High temperature mechanical behavior of tube stackings – Part II: Comparison between experiment and modeling

Materials Science & Engineering A ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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High temperature mechanical behavior of tube stackings – Part II: Comparison between experiment and modeling V. Marcadon n,1, D. Lévêque, A. Rafray, F. Popoff, D. Mézières, C. Davoine Onera - The French Aerospace Lab, F-92322 Châtillon, France

art ic l e i nf o

a b s t r a c t

Article history: Received 18 July 2016 Received in revised form 13 September 2016 Accepted 14 September 2016

This paper is the second part of a set of two papers dedicated to the mechanical behavior of cellular materials from room temperature to high temperatures. For that purpose, some monotonic and creep compression tests have been performed on a small tube stacking structure. In parallel, a finite-element model of these experiments has been proposed, using the constitutive elasto-viscoplastic behavior identified for the tube material in Part I [1] as input data. The comparison between the experiment and the modeling has provided many items of information regarding the collapse mechanisms of this kind of cellular structure at high temperatures and the role of the contacts created between the tube walls at large deformations. Especially, a competition between both relaxation and hardening phenomena locally has been revealed at high temperatures, resulting in some softening of the stacking effective behavior. & 2016 Elsevier B.V. All rights reserved.

Keywords: Metallic tube stackings Cellular materials High temperature behavior Mechanical characterization FE-modeling

1. Introduction For several years metallic cellular structures, such as metal foams, honeycombs, truss lattices, and hollow-sphere or tube stackings, for instance, have been abundantly studied because of their potential for various applications. Indeed, these materials present higher specific mechanical properties (i.e., mechanical properties divided by density) than the bulk and multi-functional capabilities are expected [2,3], which could be used to develop lightweight aeronautical frames. However, whereas quite an abundant literature exists regarding the mechanical behavior of these materials at room temperature, their behavior at high temperatures has not been investigated much. This is particularly true for tube stacking structures, considered here as a ‘model’ cellular structure. In the literature, other cellular structures that exhibit similar collapse mechanisms based on localized plasticity are metal foams and hollow-sphere stackings. Many studies deal with the mechanical behavior of metal foams or hollow-sphere structures at room temperature. For instance, among others, Friedl et al. [4] performed a very interesting study under both compressive and tensile loads. Their results provided many items of information on the mechanisms that govern the mechanical response of hollow-sphere structures. Caty

n

DOI of original article: http://dx.doi.org/10.1016/j.msea.2016.07.088 Corresponding author. E-mail address: [email protected] (V. Marcadon). 1 Metallic Materials and Structures Department.

et al. [5,6] also provided significant results on the mechanical behavior of hollow-sphere structures, especially undergoing fatigue [5,7]. Fallet et al. [8] and Lhuissier et al. [9] used X-ray tomography to characterize the damage mechanisms that govern the collapse of such stackings. Two complementary mechanisms were identified: the crushing of the hollow spheres and the creation of new contacts between neighboring spheres. Sanders and Gibson [10,11] have shown that the mechanical properties of such regular stackings strongly depend on the load direction; mechanical properties are stiffer along the densest direction of the stacking. Our previous work can also be referred to [12,13]. They have shown that the effective plastic behavior of hollow-sphere stackings depends not only on their architecture, but also to a large degree on the hardening capabilities of their constitutive material and the presence or absence of defects in the architecture. The mechanical behavior of metal foams is governed by localized plasticity and the collapse of the constitutive cells also; see for example the work of Brothers and Dunand [14] on zirconium foams, or those of Paul and Rammamurty [15] and Ruan et al. [16] on aluminum foams. All of these studies also agree with the fact that several elastic and plastic mechanical properties of the foams strongly vary with their density according to a power law; the higher the density, the stiffer the mechanical response. The influence of the loading rate on the mechanical response of metal foams is a more controversial issue. According to Paul and Rammamurty [15], the foam strength increases with an increasing loading rate due to viscosity phenomena, whereas for Ruan et al. [16] and Rakow and Waas [17] no effect of the loading rate exists;

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Please cite this article as: V. Marcadon, et al., High temperature mechanical behavior of tube stackings – Part II: Comparison between experiment and modeling, Materials Science & Engineering A (2016), http://dx.doi.org/10.1016/j.msea.2016.09.052i

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all of these works were performed on aluminum foams. Gibson and Ashby [18,19] have proposed an analytical model to predict the mechanical properties (Young modulus and yield stress essentially) of a foam as a function of both the constitutive material mechanical properties and the densities. This model, developed for both open- and close-cell foams, assumes that the foam edges are loaded under pure bending. A similar approach has been proposed by Hodge and Dunand [20] assuming that the foam edges work in compression. With regard to high-temperature applications, Andrews et al. [21,22] performed tensile tests on close-cell aluminum alloy foams between 275 °C and 350 °C. For the lowest temperatures (below 300 °C), they found that the effective Norton exponent and activation energy were the same as those of the constitutive material. On the contrary, for temperatures higher than 300 °C, a strong increase in the effective Norton parameters was identified compared to those of the constitutive material. This could be explained by stress concentration phenomena at some edges of the foam, resulting in a strong heterogeneity of the creep rates in the cellular material, which are potentially very high locally. In the case of compression creep, Cock and Ashby [23] have shown that the deformation of foams is governed by the bending and the buckling of their edges. Also, for reasons of local stress concentration, buckling occurs mainly along the edges, where the stress is the highest. Complementary works by Andrews and Gibson [24] and Huang and Gibson [25], or more recently by Su et al. [26], have also investigated the influence of both the defects and the curvature of the cell walls on the creep behavior of foams. NiAl open-cell foams have been studied by Hodge and Dunand [20] in creep under compression. In agreement with the works of Andrews et al. [21,22], the effective creep parameters appeared also similar to those of the constitutive material. More recently, Diologent and co-workers [27–29] studied open-cell replicated foams under tension creep and fatigue creep. They have shown that the creep response of foams is similar to that of its constitutive material. It is characterized by a significant primary creep regime, followed by a pronounced steady-state secondary creep regime, then followed by a final tertiary regime of accelerated creep. The effective activation energy of the foams studied remained very close to that of the constitutive aluminum, whereas the effective Norton exponent was higher than that of the constitutive aluminum. Such a difference was explained by the specific microstructure of aluminum in the foam struts. Contrary to the bulk, they consisted of few grains only. Soubielle et al. [28,29] also observed the influence of oxidation on the creep behavior of the foams in long creep duration tests. An approach mostly based on ‘material’ considerations has been proposed by Chos and Dunand [30,31], studying NiCr, NiAl and NiCrAl foams. They investigated the influence of the alloy compounds on the mechanical properties and the oxidation resistance of the foams, in order to develop superalloy foams. Comparison with the predictions of the analytical model by Gibson and Ashby [18,19] on the one hand, and that of Hodge and Dunand [20] on the other hand, has shown that the second model is better suited for their foams. More recently, De Fouw and Dunand [32] addressed the case of nickel-based superalloy foams. Various modeling approaches have been proposed to take into account the various mechanisms observed during the creep of foams, and to predict their effective properties [18–23,33–36]. Although most of the studies have investigated the creep behavior of metal foams under uni-axial compression only, the works of Fan et al. [35] and Kesler et al. [37] can be cited, who addressed the multi-axial and flexural behaviors of foams, respectively. In contrast to the case of metal foams, there are few studies addressing the mechanical behavior of hollow-sphere stackings at high temperatures, either by modeling or experimental.

