
400 ◦ C
398 1385
393 1100
356 346
115 273 −43 6 148,575 930 318 16.7
106 238 −68 10 117,500 1023 347 17.8
115 3 76 8 64,257 888 379 17.8
95 3 36 25 31,461 952 352 18.1
140 4.6
◦
105 5.3
◦
Copperc
20 C
175 4.5
◦
◦
Rolled and stress-relieved state. Precipitation-hardened state, the reference alloy: Elmedur-X (code: CuCr1Zr, Cr: 0.8%, Zr: 0.08%). Softened by annealing at 700 ◦ C for 1 h. Material parameters entering the Frederick–Armstrong constitutive model.
Fig. 6. Stresses of the tungsten armor block for an HHF load of 15 MW/m2 and a coolant temperature of 200 ◦ C at the end of HHF loading (left column) and after final cooling (right column). The heat sink tube and the interlayer are not shown. Cylindrical coordinates with their origin at the center of the tube on the plane of the right end cross-section of the tube are applied for the plots.
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M. Li et al. / Fusion Engineering and Design 89 (2014) 2716–2725 Table 2 Peak temperatures (◦ C) for different HHF loads and coolant temperatures at the two positions, as shown in Fig. 5. Coolant temperature
Fig. 7. Precracks in the radial direction through the tungsten armor block. The heat sink tube and the interlayer are not shown.
3. Thermo-mechanical simulations In this work, the heat transfer problem was solved first, then its solution was read into the corresponding mechanical simulation as a predefined temperature field.
3.1. Temperature fields In order to cover loading conditions in DEMO and ITER, thermal simulations were performed for nine loading combinations of HHF loads of 10 MW/m2 , 15 MW/m2 and 18 MW/m2 and coolant temperatures of 150 ◦ C, 200 ◦ C and 250 ◦ C. The temperature field of the tungsten armor block during steady state is illustrated for a HHF load of 15 MW/m2 and a coolant temperature of 200 ◦ C in Fig. 5. Two positions in the tungsten armor block are selected to characterize temperatures due to variations of HHF loads and coolant temperatures. The peak temperature in the tungsten armor block occurs at position 1. The temperature
150 ◦ C
200 ◦ C
250 ◦ C
Position 1 10 MW/m2 15 MW/m2 18 MW/m2
820 1221 1478
879 1279 1531
924 1315 1562
Position 2 10 MW/m2 15 MW/m2 18 MW/m2
294 362 404
339 398 434
368 415 447
at position 2 is interesting, since this region is considered to be vulnerable, as shown by the later mechanical simulation. The peak temperatures at two positions for different HHF loads and coolant temperatures are listed in Table 2. The peak temperature in the tungsten armor block ranges from 820 ◦ C to 1562 ◦ C, while the peak temperature at position 2 lies between 294 ◦ C and 447 ◦ C, indicating that this region of the PFC is mostly below the DBTT during the HHF loading. A higher HHF load enlarges the peak temperature as well as the temperature difference between the peak temperature and the coolant temperature, while a higher coolant temperature only enlarges the peak temperature. The difference between the peak temperature and the coolant temperature remains almost the same, irrespective of the coolant temperature. 3.2. Stress fields To illustrate the stress distribution, the stress field is plotted for an HHF load of 15 MW/m2 and a coolant temperature of 200 ◦ C in Fig. 6. The stress field in the tungsten armor block results from the temperature distribution in the tungsten armor block as well as the thermal mismatch between the tungsten armor block and the interlayer. Stresses are, therefore, most critical, when the temperature and the temperature gradient are highest, namely, at the end of the HHF loading. At the end of the HHF loading, compressive hoop stresses concentrate at the top surface close to the plane of symmetry and the back surface in the narrow strip-shaped domains of the tungsten armor block. Tensile hoop stresses are concentrated in the region, which is close to the interlayer in the upper part of
Fig. 8. Stress intensity factor KI for precracks of different angles and lengths at the end of the HHF loading for the HHF load of 15 MW/m2 and the coolant temperature of 200 ◦ C, see Fig. 7.
