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EUROFER97 joints
Engineering Fracture Mechanics 100 (2013) 63–75
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Considering brittleness of tungsten in failure analysis of helium-cooled divertor components with functionally graded tungsten/EUROFER97 joints q T. Weber ⇑, M. Härtelt, J. Aktaa Karlsruhe Institute of Technology, Institute for Applied Materials, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany
a r t i c l e
i n f o
Keywords: Functionally graded material Tungsten EUROFER97 Finite element analysis Failure assessment Weibull statistics
a b s t r a c t Helium-cooled divertor components made of tungsten and reduced activated ferritic martensitic steel are part of current research for future power plants. Two different failure analyses, a deterministic and a probabilistic, have been performed on the tungsten section of the considered thimble of the divertor component which is joined to a steel cartridge by a functionally graded tungsten/EUROFER97 layer. In the deterministic failure analysis, the extended finite element method (XFEM) is used providing insights concerning the initiation and the propagation of possible cracks, which depends on the layer thickness. Due to the brittleness of tungsten at low temperatures and its embrittlement in view of neutron irradiation a probabilistic failure analysis is necessary to assess the reliability of the divertor component. Failure probabilities for the tungsten thimble were calculated based on Weibull statistics using the STAU code. The influence of the thickness and inclination angle of functionally graded tungsten/EUROFER97 joints on the failure probabilities were studied and revealed a minimum required layer thickness of 3000 lm and the necessity of an inclination. The Weibull parameters for a tungsten alloy with 1 wt.% La2O3 were determined in four point bending tests and used as input data for the failure analysis of the component during production. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Divertor components of future fusion power plants will be subjected to concentrated flux of plasma particles resulting in very high surface erosion and heat loads. In the helium cooled divertor concept [1] considered here, tungsten and its alloys are selected as refractory as well as structural materials. Due to the brittleness of tungsten at low temperatures, its use as a structural material is limited to the high temperature region (>650 °C) of the component and a joint to another structural material, the ferritic martensitic high chromium steel EUROFER97 [2,3], is necessary. However, the remarkable difference in thermal expansion between tungsten (ath = 4.4e6 K1 [4]) and EUROFER97 (ath = 12.7e6 K1 [5]) causes thermal mismatch between both materials resulting in stresses which would yield failure of the joint. One strategy to reduce thermally induced stresses is to introduce a functionally graded layer between the materials to be jointed. The joint between both structural materials is located in the thimble of the divertor component. The effect of a functionally graded layer on component stresses in this issue was recently investigated by elasto-viscoplastic finite element (FE) simulations [6]. Considering application relevant loadings, the effects of variation of the thickness and
q
In Honor of Prof. Dr. Dietrich Munz, Engng. Fract. Mech. (2012).
⇑ Corresponding author.
E-mail address: [email protected] (T. Weber). 0013-7944/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2012.07.024
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T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
Nomenclature PF,V PF,A V A X
failure probability of a single thimble based on volume flaws failure probability of a single thimble based on surface flaws component volume component surface flaw orientation within volume a flaw orientation with respect to surface req equivalent stress x flaw location a flaw length Veff effective volume Aeff effective surface V0 unit volume unit surface A0 r0 normalised strength with respect to the unit volume V0/unit surface A0 rref reference stress characterising the load level YI geometric correction factor corresponding to fracture mode I equivalent mode I stress intensity factor KI eq KIc mode I fracture toughness ac critical crack size m Weibull shape parameter b Weibull scale parameter N amount of tested samples Fj assigned failure probability of specimen with rank j rj bending stress of specimen with rank j j rank of the considered bending specimen Ra arithmetic average of surface roughness Rq root mean squared surface roughness maximum height of the profile Rt Rz average distance between the highest peak and lowest valley in each sampling length ath thermal expansion coefficient f field variable indicating the gradation level i index of time increment l viscosity parameter in XFEM-simulation D height of the fine meshed region U cohesive energy JIc energy release rate for crack propagation rD threshold value for damage initiation dn, sn, Kn separation/traction/stiffness in normal direction separation in shear direction ds, dt ss, st traction in shear direction Ks, Kt stiffness in shear direction d damage of the enriched element t thickness of the enriched element dm mixed-mode relative displacement d0m
mixed-mode relative displacement corresponding to damage initiation
dfm
mixed-mode relative displacement corresponding to complete failure
dmax m E # n Pr
maximum value of mixed-mode relative displacement during loading history Young’s modulus Poisson’s ratio number of thimbles total reliability of all thimbles
the transition function within the graded layer was analysed. However, the failure of tungsten has to be accounted by further analyses, amongst others by a probabilistic failure analysis, which take the brittleness of tungsten into account. Failure of brittle materials [7] is governed by inherent flaws which can be pores, inclusions or machining cracks. A component fails once a critical flaw size, which depends on the stress state and crack model, is exceeded (weakest-link approach). The characteristic scatter in strength of brittle components is a consequence of the fact that the critical flaw size scatters. Furthermore, the strength decreases with increasing component volume (or surface) since it is more likely to find a catastrophic flaw in a big component. Based on the Weibull theory, the failure probability of components under complex
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T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
multi-axial stress states can be calculated using numerical tools (e.g. STAU [8]) which act as a post-processor of a FE-analysis. The effect of the brittle failure mode of tungsten on component design has rarely been studied for fusion applications. You and Komarova [9] presented a probabilistic analysis for a water-cooled divertor based on strength data for tungsten undergoing different heat treatments. They found that the failure probability of the considered component is acceptable but also stated that irradiation embrittlement (change in flaw size scatter) can be critical for the considered component. However, a systematic design optimisation procedure based on probabilistic methods has not been yet reported for fusion components. We will present a design study for the tungsten-EUROFER97 joint of a helium-cooled divertor component. For that purpose, the effect of a variation of the thickness of the graded layer under application relevant loading will be studied on the base of FE stress analyses, which show in the tungsten part highly inhomogeneous stress fields with localised high stresses [6]. For the assessment of failure caused by these stress fields failure criteria evaluating only local stress values are at best sufficient to predict crack initiation. As tungsten is not perfectly brittle it is not self-evident that even when a crack is initiated, it would propagate within the highly inhomogeneous stress field. Therefore, in a first approach, the stress fields appearing in the tungsten part are evaluated by investigation of their capabilities in promoting crack initiation and propagation using XFEM, as it is provided since version 6.9-1 of the commercial finite element code ABAQUS [10]. The failure probability is calculated with the STAU code in order to evaluate the joint design. The relevant material properties for the probabilistic analysis are obtained from strength measurements for a sintered tungsten grade. 2. FEM model The finite element code ABAQUS is used to perform the simulations. The simplified FE model of the divertor component considered consists of an axis symmetric shell with a thickness of 1 mm, see Fig. 1a. The thickness of the graded layer situated in the joining section is varied between 10 and 3000 lm. Therefore, a field variable f ranging from 0 to 1 is used to indicate level of gradation with f = 0 on the tungsten side and f = 1 on the steel side. Inside the joining section, the material properties are interpolated linearly between 0 and 1. Hence, the thickness of the graded layer is controlled by the field variable f. Since the graded layer thickness ranges over two orders of magnitude it became necessary to use adequate meshes for each layer thickness. Thus, six meshes were used for all scenarios with D e {20 lm, 200 lm, 500 lm, 1 mm, 2 mm, 3 mm}, see Fig. 1b. They are designed in such a way that the highest strains and stresses always lie inside the densest elemental region of the mesh, which consists of rectangular horizontal elements with a size of 20 lm 10 lm stapled in y-direction. Assuming a hot joining process, the initial temperature of the whole FE model is set to 800 °C/900 °C/1000 °C, at which it is considered to be stress free. Thereafter, the FE model is loaded by varying the temperature homogeneously with a cooling down to 0 °C. The temperature dependent Young’s modulus, yield strength and thermal expansion coefficient for the EUROFER97 and joining section are adopted from Ref. [6]. For the tungsten section the removal of plasticity below the ductile to brittle transition temperature (DBTT) [11] is necessary (Fig. 2). Exceeding the yield strength in an ideal-elasto-plastic simulation means for brittle materials the beginning of spontaneous cracking. Limiting stresses by setting yield strengths and consequently simulating plastic deformation, which is actually negligibly small for tungsten below the DBTT, would be inappropriate. So although the simulated stress values might be higher than they occur in reality, since cracking occur first, they
(b)
(a) 7.5 mm
tungsten section
10.
EUROFER97 section
4.
200μm
EUROFER97
Δ
x
joining section
y
100μm
Tungsten
6.5
9.
1. Fig. 1. (a) Sketch of the finite element model (dimensions in mm) and (b) cut-out of the mesh with D = 20 lm.
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T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
(b)
(a)
Fig. 2. (a) Yield strength and (b) Young’s modulus used in the FEM model.
are rated as poor in the deterministic and probabilistic failure analysis. The highest probability of brittle fracture in tungsten is expected at temperatures below DBTT and when stresses are at maximum. Hence, only the stress state at 0 °C is considered for the failure analysis.