Nevertheless, another of our previous works can be referred to [38], it aimed at investigating the influence of the stacking architecture on its creep behavior. This work is dedicated to both the experimental characterization and the modeling of the effective mechanical behavior of tube stackings at high temperatures. This Part II of a set of two complementary papers focuses on the behavior of the cellular structure, whereas Part I [1] addresses its constitutive material properties. After a brief description of the cellular structure and the test conditions considered (see Section 2), the experimental results obtained from both quasi-static monotonic and creep compression tests are discussed in Section 3. Various temperatures (room temperature, 600 °C and 800 °C) have been considered. Various strain rates and stress levels have been applied also for the monotonic and creep tests, respectively. Then, Section 4 addresses the finite-element modeling of the aforementioned tests. The simulations have been performed accounting for large deformations and contacts. Their predictions are compared with the experimental results, in order to assess the modeling assumptions and to discuss the experimental results obtained.

2. Mechanical test campaign The aim of the test campaign detailed here was to investigate the effective elasto-viscoplastic behavior of the tube stacking structure studied. For that purpose, two different tests were considered: monotonic compression tests and compression creep tests. All of these tests were conducted between room temperature and 800 °C (600 °C and 800 °C only for the creep tests). 2.1. Sample geometry As already described in Part I [1], the cellular structure considered here was a 5  5 stacking of tubes, brazed to each other and to two metallic skins following a square pattern (Fig. 1). As a reminder, the constitutive material of the tubes and the skins was Inconel®600 and they were brazed together using Ni-1.5B-19.0Cr7.3Si-0.08C (wt%) braze foils. Theoretically, each tube had an outer diameter of 4 mm and a wall thickness of 250 mm. The skin thickness was 1 mm and the mean braze joint length was 1.43 mm. The length of the samples was equal to 20 mm after electro-discharge cutting. Their density was 2.29 g cm−3 compared to 8.25 g cm−3 for Inconel®600. The actual geometry of the tube stacking samples, after the brazing step and before testing, is listed in Table 1, where the height includes the two brazed skins and the width was measured along the tube direction. The three letters A, B and C correspond to three contiguous samples, which were successively cut by electrodischarge machining from the same manufactured bar (associated with the two first digit numbers). The width of the sample is a very precise value, considering that it was imposed by the cutting process. The height and length show more scattered values (the standard deviation values are about 1% of the mean values) due to the irregularly spaced structure of the tube stacking, considering that it was not so obvious to keep a perfectly aligned structure during the entire brazing process. In some cases, some braze joints were missing, or some sliding between tubes rows were observed. The consequence of such defects on the mechanical behavior of the tube stacking samples is discussed subsequently when the results of the tests are detailed. 2.2. Test conditions A specific compression test device has been designed in order to conduct the monotonic tests, as well as the creep tests, under

Please cite this article as: V. Marcadon, et al., High temperature mechanical behavior of tube stackings – Part II: Comparison between experiment and modeling, Materials Science & Engineering A (2016), http://dx.doi.org/10.1016/j.msea.2016.09.052i

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Fig. 1. Cellular structure studied.

Table 1 Geometry of the tube stacking samples (all dimensions in mm). Sample 50A is missing because it is the one that has been used for the microstructural characterization (see Part I [1]). Sample number 43

48

49

50 51

52

55

56

57

A B C A B C A B C B C A B C A B C A B C A B C A B C Mean value Standard deviation

Height

Length

Width

22.44 22.46 22.53 22.28 22.28 22.25 22.45 22.38 22.34 22.42 22.42 23.09 22.83 22.88 22.33 22.30 22.31 22.28 22.29 22.28 22.95 22.91 22.96 22.41 22.42 22.47 22.50 0.26

20.10 20.11 20.11 20.46 20.40 20.32 20.23 20.38 20.46 20.22 20.15 20.10 20.15 20.19 20.27 20.28 20.25 20.16 20.17 20.22 20.48 20.49 20.40 20.33 20.20 20.26 20.27 0.13

19.98 19.98 19.99 20.00 20.00 20.00 20.01 20.03 20.04 20.00 20.01 20.00 19.99 20.00 20.01 20.01 19.97 20.00 20.04 20.03 19.99 20.04 20.00 20.00 20.02 20.10 20.01 0.03

3

samples were placed between the compression plates of the setup in a vacuum furnace to avoid any oxidation. First the furnace was heated up to the prescribed temperature, at 10 °C min−1, whereas the testing machine was controlled in force so that a slight preloading was maintained on the sample at the same level. Then, we waited about one hour and a half until the temperature measured by a sheathed Type K thermocouple located very near the sample reached and remained stabilized at the right test temperature. At last, the test controlled by the crosshead displacement could begin. Moreover, this furnace was equipped with a small window to film the experiments by means of a CCD (charge coupled device) camera associated with a macro-objective and an in-house software application that synchronized the recording of all of the images with the analog outputs of the test machine (force and crosshead displacement). The use of digital image correlation as a virtual strain gauge enabled the relative displacement of the compressive plates to be measured precisely, with no influence of the stiffness of the experimental device. All of the experimental curves presented hereafter were corrected using this technique. With regard to the creep tests, they were all conducted on an usual creep machine, the load being introduced by means of simple weights. Due to the size of the compression device, the adapted furnace enabled the tests to be conducted under air conditions only, with the problem of possible oxidation of the samples. Given that this furnace has no window, no observation was possible during the creep tests, except post-mortem. After installing the samples between the compression plates and positioning the furnace around the compression setup, the furnace was heated up to the prescribed temperature, at 10 °C min 1 too, the samples being free of loading (only a 100 g weight was set in place, in order to align the entire loading chain). After the stabilization of the temperature (with regard to the monotonic compression tests, the prescribed temperature was controlled directly on the sample), the total load was imposed at the same time, with the appropriate weights. For the creep tests, two testing temperatures were considered (600 °C and 800 °C). Two different compressive stress levels (40% and 60% of the effective yield stress) were imposed at 800 °C plus one multi-level compression test (40%, 80% and 120% of the effective yield stress) at 600 °C and (40%, 60% and 80% of the effective yield stress) at 800 °C.