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Fig. 9. Stress intensity factor KI for precracks of different angles and lengths at the end of cooling for an HHF load of 15 MW/m2 and a coolant temperature of 200 ◦ C, see Fig. 7.
the tungsten armor block. Radial and axial stresses are negligible compared to the hoop stress. At the end of cooling, the hoop stress is significantly smaller than that occurring at the end of the HHF loading. Radial and axial stresses are similar to those at the end of HHF loading. 4. Fracture simulation 4.1. Fracture mechanics parameters To capture the crack features in a quantitative way, stress intensity factors (SIFs) and J-integrals for precracks are calculated by the FEM-based VCE method. As shown in Fig. 6, the tensile hoop stress is concentrated in the upper part the tungsten armor block at the end of HHF loading, while the stresses in other directions are negligible during the whole thermal excursion. Based on this result, radial precracks for computing SIFs and J-integrals are inserted in the tungsten armor block, as shown in Fig. 7. The direction of the virtual crack tip extension is along the positive radial direction. All precracks are through-thickness cracks along the z-axis. For
the purpose of a parametric study, the length of precracks in the radial direction varies from 0.5 mm to 1.5 mm. In each simulation the length of the precrack is fixed and only one precrack is set up avoiding the interaction with other precracks. 4.1.1. Stress intensity factors Cracking is governed by the stresses in the vicinity of the crack tip and the SIF is the magnitude of the stress singularity at the tip of a mathematically sharp crack in a linear elastic material. The elasto-plastic simulations show that no plastic yielding occurred in the domain of the PFC, where precracks were inserted in the tungsten armor block. This result ensures the prerequisites for applying failure analysis based on linear elastic fracture mechanics to precracks inserted in the tungsten armor block. Modes I, II and III of SIFs are used to describe the cracking modes of normal opening, inplane shear and out-of-plane shear, respectively. Based on the stress analysis, mode I is considered to be the critical crack mode in this work. A fracture criterion for crack propagation is defined as follows: As the stress intensity factor of mode I reaches the critical value, unstable fracture occurs and cracks can propagate. The 25
20
15
10 10 MW/m2 2
5
15 MW/m
Stress intensity factor KI (MPa√m)
Stress intensity factor KI (MPa√m)
25
20
15
10 Tsf: 20°C 5
Tsf: 450°C
18 MW/m2 KIC 0
150
200 Coolant temperature (°C)
250
Fig. 10. KI for the precrack at the end of HHF loading for different HHF loads and coolant temperatures. The angle of the precrack is 4.5◦ , and the length of the precrack is 1 mm.
Tsf: 900°C 0
150
200 Coolant temperature (°C)
250
Fig. 11. KI for a precrack subjected to different stress-free temperatures for an HHF load of 10 MW/m2 and different coolant temperatures. The angle of the precrack is 4.5◦ , and the length of the precrack is 1 mm.
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J−integral (mJ/mm2)
1.5
1
0.5
direct converted from SIFs 0
150
200 Coolant temperature (ϒC)
250
Fig. 12. J-integrals for a precrack for an HHF load of 15 MW/m2 and different coolant temperatures. The angle of the precrack is 4.5◦ , the length of the precrack is 1 mm.
√ critical value of KI (KIC : 10 MPa m) refers to the data gained from ◦ fracture toughness tests at 400 C [17]. As stresses are not constant along the z-axis, KI is calculated at each plane perpendicular to the z-axis in the tungsten armor block. Figs. 8 and 9 depict KI at the end of HHF loading and cooling. KI calculated for the end plane is slightly smaller than the one calculated for the center plane, as the stresses are smaller at the free surface, as shown in Fig. 6. However, the tungsten armor block is more vulnerable at the free surface due to the possible defects during fabrication, which means that a crack is more likely to be initiated at the free surface. Therefore, in the following work, if it is not particularly mentioned, KI is evaluated at the free surface (end plane). In Fig. 8, KI decreases, as the angle of the precrack increases. While KI for the precrack in the upper part (ϕ ≤ 9◦ ) of the tungsten armor block is larger than the critical value, KI for the precrack in the lower part (ϕ > 90◦ ) is much smaller, because here the temperature is not massively influenced by the HHF loads. When the angle is 157◦ , KI even attains a small negative value, which indicates that the precrack tends to close. For precracks with small angles in the upper part KI increases, if the precrack length is increased from 0.5 mm to 1.0 mm. The difference of KI for precracks of 1 mm and 1.5 mm is not significant.
