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Triple-scale failure estimation for a composite-reinforced structure based on integrated modeling approaches. Part 2: Meso- and macroscopic scale analysis
Engineering Fracture Mechanics 76 (2009) 1437–1449
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Triple-scale failure estimation for a composite-reinforced structure based on integrated modeling approaches. Part 2: Meso- and macroscopic scale analysis J.H. You * Max-Planck-Institut für Plasmaphysik, Euratom Association, Boltzmannstr. 2, 85748 Garching, Germany
a r t i c l e
i n f o
Article history: Received 26 September 2007 Accepted 31 October 2008 Available online 17 November 2008 Dedicated to Professor Wolfgang Brocks on the occasion of his 65th birthday. Keywords: Metal matrix composite Finite element analysis Probabilistics Weibull theory Shakedown Failure assessment
a b s t r a c t In this article, the meso- and macroscopic failure features are discussed considering the identical composite component as in the foregoing article. In the meso-scale failure analysis, the risk of plastic instability of the composite tube was estimated considering the shakedown boundary as a failure criterion. The meso-scale stresses of the composite tube were computed using micromechanical homogenization and compared with the shakedown boundary of the composite obtained from the direct shakedown analysis. The stress states were close to the shakedown boundary indicating no critical danger of plastic failure. In the macro-scale failure analysis, the mechanical influence of the local composite integration was investigated with regard to the brittle failure risk of the neutron-embrittled component. To this end, a probabilistic failure analysis code was applied which was based on the fracture mechanics and the weakest-link failure theory. Various fracture criteria were considered. It was found that the failure risk of the tungsten block was strongly reduced by the composite reinforcement of the tube due to the intensification of compressive stress fields. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction In the foregoing companion article (part 1), the microscopic failure features of a composite component was treated. A plasma-facing component for fusion reactors was considered whose cooling tube was locally reinforced with fiber metal matrix composite (FMMC). The focus of the microscopic failure analysis was placed on the estimation of the fiber fracture risk and the plastic damage of the ductile matrix. Non-linear triple-scale FEM was applied using the micromechanics-based homogenization algorithm. In the present article (part 2), meso- and macroscopic failure features of the same FMMC component are discussed (see Fig. 1). The failure problems are treated in two lines, that is, plastic instability of the FMMC tube (meso-scale) and brittle failure of the tungsten armor block (macro-scale). 1.1. Meso-scale failure analysis: shakedown behavior This issue is related to the plastic flow in the ductile matrix which is locally concentrated near the fibers. The plastic behavior of the FMMC is detrimentally affected by variation of the applied loads. Although major plasma instability will
* Tel.: +49 89 3299 1373. E-mail address: [email protected] 0013-7944/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2008.10.017
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Nomenclature Latin symbols a crack size critical crack size ac b Weibull parameter (scale parameter) [B] matrix of partial derivatives of the shape function [J] Jacobian transformation matrix for the shape function of an element von Mises yield function f(rs, x) g(KIeq, 0, 0) effective failure criterion g(rn, sII, sIII) mixed-mode failure criterion Jacobians of variable transformation JV, JX K stress intensity factor mode I fracture toughness KIc equivalent mode l stress intensity factor KIeq m Weibull modulus (shape parameter) number of elements of the FEM model Ne P elastically admissible domain failure probability due to unstable propagation of surface cracks PF,A failure probability due to unstable propagation of volume cracks PF,V volume of the eth element Ve dV infinitesimal volume element X coordinate for macroscopic length scale x coordinate for microscopic length scale Y geometric correction factor Greek symbols h element coordinate in the reference configuration R macroscopic stress tensor r microscopic stress tensor rc elastic stress field in virtual elastic matrix re microscopic elastic stress state req equivalent stress rn normal projection of stress tensor on crack plane ro material parameter in a 2-parameter Weibull distribution r^ r time-independent periodic residual stress field rs safe states of stress r* reference stress characterizing a load level sII, sIII shear projection of stress tensor on crack plane n element coordinate in the reference configuration X domain occupied by the unit cell
be suppressed in fusion power plants, unexpected minor plasma instability may possibly appear generating heat flux fluctuations [1]. This thermal load perturbation will lead to a thermal stress variation in the plasma-facing component. Depending on the stress state and loading history, the plastic response of a FMMC under variable loads will show one of the following behaviors: elastic shakedown, alternating plasticity (low cycle fatigue) or progressive straining (ratcheting). Such plastic responses can be most suitably described on a mesoscopic length scale (e.g. length scale of a unit cell or a representative volume element) [2]. According to the cyclic plasticity FEM study of a conventional plasma-facing component, low cycle fatigue of the softened copper alloy cooling tube can be a critical failure problem [3]. Hence, shakedown limit may be regarded as an appropriate failure criterion in the context of failure assessment of the FMMC tube under variable loads. The FMMC, which was initially loaded in a plastic regime, eventually reach elastic state under arbitrarily varying load, provided that the peak load is bounded within the shakedown limit. In this circumstance the plastic strain ceases to develop and overall FMMC response is confined within safe elastic regime regardless of the loading path thereafter. When maximum load is located beyond this limit, the FMMC component will undergo either low cycle fatigue or incremental plastic collapse (ratcheting) [4–6]. Recently, several theoretical works were reported in the literature in which shakedown boundaries of FMMC under various loading conditions were computed using the FEM-based direct shakedown analysis method [7–12]. On the other hand, theoretical study to investigate the shakedown behavior of a composite component is not found yet.
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Fig. 1. Geometry (symmetric half) and the finite element mesh of the model composite component consisting of tungsten armor block and FMMC heat sink cooling tube. FMMC was a unidirectional copper alloy matrix composite-reinforced with tungsten fibers wound in the tube hoop direction (fiber volume fraction: 40%).
In the first part of this article, risk of plastic failure of the FMMC part is estimated in terms of the shakedown criterion. The FMMC stress and strain were homogenized for individual meso-scale lamina layers. The meso-scale stresses were compared with the shakedown boundary of the FMMC computed for the specific loading condition of the current FMMC component. 1.2. Macro-scale failure analysis: brittle fracture behavior This issue is related to the stress intensification effect caused by the FMMC integration in the tungsten block which is inherently brittle. In general, integration of a FMMC part into a component will affect the overall stress field of the component. When the FMMC has a high stiffness, the local FMMC integration will lead to intensification of thermal stress in the component system. The brittleness problem can be a crucial issue regarding structural reliability when the tungsten block is subjected to a dangerous stress state. The situation will be even more acute during nuclear fusion operation since severe neutron irradiation will cause further embrittlement of tungsten [13,14]. Therefore, the fracture mechanical impact of local FMMC integration should be estimated in the context of structural reliability of the whole composite component. The fracture mechanical failure assessment of a brittle structure requires a statistical treatment due to strong scattering of strength data. The stress analysis needed for the failure assessment is carried out in the framework of a macro-scale FEM model using normal solid elements. In the second part of this article, the impact of FMMC reinforcement for the coolant tube is investigated with regard to the risk of brittle failure of the tungsten block. To this end, a probabilistic failure analysis code was applied. This code was based on the weakest-link failure theory and the linear elastic fracture mechanics (LEFM). Fully elasto-plastic FEM analysis verified that no plastic yielding occurred in the tungsten block. This result justified the application of the weakest-link failure theory based on the LEFM.
