J Hidalgo
email: jhidalgo@ing.uc3m.es

A Jiménez-Morales

T Barriere
email: thierry.barriere@univ-fcomte.fr

J C Gelin

J M Torralba

Mechanical and functional properties of an invar alloy for μ-MIM

Keywords: Invar 36, cellulose acetate butyrate, polyethylene glycol, micro-MIM, sintering atmospheres

Micro metal injection moulding (μMIM) is a promising alternative for fabricating micro parts. Low coefficient of thermal expansion (CTE) Fex-Ni1-x alloys are suitable to meet high dimensional stability with temperature keeping acceptable mechanical properties. However these alloys are sensible of elements contamination during the debinding and sintering stages and their dimensional stability and mechanical properties could be affected.

The present work studies the effect of using combinations of debinding and sintering atmospheres in the μPIM of Invar 36 feedstocks based on cellulose acetate butyrate (CAB) binders. Micro-tensile specimens were successfully injected. Densification and microstructure at different conditions were linked with mechanical and physical properties of the parts. The processing conditions and thus the residual C, O and especially H influence severely these properties.

Introduction

Powder metallurgy (PM) of Fex-Ni1-x of low coefficient of thermal expansion (LCTE) alloys presents several processing advantages with respect casting. Energy and cost savings and a reduction of elements segregation risk could be achieved, thus increasing alloy properties 1 . Furthermore, PM can reduce tedious and time consuming machining steps that are specially complicated for Fex-Ni1-x LCTE alloys due to their high ductility and low thermal conductivity. It is also greatly complicated to maintain Invar properties when joining by fusion welding. Such welding results in solidification cracking, porosity and reheating cracking when a matching CTE filler metal is used 2 . Metal injection moulding (MIM) design versatility may also eliminate the needs of joints by welding. To that respect MIM emerges as a cost-effective solution for near net-shape fabrication of complex geometry small pieces 3 . Examples of potential MIM application of LCTE alloys are wide in the field of high precision devices like in optoelectronic or light wave communication systems, housing or structural components 4 . Currently, there is an increasing demand of miniaturised components for micro devices like micro gears or micro actuators [5][START_REF] Piotter | Microsystem Technologies-Micro-and Nanosystems-Information Storage and Processing Systems[END_REF][START_REF] Kong | [END_REF][8] in which micro metal injection moulding (μ-MIM) of low CTE alloys could be an interesting alternative.

The abnormal LCTE behaviour within metals, which normally considerably expand with heat and contract with cold, is consequence of the so-called Invar (volume INVARiance) effect. The physical mechanism explaining Invar effect still remains uncertain, but from general standpoint low expansion occurs due to two opposing effects over a specific temperature range. During heating, the alloy naturally expands while simultaneously a ferromagnetic-to-paramagnetic transformation imposes a magnetostriction that counteracts the expansion 9,10 .

The minimum CTE in the binary iron-nickel alloying system occurs in the composition Fe0.65-Ni0.35, the commercial Invar 36, which also has one of the lowest CTE of any alloy from room temperature to about 230ºC (the 4.5% cobalt modification has almost zero expansion over that range). Microstructure of Fex-Ni1-x LCTE alloys commonly consists of a substitutional solid solution of nickel in iron creating an austenitic face centred cubic lattice. The austenitic γ phase, that is present at all temperatures, shows a high ferromagnetism at temperatures below the Currie temperature, and a paramagnetic behaviour above this temperature.

Heterogeneities in the cast alloy and the presence of other elements like carbon introduce the possibility of formation of Ni segregation regions within the γ lattice [11][12][13][START_REF] Wittenauer | Factors affecting the mechanical strength of Fe-36Ni Invar[END_REF] . Therefore low moment (LM) paramagnetic Fe-rich phases could coexist with clusters of Ni-rich high moment (HM) ferromagnetic order phase negatively affecting CTE. Furthermore a martensitic phase could be formed for quenched alloys which also degrade low expansion properties. However, elements like C and Ni act as a γ phase stabilizers diminishing the martensitic temperature (Ms). Concentrations below about 33 at.% Ni lead on structural transitions from γ to a body centred cubic phase α which does not show the Invar effect. Carbon presence could stabilize γ phase for concentrations below 33 at.% Ni. Dimensional stability has to be accompanied with good mechanical properties. Table 1 shows some typical mechanical properties of Invar 36 alloys processed under different conditions. Invar 36 ordinary wrought alloy [START_REF] Asm | Metals Hadbook:Properties and Selection -Nonferrous Alloys and Pure Metals[END_REF] and high purity powder-metallurgy-route processed alloy [START_REF] Sokolowski | New HP (high purity) Invar 36[END_REF] have relatively low strength, but high ductility. Age hardening by some by addition of elements (Ti, Al, Sn, etc) is an alternative to enhance Invar 36 properties.

Nonetheless the CTE, which is the property of primary importance in these alloys, increases with both aging and concentration of additions [START_REF] Gulyaev | [END_REF] . Small amounts of carbon could be beneficial to improve mechanical properties while an acceptable low CTE is maintained [START_REF] Wittenauer | Factors affecting the mechanical strength of Fe-36Ni Invar[END_REF]20 . Carbon presumably affects mechanical properties by strengthening the interstitial solid solution. Moreover, creating ultra-fine grain structures by e.g. severe plastic deformation using equal-channel angular pressing (ECAP) has demonstrate to significantly increase mechanical properties without scarifying low CTE [START_REF] Vinogradov | [END_REF] .

The present work studies the effect of debinding and sintering atmospheres in the mechanical and physical properties of Invar 36 micro parts. A binder composed of cellulose acetate butyrate (CAB) and water soluble polyethylene glycol (PEG) was used in the μMIM process. CAB thermoplastic polymers have been successfully

employed for PIM proposes in several references 21,22 . Nevertheless CAB does not degrade completely during thermal elimination step if a protective atmosphere is employed during debinding 23 . The residual carbon could be introduced in alloying system in further steps altering alloy properties. Air atmosphere is able to completely degrade CAB due to organic compounds oxidative reactions with oxygen but also oxidises metallic powders at typical debinding temperatures. Residual carbon introduced by CAB during debinding might be eliminated in presence of hydrogen, oxides or vacuum during sintering. However there is the concern of possible carbon contamination that would affect the part properties as well as interstitial hydrogen if this atmosphere is used.

Taking into account all these facts, different debinding and sintering atmospheres have been used to study their effect in the sinterability, microstructure and elements contamination of Invar 36 micro parts. These results were linked with mechanical and physical properties. Micro tensile and hardness test were conducted and the coefficient of thermal expansion and Curie's temperature evaluated depending on the processing conditions.

Experimental processes

Gas atomized spherical INVAR 36 alloy powders were used in this study. The INVAR 36 alloy consists of 36 wt% of Ni and Fe balanced except some other elements traces. The composition of the INVAR 36 alloy is shown in Table 2 along with the particle size parameters. The powders were supplied by Sandvik Osprey Ltd. (UK).

The powder has a particle size distribution with a D90 of 5.9 μm. The small size of the powders will allow a good reproducibility of the shape with smooth surfaces and a minimum part section of about 60 μm. A feedstock with a powder solid loading fraction of 65 vol.% was prepared in a Haake Rheomex twin sigma rotor internal mixer. The feedstock was mixed at a temperature of 150ºC and a rotor speed of 50 rpm for one hour to ensure the homogenisation of all components. The feedstock was employed to perform a full μPIM process. Injection of tensile and bending test micro parts was carried out by a Battenfeld Microsystem 50 microinjection moulding machine. Green micro parts were subjected to a two steps debinding process. First a solvent debinding in water media at 30 ºC was carried out previous to a thermal debinding. Then, two types of atmospheres were used during thermal debinding, air or vacuum, trying to achieve different levels of residual carbon and oxides. During thermal debinding, specimens were heated up to 400 ºC with a heating rate of 2 ºC/min and this temperature was held during 30 minutes.

Brown parts were sintered at 1100 ºC and with a heating rate of 5 ºC/min in vacuum or hydrogen atmospheres.

Sintering temperatures were held during 60 minutes. Therefore four types of sintered parts were obtained: debinding in vacuumsintering in vacuum (V-V), debinding in vacuumsintering in hydrogen (V-H), debinding in airsintering in vacuum (A-V) and debinding in airsintering in hydrogen (A-H). Examples of different green, brown and sintered parts are shown in Figure 1. The narrowest section of the parts corresponds to the tensile test sample with 0.7mm x 0.7 mm. However the runners of both parts have a diameter of 0.3 mm. Sintered parts were characterised in terms of microstructure investigation with a Philips XL-30 scanning electron microscope (SEM) and a Philips X'Pert X-ray diffractometer. Sinterablity was assessed by three methodologies: relative density measurement by Archimedes immersion method, determination of porosity by image analysis and shrinkage by comparing green and sintered part dimensions. For the image analysis, light optical microscope (LOM) images of ten different regions within the polished part cross section were evaluated and an averaged value with the standard deviation presented. The specimens preparation, the brightness and contrast parameters used to discriminates features, the number of fields examined, the total surface examined and the magnification of the objective lenses were optimised to obtain the most reliable and comparable results possible. In this case the magnification was settled at x2000.

