Indentation strength of silicon nitride ceramics processed by spark plasma sintering technique

Materials Science & Engineering A 644 (2015) 159–170

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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea

Indentation strength of silicon nitride ceramics processed by spark plasma sintering technique N. Azeggagh a,b,c, L. Joly-Pottuz b,n, J. Chevalier b, M. Omori c, T. Hashida c, D. Nélias a a

Université de Lyon, INSA-Lyon, LaMCoS CNRS UMR5259, F-69621 Villeurbanne, France Université de Lyon, INSA-Lyon, MATEIS CNRS UMR5510, F-69621 Villeurbanne, France c Tohoku University, 6-6-11, Aza-Aoba, Aramaki, Aobaku, Sendai 980-8579, Japan b

art ic l e i nf o

a b s t r a c t

Article history: Received 1 June 2015 Received in revised form 17 July 2015 Accepted 18 July 2015 Available online 21 July 2015

We investigated the influence of the microstructure on the true stress–strain curve of silicon nitride based ceramics. The materials were processed by spark plasma sintering technique. Si3N4 with fine, average and coarse microstructures were obtained. Load versus displacement curves (P–h) were obtained by means of instrumented indentation technique using diamond coni-spherical tip. The experimental data were coupled with a minimization method based on the Levenberg–Marquardt algorithm and the non-linear part of the mechanical response was identified. Based on the obtained stress–strain curves, rolling contact simulations were performed. In addition, the nature of Hertzian contact damage was examined in the material with coarse microstructure using diamond indenters of radii 0.2 and 1 mm. The surface damage was observed under optical microscopy while Focused Ion Beam Sectioning technique permitted to image the subsurface damage. An evident size effect was noticed: fracture consisting of classical ring cracks dominated at large scale while distributed microcracks beneath the indent dominated at small scale. & 2015 Elsevier B.V. All rights reserved.

Keywords: Silicon nitride Inverse analysis Stress–strain curves Hertzian contact damage

1. Introduction

α metastable form at low temperatures and hexagonal more stable β form. Idealized α and β forms are assigned to space groups P31c

Silicon nitride based ceramics exhibit a remarkable combination of mechanical properties: good wear, oxidation and corrosive resistance [1], low density (3.2 g/cm3), excellent thermal shock resistance, resistance to impacts, high hardness (1600–2000 Hv) [2] and one of the highest fracture toughness among ceramic materials (6–10 MPa m1/2) [3]. Therefore, silicon nitride based materials have been widely used in various industrial domains since 1950s. For example, Kristic [4] mentioned applications in nuclear fusion reactors, construction of thermal conductors and gas turbines. Hampshire [5] cited structural application at high temperatures in turbocharger rotors while Bal and Rahaman [6] reported recent clinical use of silicon nitride to promote bone fusion in spinal surgery and current developments as femoral heads for hip joints. Today, rolling bearing manufacturing companies are increasingly turning to Si3N4 with a growth of about 40% per year [7], high-pressure turbopumps on NASA space shuttles are a representative example [8]. Si3N4 exists in two1 major crystallographic structures: trigonal

and P63/m, respectively [10]. An irreversible α → β phase transformation occurs above 1400 °C leading to β elongated grains and to the so-called in situ composites or self-reinforced materials exhibiting a higher fracture toughness [1]. Because of its high boundary energy and low diffusivity coefficient, sintering of fully dense Si3N4 by means of conventional or unconventional techniques requires an addition of small amounts of various additives (MgO, Al2O3,Yb2O3, La2 O3, etc.) [11–13]. During the sintering process, the additives react with the silicon nitride and the silica present at surface of each single powder to form an intergranular glassy film. This phase is few nanometres thick, it enhances precipitation and rearrangement mechanisms needed to achieve full densification but strongly impacts the mechanical and thermal behaviour of ceramic materials at room and elevated temperatures [14–17]. Depending on the application target, the quantity of sintering additive and processing conditions can be controlled to obtain tailored microstructures with small equiaxed or large elongated grains [18]. In coarse microstructures, grains with high aspect ratio enhance the toughening mechanisms such as crack bridging, deflection and grain pull-out which is found to significantly improve the fracture toughness value and the overall crack resistance [14,3,19–21]. The Hertzian damage is a crucial issue in the lifetime

n

Corresponding author. Fax: þ 33 472437930. E-mail address: [email protected] (L. Joly-Pottuz). 1 A third phase, cubic Si3N4 was synthesized under high conditions of temperature and pressure [9]. http://dx.doi.org/10.1016/j.msea.2015.07.053 0921-5093/& 2015 Elsevier B.V. All rights reserved.