3. Failure assessment procedures 3.1. Deterministic approach In addition to the stress analysis using the finite element model described above, crack initiation and crack propagation are investigated using the extended finite element method. Crack initiation and propagation can be modelled in ABAQUS in accordance with the cohesive zone model (CZM) by using ‘cohesive elements’. The concept of the CZM, first conceived by Dugdale [12] and Barenblatt [13] regards fracture as a gradual phenomenon in which separation takes place between two adjacent virtual surfaces across an extended crack tip (cohesive zone) and is resisted by the presence of cohesive forces. This is in contrast to linear elastic and elastic–plastic fracture mechanics, where an infinite sharp crack tip is assumed. The description of the failure behaviour for a given material is defined by traction-separation law. As the cohesive surfaces separate, the traction first increases until a maximum is reached, and subsequently the traction decreases to zero, which results in complete separation. The area under the traction separation curve corresponds to the cohesive energy U needed for separation. Considering the same traction-separation law the XFEM allows the simulation of crack initiation and crack propagation by using ‘enriched elements’ instead of ‘cohesive elements’. The XFEM is implemented since version 6.9-1 of ABAQUS and provides some advantages. It alleviates the shortcomings associated with meshing crack surfaces as was first introduced by Belytschko and Black [14]. Furthermore, enriched elements allow the simulation of crack propagation along an arbitrary, solution-dependent path, since the crack propagation is not tied to the element boundaries in a mesh [10]. The traction separation model used in this work, described with more details in Appendix A, assumes initially linear elastic behaviour followed by the initiation and evolution of damage. Damage is assumed to initiate when the maximum principal stress exceeds a threshold value, rD. Damage evolution follows linearly as soon as the damage initiation criterion is reached, reducing the cohesive stiffness of the enriched element. When the stiffness of the enriched element reaches at an integration point a value of zero crack initiation or, in case of an already existing crack, crack propagation appears with a length equal to the length represented by the integration point. The energy released during this process is specified by the cohesive energy U, which is provided as input parameter. Since the cohesive energy U is equal to the energy release rate for N is used for U. This value is obtained by applying the relation between JIc and crack propagation JIc, the value J Ic ¼ 0:1162 mm the fracture toughness KIc for elastic media and considering the value of KIc = 7.19 MPa m½, which has been experimentally measured for commercially available tungsten [15]. As tungsten shows strong anisotropy in its fracture behaviour, the considered KIc value here corresponds to a measurement at room temperature and to the most unfavourable orientation. Hence, it is a conservative value, which can be assumed as minimum value for tungsten fabricated by today’s methods. Material models exhibiting softening behaviour and stiffness degradation often lead to severe convergence difficulties. Thus the viscous regularization method was used to resolve this issue. The regularization process is described in [10] and can be controlled by the viscosity parameter l, which represents the relaxation time of the viscous system. For the analysis in Section 4.1, the time increments and the viscosity parameter l have been chosen sufficient small so that the results are not compromised by the viscous regularization. Interfacial fracture is not considered in this XFEM analysis, although a weak bond interface may affect the failure feature completely. Usually the interfacial bond strength depends very strongly on the fabrication method, intermetallic phases, surface roughness, etc. However, in this a work an ideally bonded interface is assumed.
T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
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3.2. Probabilistic approach The Weibull theory is based on a weakest link approach assuming that a component fails once one flaw exceeds a critical size ac. The critical size depends on the load, the location and the orientation of a crack meaning that the most unfavourable flaw will cause failure. Flaws are modelled as planar cracks and a multi-axial criterion is used as a fracture mechanics failure criterion under mixed-mode loading:
pffiffiffi K Ieq ¼ req ðx; XÞY i a:
ð3:1Þ
KIeq is the equivalent mode I stress intensity factor and req is the equivalent stress which depends on the location x and orientation X of the crack. YI is the geometry factor of the corresponding crack model. Failure occurs once KIeq exceeds the fracture toughness KIc of the material and, hence, ac can be expressed in terms of the equivalent stress req based on Eq. (3.1). For an arbitrary number of equally distributed and orientated flaws, the failure probability for volume flaws can be calculated by solving the integral in Eq. (3.2), which was derived in different ways in [16–18] and reviewed for instance in [8].
Z Z 1 1 req ðx; XÞ m PF;V ¼ 1 exp dXdV V 0 V 4p X ro
ð3:2Þ
The integration is performed over the component volume V and possible flaw orientation X. The equivalent stress req at location x and orientation X relative to the global coordinate system is mode I equivalent stress based on a fracture mechanics mixed-mode criterion [16,19] and depends on the location as well as on the orientation of the crack. A number of mixedmode criteria are implemented in STAU. If, for example, failure is governed solely by mode I loading, the equivalent stress is equal to the normal stress with respect to the crack plane (normal stress criterion) [7]. It was shown that the normal stress criterion can be applied for modelling the failure behaviour of natural flaws in ceramics [19]. Therefore, the normal stress criterion will also be used for the calculations of tungsten in this work, assuming that the failure behaviour is determined by pure mode I loading. The material specific parameters m and r0 must be determined from strength tests. r0 can be regarded as a normalised strength with respect to the unit volume V0 and can be extracted from the size-dependent Weibull scale parameter b obtained for arbitrary specimen geometries. The calculation of the failure probability in Eq. (3.2) requires solving an integral over 5 dimensions (three spatial directions and two orientations angles) numerically. STAU uses Gaussian quadrature in order to perform the integration. In the course of the integration procedure, a number of supporting points are generated by interpolating the stresses using the element form functions. Thus, accurate results can be achieved. Using Eq. (3.2), no increase in crack resistance is assumed, which is unknown for the tungsten materials considered. Eq. (3.2) can be reformulated in order to obtain a Weibull distribution for a reference stress rref with the Weibull parameters m and b [20]:
rref m : PF;V ¼ 1 exp b
ð3:3Þ
The reference stress can be for example the outer fibre stress in a 4-point bending test. In Eq. (3.3), b depends on the specimen size (geometry) and the type of loading. The normalised strength r0 is related to b by the effective volume Veff:
r0 ¼ b V eff ¼
1 V eff m Vo
Z V
1 4p
ð3:4Þ
Z X
m
req ðx; XÞ rref
dXdV
ð3:5Þ
The effective volume can be calculated by STAU for a given reference stress and a corresponding stress distribution from a FE analysis. This allows for determining r0 from the Weibull parameters m and b which can be measured in a strength test. Eq. (3.4) implies that r0 depends on whether volume or surface flaws are considered and on the mixed-mode criterion used for calculating req. The corresponding formulations for surface flaws can be found in Appendix B. To obtain the required Weibull parameters m and r0 for the failure analysis, four-point bending tests were performed at room temperature on rolled bar material made of sintered WL10 from the manufacturer Plansee AG, Austria. WL10 is a tungsten alloy with 1 wt.% La2O3 having an improved creep resistance and machinability. In the performed FEM analysis, WL10 and pure tungsten are not distinguished, since their material properties are assumed to be identical for this kind of analysis. Thus, although the considered thimble is actually planned to be manufactured out of WL10, the more familiar term tungsten is used when the results and statements are of a general manner and apply to both materials. Otherwise it is specified by WL10. However, the required Weibull parameters used in the failure analysis are fairly different for WL10 and pure tungsten. The test setup and specimen geometry were chosen in compliance with the standard DIN EN 843. While the size of the bending specimen was 3 4 45 mm, the distance of the inner support was 20 mm and 40 mm for the outer ones. The long edges of the specimen are parallel to the rotational symmetry axis of the source bar material, so that the tensile stresses in the bending test have the same direction as the max. principal stresses in the considered use case of the divertor component. The conformity of the load direction in the bending test and in the divertor application case is important in consideration of the strong anisotropy of the fracture toughness [21]. N = 40 specimens were prepared using wire-spark-erosion
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T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
ln (ln(1/(1−Fj)))
4
2
WL10 − measured data WL10 − Weibull WL10 − upper 90% conf. interval WL10 − lower 90% conf. interval W − as received [9] W − annealed [9]
0
−2
−4 6.6
6.8
7
7.2
7.4
7.6
7.8
8
ln (σj / MPa) Fig. 3. Weibull-parameters for WL10 (measured in this work) and tungsten (determined in [9]).
cutting method followed by a surface grinding step and a slight bevelling of the edges. Surface roughness was measured by a Tencor P-10 Surface Profiler, yielding the following maximum values out of four measurements: Ra = 556 nm, Rq = 739 nm, Rt = 4887 nm and Rz = 3223 nm. The strength data measured for WL10 is presented in the Weibull plot in Fig. 3, showing that the distribution obeys a 2-parameters Weibull distribution. In this diagram rj is the bending stress at rupture of the specimen with rank j and Fj is the assigned failure probability calculated as:
Fj ¼
j 0:5 N
ð3:6Þ
The rank j of the specimens is calculated by sorting their bending stress at rupture in ascending order and enumerating from 1 to N. The estimated Weibull parameters are m = 12.8 (shape parameter) and b = 1583 MPa (scale parameter). The Weibull parameters for warm rolled pure tungsten sheets before and after heat treatment at 1000 °C for 10 h have already been published by You and Komarova [9]. Higher bending strengths and a smaller scattering of pure tungsten can be noticed compared to WL10. This may be explained by the lanthanum oxide particles, which act as cracking sources. A second reason might be the influence of the surface morphology [22]. Higher shape and scale parameters are measured when decreasing surface roughness.