Figure 1

Green, Brown and sintered parts after different processing conditions: A: air, V:vacuum, H:hydrogen.

Residual carbon and oxygen content was measured by ThermoFischer Scientific Flash 2000 CHNS/O elementary analyser. Curie temperature that determines the end of the almost constant low thermal expansion temperature range in invar alloys was evaluated with a TA Instruments Q50 thermogravimeter coupled with a magnet placed always at the same distance to the sample. The ferromagnetic to paramagnetic transition is determined with a suddenly drop in the registered sample weight. The linear coefficient of thermal expansion was evaluated from room to 350 ºC at 5 ºC/min in a TA Instruments TMA. It uses a simple rod, hence the data presented are not absolute CTE but they still provide valuable information when the CTE's of the processes were compared. A Kammrath and Weiss micro-mechanical testing stage with a load cell of 1 kN was used to perform micro tensile tests at room temperature. This was coupled with a Zeiss EVOMA 15 SEM to perform in situ micro mechanical tests. A fractography analysis was carried out after the tests. Finally, the specimens' microhardness was obtained by a Time Technology Europe digital microhardness tester HVS-1000.

Results

Densification and elements analysis after sintering

After sintering in different atmospheres and at different heating rates the sinterability of the resulting parts was evaluated by different methodologies. Figure 2 shows the highly illuminated and contrasted LOM images of the cross section of the parts. In Figure 3, the relative density by Archimedes immersion method, the densification assessed by the ratio between the pore area to total area and the shrinkage values are presented. There were differences between the relative densification measured by Archimedes method and image analysis. This could be attributable to the presence of macro porosity in some areas of the parts or to the parameters chosen by image analysis. Nevertheless, the tendency was the same in both cases. The residual carbon and oxygen content were also determined and the results were collected in Table 4.

Table 4 Carbon and oxygen contents for different processing conditions

Powder Debinding

Sintering Carbon may also participate in the oxide reduction essentially in two stages 25,26 . The first involves the reduction of surface oxides which are in contact with residual binder organic compounds during the heating stage. This reaction is the so called direct carbothermal reduction. The produced carbon monoxide can further react with the existing oxides towards the formation of carbon dioxide. This reaction in turn is called indirect carbothermal reduction. Direct and indirect carbothermal reductive reactions become more important above 720 ºC since then the equilibrium in the Boudouard reaction is shifted towards the formation of carbon monoxide which is the more stable product from this temperature and above. Regarding hydrogen as a reductive atmosphere, the reductive effect will depend on the water vapour partial pressure, the heating rates, the velocity and concentration of hydrogen. Jozwiak et al. 27 evaluates these parameters influence on the reduction of different iron oxides by hydrogen and concluded that the complete reduction of these oxides into metallic iron phase can be accomplished at a relatively low temperature, below 380 ºC. However this temperature shifted to higher temperatures as the heating rates are increases (around 750 ºC at 10 ºC/min). More over the effectiveness of the reduction was is strongly influenced by their size and crystallinity and presence of other metals.

A V A-V A-H V-V V-H C [
The A-V turned out to be the most unfavourable condition in terms of densification and shrinkage due to the presence of oxide layers surrounding powder particles formed most likely during debinding. This was coherent with the high oxygen content measured in this condition in both brown and sintered parts compared to the original oxygen content of the powder. Although the vacuum condition may cause a reductive effect, the oxides layers were probably stable at sintering conditions and hindered the diffusion of the metallic atoms. However it is worth to note that there was a considerable reduction of the oxygen content comparing the brown and sintered parts. This would be explained by a direct or indirect carbo-reductive effect of the remaining carbon after debinding aided by the low pressures. In the V-V condition parts, some original powder particles were still observed as it occurred in A-V parts. The number of these original powder particles was considerably lower than in the A-V conditions as well as the proportion and size of pores observed. The oxygen measured in the V-V parts was lower than the A-V part but comparable to the corresponding brown part measured after debinding. These brown parts has slightly higher oxygen content than the original particles, thus it can be concluded that vacuum debinding atmosphere was effective to preserve particles from oxidation.

The parts sintered in hydrogen, independently of the debinding atmosphere, demonstrated to have a higher volumetric shrinkage factor and thus a better densification. They showed a homogeneous porosity distribution and the absence of original powder particles. These results were consistent with the low oxygen contents of these parts which demonstrated an efficiently reduction of the oxides by the hydrogen.

The measured carbon content was comparable in all conditions and was higher than the original powder particles. This denoted that there is a carbon contamination that cannot be avoided by any of the selected processing conditions. The carbon contents were still low but this contamination might affect the Invar properties. Ni rich phase is not discarded but was not detected by SEM inspections. Sometimes γ' precipitates are in the order of nm which is out of the SEM range. In the same way graphite precipitates were not detected. The grain size of the austenite varied depending on the processing condition with apparently the bigger grains detected for the V-H condition (Figure 4 c). The big austenite grains of V-H made thinking on a prompter initiation of the elements diffusion mechanisms compared with other conditions. This would be explained by a rapid reduction of the particle surface oxide at earlier stages of the sintering. Comparing V-V and A-H conditions, the last seemed to have lower grain sizes. Finally, A-V condition had the smallest grain sizes in the order of powder particle size.

Microstructure analysis and phases determination by XRD

The pore sizes seemed to be higher than those observed in LOM by image analysis, but the proportion of porosity was similar and more coherent with the Archimedes measurements. However, in these cases the samples were etched and the porosity may be broadened. It is worth to note the roundness and the small sizes of the pores which were related to the ending stages of the sintering. The effectiveness of the sintering at 1100 ºC may be explained by the small powder particle size that is well established to promote sintering; as powders become finer, sintering activity increases and full density can be achieved at lower temperatures. Imbaby et al. [START_REF] Imbaby | World Congress on Engineering[END_REF] reported that 99% density is obtained from 90%-4µm 316L powder at 1200 ºC. As it was already observed by LOM, in case of sintering in vacuum conditions, some original powder particles that apparently had not take part of the sintering mass transfer mechanism were observed. However, some differences could be appreciated when comparing V-V or A-V. The main difference was the thickness of the oxide layer around particles. A thick dense oxide layer without fissures surrounded powder particles in case of A-V parts (Figure 5a) whereas a thin oxide layer with some discontinuities embedded powder particles in case of V-V parts (Figure 5b); in these discontinuities sintering necks had began to be formed. The residual carbon in V-V combined with a vacuum atmosphere had a reductive effect but it was not enough to reduce all the oxides. In the A-V and V-V SEM images the number of original particles was similar. This seems to contradict Figure 2 results were more original particles were observed in A-V condition. This may be explained by a combined effect of an etching media with ultrasound bath surface cleaning that made some particles to be removed from the surface of A-V condition. The fracture surfaces in Figure 11 again showed a higher density of original particles in the case of A-V condition. The absence of oxides in hydrogen sintered parts evidenced the efficacy of the reductive effect of a hydrogen atmosphere. However, some original particles were sporadically observed in A-H conditions (see Figure 5c).

They were in a very advanced sintering stage with the entire surrounding oxide layer reduced and with lot of sintering necks. This was not observed in V-H parts. The oxide layer generated during debinding in air most likely was thicker than debinding in vacuum as it could deduced from A-V micrographs. Hence the A-H parts would need more time to end with the oxide layer reduction than V-H parts, and thus some particles are still at the end stages of the sintering. This also would explain the differences in grain sizes.

During SEM inspections SiO2 and other complex oxides inclusions were found. Silica particles are often observed sintered materials because a small amount of silicon, <0.8%, is usually added into the melt to facilitate atomization. Silica particles are difficult to reduce, even though using hydrogen atmospheres, especially at 1100 ºC.

When XRD analyses were conducted to determine the phases formed during the different consolidating processes, similarities were observed in the resulting diffractograms. In all cases, as could be observed in Figure 6, austenitic γ(FCC) phase characteristic of Fe0.64-Ni0.36 was clearly identified and almost perfectly indexed by comparing peaks positions with XDR patterns. Nevertheless there were divergences in the relative peaks intensities patterns and occurrence of double peaks. The double peaks at high angles was associated to Cu-Kα2 radiation as the second peak is displaced to higher angles and had almost the half of the intensity of the most intensive first peak. However at lower angles these double peaks of similar intensities could be associated with heterogeneities in the composition due to Ni segregations and lattice distortions caused by interstitial elements.

For the original powders and for the A-V condition, small intensity peaks associated with a α BCC phase were detected. This fact was predictable since the non reduced oxide layer likely would have impeded the elements homogenisation by diffusion mechanisms, in case of having heterogeneous particle compositions and α particles.

Coefficient of thermal expansion and Curie temperatures

Figure 7 shows a TGA weight loss (dashed line) associated with the Invar 36 ferromagnetic to paramagnetic transition, i.e. the Curie temperature. During this transition, a magnet positioned close to the sample stopped attracting it when the paramagnetic state occurred. The derivative of the curve represents the characteristic Curie temperature peak defined by its maximum, the offset and the width. Table 5 collects the results of the different processing conditions micro parts and of the original powders. Comparing the values, there was a reduction of the Curie temperatures in the sintered parts compared with the original powders. This could be associated with the presence of interstitial elements (C or H) or Ni segregations. This variation was more severe in the sintered in hydrogen parts which remarked the negative effect of the hydrogen in this property. The wide peak registered for the powder particles may be associated with heterogeneities in the composition of different powders and/or the particle size distribution. The Curie's temperature has demonstrated to be influenced by composition and by the austenite grain size. This would also explain the double peak detected in the A-V condition which apparently showed the lowest grain size and also original powder particles.

