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Simulation of cracks in tungsten under ITER specific transient heat loads
Fusion Engineering and Design 82 (2007) 1657–1663
Simulation of cracks in tungsten under ITER specific transient heat loads S.E. Pestchanyi a,∗ , J. Linke b a
Forschungszentrum Karlsruhe, Institute for Pulsed Power and Microwave Technology, P.O. Box 3640, 76021 Karlsruhe, Germany b Forschungszentrum J¨ ulich GmbH, EURATOM Association, D-52425 J¨ulich, Germany
Received 1 August 2006; received in revised form 30 January 2007; accepted 30 January 2007 Available online 23 March 2007
Abstract The results of the experiments performed in quasistationary plasma accelerator (QSPA), simulating the type I ITER ELM heat loads on the tungsten divertor tiles, has been analysed. An analytical model has been proposed to explain the crack pattern with two characteristic crack depth of ∼500 m and of ∼50 m. It was assumed that the crack pattern observed in the experiment can be explained by the thermostress, generated at the interface between thin resolidified layer at the tungsten surface and the underlying tungsten bulk. Numerical simulation using the modified PEGASUS-3D code proved that this mechanism reproduces the experimentally observed crack pattern. © 2007 Published by Elsevier B.V. Keywords: Divertor armour materials; Tungsten; Cracking; ITER ELM heat load; Simulations
1. Introduction Tungsten is a reference material for the ITER divertor cover, except for the separatrix strike point armoured with carbon fibre composite (CFC) [1]. Besides, the problem of cleaning of the ITER vacuum chamber from tritium, accumulated in co-deposited T–C layers, is still in suspense. This fact motivates ∗ Corresponding author. Tel.: +49 7247823408; fax: +49 7247824874. E-mail address: [email protected] (S.E. Pestchanyi).
0920-3796/$ – see front matter © 2007 Published by Elsevier B.V. doi:10.1016/j.fusengdes.2007.01.028
investigation of alternative materials for the divertor armour. Tungsten is most probable material for CFC replacement as the divertor armour because of high vaporisation temperature and heat conductivity. This motivates the investigation of tungsten damage under severe heat loads at disruptions, edge localized modes (ELMs) and vertical displacement events. Experiments on tungsten heating in plasma guns and electron beam facilities have shown an intense crack formation at the heated surface and the cracks goes deep inside the irradiated sample. The reason for tungsten cracking under severe heat loads is thermostress, produced in heated sample due to temperature gradient close to
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the heated surface as well as due to surface melting, resolidification and cooling down. Temperature gradient generated under the heated surface due to fast heating and the faster heating rate the stronger the gradient. So cracking of tungsten cover is absent in stationary regime and occurs under off normal events only. Temperature gradient, perpendicular to the heated surface, causes different thermal expansion of tungsten layers and generates thermostress. Various defects existing in tungsten crystal lattice like dislocations and microcracks move and grow under influence of the thermostress. Movement and concentration of dislocations produces macrocracks. To have an idea of tungsten ability to produce cracks it is useful to estimate its thermophysical parameters. Tungsten melting point is Tm ∼ 3600 ◦ K and all its thermophysical properties are varying with temperature in rather wide range, but the characteristic values are the following. The Young’s modulus E ≈ 200–400 GPa, the shear modulus G ≈ 100–160 GPa [2], the linear expansion during heating from room temperature till the melting temperature is αT ≈ 2.10−2 , [3] surface tension μ ∼ 3 N/m and Poisson ratio σ ≈ 0.3. The tensile strength of policrystallic tungsten is of σ T ≤ 1 GPa at room temperature and at high temperatures drops at least one order of the magnitude see Ref. [2]. Under the ITER type I ELM action tungsten can melt and the melted layer thickness is δm ∼ 0–80 m depending on the ELM size [4,5]. The thermostress developed in tungsten sample during surface heating up to the melting temperature or during cooling down from the melting temperature to room temperature is: σ c = E(αT) ∼ 8–16 GPa, if one assumes fixed boundary conditions at the sample sides. This value is much higher than the tensile strength value. This means that during cooling down of the resolidified layer various cracks of different characteristic scales arises at the heated surface. Taking into account this huge potential of tungsten for cracking (the maximal value of tensile stress is at least 10–100 times higher than the tensile strength) one can expect the following mechanism for the cracks development. Tungsten armour is heated from the surface to the melting temperature. Inside the melted layer all the stresses existing in the sample before heating is relaxed. Resolidification of the layer after heating stop creates a stress-less layer at the melting temperature. Then, during cooling down of the resolidified layer the ten-
sile stress is developed in the layer. When the stress overreaches the tensile strength value, cracks may arise in the resolidified layer. Developing the crack at the surface lead to the thermostress relaxation in some vicinity of the crack, so the next crack can arise at a distance from the first crack only. As a result, a mesh of cracks with some characteristic distance between cracks should arise at the cooled tungsten surface. But, further cooling down of the sample increases thermostress inside each mesh of the primary crack pattern. This give rise to further cracking of the large scale meshes. In principle, repetition of this process can lead to development of fractal pattern of cracks at the sample surface with various characteristic mesh sizes and crack depths. For simulation of this process an analytic model for the crack formation and PEGASUS-3D code development for numeric simulation of tungsten brittle destruction has been performed and described in Section 2, an analysis of experimentally observed crack pattern and comparison with the numeric simulations are given in Section 3.
