Triple-scale failure estimation for a composite-reinforced structure based on integrated modeling approaches. Part 1: Microscopic scale analysis

Engineering Fracture Mechanics 76 (2009) 1425–1436

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Triple-scale failure estimation for a composite-reinforced structure based on integrated modeling approaches. Part 1: Microscopic scale analysis q 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 Received in revised form 7 November 2008 Accepted 16 December 2008 Available online 25 December 2008 Keywords: Metal matrix composite Finite element analysis Micromechanics Failure assessment Homogenization Damage

a b s t r a c t The structural reliability of a composite component locally reinforced with a fibrous metal matrix composite is essentially affected by the micro-scale failures. The micro-scale failures such as fiber fracture or matrix damage are directly governed by the internal stress states such as mismatch thermal stress. A proper computational method is needed in order to obtain micro-scale stress data for arbitrary thermo-mechanical loads. In this work a computational scheme of microscale failure analysis is presented for a composite component. Micromechanics-based triple-scale FEM was developed using composite laminate element. The considered composite component was a plasma-facing component of fusion reactors consisting of a tungsten block and a composite cooling tube. The micro-scale stress and strain data were estimated for a fusion-relevant heat flux load. Ductile damage of the matrix was estimated by means of a damage indicator. It was shown that the risk of microscale composite failure was bounded below an acceptable level. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Nowadays fiber-reinforced metal matrix composites (FMMC) are considered as promising materials for high-temperature structural applications. Examples of potential application are aircraft engine turbine, heat sink of power electronics and plasma-facing component of nuclear fusion reactors [1–4]. The essential advantage of FMMCs is that high thermal conductivity and toughness of a metallic matrix can be combined with excellent strength and creep resistance of strong fibers [5,6]. Further, small coefficient of thermal expansion (CTE) and high elastic modulus of the fibers can be utilized to tailor the corresponding composite properties. In design practice of a composite component, structural reliability is usually evaluated in terms of composite failure on a macroscopic scale. For instance, some commercial FEM codes provide failure estimation postprocessors in which stresses are averaged for each element by means of elastic homogenization method and compared directly with the macroscopic strength data of the composite according to various composite failure criteria such as Tsai-Hill [5,6]. Although the spatial resolution of the prediction can be arbitrarily improved by mesh refinement, this approach has a macroscopic nature since the stress state is averaged in each finite element. In the framework of an empirical failure criterion it is tacitly assumed that the composite failure is controlled solely by macroscopic stress. In reality, the situation is more complex because of internal stress and matrix plasticity. It should be noted that FMMCs are fabricated usually at high temperatures and used also at elevated temperatures. Due to the mismatch of CTE between the q

Dedicated to Professor Wolfgang Brocks on the occasion of his 65th birthday. * 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.2008.12.014

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Nomenclature Latin symbols  ðpÞ instantaneous mean field strain concentration tensor of phase p A t  ðpÞ instantaneous mean field stress concentration tensor B t C Dp E Et I St

global elastic compliance matrix Rice–Tracey damage indicator global elastic stiffness matrix global instantaneous tangent stiffness fourth rank identity tensor instantaneous Eshelby tensor

Greek symbols global volume averaged stress tensor hei ðpÞ phase averaged total stress tensor in phase p heitot dheimech far-field global averaged mechanical strain rate Dec global strain increment Demech global strain increment from purely mechanical contribution DeEB Euler-backward strain ee;pl equivalent plastic strain nþ1 De applied global strain sub-increment ðmÞ Dheimech phase averaged mechanical strain increment in matrix g stress triaxiality re von Mises equivalent stress rh hydrostatic stress global volume averaged stress tensor hri ðpÞ phase averaged total stress tensor in phase p hritot ðmÞ Dhritot;el elastic predictor