3. Experimental characterization 3.1. Monotonic compression tests

the same loading conditions. An inverse tension-compression setup was chosen to improve the guidance of the compressive plates. A first setup was designed to fit our high temperature vacuum furnace, which was used for the monotonic compression tests (Fig. 2(a)). Considering the case of the creep tests, a second compression device, very similar to the first one, was fitted to be used in a classical creep test machine with a high temperature tubular furnace (Fig. 2(b)). It was slightly modified in order to be able to use our usual extensometer system, a LVDT (linear variable differential transducer) that measures the relative elongation (contraction in our case) between the ends of the sample (the compressive plates in this case) transmitted by two alumina rods ending in conical supports and sliding freely around the loading chain. The monotonic compression experiments have been performed using a tension-compression electro-mechanical testing machine. Two different rates of the crosshead displacement were used: 1 and −0.1 mm min−1. Three various test temperatures were investigated: room temperature (RT), 600 °C and 800 °C. The

As detailed before, the monotonic compression tests have been performed at two different crosshead displacement rates and for three test temperatures. The goal was to determine whether or not there was any viscous behavior in the case of such structures, and at what temperature the phenomenon was emphasized. All of the samples tested under compression are listed in Table 2, with the indication of the target temperature and the imposed displacement rate. As far as possible, we tried to combine the origin of the samples (the bar that they were extracted from), in order to be more representative of the natural discrepancy due to the manufacturing process. We chose to stop the tests when 50% crushing was reached. A total of six samples were tested at room temperature with the two different crosshead displacement rates (see Table 2). All of the corresponding mechanical responses are plotted in Fig. 3(a). In the following, the mechanical responses are always plotted in terms of compressive load F2 vs. compression displacement U2, because at large crushing levels a pronounced barrel effect is observed, as illustrated below, making the definition of the

Please cite this article as: V. Marcadon, et al., High temperature mechanical behavior of tube stackings – Part II: Comparison between experiment and modeling, Materials Science & Engineering A (2016), http://dx.doi.org/10.1016/j.msea.2016.09.052i

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Fig. 2. Compression test setups (in-situ) (a) monotonic compression test setup assembly (b) creep test setup assembly (outside the furnace).

Table 2 Tube stacking samples for monotonic compression tests. Sample number

Temperature (°C)

43A 52C 56B 48B 49B 56A 50C 57B 48A 55A 51C 57A 49A 52A

RT RT RT RT RT RT 600 600 600 600 800 800 800 800

Displacement rate (mm min

1

)

0.1 0.1 0.1 1.0 1.0 1.0 0.1 0.1 1.0 1.0 0.1 0.1 1.0 1.0

instantaneous mean section for the computation of the true stress difficult. As can be seen in Fig. 3(a), there was no evidence of any viscous effect at room temperature. Except for the discrepancy between the curves, they all looked similar with an initial fast increase in the load (‘elastic part’) followed by a long plastic plateau after 5% crushing ( 1.1 mm displacement). Small but marked bumps were observed at around 25% crushing ( 5.5 mm displacement) followed by a sharp increase in the load from 45% crushing ( 9.9 mm displacement) up until the end of the tests. In the following, the crushing level is often used instead of the compressive displacement value, which is less visual. The crushing level has been defined as the absolute value of the engineering compressive strain. It is worth noting that the two samples 56A and 56B (extracted from the same manufactured bar) tested at the two different displacement rates, exhibited more or less the same load-

displacement curves. Apart from the absence of the aforementioned viscous effect, it seems that the discrepancy observed between all of the curves was rather due to manufacturing defects. For instance, compared to Sample 52C, which exhibited a rather regular tube stacking structure, Sample 56B exhibited several defects, such as missing braze joints or tube misalignment between the second and third rows. These observations can also explain the difference in the elastic behavior of Samples 56A and 56B, which appeared softer than that of all of the other samples as from the beginning of the loading. At 600 °C, the four samples tested exhibited load-displacement curves similar to those obtained at room temperature, apart from the fact that the load levels were lower (see Fig. 3(b)). There was apparently no more a strain rate effect at this temperature. On the contrary, a strain rate effect was clearly apparent in the results at 800 °C (see Fig. 4(a)); two groups of curves were obtained, the one being homothetic from the other. The higher load levels were observed for the higher crosshead displacement rate, as expected, as a result of the viscous constitutive material behavior at this temperature. The same kind of evolution as that observed at room temperature and 600 °C was observed at 800 °C, except that there was a marked softening stage at the beginning of the plastic plateau. The mechanical responses showed a clear bump at around 25% crushing ( 5.5 mm displacement), as already observed at the two other test temperatures. Note that not only the absence of viscosity at room temperature and 600 °C but also its presence at 800 °C are in agreement with the results obtained on the constitutive material behavior (see Part I [1]). 3.2. Compression creep tests The samples used for the creep tests are listed in Table 3. Two creep durations were investigated: a short duration (∼100 h) and a long one (∼1000 h).

Please cite this article as: V. Marcadon, et al., High temperature mechanical behavior of tube stackings – Part II: Comparison between experiment and modeling, Materials Science & Engineering A (2016), http://dx.doi.org/10.1016/j.msea.2016.09.052i

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Fig. 3. Compressive load-displacement curves obtained at (a) room temperature (b) 600 °C.