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estimation of brittle materials, especially for application in rolling bearings. Previous studies in hot-pressed dense silicon nitride revealed a significant influence of grain size on the contact damage mechanisms [22]. In fact, a transition from brittle mode to quasi-ductile mode was experimentally demonstrated when grain size increases. The first mode consists of ring cracks created during loading by the radial stress which is maximum (tensile) at the surface along the contact area boundary, whereas the second mode is shear and compression-driven damage beneath the indent [23,24]. However, all these studies have not taken into account the motion of the rolling body, which radically changes the stress fields and consequently the subsurface damage. To assess the local mechanical behaviour of a large variety of materials including hard technical ceramics, instrumented indentation technique is generally used [25–28]. An indenter of a given geometry is pushed into a flat well-polished surface. The applied load, P, and the vertical displacement of the indenter, h, are simultaneously recorded during testing to obtain a P–h curve. Oliver and Pharr method [29] improved the earlier work of Doerner and Nix [30] to capture Young's modulus and hardness from the unloading part of the measured curves assumed to be purely elastic. The plastic properties of ductile materials can also be derived from the load versus displacement curves. The usual procedure consists of coupling analytical formulas with finite element simulations to derive required mechanical parameters. Based on a rigorous dimensional analysis, Dao et al. [31] proposed a set of dimensionless functions to calculate the plastic parameters of various pure and alloyed engineering metals with isotropic hardening from instrumented indentation with Vickers and Berkovich tips. The set of functions was afterwards extended by Chollacoop et al. [32] to other indenter tip geometries (50°, 60°, 80° cones) and Bucaille et al. [33] to take into account the friction coefficient in the identification process. Based on this approach, isotropic [31], anisotropic [34] and viscoplastic [35] constitutive laws were successfully identified. Regarding brittle hard materials, a different approach is generally considered to characterize the non-linear behaviour: a spherical indenter with a radius, R, mounted on a standard tensile machine is pressed on the polished surface of the specimen. Afterwards, the contact radius of the permanent indent, a, is measured under optical or Nomarski microscopy after gold coating. The indentation stress or mean pressure, p0, is defined as the maximum applied load, P, divided by the contact area after complete unloading, i.e. p0 ¼P/πa2, and the indentation strain, a/R, as the contact radius a divided by the sphere radius R [36]. In the elastic domain, p0 is proportional to a/R [37]. Using a set of indenters with various radii, a scatter-plot is obtained and p0(a/R) relations are then used to derive the mechanical behaviour of the brittle materials. Fischer-Cripps and Lawn [38] analysed the stress field in tough ceramics and proposed simple relations based on finite element calculations to determine the work hardening coefficient α (α ranges between 0 and 1: 0 for perfect plasticity without hardening and 1 for pure elasticity). In this approach, the yield stress Y corresponding to the first plastic flow is experimentally determined. Indeed, by combining the critical shear stress criterion and Hertzian theory, it is possible to correlate the yield stress Y with the mean pressure of subsurface damage initiation. From the micromechanics point of view, it is found that Y is related to the intrinsic shear stress of the different materials while α is determined by the damage intensity Nl3 where N is the density and l the size of the faults [39]. However, for hard materials such as Si3N4, the deviation from linearity in the indentation stress–strain curve is overestimated because of the flattening of the indenting spheres [22]. In fact, it is found that the yield stress and hardening coefficient of Tungsten