matrix and the fibers considerable internal thermal stress fields may appear [6,7]. In the absence of applied load this mismatch stress is a kind of residual stress. The characteristic fluctuation length of the internal stress fields spans typically over the length of a fiber diameter. In the residual stress state, tensile stress prevails in the matrix whereas the fibers are stressed compressively [7,8]. During subsequent thermo-mechanical loading, the applied mechanical stress and the mismatch thermal stress are superposed on micro-scale. As a result of superposition, the matrix and the fibers experience completely different evolution of the internal stresses. Hence, the actual internal stress state is determined by previous thermal load history as well as by currently applied load. The structural reliability of a FMMC component is essentially affected by the micro-scale failures such as fiber fracture or matrix damage. On the other hand, these micro-scale failures are directly governed by the internal stress states [9]. Hence, accurate estimation of the internal stresses is a primary design concern. In this context, a computational method is needed which is able to provide with the micro-scale data of the composite related to the micro-scale failure mechanisms. There are also failure features which can be better assessed and interpreted on higher length scales, for example, plastic instability of FMMC or stress perturbation effect in the vicinity of the embedded FMMC. The correlation between micro- and macro-scale field quantities can be established by means of micromechanics-based homogenization methods [10–12]. One of the most widely used methods is the mean field theory of Mori and Tanaka [13]. This method enables to identify the local stress (strain) state of each phase (i.e. matrix and inclusions) via stress (strain) concentration tensor which is a function of the elastic stiffness tensor of the local phases and the Eshelby tensor [14–16]. The original elastic formulation was extended to a non-linear rate form to consider matrix plasticity [17]. This scheme was initially restricted to uniformly applied loads. Afterward, it was formulated as a constitutive equation and was further implemented into a commercial FEM code as a material law subroutine [18]. It was demonstrated that this implemented algorithm worked successfully for a hierarchic stress analysis of a particulate composite [19]. The algorithm was further applied to a FMMC laminate system by the author for which the micromechanical material subroutine was combined with the 3-dimensional composite laminate element of the ABAQUS code [20]. This approach allowed elasto-plastic FEM analysis of a FMMC laminate system on three different composite length scales. Stress and strain data were estimated for matrix and fiber (micro-scale), for lamina layer (meso-scale) and for laminate element (macroscale). In the following we call it triple-scale analysis. Considering that FMMC is often locally integrated into a complicated component system, it is highly desired to establish a scale-bridging failure analysis methodology for industrial composite component systems. The aim of present work is to illustrate a scale-dependent failure estimation methodology for a FMMC component on the basis of the nonlinear triple-scale

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FEM analysis and scale-relevant failure criteria. This work consists of two articles: The first paper (Part 1) deals with the microscopic failure features while the other companion paper (Part 2) treats the failure problems on higher length scales. Although only two length scales will be treated in each part respectively, (Part1: micro- and macro, Part2: meso- and macro-scale) the overall range of the length scales considered in the whole contribution (Part1 and Part2) are triple (i.e. micro-, meso- and macro).

2. Model composite component and loading condition 2.1. Geometry, mesh and materials The model FMMC component considered for the present study was an actively cooled plasma-facing component for fusion reactors. This model component had a duplex bond structure consisting of a tungsten armor block and a FMMC heat sink cooling tube. The geometry (symmetric left half) and the finite element mesh of the model are shown in Fig. 1. The dimension of the block was 19.5  18  4 mm3. The cooling tube had inner diameter of 10 mm and thickness of 2 mm. The shortest distance from the top surface to the tube was 3.5 mm. The number of elements amounted to 10,400. The reference design of this model was originally suggested in the European Power Plant Conceptual Study (PPCS model A: Water-Cooled Lithium– Lead Divertor) where copper alloy CuCrZr was considered for the cooling tube [21]. Tungsten was chosen due to its excellent physical compatibility with fusion plasma and refractory nature. The cooling tube should satisfy two requirements simultaneously: large heat conductivity and high-temperature strength. This was the motivation to apply a FMMC for the cooling tube. The FMMC considered in this study was a unidirectional copper matrix composite reinforced with continuous tungsten fibers wound in the tube hoop direction (fiber volume fraction: 40%). The structure of composite reinforcement in this component and the homogenization scheme is schematically illustrated in Fig. 2. The matrix was assumed to be CuCrZr alloy. Selected material properties are listed in Table 1 [22]. Experimental work to fabricate this FMMC is ongoing [4].

