A new criterion for elasto-plastic transition in nanomaterials: Application to size and composite effects on Cu–Nb nanocomposite wires

Available online at www.sciencedirect.com

Acta Materialia 57 (2009) 3157–3169 www.elsevier.com/locate/actamat

A new criterion for elasto-plastic transition in nanomaterials: Application to size and composite effects on Cu–Nb nanocomposite wires Ludovic Thilly a,*, Steven Van Petegem b, Pierre-Olivier Renault a, Florence Lecouturier c, Vanessa Vidal d, Bernd Schmitt b, Helena Van Swygenhoven b a

PHYMAT, University of Poitiers, SP2MI, 86962 Futuroscope, France b Paul Scherrer Institute, CH-5232 Villigen-PSI, Switzerland c Laboratoire National des Champs Magne´tiques Pulse´s, UPS-INSA-CNRS, 31400 Toulouse, France d CROMeP, ENSTIMAC, Campus Jarlard, 81013 Albi, France Received 22 September 2008; received in revised form 12 March 2009; accepted 15 March 2009 Available online 17 April 2009

Abstract Nanocomposite wires composed of a multi-scale Cu matrix embedding Nb nanotubes are cyclically deformed in tension under synchrotron radiation in order to follow the X-ray peak profiles (position and width) during mechanical testing. The evolution of elastic strains vs. applied stress suggests the presence of phase-specific elasto-plastic regimes. The nature of the elasto-plastic transition is uncovered by the ‘‘tangent modulus” analysis and correlated to the microstructure of the Cu channels and the Nb nanotubes. Finally, a new criterion for the determination of the macroyield stress is given as the stress to which the macroscopic work hardening, ha = dra/de0, becomes smaller than one third of the macroscopic elastic modulus. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Nanocomposite; Bauschinger effect; X-ray diffraction; Line broadening; Plastic deformation

1. Introduction As materials science has entered into the ‘‘nanomaterials era” tremendous efforts have been devoted to understanding the effect of grain size reduction on the mechanical properties that were observed to obey to the now famous ‘‘smaller is stronger” principle. Experimentally, many studies have tried to uncover some universal behaviour in the strengthening recorded in several nanocrystalline materials (single- or multi-phase, in the bulk form or thin films and multilayers, etc.), in particular via the Hall–Petch plot, which represents the macroscopic yield stress (or ‘‘macroyield”), rM, as a function of the inverse of the square root of the grain size, d: most of the materials exhibit a linear *

Corresponding author. Tel.: +33 5 49 49 68 31; fax: +33 5 49 49 66 92. E-mail address: [email protected] (L. Thilly).

dependence of rM vs. d1/2 over several orders of magnitude of grain size, while others present a grain size threshold below which some softening occurs, the so-called ‘‘inverse” Hall–Petch effect [1–4]. Theoretically, several attempts have been made to explain the Hall–Petch relationship in terms of dislocation pile-ups against grain boundaries (GBs), despite pile-ups never having been observed in most of the materials studied [5,6]. Recently, the debate on the origin of the Hall–Petch relationship and its validity has been revived by a series of theoretical and experimental studies that have led to the following conclusions: (i) the Hall–Petch law could result from the collective motion of interacting dislocations via elastic energy transfer between grains withstanding deformation by dislocation avalanches [7], the breakdown of such law at very small grain size then being related to the loss of the collective behaviour as deformation proceeds in the

1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.03.021

3158

L. Thilly et al. / Acta Materialia 57 (2009) 3157–3169

nanometre range by the gliding of individual dislocations and (ii) nanocrystalline materials deform very heterogeneously because of complex internal stresses arising both from processing and from plastic incompatibilities between grains with different orientations towards an externally applied stress. The probability that a grain experiences a plastic event (i.e. the emission of one individual dislocation at a GB, its gliding in the grain interior and its absorption at the GB) is thus very small [5,6]. As a consequence, nanocrystalline materials exhibit an extended microplastic regime characterized by early strain hardening in the stress–strain curve. Hence the conventional criterion used to define the onset of macroplasticity, i.e. the value of rM when the macroyield strain, eM, is equal to 0.2%, appears meaningless since at such strain only a very small portion of grains have deformed plastically for a very small grain size [5,6]. This statement has been experimentally confirmed by in situ deformation under X-rays of nanocrystalline Ni and nanostructured Ni–Fe alloys, allowing for the measurement of elastic strain upon tensile loading: the amount of strain assigned to microplasticity was measured to significantly exceed the 0.2% definition [8,9]. A similar observation was obtained from the careful analysis of macroscopic stress–strain curves of free-standing thin films and multilayers via the extraction of the macroscopic work hardening, or ‘‘tangent modulus” ha, defined as [10]: ha ¼ dra =de0

ð1Þ

where ra and e0 are the applied stress and the applied strain, respectively. In this context, the study of mechanical properties of composite materials is of interest since they generally result from a complex interplay between the mechanical properties of individual phases and the presence of interfaces. In particular, internal stresses develop during co-deformation because of intra- and inter-granular variations of plastic strain, and have a strong impact on the strengthening mechanisms and macroscopic mechanical properties. In two-phase materials with yield stress mismatch, simple models can be used to describe and estimate the development of internal stresses during deformation [11–13]: as the composite is subjected to tensile loading, one phase enters in the plastic regime before the other one, which will eventually plastically deform at higher load. The composite stress is then the result of a rule of mixture of the stress of each phase. When the composite is then unloaded, the macroscopic stress remains higher than the stress in the soft phase until it reaches the unloaded state, where the soft phase is subjected to an (elastic) compression stress (a tensile stress in the hard phase). If the composite is subsequently subjected to compressive loading, it initially behaves elastically until the soft phase enters in the plastic regime in compression, a situation that will take place at a much lower absolute value compared to the tensile loading case because of the initial compression of the soft phase. As a consequence, an asymmetry in the forward (tensile) and reverse (compressive) composite yield stresses will occur,