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2. Theoretical background 2.1. Direct shakedown analysis The shakedown boundary of the FMMC considered in the current study was obtained using the FEM-based shakedown analysis method developed by Weichert and coworkers [8]. This was the so called direct method based on the Melan’s static shakedown theorem [15]. Detailed description of the theoretical background is found in the literature [8] (see Appendix A). To calculate the shakedown limit for a specific loading condition, following requirements have to be fulfilled: 1. The time-independent residual stress field must satisfy static equilibrium equations and boundary conditions at every nodes and for whole loading domain. 2. The Melan’s shakedown condition must be satisfied at every node and for whole loading domain. In the FEM-based direct shakedown analysis, the virtual elastic stress fields are required (i.e. elastic stress solution from an elastic FEM analysis). The analytical formulation of the shakedown theorem is transformed into a discretized form which is compatible with the finite element structure being used. The equilibrium equation for the time-independent residual stress is also discretized in a similar way (i.e., the [C] matrix in Appendix A has to be computed). Then, the shakedown conditions are checked at each Gauss integration points. The inequality constraints of the shakedown conditions constitute a constrained maximization problem together with the equilibrium constraint of the residual stress. In order to solve this maximization problem a large-scale non-linear optimization code LANCELOT was used [16]. The computational procedure is described in Appendix A. Shakedown boundary was computed for the FMMC lamina considering bi-axial combined loading condition consisting of a longitudinal (parallel to the fiber axis) and a transverse (perpendicular to the fiber axis) variable traction [17]. This specific combined bi-axial load corresponds to the plane-stress state appearing at the cooling tube wall and the fiber architecture. 2.2. Probabilistic failure analysis In this work the probabilistic failure analysis code STAU was used [18]. This code was developed as a post-processor for commercial FEM codes from which the stress data at each integration point were imported into STAU for further failure probability computation. In STAU the probabilistic description of component failure was based on the weakest-link failure theory formulated with Weibull parameters. The theoretical basis is presented in Appendix B. According to the weakest-link theory, failure of a component is triggered by unstable propagation of the most dangerous flaw which is inherently contained in that component [19]. According to the LEFM, the most critical flaw is determined by the most unfavorable combination of size a, location x and orientation x for a given stress field r. For simplicity, only sharp planar cracks are considered here. Random orientation of cracks and multi-axial stress fields necessarily cause a mixed-mode crack tip loading. In practice, mixed-mode fracture criteria are reformulated into an effective criterion in terms of an equivalent mode I stress intensity factor [20,21]. Usually, different mixed-mode criteria predict different failure risks. In this study, four different fracture criteria implemented in STAU were considered as below (see Appendix B): (1) The coplanar energy release rate criterion (coplanar G) [22]. (2) The normal stress criterion (normal stress) [23].
2 4-point bending test
LnLn(1/Ps).
1 0 -1 -2 -3
heat treated (1000 °C, 10 h) as-received
-4 -5 7.6
7.7 7.8 Ln(strength)
7.9
Fig. 2. Weibull plots of tungsten bending strength measured at room temperature (high purity warm rolled sheets). Strength data are given for two specimen groups: heat-treated specimens (heated at 1000 °C in vacuum for 10 h) and as-received ones. Dimension of the specimens was 50 5 1 mm3. (Solid circle: heat-treated specimens, open circle: as-received specimens).
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J.H. You / Engineering Fracture Mechanics 76 (2009) 1437–1449 Table 1 Measured Weibull parameters.
As-received (rolled at 350 °C) Heat-treated (1000 °C, 10 h)
Shape parameter m
Scale parameter b
19 31
2489 2353
(3) The maximum hoop stress criterion (hoop stress) [24]. (4) The maximum energy release rate criterion (max. G) [25]. In order to obtain the input material data required for the STAU analysis, the Weibull parameters of commercial high purity tungsten were experimentally estimated. Warm rolled (at 350 °C) tungsten sheets were used for specimen preparation. Strength data were obtained from four-point bending tests at room temperature. The dimension of the specimens was 50 5 1 mm3. For comparison, heat-treated specimens (annealing at 1000 °C in vacuum for 10 h) as well as the asreceived ones were tested. The measured Weibull plots of the two specimen groups are presented in Fig. 2. Both Weibull plots indicate that the strength distribution clearly obeyed 2-parameter Weibull distributions. The estimated Weibull parameters are given in Table 1. The magnitude of the shape parameter (Weibull modulus) was significantly increased by the heat treatment whereas the scale parameter (characteristic strength) was slightly decreased. But, the metallurgical reason for this effect was unclear. The measured tungsten Weibull modulus was rather larger compared to typical engineering ceramics. 3. Results and discussion 3.1. Results of shakedown analysis To evaluate the plastic behavior of the FMMC tube in terms of shakedown or low cycle fatigue, the meso-scale stress data were compared with the shakedown boundary of the FMMC. The meso-scale stresses were obtained from the micromechanical procedure as described in the previous article (part 1). In this homogenization procedure the micro-scale stresses were averaged over each lamina layer in a composite element producing meso-scale stresses. It is reminded that each composite element consisted of four lamina layers. Three components of the meso-scale stresses are presented for a cylindrical coordinate system in Figs. 3–5, respectively. The results are plotted for two distinct stages of the thermal history, that is, the residual stress state at room temperature and the heat flux loading state at steady state. Stress profiles along the three different paths (see Fig. 1) in the tube thickness direction are shown. The abscissa denotes the positions along the paths in the tube thickness direction starting from the bond interface toward the tube wall. The hoop component of the lamina stresses is plotted in Fig. 3. The hoop stresses were significantly relaxed upon heat flux loading. This stress reduction was ascribed to the strong decrease of temperature difference at the tube between the thermal loading stage (430–530 °C) and the stress free state (500 °C). The shape of the profiles showed an analogy with that of the fiber stresses. The maximum hoop stress appeared at the tube wall. This was a consequence of the volume averaging of the micro-scale stresses.
Path C-D (residual) Path E-F (residual) Path G-H (residual) Path C-D (heat loading) Path E-F (heat loading) Path G-H (heat loading)
1600
Hoop stress (MPa)
1400 1200 1000 800 600 400 200 0 -200
Tube stress
-400 0
0.5
1
1.5
2
Distance from the interface (mm) Fig. 3. Profiles of the meso-scale composite tube lamina stress (hoop component). Results are estimated for the residual stress state and for the heat flux loading state. Stress profiles are shown for the three paths indicated in Fig. 1. The abscissa denotes positions along the tube thickness direction starting from the bond interface toward the tube wall.
J.H. You / Engineering Fracture Mechanics 76 (2009) 1437–1449
Radial stress (MPa)
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1600 1400 1200 1000 800 600 400 200 0 -200 -400
Path C-D (residual) Path E-F (residual) Path G-H (residual) Path C-D (heat loading) Path E-F (heat loading) Path G-H (heat loading)
Tube stress
0
0.5
1
1.5
2
Distance from the interface (mm) Fig. 4. Profiles of the meso-scale composite tube lamina stress (radial component). Results are estimated for the residual stress state and for the heat flux loading state. Stress profiles are shown for the three paths indicated in Fig. 1. The abscissa denotes positions along the tube thickness direction starting from the bond interface toward the tube wall.
Longitudinal stress (MPa)
1600
Path C-D (residual) Path E-F (residual) Path G-H (residual) Path C-D (heat loading) Path E-F (heat loading) Path G-H (heat loading)
1400 1200 1000 800 600 400 200
Tube stress
0 -200 -400 0
0.5
1
1.5
2
Distance from the interface (mm) Fig. 5. Profiles of the meso-scale composite tube lamina stress (axial component). Results are estimated for the residual stress state and for the heat flux loading state. Stress profiles are shown for the three paths indicated in Fig. 1. The abscissa denotes positions along the tube thickness direction starting from the bond interface toward the tube wall.
The radial component of the lamina stresses is plotted in Fig. 4. The radial stresses distributed uniformly. The magnitude of the radial stresses was much smaller than the other two components and disappeared at the tube wall as it must be. The radial stresses almost disappeared (<50 MPa) in the whole tube during the heat flux loading. The longitudinal component (parallel to the tube axis) is plotted in Fig. 5. In the residual stress state the longitudinal stresses were uniform and quite strong in tension (up to 710 MPa). Upon heat flux loading, however, the longitudinal stresses became compressive (up to 370 MPa). The most critical region may be the tube wall since the hoop stress reached the peak value at the wall. On the other hand, the radial component disappeared at the tube wall. Hence, the tube wall region was loaded in a plane-stress state. In the following the problem of plastic failure at the tube wall region is treated. In Fig. 6 the lamina stress states at three tube wall positions (position D, F, and H) are compared with the computed shakedown boundary of the FMMC for the heat flux loading step. The lamina stresses at the tube wall were very close to the shakedown boundary. The stress states at the middle and lower part of the tube (position F and H) barely remained within the boundary whereas the stress at the upper tube part (position D) located just slightly beyond the boundary. Small fluctuations of thermal loads will lead to small perturbations of thermal stress so that the deviation of the stress states at the tube wall from the shakedown boundary will be also small. This result indicates that the maximum bi-axial stress states appearing at the tube wall will not cause any substantial plastic failure related to low cycle fatigue or ratcheting. 3.2. Results of probabilistic failure analysis The distribution of local risk of fracture during the heat flux loading is plotted in Fig. 8 in an arbitrary colour scale. This distribution illustrates the relative risk of local fracture which was computed as a failure probability density at individual FEM nodes with unit volume [18]. For a qualitative interpretation of this result, the thermal stress distribution in the tung-
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1200 Shakedown Elastic limit Position D Position F Position H
Axial stress (MPa).