4. Results and discussion 4.1. Deterministic approach It has already been found in a prior study on graded joints [6] that the most critical region for the tungsten section is at the lower right corner, near the outer surface and to the bonding interface of the joining layer. The stresses are strongly concentrated in these domains, being a consequence of the thermal mismatch combined with the interfacial stress singularity effect. Fig. 4 shows the maximum von Mises stress occurring in the tungsten section for both cases, with and without plasticity for tungsten below 400 °C after the cool down from 1000 °C to 0 °C. It shows that only layers thicker than 1500 lm will not deform plastically after the cool down. This circumstance indicates that either optimised design geometry of the joint should be aspired or a stress-free state at a lower temperature must be taken into account. The stress free state at 1000 °C, as considered here, corresponds to the temperature at which the joint EUROFER97/graded layer/tungsten (or WL10) is realised, for instance by diffusion bonding of EUROFER97 [23,24] or brazing of tungsten with the aid of an interlayer [25]. As the selected damage initiation criterion is based on the maximum principal stress, no cracking and separation of the enriched elements occur as long as the threshold rD is greater than the highest value of the maximum principal stress occurring in the tungsten section rmax. Within a parameter study, rD has been varied along the different layer thicknesses, while N is kept constant. For rD values right below rmax, spontaneous continuous cracking is obthe cohesive energy U ¼ 0:1162 mm served, meaning that once cracking has started, it will not be stopped by cohesive forces, although energy is dissipated. To illustrate this behaviour, Fig. 5 shows the damaging of the enriched elements when cooling down a joint with a 2 mm thick linear graded layer without inclination from 800 °C to 0 °C. While rD is varied from 766 MPa to 764 MPa, representing a difference smaller than uncertainties in the assumptions made in the FE analysis, the result spans from no cracking at all to complete propagation of the crack (see Fig. 5a–c). Fig. 6 shows the damage evolution after a different number of time
T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
maximum von Mises stress in MPa
2200
69
yield stress is limited below 400 o C o yield stress is unlimited below 400 C
2000 1800 1600 1400 1200 1000
00 2000 15 00 10
0 50
0 20
0 10
50
20
10
layer thickness in μm Fig. 4. Maximum von Mises stress in the tungsten section plotted over layer thickness for a linearly graded joint after cooling down from 1000 °C to 0 °C.
damage initiation
(a) σ D = 766 MPa
(b) σ D = 765 MPa
(c) σ D = 764 MPa 100 µm Fig. 5. Influence of rD on damage initiation in a joint with 2000 lm thick linear graded layer without inclination when cooling down from 800 °C to 0 °C: (a) no damage, (b) one element with partial damage, and (c) crack has propagated largely through the wall thickness.
damage evolution
(a) i = 12
(b) i = 100
(c) i = 200 100 µm Fig. 6. Damage evolution and crack propagation in a joint with 2000 lm thick linear graded layer without inclination when cooling down from 800 °C to 0 °C with rD = 764 MPa: (a–c) represent the states after the number of time increments i.
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T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
fieldvariable i bl f
(a)
Tungsten
(b)
(c)
EUROFER97 200 µm Fig. 7. Path of the crack in the tungsten area for three different layer thicknesses: (a) 20 lm, (b) 200 lm and (c) 2000 lm.
increments. The time increment is not constant and the simulated temperature decrease is less than 0.5 °C for the three images. In Fig. 6a the damage initiation criterion has been fulfilled along a depth of two enriched elements (green).1 Before complete separation (red) comes up, the crack tip is already driven far into the material within a very short time (see Fig. 6b and c). Accordingly, when cracking is initiated, unstable crack propagation is expected which is mainly caused by the brittleness and low fracture toughness (cohesive energy) of tungsten. For thin layers, the maximum principal stress is concentrated on a small spot in the tungsten region at the outer side of the thimble, so that cracking starts most likely at this point (see Fig. 7). Thicker layers show a more distributed maximum stress area, so that cracking might occur at different places. The crack path starts perpendicular to the outer surface and is slightly deflected upwards. Apart from the exact starting position this crack propagation behaviour is similar for all layer thicknesses. However, to observe cracking in the tungsten region for thicker layers, the rD value has to be reduced to values much lower than the mean bending strength measured for W and WL10 (ref. to Section 3.2). Due to the probabilistic nature of strength for tungsten as a brittle material this does not mean that tungsten cracking and failure in joints with thicker layers can be excluded. However, lower failure probability might be expected. For its evaluation, a probabilistic approach is used which accounts for failure probability by considering inherent flaws and their statistically distributed size and orientation.
4.2. Probabilistic approach 4.2.1. Failure probability during cooling down The failure probability at the end of the cooling step was calculated based on surface and volume flaws for different layer thicknesses as can be seen in Fig. 8. It shows that the failure probability is higher if surface flaws are considered. Indeed, this result corresponds to the fact that stresses are concentrated at the outer surface of the tungsten section. Particularly, very thin (<50 lm) linear graded layers exhibit a pronounced concentration of stresses at the outer surface, so that the effective volume is rather low compared to the effective volume of the tested bending specimen. As a result, in spite of high maximum stresses (Fig. 4) being as high as the Weibull parameter b, very thin graded layers obtain relatively low failure probabilities (less than 10%). Linearly graded joints with a thickness in the range between 50 and 500 lm are more critical. In addition to a large effective volume, the interlayer in this simulation comes up with higher yield strength than pure EUROFER97 leading to elevated stresses in the tungsten section. Very thick layers (>500 lm) can compensate this circumstance, so that smaller failure probabilities can be achieved compared to a direct joint. The great difference of the Weibull parameters between pure tungsten [9] and WL10 strongly influences the calculated failure probabilities. In order to estimate the influence of the Weibull-parameters, a comparison of the failure probabilities resulting from the experimentally determined parameters for pure tungsten and the ones for WL10 has been done. The results are presented in Fig. 9. In spite of the strongly reduced failure probabilities of a pure tungsten thimble, the use of WL10 is still very important at the head of the thimble where temperatures above 1100 °C are expected in combination with stresses around 300 MPa [26]. Considering these two materials only WL10 is able to sufficiently withstand creep under these conditions. One possibility for the reduction of strains and stresses is the inclination of the graded layer along the joint as shown in [6]. For that purpose, STAU was used to analyse the degree to which the inclination angle can reduce the failure probability of the tungsten section. The results, shown in Fig. 10, reveal that the surface failure probability can be decreased by an order of magnitude, to a minimum of 1.831e5, when choosing a negative inclination angle between 30° and 40°. A negative inclination angle means that the layer is tilted towards the outer surface. A second possibility to reduce the occurring stresses is to lower the initial temperature. Fig. 11 shows the resulting surface failure probability for three different initial temperatures. We performed preliminary diffusion bonding experiments 1
For interpretation of color in Fig. 6 , the reader is referred to the web version of this article.
T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
71
0.1
failure probability
0.01
0.001
0.0001 surface volume
1e−005
00 20 00 15 00 10
0 50
0 20
0 10
50
20
10
layer thickness in microns Fig. 8. Failure probability of the tungsten section after the cool down from 1000 °C to 0 °C.
1
surface failure probability
0.01
0.0001
1e−006
1e−008
1e−010
WL10 W − as received [9] W − annealed [9]
1e−012
00 20 0 0 15 00 10
0
50
0
20
0
10
50
20
10
layer thickness in microns Fig. 9. Material influence on surface failure probability of the tungsten section after the cool down without inclined linear graded layer and 1000 °C starting-temperature.
and proved that is possible to join EUROFER97 at 800 °C [27]. It shows that the surface failure probability can be reduced by more than one order of magnitude to 1.064e5 for 2 mm thick linear graded layers without inclination. A third option for a further reduction of the failure probability is the use of thicker interlayers. In the publication [28], the possibility to deposit tungsten coatings thicker than 2 mm by plasma spraying are postulated and in [29] functionally graded layers of W/Cu are produced with a thickness of 2–3 mm. Considering other processes, for example resistance sintering under ultra high pressure, tungsten/EUROFER97 composites of several millimetres size in each dimension can be produced easily [30]. The combination of all three improvement measures (thickness, initial temperature and inclination) leads to very low surface failure probabilities in the tungsten section, as can be seen in Fig. 12, so that a minimum surface failure probability of 1.676e7 can be achieved.
4.2.2. Influence of irradiation and operational loadings on the failure probability Neutron irradiation during operation time of a future fusion reactor leads to embrittlement of tungsten and change in the Weibull parameters. This embrittlement must be considered in a reliability analysis of the components. However, since Weibull parameters of irradiated tungsten have not yet been determined, this issue is only discussed qualitatively. The conditions depend upon the design of a Tokamak fusion reactor thermal cycling schedule. Thus, dwell time periods at high temperatures and downtimes at room temperature have to be considered. A brittle failure of tungsten during the dwell time period is less likely due to the reduced stresses, the activation of slip systems and the increase in ductility, which comes along at higher temperatures. But still of interest is the failure probability after shut downs of the fusion reactor. The
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T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
(b)
0.1
10°inclination
0.01
0.001
0.0001
1e−005 -40° -30° -20° -10°
outer surface
inner surface
surface failure probability
(a)
20μm 100μm 500μm 2000μm 0°
10°
20°
30°
tungsten
40°
inclination angle in degrees
EUROFER97
Fig. 10. (a) Surface failure probability of the tungsten section after cooling down from 1000 °C to 0 °C with inclined linear graded layer and (b) example showing a 10° inclination angle.
surface failure probability
0.1
0.01
0.001
0.0001 1000 o C 900 o C 800 o C
1e−005
00 20 0 0 15
00
10
0
50
0
20
0
10
50
20
10
layer thickness in microns Fig. 11. Surface failure probability of the tungsten section after cooling down starting from different equilibrium temperatures (without inclination of the interlayer).
0.01
failure probability
0.001
0.0001
1e−005
1e−006
1e−007 surface volume
1e−008
00 30 00 20 0 0 15 00
10
0
0
50
20
0
10
50
20
10
layer thickness in microns Fig. 12. Failure probability based on surface/volume flaws of the tungsten section after cool down with inclined linear graded layer (30°) and 800 °C starting-temperature.