2. Analytical model and PEGASUS-3D simulation for cracks in tungsten The problem of equilibrium crack formation in elastic material has a significant singularity comparing with other problems of elasticity theory. From physical point of view, the crack is a cavity in elastic media arising only when an internal tensile stress exists in the medium. After the stress relief the crack is closed. The crack size and shape are considerably dependent on the distribution of stress inside the material. The fact of the mathematical singularity of the problem is that the boundary conditions are put at the unknown surface of the crack, to be determined as a result of the problem solution. Analytical treatment for the problem is possible [6] for two-dimensional crack in isotropic medium, being infinite in one direction and symmetrical relative to the crack centre. Due to the symmetry of the problem the crack going from the flat tungsten surface into the bulk material under the constant tensile stress acting in thin resolidified surface layer is equivalent to the crack in infinite medium arising under action of the doubled tensile stress in thin plane layer as is shown in Fig. 1. Using this symmetry the analytic solution [6] gives the
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Fig. 1. Equivalence of the crack shape arising from two forces F acting in opposite directions on the crack sides at the surface of isotropic elastic medium occupying lower hemi-space and of the crack inside the same medium occupying full space produced by the doubled force.
expression for the crack depth L: 2L =
F 2 (1 − σ 2 ) πμE
(1)
with F = EαT the force acting on a unit length of the crack due to the tensile stress existing in the resolidified layer. Only the cracks arising in tungsten due to the stress in the resolidified surface layer are considered in this paper. It is assumed that the sample is heated from the surface till the melting temperature. In a thin melt layer of a few micrometers all the stresses existing in the sample before melting are relaxed, so resolidification of the layer after heating stop creates a stress-less layer at the melting temperature. In course of cooling down of the sample the thermostress inside the resolidified layer at temperature T is of the order of F = F0 = Eα(Tm − T). After arising of primary cracks, when their depths became larger than the resolidified layer depth the crack edges are diverged, thus relaxing the thermostress inside the resolidified layer. The thermostress relaxation is determined by the crack width dcr at the sample surface and the mean distance Λ between the cracks: dcr F1 = E α(Tm − T ) − (2) Λ and the secondary cracks should have smaller length then the primary ones according to the Eq. (1) with the relaxed force Eq. (2).
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For more ITER relevant simulation of the cracks in tungsten, arising due to the thermostress in thin resolidified layer the thermo-mechanical code PEGASUS-3D has been used. The code PEGASUS-3D [7] has been developed for simulation of brittle destruction of fine grain graphite and CFCs under severe surface heat loads. It has proved an adequate modelling of thermostress and brittle destruction for complicated anisotropic structures [8,9]. The modifications of the code include implementation of tungsten thermophysical parameters, creation of numerical sample simulating tungsten sample with elongated grains perpendicular to the heated surface, seen in Fig. 2 and application of special heating scenario. According to the scenario the tungsten sample is heated from the surface with the constant heat flux depositing approximately 1 MJ/m2 during 0.5 ms. These parameters correspond to the heat load of the divertor targets during moderate type I ELM in ITER. PEGASUS code simulates heat conductivity in the tungsten sample, the thermostress generation and the numerical sample cracking at the sites where the thermostress exceeds the local tensile strength value. According to the sample heating scenario, the thermostress in thin layer of molten tungsten relaxes to zero. At the cooling down phase, molten tungsten resolidified with zero stress at the melting temperature. Then, in course of further
Fig. 2. The cross section of the tungsten sample after 100 shots in QSPA facility with heat load of 0.9 MJ/m2 and 0.5 ms time duration. Meandrous pale vertical lines are the boundaries separating elongated tungsten grains, perpendicular to the sample surface. Molten tungsten layer of 3–5 m thickness is seen at the irradiated surface. Bold dark lines are the cracks.
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sample cooling down the thermostress in bulk tungsten relaxes and the tensile thermostress in the resolidified layer is growing.