800 400 0 -400 -800 -1200 -600
Stresses at the tube wall during heat flux loading -200
200
600
1000 1400 1800
Transverse stress (MPa) Fig. 6. Meso-scale lamina stress states under heat flux loading at three tube wall positions (position D, F, and H) compared with the computed shakedown boundary of the tungsten fiber-reinforced copper alloy matrix composite (unidirectional lamina). In addition, the local stress state at a laminar unit cell is illustrated schematically.
sten block under the heat flux load is presented in Fig. 7. Plotted are the fields of two stress components (rx, ry) in the tungsten block bonded to the copper alloy tube (a) and to the FMMC tube (b), respectively. The Heat sink tube is not shown. The highest risk of local fracture appeared near the free surface edges of the bond interface between the tungsten block and the copper tube due to the critical stress concentration appearing in this region. This correlation is clearly demonstrated by the exactly coinciding locations of the hot spots in Figs. 7 and 8. The multi-axial stress concentration appearing near the free surface edge of a bond interface between dissimilar materials is a well known phenomenon. The stress concentration behaviour of the tungsten/CuCrZr interface was theoretically investigated in [26] by means of a fracture mechanical approach. The estimated brittle failure probabilities of the tungsten block are summarized in Table 2 (for surface cracks) and Table 3 (for volume cracks). In general, the failure risk of surface cracks is different from that of volume cracks. Thus the maximum value of the two data sets should be taken for a conservative failure prediction. The listed results were obtained for the stress
Fig. 7. Distribution of thermal stresses under heat flux load in the tungsten block of the divertor component without (a) and with (b) the FMMC reinforcement. The Heat sink tube is not shown.
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Fig. 8. Distribution of the local risk of fracture in the tungsten block of the divertor component without (a) and with (b) the FMMC reinforcement. (Arbitrary colour scale).
Table 2 Failure probabilities under heat flux loading (estimated for surface cracks). m
b
Criteria
ro
PF,A (%) (FMMC tube)
PF,A (%) (CuCrZr tube)
19
2489
31
2353
Coplanar G Normal stress Hoop stress Max. G Coplanar G Normal stress Hoop stress Max. G
2907 2856 3097 3068 2566 2537 2705 2681
0.0059 0.0062 0.0038 0.0042 0.0008 0.0009 0.0004 0.0005
0.0239 0.0262 0.0132 0.0152 0.0065 0.0072 0.0025 0.0032
state under heat flux loading. The failure probability data computed for the residual stress state are not given here since the results were very similar to the thermal loading case. The failure probabilities were computed using four different fracture criteria. In the case of surface cracks, the coplanar energy release rate criterion and the normal stress criterion produced similar results whereas the maximum hoop stress criterion and the maximum energy release rate criterion yielded similar values. In the case of volume cracks, the results were very similar except for the normal stress criterion. The maximum of the estimated failure probabilities was 0.028%. This magnitude seems to be sufficiently small to be accepted. For comparison the failure probability data estimated for a tungsten block joined with an unreinforced copper alloy tube are also presented in the tables. It is found that the failure probability of the tungsten block was strongly reduced through the FMMC reinforcement of the tube. This positive implication of the composite reinforcement can be attributed to the significantly reduced stress concentration in the tungsten block near the free surface edges of the bond interface as demonstrated in Fig. 7. It was also demonstrated that the brittle failure risk of the embrittled tungsten block will not be excessively critical during the whole component life. The FMMC reinforcement of the tube contributed positively to the reliability of the tungsten block.
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J.H. You / Engineering Fracture Mechanics 76 (2009) 1437–1449 Table 3 Failure probabilities under heat flux loading (estimated for volume cracks). m
b
Criteria
ro
PF,V (%) (FMMC tube)
PF,V (%) (CuCrZr tube)
19
2489
31
2353
Coplanar G Normal stress Hoop stress Max. G Coplanar G Normal stress Hoop stress Max. G
2280 2134 2618 2548 2163 2046 2453 2394
0.0202 0.0279 0.0189 0.0185 0.0085 0.0169 0.0068 0.0065
0.0579 0.0793 0.0572 0.0521 0.0378 0.0733 0.0355 0.0292
4. Conclusions In this work computational schemes of the meso- and macro-scale failure analysis were presented for a fiber-reinforced composite component which was the same as in the foregoing paper. In the meso-scale failure analysis, plastic instability of the composite cooling tube could be estimated by means of the shakedown analysis of the fiber-composite. The computed shakedown boundary was considered as a plastic failure criterion. The meso-scale tube lamina stresses obtained from the micromechanical homogenization were compared with the shakedown boundary. The tube wall stresses located very close to the shakedown boundary under the assumed heat flux load. This result indicated that the maximum stress state appearing at the tube wall will not cause any substantial plastic failure related to low cycle fatigue or ratcheting. In the macro-scale failure analysis, the probabilistic failure analysis could be successfully applied for the brittle tungsten block using four different mixed-mode fracture criteria. The predictions obtained from the four fracture criteria were relatively uniform. The estimated failure probability reached up to 0.028% which could be regarded as sufficiently small for engineering practice. It was also found that the failure risk of the tungsten block was strongly reduced through the composite reinforcement of the tube. This positive effect was attributed to intensification of the compressive stress fields in the tungsten domain near the tube. Appendix A. Shakedown theorem and analysis In the formulation of the static shakedown theorem for a FMMC the relevant stress fields has to be considered on triple length scales. The applied loads in a bi-axial stress space are expressed with macroscopic stress states (for whole unit cell) whereas the actual shakedown condition is described with microscopic stress fields (matrix of unit cell). On the macroscopic scale the field quantities are averaged over the representative volume element, which in the present study corresponds to either the lamina or the laminate unit cell. We denote the coordinates for the macroscopic and microscopic length scales with X and x, respectively. According to Hill, the homogenization relations are expressed as [27]
RðXÞ ¼ hrðxÞi ¼
1
X
Z
rðxÞdX;
ðA1Þ
X
where R is the macroscopic stress tensor, r the microscopic stress tensor and X indicates the domain occupied by the unit cell. The microscopic elastic stress states of a FMMC re constitute the elastically admissible domain P of the macroscopic load space R below the elastic limit. This relation is expressed as
P ¼ fRj9re ; 8re ðxÞ 2 FðxÞg
ðA2Þ
FðxÞ ¼ fre jf ðre ; xÞ < 0; 8x 2 Xg
ðA3Þ
for the von Mises yield function f(rs, x). The static shakedown theorem for a FMMC can be formulated as follows [11]: the composite material is in the shakedown state under given domain of a load space, if there exist a real number a > 1, a time-independent periodic residual stress field r^ r satisfying given boundary conditions and an admissible domain P^ of macroscopic load space R to be defined by the safe states of stresses rs obeying the relations
^ ¼ fRj9rs ; rs ðxÞ 2 FðxÞg ^ P
ðA4Þ
hrs ðxÞi ¼ RðXÞ ^ FðxÞ ¼ frs jf ðrs ; xÞ < 0; 8x 2 Xg
ðA5Þ ðA6Þ
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Here, the safe stress states rs are defined by
rs ¼ arc þ r^ r :
ðA7Þ
^ r satisfies the equilibrium equations and boundary condition where r
^ r ¼ 0 in X divr
ðA8Þ
r^ r n ¼ 0 on @ X
ðA9Þ
^ r i ¼ 0 in X hr
ðA10Þ
and rc denotes the elastic stress field in the virtual, purely elastic counterpart of the actual FMMC unit cell under the same boundary and equilibrium conditions as in the original problem such that
rc n ¼ R n on @ X divrc ¼ 0 in X
ðA11Þ ðA12Þ
^ r (A10) leads to Discretisation of the self-equilibrium equation of the time-independent residual stress field r
^ r ðyji Þg ¼ 0 ½Cfr
ðA13Þ
where the residual stress fr g is given in a vector form at each integration point ^r
½C ¼
( NE X NGE X j¼1
yji .