We have focused the creep tests at the temperature which emphasized the viscoplastic behavior of the material, i.e., 800 °C. The multi-level creep tests were of interest to study the effect of an increasing load level on the creep behavior from the same material sample (and under the same testing conditions), without removing it from the device. The equivalent compressive stress levels were determined from the monotonic compressive curves detailed previously. More precisely, the effective yield stress was identified from the load-displacement curves obtained at 600 °C and 800 °C on two samples tested under compression at 1 mm min 1. It has been computed from the initial dimensions of the samples, since this quantity is characterized at low strain levels. Samples 48A and 49A have been chosen as references for that purpose, at 600 °C and 800 °C respectively. At 600 °C, the effective yield stress was equal to 3.20 MPa, whereas it was equal to 2.52 MPa at 800 °C. The four creep tests conducted at 800 °C for two single load levels (40% and 60% of the effective yield stress) are represented in Fig. 4(b) in terms of global compressive displacement (relative displacement between the compressive plates) versus time. We have excluded the fifth test on Sample 57C, because of a bad initial position of the sample in the experimental device, involving an apparent but artificial stiffening of the behavior not representative of the material one. Considering the curves plotted in Fig. 4(b), we observe a slight difference between the two load levels, but the displacements remain quite close whatever the load level, resulting probably from a predominant contribution of the viscosity compared to that of the plasticity. The curves appear rather chaotic in the first times, and of course some discrepancy exists between the different samples (resulting from geometrical and material defects).

Table 3 Tube stacking samples for creep compression tests. Sample number

Temperature (°C) Creep load (MPa)/duration (h)

Remark

50B

600

Multi-level test

43B 51B 43C 51A 48C

800 800 800 800 800

57C

800

1.28/385.2; 2.56/333.1; 3.84/281.7 1.0/1286.3 1.0/364.7 1.5/111.0 1.5/955.3 1.0/101.2; 1.5/94.9; 2.0/ 154.0 2.0/139.2

Multi-level test

The creep compression curves obtained from the two multilevel creep tests conducted at 600 °C and 800 °C are plotted in Fig. 5(a). At 600 °C, very small displacements were observed at the first two stages (40% and 80% of the effective yield stress), but a sudden increase appeared at the third stage (120% of the effective yield stress). Despite the fact that the influence of the plasticity at this third load level was expected, the evolution of the strain over time was more surprising because no apparent viscosity was expected at this temperature in the absence of viscous behavior at 600 °C for Inconel®600 (see Part I [1]). This issue is addressed in more detail in Section 4. At 800 °C, the creep behavior was smoother, with a rather slight influence of the load level only, as mentioned above. It is interesting to compare this multi-level creep test (Sample 48C) conducted at 800 °C in its first stage (40% of the effective yield stress for approximately 100 h) with the two

Fig. 4. (a) Compressive load-displacement curves obtained at 800 °C (b) Creep compression curves obtained from the single-level creep tests at 800 °C.

Please cite this article as: V. Marcadon, et al., High temperature mechanical behavior of tube stackings – Part II: Comparison between experiment and modeling, Materials Science & Engineering A (2016), http://dx.doi.org/10.1016/j.msea.2016.09.052i

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Fig. 5. (a) Creep compression curves obtained from the multi-level creep tests at 600 °C and 800 °C b) Creep compression curves at 800 °C for the 40% equivalent stress level (first 100 h).

equivalent single-level creep tests (Samples 43B and 51B, see Table 3) for the same test duration (see Fig. 5(b)). It is not surprising to retrieve the same global behavior, even with the same bumps in the curves around 20%, 30% and 40% crushing ( 4.4, 6.6 and 8.8 displacement, respectively). These bumps were probably related to the occurrence of the first contacts between neighboring tubes and to the consolidation stage at the end, similar to that already observed in the monotonic compression tests. 3.3. Collapse mechanisms The optical system used during the monotonic compression tests enabled some interesting features of the deformation of the tube stacking structure to be shown. The observations made hereafter, for one particular sample (sample 51C tested at 800 °C at 0.1 mm min 1), can be generalized to all of the other samples. Various views of sample 51C, with successive deformation states, are shown in Fig. 6. At 5% crushing ( 1.1 mm displacement), when the plastic plateau starts, the first collapsing tubes are observed. At 15% crushing ( 3.3 mm displacement), this is accentuated with an emerging localized deformation in a band diagonally towards the loading direction. Some new contacts between the adjacent central tubes are created at about 20% crushing ( 4.4 mm displacement), and then emphasized at 25% crushing ( 5.5 mm displacement), the value at which there is a clear bump in the loaddisplacement curves resulting from the increase of the structural stiffness. At 35% crushing ( 7.7 mm displacement), the localization of the deformation through the tube stackings is well apparent with a characteristic X-shape band. At 45% crushing ( 9.9 mm displacement), all of the tubes are deformed and the difficulty of increasing this deformation due to the confinement of each tube entailed a sharp increase in the load at the end of the load-displacement curves, the so-called consolidation stage. The global barrel-shape deformation state observed during the various

deformation steps was due to the limit conditions of the structure: free-edge conditions along the width and length directions, whereas both the first and last rows of tubes were blocked at the braze joints with the upper and lower skins, which were more rigid. Considering that we had no means to optically observe the sample deformation during the creep tests, some observations of the creep samples have been done after testing. The collapse patterns were quite similar to those observed for the same crushing levels in the monotonic compression tests, suggesting that the deformation phenomena were probably the same. The experimental results and the collapse mechanisms observed are further discussed in Section 4 through a comparison with the finite element modeling of the aforementioned tests. 3.4. Post-mortem analysis of the samples Post-mortem observations have been performed on samples compressed up to 50% crushing at 1 mm min 1. As shown in Fig. 7, whereas one third of the braze joints were broken after the tests at room temperature, only few of them and none of them were broken at 600 °C and 800 °C, respectively. Actually, the mechanical behavior of the braze joint tends to become more ductile when increasing the test temperature. Such a trend has been asserted by carrying out some Vickers hardness measurements, in both the tube and the braze materials at room temperature and 600 °C. Given that our hardness apparatus was not able to provide measurements at 800 °C, an intermediate temperature of 300 °C has been considered instead, to confirm the evolutions observed. The hardness values were equal to 1.5 70.1 GPa, 1.27 0.1 GPa and 0.9 70.1 GPa at room temperature, 300 °C and 600 °C, respecInconel®600. The corresponding values were tively, for 8.4 70.9 GPa, 4.3 7 0.8 GPa and 2.6 70.5 GPa for the braze material. Despite the fact that a decrease in the hardness value was observed when the temperature was increased for both materials,

Fig. 6. Successive deformation views for sample 51C for a (a) 0% crushing (b) 5% crushing (c) 15% crushing (d) 20% crushing (e) 25% crushing (f) 35% crushing (g) 45% crushing.