3. Simulation of tungsten damage with ITER ELMs The divertor target heat loads expected for the type I ELMs in ITER and disruptions are achievable using plasma guns. However, some important parameters of the guns, like magnetic field induction and plasma temperature differ from those of ITER off-normal events. Nevertheless, the results of the plasma gun experiments can be extrapolated to ITER by means of numerical simulations. The experiments were carried out in QSPA facility located in SRC RF TRINITI, Moscow on tungsten targets similar to those of the ITER divertor. The QSPA facility provides realistic heat loads (i.e., adequate pulse duration up to 0.5 ms and energy density up to 5 MJ/m2 ) to simulate the expected type I ELM loads in ITER. The results of the experiment have been reported in [10]. Analysis of the tungsten surface, performed after 100 plasma shots in QSPA facility has revealed that the molten tungsten layer of
3–5 m thickness developed at the surface irradiated with heat load of 0.9 MJ/m2 during 0.5 ms. The sample surface was covered by dense irregular grid of cracks going from the surface deep inside the bulk tungsten as is shown in Fig. 3. Metallographic analysis of the cracks, performed in Forschungszentrum Julich, has shown that the cracks have two different scales seen in Fig. 2. The primary cracks with characteristic depth of ∼500 m construct a coarse grid with characteristic mesh size of 1–2 mm. Each mesh of the primary grid on the tungsten surface, separated from the neighbouring primary meshes with large primary cracks, is in turn subdivided on smaller, secondary meshes of 100–200 m characteristic size by the smaller cracks of ∼50 m depth. As was already mentioned above, the only reason for crack formation in the tungsten sample under severe pulse heat load is thermostress. But the thermostress in tungsten sample under the heated surface can arise due to two different reasons: the first one is the temperature gradient and the second one is the interaction of thin molten and resolidified layer with unmolten tungsten bulk. The results of the experiment in QSPA facility [10] has shown that the cracks at the tungsten surface arise as due to the temperature gradient, when the sam-
Fig. 3. Tungsten tile surface after 100 plasma shots in QSPA facility irradiated with heat load of 0.9 MJ/m2 during 0.5 ms. The sample surface is covered by irregular grid of cracks going from the surface deep inside the bulk tungsten. The cracks have two different scales. The primary cracks at the left panel with characteristic depth of ∼500 m construct a coarse grid with characteristic mesh size of 1–2 mm. Each mesh of the primary grid, as is seen in the right panel, is in turn subdivided on secondary meshes of 100–200 m characteristic size by the cracks of ∼50 m depth.
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ple remains unmolten, as at higher heat loads when thin surface layer of tungsten was molten and resolidified. For the first simulation of the tungsten cracking the last case has been chosen. It seems that the tensile stress arising in the resolidified layer is more dangerous for the crack formation than the compressive stress from the temperature gradient close to the sample surface. Another reason for priority investigation of the cracks produced by the resolidified layer is rather complicated crack pattern having at least two very different length scales. First simulations of the cracks development using PEGASUS-3D code have been performed. For the simulation it has been assumed that the tungsten sample has no residual stress at initial temperature. Then, the heat conduction due to the surface heating was simulated in the sample simultaneously with the thermostress calculation. Heating of the sample leads to melting of a thin surface layer of a few micrometers thickness. The thermostress in the molten layer relaxed to zero and then, after stop of the surface heating the melt is resolidified. In our model it was assumed that the resolidified layer is stress-less at the solidification temperature Tm . The calculations of PEGASUS-3D code continue further and simulate cooling down of the sample, compressive thermostress relaxation in the unmolten bulk and ten-
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sile thermostress generation in the resolidified layer. When the local thermostress value exceeds the local tensile strength value an elementary crack is produced. Stress concentration at the crack vertex and the crack propagation simulated self-consistently in PEGASUS. The fitting parameters of the model, the mean tengr sile strength for tungsten grains, σT and for the grain b boundaries, σT have been selected to reproduce the experimentally observed cracks pattern. Calculations with the mean tensile strength for σTb = 0.5–1.5 GPa gr and σT /σTb = 3 has shown similar crack patterns with two different crack scales, the primary and the secondary ones. The characteristic primary crack mesh size increased with the σTb value. The view on the simulated crack pattern from the sample surface is shown in Fig. 4. The crack pattern at the depth of approximately 200 m is shown in the right hand side of the figure to separate the deep primary cracks from the secondary ones. These results qualitatively reproduce the experimentally observed pattern, seen in Fig. 3. Fig. 5 illustrates the similarity of the simulated and experimentally produced crack depths distribution. The panel (a) shows the cross section of the tungsten sample irradiated in QSPA facility. The primary cracks of the characteristic depth of ∼500 m and the mean distance between the cracks of 1–2 mm are seen in this panel.