Fig. 13. XFEM simulations for different HHF loads and coolant temperatures with an MPS of 400 MPa. The STATUSXFEM of an enriched element is 1.0, if the element is completely cracked and 0.0, if the element contains no crack. If the element is partially cracked (the cohesive stiffness is degraded), the value of STATUSXFEM lies between 1.0 and 0.0. The heat sink tube and the interlayer are not shown in the plots.
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Fig. 14. XFEM simulations for different HHF loads and coolant temperatures with an MPS of 600 MPa. The heat sink tube and the interlayer are not shown in the plots.
At the end of cooling, KI of all precracks is far smaller than the critical value (see Fig. 9), which suggests that radial cracks of mode I will not propagate in the tungsten armor block at the end of cooling. This result is in accordance with the stress analysis. Fig. 10 shows KI for the precrack at the end of HHF loading for different HHF loads and coolant temperatures. KI increases as HHF loads and coolant temperatures increase. The increase of KI is more significant as the HHF load changes from 10 MW/m2 to 18 MW/m2 than when the coolant temperature changes from 150 ◦ C to 250 ◦ C. KI for the HHF load of 10 MW/m2 is about the critical value and KI for the HHF loads of 15 MW/m2 and 18 MW/m2 is far larger than the critical value. 4.1.1.1. Effect of stress-free temperature. In this study, the effective stress-free temperature is set equal to the heat treatment temperature of the joined mock-up that is assumed for prime hardening of the CuCrZr alloy (i.e. for precipitation hardening of the tube). This annealing shall take place after joining at higher temperature and subsequent cooling to room temperature. It is assumed that the previous residual stress disappears during annealing, and the final residual stress is solely dictated by the thermal expansion mismatch stress due to cooling from this annealing temperature (450 ◦ C) to room temperature. To study the effect of the stress-free temperature, simulations with different stress-free temperatures (20 ◦ C, 450 ◦ C and 900 ◦ C) were conducted for heat flux load of 10 MW/m2 . KI for the precrack subjected to different stress-free
temperatures is depicted in Fig. 11. KI is much larger for a stressfree temperature of 20 ◦ C than for the other two. When the PFC is cooled to room temperature from the stress-free temperature, elastic deformation, which could be used to counteract the deformation resulting from preheating and HHF loading, is formed in the tungsten block. Therefore, when there is no such elastic deformation in the tungsten armor block (e.g. with a stress-free temperature of 20 ◦ C), the deformation resulting from the preheating and HHF loading is larger and accordingly, KI is larger. However, the stored elastic deformation is limited by the onset of the plastic deformation in the interlayer. When the interlayer behavior is dominated by plasticity, the increase of stored elastic deformation in the tungsten armor block due to an increase of the stress-free temperature is limited. This is the reason why the decrease of KI resulting from the stress-free temperature change from 450 ◦ C to 900 ◦ C is not significant, compared to the decrease of KI resulting from the stress-free temperature change from 20 ◦ C to 450 ◦ C. 4.1.2. J-integrals The J-integral as introduced by Rice [18] is defined in terms of the energy release rate associated with crack advance and it can be evaluated by a path independent contour integral. In the theory of elasticity, SIFs can be converted to J-integrals. Fig. 12 shows Jintegrals for the specified precrack at the end of the HHF loading for an HHF load of 15 MW/m2 and different coolant temperatures. J-integrals show the same tendency as KI : the J-integral increases as
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the coolant temperature increases. The J-integral results calculated from the SIFs and directly with ABAQUS are generally in accordance with each other. The minor difference between J-integrals calculated from SIFs and directly with ABAQUS is expected as well, since the method of calculating J-integrals from the SIF is more sensitive to numerical precision than calculating J-integrals in ABAQUS [19]. Moreover, the J-integral could be used as an elastic-plastic fracture mechanics parameter, with which it is convenient to study the fracture behavior of tungsten, when plasticity occurs due to softening of the material by recrystallization or with long term cyclic loading. 