The matrix [C] is expressed as
) wi det½J j ðni Þ½Bðyji Þt
ðA14Þ
i¼1
where [J] is the Jacobian transformation matrix for the shape function of finite elements and [B] is the matrix of partial derivatives of the shape function. NE and NGE stand for the number of finite elements and the number of Gauss integration points in an element, respectively. The computational procedure to determine the shakedown boundary of the considered FMMC consists of four steps: 1. Computation of elastic stress solutions by FEM for the FMMC unit cell models to be considered. 2. Setting up a system of equations of constraints for time-independent residual stress for the used FEM models considering prescribed kinematic boundary conditions. 3. Formulation of an optimisation problem by combining the shakedown conditions and the equilibrium constraints of residual stress at each Gauss integration point and for each load corner in a loading space. 4. Determination of the safety factors by means of mathematical programming process. By multiplying the safety factors with the initially assumed load corners a shakedown boundary was obtained.
Appendix B. Weibull theory of brittle failure [18] In STAU, volume flaws and surface flaws are modeled as penny-shaped cracks and through-wall cracks, respectively. The singular crack tip stress fields under mixed-mode loads are expressed in terms of the stress intensity factors K corresponding to three fracture modes
pffiffiffi K I ¼ rn Y I a;
pffiffiffi K II ¼ sII Y II a;
pffiffiffi K III ¼ sIII Y III a;
ðB1Þ
where Y denotes geometric correction factor and rn and sII, sIII the normal and shear projections of the stress tensor on the crack plane which are computed by the tensor transformation rules. The mixed-mode failure criterion for a multi-axial stress field has a generic form
gðrn ; sII ; sIII Þ P g c
ðB2Þ
In STAU, the effective failure criterion is employed which is formulated with equivalent mode I stress intensity factor as follows:
gðK Ieq ; 0; 0Þ ¼ gðK I ; K II ; K III Þ
ðB3Þ
Then the failure criterion is reformulated as
K Ieq P K Ic ;
ðB4Þ
where KIc stands for the mode I fracture toughness. By analogy to Eq. (B1), an equivalent stress req is defined by
pffiffiffi K Ieq req Y I a
ðB5Þ
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Finally, the critical crack size ac is given by
ac ¼
K Ic req Y I
2 ðB6Þ
Failure by spontaneous crack extension occurs if the crack size a exceeds ac. As the size, the location and the orientation are random variables, the strength is also a random variable for which statistical treatment is required. For statistically independent infinitesimal volume elements dV the actual number n of cracks contained in a component is a Poisson distributed random variable. The probability density function of the crack size a is expressed by a Pareto distribution by
fa ðaÞ ¼ 1
aor1 ar1
ðB7Þ
where a P ao. For a multi-axially loaded brittle component containing randomly oriented sharp volume cracks of an arbitrary (large) number, the failure probability due to the unstable crack propagation is formulated as
Z Z 1 1 req ðx; xÞ PF;V ¼ 1 exp dXdV V o V 4p X ro
ðB8Þ
where Vo is the unit volume containing an average number of Mo cracks. m and ro are material parameters expressed as
m ¼ 2ðr 1Þ
ro ¼
1 1 m
ðMo Þ
ðB9Þ
K Ic pffiffiffiffiffi Y I ao
ðB10Þ
Further, Eq. (A8) can be rewritten in terms of a 2-parameter Weibull distribution
m r PF;V ¼ 1 exp b
ðB11Þ
where r* denotes a reference stress characterizing the load level. The Weibull parameter b is related to ro by
b ¼ ro
1 Vo
Z V
Z
1 4p
X
req ðx; xÞ r
m
dXdV
m1 ðB12Þ
Eqs. (B11) and (B12) suggest that the required parameters m and ro can be obtained from a series of uniaxial tests where the shape parameter m and the scale parameter b are estimated from the measured strength data of simple specimens using the Weibull statistics. At the stage of parameter calibration the effective stress in Eq. (B12) refers to the stress state produced by the testing. In the case of surface cracks of the length 2a normal to the surface, the failure probability PF,A of a component containing arbitrary number of randomly oriented sharp surface cracks of random size and random location is given in the same line as PF,V
Z Z 1 1 req ðx; xÞ m PF;A ¼ 1 exp dXdA Ao V 2 p X ro
ðB13Þ
Then the Weibull parameter b is related to ro by
b ¼ ro
1 Ao
Z A
1 4p
Z X
req ðx; xÞ r
m
m1 dXdA
ðB14Þ
The domain and the orientation integrals are estimated numerically by means of Gauss quadrature method for each finite element. To this end, the analytical formulation for PF,V and PF,A has to be converted into a discretized form, for instance
" PF;V ¼ 1 exp "
# Z Ne Z ~Þ m 1 X 1 req ð~x; x dXdV V o e¼1 V e 4p X ro
Z 1 Z 1 Ne Z 1 Z 1 Z 1 1 X 1 ¼ 1 exp V o e¼1 1 1 1 4p 1 1
req ð~n; ~hÞ ro
!m
# ~ ~ ~ ~ J V ðnÞ JX ðhÞ dh dn
ðB15Þ
where Ne is the number of elements of the FEM model, Ve the volume of the e th element, n and h are the coordinates in the reference configuration considered for the numerical integration using the Gauss quadrature and JV and JX denote Jacobians of the corresponding variable transformations.
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The effective fracture criteria and the equivalent stresses are listed below [18] (1) The coplanar energy release rate criterion [22]
K Ieq ¼ ½K 2I þ K 2II þ "
req ¼ r2I þ s2II
1 K 2 0:5 1 m III
ðB16Þ
2 #0:5 Y II s2 Y III 2 þ III YI 1 m YI
ðB17Þ
(2) The normal stress criterion [23]
K Ieq ¼ K I
ðB18Þ
req ¼ rn
ðB19Þ
(3) The maximum hoop stress criterion [24]
K Ieq
req
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 8 2K I þ 6 K 2I þ 8K 2II K 3II ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:5 K 2I þ 12K 2II K I K 2I þ 8K 2II
ðB20Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 8 2rn þ 6 r2n þ 8s2II ðY II =Y I Þ2 s3II ðY II =Y I Þ3 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:5 r2n þ 12s2II ðY II =Y I Þ2 rn r2n þ 8s2II ðY II =Y I Þ2
ðB21Þ
(4) The maximum energy release rate criterion [25]
h i0:25 K Ieq ¼ K 4I þ 6K 2I K 2II þ K 4II h
req ¼ r4n þ 6r2n s2II ðY II =Y I Þ2 þ s4II ðY II =Y I Þ4
ðB22Þ i0:25
ðB23Þ
References [1] A conceptual study of commercial fusion power plants. Final report of the European Fusion Power Plant Conceptual Study (PPCS), EFDA-RP-RE-5.0; 2004. [2] Chen HF, Ponter ARS. On the behaviour of a particulate metal matrix composite subjected to cyclic temperature and constant stress. Comp Mater Sci 2005;34(4):425–41. [3] Miskiewicz M, You JH. Impact of plastic softening of over-aged CuCrZr alloy heat sink tube on the structural reliability of a plasma-facing component. Fus Engng Des (in press) doi:10.1016/j.fusengdes.2007.06.002. [4] König JA, Maier G. Shakedown analysis of elastic-plastic structures: a review of recent developments. Nucl Engng Des 1981;66(1):81–95. [5] Stein E, Zhang G, Huang Y. Modeling and computation of shakedown problems for nonlinear hardening materials. Comput Meth Appl Mech Engng 1993;103(1–2):247–72. [6] Staat M, Heitzer M. LISA a European project for FEM-based limit and shakedown analysis. Nucl Engng Des 2001;206(2-3):151–66. [7] Tirosh J. The dual shakedown conditions for dilute fibrous composites. J Mech Phys Solids 1998;46(1):167–85. [8] Weichert D, Hachemi A, Schwabe F. Application of shakedown analysis to the plastic design of composites. Arch Appl Mech 1999;69:623–33. [9] Magoariec H, Bourgeois S, Debordes O. Elastic plastic shakedown of 3D periodic heterogeneous media: a direct numerical approach. Int J Plasticity 2004;20(1–9):1655–75. [10] Carvelli V. Shakedown analysis of unidirectional fiber reinforced metal matrix composites. Comp Mater Sci 2004;31(1):24–32. [11] Hachemi A, Weichert D. On the problem of interfacial damage in fibre-reinforced composites under variable loads. Mech Res Commun 2005;32(1):15–23. [12] Hachemi A, Mouhtamid S, Weichert D. Progress in shakedown analysis with applications to composites. Arch Appl Mech 2005;74:762–72. [13] Baluc N. Assessment report on W. Final report on EFDA task TW1-TTMA-002 Deliverable 5; 2002. [14] Ishijima Y, Kurishita H, Yubuta K, Arakawa H, Hasegawa M, Hiraoka Y, et al. Current status of ductile tungsten alloy development by mechanical alloying. J Nucl Mater 2004;329–333:775–9. [15] Melan E. Zur Plastizität des räumlichen Kontinuums. Ingenieur Archiv 1938;9:116–26. [16] Conn AR, Gould NIM, Toint PL. LANCELOT: a FORTRAN package for large-scale nonlinear optimization (release A). Berlin: Springer-Verlag; 1992. [17] You J-H, Kim BY, Miskiewicz M. Shakedown analysis of fibre-reinforced copper matrix composites by direct and incremental approaches. Mech Mater, in press, doi:10.1016/j.mechmat.2008.10.007. [18] Brückner-Foit A, Heger A, Heiermann K, Hülsmeier P, Mahler A, Mann A, et al. STAU 4 User’s manual, Forschungszentrum Karlsruhe, Karlsruhe; 2003. [19] Batdorf SB, Heinisch HL. Weakest link theory reformulated for arbitrary fracture criterion. J Am Ceram Soc 1978;61(7–8):355–8. [20] Thiemeier T, Brückner-Foit A, Kölker HJ. Influence of the fracture criterion on the failure prediction of ceramics loaded in biaxial flexure. J Am Ceram Soc 1991;74(1):48–52. [21] Munz D, Fett T. Ceramics mechanical properties, failure behaviour, materials selection. Berlin-Heidelberg: Springer-Verlag; 1999.