Fig. 4. Result of the PEGASUS-3D simulation of the cracks developed in tungsten sample after irradiation with heat load of 0.9 MJ/m2 and 0.5 ms time duration. Primary cracks on depth of 200 m are shown in the left hand panel. Right hand panel illustrates both primary and secondary crack pattern at the sample surface.
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Fig. 5. Comparison of the cracks pattern in the perpendicular cross section of tungsten sample experimentally observed in the QSPA facility and simulated using PEGASUS-3D code. The cracks developed in tungsten sample after irradiation by 100 shots with heat load of 0.9 MJ/m2 and 0.5 ms time duration. Two primary cracks are shown in the panel (a). The panel (b) is the closer view of the sample showing the secondary cracks. The panel (c) illustrates the PEGASUS result with both primary and secondary cracks.
The panel (b) is the closer view of the sample cross section to illustrate the secondary cracks of ∼50 m mean depth and 100–200 m average distance between the cracks. The panel (c) of the Fig. 5 is the cross section of the numerical sample after simulation of QSPA shot. The heating parameters correspond to the experimental ones. The fact that the simulations adequately reproduce the characteristic features of the cracks pattern proves that the physics of the cracking process was described appropriately.
4. Conclusions The results of the experiments in the QSPA facility simulating the type I ITER ELM heat loads on the tungsten divertor tiles, has been analysed. The experiments have shown an intense tungsten surface cracking under the ITER ELM conditions at the divertor. Analysis of the tungsten thermophysical parameters revealed its large potential for cracking: the thermostress due to thermal expansion from room temperature to the melting temperature is more than one order of the mag-
nitude larger then the tungsten tensile strength. The cracks at the tungsten surface arise due to the thermostress, developed in the sample under severe heat loads characteristic for ITER ELMs and disruptions. The thermostress generated in tungsten sample under the heated surface due to two different reasons. The first one is the temperature gradient, which produces compressive stress at the surface. The second one is the tensile stress generated during cooling down of the resolidified layer if the surface has been melted. In this paper, the second mechanism of crack generation was investigated, because the tensile stress seems more dangerous for development of cracks. Cracking of tungsten under the compressive stress due to the temperature gradient without surface melting and associated tungsten embrittlement and thermoconductivity deterioration are subjects for further investigations. An analytical model for cracks formation in tungsten under the tensile stress in the resolidified layer has been proposed. The model links the crack depth with the characteristic size of the crack mesh at the tungsten surface. The model predicts development of cracks of several characteristic scales. The primary,
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rough mesh of cracks is developed during cooling of the sample to some intermediate temperature, higher than the final sample temperature. The primary cracks relax the thermostress inside the primary meshes, so the secondary cracks inside the primary meshes should have smaller depth than the primary ones and so on. At least two different characteristic sizes of cracks have been observed experimentally. An analytical model explains existence of the secondary fine grain crack grid by the thermostress relaxation due to the primary cracks in the resolidified layer. The thermo-mechanical code PEGASUS-3D has been developed for numerical simulation of tungsten surface cracking under the action of the thermostress arising in thin resolidified surface layer. Numerical simulation of the experiments with melting of thin surface layer of tungsten resulted in qualitative agreement between crack patterns obtained in the experiments with the pattern seen in the simulations. Both, experimentally observed and simulated crack grids consist of a coarse primary meshes formed of deep cracks and each coarse mesh is covered by shallower secondary crack grid with one order of the magnitude smaller depth.
Acknowledgements This work, supported by the European Communities under the contract of Association between EURATOM and Forschungszentrum Karlsruhe, was carried out within the framework of the European Fusion
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Development Agreement. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
References [1] G. Federici, P. Andrew, P. Barabaschi, J. Brooks, R. Doerner, A. Geier, et al., Key ITER plasma edge and plasmamaterial interaction issues, J. Nucl. Mater. 313-316 (2003) 11–22. [2] Gmelin Handbook of Inorganic and Organometallic Chemistry. W suppl., vol. A3, Springer-Verlag, New York, Heidelberg, 1992. [3] Y.S. Touloukian (Ed.), Thermophysical Properties of High Temperature Solid Materials, vol. 1, The Macmillan company, New York. [4] B.N. Bazylev, G. Janeschitz, I.S. Landman, S.E. Pestchanyi, Europhys. Conf. Abstr., vol. 27A, 2003, p. P-2.44. [5] B.N. Bazylev, G. Janeschitz, I.S. Landman, S.E. Pestchanyi, Erosion of tungsten armor after multiple intense transient events in ITER, J. Nucl. Mater. 337-339 (2005) 766–770. [6] L.D. Landau, E.M. Lifshitz, The Theory of Elasticity, Nauka, Moscow, 1965 (in Russian). [7] S. Pestchanyi, H. Wuerz, Brittle destruction of carbon based materials under off-normal ITER-FEAT conditions, Phys./Scr. T91 (2001) 84–89. [8] S. Pestchanyi, V. Safronov, I. Landman, Estimation of carbon fibre composites as ITER divertor armour, J. Nucl. Mater. 329–333 (2003) 697–701. [9] S. Pestchanyi, H. Wuerz, 3-D simulation of macroscopic erosion of CFC under ITER off-normal heat loads, Fusion Eng. Des. 66–68 (2003) 271–276. [10] A. Zhitlukhin, N. Klimov, I. Landman, J. Linke, A. Loarte, M. Merola, et al., Effects of ELMs and Disruptions on ITER Divertor Armour Materials an Invited Talk at the PSI-17 Conference, May 22–26, Hefei, China, 2006.