4.2. XFEM simulations2 For computing the fracture mechanics parameters, the lengths and locations of the cracks are fixed and predefined in each simulation. Of course, the driving force for cracking is strongly dependent on the location and orientation of a initial crack. In this regard, a question arises as to how to determine the most probable location of crack initiation and the preferred orientation of initial crack growth. This issue can be treated by means of XFEM technique. With XFEM one is able to study crack growth along an arbitrary, solution-dependent path without needing to remesh the model. For predicting progressive cracking using XFEM, the maximum principal stress criterion (MPS) and the linear traction–separation response are defined as the damage initiation criterion and the damage evolution law, respectively. Once the principal stress exceeds its maximum allowable value, a crack is initiated. At the same time, the cohesive stiffness in the elements in which crack formation occurs, is degraded. The cohesive stiffness degradation is described by the damage evolution law. If the energy dissipation associated with crack extension is larger than the fracture energy, the cohesive stiffness is reduced to zero and the crack is completely opened. The value of the ultimate tensile strength could in principle be used for the MPS. In Table 2, the peak temperature in the region, where the hoop stress is concentrated, is about 400 ◦ C and the ultimate tensile strength of tungsten at 400 ◦ C is about 950 MPa [20]. Strictly speaking, the ultimate tensile strength cannot be reached in the simulation. However, to study a possible crack propagation pattern and to take the possible reduction of the strength of tungsten during the long term operation into consideration, we make a parametric study of the MPS assuming values of 400 MPa, 600 MPa and 800 MPa. The fracture energy is set to be 0.25 mJ/mm2 ,a value which is gained from fracture toughness tests of tungsten at 400 ◦ C [17]. In this work, we focus on the fracture behavior of the tungsten armor block, and cracking is only allowed in the tungsten armor block in the XFEM simulations. Fig. 13 shows the results of XFEM simulations with an MPS of 400 MPa for different HHF loads and coolant temperatures. Under the HHF load of 10 MW/m2 , no crack appears. When the HHF load is above 15 MW/m2 , radial cracks occur in the critical region as predicted by the stress analysis: Cracks are initiated near the interlayer in the upper part of the tungsten armor block and propagate along the radial direction. The influence of multiple cracks is obvious in the simulations. The most critical region for cracking, as predicted by SIF, lies in the region, which is close to the interlayer and the plane of symmetry in the upper part of the tungsten armor block. However, when multiple cracks occur simultaneously in this region, the stress concentration is largely relaxed and cracks cannot grow significantly. At the same time, the radial crack at a larger angle has the possibility to be initiated and propagate, as shown in the HHF loading cases of 15 MW/m2 and 18 MW/m2 in Fig. 13. If
2 In this work, linear brick elements were used for the XFEM simulations, as quadratic brick elements are not supported in the ABAQUS XFEM code.
the MPS is set to be 600 MPa, as shown in Fig. 14, no opened crack appears for the HHF loads of 10 MW/m2 and 15 MW/m2 . When the MPS is set to 800 MPa, no opened crack is predicted for any HHF load studied in this work. 5. Conclusions In this study, an extensive finite element analysis of the fracture mechanical behavior of a tungsten monoblock divertor target was carried out considering ITER- and DEMO-relevant heat flux loading conditions. Two different computational approaches were employed for the fracture simulation, namely, the FEM-based VCE method for computing KI (or J-integral) and XFEM for predicting progressive cracking. A comparative parametric investigation was conducted for three different heat flux loads and three different coolant temperatures (in total, nine load case combinations). The tungsten armor block exhibited several unique fracture mechanical features allowing to draw several conclusions: 1. The most probable pattern of crack formation is radial cracking in the tungsten armor block starting from the interface with the copper interlayer. 2. The most probable site of cracking is the upper interfacial region of the tungsten armor block adjacent to the top position of the copper interlayer. 