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Paris PC, Sih GC. ASTM STP 381, Am. Soc. Test. Mater; 1965. p. 30–83. Broek D. Elementary engineering fracture mechanics. Dordrecht: Martinus Nijhoff Publishers; 1986. Erdogan F, Sih GC. J Basic Engng 1963;85:519–27. Hellen TK, Blackburn WS. The calculation of stress intensity factors for combined tensile and shear loading. Int J Fracture 1975;11(4):605–17. You JH, Bolt H. Analysis of singular interface stress in dissimilar material joints for plasma facing components. J Nucl Mater 2001;299:1–8. Hill R. Elastic properties of reinforced solids some theoretical principles. J Mech Phys Solid 1963;11(5):357–72.
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Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Triple-scale failure estimation for a composite-reinforced structure based on integrated modeling approaches. Part 2: Meso- and macroscopic scale analysis J.H. You * Max-Planck-Institut für Plasmaphysik, Euratom Association, Boltzmannstr. 2, 85748 Garching, Germany
a r t i c l e
i n f o
Article history: Received 26 September 2007 Accepted 31 October 2008 Available online 17 November 2008 Dedicated to Professor Wolfgang Brocks on the occasion of his 65th birthday. Keywords: Metal matrix composite Finite element analysis Probabilistics Weibull theory Shakedown Failure assessment
a b s t r a c t In this article, the meso- and macroscopic failure features are discussed considering the identical composite component as in the foregoing article. In the meso-scale failure analysis, the risk of plastic instability of the composite tube was estimated considering the shakedown boundary as a failure criterion. The meso-scale stresses of the composite tube were computed using micromechanical homogenization and compared with the shakedown boundary of the composite obtained from the direct shakedown analysis. The stress states were close to the shakedown boundary indicating no critical danger of plastic failure. In the macro-scale failure analysis, the mechanical influence of the local composite integration was investigated with regard to the brittle failure risk of the neutron-embrittled component. To this end, a probabilistic failure analysis code was applied which was based on the fracture mechanics and the weakest-link failure theory. Various fracture criteria were considered. It was found that the failure risk of the tungsten block was strongly reduced by the composite reinforcement of the tube due to the intensification of compressive stress fields. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction In the foregoing companion article (part 1), the microscopic failure features of a composite component was treated. A plasma-facing component for fusion reactors was considered whose cooling tube was locally reinforced with fiber metal matrix composite (FMMC). The focus of the microscopic failure analysis was placed on the estimation of the fiber fracture risk and the plastic damage of the ductile matrix. Non-linear triple-scale FEM was applied using the micromechanics-based homogenization algorithm. In the present article (part 2), meso- and macroscopic failure features of the same FMMC component are discussed (see Fig. 1). The failure problems are treated in two lines, that is, plastic instability of the FMMC tube (meso-scale) and brittle failure of the tungsten armor block (macro-scale). 1.1. Meso-scale failure analysis: shakedown behavior This issue is related to the plastic flow in the ductile matrix which is locally concentrated near the fibers. The plastic behavior of the FMMC is detrimentally affected by variation of the applied loads. Although major plasma instability will
* Tel.: +49 89 3299 1373. E-mail address: [email protected] 0013-7944/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2008.10.017
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Nomenclature Latin symbols a crack size critical crack size ac b Weibull parameter (scale parameter) [B] matrix of partial derivatives of the shape function [J] Jacobian transformation matrix for the shape function of an element von Mises yield function f(rs, x) g(KIeq, 0, 0) effective failure criterion g(rn, sII, sIII) mixed-mode failure criterion Jacobians of variable transformation JV, JX K stress intensity factor mode I fracture toughness KIc equivalent mode l stress intensity factor KIeq m Weibull modulus (shape parameter) number of elements of the FEM model Ne P elastically admissible domain failure probability due to unstable propagation of surface cracks PF,A failure probability due to unstable propagation of volume cracks PF,V volume of the eth element Ve dV infinitesimal volume element X coordinate for macroscopic length scale x coordinate for microscopic length scale Y geometric correction factor Greek symbols h element coordinate in the reference configuration R macroscopic stress tensor r microscopic stress tensor rc elastic stress field in virtual elastic matrix re microscopic elastic stress state req equivalent stress rn normal projection of stress tensor on crack plane ro material parameter in a 2-parameter Weibull distribution r^ r time-independent periodic residual stress field rs safe states of stress r* reference stress characterizing a load level sII, sIII shear projection of stress tensor on crack plane n element coordinate in the reference configuration X domain occupied by the unit cell
be suppressed in fusion power plants, unexpected minor plasma instability may possibly appear generating heat flux fluctuations [1]. This thermal load perturbation will lead to a thermal stress variation in the plasma-facing component. Depending on the stress state and loading history, the plastic response of a FMMC under variable loads will show one of the following behaviors: elastic shakedown, alternating plasticity (low cycle fatigue) or progressive straining (ratcheting). Such plastic responses can be most suitably described on a mesoscopic length scale (e.g. length scale of a unit cell or a representative volume element) [2]. According to the cyclic plasticity FEM study of a conventional plasma-facing component, low cycle fatigue of the softened copper alloy cooling tube can be a critical failure problem [3]. Hence, shakedown limit may be regarded as an appropriate failure criterion in the context of failure assessment of the FMMC tube under variable loads. The FMMC, which was initially loaded in a plastic regime, eventually reach elastic state under arbitrarily varying load, provided that the peak load is bounded within the shakedown limit. In this circumstance the plastic strain ceases to develop and overall FMMC response is confined within safe elastic regime regardless of the loading path thereafter. When maximum load is located beyond this limit, the FMMC component will undergo either low cycle fatigue or incremental plastic collapse (ratcheting) [4–6]. Recently, several theoretical works were reported in the literature in which shakedown boundaries of FMMC under various loading conditions were computed using the FEM-based direct shakedown analysis method [7–12]. On the other hand, theoretical study to investigate the shakedown behavior of a composite component is not found yet.
J.H. You / Engineering Fracture Mechanics 76 (2009) 1437–1449
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Fig. 1. Geometry (symmetric half) and the finite element mesh of the model composite component consisting of tungsten armor block and FMMC heat sink cooling tube. FMMC was a unidirectional copper alloy matrix composite-reinforced with tungsten fibers wound in the tube hoop direction (fiber volume fraction: 40%).