carbide (WC) indenters are below the identified values of tested specimens. In addition, the described technique requires optical measurements of the remaining contact radius and subsurface damage observation increasing the uncertainty of parameter values. Alternative methods taking advantage of the recent development of high precision indentation instruments are then needed. In this context, an approach was recently proposed by Luo et al. [40] to calculate the true σ –ϵ curve of Si2N2O–Si3N4 composites based on nanoindentation tests using a Berkovich rounded tip and FEM calculations. Because of the size of the indenter considered by the authors, Rberk ≃ 500 nm , only ultra-fine-grained microstructures were tested. Other approaches based on TEM nanoindentation of ceramic nanoparticles [41] or nanocubes [42] and compression of micropillars [43] were also proposed. But in each case, a sophisticated in situ nanomechanical testing system is required and the derivation of behaviour laws at larger scale, relevant for macroscopic products, has not been addressed yet. The purpose of the present work on silicon nitride based materials is twofold: (i) to use instrumented indentation testing with single spherical indenter available commercially and inverse identification to obtain the true stress–strain relations. (ii) To investigate the size effect on the nature of Hertzian contact damage mechanisms of silicon nitride ceramics with coarse microstructure under indentation. Numerical simulations using in-house software were then performed to simulate the development of this damaged zone under pure rolling loading.

2. Materials and techniques 2.1. Material elaboration A commercial nanosized Si3N4 powder (SN-ESP, UBE Industries, Japan) with high α/β ratio ( ≥ 95%) was used as a starting material. Powders with high α-phase content are known to lead to materials with better mechanical properties thanks to the β elongated grains resulting from the phase transformation at high temperatures [44]. Physical properties of SN-ESP powder are summarized in Table 1. Various amounts of yttrium oxide (Y2O3, 99.99% high purity, Wako Pure Chemical Industries, Japan) ranging from 1% to 5% were added to the starting silicon nitride powder in order to obtain different microstructures (from fine to coarse). The powder mixtures were ball milled in ethanol with alumina balls as grinding media and finally dried in an electric oven at 80 °C. The samples were sintered into disks of diameter 30 mm and thickness 3 mm in nitrogen (N2) gas atmosphere using spark plasma technique (SPS 1050, Syntex Inc., Japan ). The technique has the capability to densify the starting powders in a drastically shorter time and at lower temperature than other conventional processes. Heating rates during sintering were as follows: 120 °C/ min from 0 to 600 °C, 100 °C/min from 600 °C to 1600 °C, then 20 °C/min from 1600 °C to 1700 °C. This latter one is held constant for 15 min. The applied mechanical pressure was in all cases of 40 MPa. All the materials and corresponding processing conditions are listed in Table 2. Table 1 Physical properties of starting silicon nitride SN-ESP powder. Grade SN-ESP Purity

Phase

SSA (BET) 6–8 m2/g O ≤ 2.0%

Density(g cm  3) 3.19 C ≤ 0.2%, Cl ≤ 100 ppm

Cl ≤ 100 ppm

Fe ≤ 100 ppm, Al, Ca ≤ 50 ppm

α-Phase crystal ≥ 95%

N. Azeggagh et al. / Materials Science & Engineering A 644 (2015) 159–170

Table 2 List of the sintered materials and corresponding processing conditions. Yttria (wt%)

Temperature (°C)

Pressure (MPa)

Relative density (%)

1 2 3 4 5

1700 1700 1700 1700 1700

40 40 40 40 40

98.3 98.9 >99 >99 >99

2.2. Microstructural and mechanical characterization The bulk densities ρ of the sintered disks were measured using Archimedes method according to ASTM-C20 standard [45]. The relative density reported in Table 2 is the experimental density ρ divided by the theoretical one ρthe. An approximate value of ρthe is calculated using the mixture law from the theoretical densities of the starting α − Si3N4 and Y2O3 powders. Five measurements were performed for each material to calculate the average value and the standard deviation. The elastic properties E and ν were measured on the SPSed disks using ultrasonic method (TM506A, Hitachi Construction Machinery, Japan). The top surfaces of Si3N4 samples were mechanically polished with 1 μm diamond pastes then ultrasonically cleaned and tried. Vickers hardness measurements were carried out on the polished specimen using a Buehler Micromet 5104 machine, the final imprints were observed under magnification of  10 and  40. Preliminary tests were completed to investigate load and dwell time effects on hardness values, then they were set to 19.6 N (2 kgf) and 15 s, respectively. Series of ten tests were performed for each material in order to improve the confidence in the results. To highlight the grain structures, the samples were chemically etched with sodium hydroxide (NaOH) at 500 °C (FM27, Yamato Scientific Co, Japan). Etching time was optimized for each considered material. After etching and carbon coating, the surfaces were observed under scanning electron microscopy (SEM 7500F,