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 composite consisting of copper alloy matrix reinforced with tungsten fibers wound in the tube hoop direction (fiber volume fraction: 40%).

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a

b

hoop Fig. 2. Schematic structure of composite reinforcement in the component (a) and the homogenization scheme (b).

Table 1 Selected material properties used for the FEM simulation [18]. Tungsten block Young’s modulus (GPa) Yield stress (MPa) Heat conductivity (W/mK) CTE (106/K) a b

400 (20 °C) 393 (400 °C) 1385 (20 °C) 950 (400 °C) 175 (20 °C) 140 (400 °C) 4.5 (20 °C) 4.6 (400 °C)

a

b

CuCrZr matrix

W wire

128 (20 °C) 110 (400 °C) 300 (20 °C) 273 (400 °C) 380 (20 °C) 350 (400 °C) 15.7 (20 °C) 19.3 (400 °C)

400 (20 °C) 393 (400 °C) 2000 (20 °C) 175 (20 °C) 140 (400 °C) 4.5 (20 °C) 4.6 (400 °C)

Cold worked and annealed (stress relieved) tungsten. Measured by tensile test using laser speckle strain gauge.

2.2. Loading and boundary conditions The considered thermal load history includes fabrication process, heat flux loading and cooling phase. The effective stressfree temperature was assumed to be 500 °C. The actual processing temperature may be still higher than this. The residual stresses were generated by a uniform cooling from the stress-free temperature to room temperature. Subsequently, highheat-flux loading was simulated assuming a heat flux load of 15 MW/m2 and coolant water temperature of 320 °C. The assumed heat transfer coefficient was 0.156 MW/m2 K and the coolant pressure was 15.5 MPa. These load parameters were defined from the PPCS Model A [21]. The displacement of the edge cross-sections of the coolant tube and of the interlayer were fully constrained in the tube axis direction. 3. Computation scheme: micromechanics model In the framework of the non-linear triple-scale FEM, the stresses and the strains are computed on the micro- as well as on the macro-scale simultaneously. The computational procedure consisted of two steps: (1) FEM analysis of the global stress states of the whole composite component, (2) micromechanical computation of the local phase stresses (matrix and fibers) in the FMMC tube. For the micromechanical computation, the algorithm of Pettermann was used which was implemented into the FEM code ABAQUS as a user-defined material subroutine [18,23]. This was a FMMC material law based on the incremental formulation of the Mori–Tanaka mean field homogenization theory [24]. The material subroutine was further coupled with the threedimensional composite laminate element of the ABAQUS element library. This coupling enabled meso- and macro-scale homogenization of composite stress and strain allowing arbitrary laminate architecture. Since the FMMC tube of the current structure was assumed to consist of unidirectional laminae, the laminate element was reduced to a unidirectional stacking of four composite laminae reinforced in the hoop direction by defining the stacking orientation correspondingly [0°/0°]. For the tungsten block quadratic reduced-integration solid elements were used. A schematic flow chart of the computational procedure is illustrated in Fig. 3. The Prandtl–Reuss flow rule was used for the matrix plasticity. Internal iterations had to be performed at each increment to correct the characteristic Euler-backward

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Global equilibrium iteration

Boundary condition

Strain increment, Temperature increment

Elastic computation of local state variables

Computation of global state variables

Plasticity return mapping, Updating of state variables

Euler backward iteration FEM code (ABAQUS)

Subroutine (UMAT)

Fig. 3. A schematic flow chart of the triple-scale finite element analysis illustrating the coupling between the global FEM structure and the micromechanics-based computational procedure in the framework of a user-defined material subroutine.