In the first part of this article, risk of plastic failure of the FMMC part is estimated in terms of the shakedown criterion. The FMMC stress and strain were homogenized for individual meso-scale lamina layers. The meso-scale stresses were compared with the shakedown boundary of the FMMC computed for the specific loading condition of the current FMMC component. 1.2. Macro-scale failure analysis: brittle fracture behavior This issue is related to the stress intensification effect caused by the FMMC integration in the tungsten block which is inherently brittle. In general, integration of a FMMC part into a component will affect the overall stress field of the component. When the FMMC has a high stiffness, the local FMMC integration will lead to intensification of thermal stress in the component system. The brittleness problem can be a crucial issue regarding structural reliability when the tungsten block is subjected to a dangerous stress state. The situation will be even more acute during nuclear fusion operation since severe neutron irradiation will cause further embrittlement of tungsten [13,14]. Therefore, the fracture mechanical impact of local FMMC integration should be estimated in the context of structural reliability of the whole composite component. The fracture mechanical failure assessment of a brittle structure requires a statistical treatment due to strong scattering of strength data. The stress analysis needed for the failure assessment is carried out in the framework of a macro-scale FEM model using normal solid elements. In the second part of this article, the impact of FMMC reinforcement for the coolant tube is investigated with regard to the risk of brittle failure of the tungsten block. To this end, a probabilistic failure analysis code was applied. This code was based on the weakest-link failure theory and the linear elastic fracture mechanics (LEFM). Fully elasto-plastic FEM analysis verified that no plastic yielding occurred in the tungsten block. This result justified the application of the weakest-link failure theory based on the LEFM.
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2. Theoretical background 2.1. Direct shakedown analysis The shakedown boundary of the FMMC considered in the current study was obtained using the FEM-based shakedown analysis method developed by Weichert and coworkers [8]. This was the so called direct method based on the Melan’s static shakedown theorem [15]. Detailed description of the theoretical background is found in the literature [8] (see Appendix A). To calculate the shakedown limit for a specific loading condition, following requirements have to be fulfilled: 1. The time-independent residual stress field must satisfy static equilibrium equations and boundary conditions at every nodes and for whole loading domain. 2. The Melan’s shakedown condition must be satisfied at every node and for whole loading domain. In the FEM-based direct shakedown analysis, the virtual elastic stress fields are required (i.e. elastic stress solution from an elastic FEM analysis). The analytical formulation of the shakedown theorem is transformed into a discretized form which is compatible with the finite element structure being used. The equilibrium equation for the time-independent residual stress is also discretized in a similar way (i.e., the [C] matrix in Appendix A has to be computed). Then, the shakedown conditions are checked at each Gauss integration points. The inequality constraints of the shakedown conditions constitute a constrained maximization problem together with the equilibrium constraint of the residual stress. In order to solve this maximization problem a large-scale non-linear optimization code LANCELOT was used [16]. The computational procedure is described in Appendix A. Shakedown boundary was computed for the FMMC lamina considering bi-axial combined loading condition consisting of a longitudinal (parallel to the fiber axis) and a transverse (perpendicular to the fiber axis) variable traction [17]. This specific combined bi-axial load corresponds to the plane-stress state appearing at the cooling tube wall and the fiber architecture. 2.2. Probabilistic failure analysis In this work the probabilistic failure analysis code STAU was used [18]. This code was developed as a post-processor for commercial FEM codes from which the stress data at each integration point were imported into STAU for further failure probability computation. In STAU the probabilistic description of component failure was based on the weakest-link failure theory formulated with Weibull parameters. The theoretical basis is presented in Appendix B. According to the weakest-link theory, failure of a component is triggered by unstable propagation of the most dangerous flaw which is inherently contained in that component [19]. According to the LEFM, the most critical flaw is determined by the most unfavorable combination of size a, location x and orientation x for a given stress field r. For simplicity, only sharp planar cracks are considered here. Random orientation of cracks and multi-axial stress fields necessarily cause a mixed-mode crack tip loading. In practice, mixed-mode fracture criteria are reformulated into an effective criterion in terms of an equivalent mode I stress intensity factor [20,21]. Usually, different mixed-mode criteria predict different failure risks. In this study, four different fracture criteria implemented in STAU were considered as below (see Appendix B): (1) The coplanar energy release rate criterion (coplanar G) [22]. (2) The normal stress criterion (normal stress) [23].
2 4-point bending test
LnLn(1/Ps).
1 0 -1 -2 -3
heat treated (1000 °C, 10 h) as-received
-4 -5 7.6
7.7 7.8 Ln(strength)
7.9
Fig. 2. Weibull plots of tungsten bending strength measured at room temperature (high purity warm rolled sheets). Strength data are given for two specimen groups: heat-treated specimens (heated at 1000 °C in vacuum for 10 h) and as-received ones. Dimension of the specimens was 50 5 1 mm3. (Solid circle: heat-treated specimens, open circle: as-received specimens).
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J.H. You / Engineering Fracture Mechanics 76 (2009) 1437–1449 Table 1 Measured Weibull parameters.
As-received (rolled at 350 °C) Heat-treated (1000 °C, 10 h)
Shape parameter m
Scale parameter b
19 31
2489 2353
(3) The maximum hoop stress criterion (hoop stress) [24]. (4) The maximum energy release rate criterion (max. G) [25]. In order to obtain the input material data required for the STAU analysis, the Weibull parameters of commercial high purity tungsten were experimentally estimated. Warm rolled (at 350 °C) tungsten sheets were used for specimen preparation. Strength data were obtained from four-point bending tests at room temperature. The dimension of the specimens was 50 5 1 mm3. For comparison, heat-treated specimens (annealing at 1000 °C in vacuum for 10 h) as well as the asreceived ones were tested. The measured Weibull plots of the two specimen groups are presented in Fig. 2. Both Weibull plots indicate that the strength distribution clearly obeyed 2-parameter Weibull distributions. The estimated Weibull parameters are given in Table 1. The magnitude of the shape parameter (Weibull modulus) was significantly increased by the heat treatment whereas the scale parameter (characteristic strength) was slightly decreased. But, the metallurgical reason for this effect was unclear. The measured tungsten Weibull modulus was rather larger compared to typical engineering ceramics. 3. Results and discussion 3.1. Results of shakedown analysis To evaluate the plastic behavior of the FMMC tube in terms of shakedown or low cycle fatigue, the meso-scale stress data were compared with the shakedown boundary of the FMMC. The meso-scale stresses were obtained from the micromechanical procedure as described in the previous article (part 1). In this homogenization procedure the micro-scale stresses were averaged over each lamina layer in a composite element producing meso-scale stresses. It is reminded that each composite element consisted of four lamina layers. Three components of the meso-scale stresses are presented for a cylindrical coordinate system in Figs. 3–5, respectively. The results are plotted for two distinct stages of the thermal history, that is, the residual stress state at room temperature and the heat flux loading state at steady state. Stress profiles along the three different paths (see Fig. 1) in the tube thickness direction are shown. The abscissa denotes the positions along the paths in the tube thickness direction starting from the bond interface toward the tube wall. The hoop component of the lamina stresses is plotted in Fig. 3. The hoop stresses were significantly relaxed upon heat flux loading. This stress reduction was ascribed to the strong decrease of temperature difference at the tube between the thermal loading stage (430–530 °C) and the stress free state (500 °C). The shape of the profiles showed an analogy with that of the fiber stresses. The maximum hoop stress appeared at the tube wall. This was a consequence of the volume averaging of the micro-scale stresses.
Path C-D (residual) Path E-F (residual) Path G-H (residual) Path C-D (heat loading) Path E-F (heat loading) Path G-H (heat loading)
1600
Hoop stress (MPa)
1400 1200 1000 800 600 400 200 0 -200
Tube stress
-400 0
0.5
1
1.5
2
Distance from the interface (mm) Fig. 3. Profiles of the meso-scale composite tube lamina stress (hoop component). Results are estimated for the residual stress state and for the heat flux loading state. Stress profiles are shown for the three paths indicated in Fig. 1. The abscissa denotes positions along the tube thickness direction starting from the bond interface toward the tube wall.
J.H. You / Engineering Fracture Mechanics 76 (2009) 1437–1449
Radial stress (MPa)
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1600 1400 1200 1000 800 600 400 200 0 -200 -400
Path C-D (residual) Path E-F (residual) Path G-H (residual) Path C-D (heat loading) Path E-F (heat loading) Path G-H (heat loading)
Tube stress
0
0.5
1
1.5
2
Distance from the interface (mm) Fig. 4. Profiles of the meso-scale composite tube lamina stress (radial component). Results are estimated for the residual stress state and for the heat flux loading state. Stress profiles are shown for the three paths indicated in Fig. 1. The abscissa denotes positions along the tube thickness direction starting from the bond interface toward the tube wall.