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JEOL, Japan). Electron backscattering diffraction observations (EBSD, TSL, USA and SEM 8503F, JEOL, Japan) were also performed to examine crystallographic orientation of the grains. Finally, the different crystalline phases were examined by X-ray diffraction (XRD) measurements with Cu K α radiation at 40 kV–40 mA (Ultima IV, Rigaku). Continuous scanning was performed from 10° to 100° with an angle increment of 0.02°. The acquired data were analysed by Jade 5.0 (Materials Data, Inc., USA) and Topas (Bruker AXS, USA) softwares for Rietveld refinement. Rietveld refinement was performed to determine precisely the α/β ratio, considering some textural effects. 2.3. Circular contact testing The Hertzian contact tests were conducted on polished silicon nitride specimens using standard testing machine (Model 5965, Instron Corporation, USA). Diamond spherical balls were pressed against the polished samples under loads ranging from 0 to 1500 N under a constant cross-head speed of 0.05 mm/min. The permanent indents after complete unloading were observed with laser microscope (VK100, Keyence, Japan) and SEM while examination of the subsurface damage was performed using Focused Ion Beam (Versa 3D LoVac FEI, USA) Sectioning technique from the centre of the indents. FIB combines a high resolution imaging step and cross-sectioning analysis. The techniques consist of three successive steps: first, a thin layer of carbon is deposited in order to protect the original surface and prevent erosion during milling and image acquisition. Second, an ion beam is used for digging and cleaning beneath the contact area. Finally, the zones of interest are imaged by electronic microscopy. Also in relation to subsurface damage observation, cross-sectioning of the opposite side is made to machine thin foils from the damaged area under the residual impressions, see Fig. 1. A manipulator probe is then introduced in the chamber and the last specimen-bulk body connection is cut. The thin foil is glued to a TEM grid using carbon deposition and its connection with the probe is cut. Next, the sample is tilted for final thinning under acceleration voltage of 5 kV and beam current of

Fig. 1. Transmission Electron Microscopy thin foils preparation from the damaged area (A) edges and bottom cutting, (B) lift-out, (C) TEM grid, (D) rotation and final thinning.

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Fig. 2. Circular contact of a sphere on a flat surface.

16 pA with an incidence angle of 72°. The final specimen dimensions are 20 μm length × 20 μm width × 100 nm thickness. The sample is then introduced into the TEM instrument (Titan3™ G2 60-300, FEI company) for the observations. 2.4. Size effect To investigate the size effect on the damage mode, indentation tests were performed with spheres of two different radii, R1 ¼0.2 and R2 ¼1.0 mm, on a silicon nitride sample with 5% of Y2O3. The tests were performed under the same maximum contact pressure, also called the Hertz pressure Phertz. We briefly recall thereafter the main parameters of the contact problem between two elastic bodies in the elastic domain [37]. Let P be the maximum applied vertical load, all the geometrical parameters are presented in Fig. 2

 Phertz value is defined as follows: Phertz =

⎡ 6PE 2 ⎤1/3 ⎢ 3 r2 ⎥ ⎣ π Rr ⎦

(1)

where Rr is the equivalent radius of the bodies in contact −1 ⎡1 1⎤ Rr = ⎢ + ⎥ ⎣ Rs Ri ⎦

(2)

It is simply equal to the radius of the ball for the contact of a sphere over a flat surface  Er the reduced Young's modulus defined as −1 ⎡1 − ν2 1 − νi2 ⎤ s Er = ⎢ ⎥ + Ei ⎦ ⎣ Es

(3)

where indices s and i refer to the ceramic specimens and the diamond indenters, respectively.  Concerning the stress field, the tensile component srr is maximum at the surface of the specimen at radial distance r ¼a from the contact centre