behavior of the global composite stress caused by the updating of the matrix stress after plastic yield [25]. The theoretical background and detailed formulation are presented in Appendix A. The tungsten fiber was assumed to be elastic to consider the neutron embrittlement effect [26]. For the verification of this novel numerical technique, two test simulations were carried out for a typical titanium base FMMC cross-ply laminate (matrix: Timetal21S, fibers: SiC SCS6, fiber volume fraction: 35%, laminate stacking: [0°/90°], reference temperature: 815 °C). Two different thermo-mechanical load cycles were considered: (1) uniaxial in-plane tensile loading–unloading cycle at 650 °C (isothermal loading) and (2) uniaxial in-plane tension-heating followed by cooling– unloading cycle ranging from 20 to 650 °C (in-phase loading). Results are shown in Figs. 4 and 5. The experimental data found in the literature are also compared [27]. It is found that the simulated stress–strain responses from the present procedure are in good agreement with the experimental data. 4. Results and discussion 4.1. General remarks In the following results of the micromechanical analysis and interpretations are presented. The estimated micro-scale data are presented in Figs. 6 and 8–10 for a cylindrical coordinate system. The results are plotted for two distinct stages of the thermal history, that is, the residual stress state at room temperature and the steady state heat flux loading stage.

Composite stress (MPa)

700 600

Micromechanics simulation Experiment

500 400 300 200 100 0 -0.2

0

0.2

0.4

0.6

0.8

Composite strain (%) Fig. 4. Comparison between the computed and measured stress–strain curve of a SiC fiber-reinforced titanium matrix composite laminate (matrix: Timetal21S, fiber: SCS6, fiber volume fraction: 35%, stacking: [0°/90°] cross-ply, reference temperature: 815 °C). Uniaxial isothermal tension test at 650 °C was simulated for a loading-unloading cycle. The experimental data was obtained from the literature [5].

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Composite stress (MPa).

600

Micromechanics simulation 500

Experiment

400 300 200 100 0 -0.2

0

0.2

0.4

0.6

0.8

Composite strain (%) Fig. 5. Comparison between the computed and measured stress–strain curve of a SiC fiber-reinforced titanium matrix composite laminate (matrix: Timetal21S, fiber: SCS6, fiber volume fraction: 35%, stacking: [0°/90°] cross-ply, reference temperature: 815 °C). Uniaxial in-phase tension-heating and unloading–cooling cycle for temperature range between 20 and 650 °C. The experimental data was obtained from the literature [5].

2500

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)

Stress (MPa)

2000 1500 1000 500 0 -500

Fiber stress (axial component)

-1000 0

0.5

1

1.5

2

Distance from the interface (mm) Fig. 6. Micro-scale stresses of the tungsten fibers (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 the positions along the paths in the tube radial direction starting from the bond interface toward the tube wall.

Stress profiles along three different paths in the tube thickness direction are shown. These three paths are indicated in Fig. 1. The abscissa denotes the positions along the paths in the tube thickness direction starting from the bond interface toward the tube wall. 4.2. Temperatures The model component reached thermal steady state in 3 s after the start of heat flux loading. The steady state surface temperature ranged from 984 (position B) to 1204 °C (position A). The maximum temperature of the cooling tube was 536 °C at the upper interface (position C) and 443 °C at the inner wall (position D). 4.3. Evaluation of fiber failure risk by micromechanics analysis In Fig. 6 the stresses of the tungsten fibers (axial component) are plotted. Three features are notable. Firstly, the fibers experienced significant stress relaxation upon heat flux loading. This stress reduction effect was ascribed to the increased tube temperature approaching the stress-free temperature. Secondly, the fiber stress in the residual stress state (at RT) was compressive in the inner half region of the tube near the bond interface while it changed to tensile in the outer half and increased toward the tube wall. The peak tensile stress at the tube wall reached 2385 MPa. This value amounts to 80–95% of the tensile strength of commercially available tungsten wires (strength range: 2.5–3.0 GPa). During the thermal loading the fiber stress was fully relieved near the bond interface and increased monotonically in tension toward the tube wall. The maximum fiber stress at the tube wall reached 1560 MPa in tension under the heat flux load.