Longitudinal stress (MPa)
1600
Path C-D (residual) Path E-F (residual) Path G-H (residual) Path C-D (heat loading) Path E-F (heat loading) Path G-H (heat loading)
1400 1200 1000 800 600 400 200
Tube stress
0 -200 -400 0
0.5
1
1.5
2
Distance from the interface (mm) Fig. 5. Profiles of the meso-scale composite tube lamina stress (axial component). Results are estimated for the residual stress state and for the heat flux loading state. Stress profiles are shown for the three paths indicated in Fig. 1. The abscissa denotes positions along the tube thickness direction starting from the bond interface toward the tube wall.
The radial component of the lamina stresses is plotted in Fig. 4. The radial stresses distributed uniformly. The magnitude of the radial stresses was much smaller than the other two components and disappeared at the tube wall as it must be. The radial stresses almost disappeared (<50 MPa) in the whole tube during the heat flux loading. The longitudinal component (parallel to the tube axis) is plotted in Fig. 5. In the residual stress state the longitudinal stresses were uniform and quite strong in tension (up to 710 MPa). Upon heat flux loading, however, the longitudinal stresses became compressive (up to 370 MPa). The most critical region may be the tube wall since the hoop stress reached the peak value at the wall. On the other hand, the radial component disappeared at the tube wall. Hence, the tube wall region was loaded in a plane-stress state. In the following the problem of plastic failure at the tube wall region is treated. In Fig. 6 the lamina stress states at three tube wall positions (position D, F, and H) are compared with the computed shakedown boundary of the FMMC for the heat flux loading step. The lamina stresses at the tube wall were very close to the shakedown boundary. The stress states at the middle and lower part of the tube (position F and H) barely remained within the boundary whereas the stress at the upper tube part (position D) located just slightly beyond the boundary. Small fluctuations of thermal loads will lead to small perturbations of thermal stress so that the deviation of the stress states at the tube wall from the shakedown boundary will be also small. This result indicates that the maximum bi-axial stress states appearing at the tube wall will not cause any substantial plastic failure related to low cycle fatigue or ratcheting. 3.2. Results of probabilistic failure analysis The distribution of local risk of fracture during the heat flux loading is plotted in Fig. 8 in an arbitrary colour scale. This distribution illustrates the relative risk of local fracture which was computed as a failure probability density at individual FEM nodes with unit volume [18]. For a qualitative interpretation of this result, the thermal stress distribution in the tung-
J.H. You / Engineering Fracture Mechanics 76 (2009) 1437–1449
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1200 Shakedown Elastic limit Position D Position F Position H
Axial stress (MPa).
800 400 0 -400 -800 -1200 -600
Stresses at the tube wall during heat flux loading -200
200
600
1000 1400 1800
Transverse stress (MPa) Fig. 6. Meso-scale lamina stress states under heat flux loading at three tube wall positions (position D, F, and H) compared with the computed shakedown boundary of the tungsten fiber-reinforced copper alloy matrix composite (unidirectional lamina). In addition, the local stress state at a laminar unit cell is illustrated schematically.
sten block under the heat flux load is presented in Fig. 7. Plotted are the fields of two stress components (rx, ry) in the tungsten block bonded to the copper alloy tube (a) and to the FMMC tube (b), respectively. The Heat sink tube is not shown. The highest risk of local fracture appeared near the free surface edges of the bond interface between the tungsten block and the copper tube due to the critical stress concentration appearing in this region. This correlation is clearly demonstrated by the exactly coinciding locations of the hot spots in Figs. 7 and 8. The multi-axial stress concentration appearing near the free surface edge of a bond interface between dissimilar materials is a well known phenomenon. The stress concentration behaviour of the tungsten/CuCrZr interface was theoretically investigated in [26] by means of a fracture mechanical approach. The estimated brittle failure probabilities of the tungsten block are summarized in Table 2 (for surface cracks) and Table 3 (for volume cracks). In general, the failure risk of surface cracks is different from that of volume cracks. Thus the maximum value of the two data sets should be taken for a conservative failure prediction. The listed results were obtained for the stress
Fig. 7. Distribution of thermal stresses under heat flux load in the tungsten block of the divertor component without (a) and with (b) the FMMC reinforcement. The Heat sink tube is not shown.
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Fig. 8. Distribution of the local risk of fracture in the tungsten block of the divertor component without (a) and with (b) the FMMC reinforcement. (Arbitrary colour scale).
Table 2 Failure probabilities under heat flux loading (estimated for surface cracks). m
b
Criteria
ro
PF,A (%) (FMMC tube)
PF,A (%) (CuCrZr tube)
19
2489
31
2353
Coplanar G Normal stress Hoop stress Max. G Coplanar G Normal stress Hoop stress Max. G
2907 2856 3097 3068 2566 2537 2705 2681
0.0059 0.0062 0.0038 0.0042 0.0008 0.0009 0.0004 0.0005
0.0239 0.0262 0.0132 0.0152 0.0065 0.0072 0.0025 0.0032
state under heat flux loading. The failure probability data computed for the residual stress state are not given here since the results were very similar to the thermal loading case. The failure probabilities were computed using four different fracture criteria. In the case of surface cracks, the coplanar energy release rate criterion and the normal stress criterion produced similar results whereas the maximum hoop stress criterion and the maximum energy release rate criterion yielded similar values. In the case of volume cracks, the results were very similar except for the normal stress criterion. The maximum of the estimated failure probabilities was 0.028%. This magnitude seems to be sufficiently small to be accepted. For comparison the failure probability data estimated for a tungsten block joined with an unreinforced copper alloy tube are also presented in the tables. It is found that the failure probability of the tungsten block was strongly reduced through the FMMC reinforcement of the tube. This positive implication of the composite reinforcement can be attributed to the significantly reduced stress concentration in the tungsten block near the free surface edges of the bond interface as demonstrated in Fig. 7. It was also demonstrated that the brittle failure risk of the embrittled tungsten block will not be excessively critical during the whole component life. The FMMC reinforcement of the tube contributed positively to the reliability of the tungsten block.
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J.H. You / Engineering Fracture Mechanics 76 (2009) 1437–1449 Table 3 Failure probabilities under heat flux loading (estimated for volume cracks). m
b
Criteria
ro
PF,V (%) (FMMC tube)
PF,V (%) (CuCrZr tube)
19
2489
31
2353
Coplanar G Normal stress Hoop stress Max. G Coplanar G Normal stress Hoop stress Max. G
2280 2134 2618 2548 2163 2046 2453 2394
0.0202 0.0279 0.0189 0.0185 0.0085 0.0169 0.0068 0.0065
0.0579 0.0793 0.0572 0.0521 0.0378 0.0733 0.0355 0.0292
4. Conclusions In this work computational schemes of the meso- and macro-scale failure analysis were presented for a fiber-reinforced composite component which was the same as in the foregoing paper. In the meso-scale failure analysis, plastic instability of the composite cooling tube could be estimated by means of the shakedown analysis of the fiber-composite. The computed shakedown boundary was considered as a plastic failure criterion. The meso-scale tube lamina stresses obtained from the micromechanical homogenization were compared with the shakedown boundary. The tube wall stresses located very close to the shakedown boundary under the assumed heat flux load. This result indicated that the maximum stress state appearing at the tube wall will not cause any substantial plastic failure related to low cycle fatigue or ratcheting. In the macro-scale failure analysis, the probabilistic failure analysis could be successfully applied for the brittle tungsten block using four different mixed-mode fracture criteria. The predictions obtained from the four fracture criteria were relatively uniform. The estimated failure probability reached up to 0.028% which could be regarded as sufficiently small for engineering practice. It was also found that the failure risk of the tungsten block was strongly reduced through the composite reinforcement of the tube. This positive effect was attributed to intensification of the compressive stress fields in the tungsten domain near the tube. Appendix A. Shakedown theorem and analysis In the formulation of the static shakedown theorem for a FMMC the relevant stress fields has to be considered on triple length scales. The applied loads in a bi-axial stress space are expressed with macroscopic stress states (for whole unit cell) whereas the actual shakedown condition is described with microscopic stress fields (matrix of unit cell). On the macroscopic scale the field quantities are averaged over the representative volume element, which in the present study corresponds to either the lamina or the laminate unit cell. We denote the coordinates for the macroscopic and microscopic length scales with X and x, respectively. According to Hill, the homogenization relations are expressed as [27]
RðXÞ ¼ hrðxÞi ¼
1
X
Z
rðxÞdX;
ðA1Þ
X
where R is the macroscopic stress tensor, r the microscopic stress tensor and X indicates the domain occupied by the unit cell. The microscopic elastic stress states of a FMMC re constitute the elastically admissible domain P of the macroscopic load space R below the elastic limit. This relation is expressed as
P ¼ fRj9re ; 8re ðxÞ 2 FðxÞg
ðA2Þ
FðxÞ ¼ fre jf ðre ; xÞ < 0; 8x 2 Xg
ðA3Þ
for the von Mises yield function f(rs, x). The static shakedown theorem for a FMMC can be formulated as follows [11]: the composite material is in the shakedown state under given domain of a load space, if there exist a real number a > 1, a time-independent periodic residual stress field r^ r satisfying given boundary conditions and an admissible domain P^ of macroscopic load space R to be defined by the safe states of stresses rs obeying the relations
^ ¼ fRj9rs ; rs ðxÞ 2 FðxÞg ^ P
ðA4Þ
hrs ðxÞi ¼ RðXÞ ^ FðxÞ ¼ frs jf ðrs ; xÞ < 0; 8x 2 Xg
ðA5Þ ðA6Þ
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Here, the safe stress states rs are defined by
rs ¼ arc þ r^ r :
ðA7Þ
^ r satisfies the equilibrium equations and boundary condition where r
^ r ¼ 0 in X divr
ðA8Þ
r^ r n ¼ 0 on @ X
ðA9Þ
^ r i ¼ 0 in X hr
ðA10Þ
and rc denotes the elastic stress field in the virtual, purely elastic counterpart of the actual FMMC unit cell under the same boundary and equilibrium conditions as in the original problem such that
rc n ¼ R n on @ X divrc ¼ 0 in X
ðA11Þ ðA12Þ
^ r (A10) leads to Discretisation of the self-equilibrium equation of the time-independent residual stress field r
^ r ðyji Þg ¼ 0 ½Cfr
ðA13Þ
where the residual stress fr g is given in a vector form at each integration point ^r
½C ¼
( NE X NGE X j¼1
yji .