σrrmax =

1 − 2νs Phertz 3

(4)

 While the maximum magnitude of the principal shear stress lies at depth approximately equal to one-half the contact radius, z¼0.48a along the vertical contact axis z

τm (νs = 0.3) = 0.31Phertz

(5)

2.5. Indentation testing A coni-spherical diamond tip manufactured by Synton-MDP (Switzerland) was used to perform indentation tests at constant strain rate of 0.05 s  1 and allowed thermal drift rate of

0.05 nm s  1. The radius communicated by the manufacturer was 50 μm. However, further measurements with a high resolution KH-7700 digital microscope (Hirox company, Japan) and PLu neox profiler (Sensofar company, USA) led to a better estimation of the radius value. Recorded images of the tip geometry were analysed using MountainsMap® Universal software (Digital Surf company, France). Ri = 42 μm was the optimal value in the displacement range considered in this study, up to 3 μm. This radius value will be used in the subsequent work. The deviation between the suitable and measured indenter size is due to the difficulty in polishing a hard anisotropic diamond crystal into exact spherical shape. The tip was afterwards mounted on Agilent Nano Indenter G200 system. In order to get representative P–h curves, arrays of 5  5 with an inter-indentation spacing of 100 μm along the x and y directions were performed on each material. Surface roughness is a crucial parameter for a good determination of load versus displacement curve. Scratch tests with a three-sided pyramid diamond Berkovich tip were performed to estimate the arithmetical mean deviation of the roughness profile, Ra. Mean value of 0.08 μm was obtained for the considered materials. The initial distance between the tip and the specimen surface is 1000 nm and the surface approach velocity was fixed to 10 nm s  1. The calibration of the system was performed by indenting a reference material with well-known elastic modulus and hardness. Standard fused silica specimen (E¼ 69–72 GPa, ν ¼0.17 and H ¼10 GPa) was used here to calibrate the system compliance for a better accuracy. 2.6. Finite element model To simulate the indentation response of the Si3N4 ceramics, an axisymmetric two-dimensional Finite Element Model (FEM) was constructed with the commercial package Abaqus v6.11 (Dassault Systèmes, France) [46]. The model represents the diamond (Ei ¼1140 GPa, νi ¼0.07) coni-spherical indenter of tip radius Ri = 42 μm and a large homogeneous half-space. The indenter tip was assumed to be purely elastic while the specimen was considered as an elastoplastic material with an isotropic hardening. Bilinear four-node axisymmetric elements (CAX4R) with reduced integration and hourglass control were used to mesh the two bodies in contact. A fine mesh was considered in the contact area to ensure the convergence of the contact simulations, the mesh being progressively coarser when moving away from the contact, see Fig. 3. In order to optimize the computation time, the sensitivity to the mesh refinement was studied. It was found that 5000 elements in the contact zone were enough to reproduce well the indentation response of the model within a large interval of the penetration depth. The Von Mises yield criterion (J2 flow theory of plasticity) is assumed in all the numerical part of the study to represent the yield surface. The contact between the diamond tip and ceramic specimen was assumed to be frictionless (the same numerical indentation curve was obtained with values of friction coefficient μ ranging between 0 and 0.8). It was modelled using augmented Lagrange multiplier algorithm while the applied boundary conditions are based on the work of Jacq et al. [47]. The simulation consists of three successive steps: pre-contact, loading to a maximum depth and finally complete unloading to zero. During the second and third steps, the displacement of a reference node attached to the tip was controlled and the vertical reaction force was reported as the indentation load. Excellent agreement between the FE simulations and Hertz analytical results was obtained in the elastic domain confirming the validity of the numerical model.

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Fig. 3. FE model (right) and magnification of the mesh under the indenter (left).

2.7. Identification of the material parameters In this study, the indented ceramic specimens are considered to behave as elastoplastic materials. A bilinear law with isotropic hardening was used for the plastic part:

⎧ σ = Eε if σ ≤ σy ⎨ ⎪ ⎩ σ = σy + Kεp if σ ≥ σy ⎪

(6) 2.8. Rolling

The total strain consists of two parts

ε = εe + εp

varying the initial parameters sy0 and K of fictive elastoplastic material (E¼ 320 GPa, sy ¼6.0 GPa, K¼ 16 GPa). An illustrating example of the convergence trend of sy as a function of the initial values and the convergence behaviour of sy0 and K is presented in Fig. 4. One can notice that the same final value syf is reached after ten iterations independently of the starting value confirming the unicity of identified parameters.