and this characterizes the development of internal stress. If multiple tension–compression cycles are applied to the composite, the internal stress will develop accordingly to the load transfer between the soft and the hard phase and lead to differences in the forward and reverse loading curves. Experimentally, such a phenomenon was observed for the first time by Bauschinger [14]: materials containing internal stress exhibit a decrease in the reverse yield stress and a rounding of the stress–strain curve during compression compared to the tensile stress–strain curve. This socalled ‘‘Bauschinger effect” has been widely used to characterize the internal stresses in materials. However, the use of diffraction techniques has supplanted this macroscopic mechanical characterization of internal stresses by the measure of lattice strain in individual phases. Recently, modified Bauschinger tests have been reintroduced via simple load–unload tests for the study of internal stresses in thin films or composite wires where compression cannot be applied [15–17]. In this paper we combine the diffraction technique and tangent modulus analysis to study the size and composite effects on the elasto-plastic transition in nanocomposite wires composed of a multi-scale copper matrix containing niobium nanotubes. In the first part, the in situ tests during continuous deformation of the wires under synchrotron radiation are presented. The diffraction peak positions and profiles are studied during multiple loading–unloading tests to gain insight into the internal stress build-up as well as dislocation storage in the different phases of these materials. After presenting and adapting the tangent modulus method to the studied materials, an analysis of the elastoplastic transition in each phase of the materials is given which shows that the different regimes are correlated to the complex microstructure. Finally, a general criterion on the work hardening rate value is defined at the transition between microplastic and macroplastic regimes. 2. Experimental details The materials studied here are members of the family of high strength–high conductivity materials for non-destructive high field magnets [4,18–20]. They are fabricated via a severe plastic deformation (SPD) process, based on accumulated drawing and bundling, that produces copperbased nanostructured wires (so-called ‘‘co-cylindrical Cu/ Nb/Cu” wires) composed of a multi-scale oxygen-free high conductivity (OFHC) Cu matrix embedding Nb nanotubes [21]: a series of hot-extrusion/cold drawing/bundling cycles is repeated three times to obtain conductors containing N = 853 Nb nanotubes, as illustrated in Fig. 1a. In the following, all dimensions are given in the cross-section, i.e. perpendicular to the wire axis. For a wire with a total diameter of 0.5 mm, Nb nanotubes (average thickness tNb = 88 nm and total Nb volume fraction XNb = 20.8%) are filled with ‘‘Cu–f” copper nanofilaments (diameter dCu–f = 130 nm and volume fraction XCu–f = 4.5%), separated by the finest ‘‘Cu-0” copper channels (width

L. Thilly et al. / Acta Materialia 57 (2009) 3157–3169

Fig. 1. (a) Successive views of the multi-scale structure of a Cu/Nb/Cu nanocomposite wire with 1.5 mm diameter. At the highest magnification, the TEM micrograph shows Nb nanotubes filled with Cu (Cu–f) and embedded in Cu channels (Cu-0) (scale bar is 1 lm). (b) Schematic of half of a sample with a reduced section. (c) Geometry for X-ray diffraction with fixed incoming angle and beam scattering at crystallographic planes parallel to the tensile axis, (h k l)transverse (the tensile loading is applied perpendicular to the drawing plane).

dCu-0 = 93 nm and volume fraction XCu-0 = 17.7%); groups of 85 Nb nanotubes are separated by ‘‘Cu-1” copper channels (width dCu-1 = 360 nm and XCu-1 = 9.6%), etc. Finally, the group of 853 Nb nanotubes is embedded in an external ‘‘Cu-3” copper jacket. The width of ‘‘Cu-2” and ‘‘Cu-3” copper channels are dCu-2 = 3.9 lm and dCu-3 = 21.1 lm (XCu-2 = 19.9% and XCu-3 = 27.5%), respectively. For the wire with a 0.5 mm diameter, the cold work rate (RA) is 99.99% (RA = (Si  Sf)/Si, where Si and Sf are the wire section at the last recrystallization heat treatment and the final drawn section, respectively) and the total true strain g is 23.2 (g = ln (S0/Sf), where Sf is the final drawn section

3159

and S0 is the initial section of the precursor Cu–Nb billet, respectively). Transmission electron microscope (TEM) investigations on the severely cold drawn wires revealed that Nb nanotubes are composed of nanograins with diameters in the 50–200 nm range and elongated along the wire axis with sharp h1 1 0i texture in the wire axis direction. Here the Nb grain size is equal to the nanotubes thickness tNb. A double texture with major h1 1 1i and minor h2 0 0i orientations in the wire axis direction was observed in the copper matrix (with a minimum fraction of 70% for the h1 1 1i component as estimated by laboratory X-ray diffraction at a diameter of 3.5 mm [21]). Because of the multi-scale structure, different types of copper channels are present in the matrix: (i) ‘‘fine” Cu channels (Cu–f, Cu-0 and Cu1 channels, corresponding to 40% of the Cu matrix), which are composed of individual grains of the size of the channel width dCu–i (i = f, 0, 1), elongated along the wire axis. In particular, the Cu–f fibres tend to a single crystalline structure and (ii) ‘‘large” Cu channels (Cu–i, i = 2, 3), which are similar to ultrafine-grained Cu, with grains from 200 nm to the micrometer range and with a high dislocation density, and are elongated along the wire axis [21]. The ultimate tensile strength (UTS) of the 0.5 mm wire is 1.2 GPa at room temperature [21]. Such high strength is explained by specific strengthening mechanisms operating in the different nanometre scale regions of the nanocomposite wire: Orowan-type strengthening in the Cu-0 channels, the Cu-1 channels and the Nb nanotubes, and whisker-type strengthening in the Cu–f nanofilaments [21,22]. The cylindrical 0.5 mm wire was mechanically polished on two parallel sides using a tripod device at very low speed to limit the formation of cold worked layers. The section of the wire was then reduced to obtain a gauge section below 0.14 mm2 over several millimetres (Fig. 1b). Such preparation (i) enables probing of the nanocomposite regions that would be screened by the large external Cu jacket because of the limited penetration depth of X-rays and (ii) forces the plastic deformation to appear under the synchrotron beam, where the section is reduced. The in situ tensile tests were performed at room temperature at the Materials Science Beamline at the Swiss Light Source of the Paul Scherrer Institute, Switzerland, because of the combination of a miniaturized tensile machine, a high-flux and high-energy beam and a one-dimensional position-sensitive microstrip detector that allows for fast measurements of diffraction patterns over a 2h range of 60° with an angular resolution of 0.004° [23]. This setup was used to study the diffraction peak positions and profiles during multiple loading– unloading tests, with fixed incoming beam angle and beam scattering at crystallographic planes parallel to the tensile axis using X-rays with energy of 24.2 keV. In such a configuration, the observed diffraction peaks for the Cu and Nb phases correspond to reflections that are perpendicular to the axial texture components and the measured lattice strains correspond to transverse strains, as illustrated by