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Fig. 7. Micro-scale stresses of the tungsten fibers (axial component) computed by the sub-modeling analysis. Shown are the stress distributions plotted in the sub-model unit consisting of four fiber layers embedded in the matrix (above) and for the array of the isolated fibers exhibiting the internal stress field (below).

Plastic flow in matrix

Equivalent plastic strain (%)

4.0 3.5 3.0 2.5 2.0

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)

1.5 1.0 0.5 0.0 0

0.5 1 1.5 Distance from the interface (mm)

2

Fig. 8. Equivalent plastic strains generated in the FMMC matrix. Results are estimated for the residual stress state and for the heat flux loading state. Plastic strain profiles are shown for the three paths indicated in Fig. 1. The abscissa denotes the positions along the paths in the tube radial direction starting from the bond interface toward the tube wall.

This value was 65% of the residual fiber stress. The Weibull modulus of the tungsten wires measured in the author’s laboratory was fairly large: it ranged from 220 (as-received wires) to 340 (heat-treated wires at 1000 °C for 10 h). This means that once the fibers have survived the fabrication process, their failure will hardly occur during the subsequent heat flux loading due to the stress relaxation and the small scattering of strength. Therefore, the fabrication process itself may be regarded as a proof test. Thirdly, the fiber stress was nearly uniform along the tube hoop direction except the upper part of the tube (path C–D). Regarding fiber failure risk, the most critical region was identified to be the upper part of the tube wall. 4.4. Evaluation of fiber failure risk by sub-modeling technique In order to verify the above results a completely independent FEM analysis based on the sub-modeling technique was performed. In this approach the FEM analyses were carried out in two consecutive steps: (1) FEM modeling of whole

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2.5

Stress triaxiality

2

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)

1.5 1 0.5 0 -0.5 -1

Triaxiality of matrix stress

-1.5 0

0.5

1

1.5

2

Distance from the interface (mm) Fig. 9. Profiles of stress triaxiality in the FMMC matrix. Results are estimated for the residual stress state and for the heat flux loading state. Stress triaxiality profiles are shown for the three paths indicated in Fig. 1. The abscissa denotes the positions along the paths in the tube radial direction starting from the bond interface toward the tube wall.

1.6

Damage indicator

1.5 1.4 Plastic damage in matrix

1.3

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)

1.2 1.1 1 0

0.5 1 1.5 Distance from the interface (mm)

2

Fig. 10. Profiles of the estimated damage indicator defined by Rice and Tracey. Results are estimated for the residual stress state and for the heat flux loading state. Damage profiles are shown for the three paths indicated in Fig. 1. The abscissa denotes the positions along the paths in the tube radial direction starting from the bond interface toward the tube wall.

component structure using a global mesh and homogenized effective properties, (2) FEM modeling of local FMMC structure for a selected composite element considering the actual fiber architecture. In the second step the internal stresses of the chosen element were computed for the actual geometry of the fiber array using the previously computed displacement data which were then imposed as boundary conditions at the sub-model domain edges. Further details of this standard technique are referred to literature [23]. In the present analysis one element at the upper part of the tube wall was chosen for the submodeling analysis. The geometry of the FEM sub-model and the computed stress distribution in it are plotted in Fig. 7. For better visual appreciation the internal fiber stresses are plotted again for the isolated fiber array. The estimated maximum fiber stresses at the tube wall in the residual stress state are listed in Table 2. The fiber stresses obtained from the two different approaches are compared at each lamina layer. Relatively good agreement was achieved between the two results although the two compared approaches were entirely independent each other.