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maximum stresses at room temperature are lower after thermal cycling compared to the situation after the first cool down after joining, but the failure probability might increase as consequence of the irradiation embrittlement of tungsten. Another dynamic effect in the area of probabilistic failure analysis is the subcritical crack growth, mainly induced by stress corrosion. However, in the helium atmosphere this effect is negligible compared to the embrittlement by neutron irradiation. 4.2.3. Estimation of the required failure probability Considering a raw guess of n = 300,000 thimbles [31] in a future fusion reactor, a required failure probability Pf for each single thimble can be estimated by the formula 1
Pf ¼ 1 ðPr Þn
ð4:1Þ
where Pr is the specified reliability of all thimbles. Assuming Pr = 0.95, meaning that with 95% probability no thimble will fail, the failure probability Pf for a single thimble must be lower than 1.7097 107. This failure probability is regarded as acceptance level in the conclusion. 5. Summary and conclusion In this work, a deterministic and a probabilistic failure analysis for the tungsten section of the thimble of a helium-cooled divertor component with a linearly graded tungsten/EUROFER97 joint was performed. A stress-free state at 800 °C/900 °C/ 1000 °C, followed by a cool down to 0 °C is considered in the FEM simulations. In the deterministic failure analysis the stress analyses performed in recent works were broadened through an investigation of the capability of stress fields, induced by thermal mismatch, in promoting crack initiation and crack propagation. Therefore, crack initiation and crack propagation are simulated using the extended finite element method (XFEM) and the fracture mechanical properties of tungsten. For the traction separation law, the considered threshold for damage initiation and the cohesive energy are required as input parameters. While the cohesive energy is directly determined by the fracture toughness of tungsten, the threshold for damage initiation is unknown for tungsten. Therefore, it is systematically varied for the thimble component with different thicknesses of the graded tungsten/EUROFER97 joint. Values right below the maximum principal stress occurring in the tungsten section yield fast crack initiation and unstable crack propagation, which is mainly caused by the low fracture toughness of tungsten. In conclusion, the maximum principle stress failure criterion is sufficient for the failure assessment of the tungsten section in the considered thimble design. Concerning the probabilistic approach, the failure risk is calculated by means of the STAU code. The gain of a linear graded interlayer with a thickness ranging from 10 lm to 3000 lm as well as the advantage of its inclination are assessed. Experimentally determined Weibull parameters are used as material data. The determined Weibull parameters for WL10 are m = 12.8 and b = 1583 MPa. Considering the total reliability of 300,000 thimbles, the calculated failure probability of a single thimble is high for thimbles with linear graded layers without inclination and thinner than 3000 lm. A global reliability of 95.1% can be achieved for the case where the graded layer is 3000 lm thick, the inclination angle equal 30° and the starting temperature equal 800 °C. However, the use of pure tungsten instead of WL10 and/or a smaller surface roughness can spare the inclination of the graded interlayer and/or admit thinner interlayers. The technical feasibility of the recommended graded layer design has been already discussed and approved for vacuum plasma spraying and resistance sintering under ultra high pressure in [28–30]. Our study provides an extensive analysis of the failure behaviour of tungsten in a divertor component based on two different methods. While the Weibull approach considers the statistical nature of pre-existing inherent flaws, the deterministic approach incorporates micro-mechanical damage and subsequent crack initiation and propagation. The complementary application of both methods is necessary for a comprehensive design optimisation of future fusion rector components made of tungsten. Acknowledgements This work, supported by the European Communities under the contract of Association between EURATOM and Karlsruhe Institute of Technology, was carried out within the framework of the European Fusion Development Agreement. The views and opinions expressed herein do not necessarily reflect those of the European Commission. Appendix A. The traction separation model In the extended finite element method (XFEM) enriched elements are used in which cracks can initiate and propagate along contact surfaces according to a traction separation model. The position and orientation of the contact surfaces inside an enriched element are also determined by the traction separation model. The following description of the traction separation model used in this work is mainly taken from the manual of the FEM software ABAQUS [10] and from [32]. This model considers a process zone or cohesive zone ahead of the crack tip, represented in Fig. 13 for a crack loaded in pure Mode I, pure Mode II or pure Mode III. The relative displacement, also called separation, of the contact surfaces of a considered enriched
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T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
Fig. 13. Traction separation curve for pure mode loading (sketch is similar to [32]).
element in normal direction is denoted by dn, and in the tangential directions by ds and dt. The traction stress acting on and transferred by the contact surfaces of the considered enriched element in normal direction is denoted by sn and for the two shear directions with ss and st. An initial stiffness Ki, the index i can be n, s or t, is used to hold the contact surfaces of the enriched element together in the linear elastic range (point 1 in Fig. 13). Damage initiation is assumed to occur when the principal stress reaches the maximum traction stress value rD (value at point 2 in Fig. 13). The softening behaviour of the element due to increasing damage d and increased separation of the contact surfaces is simulated by applying the factor (1-d) on the stiffness Ki. Once the contact surfaces are unable to transfer any further load (points 4 and 5 in Fig. 13), all the penalty stiffness revert to zero, resulting in initiation of a crack or propagation of an already existing crack. The constitutive equation for the relation between traction and separation is defined as:
si ¼
8 > < > :
dmax 6 d0m m
K i di ð1 dÞK i di
for
d0m
< dmax < dfm ; m dfm
0
6
i 2 fn; t; sg
ðA:1Þ
dmax m
To describe the evolution of damage under a combination of normal and shear deformation across the interface the mixed-mode relative displacement defined as
dm ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2n þ d2t þ d2s
ðA:2Þ
and the following damage evolution function are used [32]:
d¼
0 dfm ðdmax m dm Þ f 0 dmax m ðdm dm Þ
;
d 2 ½0; 1:
ðA:3Þ
The mixed-mode relative displacement corresponding to damage initiation d0m is calculated by Eq. (A.2), when the damage initiation criterion is fulfilled. The mixed-mode relative displacement corresponding to initiation of a crack or propagation of an already existing crack is given as function of the cohesive energy U and the effective traction at damage initiation, which is equal rD in this model:
dfm ¼ 2
U
ðA:4Þ
rD
The value dmax refers to the maximum value of the mixed-mode relative displacement attained during the loading history. m The stiffness components are calculated based on the elastic properties for an enriched element. In this work an isotropic behaviour is assumed, so that their calculation reduces to
Kn ¼
E t
and K t ¼ K s ¼
E 2ð1 þ #Þt
ðA:5Þ
with t being the thickness of the enriched element, E the Young’s modulus and # the Poisson’s ratio. It is worth noting that Eqs. (A.1) and (A.2) do not account for compressive states (dn < 0), which however are regarded in [32] and in the FEM code.
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75
Appendix B. Calculation of the failure probability If surface flaws are considered, we assume that natural flaws can be represented by edge cracks perpendicular to the surface with a certain orientation angle a on the surface. The calculation of the failure probability then reduces to an integral over the component surface A and the orientation angle a:
Z Z 1 1 req ðx; aÞ m PF;A ¼ 1 exp dadA A0 A 2p a r0
ðB:1Þ
The numerical integration must be performed over three dimensions: 2 spatial directions and one orientation angle. It is important that r0 refers to the unit surface A0 and is different from the value obtained in the case of volume flaws. For the chosen normal stress criterion the equivalent stress req is for both, volume and surface flaws, equal to the perpendicular stress value acting on the flaw. Similar to Eq. (3.4), the Weibull parameter b is related to r0 by the effective surface Aeff:
r0 ¼ b Aeff ¼
m1 Aeff A0
Z A
1 2p
ðB:2Þ
Z a
m
req ðx; aÞ rref
dadV
ðB:3Þ
References [1] Norajitra P, Gervash A, Giniyatulin R, Ihli T, Krauss W, Kruessmann R, et al. He-cooled divertor for DEMO: experimental verification of the conceptual modular design. Fusion Eng Design 2006;81:341–6. [2] Klueh RL, Gelles DS, Jitsukawa S, Kimura A, Odette GR, van der Schaaf B, et al. Ferritic/martensitic steels – overview of recent results. J Nucl Mater 2002;307–311:455–65. [3] Lindau R, Schirra M. First results on the characterisation of the reduced-activation-ferritic–martensitic steel EUROFER. Fusion Eng Design 2001;58– 59:781–5. [4] Plansee. Tungsten – material properties and applications. Material Catalogue; 2010. [5] Paúl A, Beirante A, Nuno Franco, Eduardo Alves, José Antonio Odriozola. Phase transformation and structural studies of EUROFER RAFM alloy. Mater Sci Forum 2006;514–516:500–4. [6] Weber T, Aktaa J. Numerical assessment of functionally graded tungsten/steel joints for divertor applications. Fusion Eng Design 2011;86:220–6. [7] Munz D, Fett T. Ceramics mechanical properties failure behaviour, materials selection. Berlin Heidelberg: Springer Verlag; 1999. [8] Riesch-Oppermann H, Härtelt M, Kraft O. STAU – a review of the Karlsruhe weakest link finite element postprocessor with extensive capabilities. Int J Mater Res 2008;99:1055–65. [9] You J-H, Komarova I. Probabilistic failure analysis of a water-cooled tungsten divertor: impact of embrittlement. J Nucl Mater 2008;375:283–9. [10] ABAQUS 6.9-1 User’s manual analysis. Providence RI: ABAQUS Inc.; 2009. [11] Rupp D, Weygand SM. Loading rate dependence of the fracture toughness of polycrystalline tungsten. J Nucl Mater 2011;417:477–80. [12] Dugdale DS. Yielding of steel sheets containing slits. J Mech Phys Solids 1960;8:100–4. [13] Barenblatt GI. The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 1962;7:55–129. [14] Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 1999;45:601–20. [15] E. Gaganidze, Karlsruhe Institute of Technology – Institute for Applied Materials. Karlsruhe, Germany; 2011 [private communication]. [16] Batdorf SB, Heinisch HL. J Am Ceram Soc 1978;61:355. [17] Evans AG. A general approach for the statistical analysis of multiaxial fracture. J Am Ceram Soc 1978;61:302–8. [18] Matsuo Y. Probabilistic analysis of the brittle fracture loci under bi-axial stress state. Bull JSME 1981;24:290–4. [19] Brückner-Foit A, Fett T, Munz D, Schirmer K. Discrimination of multiaxiality criteria with the Brazilian disc test. J Eur Ceram Soc 1997;17:689–96. [20] Brückner-Foit A, Heger A, Munz D. On the contribution of notches to the failure probability of ceramic components. J Eur Ceram Soc 1996;16:1027–34. [21] Rupp D, Weygand SM. Anisotropic fracture behaviour and brittle-to-ductile transition of polycrystalline tungsten. Philos Mag 2010;90:4055–69. [22] Fernandes JJ, Da Silva CP, Rosa LG, Martins CS. Effects of surface roughness on the flexural strength of a hardmetal. J Mater Sci 1994;29:2008–12. [23] Basuki WW, Aktaa J. Investigation on the diffusion bonding of tungsten and EUROFER97. J Nucl Mater 2011;86:2585–8. [24] von der Weth A, Kempe H, Aktaa J, Dafferner B. Optimization of the EUROFER uniaxial diffusion weld. J Nucl Mater 2007;367–370:1203–7. [25] Chehtov T, Aktaa J, Kraft O. Mechanical characterization and modeling of brazed EUROFER-tungsten-joints. J Nucl Mater 2007;367–370:1228–32. [26] Widak V, Norajitra P. Optimization of He-cooled divertor cooling fingers using a CAD-FEM method. Fusion Eng Design 2009;84:1973–8. [27] Basuki W. Karlsruhe Institute of Technology – Institute for Applied Materials. Karlsruhe, Germany; 2011 [private communication]. [28] Bolt H, Barabash V, Krauss W, Linke J, Neu R, Suzuki S, et al. Materials for the plasma-facing components of fusion reactors. J Nucl Mater 2004;329– 333:66–73. [29] Pintsuk G, Smid I, Döring J-E, Hohenauer W, Linke J. Fabrication and characterization of vacuum plasma sprayed W/Cu-composites for extreme thermal conditions. J Mater Sci 2007;42:30–9. [30] Weber T, Zhou Z, Qu D, Aktaa J. Resistance sintering under ultra high pressure of tungsten/EUROFER97 composites. J Nucl Mater 2011;414:19–22. [31] Kruessmann R, Norajitra P. Conceptual design of a He-cooled divertor with integrated flow and heat transfer promoters (PPCS Subtask TW3-TRP-001D2) Part II: Detailed version, FZKA 6975; 2004. [32] Camanho PP, Davila CG. Mixed-mode decohesion finite elements for the simulation of delamination in composite materials. NASA technical paper, 211737; 2002.