Please cite this article as: V. Marcadon, et al., High temperature mechanical behavior of tube stackings – Part II: Comparison between experiment and modeling, Materials Science & Engineering A (2016), http://dx.doi.org/10.1016/j.msea.2016.09.052i

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Fig. 7. Post-mortem observations of samples tested at 1 mm min 1 until a 50% crushing at (a) room temperature (b) 600 °C (c) 800 °C. Cracks are indicated with red circles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

it is worth noting that this decrease was considerably more significant for the braze material. At high temperature, the behavior tends to be uniform and more ductile over the entire structure. The homogenization of the material properties when increasing the test temperature has been confirmed by the post-mortem analysis of the creep samples. Scanning electron microscopy (SEM) analyses performed on several creep samples have shown that some cracks were observed in the tube walls only, whereas the braze joints were undamaged (Fig. 8). The presence of oxides was detected inside some cracks. Both oxidized and non-oxidized cracks were observed; the oxidized cracks probably occurred earlier during the test than the non-oxidized ones. Oxidation was not limited to the cracks, but rather it occurred at the surface of the tubes as well, especially for the samples whose test duration was the longest. The evolution of the microstructures during the creep tests performed under air at 800 °C has been addressed. Initially, the microstructure of the NiBCrSi braze joints (see Part I [1] for sample processing) consisted of a mixture of a Ni solid solution phase, a NixSiy phase and some chromium borides CrxBy (see Fig. 9(a)). The chromium borides had elongated and interconnected shapes. After the creep tests at 800 °C, this microstructure still persists. However, during the tests, the formation of some additional chromium silicide CrxSiy phases and chromium carbides CrxCy occurs at the interface between the braze joints and the tubes (see Fig. 9(b)). A decreasing gradient of iron concentration from the tubes towards the braze joints is observed. The elemental distributions according to the X-ray mapping show the presence of oxygen and chromium at the outer surface of the braze joints forming a thin layer of chromia Cr2O3. Thus, good oxidation resistance properties can be expected for the tube stacking because a strong concentration of chromium at the surface of the walls is needed to create a protective layer under severe atmospheres.

Fig. 8. Post-mortem observations of Sample 43B tested at 800 °C and 1 MPa for more than 1200 h of creep.

4. Modeling of the mechanical behavior of tube stackings This modeling part aimed at developing a predictive modeling of the mechanical behavior of cellular structures, based on the use of both the mechanical and geometrical properties of the constitutive material and cells measured experimentally as input data and, if possible, by not adjusting any of them to achieve a good fit between experiment and modeling. According to the literature [6,39,40], this is a challenging issue even for simplified model cellular structures, such as the tube stacking studied here, because of many non-linear and complex phenomena that contribute to the overall mechanical behavior of cellular structures, especially when high compaction levels are concerned. 4.1. Modeling assumptions Both the monotonic and creep compression tests described in the previous section were modeled in 2D using the finite-element suite Z-Set (http://www.zset-software.com/) and by applying plain strain conditions. Thanks to the symmetries, only one quarter of the 5  5-tube stacking core sandwich structure was simulated applying planar boundary conditions on the symmetrical faces. The structure was meshed using three quadratic tetrahedrons in the tube thickness. The mesh was refined around the braze joint extremities, in order to correctly capture the stress localization (see Fig. 10). A reduced integration was used. The convergence of the numerical problem was checked by varying the mesh size. The new contacts created between neighboring tubes during the compaction were modeled through a frictionless Coulomb contact, without any penalty condition. The calculations were conducted up to their divergence by using a polar decomposition for the finite strain Lagrangian formulation. For the sake of simplicity, the constitutive material of the sandwich structure was assumed to be homogeneous in the tube walls, the braze joints and the skins, being the one identified previously from the tensile tests on single tubes (see Part I [1]). Indeed, our previous works [12,39] have shown that the compression behavior of tube stackings or hollow-sphere structures is mainly governed by a mechanism of plastic hinges in the cell walls close to the rigid braze joints. Therefore, the length of the braze joints seems to be a more critical parameter than its mechanical properties. This was particularly true here, because the braze joints were rather large, whereas the tube walls were rather thin compared to the tube size (see Fig. 10). The constitutive material was supposed to be elasto-viscoplastic based on a bi-potential model with an isotropic elasticity, an isotropic non-linear plasticity and a Norton-type viscosity. For the sake of brevity, the constitutive equations and the values of the model parameters are not recalled here, but they are given in Section 3.2 of Part I.

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Fig. 9. SEM images of a brazed joint between two neighboring tubes and elemental distributions of Cr, Si, Ni and Fe according to the X-ray mapping (a) initial microstructure before testing (b) microstructure after creep testing at 800 °C and 1 MPa during more than 1200 h (Sample 43B). The format and length-scale of the micrographs are not the same in Figures (a) and (b) because they have been obtained from two different SEMs, but the magnification is the same  200.

According to the post-mortem observations, the crack occurrence has also been neglected in the modeling, because it appeared as a marginal phenomenon. The tube stacking was modeled with 1 mm thick skins, 4 mm large and 250 mm thick tubes, and 1.43 mm large braze joints. It is worth noting that the experimental scattering observed between the vertical and the horizontal braze joints (see Part I [1]) has been neglected here, and only their mean length has been considered. Similarly, architectural defects, such as tube misalignment and missing braze joints, have been neglected. In order to account for such architectural dispersions, real geometries should have been captured from image analysis techniques and meshed [41,42]. The model proposed here is thus unable to explain the experimental scattering observed. We can only suppose that, on the basis of our previous works [13], some samples showed a particular distribution of their geometrical defects towards the load direction that was more or less detrimental for the effective behavior. 4.2. Comparison between experiment and modeling 4.2.1. Monotonic compression tests The monotonic compression tests have been considered first

for the comparison between the experimental results and the modeling predictions. The three different temperatures have been simulated (room temperature, 600 °C and 800 °C), as well as the two different displacement rates ( 0.1 and 1 mm min 1), similarly to the experimental campaign. The mechanical responses are plotted in Figs. 11(a) and (b). They show a very good agreement between the experimental force-displacement curves and their predictions. Indeed, it is worth noting that the stress plateau and the effective hardening are very well predicted by the modeling, as well as the contribution of the viscosity at 800 °C (see Fig. 11(b)). Obviously, as expected, the viscous contribution predicted at room temperature and 600 °C is negligible (see Fig. 11(a)). If the curves are further analyzed, the bump observed in the numerical curves for a displacement of about 4.5 mm corresponds to the activation of the first internal contacts between neighboring tubes. Thus, due to the symmetries and the perfect geometry, this occurred suddenly and simultaneously in several places numerically. Experimentally, successive bumps were observed rather, but the first activation of the contacts was very close to that simulated. On the contrary, the final consolidation stage observed in the experimental curves (i.e., the final increase in force), occurs for lower

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At the end, during the consolidation stage, the compaction becomes more difficult and mainly a rapid increase in the Von Mises stress locally is observed.