Table 2 Estimated maximum tungsten fiber stresses at the tube wall (elastic fiber).

1st layer 2nd layer 3rd layer 4th layer

Triple-scale FEM based on micromechanics

FEM sub-modeling

1850 2000 2150 2300

2000 2300 2500 2700

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To assess the effect of fiber plastic yield, fully elasto-plastic FEM sub-modeling analysis was also carried out using the stress–strain curve of tungsten fibers measured with a laser speckle strain sensor. Significant stress reduction up to 40% was predicted. However, it should be noted that extensive plastic yield appears only in the unirradiated fibers. Hence, our assumption of elastic fibers was a reasonable and conservative condition to consider neutron embrittled fibers. On the other hand, the copper alloy matrix exhibits negligible embrittlement effect at the operation temperature above 400 °C due to thermally activated recovery. 4.5. Plastic deformation in the matrix In Fig. 8 the equivalent plastic strains generated in the FMMC matrix (copper alloy) are plotted. Substantial plastic flow occurred already throughout the fabrication. The strains were distributed uniformly along the tube hoop direction and ranged from 1.7% to 2.8% along the tube thickness. During the heat flux loading plastic strains increased strongly up to 50%. The plastic strain at the middle and the lower part of the tube ranged from 2.0% to 3.8%. The largest plastic strains occurred along the path C–D ranging from 2.7% to 4.2%. This result can be easily understood considering that the largest thermal softening of the matrix occurred at the path C–D due to the highest tube temperature at this path. The heavy plastic straining in the matrix during the heat flux loading could be a major cause of ductile damage. The most critical region regarding plastic deformation was the upper part of the tube wall. 4.6. Stress state in the matrix Another factor controlling the damage evolution is the hydrostatic stress term. Growth and coalescence of micro-pores are driven by the tensile hydrostatic stress state. In the most damage mechanics models a normalized stress quantity is used called stress triaxiality. This is defined as a normalized hydrostatic stress divided by von Mises equivalent stress. The profiles of stress triaxiality in the FMMC matrix are plotted in Fig. 9. Three notable features are found: Firstly, in the residual stress state the matrix was stressed under strong hydrostatic tension. The hydrostatic stress was twice as strong as the equivalent stress in the whole tube region. The matrix hydrostatic stress was nearly uniform throughout the whole tube volume. The strong tensile hydrostatic stress essentially contributed to progress of ductile damage as will be discussed in the next section. Secondly, the stress triaxiality turned to a compressive state upon heat flux loading. This change indicates that further growth of micro-damage would be suppressed during the heat flux loading due to the hydrostatic pressure. Thirdly, the path C–D showed the strongest compression during the thermal loading. On the other hand, the largest plastic strains were also generated at this path. These two responses would make antagonistic contributions to damage development. 4.7. Evaluation of matrix damage For a quantitative assessment of matrix damage a damage indicator defined by Rice and Tracey was estimated [18,19]. The formulation is explained in Appendix B. This damage parameter was so calibrated that the critical value of unity would correspond to the onset of local failure where micro-cracks are initiated. The profiles of the estimated damage indicator are plotted in Fig. 10. Three important features are found: Firstly, the damage indicator exceeded the critical value in the whole tube volume already before the thermal loading. It ranged from 1.24 to 1.54. As predicted above, this early development of damage can be attributed to combination of the substantial plastic strains and the tensile hydrostatic stress state in the FMMC matrix during the fabrication. At this damage level a moderate population of micro-cracks would be produced. Secondly, the magnitude of the damage indicator increased just slightly upon the heat flux loading. It ranged from 1.28 to 1.58. As predicted above, such negligible change was ascribed to the hydrostatic compression of the matrix during the thermal loading. Thirdly, the damage distributed almost homogeneously throughout the tube despite of the concentrated plastic strain at the path C–D. As already mentioned, this feature was due to the stronger antagonistic effect caused by the stronger hydrostatic compression at the upper part of the tube. The peak magnitude of the damage indicator was bounded below 1.6 which may be regarded as acceptable. An experimental investigation is necessary to approve the estimated damage data. 4.8. Comments on experimental tests In order to realize the present design concept as described in this study two German research institutes, IPP (Max-PlanckInstitute of Plasma Physics) and DLR (German Aerospace Center) are developing now prototypical mock-up components in collaboration with an industrial partner, Ansaldo in Italy for fabrication. These activities have been performed in the framework of an European integrated research project ExtreMat [28–30]. Two kinds of tungsten wire reinforced composites with copper and CuCrZr alloy matrix, respectively, were considered for the heat sink tube of the water-cooled plasma-facing component. High heat flux cyclic load test will be conducted for proof of principle.