3160

L. Thilly et al. / Acta Materialia 57 (2009) 3157–3169

Fig. 1c. For good statistics, the diffracted X-rays are collected during runs of 30 s. Therefore, a strain rate of 105 s1 was chosen to continuously follow the diffraction pattern during the loading and unloading cycles. Si powder was fixed on the sample’s surface for angular calibration. The position and profile of several Nb and Cu diffraction peaks were followed vs. the applied load: (1 1 0)Nb, (2 0 0)Nb, (2 1 1)Nb and (2 0 0)Cu, (2 2 0)Cu, (1 1 1)Cu (the latter reflection exhibits low intensity because it corresponds to non textured Cu grains). For the data analysis, i.e. peak fitting, Pearson VII functions were used. The Nb peaks and the (1 1 1)Cu, (2 0 0)Cu peaks were found to be properly fitted by single symmetric functions. As demonstrated in previous in situ experiments, it was shown that the (2 2 0)Cu reflection results from the superposition of two peaks, the first coming from the large Cu channels (‘‘large Cu peak”), the second from the fine Cu channels (‘‘fine Cu peak”) [17,22,24,25]. It was also found that, compared to the large Cu peak, the fine Cu peak is broadened because of the smaller grain size and is less intense because of the smaller volume fraction. Consequently, the (2 2 0)Cu peak was fitted by two symmetric functions, as illustrated by Fig. 2. The best fitting results were obtained with an integrated intensity ratio I2/I1 of 0.8, which is in good agreement with the ratio of fine to large Cu channels volume fractions in the probed region, where the external Cu-3 jacket was removed after polishing. The experimental errors for the peak positions are estimated to ±0.01% and for the full width at half maximum (FWHM) they are estimated to ±1%. In the following, the analysis is focused on the (2 2 0)Cu and the (1 1 0)Nb reflections because they are representative of the different regions of the material since they arise from

Fig. 2. Example of decomposition of the (2 2 0)Cu peak into two symmetric Pearson VII functions, the first coming from the large Cu channels (‘‘large Cu peak”), the second from the fine Cu channels (‘‘fine Cu peak”). This reflection corresponds to the unloaded state at the start of cycle 7 (see Fig. 4).

the Cu grains with h1 1 1i or h2 0 0i axial textures and from the Nb grains with h1 1 0i axial texture, respectively. 3. General features of the deformation Fig. 3 presents the macroscopic true stress–true strain curve of the Cu/Nb/Cu sample with d = 0.5 mm: the development of increasing hysteresis at each cycle is the signature of the build-up of large internal stresses. To study the origin of such hysteresis, the evolution of diffraction peak position vs. run numbers (i.e. vs. time) has been reported in Fig. 4 for the (2 2 0) reflection in large and fine Cu channels (Fig. 4b) and the (1 1 0) reflection in the Nb nanotubes (Fig. 4c), in correspondence with the evolution of true stress (Fig. 4a). The stress-free position of the (2 2 0)Cu peak is 23.12°, while that of the stress-free (1 1 0)Nb peak is 12.60°. Therefore, the different regions of the sample show, in the as-processed state, a certain amount of internal strain, with the fine Cu channels being under larger axial elastic compression than the large Cu channels (2h220(fine Cu) < 2h220 (large Cu) < 23.12° for (2 2 0) planes parallel to tensile axis), while the Nb nanotubes are under axial elastic tension (2h110(Nb) > 12.60° for (1 1 0) planes parallel to tensile axis). This internal stress state results from the fabrication process based on SPD [21,22,24]. It is interesting to note that the internal stress state evolves after each loading– unloading cycle, with an increase in the axial compression in Cu channels and in the axial tension in the Nb nanotubes (Fig. 4b and c). As it clearly appears that the Cu matrix is not in a stressfree state, few words can be said about the possible impact of elastic distortion of the lattice on the Bragg reflections: as stated before, the Cu matrix is heavily textured in the wire axis direction, with a major h1 1 1i texture component and a minor h2 0 0i component. Let us consider a set of grains that all exhibit a [0 0 1] orientation in the wire axis (the minor texture component): if the lattice is slightly com-

Fig. 3. Macroscopic true stress–true strain curve of one wire cyclically tested in tension.

L. Thilly et al. / Acta Materialia 57 (2009) 3157–3169

3161

Fig. 4. Evolution vs. run numbers (i.e. time) and loading–unloading cycles of: (a) true stress; (b) position of the (2 2 0) large Cu and fine Cu peaks; and (c) position of the (1 1 0) Nb peak.

pressed in the [0 0 1] direction (the c-axis of the unit cell), because of the Poisson effect the lattice is slightly expanded in the (0 0 1) plane resulting in a tetragonal lattice with a = b > c and a = b = c = 90°. Calculating the lattice spacing of the different {2 2 0} planes gives d220 = d220 = d220 = d220 = d1 and d202 = d202 = d022 = d202 = d202 = d202 = d022 = d022 = d2, with d1 > d2. However, we only record transverse reflections, therefore only the (2 2 0), (2 2 0), (2 2 0) and (2 2 0) reflections will be observed: they are at the same angular position. Let us now consider a set of grains that all exhibit a [1 1 1] orientation in the wire axis (the major texture component): if the lattice is slightly compressed in the [1 1 1] direction, because of the Poisson effect the lattice is slightly expanded in the (1 1 1) plane, resulting in a rhombohedral (or trigonal) lattice with a = b = c and a = b = c > 90°. Calculating the lattice spacing of the different {2 2 0} planes gives d220 = d220 = d202 = d202 = d022 = d022 = d3 and d220 = d220 = d202 = d202 = d022 = d022 = d4, with d3 < d4. Again, because we only record transverse reflections, only the (2 2 0), (220), (2 0 2), (202), (0 2 2) and (022) reflections will be observed: they are at the same angular position. Since h1 1 1i is the major texture component, only the latter case can be considered: as demonstrated in Ref. [22], the combination of lattice distortion, texture and elastic anisotropy of copper has no impact on the interpretation of the decomposition of the transverse (2 2 0)Cu reflection, which is solely the signature of two distinct residual stress states (and two distinct microstructures) in the Cu matrix. A deeper analysis of the phase-specific deformation can be performed by plotting the measured transverse elastic

strain of phase i, eti, as a function of the applied stress. The elastic strain is calculated from the peak position compared to the stress-free state. Fig. 5 is an example of such plot for cycle 7: upon macroscopic loading, the transverse elastic strain of the Cu matrix (both large and fine Cu channels) first decreases from positive values, until the stressfree position is reached at zero elastic strain (from A to A0 for the large Cu, from a to a0 for the fine Cu): the Cu channels first unload from axial compression; after this point, the Cu channels are loaded into axial tension (negative transverse strain). For the large Cu (2 2 0) reflection, the slope detCu–large/dra decreases (and tends to zero) upon loading, suggesting the development of non-negligible plasticity, a phenomenon that is repeated upon macroscopic unloading (from C to E), leading to a wide loop in such a representation. For the fine Cu (2 2 0) peak, the slope detCu–fine/dra appears relatively constant during the loading–unloading cycle, leading to a narrow loop. The case of the Nb is opposite to the large Cu: for the (1 1 0)Nb reflection, the slope detNb/dra is rather constant in the first part of the loading curve, as a footprint of elastic loading, then it continuously increases until maximum stress is reached; the unloading part of the curve is then ‘‘above” the loading curve. Such a phenomenon is the result of a strong load transfer occurring from the yielding large Cu channels onto the elastic Nb nanotubes, as observed in coarser structures during in situ deformation under neutrons [22,24]. From these features, it appears that the nanocomposite wires are composed of phases with distinct elasto-plastic behaviour. To understand the different deformation regimes and to characterize the elasto-plastic transition