Contents lists available at SciVerse ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Considering brittleness of tungsten in failure analysis of helium-cooled divertor components with functionally graded tungsten/EUROFER97 joints q T. Weber ⇑, M. Härtelt, J. Aktaa Karlsruhe Institute of Technology, Institute for Applied Materials, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany
a r t i c l e
i n f o
Keywords: Functionally graded material Tungsten EUROFER97 Finite element analysis Failure assessment Weibull statistics
a b s t r a c t Helium-cooled divertor components made of tungsten and reduced activated ferritic martensitic steel are part of current research for future power plants. Two different failure analyses, a deterministic and a probabilistic, have been performed on the tungsten section of the considered thimble of the divertor component which is joined to a steel cartridge by a functionally graded tungsten/EUROFER97 layer. In the deterministic failure analysis, the extended finite element method (XFEM) is used providing insights concerning the initiation and the propagation of possible cracks, which depends on the layer thickness. Due to the brittleness of tungsten at low temperatures and its embrittlement in view of neutron irradiation a probabilistic failure analysis is necessary to assess the reliability of the divertor component. Failure probabilities for the tungsten thimble were calculated based on Weibull statistics using the STAU code. The influence of the thickness and inclination angle of functionally graded tungsten/EUROFER97 joints on the failure probabilities were studied and revealed a minimum required layer thickness of 3000 lm and the necessity of an inclination. The Weibull parameters for a tungsten alloy with 1 wt.% La2O3 were determined in four point bending tests and used as input data for the failure analysis of the component during production. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Divertor components of future fusion power plants will be subjected to concentrated flux of plasma particles resulting in very high surface erosion and heat loads. In the helium cooled divertor concept [1] considered here, tungsten and its alloys are selected as refractory as well as structural materials. Due to the brittleness of tungsten at low temperatures, its use as a structural material is limited to the high temperature region (>650 °C) of the component and a joint to another structural material, the ferritic martensitic high chromium steel EUROFER97 [2,3], is necessary. However, the remarkable difference in thermal expansion between tungsten (ath = 4.4e6 K1 [4]) and EUROFER97 (ath = 12.7e6 K1 [5]) causes thermal mismatch between both materials resulting in stresses which would yield failure of the joint. One strategy to reduce thermally induced stresses is to introduce a functionally graded layer between the materials to be jointed. The joint between both structural materials is located in the thimble of the divertor component. The effect of a functionally graded layer on component stresses in this issue was recently investigated by elasto-viscoplastic finite element (FE) simulations [6]. Considering application relevant loadings, the effects of variation of the thickness and
q
In Honor of Prof. Dr. Dietrich Munz, Engng. Fract. Mech. (2012).
⇑ Corresponding author.
E-mail address: [email protected] (T. Weber). 0013-7944/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2012.07.024
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T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
Nomenclature PF,V PF,A V A X
failure probability of a single thimble based on volume flaws failure probability of a single thimble based on surface flaws component volume component surface flaw orientation within volume a flaw orientation with respect to surface req equivalent stress x flaw location a flaw length Veff effective volume Aeff effective surface V0 unit volume unit surface A0 r0 normalised strength with respect to the unit volume V0/unit surface A0 rref reference stress characterising the load level YI geometric correction factor corresponding to fracture mode I equivalent mode I stress intensity factor KI eq KIc mode I fracture toughness ac critical crack size m Weibull shape parameter b Weibull scale parameter N amount of tested samples Fj assigned failure probability of specimen with rank j rj bending stress of specimen with rank j j rank of the considered bending specimen Ra arithmetic average of surface roughness Rq root mean squared surface roughness maximum height of the profile Rt Rz average distance between the highest peak and lowest valley in each sampling length ath thermal expansion coefficient f field variable indicating the gradation level i index of time increment l viscosity parameter in XFEM-simulation D height of the fine meshed region U cohesive energy JIc energy release rate for crack propagation rD threshold value for damage initiation dn, sn, Kn separation/traction/stiffness in normal direction separation in shear direction ds, dt ss, st traction in shear direction Ks, Kt stiffness in shear direction d damage of the enriched element t thickness of the enriched element dm mixed-mode relative displacement d0m
mixed-mode relative displacement corresponding to damage initiation
dfm
mixed-mode relative displacement corresponding to complete failure
dmax m E # n Pr
maximum value of mixed-mode relative displacement during loading history Young’s modulus Poisson’s ratio number of thimbles total reliability of all thimbles
the transition function within the graded layer was analysed. However, the failure of tungsten has to be accounted by further analyses, amongst others by a probabilistic failure analysis, which take the brittleness of tungsten into account. Failure of brittle materials [7] is governed by inherent flaws which can be pores, inclusions or machining cracks. A component fails once a critical flaw size, which depends on the stress state and crack model, is exceeded (weakest-link approach). The characteristic scatter in strength of brittle components is a consequence of the fact that the critical flaw size scatters. Furthermore, the strength decreases with increasing component volume (or surface) since it is more likely to find a catastrophic flaw in a big component. Based on the Weibull theory, the failure probability of components under complex
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T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
multi-axial stress states can be calculated using numerical tools (e.g. STAU [8]) which act as a post-processor of a FE-analysis. The effect of the brittle failure mode of tungsten on component design has rarely been studied for fusion applications. You and Komarova [9] presented a probabilistic analysis for a water-cooled divertor based on strength data for tungsten undergoing different heat treatments. They found that the failure probability of the considered component is acceptable but also stated that irradiation embrittlement (change in flaw size scatter) can be critical for the considered component. However, a systematic design optimisation procedure based on probabilistic methods has not been yet reported for fusion components. We will present a design study for the tungsten-EUROFER97 joint of a helium-cooled divertor component. For that purpose, the effect of a variation of the thickness of the graded layer under application relevant loading will be studied on the base of FE stress analyses, which show in the tungsten part highly inhomogeneous stress fields with localised high stresses [6]. For the assessment of failure caused by these stress fields failure criteria evaluating only local stress values are at best sufficient to predict crack initiation. As tungsten is not perfectly brittle it is not self-evident that even when a crack is initiated, it would propagate within the highly inhomogeneous stress field. Therefore, in a first approach, the stress fields appearing in the tungsten part are evaluated by investigation of their capabilities in promoting crack initiation and propagation using XFEM, as it is provided since version 6.9-1 of the commercial finite element code ABAQUS [10]. The failure probability is calculated with the STAU code in order to evaluate the joint design. The relevant material properties for the probabilistic analysis are obtained from strength measurements for a sintered tungsten grade. 2. FEM model The finite element code ABAQUS is used to perform the simulations. The simplified FE model of the divertor component considered consists of an axis symmetric shell with a thickness of 1 mm, see Fig. 1a. The thickness of the graded layer situated in the joining section is varied between 10 and 3000 lm. Therefore, a field variable f ranging from 0 to 1 is used to indicate level of gradation with f = 0 on the tungsten side and f = 1 on the steel side. Inside the joining section, the material properties are interpolated linearly between 0 and 1. Hence, the thickness of the graded layer is controlled by the field variable f. Since the graded layer thickness ranges over two orders of magnitude it became necessary to use adequate meshes for each layer thickness. Thus, six meshes were used for all scenarios with D e {20 lm, 200 lm, 500 lm, 1 mm, 2 mm, 3 mm}, see Fig. 1b. They are designed in such a way that the highest strains and stresses always lie inside the densest elemental region of the mesh, which consists of rectangular horizontal elements with a size of 20 lm 10 lm stapled in y-direction. Assuming a hot joining process, the initial temperature of the whole FE model is set to 800 °C/900 °C/1000 °C, at which it is considered to be stress free. Thereafter, the FE model is loaded by varying the temperature homogeneously with a cooling down to 0 °C. The temperature dependent Young’s modulus, yield strength and thermal expansion coefficient for the EUROFER97 and joining section are adopted from Ref. [6]. For the tungsten section the removal of plasticity below the ductile to brittle transition temperature (DBTT) [11] is necessary (Fig. 2). Exceeding the yield strength in an ideal-elasto-plastic simulation means for brittle materials the beginning of spontaneous cracking. Limiting stresses by setting yield strengths and consequently simulating plastic deformation, which is actually negligibly small for tungsten below the DBTT, would be inappropriate. So although the simulated stress values might be higher than they occur in reality, since cracking occur first, they
(b)
(a) 7.5 mm
tungsten section
10.