Simulation of cracks in tungsten under ITER specific transient heat loads S.E. Pestchanyi a,∗ , J. Linke b a
Forschungszentrum Karlsruhe, Institute for Pulsed Power and Microwave Technology, P.O. Box 3640, 76021 Karlsruhe, Germany b Forschungszentrum J¨ ulich GmbH, EURATOM Association, D-52425 J¨ulich, Germany
Received 1 August 2006; received in revised form 30 January 2007; accepted 30 January 2007 Available online 23 March 2007
Abstract The results of the experiments performed in quasistationary plasma accelerator (QSPA), simulating the type I ITER ELM heat loads on the tungsten divertor tiles, has been analysed. An analytical model has been proposed to explain the crack pattern with two characteristic crack depth of ∼500 m and of ∼50 m. It was assumed that the crack pattern observed in the experiment can be explained by the thermostress, generated at the interface between thin resolidified layer at the tungsten surface and the underlying tungsten bulk. Numerical simulation using the modified PEGASUS-3D code proved that this mechanism reproduces the experimentally observed crack pattern. © 2007 Published by Elsevier B.V. Keywords: Divertor armour materials; Tungsten; Cracking; ITER ELM heat load; Simulations
1. Introduction Tungsten is a reference material for the ITER divertor cover, except for the separatrix strike point armoured with carbon fibre composite (CFC) [1]. Besides, the problem of cleaning of the ITER vacuum chamber from tritium, accumulated in co-deposited T–C layers, is still in suspense. This fact motivates ∗ Corresponding author. Tel.: +49 7247823408; fax: +49 7247824874. E-mail address: [email protected] (S.E. Pestchanyi).
0920-3796/$ – see front matter © 2007 Published by Elsevier B.V. doi:10.1016/j.fusengdes.2007.01.028
investigation of alternative materials for the divertor armour. Tungsten is most probable material for CFC replacement as the divertor armour because of high vaporisation temperature and heat conductivity. This motivates the investigation of tungsten damage under severe heat loads at disruptions, edge localized modes (ELMs) and vertical displacement events. Experiments on tungsten heating in plasma guns and electron beam facilities have shown an intense crack formation at the heated surface and the cracks goes deep inside the irradiated sample. The reason for tungsten cracking under severe heat loads is thermostress, produced in heated sample due to temperature gradient close to
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the heated surface as well as due to surface melting, resolidification and cooling down. Temperature gradient generated under the heated surface due to fast heating and the faster heating rate the stronger the gradient. So cracking of tungsten cover is absent in stationary regime and occurs under off normal events only. Temperature gradient, perpendicular to the heated surface, causes different thermal expansion of tungsten layers and generates thermostress. Various defects existing in tungsten crystal lattice like dislocations and microcracks move and grow under influence of the thermostress. Movement and concentration of dislocations produces macrocracks. To have an idea of tungsten ability to produce cracks it is useful to estimate its thermophysical parameters. Tungsten melting point is Tm ∼ 3600 ◦ K and all its thermophysical properties are varying with temperature in rather wide range, but the characteristic values are the following. The Young’s modulus E ≈ 200–400 GPa, the shear modulus G ≈ 100–160 GPa [2], the linear expansion during heating from room temperature till the melting temperature is αT ≈ 2.10−2 , [3] surface tension μ ∼ 3 N/m and Poisson ratio σ ≈ 0.3. The tensile strength of policrystallic tungsten is of σ T ≤ 1 GPa at room temperature and at high temperatures drops at least one order of the magnitude see Ref. [2]. Under the ITER type I ELM action tungsten can melt and the melted layer thickness is δm ∼ 0–80 m depending on the ELM size [4,5]. The thermostress developed in tungsten sample during surface heating up to the melting temperature or during cooling down from the melting temperature to room temperature is: σ c = E(αT) ∼ 8–16 GPa, if one assumes fixed boundary conditions at the sample sides. This value is much higher than the tensile strength value. This means that during cooling down of the resolidified layer various cracks of different characteristic scales arises at the heated surface. Taking into account this huge potential of tungsten for cracking (the maximal value of tensile stress is at least 10–100 times higher than the tensile strength) one can expect the following mechanism for the cracks development. Tungsten armour is heated from the surface to the melting temperature. Inside the melted layer all the stresses existing in the sample before heating is relaxed. Resolidification of the layer after heating stop creates a stress-less layer at the melting temperature. Then, during cooling down of the resolidified layer the ten-