3162

L. Thilly et al. / Acta Materialia 57 (2009) 3157–3169

Fig. 5. Evolution of transverse elastic strain vs. applied stress during cycle 7 in large Cu channels (from the (2 2 0) large Cu peak), in fine Cu channels (from the (2 2 0) fine Cu peak) and in Nb nanotubes (from the (1 1 0)Nb peak).

that occurs within each phase during the deformation of the whole wire, the tangent modulus method is applied with an adaptation suited to the present structure. 4. The tangent modulus model Initially, the tangent modulus model was developed to study the strain hardening behaviour of copper–chromium composite wires deformed under a neutron beam [16]. It was then extended to the study of thin films and multilayers [10]. Here, this model has been modified to take into account the complex multi-scale structure of the Cu/Nb/ Cu wires, a microstructure that can be simplified as composed of three ‘‘phases”: large Cu channels (Cu–L), fine Cu channels (Cu–F) and Nb nanotubes that are assumed to be uniform in properties and dimensions; they are continuous and parallel throughout the entire sample and form a unidirectional composite material. Perfect bonding exists between the three phases. Since the three phases are in parallel and the stress is applied in the wire axis direction, the deformation of the wire follows an iso-total axial strain model, with identical total axial strain in each phase [16,26]. Let us call ei the axial elastic strain in phase i and epi the axial plastic strain in phase i; the total axial applied strain e0 is then related to the phase-related elastic and plastic strains as: e0 ¼ eCu–L þ epCu–L ¼ eCu–F þ epCu–F ¼ eNb þ epNb

ð2Þ

It is worth remembering that ei can be measured from diffraction experiments. In the present samples, all Nb grains have a h1 1 0i orientation in the wire axis: experimentally, eNb is therefore equivalent to e110Nb. The Cu grains exhibit either a h1 1 1i or a h2 0 0i orientation in the wire axis; since the h2 0 0i texture component is negligible here, eCu is taken as equivalent to e111Cu.

However, in the present setup, we can only measure the transverse elastic strain in each phase, eti: to apply the tangent modulus analysis, the phase-related axial elastic strains ei need first to be calculated from the transverse elastic strains. Such calculation is done using the following equation: ei ¼ eti =Ai

ð3Þ

where Ai is analogous to the Poisson ratio. Ai was experimentally obtained from previous experiments on in situ deformation under neutrons of similar nanocomposite materials [24,25] and calculated as the ratio of transverse strain to axial strain for the same Cu or Nb grain families (i.e. from transverse (2 2 0) and axial (1 1 1) reflections for Cu and from transverse (1 1 0) and axial (1 1 0) reflections for Nb): it was found that ACu–L = 0.340 ± 0.044, ACu–F = 0.340 ± 0.084 and ANb = 0.624 ± 0.062. The use of Eq. (3) (again experimentally verified during in situ deformation under neutrons) implies that the elastic constants of Cu and Nb remain constant all along the tensile test (the generalized Hooke’s law can be applied) and that the crystallographic symmetries of the different phases (face-centred cubic for Cu and body-centred cubic for Nb) are kept all along the test. Lattice-statics calculations and isostress molecular-dynamics simulations of the mechanical response of model cubic metallic crystals to applied uniaxial loading (tension and compression) have shown that these assumptions are correct within the range of elastic strains applied in the present experiments, which are always (much) smaller than a few percents [27,28]. Pure OFHC Cu and pure Nb do not exhibit any phase transition in the present experimental conditions (room temperature and stress intensity smaller than 2 GPa). The applied axial load carried by the composite is the sum of the axial loads carried by each constituent: the mac-

L. Thilly et al. / Acta Materialia 57 (2009) 3157–3169

3163

roscopic applied axial stress ra can be expressed as a rule of mixture of the phase-related axial stresses, ri, and the volume fractions, fi [16,26]: ra ¼ fCu–L rCu–L þ fCu–F rCu–F þ fNb rNb

ð4Þ

If one considers the macroscopic axial elastic modulus, E, a similar expression applies for E as a function of the phaserelated axial elastic moduli, Ei (by differentiating Eq. (4) in the elastic regime or applying Hooke’s law) [16,26]: E ¼ fCu–L ECu–L þ fCu–F ECu–F þ fNb ENb

ð5Þ

From Eqs. (4) and (1), which gives the definition of the macroscopic work hardening (or tangent modulus), ha, the tangent modulus can be expressed as a function of the strain hardening rate in phase i, hi = dri/de0: ha ¼ dra =de0 ¼ fCu–L hCu–L þ fCu–F hCu–F þ fNb hNb

ð6Þ

In addition, the axial stress carried by the phase i may be written as: ri ¼ E i ei

ð7Þ

Then the strain hardening of phase i becomes: hi ¼ dðEi ei Þ=de0 ¼ Ei dei =de0

ð8Þ

Let us define the quantity xi as: xi ¼ dei =de0

ð9Þ

Eq. (8) shows that if xi is equal to one, the ith phase deforms in a purely elastic regime (hi = Ti); if xi is equal to zero, the ith phase deforms in a purely plastic regime (hi = 0). Combining Eqs. (6)–(9) allows the tangent modulus to be written as a function of xi: ha ¼ fCu–L ECu–L xCu–L þ fCu–F ECu–F xCu–F þ fNb ENb xNb

ð10Þ

Moreover, the tangent modulus can be written as: ha ¼ dra =de0 ¼ ðdei =de0 Þðdra =dei Þ ¼ xi dra =dei

ð11Þ

From Eqs. (3) and (11), one obtains a simple relation that allows for the calculation of xi: xi ¼ ð1=Ai Þha ðdeti =dra Þ

ð12Þ

Eq. (12) is of particular interest because it shows that, for each phase i, the quantity xi can be calculated with the knowledge of the macroscopic work hardening, ha, and the slope deti/dra of the curve showing the evolution of the transverse elastic strain in phase i, eti, as a function of applied stress ra. In other words, ha can be computed for each loading–unloading cycle from Fig. 3 while deti/dra can be calculated from Fig. 5 (for cycle 7). For more clarity, the analysis will be focused again on cycle 7 but the study of the other cycles lead to similar conclusions. Fig. 6 is a plot of xCu–L, xCu–F and xNb as a function of applied stress, during the loading phase of cycle 7. First, from the start of the loading, the value of xCu–L decreases rapidly from its initial value of 1. One has to remember that the large Cu channels are initially in axial compression so the first part of the loading corresponds to the unloading