The matrix [C] is expressed as
) wi det½J j ðni Þ½Bðyji Þt
ðA14Þ
i¼1
where [J] is the Jacobian transformation matrix for the shape function of finite elements and [B] is the matrix of partial derivatives of the shape function. NE and NGE stand for the number of finite elements and the number of Gauss integration points in an element, respectively. The computational procedure to determine the shakedown boundary of the considered FMMC consists of four steps: 1. Computation of elastic stress solutions by FEM for the FMMC unit cell models to be considered. 2. Setting up a system of equations of constraints for time-independent residual stress for the used FEM models considering prescribed kinematic boundary conditions. 3. Formulation of an optimisation problem by combining the shakedown conditions and the equilibrium constraints of residual stress at each Gauss integration point and for each load corner in a loading space. 4. Determination of the safety factors by means of mathematical programming process. By multiplying the safety factors with the initially assumed load corners a shakedown boundary was obtained.
Appendix B. Weibull theory of brittle failure [18] In STAU, volume flaws and surface flaws are modeled as penny-shaped cracks and through-wall cracks, respectively. The singular crack tip stress fields under mixed-mode loads are expressed in terms of the stress intensity factors K corresponding to three fracture modes
pffiffiffi K I ¼ rn Y I a;
pffiffiffi K II ¼ sII Y II a;
pffiffiffi K III ¼ sIII Y III a;
ðB1Þ
where Y denotes geometric correction factor and rn and sII, sIII the normal and shear projections of the stress tensor on the crack plane which are computed by the tensor transformation rules. The mixed-mode failure criterion for a multi-axial stress field has a generic form
gðrn ; sII ; sIII Þ P g c
ðB2Þ
In STAU, the effective failure criterion is employed which is formulated with equivalent mode I stress intensity factor as follows:
gðK Ieq ; 0; 0Þ ¼ gðK I ; K II ; K III Þ
ðB3Þ
Then the failure criterion is reformulated as
K Ieq P K Ic ;
ðB4Þ
where KIc stands for the mode I fracture toughness. By analogy to Eq. (B1), an equivalent stress req is defined by
pffiffiffi K Ieq req Y I a
ðB5Þ
J.H. You / Engineering Fracture Mechanics 76 (2009) 1437–1449
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Finally, the critical crack size ac is given by
ac ¼
K Ic req Y I
2 ðB6Þ
Failure by spontaneous crack extension occurs if the crack size a exceeds ac. As the size, the location and the orientation are random variables, the strength is also a random variable for which statistical treatment is required. For statistically independent infinitesimal volume elements dV the actual number n of cracks contained in a component is a Poisson distributed random variable. The probability density function of the crack size a is expressed by a Pareto distribution by
fa ðaÞ ¼ 1
aor1 ar1
ðB7Þ
where a P ao. For a multi-axially loaded brittle component containing randomly oriented sharp volume cracks of an arbitrary (large) number, the failure probability due to the unstable crack propagation is formulated as
Z Z 1 1 req ðx; xÞ PF;V ¼ 1 exp dXdV V o V 4p X ro
ðB8Þ
where Vo is the unit volume containing an average number of Mo cracks. m and ro are material parameters expressed as
m ¼ 2ðr 1Þ
ro ¼
1 1 m
ðMo Þ
ðB9Þ
K Ic pffiffiffiffiffi Y I ao
ðB10Þ
Further, Eq. (A8) can be rewritten in terms of a 2-parameter Weibull distribution
m r PF;V ¼ 1 exp b
ðB11Þ
where r* denotes a reference stress characterizing the load level. The Weibull parameter b is related to ro by
b ¼ ro
1 Vo
Z V
Z
1 4p
X
req ðx; xÞ r
m
dXdV
m1 ðB12Þ
Eqs. (B11) and (B12) suggest that the required parameters m and ro can be obtained from a series of uniaxial tests where the shape parameter m and the scale parameter b are estimated from the measured strength data of simple specimens using the Weibull statistics. At the stage of parameter calibration the effective stress in Eq. (B12) refers to the stress state produced by the testing. In the case of surface cracks of the length 2a normal to the surface, the failure probability PF,A of a component containing arbitrary number of randomly oriented sharp surface cracks of random size and random location is given in the same line as PF,V
Z Z 1 1 req ðx; xÞ m PF;A ¼ 1 exp dXdA Ao V 2 p X ro
ðB13Þ
Then the Weibull parameter b is related to ro by
b ¼ ro
1 Ao
Z A
1 4p
Z X
req ðx; xÞ r
m
m1 dXdA
ðB14Þ
The domain and the orientation integrals are estimated numerically by means of Gauss quadrature method for each finite element. To this end, the analytical formulation for PF,V and PF,A has to be converted into a discretized form, for instance
" PF;V ¼ 1 exp "
# Z Ne Z ~Þ m 1 X 1 req ð~x; x dXdV V o e¼1 V e 4p X ro
Z 1 Z 1 Ne Z 1 Z 1 Z 1 1 X 1 ¼ 1 exp V o e¼1 1 1 1 4p 1 1
req ð~n; ~hÞ ro
!m
# ~ ~ ~ ~ J V ðnÞ JX ðhÞ dh dn
ðB15Þ
where Ne is the number of elements of the FEM model, Ve the volume of the e th element, n and h are the coordinates in the reference configuration considered for the numerical integration using the Gauss quadrature and JV and JX denote Jacobians of the corresponding variable transformations.
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The effective fracture criteria and the equivalent stresses are listed below [18] (1) The coplanar energy release rate criterion [22]
K Ieq ¼ ½K 2I þ K 2II þ "
req ¼ r2I þ s2II
1 K 2 0:5 1 m III
ðB16Þ
2 #0:5 Y II s2 Y III 2 þ III YI 1 m YI
ðB17Þ
(2) The normal stress criterion [23]
K Ieq ¼ K I
ðB18Þ
req ¼ rn
ðB19Þ
(3) The maximum hoop stress criterion [24]
K Ieq
req
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 8 2K I þ 6 K 2I þ 8K 2II K 3II ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:5 K 2I þ 12K 2II K I K 2I þ 8K 2II
ðB20Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 8 2rn þ 6 r2n þ 8s2II ðY II =Y I Þ2 s3II ðY II =Y I Þ3 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:5 r2n þ 12s2II ðY II =Y I Þ2 rn r2n þ 8s2II ðY II =Y I Þ2
ðB21Þ
(4) The maximum energy release rate criterion [25]
h i0:25 K Ieq ¼ K 4I þ 6K 2I K 2II þ K 4II h
req ¼ r4n þ 6r2n s2II ðY II =Y I Þ2 þ s4II ðY II =Y I Þ4
ðB22Þ i0:25
ðB23Þ
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