Figure 7 TGA weight loss (dashed line) associated with the Invar 36 ferromagnetic to paramagnetic transition. The derivative of the curve represents the characteristic Curie's temperature peak defined by its maximum, the offset and the width. Figure 8 plots the linear coefficient of thermal expansion (CTE) at different temperatures of different debinding and sintering conditions. The small dimensions of the samples caused noise in the measurements of the expansion with temperature curves, thus smoothing had to be carried out to determine the linear CTE by the following expression:

𝐶𝑇𝐸 𝑙𝑖𝑛𝑒𝑎𝑟 = 1 𝐿 0 • ∆𝐿 ∆𝑇 (1)
where L0 is the initial length of the sample and ΔL represents the variation of length at a certain increment of temperature ΔT. Smoothing and the use of a simple rod dilatometer may produce some errors in the assessment of the linear CTE. Therefore the data presented are not absolute CTE but they still provide valuable information when the CTE's of the four processes were compared. The values obtained and the behaviour with temperature and processing conditions seemed to be consistent with the Curies' temperature results and with the intrinsic behaviour of an Invar 36 alloy. At temperatures below the Tc, the CTE was low (in the order of 1. 3 10 -6 •ºC -1 to 3 10 -6 •ºC -1 ) and relatively steady. Near the Tc this tendency changed and the CTE values began to raise with temperature up to values were the curve seemed to stabilise again at temperatures above the Tc. The CTE at high temperatures was in the order of those for stainless steels. As it occurred with the Tc there was a clear effect of the sintering in hydrogen conditions in the CTE values. These conditions showed the worst values of CTE at all the temperature range.

A-V condition resulted to give the best CTE results. The reason explaining this fact was unclear, especially considering that some alpha phase was developed in these parts (as reflected in XRD analysis). A combined effect of Invar 36 low CTE until around 200 ºC and the low CTE of non-reduced oxides at higher temperature may be a plausible explanation. Moreover, the small grain size may be also influencing the results.

In situ tensile and hardness micro mechanical test

Figure 9 shows the representative strain-stress curves of different processing conditions. Table 6 collects some characteristic mechanical parameters according to these curves. Zhao et al. [START_REF] Zhao | [END_REF] discussed the limitations and issues in correctly interpreting micro tensile experiments. The micro test experimental studies assume errors in the strain measurements as the data were collected directly from the cross head displacement without any extensometers, digital image correlation or laser interferometry. Thus, only engineering results are presented in this work. However, as the tests were performed on the same device, the results are considered valid for comparative purposes.

The ASTM Designation A658 covers 36 percent nickel-iron alloy and specifies a tensile strength of 448-552

MPa, a minimum yield strength (0.2% offset) of 241 MPa and a minimum elongation of 30%. Although these values are representative of the wrought Invar 36, the results derived from the micro-tests of powder consolidated specimens were consistent with these specifications and the values collected in Table 1. However there was a great diminution of the strain and thus of the ductility in the sintered in hydrogen parts. This coincided with the highest yield strength values registered. Considering that the porosity values in V-V, V-H and A-H conditions were similar as well as the carbon content, it could be stated that hydrogen was the responsible of this embrittlement effect. However, the ultimate tensile stress (UTS) values were similar for those conditions. The lowest UTS was registered by the A-V condition and it is probably associated with the high porosity of these parts. The presence of oxides also limits the elongation values in comparison with the V-V condition. These results were consistent with the microhardness results. The highest HV corresponded to the sintered in hydrogen part evidencing the low ductility results. The lowest value corresponded to the A-V condition, most likely due to the high porosity. The micro hardness values were also in the range of the Invar 36 materials (see Table 1).

A-H A-V V-H V-V UTS [MPa] 466±15 436±22 458±15 462±19 Strain to fracture [%] 17±1 22±2 18±1 28±2 Yield strength (0.2% offset) [MPa] 290±12 252±21 267±15 212±19 HV/9.8N [GPa]
1.70±0.021 1.2±0.023 1.69±0.020 1.63±0.021

A representative behaviour of the microstructure evolution with the applied stress in Invar 36 micro parts is shown in Figure 10, corresponding to V-V condition. No significant differences were observed for the rest of the conditions. At stresses near the yield point, a small deformation of the austenitic grain was observed but no other differences were observed in the microstructure comparing the images with the uncharged condition. Above the yield point the austenitic grain deformation became appreciable as well as the formation of slip bands. The deformation and the density of the slip bands were considerable near the UTS. The round pores were elongated in the strain direction and some of them begin to coalesce. At fracture, slip bands moving from different sliding directions were observed crossing over. The fracture surfaces of all the tensile tested specimens were examined by SEM. Resultant fractography images are shown in Figure 11. In principal, the sintered in vacuum samples exhibited a typical ductile fracture by microvoids coalescence (MVC) based on the presence of dimples initiated at inclusions. These inclusions were sometimes original powder particles that were not sintered due to the presence of an oxide coat that was not reduced with the sintering conditions. This was more evident in the A-V condition (Figure 11b) with dimple sizes around 5μm. Spherical particles were clearly observed in the middle of the rounded dimple with the same size of powder particles. Deformation slip bands were observed in many dimples. Figure 12 shows a magnified image of A-V fracture surface with EDX semi-quantitative elemental analysis at different regions. The element estimation may fall in errors since the surface is rough. However, spherical Invar 36 particles were identified in the middle of the dimple (Figure 12(1)) with a broken oxide shell surrounding the particles (Figure 12(2)).

Some silica inclusions were also identified in some dimples (Figure 12(3)).

Conclusions

The effect of debinding and sintering atmospheres of INVAR 36 micro parts was investigated. The following conclusions were drawn:

 Air degraded more effectively CAB during sintering but is responsible of metallic particles oxidation.

However these oxides could be reduced employing reductive hydrogen or vacuum atmospheres also aided by carboreductive effect of remaining carbon after debinding.

 The debinding in vacuum atmosphere leaved a high percentage of contaminating carbon source.

Residual carbon could be eliminated by pyrolysis with vacuum media or by reductive hydrogen during sintering.

 All combined processing condition leaded to the formation of desirable austenitic microstructure responsible of the low CTE of INVAR 36 and produces similar levels of carbon contamination. This was contrasted by microstructure investigations by SEM, by XRD and element analysis.

 To some extent, resulting mechanical properties meet the standards of an Invar 36 alloy. Presence of none reduced oxides influenced the densification of the micro parts and leaded on a detriment of the mechanical properties.

 The hydrogen atmosphere have demonstrated to be effective for the reduction of oxides and the part densification, but this element has been easily introduced as an interstitial element worsening the ductility, the Curie's temperature and the coefficient of thermal expansion.

 The debinding and sintering in vacuum produced the most equilibrated physical and mechanical properties.

Figure 2 .

 2 Figure 2. Highly contrasted LOM images of different grinded and polished Invar 36 micro parts at the same magnification for the evaluation of the densification and the remaining porosity. The specimens were not etched in this case.

Figure 3 .

 3 Figure 3.Sinterability study of the micro parts by different methodologies. The percentage of porosity was assessed by image analysis software. The relative density was measured comparing Archimedes method density with respect to the Invar 36 powders density. The shrinkage of the part was evaluated by direct measurement of the sintered part dimensions and comparing them with green part dimensions.

Figure 4

 4 Figure 4 shows back scattering electron detector SEM images of INVAR 36 micro-parts cross sections representing different debinding and sintering conditions. With the exception of A-V parts (Figure 4 b) the rest of conditions developed a clear austenitic microstructure with typical austenite twin bands. The presence of a γ'

Figure 4

 4 Figure 4 Secondary scattered electron detector SEM images of cross sections of different processing conditions INVAR 36 micro-parts: a) Vacuum-vacuum, b) air-vacuum, c) vacuum-hydrogen, d) air-hydrogen. The parts were etched with aqua regia a solution of concentrate nitric acid and concentrate hydrochloric acid in a volume ratio of 1:3.

Figure 5

 5 Figure 5 Secondary electrons detector SEM micrographs of cross sections of INVAR 36 sintered micro-part: a) A-V, b) V-V c) A-H. (i. oxide layer, ii. sintering neck, iii. micro void formed due to elements diffusion )

Figure 6

 6 Figure 6 XRD diffractograms and phases identification of sintered parts subjected to different debinding and sintering conditions: a) vacuum-vacuum, b) air-hydrogen, c) vacuum-hydrogen, d) air-vacuum.

Figure 8

 8 Figure 8 Linear coefficient of thermal expansion at different temperatures of different debinding and sintering conditions.

Figure 9

 9 Figure 9 Representative stress-strain curves of different processing conditions.

Figure 10

 10 Figure 10From left to right: evolution of the V-V parts microstructure with strain at approximately the yield point, the ultimate tensile stress and the rupture strength.