Fig. 6. Evolution of parameter xi (see text) vs. applied stress during the loading stage of cycle 7 for the large Cu channels, the fine Cu channels and the Nb nanotubes.

of these channels: in detail, this unloading from axial compression occurs up to an applied stress of 300 MPa (from A to A0 in Figs. 5 and 6); during this ‘‘internal unloading” stage, xCu–L decreases from 1 to 0.8, suggesting that the large Cu channels do not unload purely elastically. Upon further loading, the large Cu channels are then put into axial tension up to the maximum applied stress of the cycle; during this stage, xCu–L continuously decreases down to 0.2 (point C), suggesting that plasticity occurs in this phase. Secondly, for the fine Cu channels, xCu–F remains close to 1 upon loading up the point that corresponds to the stress–free state of these channels (point a0 in Figs. 5 and 6): the ‘‘internal unloading” from axial tension seems to occur almost purely elastically. Then xCu–F starts to deviate from unity and decreases to 0.5 at maximum applied load (point v). Thirdly, xNb remains close to 1 during all the loading stage and even increases above 1 at high applied stress: the increase in xNb above 1 is another signature of the load transfer occurring between the yielding Cu channels and the elastic regions of the sample, in particular the elastically loaded Nb nanotubes. Fig. 7 is a plot of xCu–L, xCu–F and xNb as a function of applied stress, during the unloading phase of cycle 7: all the previous remarks on the evolution of x for the three phases can be repeated. We see from the evolution of the parameter x that this quantity may be used as a tool to characterize the different deformation regime in each phase: if xi is equal to one, the ith phase deforms in a purely elastic regime; if xi tends to zero, the ith phase deforms in a purely plastic regime. However, there is no criterion on the definition of the elastoplastic transition, i.e. a value for x to define the transition from elastic to plastic deformation. To obtain such a criterion, the tangent modulus model can be completed by the analysis of X-ray diffraction data, in particular the peak profile evolution.

3164

L. Thilly et al. / Acta Materialia 57 (2009) 3157–3169

Fig. 7. Evolution of parameter xi (see text) vs. applied stress during the unloading stage of cycle 7 for the large Cu channels, the fine Cu channels and the Nb nanotubes.

to the decrease in the peak FWHM. Meanwhile, the fine Cu (2 2 0) peak FWHM increases from point a to point v (Fig. 8c): since the value of xCu–F remains always greater than the value x = 0.33 (Fig. 6), such an FWHM increase may be related to microplasticity events. Similarly, the small changes in FWHM observed for the Nb (1 1 0) peak upon loading can be mostly attributed to changes in RMS strain in the elastic Nb nanotubes. During the unloading stage, it is worth focusing on the macroyield threshold previously defined at x = 0.33: only xCu–L decreases below this value, which is attained at an applied stress close to 500 MPa, at point D (Fig. 7). This applied stress corresponds exactly to the stress at which the large Cu (2 2 0) peak FWHM reaches a minimum (point D in Fig. 8c) before increasing until the macroscopic unloaded state is reached. The D to E regime of Fig. 7 is therefore correlated to a macroplastic regime in the large Cu channels. As a consequence, the 0.33 value for x as a definition of the macroplastic threshold holds independently for the loading and the unloading parts.

5. Tangent modulus model confronted to X-ray diffraction

6. Discussion

Fig. 8 shows in detail the loading–unloading cycle previously studied, cycle 7, with additional data on the FWHM values of the large and fine Cu (2 2 0) peaks (Fig. 8c) and (1 1 0)Nb peak (Fig. 8e). Upon loading from the unloaded state (point A), the FWHM of the large Cu (2 2 0) peak decreases (Fig. 8c) until it reaches a local minimum (point B), then increases again until the maximum applied stress is attained (point C). During the unloading regime, the FWHM exhibits a similar trend with a decreasing phase from point C to point D followed by an increasing phase from point D to point E. For the fine Cu peak, the situation is different: the FWHM increases upon loading (from a to v), reaches a plateau when entering into axial tension (from a0 to b)), and decreases slowly when unloading in axial tension (from point v to point v0 ) then rapidly when loading in axial compression (from point v0 to point e). For the Nb (Fig. 8e), the FWHM change is always less than 2%. These results can be correlated with the evolution of the x parameter in each phase during the loading stage. For the large Cu channels, xCu–L decreases continuously (from A to C, Fig. 6) while the FWHM of the large Cu (2 2 0) peak reaches a minimum at point B, corresponding to an applied stress of 650 MPa, and increases to a maximum at point C (Fig. 8c). If one considers that the increase in FWHM at point B defines the onset of macroplasticity, i.e. when a majority of grains in the large Cu channels have experienced a plastic event leading to the increase in the dislocation content and therefore FWHM, such a macroyield threshold corresponds to x = 0.33 (point B, Fig. 6). In this context, the decrease in xCu–L upon loading from point A to point B may be attributed to a microplastic regime where discrete dislocation movements/rearrangements and a reduction in root mean square (RMS) strain lead

The consequences of the correlation between the in situ deformation under X-rays (allowing for the following of diffraction peaks and profiles upon deformation) and the tangent modulus analysis is essentially twofold: first, it enables the determination of phase-specific elasto-plastic regimes in the complex Cu/Nb/Cu nanocomposite wires; and secondly, it allows for defining a general criterion on the tangent modulus value at the transition between microto macroplastic regimes. Let us first address the detailed deformation scheme of the Cu/Nb/Cu wires during one loading–unloading cycle, with respect to the three phases present. The large Cu channels exhibit a microstructure that is typical of the ultrafine-grained copper obtained by SPD processing, i.e. they comprise grains with a typical diameter of around 200–400 nm, with a few grains still in the micrometer range but containing low angle GBs. Such cold worked Cu contains an elevated dislocation density (up to 1016 m2 [29,30]) that is not homogeneously distributed, with the grain interior rather poor in dislocations because of dynamic recrystallization and GB regions containing a lot of dislocation segments bowing out of the GBs or low angle GBs [21]. This heterogeneous microstructure is most likely to lead to discrete dislocation movements and changes in the RMS strain even at small applied stress. Such behaviour would explain the decrease in FWHM as soon as the sample is loaded, i.e. when the large Cu channels are unloaded from axial compression. At an applied stress of 300 MPa, the large Cu channels are then in a stress-free state. When the sample is further loaded, the large Cu channels are put into axial tension. During the first part of the axial loading, the large Cu channels do not deform purely elastically again because of their complex microstructure: it is supposed that, because of the