sile stress is developed in the layer. When the stress overreaches the tensile strength value, cracks may arise in the resolidified layer. Developing the crack at the surface lead to the thermostress relaxation in some vicinity of the crack, so the next crack can arise at a distance from the first crack only. As a result, a mesh of cracks with some characteristic distance between cracks should arise at the cooled tungsten surface. But, further cooling down of the sample increases thermostress inside each mesh of the primary crack pattern. This give rise to further cracking of the large scale meshes. In principle, repetition of this process can lead to development of fractal pattern of cracks at the sample surface with various characteristic mesh sizes and crack depths. For simulation of this process an analytic model for the crack formation and PEGASUS-3D code development for numeric simulation of tungsten brittle destruction has been performed and described in Section 2, an analysis of experimentally observed crack pattern and comparison with the numeric simulations are given in Section 3.
2. Analytical model and PEGASUS-3D simulation for cracks in tungsten The problem of equilibrium crack formation in elastic material has a significant singularity comparing with other problems of elasticity theory. From physical point of view, the crack is a cavity in elastic media arising only when an internal tensile stress exists in the medium. After the stress relief the crack is closed. The crack size and shape are considerably dependent on the distribution of stress inside the material. The fact of the mathematical singularity of the problem is that the boundary conditions are put at the unknown surface of the crack, to be determined as a result of the problem solution. Analytical treatment for the problem is possible [6] for two-dimensional crack in isotropic medium, being infinite in one direction and symmetrical relative to the crack centre. Due to the symmetry of the problem the crack going from the flat tungsten surface into the bulk material under the constant tensile stress acting in thin resolidified surface layer is equivalent to the crack in infinite medium arising under action of the doubled tensile stress in thin plane layer as is shown in Fig. 1. Using this symmetry the analytic solution [6] gives the
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Fig. 1. Equivalence of the crack shape arising from two forces F acting in opposite directions on the crack sides at the surface of isotropic elastic medium occupying lower hemi-space and of the crack inside the same medium occupying full space produced by the doubled force.
expression for the crack depth L: 2L =
F 2 (1 − σ 2 ) πμE
(1)
with F = EαT the force acting on a unit length of the crack due to the tensile stress existing in the resolidified layer. Only the cracks arising in tungsten due to the stress in the resolidified surface layer are considered in this paper. It is assumed that the sample is heated from the surface till the melting temperature. In a thin melt layer of a few micrometers all the stresses existing in the sample before melting are relaxed, so resolidification of the layer after heating stop creates a stress-less layer at the melting temperature. In course of cooling down of the sample the thermostress inside the resolidified layer at temperature T is of the order of F = F0 = Eα(Tm − T). After arising of primary cracks, when their depths became larger than the resolidified layer depth the crack edges are diverged, thus relaxing the thermostress inside the resolidified layer. The thermostress relaxation is determined by the crack width dcr at the sample surface and the mean distance Λ between the cracks: dcr F1 = E α(Tm − T ) − (2) Λ and the secondary cracks should have smaller length then the primary ones according to the Eq. (1) with the relaxed force Eq. (2).