Figure 11

 11 Figure 11 Fracture surfaces of different tensile specimens after micro tensile test: a) V-V, b) A-V, c) V-H and d) A-H.By contrast the sintered in hydrogen specimen exhibited two modes of fracture: a predominant MVC and a second less ductile mode with some transgranular facets sometimes roughened by slip steps. The dimple sizes in those cases were smaller than sintered in vacuum specimens (around 1μm) most likely they were initiated by sub-micrometer inclusions. Moreover, there were less clear slip bands in the dimples or directly they were not observed. The presence of transgranular facets may be linked with the embrittlement observed in the stress-strain curves of these conditions. Winding regions of apparently dimples clusters were observed in all specimens independently of the sintering atmosphere.

Figure 12

 12 Figure 12 Magnified image of A-V fracture surface accompanied of EDX analysis at different points. (1) Original Invar 36 particle, (2) Possible complex FeNiSiMn oxide shell sourronding original Invar 36 particle, (3) Silica inclusions with a small amount of Mn according to EDX.

Table 1

 1 Mechanical properties of different Invar 36 materials.

	Material	YS [MPa] UTS [Mpa]	Strain to breakage [%]	Hv [Gpa]
	Wrought Invar 36 (ASTM A658)	> 241	448-552	> 30	N/A
	Wrought Invar 36 15	275	490	40	1.30
	High purity PM Invar 36 extruded 16	271	468	70	N/A
	Invar 36 (0.06 C)	310	N/A	N/A	1.5-1.6
	Invar 36 ECAP 2 passes 17	570	732	47	2.42
	Invar 36 ECAP 12 passes 17	835	912	52	2.60

Table 2

 2 Composition and particle size distribution parameters for Invar 36 powders

	Composition	Ni	Mn	Si	C	O	Fe	Particle Size Distribution	d10 d50 d90
	[mass%]	36.50 0.20 0.02 0.01 0.13 Balanced	[μm]	1.9 3.4 5.9
	Each powder was mixed with a binder system based on a combination of several thermoplastic polymers,
	employing poly (ethylene glycol) (PEG) (supplied by Sigma-Aldrich) of different molecular weights and two

Table 3

 3 Binder composition

	Component	Acetyl* Butyryl* Hydroxyl* %vol
	CAB381-0.1	13	37	1,5	30
	CAB551-0.01	2	53	1,5	10
	PEG 20k	-	-	-	58
	PEG 4k	-	-	-	2
	* Percentage of side groups in chain

  Labrador24 studied the oxidation kinetics of a wrought Invar 36 alloy in air. The oxidation in terms of mass gains begins to be appreciable above 580 ºC, with the exclusive formation of porous iron oxides specially Fe2O3 but also Fe3O4 and FeO. Metallic oxide layers in the powder surface inhibit the atom diffusions and thus the sintering mechanism being this fact detrimental for densification of the sintered parts. Reductive conditions

	mass%] 0.01	0.12 0.73 0.06 0.06 0.06 0.05
	O [mass%] 0.12	0.57 0.15 0.38 0.07 0.13 0.10

(hydrogen, vacuum or CO atmospheres) during sintering could act in the oxide layer enhancing sintering mechanisms. The thermodynamic data calculated using thermodynamic modelling software (HSC Chemistry 7.1) reveal that at 1100 ºC oxygen partial pressures of approximately 10 -13 bar are needed to decompose the different iron oxides. Below this temperature, partial pressures have to be lowered to achieve the same effect.

Table 5

 5 Curie temperatures for different parts processed with different atmospheres

	Tc [ºC] INVAR Powder V-V	V-H	A-V	A-H
	Peak	221.2	199.4 185.7 213.7 236.7 186.7
	Offset	282.2	216.0 196.4	245.4	201.6
	Width	89.1	27.3	20.4	42.4	25.8

Table 6

 6 Micro-mechanical properties of different processing conditions

Acknowledgements

The authors wish to thank GUZMÁN GLOBAL S.L. and MIMTECH ALFA for their collaboration on the ECOPIM project (ref. IPT-2011-0931-20000) that was funded by the Spanish Ministry of the Economy and Competitiveness. Furthermore, the authors would like to acknowledge the funding provided by CAM-Consejeria de Educación Dir. Gral. Universidades e Investigación in the project ESTRUMAT (ref. S2009/MAT-1585) and the Nacional Project (MAT2012/38650-C02-01).


ABSTRACT

In a woman presenting a pervasive denial of pregnancy, it happens that abdominal swelling is significantly reduced if not completely absent. In such cases, the announcement of the pregnancy to the unaware woman by a physician is often sufficient to trigger the spontaneous appearance of a normal silhouette of pregnancy with abdominal prominence. The pregnancy announcement acts as an input signal applied to the regulation mechanisms of the body with the consequence that the output signal; i.e. the silhouette of the pregnant woman; turns spontaneously from an abnormal shape to a normal one. By applying a system-based analysis to these observations, the brain appears clearly to be responsible for this disorder through instructions sent to the body by the nervous system.

The silhouette effect is thus shown to be a cybernetic disorder as opposed to a functional one.

The existence of unexpected information-and-semantic pathways throughout the human body is also revealed by this spontaneous transition.

A reactive-homeostasis model is then proposed in which symptoms result from an abnormal but coherent regulation of the functioning of the body caused by an unconscious mechanism by the brain to escape paradoxical realities. Spontaneous recovery is then a consequence of the paradoxical situation dissolution induced by the pregnancy announcement. This concept of reactive-homeostasis prompts us to investigate human disorders in complementary categories; i.e. cybernetic versus functional. Such an enlarged point of view may improve our knowledge of aetiology and suggest new perspectives for therapeutics research.

DENIAL OF PREGNANCY AND THE SILHOUETTE EFFECT

Introduction to denial of pregnancy

Denial of pregnancy is a woman's subjective lack of awareness of being pregnant [START_REF] Wessel | Denial of pregnancy-characteristics of women at risk[END_REF]. Denial is defined as a primitive defence mechanism found not only in psychotic patients but also in healthy individuals under unusually severe stress. Its aim is to alleviate unpleasant affects by disavowing aspects of reality [START_REF] Vaillant | Theoretical hierarchy of adaptative ego mechanisms[END_REF][START_REF] Brezinka | Denial of pregnancy: obstetrical aspects[END_REF]. Denial of pregnancy can last for months or even until the unexpected 'sudden' delivery of the baby. In the latter case, significant risk to the infant and mother may result, such as poor nutrition, foetal abuse, unattended or precipitous delivery, maternal postpartum emotional disturbance, and neonaticide(4). Wessel et al. report an incidence of 1/475 for unawareness of pregnancy after the 20th week of gestation and 1/2455 for totally unexpected births from a one year study in the Metropolitan Berlin area [START_REF] Wessel | Frequency of denial of pregnancy: results and epidemiological significance of a 1-year prospective study in Berlin[END_REF]. Miller describes denial of pregnancy in three different categories: affective, pervasive and psychotic [START_REF] Miller | Denial of pregnancy, in Infanticide: psychosocial and legal perspectives on mothers who kill[END_REF]. There is no clear dividing line between conscious coping and unconscious defence mechanisms but there is a fluid transition described as "middle knowledge" [START_REF] Brezinka | Denial of pregnancy: obstetrical aspects[END_REF][START_REF] Brezinka | A propos des données épidémiologiques[END_REF]. Spielvogel confirms that persistent denial of pregnancy occurs in women with otherwise intact reality testing(4).

In the case of pervasive denial of pregnancy, women frequently do not experience the physical changes of pregnancy such as morning sickness, cessation of menstruation, excessive weight gain and foetal movements [START_REF] Brezinka | Denial of pregnancy: obstetrical aspects[END_REF][START_REF] Spielvogel | Denial of pregnancy: A review and case reports[END_REF][START_REF] Miller | Denial of pregnancy, in Infanticide: psychosocial and legal perspectives on mothers who kill[END_REF][START_REF] Brezinka | A propos des données épidémiologiques[END_REF][START_REF] Brozowsky | Neonaticide: Clinical and psychodynamic considerations[END_REF][START_REF] Bascom | Women who refuse to believe: persistent denial of pregnancy[END_REF][START_REF] Wessel | Cyclic menstruation-like bleeding during denied pregnancy. Is there a particular hormonal cause?[END_REF][START_REF] Afrdg | déni de grossesse ; 99 histoires vécues[END_REF]. For instance, an 18 year-old woman delivered a 3kg newborn with breach presentation when she came to the hospital complaining of appendicitis. Two weeks earlier, this woman spent a holiday on a naturist beach. In spite of being naked, no-one in her surroundings suspected she was pregnant [START_REF] Brezinka | A propos des données épidémiologiques[END_REF].

Silhouette effect and clinical observations

Throughout this work, we are interested in particular cases of pervasive denial of pregnancy in which the three following conditions are present:

• C1: absence of abdominal swelling at the time of denial,

• C2: pregnancy revealed before delivery during a visit to a physician,

• C3: abdominal swelling taking place spontaneously after the pregnancy announcement.

Nisand explains how abdominal swelling can be absent, even at the end of pregnancy [START_REF] Nisand | Déni de grossesse, ces bébés clandestins, documentary[END_REF].

Instead of taking place horizontally and thus causing abdomen prominence, the fetus expands itself in the vertical direction to the detriment of the diaphragm, thus allowing the silhouette to remain slim.