L. Thilly et al. / Acta Materialia 57 (2009) 3157–3169

3165

Fig. 8. Evolution vs. run numbers during cycle 7 of: (a) true stress; (b); and (c) position and FWHM of (2 2 0) large Cu and fine Cu peaks, respectively; and (d) and (e) position and FWHM of (1 1 0) Nb peak, respectively.

inversion of applied stress, interactions between dislocations and microstructure lead to a further reduction in the RMS strain. The macroyield threshold is then reached at an applied stress of 650 MPa. If one takes into account the initial axial compression, this corresponds to an effective axial stress of the order of 350 MPa, which is indeed usually recorded in cold worked Cu. It is therefore not surprising at this point to observe an increase in the large Cu peak FWHM. Upon macroscopic unloading of the sample, the same scenario applies: during unloading from axial tension and in the first part of the loading into axial compression, the large Cu peak FWHM decreases because of a

reduction in the RMS strain accumulated during the previous macroplastic regime. However, in the unloaded state after tensile yielding (point C0 in Fig. 8c), the large Cu peak FWHM is greater than the one before tensile yielding (FWHM(C0 ) > FWHM(A0 )): the storage of dislocations within the grain interior is indeed possible in ultrafinegrained Cu. Then the large Cu channels reach the macroyield threshold but this time in compression. The corresponding macroscopic stress is close to 500 MPa, a value that is significantly smaller that the applied stress at tensile macroyielding. This macroyield stress asymmetry (the socalled ‘‘Bauschinger effect”) is usually related to long-range

3166

L. Thilly et al. / Acta Materialia 57 (2009) 3157–3169

effects (long-range internal stresses (LRIS) due to dislocation–microstructure interactions) and/or short-range effects (directionality of mobile dislocations or annihilation with reverse strain) [15,31]. The exact origin of this effect remains unclear. For example, the concept and presence of LRIS is still under debate in cyclically deformed fcc metals developing heterogeneous microstructures, such as Al, Ni and Cu [32–36]: direct estimation of internal stress in TEM thin foils via dislocation dipole spacing measurements or local convergent beam electron diffraction far or near dislocation walls suggests the absence of LRIS in cyclically deformed Cu [34,36]. In contrast, line profile analysis during forward deformation of Cu under X-rays advocates for a build-up of LRIS along with the increase in dislocation density (increase in FWHM and peak asymmetry) and a relaxation of LRIS and decrease in dislocation density upon unloading (decrease in FWHM and peak asymmetry) [32,35]. In this controversy, a further argument for the absence of LRIS in heterogeneous microstructures comes from in situ deformation in a TEM of precyclically deformed metals: no reversed motion or unbowing of dislocations is evidenced with unloading [33]. Again, this argument is controversial, since recent experiments of in situ forward loading in a TEM of thin foil made from SPD-deformed Cu (similar to the here studied large Cu channels) show that the mobile dislocations experience reverse motion as soon as the thin foil starts to be unloaded because of LRIS [37]. Such behaviour is most likely to occur here in the bulk large Cu channels, as evidenced by the decrease in FWHM each time the stress direction is inverted. In summary, the large Cu channels are subjected not only to the applied stress but also to LRIS arising from their own microstructural heterogeneity and the back stress from the Nb nanotubes in axial tension. In other words, because of the presence of complex internal stresses, the large Cu channels undergo compression–tension–compression cycles during the multiple tensile loading–unloading cycles and exhibit two macroplastic stages during each cycle. The microstructure of the fine Cu channels is radically different: first, they are composed of only one grain between two Cu–Nb interfaces over several micrometres in the wire axis direction. From the dislocation glide viewpoint, they can be seen locally as single crystals. Moreover, their initial dislocation content is relatively small because of the dynamic recrystallization arising after a total true strain of 23.2, i.e. beyond stage V of the deformation of Cu [38]. Secondly, their size is in the nanometre range, with the diameter of the Cu fibres (Cu–f) inside the Nb nanotubes dCu–f = 130 nm, the width of the inter-tube Cu (Cu0) channels dCu-0 = 93 nm and the thickness of the Cu channels around 85 nanotubes (Cu-1) dCu-1 = 360 nm. At such dimensions, particularly in the Cu–f and the Cu-0 channels, the macroyield stress is enhanced by the size effect via a modification of the deformation mechanisms. In detail, in situ deformation in a TEM has been performed on a nanofilamentary thin foil made from a Cu/Nb wire

containing Nb nanofilaments with a diameter dNb = 264 nm and Cu-0 interfilamentary channels with dCu-0 = 136 nm [39]. It was observed unambiguously that plastic deformation proceeds in the Cu-0 channels by emission of individual dislocation loops at one Cu–Nb interface, then gliding in the Cu-0 grain interior and absorption in the neighbouring Cu–Nb interface. As a result, edge segments are absorbed in the Cu–Nb interfaces and screw segments bow out between two interfaces, eventually propagating by further bowing and deposition of dislocation debris at interfaces. Such a ‘‘single dislocation” regime has been observed in other nanocrystalline materials [40– 42] and is usually related to the so-called Orowan-type behaviour, where the enhanced yield stress can be related to the microstructure size, d, as [39]: ry ¼ r0y þ alðb=dÞ lnðd=bÞ

ð13Þ

where r0y is the yield stress of coarse grain material, a is a constant depending on the dislocation character and the Taylor factor, l is the shear modulus and b is the length of Burgers vector. Moreover, previous studies have shown that the Cu–f fibres behave as whiskers with an extremely high elastic limit; their yield stress can be described by an exponential law [21,22]: ry ¼ r0y þ A expðd Cu–f =BÞ

ð14Þ

where A = 15.103 MPa and B = 156 nm. In addition, nanocrystalline materials have been observed to exhibit an extended microplastic regime characterized by early strain hardening in the stress–strain curve [8,9]. Together, these elements allow an understanding of the behaviour of the fine Cu channels, in particular the extended elasto-microplastic regime upon loading and unloading. The fact that the macroplasticity regime is barely reached is related to the elevated yield stress of these regions: in the case of the Cu-0 channels, Eq. (13) gives a yield stress of 760 MPa (with r0y ¼ 350 MPa, a = 0.6, l = 42.5 GPa, bCu = 0.256 nm and d = dCu-0 = 93 nm), while for the Cu-1 channels it gives a yield stress of 480 MPa. For the Cu–f fibres, Eq. (14) gives a yield stress of 6.8 GPa. The fine Cu channels are initially in strong axial compression and they reach the stress-free state at an applied stress of 500 MPa. Since the UTS of the sample is 1.2 GPa, the maximum tensile stress carried by the fine Cu channels may reach the macroyield stress only in the Cu-1 channels before the fracture of the sample. As a consequence, most of the fine Cu channels remain in the microplastic regime. Therefore, the observed increase in the fine Cu peak FWHM upon loading is mainly related to changes in the RMS strains and/or discrete movements of dislocations that are bowing out of Cu–Nb interfaces. This latter aspect can be studied in more detail by looking at the evolution of FWHM during one loading–unloading cycle (Fig. 8c): upon loading from the macroscopic unloaded