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For more ITER relevant simulation of the cracks in tungsten, arising due to the thermostress in thin resolidified layer the thermo-mechanical code PEGASUS-3D has been used. The code PEGASUS-3D [7] has been developed for simulation of brittle destruction of fine grain graphite and CFCs under severe surface heat loads. It has proved an adequate modelling of thermostress and brittle destruction for complicated anisotropic structures [8,9]. The modifications of the code include implementation of tungsten thermophysical parameters, creation of numerical sample simulating tungsten sample with elongated grains perpendicular to the heated surface, seen in Fig. 2 and application of special heating scenario. According to the scenario the tungsten sample is heated from the surface with the constant heat flux depositing approximately 1 MJ/m2 during 0.5 ms. These parameters correspond to the heat load of the divertor targets during moderate type I ELM in ITER. PEGASUS code simulates heat conductivity in the tungsten sample, the thermostress generation and the numerical sample cracking at the sites where the thermostress exceeds the local tensile strength value. According to the sample heating scenario, the thermostress in thin layer of molten tungsten relaxes to zero. At the cooling down phase, molten tungsten resolidified with zero stress at the melting temperature. Then, in course of further
Fig. 2. The cross section of the tungsten sample after 100 shots in QSPA facility with heat load of 0.9 MJ/m2 and 0.5 ms time duration. Meandrous pale vertical lines are the boundaries separating elongated tungsten grains, perpendicular to the sample surface. Molten tungsten layer of 3–5 m thickness is seen at the irradiated surface. Bold dark lines are the cracks.
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sample cooling down the thermostress in bulk tungsten relaxes and the tensile thermostress in the resolidified layer is growing.
3. Simulation of tungsten damage with ITER ELMs The divertor target heat loads expected for the type I ELMs in ITER and disruptions are achievable using plasma guns. However, some important parameters of the guns, like magnetic field induction and plasma temperature differ from those of ITER off-normal events. Nevertheless, the results of the plasma gun experiments can be extrapolated to ITER by means of numerical simulations. The experiments were carried out in QSPA facility located in SRC RF TRINITI, Moscow on tungsten targets similar to those of the ITER divertor. The QSPA facility provides realistic heat loads (i.e., adequate pulse duration up to 0.5 ms and energy density up to 5 MJ/m2 ) to simulate the expected type I ELM loads in ITER. The results of the experiment have been reported in [10]. Analysis of the tungsten surface, performed after 100 plasma shots in QSPA facility has revealed that the molten tungsten layer of
3–5 m thickness developed at the surface irradiated with heat load of 0.9 MJ/m2 during 0.5 ms. The sample surface was covered by dense irregular grid of cracks going from the surface deep inside the bulk tungsten as is shown in Fig. 3. Metallographic analysis of the cracks, performed in Forschungszentrum Julich, has shown that the cracks have two different scales seen in Fig. 2. The primary cracks with characteristic depth of ∼500 m construct a coarse grid with characteristic mesh size of 1–2 mm. Each mesh of the primary grid on the tungsten surface, separated from the neighbouring primary meshes with large primary cracks, is in turn subdivided on smaller, secondary meshes of 100–200 m characteristic size by the smaller cracks of ∼50 m depth. As was already mentioned above, the only reason for crack formation in the tungsten sample under severe pulse heat load is thermostress. But the thermostress in tungsten sample under the heated surface can arise due to two different reasons: the first one is the temperature gradient and the second one is the interaction of thin molten and resolidified layer with unmolten tungsten bulk. The results of the experiment in QSPA facility [10] has shown that the cracks at the tungsten surface arise as due to the temperature gradient, when the sam-
Fig. 3. Tungsten tile surface after 100 plasma shots in QSPA facility irradiated with heat load of 0.9 MJ/m2 during 0.5 ms. The sample surface is covered by irregular grid of cracks going from the surface deep inside the bulk tungsten. The cracks have two different scales. The primary cracks at the left panel with characteristic depth of ∼500 m construct a coarse grid with characteristic mesh size of 1–2 mm. Each mesh of the primary grid, as is seen in the right panel, is in turn subdivided on secondary meshes of 100–200 m characteristic size by the cracks of ∼50 m depth.
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ple remains unmolten, as at higher heat loads when thin surface layer of tungsten was molten and resolidified. For the first simulation of the tungsten cracking the last case has been chosen. It seems that the tensile stress arising in the resolidified layer is more dangerous for the crack formation than the compressive stress from the temperature gradient close to the sample surface. Another reason for priority investigation of the cracks produced by the resolidified layer is rather complicated crack pattern having at least two very different length scales. First simulations of the cracks development using PEGASUS-3D code have been performed. For the simulation it has been assumed that the tungsten sample has no residual stress at initial temperature. Then, the heat conduction due to the surface heating was simulated in the sample simultaneously with the thermostress calculation. Heating of the sample leads to melting of a thin surface layer of a few micrometers thickness. The thermostress in the molten layer relaxed to zero and then, after stop of the surface heating the melt is resolidified. In our model it was assumed that the resolidified layer is stress-less at the solidification temperature Tm . The calculations of PEGASUS-3D code continue further and simulate cooling down of the sample, compressive thermostress relaxation in the unmolten bulk and ten-
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sile thermostress generation in the resolidified layer. When the local thermostress value exceeds the local tensile strength value an elementary crack is produced. Stress concentration at the crack vertex and the crack propagation simulated self-consistently in PEGASUS. The fitting parameters of the model, the mean tengr sile strength for tungsten grains, σT and for the grain b boundaries, σT have been selected to reproduce the experimentally observed cracks pattern. Calculations with the mean tensile strength for σTb = 0.5–1.5 GPa gr and σT /σTb = 3 has shown similar crack patterns with two different crack scales, the primary and the secondary ones. The characteristic primary crack mesh size increased with the σTb value. The view on the simulated crack pattern from the sample surface is shown in Fig. 4. The crack pattern at the depth of approximately 200 m is shown in the right hand side of the figure to separate the deep primary cracks from the secondary ones. These results qualitatively reproduce the experimentally observed pattern, seen in Fig. 3. Fig. 5 illustrates the similarity of the simulated and experimentally produced crack depths distribution. The panel (a) shows the cross section of the tungsten sample irradiated in QSPA facility. The primary cracks of the characteristic depth of ∼500 m and the mean distance between the cracks of 1–2 mm are seen in this panel.