(7)

where εe and εp are the elastic and plastic (inelastic) strain, respectively. For ductile materials, tensile tests help to completely identify such constitutive law. However, another approach is required in case of brittle materials. In the present work, the identification of the parameters E, sy and K of the silicon nitride materials was performed with MIC2M [48]. The algorithm was already successfully exploited to characterize different types of materials [49,50]. The approach consists in minimizing the cost function that quantifies the gap between the numerical load–displacement curve and experimental one by generating different combinations for E, sy and K. Both the loading and unloading parts of the indentation curves are taken into account. The optimization problem in least-square sense is solved by the Levenberg–Marquardt (LM) algorithm [51,52]. The convergence criterion in this study was set at 1%, a higher accuracy can be reached but requires a larger number of iterations which considerably extends the total computation time. We studied the convergence of the LM algorithm by

Once all the material parameters are obtained, it was possible to simulate the evolution of the permanent subsurface deformation under frictionless rolling contact loading. An elastic diamond ball (Ei ¼1140 GPa, νi ¼0.07) of radius 0.2 mm was considered as rolling element over a flat homogeneous Si3N4 body. A maximum load of 45 N was applied corresponding to a contact pressure Phertz of about 24 GPa. The elastoplastic behaviour of the studied ceramic specimens is described by the corresponding sy and K. Von Mises criterion and isotropic hardening were assumed in all the computations. For a circular contact and νs ¼ 0.3, the plastic deformation occurs for Phertz ≥ 1.60σy [37]. It is very important to emphasize that only the evolution of the plastic deformation was considered in the simulations without taking into account any possible crack initiation at the surface. Because of this very restrictive hypothesis, the results about the permanent deformations in the ceramic materials can therefore be considered as an upper estimation. The simulations were performed using in-house developed software based on semi-analytical approach of the contact problem [47,53]. The conjugate gradient method (CGM) and Fast

Fig. 4. Convergence trend of the yield stress sy.

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Fig. 5. Steps of rolling simulation of a spherical ball over a flat Si3N4 specimen. Fig. 7. Evolution of β phase in Si3N4 samples depending on the Y2O3 content.

Fourier Transform (FFT) drastically reduce the computation time when compared to finite element software packages. The performed simulations consist of three successive steps: (1) vertical loading to the maximum contact pressure, (2) rolling over a distance of 6 times the elastic contact radius and (3) unloading to zero (Fig. 5).

3. Results 3.1. Microstructure Fully dense materials were obtained when the volume of Y2O3 was at least 3%, and more than 98% of the theoretical density was reached when only 1% of yttrium oxide was added. Fig. 6 shows EBSD maps of the Si3N4 materials sintered with an addition of 1%, 3% and 5% oxide additive at 1700 °C. As expected, the sintered materials have microstructures with significantly different grain sizes and morphologies. In fact, quantitative measurements using commercial software ImageJ combined to phase identification and Rietveld refinement provide a complete overview on the grain sizes and phase compositions: (i) F-Si3N4 (Fig. 6A): the bimodal fine microstructure consists of 45.3 vol% of equiaxed α grains and 54.7 vol% of fine β grains; (ii) M-Si3N4 (Fig. 6B): the microstructure is bimodal and consists of 30 vol% of equiaxed α grains and 70 vol% of β with mean of length 4.4 μm and diameter 1.0 μm ; (iii) C-Si3N4 (Fig. 6C): the microstructure is coarse and consists of 15.5 vol% of equiaxed α grains and 84.5 vol% of β with mean of length 7.7 μm and diameter 1.5 μm . X-ray diffraction revealed that the sintered materials have different phase compositions, see Fig. 7. As expected, more β-Si3N4 was formed with increasing the amount of sintering additive. It