Abdominal muscles are involved in this disorder to maintain a flat abdominal shape. After pregnancy denial disclosure, Nisand explains that the fetus takes up a normal position; horizontally. Then abdominal swelling appears as the muscles relax. This spontaneous bodily transformation following the denial disclosure is what we call here the silhouette effect of denial of pregnancy. This silhouette effect has been observed by different clinicians [START_REF] Bascom | Women who refuse to believe: persistent denial of pregnancy[END_REF][START_REF] Nisand | Déni de grossesse, ces bébés clandestins, documentary[END_REF][START_REF] Lolo | Déni de grossesse : Quand le psychisme occulte une partie du corps ; Discussion anthropopsychanalytique[END_REF][START_REF] Rozan | L'obstétricien et le déni de grossesse[END_REF] and seems to be a frequent consequence of announcing her pregnancy to a woman in denial of pregnancy. Evidence has been reported by more than ten women who have experienced this [START_REF] Afrdg | déni de grossesse ; 99 histoires vécues[END_REF][START_REF] Luna | T'étais déjà là mon fils, mais[END_REF].

Let us describe two cases as typical examples: Rozan reports on a young woman who consulted him while in denial of a 7 month pregnancy without any silhouette transformation [START_REF] Rozan | L'obstétricien et le déni de grossesse[END_REF]. The practitioner detected her pregnant state and confirmed his diagnosis by means of an ultrasound examination. After the announcement the woman was surprised but happy. Still naked she moved from the examination table to sit in front of the physician. Then she asked for confirmation: Is it actually mine, the baby? Rozan confirmed her gravid state and when the woman stood up again to get dressed he had almost a different woman before him. A few minutes before, the pregnancy had not been noticeable from her silhouette while now, standing in a different posture, leaning backwards slightly, her abdomen was prominent. Rozan reports with surprise this complete physical metamorphosis that had taken place so quickly because of her psyche.

The case of S. Luna is another striking example [START_REF] Luna | T'étais déjà là mon fils, mais[END_REF]. This woman participated in a TV program that she eventually won, in which young singers competed. In the final TV show, she was singing and dancing while already 5 months pregnant. She was still unaware of her pregnancy as were the production team and millions of TV viewers. Her pregnancy was discovered in the 6th month while she still had not gained any weight. Between denial disclosure and delivery, she finally put on 18kg of which 5kg during the week following the announcement. She also felt foetal movements for the first time during the night following her pregnancy announcement.

In scientific work, denial of pregnancy has been first and foremost studied because of the risks for the mother and the newborn's health, especially in the case of unexpected deliveries [START_REF] Friedman | Characteristics of woman who deny or conceal pregnancy[END_REF](17)[START_REF] Spinelli | Denial of Pregnancy: A Psychodynamic Paradigm[END_REF]. From a medical point of view such as this and unlike in unexpected deliveries, the silhouette effect is only a curiosity when it has no actual consequences. This is probably the reason why it is usually not reported in the literature on denial of pregnancy. In practice, it may be more difficult to source accounts of the silhouette effect in scientific papers than in memoirs [START_REF] Afrdg | déni de grossesse ; 99 histoires vécues[END_REF][START_REF] Luna | T'étais déjà là mon fils, mais[END_REF] or in videos available on the internet.

SYSTEM-BASED ANALYSIS OF THE SILHOUETTE EFFECT

Cybernetic versus functional disorders

Let us apply an elementary system-based analysis to the set of conditions C1 through C3. C1 and C3 are the successive output states of the body; respectively abnormal and normal; while C2 is the input signal triggering the spontaneous transition from C1 to C3. From a physiological point of view, let us consider the case of abdominal muscles that are in a steady state of contraction in C1 whereas they are relaxed in C3. The presence of C3 demonstrates unambiguously that abdominal muscles in this case are perfectly functional. Any functional inability of the body to behave normally has thus to be excluded as a cause of C1. It is therefore evident that the abdominal muscle contraction observed in C1 results from instructions received from the nervous system that ask them to remain contracted. These elementary considerations of clinical observations establish clearly that the silhouette effect of denial of pregnancy corresponds to a cybernetic disorder rather than to a functional one.

According to Wiener's book title 60 years ago, cybernetics stands here for communication between elements of a controlled system [START_REF] Wiener | Cybernetics or control and communication in the animal and in the machine[END_REF]. It is remarkable that this cybernetic disorder is cured by a piece of information; i.e. the pregnancy announcement. This spontaneous recovery, reported many times [START_REF] Afrdg | déni de grossesse ; 99 histoires vécues[END_REF][START_REF] Nisand | Déni de grossesse, ces bébés clandestins, documentary[END_REF][START_REF] Lolo | Déni de grossesse : Quand le psychisme occulte une partie du corps ; Discussion anthropopsychanalytique[END_REF][START_REF] Rozan | L'obstétricien et le déni de grossesse[END_REF][START_REF] Luna | T'étais déjà là mon fils, mais[END_REF], is evidence that, at least in this case, the assimilation of a piece of verbal information can induce involuntary bodily transformations and heal the disorder. This distinction between cybernetic and functional disorders is somewhat similar to the distinction between software and hardware problems in the computer domain. Just as software problems can be solved by software solutions, the silhouette effect is a cybernetic disorder that can be cured by a cybernetic solution.

Information pathways throughout the human body

The silhouette effect of denial of pregnancy reveals the existence of active information links between different levels of the human nervous system: i) the psyche responsible for pregnancy denial [START_REF] Spinelli | Denial of Pregnancy: A Psychodynamic Paradigm[END_REF], ii) the autonomous nervous system controlling the abdominal muscles, iii) and the functions of the cerebral cortex in charge of language.

The physiological impact of denial of pregnancy is even deeper since pregnancy symptoms are commonly absent in those women [START_REF] Brezinka | Denial of pregnancy: obstetrical aspects[END_REF][START_REF] Spielvogel | Denial of pregnancy: A review and case reports[END_REF][START_REF] Miller | Denial of pregnancy, in Infanticide: psychosocial and legal perspectives on mothers who kill[END_REF][START_REF] Brozowsky | Neonaticide: Clinical and psychodynamic considerations[END_REF][START_REF] Wessel | Cyclic menstruation-like bleeding during denied pregnancy. Is there a particular hormonal cause?[END_REF][START_REF] Afrdg | déni de grossesse ; 99 histoires vécues[END_REF]. In the Brezinka study(3) for instance, of the 11 women whose denial ended only during labour or delivery, none experienced nausea or related symptoms. This 0% observation is far from the normal rate of between 56% and 85% [START_REF] Klebanoff | Epidemiology of vomiting in early pregnancy[END_REF][START_REF] Davis | Nausea and vomiting of pregnancy: an evidence-based review[END_REF][START_REF] Pepper | Rates of nausea and vomiting in pregnancy and dietary characteristics across populations[END_REF]. While the aetiology of nausea of pregnancy is not fully agreed upon, it is commonly accepted that it results from human adaptation and involves the human chorionic gonadotropin hormone secreted by the placenta [START_REF] Davis | Nausea and vomiting of pregnancy: an evidence-based review[END_REF][START_REF] Sherman | Nausea and vomiting of pregnancy in an evolutionary perspective[END_REF]. Nausea as a pregnancy symptom is thus associated with hormonal processes that are among the most primitive and subtle physiological mechanisms occurring within the human body. The abnormal observations of Brezinka suggest that either these subtle processes or their effects on the human body might be inhibited in the case of denial of pregnancy where the origin is psychic. This result is surprising since no psychic problem would have been anticipated to act at such a basic physiological level. A corollary implication is that it would be very speculative to exclude a priori a psychic cause from any physiological disorder. It is therefore important to consider the cybernetic approach discussed above. This recommendation was already expressed differently thirty years ago by H. Laborit: "All dichotomy between man and his environment, between the psychic and the somatic, seems to hark back to some prehistoric era of biology, to a time when thermodynamics dominated scientific research and a formal notion of information did not yet exist" [START_REF] Laborit | Toute dichotomie de l'homme et de son environnement, du psychique et du somatique, nous paraît ressortir d'une préhistoire de la biologie[END_REF].

REACTIVE-HOMEOSTASIS AS A CYBERNETIC MODEL

There is an obvious relationship in meaning between the pregnancy announcement and the disorder being cured spontaneously. However at this point we do not have any explanation as to why or how this announcement triggers the spontaneous bodily transformation observed. To explore this issue, let us consider the actual significance of the pregnancy announcement: At some level of the autonomous nervous system, pregnancy is necessarily "known" through information issued by the sensory system. Therefore the pregnancy announcement cannot be considered directly as "new information" liable to trigger changes. For the conscious mind, on the contrary, the woman who was unaware in C1 becomes aware of her gravid state through the pregnancy announcement. This is indeed an actual change but the latter concerns the conscious mind rather than the autonomous nervous system controlling the abdominal muscles. Therefore abdominal swelling appearing spontaneously after denial disclosure cannot be a direct consequence of this announcement. Some kind of communication between the conscious mind and the autonomous nervous system is necessarily involved in the bodily transformation observed. We do not have a watertight explanation for this phenomenon. However, let us introduce a reactive-homeostasis model that is consistent with the observations. The aim of denial is to alleviate unpleasant affects by disavowing aspects of reality [START_REF] Vaillant | Theoretical hierarchy of adaptative ego mechanisms[END_REF][START_REF] Brezinka | Denial of pregnancy: obstetrical aspects[END_REF]. In the case of pregnancy denial, we may say that the women concerned are not "allowed" by their psyches to be pregnant even though they physiologically are pregnant. Denial may thus be seen as a contrived way to respect both of these paradoxical realities. The solution to this is a "secret" pregnancy, one that exists below the woman's awareness. In this model the absence of pregnancy symptoms have to be considered as resulting from an unconscious but coherent mechanism aimed at hiding the gravid state rather than from any functional failure. The absence of abdominal swelling is thus compatible with the ability of the body to function normally by responding to a reactive-homeostasis. The latter is induced by the central nervous system in order to comply with these paradoxical realities.