Fig. 4. Result of the PEGASUS-3D simulation of the cracks developed in tungsten sample after irradiation with heat load of 0.9 MJ/m2 and 0.5 ms time duration. Primary cracks on depth of 200 m are shown in the left hand panel. Right hand panel illustrates both primary and secondary crack pattern at the sample surface.
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Fig. 5. Comparison of the cracks pattern in the perpendicular cross section of tungsten sample experimentally observed in the QSPA facility and simulated using PEGASUS-3D code. The cracks developed in tungsten sample after irradiation by 100 shots with heat load of 0.9 MJ/m2 and 0.5 ms time duration. Two primary cracks are shown in the panel (a). The panel (b) is the closer view of the sample showing the secondary cracks. The panel (c) illustrates the PEGASUS result with both primary and secondary cracks.
The panel (b) is the closer view of the sample cross section to illustrate the secondary cracks of ∼50 m mean depth and 100–200 m average distance between the cracks. The panel (c) of the Fig. 5 is the cross section of the numerical sample after simulation of QSPA shot. The heating parameters correspond to the experimental ones. The fact that the simulations adequately reproduce the characteristic features of the cracks pattern proves that the physics of the cracking process was described appropriately.
4. Conclusions The results of the experiments in the QSPA facility simulating the type I ITER ELM heat loads on the tungsten divertor tiles, has been analysed. The experiments have shown an intense tungsten surface cracking under the ITER ELM conditions at the divertor. Analysis of the tungsten thermophysical parameters revealed its large potential for cracking: the thermostress due to thermal expansion from room temperature to the melting temperature is more than one order of the mag-
nitude larger then the tungsten tensile strength. The cracks at the tungsten surface arise due to the thermostress, developed in the sample under severe heat loads characteristic for ITER ELMs and disruptions. The thermostress generated in tungsten sample under the heated surface due to two different reasons. The first one is the temperature gradient, which produces compressive stress at the surface. The second one is the tensile stress generated during cooling down of the resolidified layer if the surface has been melted. In this paper, the second mechanism of crack generation was investigated, because the tensile stress seems more dangerous for development of cracks. Cracking of tungsten under the compressive stress due to the temperature gradient without surface melting and associated tungsten embrittlement and thermoconductivity deterioration are subjects for further investigations. An analytical model for cracks formation in tungsten under the tensile stress in the resolidified layer has been proposed. The model links the crack depth with the characteristic size of the crack mesh at the tungsten surface. The model predicts development of cracks of several characteristic scales. The primary,
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rough mesh of cracks is developed during cooling of the sample to some intermediate temperature, higher than the final sample temperature. The primary cracks relax the thermostress inside the primary meshes, so the secondary cracks inside the primary meshes should have smaller depth than the primary ones and so on. At least two different characteristic sizes of cracks have been observed experimentally. An analytical model explains existence of the secondary fine grain crack grid by the thermostress relaxation due to the primary cracks in the resolidified layer. The thermo-mechanical code PEGASUS-3D has been developed for numerical simulation of tungsten surface cracking under the action of the thermostress arising in thin resolidified surface layer. Numerical simulation of the experiments with melting of thin surface layer of tungsten resulted in qualitative agreement between crack patterns obtained in the experiments with the pattern seen in the simulations. Both, experimentally observed and simulated crack grids consist of a coarse primary meshes formed of deep cracks and each coarse mesh is covered by shallower secondary crack grid with one order of the magnitude smaller depth.
Acknowledgements This work, supported by the European Communities under the contract of Association between EURATOM and Forschungszentrum Karlsruhe, was carried out within the framework of the European Fusion
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Development Agreement. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
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