L. Thilly et al. / Acta Materialia 57 (2009) 3157–3169

state a, FWHM first increases linearly until the unloaded state is reached (point a0 ), when it suddenly levels off. As long as the peak position of the fine Cu phase increases linearly (with time but also with applied stress, as shown in Fig. 5), the FWHM stays approximately constant. When the peak position starts deviating from linear behaviour (point b) the FWHM starts increasing again until the maximum macroscopic tensile load is reached at v. During unloading, the FWHM decreases very slowly until the unloaded state is reached (point v0 ), then decreases rapidly until the macroscopic unloaded state (point e). These trends are observed during all cycles. In the initial unloaded state, the fine Cu channels are in strong axial compression: the dislocation segments that were bowing out of the Cu–Nb interfaces in the previous forward loading regime are pushed back into the interfaces. When the macroscopic load is increased, the fine Cu channels gradually unload from compression and the dislocations return to an equilibrium configuration, i.e. bow out again from the interfaces. The FWHM increases because the channel dislocation content increases and the RMS strains build-up at the interfaces. Then, when the fine Cu phase is loaded into tension, the FWHM stays approximately constant (point a0 –point b) before eventually increasing when the stress carried by the fine Cu channels is high enough to generate further expansion of existing dislocations loops between the Cu–Nb interfaces (from point b to point v), a situation that is characteristic of a microplastic regime with modifications of RMS strains at Cu–Nb interfaces. That FWHM does not decrease substantially during the first part of the tensile unloading (point v–point v0 ) suggests the necessity of a force to unpin the dislocations from Cu–Nb interfaces to initiate their reverse motion in the fine Cu channels. This can be easily understood, bearing in mind that the forward expansion of the dislocation loops is accompanied by the absorption of edge segments in the interfaces, a mechanism that is associated with atomic rearrangements and formation of ledges that have to be overcome during reverse motion. For the Nb nanotubes, the deformation history during one loading–unloading cycle is simpler: they are initially in axial tension and remain in this state all along the tensile cycling. Moreover, because of their microstructure, they can be seen as nanocrystalline layers (similar to the Cu-0 channels). Therefore, their yield stress can be estimated from Eq. (13) with r0y ¼ 1400 MPa, a = 1, l = 42.5 GPa, bNb = 0.286 nm and d = tNb = 88 nm: the Nb nanotubes should start to yield under a stress of 2.2 GPa. At maximum forward tensile load, this stress is not attained in the Nb even with the help of load transfer occurring from the Cu matrix: the Nb nanotubes remain fully elastic all along the tensile cycling and the slight changes in FWHM observed during the different cycles are mostly related to RMS strain modifications, probably concentrated at Cu– Nb interfaces. Let us now return to the definition of a criterion on the tangent modulus value at the transition between micro-

3167

and macroplastic regimes. From the correlation between the tangent modulus analysis and the in situ diffraction data, the parameter xi, defined by Eq. (12), has been calculated for the three phases and was shown to decrease below 0.33 when the corresponding phase reaches the macroplastic regime, i.e. when this phase experiences plastic events in all grains. Let us first consider a composite material (or a single phase material with several grain families) where all phases (or all grain families) reach the macroplastic regime simultaneously, i.e. xi = 0.33. At macroscopic macroyield, Eq. (10) becomes: ! X fi Ei ð15Þ haM ¼ 0:33 i

haM is then the value of the tangent modulus or macroscopic work hardening at macroscopic yielding. Because the macroscopic axial elastic modulus, E, is a function of the phase-related axial elastic moduli, Ei, via Eq. (5), the tangent modulus at the onset of macroplasticity is defined as: haM ¼ 0:33E

ð16Þ

This new criterion for defining the onset of macroplasticity can be compared to the conventional criterion i.e. the macroyield strain, eM, usually taken to be 0.2%. Let us compare two opposite cases that are schematized, respectively, in Fig. 9a and b: case 1, a material (singleor multi-phased) composed of coarse grains; and case 2, a nanocrystalline material. In case 1, with a limited microplastic regime, the tangent modulus decreases rapidly once the macroplastic regime is reached; the criterion haM = 0.33E is reached at moderate macroscopic strain and coincides with the conventional criterion eM = 0.2%. In case 2 an extended microplastic regime is observed, with early strain hardening in the stress–strain curve. Because of the strong rounding of the stress–strain curve, the criterion haM = 0.33E is reached at a macroscopic strain much larger than the conventional 0.2% criterion. As already pointed out in the introduction, the conventional criterion is meaningless, but the tangent modulus criterion haM = 0.33E can still be applied. For validation, this can be matched with experimental data on nanocrystalline Ni that was deformed under X-rays: the peak profile analysis during mechanical testing shows that the macroscopic strain at the onset of macroplasticity exceeds the usual 0.2% and is situated around eM = 0.7% [8]. Looking at the evolution of macroscopic work hardening from the stress–strain curve (Fig. 3b of Ref. [8]), the tangent modulus criterion haM = 0.33E is in good agreement with eM = 0.7%. However, in the case of a composite material, the tangent modulus criterion should be taken with care, since the tangent modulus reflects a macroscopic average of the phase-related strain hardening rates, hi (Eq. (6)), and can be also seen as an average of the phase-related xi parameters (Eq. (10)). In an extreme case, the x parameter may decrease strongly only in the softest phase and the macroscopic tangent modulus reaches the

3168

L. Thilly et al. / Acta Materialia 57 (2009) 3157–3169

Fig. 9. Schematics of macroscopic stress–strain curves in the case of (a) a material (single- or multi-phased) composed of coarse grains and (b) a nanocrystalline material. The conventional criterion for macroyield stress at a macroscopic strain of 0.2% is compared to the criterion for macroyield stress at a tangent modulus equal to one third of the elastic modulus.

macroyield criterion of E/3: in such a case, the tangent modulus criterion would not reflect that the whole composite has reached the macroyield regime but only that this regime has been reached in some of the constituents. It is interesting to note that, in the case of the Cu/Nb/Cu nanocomposite wires studied in this work, and in particular during loading–unloading cycle 7, the tangent modulus reaches the E/3 value upon loading only at an applied stress much larger than the applied stress for which xCu–L reaches 0.33 (see Fig. 10) and does not reach this value during unloading. This illustrates that the extreme case previously described is specific to composite materials with an extremely large internal yield stress mismatch, a situation that is rarely experienced. 7. Conclusions Nanocomposite wires composed of a multi-scale Cu matrix embedding Nb nanotubes have been cyclically deformed in tension under synchrotron radiation in order