Fig. 10. Finite element mesh used for the modeling of the compression samples.

displacements numerically. This limitation of the modeling approach has already been described in [39] and results from some locking problems of the mesh elements around the braze joints at very high compaction levels. The prediction of the initial elastic domain is also not so good. The effective stiffness of the structure is significantly overestimated by the modeling, but it is a very classical result because of some localized microplasticity (emphasized by the presence of architectural defects) and initial backlash of the experimental device, which make difficult the experimental characterization of the effective elastic properties [6,41,42]. For the sake of brevity, only few finite element maps are illustrated here, mainly due to the fact that the collapse mechanisms observed were very similar, whatever the type of test simulated (monotonic or creep), the test temperature, the strain rate or the load level applied. Fig. 12 shows some particular frames obtained at various compaction levels during the simulation of the monotonic compression test at room temperature and 0.1 mm min 1. According to the experimental observations, the first contacts are observed at the center of the structure at about 20% crushing ( 4.4 mm displacement). Then they are generalized progressively to the whole core, up to ∼40% crushing ( 8.8 mm displacement).

4.2.2. Creep compression tests Similarly to the monotonic tests, the different creep tests have been simulated. As a reminder, at 800 °C single-level tests at 40% and 60% of the effective yield stress were performed, plus one multi-level test at 40%, 60% and 80% of the effective yield stress. At 600 °C, only one multi-level test was performed, but for stress levels of 40%, 80% and 120% of the effective yield stress, because at this temperature the displacements remained quite small. The comparison between the experimental results and the simulated ones is presented in Fig. 13(a) to Fig. 14(a). It is not as good as that obtained for the monotonic tests. Figs. 13(a) and (b) show this comparison in the case of the multi-level compression creep tests. It can be noticed that at 600 °C the simulation is not able to capture the viscous effect revealed and the considerable increase in the displacement as from the beginning of the third stage. Indeed, numerically, no evolution of the displacement over time was possible because of the constitutive material behavior, and only the initial gap in the displacement when changing the load level is predicted correctly. The model is able to account for plasticity only, since increasing the load by a factor of three resulted in a displacement value multiplied by about ten numerically. This extradisplacement results from a strong contribution of the plasticity to the overall displacement as soon as the effective yield stress is exceeded. This can be explained by the role of the localized plasticity, which starts in many places of the architecture, even for stress levels lower than the effective yield stress, because of stress concentration phenomena. However, some coupling between plasticity and viscosity may exist at this longer time scale, which can no longer be neglected as an important part of the total viscous contribution. At 800 °C, the prediction of the compression creep curves was significantly better (see Fig. 13(b) and Fig. 14(a)). Although some gap between both the experimental and numerical curves is still observed, it remains rather slight and the displacement levels are correctly estimated. This can be explained by the fact that, at this temperature, the viscous contribution was significant and it was easier to capture its main features thanks to the constitutive behavior model. Moreover, the displacements observed at 800 °C were considerably higher than those observed at 600 °C. At 800 °C, the main limitation for the comparison was the locking problem of the mesh elements evoked before. The displacement levels being rather significant even for the lowest load level (at least 8 mm, equivalent to 36% crushing), the creep durations simulated were rather short.

Fig. 11. Comparison between experiment and modeling for the monotonic compression tests (a) at room temperature and 600 °C (b) at 800 °C.

Please cite this article as: V. Marcadon, et al., High temperature mechanical behavior of tube stackings – Part II: Comparison between experiment and modeling, Materials Science & Engineering A (2016), http://dx.doi.org/10.1016/j.msea.2016.09.052i

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Fig. 12. Equivalent Von Mises stress map obtained during the simulation of the monotonic compression test, at room temperature and 0.1 mm min 1, for a (a) 0% crushing (b) 10.3% crushing (c) 19.5% crushing (d) 29.7% crushing (e) 40% crushing (f) 45.8% crushing. The color scale varies between 0 MPa (blue) and 750 MPa (red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 14(b) proposes a zoom of Fig. 14(a) on the first fifty hours of creep, at 800 °C and for the single-level compression creep tests, for a deeper analysis of the inflections observed in the curves. A first inflection is observed for a displacement level of about 0.5 mm (2.3% crushing). This corresponds to the beginning of

the plasticity. A second inflection appears for a displacement of approximately 5 mm (22.7% crushing), resulting from the occurrence of the first internal contacts at this time and then followed by some other oscillations. These inflections are visible in the numerical curves also, and the associated displacement levels

Please cite this article as: V. Marcadon, et al., High temperature mechanical behavior of tube stackings – Part II: Comparison between experiment and modeling, Materials Science & Engineering A (2016), http://dx.doi.org/10.1016/j.msea.2016.09.052i

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Fig. 13. Comparison between experiment and modeling for the multi-level compression creep tests (a) at 600 °C (b) at 800 °C.

Fig. 14. Comparison between experiment and modeling for the single-level compression creep tests at 800 °C (a) total curves (b) zoom on the first fifty hours.

are very close to those estimated from the monotonic compression creep tests. However, the great scattering observed experimentally for the tests performed at 40% of the effective yield stress, compared with that measured for the tests at 60% of the effective yield stress, cannot be clearly explained. As already mentioned in Section 2, it might be due to the presence of some defects in the sample architecture. Unfortunately, due to a low number of samples and quite long test durations there is a lack of creep results to discuss further, namely with regard to their repeatability. 4.2.3. Discussion about modeling results The main contribution of the modeling has concerned the explanation of the softening observed at 800 °C at the beginning of the stress plateau for the monotonic compression tests (see Fig. 11 (b)). Actually, this phenomenon did not result from some experimental problems, but rather, it arises from the competition between the plastic and viscous effects observed in the constitutive material behavior at this temperature (see Part I [1]). The monotonic compression tests being governed in displacement, because of the strain localization in the architecture some parts of the stacking are subjected to a constant strain locally and thus they can relax, all the more so because the viscosity is activated at the very beginning of the loading (no activation threshold in the model in the viscous potential, see Part I). Hence, an overall softening is observed while the plastic hardening is not yet sufficiently spread to the whole structure to counterbalance the local relaxation.