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5. Conclusions In this work a computational scheme and interpretation of micro-scale failure analysis for a fiber-reinforced composite component was presented. The developed micromechanics-based triple-scale FEM technique could be successfully applied to a duplex fusion reactor component where an elasto-plastic composite cooling tube was integrated. The micro-scale structural reliability of the considered composite structure could be quantitatively assessed for a typical loading history. The microscopic failure features which could be estimated were fiber fracture and matrix damage which were controlled by fiber stress and the Rice–Tracy damage parameter, respectively. The matrix damage behavior was explained considering plastic strain and hydrostatic stress. This computational method was verified by comparing the predicted fiber stresses with those of the fully independent FEM sub-modeling analysis. It was shown that the failure risk of the fusion composite component was bounded below an acceptable level.

Acknowledgement The author is grateful to Ms. Irina Komarova, Institute of Mechanics, The University of Stuttgart, Germany, for her support in the FEM sub-modeling analysis.

Appendix A. Constitutive modeling based on the incremental Mori–Tanaka method A.1. Mean field elastic constitutive relations [18] In this chapter, some fundamental relationships for the linear elastic case are shortly summarized. For elastic composite materials, the global (macroscopic) constitutive relation is written as:

hri ¼ Ehei;

or hei ¼ Chri

ðA1Þ

where E and C are the global elastic stiffness and compliance matrices, respectively. hi denotes the global volume averaged field quantities. Similarly, the elastic constitutive relations for each phase p in micro-scale are written as: ðpÞ ðpÞ hriðpÞ tot ¼ E heitot ;

ðpÞ ðpÞ or heiðpÞ tot ¼ C hritot

ðA2-a; bÞ

where hiðpÞ tot denotes the phase averaged total stress and strain state in the phase p. The superscript p can be either matrix m or fiber i, respectively. A.2. Incremental Mori–Tanaka method In the following, the incremental formulations of the Mori–Tanaka scheme and the corresponding algorithm to estimate the instantaneous state quantities for the elasto-plastic case are reviewed. The term ‘rate’ (denoted by d) is used for time derivatives whereas the term ‘increment’ (denoted by D) is employed for a sufficiently small finite increment in the framework of numerical integration algorithm. The total averaged strain rates in micro-structural phase p can be related by far-field global averaged strain rate and instantaneous mean field strain concentration tensors:

 dheiðpÞ tot ¼ At dheimech ðpÞ

ðA3Þ

 ðpÞ is the instantaneous where the subscript t denotes instantaneous state and subscript mech mechanical contribution. At mean field strain concentration tensor. The global instantaneous tangent stiffness Et can be written as:

   ðmÞ Et ¼ EðiÞ þ ð1  f Þ EðmÞ  EðiÞ A t t

ðA4Þ

where f is the volume fraction of fibers. A.3. Benveniste’s formulation of the Mori–Tanaka type mean field theory According to the Benveniste’s formulation of the Mori–Tanaka mean field theory, the instantaneous mean field concentration tensors are written as,

 h  i1 1  ðmÞ ¼ ð1  f ÞI þ f I þ St C ðmÞ EðiÞ  EðmÞ A t t t

ðA5Þ

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in which the original elastic formulation is re-expressed in the instantaneous terms for each global strain increment. I denotes the fourth rank identity tensor and St represents the instantaneous Eshelby tensor which should be determined by numerical integration [24]. Similar expressions exist for the instantaneous concentration tensors for the fiber. A.4. Computational strategy This algorithm uses the radial return mapping method for stress updating in the plastic region and an implicit iteration loop controlled by Euler backward step. For the return mapping process, Prandtl–Reuss flow rule is used. Since we assumed an isothermal load case, the prescribed global strain increment Dec can have a purely mechanical contribution Demech,

Dec ¼ Demech

ðA6Þ

In the same way, the phase averaged total matrix strain increment can have only a mechanical part as, ðmÞ DheiðmÞ tot ¼ Dheimech

ðA7Þ

For the prescribed global strain increment Dec, the phase averaged mechanical matrix strain increment Dhe mined by

 ðmÞ DheiðmÞ mech ¼ At Dec

ðmÞ imech

is deter-

ðA8Þ ðmÞ imech

The purely mechanical contribution Dhe is used to estimate the elasto-plastic behavior of the matrix further. In the ðmÞ  ðmÞ are updated, if the current stress state reaches the yield surface with the elastic predictor and A next step, hriðmÞ t tot , Et DhriðmÞ obtained using Eqs. (A2), (A4) and (A8). To check the Euler backward response of the composite to the matrix strain tot;el  ðmÞ , Eq. (A8) is recalled again. For the current (e.g. n th) iteration step, the Euler backand A increment after updating of EðmÞ t t ward strain DeEB is obtained as follows: n

h i1 h i  ðmÞ DeEB ¼ A DeðmÞ t mech

ðA9Þ

The global sub-increment for the next iteration is obtained as nþ1

De ¼ Dec 

n X

n

DeEB

ðA10Þ

i¼1

This Euler backward implicit iteration loop for current global strain increment is repeated until nþ1 De becomes smaller than a given tolerance. Then, the same procedure for the next global strain increment begins. This successively applied global strain sub-increment nþ1 De can be considered as a corrector increment for the Euler backward response. For nth iteration, the local fiber stress increment and global stress increment can be obtained as n n

iðmÞ tot ;

n DhriðiÞ ¼ EðiÞ AðiÞ t DeEB ðiÞ

n

Drc ¼ f Dhri þ ð1 

EðmÞ t ,

ðA11Þ n f Þn EðmÞ Dh t

ei

ðmÞ

ðA12Þ

 ðmÞ , n D A t

hr rc and n DrðiÞ as well as nþ1 De and DeEB are updated in every iteration step of each increment, the most recent values of which are taken. The total averaged stress rates in the micro-structural phase p can also be related to the far-field global stress rate by instantaneous mean field stress concentration tensors, ðpÞ

 dhriðpÞ tot ¼ Bt dhri where

 ðpÞ B t

ðA13Þ

is the instantaneous mean field stress concentration tensor, which is given by

h   i1  ðmÞ ¼ ð1  f ÞI þ f ½I þ EðmÞ ðI  St Þ C ðiÞ  C ðmÞ 1 B t t t The global instantaneous tangent stiffness Et can be expressed in terms of

ðA14Þ  ðpÞ B t

as well:

Appendix B. Rice–Tracey damage indicator The Rice–Tracey damage indicator Dp is defined with a material parameter eo as follows [19]:

Dp ¼

1 1:65eo

Z

ee;pl

expð1:5gÞdee;pl

ðB1Þ

o

where ee,pl denotes the equivalent plastic strain defined as

ee;pl 

Z t rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 e_ : e_ 0 dt 3 pl pl 0

ðB2Þ

1436

J.H. You / Engineering Fracture Mechanics 76 (2009) 1425–1436

and g the stress triaxiality defined as

g

rh TrðrÞ ¼ re 3re

ðB3Þ

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