should be added that the elongated β grains are of random spatial distribution. The mean aspect ratio (length/diameter) is 1, 4.4 and 5.1 for F-Si3N4, M-Si3N4 and C-Si3N4, respectively. 3.2. Mechanical properties As for the grain size, the influence of processing conditions is significant on the mechanical properties, see Table 3. Young's modulus decreases from 336 to 317 GPa when the silicon nitride microstructure shifted from fine to coarse. Concerning Poisson's ratio, a constant value of 0.277 0.01 is obtained for all the materials. F-Si3N4 exhibited the highest Vickers hardness (18.2 GPa), then this latter sensibly decreases when increasing the grain sizes. The hardness reached its minimum value of 15.5 GPa for C-Si3N4. 3.3. Hertzian contact damage Fig. 8 reveals the optical and SEM views of contact sites on coarse silicon nitride specimen (C-Si3N4) after indentation at load 1000 N and 45 N with spheres of 1 mm and 0.2 mm radius, respectively. The corresponding maximum pressures from Eq. (1) is 23.8 and 24.8 GPa, respectively. The optical observations revealed initiation of well developed ring cracks at the surface of the indented specimens with 1 mm radius sphere (Fig. 8A). For the small sphere of radius 0.2 mm, no cracking was observed around the indents at the surface of the specimen (Fig. 8B). The indent profile is shown in Fig. 8C. At the subsurface level, one can observe a well-defined deformed zone with distributed short cracks (Fig. 9A). High resolution TEM observations (Titan3 G2 60300, FEI Company) under acceleration voltage of 300 kV and beam current of 0.060 nA revealed both intergranular and transgranular propagation of the microcracks (Fig. 9B). These deformation and cracks were not observed in the matrix away from the indents or in blank specimens.

Fig. 6. EBSD micrographs of microstructure of Si3N4 SPS-sintered at 1700 °C with (A) 1%, (B) 3% and (C) 5% Y2O3.

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Table 3 The elastic modulus and Vickers hardness of Si3N4 with fine, medium and coarse microstructures. Material

E (GPa)

H (GPa)

F-Si3N4 M-Si3N4 C-Si3N4

336 327 317

18.2 7 0.2 17.6 7 0.4 15.5 7 0.4

3.4. Plastic parameters Typical load–displacement curves for the three materials obtained using the spherical tip of 42 μm radius up to a maximum load of 9 N are plotted in Fig. 10. For illustration purpose, the Hertz solution (elastic) is plotted for each material. The fits were calculated using the identified elastic moduli. A good correlation between the initial loading stage of the experimental curves and the Hertz fits was observed. Substantial deviation from linearity in the high load region indicates plastic deformations. In fact, despite the large elastic recovery in the unloading part of the indentation curves, residual irreversible displacements after unloading still subsisted. Furthermore, the residual displacement is more pronounced and the deviation from linearity occurs earlier with increasing the heterogeneity of the microstructure, i.e. the grain sizes. Table 4 summarizes the identified yield strength and hardening parameters of sintered silicon nitride specimens. For comparison purpose, we have also reported the elastic modulus measured by ultrasonic technique Eus and yield stress Y given by Tabor's relation [54]:

Y = Hv/c

(8)

where Hv is the Vickers hardness and c ≃ 3. One can see a good agreement between the identified elastic modulus Eid and the experimental one for F-Si3N4 and M-Si3N4. The gap between the experimental values and numerical simulations is more pronounced for C-Si3N4 for which a difference of about 12% was observed. The yield stress sy decreases with increasing the heterogeneity of the microstructure through the sequence F − Si3N4 → M − Si3N4 → C − Si3N4 . Tabor's relation with c ≃3 gives an overestimated value for sy especially in the case of the fine microstructure. From the Vickers hardness and yield stress, the estimated constraint factor H/sy was calculated and reported. Values obtained range between 1.8 and 2.0 depending on the grain sizes. We calculated about 10% increase through the sequence F − Si3N4 → M − Si3N4 → C − Si3N4 . The hardening parameter K does not point to a clear trend. However, typical high values superior to 20 GPa were obtained regardless of the grain sizes.