EUROFER97 section
4.
200μm
EUROFER97
Δ
x
joining section
y
100μm
Tungsten
6.5
9.
1. Fig. 1. (a) Sketch of the finite element model (dimensions in mm) and (b) cut-out of the mesh with D = 20 lm.
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T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
(b)
(a)
Fig. 2. (a) Yield strength and (b) Young’s modulus used in the FEM model.
are rated as poor in the deterministic and probabilistic failure analysis. The highest probability of brittle fracture in tungsten is expected at temperatures below DBTT and when stresses are at maximum. Hence, only the stress state at 0 °C is considered for the failure analysis.
3. Failure assessment procedures 3.1. Deterministic approach In addition to the stress analysis using the finite element model described above, crack initiation and crack propagation are investigated using the extended finite element method. Crack initiation and propagation can be modelled in ABAQUS in accordance with the cohesive zone model (CZM) by using ‘cohesive elements’. The concept of the CZM, first conceived by Dugdale [12] and Barenblatt [13] regards fracture as a gradual phenomenon in which separation takes place between two adjacent virtual surfaces across an extended crack tip (cohesive zone) and is resisted by the presence of cohesive forces. This is in contrast to linear elastic and elastic–plastic fracture mechanics, where an infinite sharp crack tip is assumed. The description of the failure behaviour for a given material is defined by traction-separation law. As the cohesive surfaces separate, the traction first increases until a maximum is reached, and subsequently the traction decreases to zero, which results in complete separation. The area under the traction separation curve corresponds to the cohesive energy U needed for separation. Considering the same traction-separation law the XFEM allows the simulation of crack initiation and crack propagation by using ‘enriched elements’ instead of ‘cohesive elements’. The XFEM is implemented since version 6.9-1 of ABAQUS and provides some advantages. It alleviates the shortcomings associated with meshing crack surfaces as was first introduced by Belytschko and Black [14]. Furthermore, enriched elements allow the simulation of crack propagation along an arbitrary, solution-dependent path, since the crack propagation is not tied to the element boundaries in a mesh [10]. The traction separation model used in this work, described with more details in Appendix A, assumes initially linear elastic behaviour followed by the initiation and evolution of damage. Damage is assumed to initiate when the maximum principal stress exceeds a threshold value, rD. Damage evolution follows linearly as soon as the damage initiation criterion is reached, reducing the cohesive stiffness of the enriched element. When the stiffness of the enriched element reaches at an integration point a value of zero crack initiation or, in case of an already existing crack, crack propagation appears with a length equal to the length represented by the integration point. The energy released during this process is specified by the cohesive energy U, which is provided as input parameter. Since the cohesive energy U is equal to the energy release rate for N is used for U. This value is obtained by applying the relation between JIc and crack propagation JIc, the value J Ic ¼ 0:1162 mm the fracture toughness KIc for elastic media and considering the value of KIc = 7.19 MPa m½, which has been experimentally measured for commercially available tungsten [15]. As tungsten shows strong anisotropy in its fracture behaviour, the considered KIc value here corresponds to a measurement at room temperature and to the most unfavourable orientation. Hence, it is a conservative value, which can be assumed as minimum value for tungsten fabricated by today’s methods. Material models exhibiting softening behaviour and stiffness degradation often lead to severe convergence difficulties. Thus the viscous regularization method was used to resolve this issue. The regularization process is described in [10] and can be controlled by the viscosity parameter l, which represents the relaxation time of the viscous system. For the analysis in Section 4.1, the time increments and the viscosity parameter l have been chosen sufficient small so that the results are not compromised by the viscous regularization. Interfacial fracture is not considered in this XFEM analysis, although a weak bond interface may affect the failure feature completely. Usually the interfacial bond strength depends very strongly on the fabrication method, intermetallic phases, surface roughness, etc. However, in this a work an ideally bonded interface is assumed.
T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
67
3.2. Probabilistic approach The Weibull theory is based on a weakest link approach assuming that a component fails once one flaw exceeds a critical size ac. The critical size depends on the load, the location and the orientation of a crack meaning that the most unfavourable flaw will cause failure. Flaws are modelled as planar cracks and a multi-axial criterion is used as a fracture mechanics failure criterion under mixed-mode loading:
pffiffiffi K Ieq ¼ req ðx; XÞY i a:
ð3:1Þ
KIeq is the equivalent mode I stress intensity factor and req is the equivalent stress which depends on the location x and orientation X of the crack. YI is the geometry factor of the corresponding crack model. Failure occurs once KIeq exceeds the fracture toughness KIc of the material and, hence, ac can be expressed in terms of the equivalent stress req based on Eq. (3.1). For an arbitrary number of equally distributed and orientated flaws, the failure probability for volume flaws can be calculated by solving the integral in Eq. (3.2), which was derived in different ways in [16–18] and reviewed for instance in [8].
Z Z 1 1 req ðx; XÞ m PF;V ¼ 1 exp dXdV V 0 V 4p X ro
ð3:2Þ
The integration is performed over the component volume V and possible flaw orientation X. The equivalent stress req at location x and orientation X relative to the global coordinate system is mode I equivalent stress based on a fracture mechanics mixed-mode criterion [16,19] and depends on the location as well as on the orientation of the crack. A number of mixedmode criteria are implemented in STAU. If, for example, failure is governed solely by mode I loading, the equivalent stress is equal to the normal stress with respect to the crack plane (normal stress criterion) [7]. It was shown that the normal stress criterion can be applied for modelling the failure behaviour of natural flaws in ceramics [19]. Therefore, the normal stress criterion will also be used for the calculations of tungsten in this work, assuming that the failure behaviour is determined by pure mode I loading. The material specific parameters m and r0 must be determined from strength tests. r0 can be regarded as a normalised strength with respect to the unit volume V0 and can be extracted from the size-dependent Weibull scale parameter b obtained for arbitrary specimen geometries. The calculation of the failure probability in Eq. (3.2) requires solving an integral over 5 dimensions (three spatial directions and two orientations angles) numerically. STAU uses Gaussian quadrature in order to perform the integration. In the course of the integration procedure, a number of supporting points are generated by interpolating the stresses using the element form functions. Thus, accurate results can be achieved. Using Eq. (3.2), no increase in crack resistance is assumed, which is unknown for the tungsten materials considered. Eq. (3.2) can be reformulated in order to obtain a Weibull distribution for a reference stress rref with the Weibull parameters m and b [20]:
rref m : PF;V ¼ 1 exp b
ð3:3Þ
The reference stress can be for example the outer fibre stress in a 4-point bending test. In Eq. (3.3), b depends on the specimen size (geometry) and the type of loading. The normalised strength r0 is related to b by the effective volume Veff:
r0 ¼ b V eff ¼
1 V eff m Vo
Z V
1 4p
ð3:4Þ
Z X
m
req ðx; XÞ rref
dXdV
ð3:5Þ
The effective volume can be calculated by STAU for a given reference stress and a corresponding stress distribution from a FE analysis. This allows for determining r0 from the Weibull parameters m and b which can be measured in a strength test. Eq. (3.4) implies that r0 depends on whether volume or surface flaws are considered and on the mixed-mode criterion used for calculating req. The corresponding formulations for surface flaws can be found in Appendix B. To obtain the required Weibull parameters m and r0 for the failure analysis, four-point bending tests were performed at room temperature on rolled bar material made of sintered WL10 from the manufacturer Plansee AG, Austria. WL10 is a tungsten alloy with 1 wt.% La2O3 having an improved creep resistance and machinability. In the performed FEM analysis, WL10 and pure tungsten are not distinguished, since their material properties are assumed to be identical for this kind of analysis. Thus, although the considered thimble is actually planned to be manufactured out of WL10, the more familiar term tungsten is used when the results and statements are of a general manner and apply to both materials. Otherwise it is specified by WL10. However, the required Weibull parameters used in the failure analysis are fairly different for WL10 and pure tungsten. The test setup and specimen geometry were chosen in compliance with the standard DIN EN 843. While the size of the bending specimen was 3 4 45 mm, the distance of the inner support was 20 mm and 40 mm for the outer ones. The long edges of the specimen are parallel to the rotational symmetry axis of the source bar material, so that the tensile stresses in the bending test have the same direction as the max. principal stresses in the considered use case of the divertor component. The conformity of the load direction in the bending test and in the divertor application case is important in consideration of the strong anisotropy of the fracture toughness [21]. N = 40 specimens were prepared using wire-spark-erosion
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T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
ln (ln(1/(1−Fj)))
4
2
WL10 − measured data WL10 − Weibull WL10 − upper 90% conf. interval WL10 − lower 90% conf. interval W − as received [9] W − annealed [9]
0
−2
−4 6.6
6.8
7
7.2
7.4
7.6
7.8
8
ln (σj / MPa) Fig. 3. Weibull-parameters for WL10 (measured in this work) and tungsten (determined in [9]).