The important gap observed between experiment and modeling as concerns the creep tests at 600 °C let suppose that some characteristic times of the creep phenomena were not captured from the monotonic tensile tests on single tubes, even though various strain rates were applied. Especially, due to the lack of creep characterization on the constitutive material alone (such an experimental device would have been rather complex to develop in the case of the tubes, see Part I [1]), we had no information concerning the primary creep of the constitutive material, despite the fact that the primary creep seems to have a significant contribution to the effective creep response of cellular structures [27– 29]. If experimental data would have been available, such primary creep could have been accounted for quite easily by considering strain hardening models, for instance [43], but here plasticity alone cannot explain displacement evolution at this particular temperature. At 800 °C there is still probably a contribution of primary creep, but it remains slight compared with the important overall viscosity, enabling a better comparison between experiment and modeling. As concerns the prediction of the collapse mechanism observed, it can be noticed that it is rather accurate when modeling only a quarter of the structure, despite collapse bands being observed experimentally at the scale of the whole structure. Actually, on the basis of the experimental observations, the collapse pattern of the square stacking is very stable and does not depend significantly on architectural defects, unlike the level and the chaotic aspect of the stress plateau. Such a conclusion would have

Please cite this article as: V. Marcadon, et al., High temperature mechanical behavior of tube stackings – Part II: Comparison between experiment and modeling, Materials Science & Engineering A (2016), http://dx.doi.org/10.1016/j.msea.2016.09.052i

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probably been different in the case of dynamic compression tests, or thinner tubes, which would have resulted in some instabilities and buckling. In this case, the collapse of the tube is governed by localized plasticity.

5. Conclusions This work has concerned the characterization of tube stacking mechanical behavior from room temperature to 800 °C. For that purpose, both monotonic and creep compression tests have been performed and simulated. These results are also based on a complementary work dedicated to the mechanical behavior of the constitutive material (see Part I [1]). The characterization of the compression behavior of the tube stacking studied has assessed the absence of viscous effects at room temperature and 600 °C. On the contrary, at 800 °C the competition between both plastic and viscous effects has been revealed from the monotonic compression tests. The finite-element modeling of these tests, based on the use of the elasto-viscoplastic law identified in Part I [1] and the knowledge of the actual geometry of the architecture, has improved the comprehension of the phenomena observed. Due to the strain localization in the architecture, some parts of the structure can relax inducing an overall softening as long as the plastic hardening is not spread enough to the whole structure to counterbalance this local relaxation. The use of a finite strain formulation has also enabled high compaction levels and internal contact issues to be investigated. The comparison between the experimental curves and those predicted has shown a very good agreement in that case, whatever the test temperature considered. Although this comparison has been not so good in the case of the compression creep tests, some encouraging results have been obtained. Overall, in the case of the creep tests at 800 °C, the experimental creep curves and those simulated are rather close. The displacement levels associated with the beginning of the plasticity and the occurrence of the first internal contacts are well predicted. They show a good consistency with those estimated from the monotonic tests. However, at 600 °C, the multi-level compression creep test performed has revealed some plastic and viscous contributions, which have not been captured correctly by the modeling. The case of the creep behavior must be further investigated. Mainly, there is a lack of experimental data related to both the creep behavior of the constitutive material, especially primary creep, and the reproducibility of the tests on the structure.

Acknowledgments This work was performed within the ‘Aero-Thermodynamic Loads on Lightweight Advanced Structures II’ project, investigating high-speed transport. ATLLAS II, coordinated by ESA-ESTEC, was supported by the European Union within the 7th Framework Program Theme 7 Transport, Contract no.: ACP0-GA-2010-263913. Further information on ATLLAS II can be found at http://www..esa. int/techresources/atllas_II. We are also indebted to our colleagues M. Thomas, P. Kanouté and S. Kruch for many fruitful discussions. J.-F. Caudrelier is thanked for the hardness measurements.

References [1] V. Marcadon, C. Davoine, D. Lévêque, A. Rafray, F. Popoff, N. Horezan, D. Boivin, Hightemperature mechanical behavior of tube stackings - Part I: Microstructural and mechanicalcharacterization of Inconels600 constitutive material, Mater. Sci. Eng. A (2016), In press, http://dx.doi.org/10.1016/j. msea.2016.07.088. [2] A.G. Evans, J.W. Hutchinson, M.F. Ashby, Multifunctionality of Cellular Metal