The pregnancy announcement constitutes an infringement of the psychic "ban" on being pregnant. The psychic problem is thus transformed. Instead of respecting the pregnancy prohibition, the mind has then to deal with its transgression. The new situation is not necessarily easier to face but the paradoxical block has been removed. The escape mechanism thereby loses its relevance and can then be released, allowing abdominal swelling to take place. From this perspective, the cybernetic disorder has to be seen as an adaptive mechanism of the nervous system to manage an unsolvable situation. Thus the body, as a way of dealing with its paradoxical realities, enters a state of reactivehomeostasis, inducing the appearance of symptoms.

Let us consider additional observations on this model of reactive-homeostasis:

In the case of pregnancy denials lasting until unexpected deliveries, the abdominal and thoracic cavities are reduced by several litres in available volume while hosting several kilograms of weight.

Despite this level of bodily transformation, much larger than a heavy meal, the women concerned do not feel poorly and can keep up their usual lifestyle. Fatigue and irritability are the only symptoms reported in the final weeks of gestation. For instance, since the fetus is lying vertically, the diaphragm is displaced upwards to a significant degree. With such a constraint imposed on this respiratory muscle, no breathing difficulties have been reported, neither in scientific papers nor in women's life stories. These observations would suggest that these internal bodily transformations are managed by some kind of adaptive process to allow a satisfactory functioning; thus supporting the reactivehomeostasis model.

Since pregnancy has a finite duration, this tactic of hiding it will necessarily fail at some point; i.e. when labour and delivery take place. While this final failure is certain, denial is maintained until the very end in many cases. This contradiction seems incompatible with a coherent mechanism. However this is no longer the case if the reactive-homeostasis state is considered as a reflexive mechanism enabling the immediate escape from an emergency situation. As with all reflexes, the reactive-homeostasis state has to be considered as an automatic and uncontrolled response to a particular emergency state, without any temporal or future considerations. From this perspective, denial persistence results from the emergency state being upheld, which itself results from the persistence of paradoxical realities. The continued denial of pregnancy until an unexpected delivery is thus compatible with the reactive-homeostasis model proposed. We may observe that this configuration is comparable to what is called "standby in tension" as proposed by Laborit in the concept of inhibition of action [START_REF] Laborit | Toute dichotomie de l'homme et de son environnement, du psychique et du somatique, nous paraît ressortir d'une préhistoire de la biologie[END_REF][START_REF] Laborit | The major mechanisms of stress, in Stress revisited 2. Systemic effects of stress[END_REF] (or behavioural inhibition).

DISCUSSION

Since its introduction in the middle of the last century, cybernetics has been applied very successfully within the domain of machines. One direct consequence is that our everyday life has been deeply transformed in western countries. From the very beginning, cybernetics has been considered as beneficially applicable to animals and humans [START_REF] Wiener | Cybernetics or control and communication in the animal and in the machine[END_REF]. However, many fewer advances have been reported in the domain of living organisms. One obvious reason is that the study of interactions within a system assumes an extensive knowledge of its internal states. In the case of living systems, this condition is hard to meet due to the huge number of parameters to consider and to the impossibility of accessing most of them. Furthermore, inter-individual differences, either due to genetics, lifeexperiences or environment, mean that comparisons between living individuals are much less representative that in the case of identical machines. In this context, the silhouette effect of denial of pregnancy is a notable exception. Since the transition between C1 and C3 occurs in a short period of time starting at the doctor's office, unknown parameters remain necessarily unchanged and can thus be discarded from the cybernetic analysis. Thanks to this particularity of the silhouette effect, a clear cause-and-effect relationship can be established between the pregnancy announcement and the spontaneous appearance of abdominal swelling. It is thus clear that the silhouette effect is a cybernetic disorder induced by the brain instead of a functional one. Furthermore, the few words announcing the pregnancy to the formerly unaware woman act as a cybernetic solution; i.e. an effective cure triggering spontaneous recovery. The existence of information-and-semantic pathways throughout the human body is also revealed by this analysis.

A reactive-homeostasis model has been proposed to explain this phenomenon. From this perspective, the symptoms observed result from a reflexive mechanism induced by the unconscious mind in response to a paradoxical situation. This concept is analogous to that of "standby in tension" [START_REF] Laborit | Toute dichotomie de l'homme et de son environnement, du psychique et du somatique, nous paraît ressortir d'une préhistoire de la biologie[END_REF][START_REF] Laborit | The major mechanisms of stress, in Stress revisited 2. Systemic effects of stress[END_REF]. It can also be compared to the "double-bind" concept introduced by Bateson et al. [START_REF] Bateson | Toward a theory of schizophrenia[END_REF] in the study of schizophrenia. Instead of the paradoxical injunctions considered by Bateson at an interindividual communication level, our model involves paradoxical realities at the intra-individual level.

Physical and psychosocial stresses are known to induce somatic disorders for a long time, especially on the digestive tract [START_REF] Selye | The general adaptation syndrome and the diseases of adaptation[END_REF][START_REF] Caso | The effects of physical and psychological stress on the gastrointestinal tract: lessons from animal models[END_REF]. The silhouette effect of denial of pregnancy discussed here may be considered as being in the same family of processes. However its capacity to be solved by a cybernetic solution consisting only of a few words is a new and specific property of which the implications have to be explored and discussed in the future.

In a time lapse of only a few generations, human lifestyles have changed considerably in western countries. While physically demanding activities are fortunately diminishing, professional and psychosocial stresses are increasing. Fatigue of the human condition is thus progressively shifting from physical pains affecting the body directly towards environmental stresses affecting the body indirectly through the central nervous system. Consequently, one may expect public health systems nowadays to address a decreasing proportion of functional disorders but an increasing proportion of cybernetic disorders. Because of this modern evolution of humanity, health and its mental or physiological disorders have to be considered through supplementary approaches. These should be global, exploring the human body as an integrated system controlled by complex information-andcommunication processes subject to weaknesses, failures or paradoxical blocks. The basis of such a cybernetics approach can be found in previous works and concepts set out by such prestigious pioneers as Bernard [START_REF] Bernard | Leçons sur les phénomènes de la vie communs aux animaux et aux végétaux[END_REF], Cannon [START_REF] Cannon | The wisdom of the body[END_REF], Selye [START_REF] Laborit | On the mechanism of activation of the hypothalamo-pituitary-adrenal reaction to changes in the environment (the "alarm reaction[END_REF] or Laborit [START_REF] Laborit | Toute dichotomie de l'homme et de son environnement, du psychique et du somatique, nous paraît ressortir d'une préhistoire de la biologie[END_REF][START_REF] Laborit | On the mechanism of activation of the hypothalamo-pituitary-adrenal reaction to changes in the environment (the "alarm reaction[END_REF]. This way of thinking is however very recent in comparison with the thousand-year-old history of medicine and further investigation of how cybernetics can help to understand human disorders may be beneficial.

Acknowledgements

The author acknowledges Belinda Lafon for bibliographic assistance and Sharon Black for manuscript revision.


F Amiot

An Euler-Bernoulli second-strain gradient beam theory for cantilever sensors

Introduction

Because of their high surface over volume ratio, the mechanical behavior of micrometer sized structures differs from that of usual macroscopic objects. Their surface plays a key role, and this property has been proposed to devise micromechanical sensors of environmental changes [1]. In particular, a significant effort has been put on the development of biological sensors [2], thus highlighting the need for a more basic understanding of coupled surface phenomena [3]. These micromechanical sensors are usually operated in either static or dynamic mode. The latter allows one to detect the change in mass experienced by a micromechanical resonator. One will focus herein on cantilevers operated in static mode : a microcantilever which is typically few hundreds of micrometer long and one micrometer thick is chemically functionalized on one side to react with a specific target molecule. It has been experimentally reported that when the chemical recognition mechanism occurs (on the modified side), it induces some micro-cantilever bending which may be detected through different ways. The available experimental material has been mostly obtained using the optical lever technique [4] which provides the (quasi-static) deflection at the cantilever tip. Such an arrangement allowed many groups to demonstrate such a chemo-mechanical coupling for a large variety of molecular interactions (see [5] for biological applications for instance). The quantitative (and sometimes qualitative) interpretation of the retrieved experimental data is however challenging. Several reasons make the investigations of these chemicallyinduced mechanical effects particularly difficult :

• These results are usually interpreted through Stoney's equation [6], which was developed to describe the mechanical state of thin metallic films deposited onto free substrates. This formulation makes use of many assumptions, the validity of which is questionable when dealing with chemically-induced effects.