cutting method followed by a surface grinding step and a slight bevelling of the edges. Surface roughness was measured by a Tencor P-10 Surface Profiler, yielding the following maximum values out of four measurements: Ra = 556 nm, Rq = 739 nm, Rt = 4887 nm and Rz = 3223 nm. The strength data measured for WL10 is presented in the Weibull plot in Fig. 3, showing that the distribution obeys a 2-parameters Weibull distribution. In this diagram rj is the bending stress at rupture of the specimen with rank j and Fj is the assigned failure probability calculated as:
Fj ¼
j 0:5 N
ð3:6Þ
The rank j of the specimens is calculated by sorting their bending stress at rupture in ascending order and enumerating from 1 to N. The estimated Weibull parameters are m = 12.8 (shape parameter) and b = 1583 MPa (scale parameter). The Weibull parameters for warm rolled pure tungsten sheets before and after heat treatment at 1000 °C for 10 h have already been published by You and Komarova [9]. Higher bending strengths and a smaller scattering of pure tungsten can be noticed compared to WL10. This may be explained by the lanthanum oxide particles, which act as cracking sources. A second reason might be the influence of the surface morphology [22]. Higher shape and scale parameters are measured when decreasing surface roughness.
4. Results and discussion 4.1. Deterministic approach It has already been found in a prior study on graded joints [6] that the most critical region for the tungsten section is at the lower right corner, near the outer surface and to the bonding interface of the joining layer. The stresses are strongly concentrated in these domains, being a consequence of the thermal mismatch combined with the interfacial stress singularity effect. Fig. 4 shows the maximum von Mises stress occurring in the tungsten section for both cases, with and without plasticity for tungsten below 400 °C after the cool down from 1000 °C to 0 °C. It shows that only layers thicker than 1500 lm will not deform plastically after the cool down. This circumstance indicates that either optimised design geometry of the joint should be aspired or a stress-free state at a lower temperature must be taken into account. The stress free state at 1000 °C, as considered here, corresponds to the temperature at which the joint EUROFER97/graded layer/tungsten (or WL10) is realised, for instance by diffusion bonding of EUROFER97 [23,24] or brazing of tungsten with the aid of an interlayer [25]. As the selected damage initiation criterion is based on the maximum principal stress, no cracking and separation of the enriched elements occur as long as the threshold rD is greater than the highest value of the maximum principal stress occurring in the tungsten section rmax. Within a parameter study, rD has been varied along the different layer thicknesses, while N is kept constant. For rD values right below rmax, spontaneous continuous cracking is obthe cohesive energy U ¼ 0:1162 mm served, meaning that once cracking has started, it will not be stopped by cohesive forces, although energy is dissipated. To illustrate this behaviour, Fig. 5 shows the damaging of the enriched elements when cooling down a joint with a 2 mm thick linear graded layer without inclination from 800 °C to 0 °C. While rD is varied from 766 MPa to 764 MPa, representing a difference smaller than uncertainties in the assumptions made in the FE analysis, the result spans from no cracking at all to complete propagation of the crack (see Fig. 5a–c). Fig. 6 shows the damage evolution after a different number of time
T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
maximum von Mises stress in MPa
2200
69
yield stress is limited below 400 o C o yield stress is unlimited below 400 C
2000 1800 1600 1400 1200 1000
00 2000 15 00 10
0 50
0 20
0 10
50
20
10
layer thickness in μm Fig. 4. Maximum von Mises stress in the tungsten section plotted over layer thickness for a linearly graded joint after cooling down from 1000 °C to 0 °C.
damage initiation
(a) σ D = 766 MPa
(b) σ D = 765 MPa
(c) σ D = 764 MPa 100 µm Fig. 5. Influence of rD on damage initiation in a joint with 2000 lm thick linear graded layer without inclination when cooling down from 800 °C to 0 °C: (a) no damage, (b) one element with partial damage, and (c) crack has propagated largely through the wall thickness.
damage evolution
(a) i = 12
(b) i = 100
(c) i = 200 100 µm Fig. 6. Damage evolution and crack propagation in a joint with 2000 lm thick linear graded layer without inclination when cooling down from 800 °C to 0 °C with rD = 764 MPa: (a–c) represent the states after the number of time increments i.
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T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
fieldvariable i bl f
(a)
Tungsten
(b)
(c)
EUROFER97 200 µm Fig. 7. Path of the crack in the tungsten area for three different layer thicknesses: (a) 20 lm, (b) 200 lm and (c) 2000 lm.
increments. The time increment is not constant and the simulated temperature decrease is less than 0.5 °C for the three images. In Fig. 6a the damage initiation criterion has been fulfilled along a depth of two enriched elements (green).1 Before complete separation (red) comes up, the crack tip is already driven far into the material within a very short time (see Fig. 6b and c). Accordingly, when cracking is initiated, unstable crack propagation is expected which is mainly caused by the brittleness and low fracture toughness (cohesive energy) of tungsten. For thin layers, the maximum principal stress is concentrated on a small spot in the tungsten region at the outer side of the thimble, so that cracking starts most likely at this point (see Fig. 7). Thicker layers show a more distributed maximum stress area, so that cracking might occur at different places. The crack path starts perpendicular to the outer surface and is slightly deflected upwards. Apart from the exact starting position this crack propagation behaviour is similar for all layer thicknesses. However, to observe cracking in the tungsten region for thicker layers, the rD value has to be reduced to values much lower than the mean bending strength measured for W and WL10 (ref. to Section 3.2). Due to the probabilistic nature of strength for tungsten as a brittle material this does not mean that tungsten cracking and failure in joints with thicker layers can be excluded. However, lower failure probability might be expected. For its evaluation, a probabilistic approach is used which accounts for failure probability by considering inherent flaws and their statistically distributed size and orientation.
4.2. Probabilistic approach 4.2.1. Failure probability during cooling down The failure probability at the end of the cooling step was calculated based on surface and volume flaws for different layer thicknesses as can be seen in Fig. 8. It shows that the failure probability is higher if surface flaws are considered. Indeed, this result corresponds to the fact that stresses are concentrated at the outer surface of the tungsten section. Particularly, very thin (<50 lm) linear graded layers exhibit a pronounced concentration of stresses at the outer surface, so that the effective volume is rather low compared to the effective volume of the tested bending specimen. As a result, in spite of high maximum stresses (Fig. 4) being as high as the Weibull parameter b, very thin graded layers obtain relatively low failure probabilities (less than 10%). Linearly graded joints with a thickness in the range between 50 and 500 lm are more critical. In addition to a large effective volume, the interlayer in this simulation comes up with higher yield strength than pure EUROFER97 leading to elevated stresses in the tungsten section. Very thick layers (>500 lm) can compensate this circumstance, so that smaller failure probabilities can be achieved compared to a direct joint. The great difference of the Weibull parameters between pure tungsten [9] and WL10 strongly influences the calculated failure probabilities. In order to estimate the influence of the Weibull-parameters, a comparison of the failure probabilities resulting from the experimentally determined parameters for pure tungsten and the ones for WL10 has been done. The results are presented in Fig. 9. In spite of the strongly reduced failure probabilities of a pure tungsten thimble, the use of WL10 is still very important at the head of the thimble where temperatures above 1100 °C are expected in combination with stresses around 300 MPa [26]. Considering these two materials only WL10 is able to sufficiently withstand creep under these conditions. One possibility for the reduction of strains and stresses is the inclination of the graded layer along the joint as shown in [6]. For that purpose, STAU was used to analyse the degree to which the inclination angle can reduce the failure probability of the tungsten section. The results, shown in Fig. 10, reveal that the surface failure probability can be decreased by an order of magnitude, to a minimum of 1.831e5, when choosing a negative inclination angle between 30° and 40°. A negative inclination angle means that the layer is tilted towards the outer surface. A second possibility to reduce the occurring stresses is to lower the initial temperature. Fig. 11 shows the resulting surface failure probability for three different initial temperatures. We performed preliminary diffusion bonding experiments 1
For interpretation of color in Fig. 6 , the reader is referred to the web version of this article.
T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
71
0.1
failure probability
0.01
0.001
0.0001 surface volume
1e−005
00 20 00 15 00 10
0 50
0 20
0 10
50
20
10
layer thickness in microns Fig. 8. Failure probability of the tungsten section after the cool down from 1000 °C to 0 °C.
1
surface failure probability
0.01
0.0001
1e−006
1e−008
1e−010
WL10 W − as received [9] W − annealed [9]
1e−012
00 20 0 0 15 00 10
0
50
0
20
0
10
50
20
10
layer thickness in microns Fig. 9. Material influence on surface failure probability of the tungsten section after the cool down without inclined linear graded layer and 1000 °C starting-temperature.
and proved that is possible to join EUROFER97 at 800 °C [27]. It shows that the surface failure probability can be reduced by more than one order of magnitude to 1.064e5 for 2 mm thick linear graded layers without inclination. A third option for a further reduction of the failure probability is the use of thicker interlayers. In the publication [28], the possibility to deposit tungsten coatings thicker than 2 mm by plasma spraying are postulated and in [29] functionally graded layers of W/Cu are produced with a thickness of 2–3 mm. Considering other processes, for example resistance sintering under ultra high pressure, tungsten/EUROFER97 composites of several millimetres size in each dimension can be produced easily [30]. The combination of all three improvement measures (thickness, initial temperature and inclination) leads to very low surface failure probabilities in the tungsten section, as can be seen in Fig. 12, so that a minimum surface failure probability of 1.676e7 can be achieved.