Systems, Cambridge University Press, Cambridge, 1997. [3] A. Ochsner, C. Augustin, et al., Multifunctional Metallic Hollow Sphere Structures: Manufacturing, Properties and Application, Springer, Berlin, 2009. [4] O. Friedl, C. Motz, H. Peterlik, S. Puchegger, N. Reger, R. Pippan, Experimental investigation of mechanical properties of metallic hollow sphere structures, Metall. Mater. Trans. B 39 (2008) 135–146. [5] O. Caty, E. Maire, R. Bouchet, Fatigue of metal hollow spheres structures, Adv. Eng. Mater. 10 (2008) 179–184. [6] O. Caty, E. Maire, S. Youssef, R. Bouchet, Modeling the properties of closed-cell cellular materials from tomography images using finite shell elements, Acta Mater. 56 (2008) 5524–5534. [7] O. Caty, E. Maire, T. Douillard, P. Bertino, R. Dejaeger, R. Bouchet, Experimental determination of the macroscopic fatigue properties of metal hollow sphere structures, Mater. Lett. 63 (2009) 1131–1134. [8] A. Fallet, P. Lhuissier, L. Salvo, Y. Bréchet, Mechanical behaviour of metallic hollow spheres foam, Adv. Eng. Mater. 10 (2008) 858–862. [9] P. Lhuissier, A. Fallet, L. Salvo, Y. Bréchet, Quasistatic mechanical behaviour of stainless steel hollow sphere foam: macroscopic properties and damage mechanisms followed by X-ray tomography, Mater. Lett. 63 (2009) 1113–1116. [10] W.S. Sanders, L.J. Gibson, Mechanics of hollow sphere foams, Mater. Sci. Eng. A 347 (2003) 70–85. [11] W.S. Sanders, L.J. Gibson, Mechanics of BCC and FCC hollow- sphere foams, Mater. Sci. Eng. A 352 (2003) 150–161. [12] V. Marcadon, F. Feyel, Mechanical modelling of the compression behaviour of metallic hollow-sphere structures: about the influence of their architecture and their constitutive material's equations, Comput. Mater. Sci. 47 (2009) 599–610. [13] V. Marcadon, S. Kruch, Influence of geometrical defects on the mechanical behaviour of hollow-sphere structures, Int. J. Solids Struct. 50 (2013) 498–510. [14] A.H. Brothers, D.C. Dunand, Plasticity and damage in cellular amorphous metals, Acta Mater. 53 (2005) 4427–4440. [15] A. Paul, U. Ramamurty, Strain rate sensitivity of a closed-cell aluminium foam, Mater. Sci. Eng. A 281 (2000) 1–7. [16] D. Ruan, G. Lu, F.L. Chen, E. Siores, Compressive behaviour of aluminium foams at low and medium strain rates, Comp. Struct. 57 (2002) 331–336. [17] J.F. Rakow, A.M. Waas, Size effects and the shear response of aluminium foam, Mech. Mater. 37 (2005) 69–82. [18] L.J. Gibson, M.F. Ashby, The mechanics of three-dimensional cellular materials, Proc. R. Soc. Lond. A 382 (1982) 43–59. [19] L.J. Gibson, M.F. Ashby, Cellular solids: structure and properties, Cambridge University Press, Cambridge, 1997. [20] A.M. Hodge, D.C. Dunand, Measurement and modeling of creep in open-cell NiAl foams, Metall. Mater. Trans. A 34 (2003) 2353–2363. [21] E.W. Andrews, L.J. Gibson, M.F. Ashby, The creep of cellular solids, Acta Mater. 47 (1999) 2853–2863. [22] E.W. Andrews, J.S. Huang, L.J. Gibson, Creep behavior of a closed-cell aluminium foam, Acta Mater. 47 (1999) 2927–2935. [23] A.C.F. Cocks, M.F. Ashby, Creep-buckling of cellular solids, Acta Mater. 48 (2000) 3395–3400. [24] E.W. Andrews, L.J. Gibson, The role of cellular structure in creep of two-dimensional cellular solids, Mater. Sci. Eng. A 303 (2001) 120–126. [25] J.S. Huang, L.J. Gibson, Creep of open-cell Voronoi foams, Mater. Sci. Eng. A 339 (2003) 220–226. [26] B. Su, Z. Zhou, Z. Wang, Z. Li, X. Shu, Effect of defects on creep behaviour of cellular materials, Mater. Lett. 136 (2014) 37–40. [27] F. Diologent, R. Goodall, A. Mortensen, Creep of aluminium-magnesium opencell foam, Acta Mater. 57 (2009) 830–837. [28] S. Soubielle, F. Diologent, L. Salvo, A. Mortensen, Creep of replicated microcellular aluminium, Acta Mater. 59 (2011) 440–450. [29] S. Soubielle, L. Salvo, F. Diologent, A. Mortensen, Fatigue and cyclic creep of replicated microcellular aluminium, Mater. Sci. Eng. A 528 (2011) 2657–2663. [30] H. Chos, D.C. Dunand, Mechanical properties of oxidation-resistant Ni-Cr foams, Mater. Sci. Eng. A 384 (2004) 184–193. [31] H. Chos, D.C. Dunand, Synthesis, structure and mechanical properties of Ni-Al and Ni-Cr-Al superalloy foams, Acta Mater. 52 (2004) 1283–1295. [32] J.D. DeFouw, D.C. Dunand, Processing and compressive creep of cast replicated IN792 Ni-base superalloy foams, Mater. Sci. Eng. A 558 (2012) 129–133. [33] S.M. Oppenheimer, D.C. Dunand, Finite element modeling of creep deformation in cellular materials, Acta Mater. 55 (2007) 3825–3834. [34] T.J. Chen, J.S. Huang, Creep-buckling of open-cell foams, Acta Mater. 57 (2009) 1497–1503. [35] Z.G. Fan, C.Q. Chen, T.J. Lu, Multiaxial creep of low density open-cell foams, Mater. Sci. Eng. A 540 (2012) 83–88. [36] A. Burteau, J.D. Bartout, Y. Bienvenu, S. Forest, On the creep deformation of nickel foams under compression, CR Phys. 15 (2014) 705–718. [37] O. Kesler, L.K. Crews, L.J. Gibson, Creep of sandwich beams with metallic foam cores, Mater. Sci. Eng. A 341 (2003) 264–272. [38] V. Marcadon, Mechanical modelling of the creep behaviour of hollow-sphere structures, Comput Mater. Sci. 50 (2011) 3005–3015. [39] V. Marcadon, C. Davoine, B. Passilly, D. Boivin, F. Popoff, A. Rafray, S. Kruch, Mechanical behaviour of hollow-tube stackings: experimental characterization and modelling of the role of their constitutive material behaviour, Acta Mater. 60 (2012) 5626–5644. [40] K.R. Mangipudi, S.W.v. anBuuren, P.R. Onck, The microstructural origin of strain hardening in two-dimensional open-cell metal foams, Int. J. Solids Struct. 47 (2010) 2081–2096. [41] S. Youssef, E. Maire, R. Gaertner, Finite element modelling of the actual

Please cite this article as: V. Marcadon, et al., High temperature mechanical behavior of tube stackings – Part II: Comparison between experiment and modeling, Materials Science & Engineering A (2016), http://dx.doi.org/10.1016/j.msea.2016.09.052i

V. Marcadon et al. / Materials Science & Engineering A ∎ (∎∎∎∎) ∎∎∎–∎∎∎ structure of cellular materials determined by X-ray tomography, Acta Mater. 53 (2005) 719–730. [42] A. Burteau, F. N'Guyen, J.D. Bartout, S. Forest, Y. Bienvenu, S. Saberi, D. Naumann, Impact of material processing and deformation on cell morphology and mechanical behavior of polyurethane and nickel foams, Int. J.

13

Solids Struct. 49 (2012) 2714–2732. [43] J. Besson, G. Cailletaud, J.L. Chaboche, S. Forest, M. Blétry, Non-Linear Mechanics of Materials (Solid Mechanics and Its Applications), Springer, Dordrecht, 2010.

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