• More elaborated frameworks are available in the literature. The concept of surface stress as presented in [7] for solids does not elucidate the connection between surface and bulk (Cauchy) stresses. This is however a key point for describing the translation from surface (chemical) to bulk (mechanical) effects. [8] introduced a rigorous formulation for the mechanics of surfaces, making use of surface elasticity moduli. These can however be negative, and the meaning of such a situation is not clear. Asymptotic analysis is a thermodynamically grounded alternative approach [9], but which does not yield closed-form solutions for the displacement of chemicallymodified cantilevers.

Gurtin and Murdoch

• In addition to this conceptual difficulty, one should add that some systems (such as DNA-DNA hybridization) yields somehow controversial experimental results [4] : for similar chemical conditions and mechanisms, the bending may be seen to occur towards or away the functionalized surface.

Starting from a insufficiently known contribution by Mindlin [10], a linear theory of deformation of elastic beams in which the strain-energy density is a function of strain and its first and second gradients is adopted. As Mindlin demonstrated that such a theory is able to describe the surface tension for solids, this paper is intended to derive a tractable beam theory based on such a higher order material description. This contribution may therefore be seen as an extension of the recent propositions to describe the behavior of tiny Euler-Bernoulli [11][12][13] or Timoshenko [14] beams. The basic assumptions regarding the beam behavior are first presented. The virtual work principle [15] is then applied to yield the differential equations governing the mechanical behavior of chemically-modified isotropic beams. These equations are solved for cases of interest and the results are discussed.

Basic assumptions

Considering a beam lying along x direction and using the Euler-Bernoulli assumption, the displacement d for a loading in the (x, y) plane reads

d x = u(x) -y dv(x) dx ; d y = v(x)
Following Mindlin [10], the free energy density is assumed to depend on the classical infinitesimal strain ǫ 1 , as well as on the triadic ǫ 2 = ∇∇d (symmetric in the first two positions) and on ǫ 3 = ∇∇∇d (symmetric in the first three positions). ǫ 1 has therefore a single non-vanishing component

ǫ xx = -y d 2 v dx 2 + du dx
The non-zero components of ǫ 2 read

ǫ xxx = -y d 3 v dx 3 + d 2 u dx 2 ; ǫ xxy = d 2 v dx 2 = -ǫ xyx = -ǫ yxx
The components of ǫ 3 read

ǫ xxxx = -y d 4 v dx 4 + d 3 u dx 3 ; ǫ xxxy = d 3 v dx 3 = -ǫ yxxx = -ǫ xyxx = -ǫ xxyx
The elastic behavior is assumed to be isotropic and is modeled according to the constitutive law derived by Mindlin [10]. The elastic behavior is thus described by five coefficients a n (found in Toupin's theory describing surface effects for non-centrosymmetric materials [START_REF] Toupin | Proc. Int. Conf. on lattice dynamics[END_REF]), a cohesion modulus b 0 , seven coefficients b n and three coefficients c n in addition to the Lamé coefficients λ and µ.

The ratios a n /µ, b 0 /µ and c n /µ scale as a squared length, and b n /µ as a length to the fourth power.

W (d ⋆ ) = τ xx ǫ ⋆ xx + τ xxx ǫ ⋆ xxx + 2τ yxx ǫ ⋆ yxx + τ xxy ǫ ⋆ xxy + τ xxxx ǫ ⋆ xxxx +3τ yxxx ǫ ⋆ yxxx + τ xxxy ǫ ⋆ xxxy with τ xx = (λ + 2µ) ǫ xx + (c 1 + c 2 + c 3 ) ǫ xxxx τ xxx = 2 (a 1 + a 2 + a 3 + a 4 + a 5 ) ǫ xxx τ yxx = (a 1 + 2a 4 + a 5 ) ǫ yxx + a 2 2 + a 5 ǫ xxy τ xxy = (a 2 + 2a 5 ) ǫ yxx + 2 (a 3 + a 4 ) ǫ xxy τ xxxx = 2 (b 1 + b 2 + b 3 + b 4 + b 5 + b 6 + b 7 ) ǫ xxxx + b 3 2 (ǫ xxyx + ǫ xxxy ) + (c 1 + c 2 + c 3 ) ǫ xx + b 0 (y) τ yxxx = 2 3 (2b 2 + b 3 + b 5 + 3b 6 + 2b 7 ) ǫ yxxx + 1 3 (b 3 + 2b 4 + 2b 7 ) ǫ xxxy τ xxxy = (b 3 + 2b 4 + 2b 7 ) ǫ yxxx + 2 (b 5 + b 6 ) ǫ xxxy
Making use of the following substitutions

A = a 1 + a 2 + a 3 + a 4 + a 5 = (λ + 2µ) l 2 A ≥ 0 (1) B = b 1 + b 2 + b 3 + b 4 + b 5 + b 6 + b 7 = (λ + 2µ) l 4 B ≥ 0 (2) C = c 1 + c 2 + c 3 = (λ + 2µ) l 2 C ≥ 0 (3) Ã = a 1 -a 2 + a 3 + 3a 4 -a 5 = (λ + 2µ) l 2 Ã (4) B = b 2 -b 4 + b 5 + 2b 6 = (λ + 2µ) l 4 B (5)
and integrating W (d ⋆ ) over a cross-section defined by -t 2 ≤ y ≤ t 2 and -h 2 ≤ z ≤ h 2 and integrating the result by part yields

h -1 t -1 L 0 h 2 -h 2 t 2 -t 2 W (d ⋆ )dydzdx = - L 0   2B d 6 u dx 6 + 2 (C -A) d 4 u dx 4 + (λ + 2µ) d 2 u dx 2   u ⋆ (x)dx +   u ⋆ (x)   2B d 5 u dx 5 + 2 (C -A) d 3 u dx 3 + (λ + 2µ) du dx     L 0 -   du ⋆ (x) dx   2B d 4 u dx 4 + (C -2A) d 2 u dx 2     L 0 +   d 2 u ⋆ (x) dx 2   2B d 3 u dx 3 + C du dx + t b0     L 0 + L 0   Bt 2 6 d 8 v dx 8 + t 2 C -A 6 -4 B d 6 v dx 6 +   2 Ã + (λ + 2µ) t 2 12   d 4 v dx 4   v ⋆ (x)dx -   v ⋆ (x)   Bt 2 6 d 7 v dx 7 + t 2 C -A 6 -4 B d 5 v dx 5 +   2 Ã + (λ + 2µ) t 2 12   d 3 v dx 3     L 0 7 hal-00783801, version 1 -1 Feb 2013 +   dv ⋆ (x) dx   Bt 2 6 d 6 v dx 6 + t 2 C -A 6 -4 B d 4 v dx 4 +   2 Ã + (λ + 2µ) t 2 12   d 2 v dx 2     L 0 -   d 2 v ⋆ (x) dx 2   Bt 2 6 d 5 v dx 5 + t 2 C -2A 12 -4 B d 3 v dx 3     L 0 +t   d 3 v ⋆ (x) dx 3   Bt 6 d 4 v dx 4 + Ct 12 d 2 v dx 2 -b0     L 0 (6)
where the through-thickness b 0 distribution appears through its projections :

t 2 -t 2 b 0 (y)dy = t b0 ; t 2 -t 2 b 0 (y)ydy = t 2 b0 ( 7
)
The chosen description calls for two remarks :

• Looking at Eq.6 it should be highlighted that the chosen parametrization yields decoupled bending and tension problems. It should however be kept in mind that most of the cantilever sensors applications reported in the literature keep one side of the cantilever unmodified, so that these situations should be described by a change in both b0 and b0 . As cantilever sensors are commonly based on the detection of the out-of-plane displacement, the tension problem will be discarded in the following.

• The chosen kinematic does not allow for the Poisson effect to develop. As a consequence, the Lamé coefficients appear only through the combination

λ + 2µ = E 1 1 -2ν 2 1-ν (8)
where E and ν stand for the Young's modulus and the Poisson ratio, respectively. In order to approach the Poisson effect without mixing kinematic and static assumptions (as for classical beam theories), it is proposed for the sake of simplicity to replace the combination λ + 2µ by the Young's modulus E everywhere in the following, the identity being obtained for ν = 0 [14]. Mixing kinematic and static assumptions makes here the resulting description useless, especially if one wishes to get closed-form solutions. Any enrichment should therefore be motivated by experimental results.