Fig. 10. Evolution of the macroscopic tangent modulus vs. applied stress during loading–unloading cycle 7. The criterion of tangent modulus equal to one third of the elastic modulus is given.

to follow the X-ray peak positions and profiles during mechanical testing. Thanks to the selectivity of diffraction, the transverse elastic strain has been measured for the three phases present: the large and fine Cu channels and the Nb nanotubes. The evolution of elastic strains vs. applied stress indicated phase-specific elasto-plastic regimes, the exact natures of which were discriminated by the tangent modulus analysis based on the calculation of the parameters xi that derive from the macroscopic work hardening ha and the variations of the ith phase elastic strains with respect to applied stress: (1) The presence of long-range internal stresses is demonstrated in the cold worked ultrafine-grained large Cu channels that never deform purely elastically and experience macroplasticity (with dislocation storage) upon loading (in tension) and unloading (in compression). (2) The nanocrystalline fine Cu channels exhibit an extended elasto-microplastic regime upon loading and unloading without experiencing macroplasticity. (3) The Nb nanotubes respond only elastically to the external load: they mainly store elastic energy in the form of residual axial tensile stress, a stress that is at the origin of the axial compression imposed on the Cu matrix (both large and fine Cu channels) in order to minimize the stored elastic energy in the Nb. From a more fundamental point of view, the combination of the tangent modulus analysis and the X-ray diffraction data (peak profiles) led to the definition of a new criterion for determining the macroyield stress as the stress at which the macroscopic work hardening becomes smaller than one third of the macroscopic elastic modulus. In particular, this criterion may be more reliable in the case of materials exhibiting strain hardening in the microplastic regime, such as nanocrystalline materials, where the conventional criterion eM = 0.2% was recently shown to be irrelevant.

L. Thilly et al. / Acta Materialia 57 (2009) 3157–3169

References [1] Arzt E. Acta Mater 1998;46:5611. [2] Gleiter H. Acta Mater 2000;48:1. [3] Kumar KS, Van Swygenhoven H, Suresh S. Acta Mater 2003;51:5743. [4] Thilly L, Lecouturier F, Von Stebut J. Acta Mater 2002;50:5049. [5] Saada G. Mater Sci Eng A 2005;400–401:146. [6] Saada G. Philos Mag 2005;85:3003. [7] Louchet F, Weiss J, Richeton T. Phys Rev Lett 2006;97:075504. [8] Brandstetter S, Van Swygenhoven H, Van Petegem S, Schmitt B, Maaß R, Derlet PM. Adv Mater 2006;18:1545. [9] Li H, Choo H, Ren Y, Saleh TA, Lienert U, Liaw PK, et al. Phys Rev Lett 2008;101:015502. [10] Saada G, Verdier M, Dirras GF. Philos Mag 2007;87:4875. [11] Ashby M. Philos Mag 1970;21:399. [12] Asaro R. Acta Metall 1975;23:271. [13] Eshelby J. Proc Roy Soc London 1957;A241:376. [14] Bauschinger J. Civilingenieur 1881;27:289. [15] Xiang Y, Vlassak JJ. Acta Mater 2006;54:5449. [16] Sinclair C, Saada G, Embury J. Philos Mag 2006;86:4081. [17] Thilly L, Van Petegem S, Renault PO, Vidal V, Lecouturier F, Schmitt B, et al. Appl Phys Lett 2007;90:241907. [18] Spencer K, Lecouturier F, Thilly L, Embury JD. Adv Eng Mater 2004;6(5):290. [19] Thilly L, Lecouturier F, Coffe G, Peyrade JP, Aske´nazy S. Physica B 2001;294–295:648. [20] Lecouturier F, Spencer K, Thilly L, Embury JD. Physica B 2004;346347:582. [21] Vidal V, Thilly L, Lecouturier F, Renault PO. Scripta Mater 2007;57(3):245. [22] Vidal V, Thilly L, Van Petegem S, Stuhr U, Lecouturier F, Renault PO, et al. Scripta Mater 2009;60:171.

3169

[23] Van Swygenhoven H, Schmitt B, Derlet PM, Van Petegem S, Cervellino A, Budrovic Z, et al. Rev Sci Instrum 2006;77:013902. [24] Thilly L, Vidal V, Van Petegem S, Stuhr U, Lecouturier F, Renault PO, et al. Appl Phys Lett 2006;88:191906. [25] Vidal V, Thilly L, Van Petegem S, Stuhr U, Lecouturier F, Renault PO, et al. Mater Res Soc Symp Proc 2007;977:191906 [0977-FF07-07EE07-07]. [26] Agarwal BD, Broutman LJ, editors. Analysis and performances of fiber composites, vol. 21. New York: John Wiley & Sons; 1980. [27] Milstein F, Zhao J, Chantasiriwan S, Maroudas D. Appl Phys Lett 2005;87:251919. [28] Milstein F, Marshall J, Fang HE. Phys Rev Lett 1995;74:2977. [29] Thilly L. Doctoral Thesis, INSA-France; 2000. [30] Gil Sevillano J. J Phys 1991;1:967. [31] Brechet Y, Jarry P. J Phys III France 1991;1:1985. [32] Straub S, Blum W, Maier HJ, Ungar T, Borbe´ly A, Renner H. Acta Mater 1996;44:4337. [33] Kassner ME. Mater Sci Eng A 1997;234–236:110. [34] Kassner ME, Perez-Prado M-T, Vecchio KS, Wall MA. Acta Mater 2000;48:4247. [35] Borbe´ly A, Blum W, Ungar T. Mater Sci Eng A 2000;276:186. [36] Kassner ME, Wall MA, Delos-Reyes MA. Mater Sci Eng A 2001;317:28. [37] Mompiou F, Thilly L; 2009, submitted for publication. [38] Rollet AD, Kocks UF. Solid State Phenom 1994;35–36:1. [39] Thilly L, Ludwig O, Ve´ron M, Lecouturier F, Peyrade JP, Aske´nazy S. Philos Mag A 2002;82(5):925. [40] Embury JD, Hirth JP. Acta Mater 1994;42:2051. [41] Misra A, Verdier M, Lu YC, Kung H, Mitchell TE, Nastasi M, et al. Scripta Mater 1998;39:555. [42] Misra A, Hirth JP, Hoagland RG. Acta Mater 2005;53:4817.