3. The driving force of cracking turns out to be heavily dependent on the thermal boundary condition. The driving force increases as either the heat flux load or the coolant temperature increase. 4. The initiation of a major crack becomes likely, only when the strength of tungsten armor block is significantly reduced from its original strength (e.g. due to embrittlement by neutron irradiation or recrystallization). 5. The threshold strength for cracking depends on the specific load case. For heat flux loads below 10 MW/m2 , the threshold value of the failure stress is about 400 MPa. Under heat flux loads of 15 MW/m2 and 18 MW/m2 , the threshold values of the failure stress are about 600 MPa and 800 MPa, respectively. 6. A lower stress-free temperature increases the risk of radial crack growth. 7. Both simulation approaches yielded qualitatively identical predictions. The methodology of the current study could be applied to design-by-analysis of a divertor target on the basis of the ITER Structural Design Criteria, even though the tungsten monoblock armor itself does not belong to the structural component. Acknowledgements The authors are grateful to Mr. Simon McIntosh and Mr. Tom Barrett at Culham Centre for Fusion Energy (CCFE), UK, for providing the data of heat transfer coefficients. References [1] A.R. Raffray, R. Nygren, D.G. Whyte, S. Abdel-Khalik, R. Doerner, F. Escourbiac, et al., High heat flux components – readiness to proceed from near term fusion systems to power plants, Fusion Eng. Design 85 (2010) 93–108. [2] J. Linke, High heat flux performance of plasma facing materials and components under service conditions in future fusion reactors, Fusion Sci. Technol. 49 (2006) 455–464. [3] H. Bolt, V. Barabash, W. Krauss, J. Linke, R. Neu, S. Suzuki, et al., Materials for the plasma-facing components of fusion reactors, J. Nucl. Mater. 66 (2004) 329–333. [4] N. Baluc, Assessment Report on W, Final Report on EFDA Task TW1-TTMA-002 Deliverable 5, 2002. [5] Y. Ishijima, H. Kurishita, K. Yubuta, H. Arakawa, M. Hasegawa, Y. Hiraoka, et al., Current status of ductile tungsten alloy development by mechanical alloying, J. Nucl. Mater. 775 (2004) 329–333. [6] D. Rupp, S.M. Weygand, Loading rate dependence of the fracture toughness of polycrystalline tungsten, J. Nucl. Mater. 417 (2011) 477–480.
M. Li et al. / Fusion Engineering and Design 89 (2014) 2716–2725 [7] J.-H. You, I. Komarova, Probabilistic failure analysis of a water-cooled tungsten divertor: impact of embrittlement, J. Nucl. Mater. 375 (2008) 283–289. [8] M. Miskiewicz, J.-H. You, Impact of plastic softening of over-aged CuCrZr alloy heat sink tube on the structural reliability of a plasma-facing component, Fusion Eng. Design 83 (2008) 66–71. [9] J.P. Blanchard, C. Martin, Fracture and creep in an all-tungsten divertor for ARIES, Fusion Sci. Technol. 64 (2013) 435–439. [10] F. Crescenzi, A. Moriani, S. Roccella, E. Visca, M. Richou, Water-cooled Divertor Target Design Study CuCrZr/W Monoblock Concept, EFDA Report: WP13DAS02-T02-D02, 2013. [11] D. Maisonnier, I. Cook, P. Sardain, R. Andreani, L. Di Pace, R. Forrest, et al., A Conceptual Study of Commercial Fusion Power Plants, EFDA(05)-27/4.10, 2005. [12] E. Visca, B. Riccardi, A. Orsini, C. Testani, Manufacturing and testing of monoblock tungsten small-scale mock-ups, Fusion Eng. Design 56-57 (2001) 343–347. [13] J. Lemaitre, J.-L. Chaboche, Mechanics of Solid Materials, 1st ed., Cambridge University Press, 1994.
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[14] P.J. Armstrong, C.O. Frederick, A Mathematical Representation of the Multiaxial Bauschinger Effect, G.E.G.B. Report RD/B/N, 1966, pp. 731. [15] J. Chaboche, Constitutive equations for cyclic plasticity and cyclic viscoplasticity, Int. J. Plast. 5 (1989) 247–302. [16] ITER structural design criteria for in-vessel components (SDC-IC) appendix A: materials design limit data, ITER, 2001. [17] B. Gludovatz, S. Wurster, A. Hoffmann, R. Pippan, Fracture toughness of polycrystalline tungsten alloys, Int. J. Refract. Metals Hard Mater. 28 (2010) 674–678. [18] J.R. Rice, A path independent integral and approximate analysis of strain concentration by notches and cracks, J. Appl. Mech. 35 (1968) 379–386. [19] Abaqus Analysis User’s Manual 6.12, Dassault Systémes Simulia Corp., Providence, RI, USA. 2012. [20] PLANSEE, Tungsten Material Properties and Applications, 2014 http://www. plansee.com/en/Materials-Tungsten-403.htm [21] J.-H. You, M. Miskiewicz, Material parameters of copper and CuCrZr alloy for cyclic plasticity at elevated temperatures, J. Nucl. Mater 373 (2008) 269–274.