4 Solutions for the bending problem of a cantilever beam

General solution

For a cantilever beam clamped at x = 0, Eq. 6 (restricted to the tension part) should be satisfied for any test field v ⋆ [15] so that

v ⋆ (x) ∈ V ad , V ad = v(x) ∈ H 1 ([0, L]) \v(0) = 0, dv dx (0) = 0 (9)
The displacement field should therefore satisfy ∀x ∈ [0, L]

Bt 2 6 d 8 v dx 8 + t 2 C -A 6 -4 B d 6 v dx 6 +   2 Ã + E t 2 12   d 4 v dx 4 = 0 (10)
Looking for solutions of the form

v(x) = exp Γ x l B ( 11
)
Making use of the characteristic lengths defined by Eqs.(1-5) and

setting τ = t 2 6l 2 B
the characteristic polynomial in Γ obtained from Eq.( 10) reads

l 2 B τ Γ 4 +   τ l 2 C -l 2 A -4 l 4 B l 2 B   Γ 2 + 2l 2 Ã + τ l 2 B 2 = 0 (12)
The general solution for Eq.( 10) therefore reads

v(x) = 3 i=0 q i x L i + 4 j=1 γ j exp Γ j x l B = 3 i=0 q i x L i + z(x)(13)
where the four Γ j are the solutions of Eq.( 12). The coefficients are to be obtained from the boundary conditions (obtained from both the known displacements and Eq.( 6)). The nature (real or complex) of the solutions Γ j is dictated by the sign of

∆ = l 2 C -l 2 A 2 -2l 4 B τ 2 -8 l 4 Bl -2 B l 2 C -l 2 A -l 2 Ãl 2 B τ + 16l 8 B l 4 B ( 14
)
10 and thus depends on the cantilever thickness. It should be highlighted that whatever the material parameters, lim τ →0 ∆ =

16l 8 B l 4
B > 0, so that the zero-thickness limit therefore always corresponds to real solutions for Γ 2 . The discussion of the sign of ∆ for a nonvanishing thickness would require the knowledge of the material parameters and is therefore beyond the scope of that paper. One should keep that for a given material, the shape of the general solution may therefore depend on the cantilever's thickness.

Transverse point-loading

The general solution being defined by Eq.( 13), the set of boundary conditions is now solved for a cantilever beam (clamped at x = 0) under the action of a transverse point loading F at its tip (no chemical surface change). Adding the corresponding external virtual work to Eq.( 6) [15] yields at x = L :

l 4 B t 2 6 d 7 v dx 7 +   t 2 l 2 C -l 2 A 6 -4l 4 B  d 5 v dx 5 +   2l 2 Ã + t 2 12   d 3 v dx 3 = F h -1 t -1 E ( 15
)
l 4 B t 2 6 d 6 v dx 6 +   t 2 l 2 C -l 2 A 6 -4l 4 B  d 4 v dx 4 +   2l 2 Ã + t 2 12   d 2 v dx 2 = 0 (16) l 4 B t 2 6 d 5 v dx 5 +   t 2 l 2 C -2l 2 A 12 -4l 4 B  d 3 v dx 3 = 0 (17) l 4 B t 6 d 4 v dx 4 + t l 2 C 12 d 2 v dx 2 = 0 (18)
Arranging the Γ j so that Γ 3 = -Γ 1 and Γ 4 = -Γ 2 and using the facts the Γ j are solutions for Eq.( 12), the system (15-18) is seen to feature six unknowns : q 2 , q 3 , as well as the four γ j . For a cantilever beam, Eq.( 6) also yields in x = 0

l 4 B t 2 6 d 5 v dx 5 +   t 2 l 2 C -2l 2 A 12 -4l 4 B  d 3 v dx 3 = 0 ; l 4 B t 6
d 4 v dx 4 + t l 2 C 12 d 2 v dx 2 = 0(19)
This linear system is solved to yield the displacement field

v(x) = q 0 + q 1 x L + F L 3 2htE   2l 2 Ã + t 2 12   -1   - x L 2 + 1 3 x L 3   + z(x)(20)
where z(x) is a function vanishing if the lengths defined by Eqs. (1)(2)(3)(4)(5) are set to 0. It should be highlighted that l 2 Ã is expected to be positive in order to maintain solutions close to the classical solution for any cantilever thickness. The terms q 0 and q 1 are finally set using the clamping conditions :

q 0 = - i γ i ; q 1 = - i γ i Γ i l B (21)
It can be easily checked that setting the characteristic lengths to 0 yields the classical solution for an isotropic material. The displacement field depend on all the 6 material parameters, so that these should be identifiable from full-field measurements along tip-loaded cantilever beams of different thicknesses. The defini-tion of a robust experimental procedure should be investigated.

It is however expected the accessible thickness range to drive the identifiability of the involved lengths. This will in turn define some hierarchy among the lengths for a given thickness range.

Pure chemical loading

Let us now consider a cantilever beam (clamped at x = 0) under the action of a heterogeneous chemical surface modification.

Eq.( 6) yields at x = L :

l 4 B t 2 6 d 7 v dx 7 +   t 2 l 2 C -l 2 A 6 -4l 4 B  d 5 v dx 5 +   2l 2 Ã + t 2 12   d 3 v dx 3 = 0 l 4 B t 2 6 d 6 v dx 6 +   t 2 l 2 C -l 2 A 6 -4l 4 B  d 4 v dx 4 +   2l 2 Ã + t 2 12   d 2 v dx 2 = 0 l 4 B t 2 6 d 5 v dx 5 +   t 2 l 2 C -2l 2 A 12 -4l 4 B  d 3 v dx 3 = 0 l 4 B t 6
d 4 v dx 4 + t l 2 C 12 d 2 v dx 2 = b0 E
Using the same Γ j arrangement, and using in x = 0

l 4 B t 2 6 d 5 v dx 5 +   t 2 l 2 C -2l 2 A 12 -4l 4 B  d 3 v dx 3 = 0 ; l 4 B t 6
d 4 v dx 4 + t l 2 C 12 d 2 v dx 2 = b0 E (22)
the displacement field reads

v(x) = q 0 + q 1 x L + b0 a 1 sinh Γ 1 L 2l B cosh Γ 2 2x-L 2l B -a 2 sinh Γ 2 L 2l B cosh Γ 1 2x-L 2l B tE a 1 b 2 sinh Γ 1 L 2l B cosh Γ 2 L 2l B + a 2 b 1 sinh Γ 2 L 2l B cosh Γ 1 L 2l B (23)
where q 0 and q 1 result from the clamping condition at x = 0 and where

a i = l 4 B t 2 6 Γ i l B 5 +   t 2 l 2 C -2l 2 A 12 -4l 4 B  Γ i l B 3 ; b i = l 4 B 6 Γ i l B 4 + l 2 C 12 Γ i l B 2 (24)
. The closed-form solution (23) for the out-of-plane displacement field induced by the chemical modification of one cantilever side calls for several comments :

• It should first be highlighted that this solution, in its generality, may significantly depart from the field resulting from Stoney's assumptions (homogeneous curvature). Besides the rigid-body motion, it may be decomposed into the sum of two hyperbolic cosines with length scales depending on the higher-order elasticity constants. As setting b n = c n = 0 yields a trivial solution (q i = 0, γ i = 0), it is confirmed that a first strain gradient theory is not sufficient to describe surface effects for isotropic materials. This pleads for the use of Mindlin's description, similarly to the elastic fluid described in [START_REF] Forest | [END_REF].

• These higher-order elasticity constants necessary reflect the length scales characterizing the material under scrutiny, and are thus expected to strongly depend on grain size or degree of crystallinity, or more generally, on the processing conditions.

The proposed framework therefore seems particularly suited to include the observed dependence on surface morphology [18].

• The role of the cantilever's thickness is much more complicated than it could be envisioned from simple beam theories. Besides the t -1 scaling factor, the thickness drives the shape of the displacement field through the solutions of the characteristic polynomial (12), possibly switching the field from hyperbolic to oscillatory (for imaginary solutions). One could easily imagine that such a situation experimentally observed using the (single-point) optical lever technique could lead to some data misinterpretation or ambiguity.

• Besides the thickness, the solution (23) highlights the role of the ratio l B /L , thus indicating that the cantilever's length could act as a filtering parameter in order to control the amplitude of the component added to the displacement field.

Conclusion

An Euler-Bernoulli beam theory for isotropic elastic materials based on a second strain gradient description with a through-thickness cohesion modulus gradient has been derived. This is thought to accurately describe the mechanical behavior of microcantilever sensors, and closed-form solutions are obtained for mechanical and chemical loadings. The proposed modelling involves 6 material parameters which seem to be identifiable from full-field measurements. The shape of the displacement field resulting from a chemical loading is found to possibly significantly depart from the homogeneous curvature assumption resulting from Stoney's assumptions, and depends on the cantilever's thickness as well as on the material parameters. Such a theory may then potentially contribute to explain some of the controversial experimental results found in the literature (when dealing with amorphous or polymeric materials). These potentialities are to be experimentally confirmed and thus require a robust experimental identification procedure of the involved material parameters. The comparison with experimental results should also drive the further developments of such an higher order description.

thor's attention by Samuel Forest through the MECANO research group activities http://www.im2np.fr/GDR-Mecano/.

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  Virtual work principle for a beam featuring a through-thickness modulus of cohesion gradientMindlin demonstrated that the cohesion modulus b 0 contributes to the free energy density through a term b 0 ǫ iijj which is then linear with respect to the kinematic variables. It thus defines a surface energy (without any external loading), so that one will assume in the following that all the material parameters are homogeneous in the cantilever beam, except the cohesion modulus b 0 which is considered to possibly depend on y in order to describe the chemical modification of one cantilever side compared to the other. Considering a virtual displacement field d ⋆ , the virtual change in the potential energy density W (d ⋆ )[15] reads

Acknowledgments

This work was partially funded through the NRA grant CheMeCo ANR-11-JS09-019-01. Mindlin's paper was brought to the au-