4.2.2. Influence of irradiation and operational loadings on the failure probability Neutron irradiation during operation time of a future fusion reactor leads to embrittlement of tungsten and change in the Weibull parameters. This embrittlement must be considered in a reliability analysis of the components. However, since Weibull parameters of irradiated tungsten have not yet been determined, this issue is only discussed qualitatively. The conditions depend upon the design of a Tokamak fusion reactor thermal cycling schedule. Thus, dwell time periods at high temperatures and downtimes at room temperature have to be considered. A brittle failure of tungsten during the dwell time period is less likely due to the reduced stresses, the activation of slip systems and the increase in ductility, which comes along at higher temperatures. But still of interest is the failure probability after shut downs of the fusion reactor. The
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T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
(b)
0.1
10°inclination
0.01
0.001
0.0001
1e−005 -40° -30° -20° -10°
outer surface
inner surface
surface failure probability
(a)
20μm 100μm 500μm 2000μm 0°
10°
20°
30°
tungsten
40°
inclination angle in degrees
EUROFER97
Fig. 10. (a) Surface failure probability of the tungsten section after cooling down from 1000 °C to 0 °C with inclined linear graded layer and (b) example showing a 10° inclination angle.
surface failure probability
0.1
0.01
0.001
0.0001 1000 o C 900 o C 800 o C
1e−005
00 20 0 0 15
00
10
0
50
0
20
0
10
50
20
10
layer thickness in microns Fig. 11. Surface failure probability of the tungsten section after cooling down starting from different equilibrium temperatures (without inclination of the interlayer).
0.01
failure probability
0.001
0.0001
1e−005
1e−006
1e−007 surface volume
1e−008
00 30 00 20 0 0 15 00
10
0
0
50
20
0
10
50
20
10
layer thickness in microns Fig. 12. Failure probability based on surface/volume flaws of the tungsten section after cool down with inclined linear graded layer (30°) and 800 °C starting-temperature.
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maximum stresses at room temperature are lower after thermal cycling compared to the situation after the first cool down after joining, but the failure probability might increase as consequence of the irradiation embrittlement of tungsten. Another dynamic effect in the area of probabilistic failure analysis is the subcritical crack growth, mainly induced by stress corrosion. However, in the helium atmosphere this effect is negligible compared to the embrittlement by neutron irradiation. 4.2.3. Estimation of the required failure probability Considering a raw guess of n = 300,000 thimbles [31] in a future fusion reactor, a required failure probability Pf for each single thimble can be estimated by the formula 1
Pf ¼ 1 ðPr Þn
ð4:1Þ
where Pr is the specified reliability of all thimbles. Assuming Pr = 0.95, meaning that with 95% probability no thimble will fail, the failure probability Pf for a single thimble must be lower than 1.7097 107. This failure probability is regarded as acceptance level in the conclusion. 5. Summary and conclusion In this work, a deterministic and a probabilistic failure analysis for the tungsten section of the thimble of a helium-cooled divertor component with a linearly graded tungsten/EUROFER97 joint was performed. A stress-free state at 800 °C/900 °C/ 1000 °C, followed by a cool down to 0 °C is considered in the FEM simulations. In the deterministic failure analysis the stress analyses performed in recent works were broadened through an investigation of the capability of stress fields, induced by thermal mismatch, in promoting crack initiation and crack propagation. Therefore, crack initiation and crack propagation are simulated using the extended finite element method (XFEM) and the fracture mechanical properties of tungsten. For the traction separation law, the considered threshold for damage initiation and the cohesive energy are required as input parameters. While the cohesive energy is directly determined by the fracture toughness of tungsten, the threshold for damage initiation is unknown for tungsten. Therefore, it is systematically varied for the thimble component with different thicknesses of the graded tungsten/EUROFER97 joint. Values right below the maximum principal stress occurring in the tungsten section yield fast crack initiation and unstable crack propagation, which is mainly caused by the low fracture toughness of tungsten. In conclusion, the maximum principle stress failure criterion is sufficient for the failure assessment of the tungsten section in the considered thimble design. Concerning the probabilistic approach, the failure risk is calculated by means of the STAU code. The gain of a linear graded interlayer with a thickness ranging from 10 lm to 3000 lm as well as the advantage of its inclination are assessed. Experimentally determined Weibull parameters are used as material data. The determined Weibull parameters for WL10 are m = 12.8 and b = 1583 MPa. Considering the total reliability of 300,000 thimbles, the calculated failure probability of a single thimble is high for thimbles with linear graded layers without inclination and thinner than 3000 lm. A global reliability of 95.1% can be achieved for the case where the graded layer is 3000 lm thick, the inclination angle equal 30° and the starting temperature equal 800 °C. However, the use of pure tungsten instead of WL10 and/or a smaller surface roughness can spare the inclination of the graded interlayer and/or admit thinner interlayers. The technical feasibility of the recommended graded layer design has been already discussed and approved for vacuum plasma spraying and resistance sintering under ultra high pressure in [28–30]. Our study provides an extensive analysis of the failure behaviour of tungsten in a divertor component based on two different methods. While the Weibull approach considers the statistical nature of pre-existing inherent flaws, the deterministic approach incorporates micro-mechanical damage and subsequent crack initiation and propagation. The complementary application of both methods is necessary for a comprehensive design optimisation of future fusion rector components made of tungsten. Acknowledgements This work, supported by the European Communities under the contract of Association between EURATOM and Karlsruhe Institute of Technology, was carried out within the framework of the European Fusion Development Agreement. The views and opinions expressed herein do not necessarily reflect those of the European Commission. Appendix A. The traction separation model In the extended finite element method (XFEM) enriched elements are used in which cracks can initiate and propagate along contact surfaces according to a traction separation model. The position and orientation of the contact surfaces inside an enriched element are also determined by the traction separation model. The following description of the traction separation model used in this work is mainly taken from the manual of the FEM software ABAQUS [10] and from [32]. This model considers a process zone or cohesive zone ahead of the crack tip, represented in Fig. 13 for a crack loaded in pure Mode I, pure Mode II or pure Mode III. The relative displacement, also called separation, of the contact surfaces of a considered enriched
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T. Weber et al. / Engineering Fracture Mechanics 100 (2013) 63–75
Fig. 13. Traction separation curve for pure mode loading (sketch is similar to [32]).
element in normal direction is denoted by dn, and in the tangential directions by ds and dt. The traction stress acting on and transferred by the contact surfaces of the considered enriched element in normal direction is denoted by sn and for the two shear directions with ss and st. An initial stiffness Ki, the index i can be n, s or t, is used to hold the contact surfaces of the enriched element together in the linear elastic range (point 1 in Fig. 13). Damage initiation is assumed to occur when the principal stress reaches the maximum traction stress value rD (value at point 2 in Fig. 13). The softening behaviour of the element due to increasing damage d and increased separation of the contact surfaces is simulated by applying the factor (1-d) on the stiffness Ki. Once the contact surfaces are unable to transfer any further load (points 4 and 5 in Fig. 13), all the penalty stiffness revert to zero, resulting in initiation of a crack or propagation of an already existing crack. The constitutive equation for the relation between traction and separation is defined as:
si ¼
8 > < > :
dmax 6 d0m m
K i di ð1 dÞK i di
for
d0m
< dmax < dfm ; m dfm
0
6
i 2 fn; t; sg
ðA:1Þ
dmax m
To describe the evolution of damage under a combination of normal and shear deformation across the interface the mixed-mode relative displacement defined as
dm ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2n þ d2t þ d2s
ðA:2Þ
and the following damage evolution function are used [32]:
d¼
0 dfm ðdmax m dm Þ f 0 dmax m ðdm dm Þ
;
d 2 ½0; 1:
ðA:3Þ
The mixed-mode relative displacement corresponding to damage initiation d0m is calculated by Eq. (A.2), when the damage initiation criterion is fulfilled. The mixed-mode relative displacement corresponding to initiation of a crack or propagation of an already existing crack is given as function of the cohesive energy U and the effective traction at damage initiation, which is equal rD in this model:
dfm ¼ 2
U
ðA:4Þ
rD
The value dmax refers to the maximum value of the mixed-mode relative displacement attained during the loading history. m The stiffness components are calculated based on the elastic properties for an enriched element. In this work an isotropic behaviour is assumed, so that their calculation reduces to
Kn ¼
E t
and K t ¼ K s ¼
E 2ð1 þ #Þt
ðA:5Þ
with t being the thickness of the enriched element, E the Young’s modulus and # the Poisson’s ratio. It is worth noting that Eqs. (A.1) and (A.2) do not account for compressive states (dn < 0), which however are regarded in [32] and in the FEM code.
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Appendix B. Calculation of the failure probability If surface flaws are considered, we assume that natural flaws can be represented by edge cracks perpendicular to the surface with a certain orientation angle a on the surface. The calculation of the failure probability then reduces to an integral over the component surface A and the orientation angle a:
Z Z 1 1 req ðx; aÞ m PF;A ¼ 1 exp dadA A0 A 2p a r0
ðB:1Þ
The numerical integration must be performed over three dimensions: 2 spatial directions and one orientation angle. It is important that r0 refers to the unit surface A0 and is different from the value obtained in the case of volume flaws. For the chosen normal stress criterion the equivalent stress req is for both, volume and surface flaws, equal to the perpendicular stress value acting on the flaw. Similar to Eq. (3.4), the Weibull parameter b is related to r0 by the effective surface Aeff:
r0 ¼ b Aeff ¼
m1 Aeff A0
Z A
1 2p
ðB:2Þ
Z a
m
req ðx; aÞ rref
dadV
ðB:3Þ
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