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3.5. Simulation of rolling contact The plastic zones at the maximum of vertical load and after rolling over 6a is shown in Fig. 11 while the maximum of the equivalent plastic strain at each time increment is plotted in Fig. 12. One can notice four distinct parts in these curves: first, the deformation in the solids is purely elastic. When the Von Mises equivalent stress is higher than the material yield stress, then maximum of εp increases with increasing the applied load. Once the maximum load is reached, the motion of the ball starts and εp increases further until a steady state is obtained when the rolling distance is higher than 4 times the contact radius. Finally, the unloading step is carried with no variation of the plastic strain. Even if the curve profiles are similar, the residual deformation strongly depends on the grain sizes. In fact, more than 6% of plastic strain is achieved for C-Si3N4 while the maximum is only about 3.5% for F-Si3N4. Finally, M-Si3N4 behaviour is intermediate to these two cases with 4.5% of maximum plastic strain.

4. Discussion 4.1. Microstructure and mechanical properties The experimental results pointed out a significant effect of the yttria amount on the microstructure of the sintered specimens. In fact, the addition of 1% amount of yttria led to a microstructure containing exclusively fine equiaxed grains. The residual porosity ( ≤ 2%) in this material is expected to have a minor role in the contact damage which is the main objective of this study [24]. The X-ray patterns of this material exhibited large fraction of α phase. It is due to a short supply of intergranular liquid phase through which the mass transport takes place during sintering. As the amount of yttria increased up to 3%, elongated β grains were formed and represent the major phase, but their size remains relatively small and their density low. When the amount of Y2O3 reached its maximum value of 5%, the microstructure of the obtained material is coarse, mostly formed of β-Si3N4 (elongated grains with high aspect ratio). The X-ray diffraction pattern of the specimen containing larger amount of sintering aid exhibited more β-phase due to the enhancement of α → β phase transformation. Measured values of elastic modulus for the three materials (336–317 GPa) are coherent with those reported in other studies for fine, medium and coarse silicon nitride microstructures [55]. The decrease is mostly related to the difference of intrinsic elastic properties of α and β phases. In fact, it was reported that Eα ≃ 362 GPa and Eβ ≃ 312 GPa [56]. Therefore, an empirical mixture law can be used to relate the phase composition to the

Fig. 8. Hertzian contact sites in coarse Si3N4, from diamond sphere of radius (A) R1 ¼1 mm at P ¼1000 N and (B) R2 ¼ 0.2 mm at load P¼ 45 N, (C) experimental indent profile.

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Fig. 9. FIB cross section of the Hertzian contact damage in C-Si3N4 from diamond sphere of 0.2 mm radius at load P¼ 45 N, (B) TEM observation of the damaged area.

Fig. 10. Experimental and numerical P–h curves for fine, medium and coarse silicon nitride specimen. Hertz fit (linear behaviour) is also plotted for comparison.

Table 4 Plastic parameters for F-Si3N4, M-Si3N4 and C-Si3N4. The values and standard deviations are based on 4 different identifications for each material. Material Eus (GPa) Eid (GPa) Y (GPa) Yield stress sy (GPa) Hardening K (GPa) H/sy

F-Si3N4 336 3307 3.0 5.9 7 0.1 10.0 7 0.3 41.5 7 1.0 1.82

M-Si3N4 327 322 7 2.3 5.7 7 0.2 9.4 7 0.1 25.8 7 1.1 1.87

C-Si3N4 317 357 71.9 5.1 70.2 7.8 7 0.1 31.4 75.9 1.99

Ref. [22] 315–335

7.3–11.7 1.79–2.16

elastic modulus of Si3N4 : E = fα Eα + (1 − fα ) Eβ , where fα is the α phase ratio. The maximum gap between the values given by the previous relation and the experimentally measured ones was found to be about 3% (results not shown). Regarding the Vickers hardness, it is strongly related to the phase composition of silicon nitride materials thus, to the amount of sintering aid. In fact, it is found that the α-Si3N4 grains have an intrinsic hardness of 21 GPa which is about 25% higher than the hardness of β-Si3N4 grains (about 16 GPa) [57,58]. Therefore, Si3N4 materials containing larger fraction of α-phase have a higher Vickers hardness.

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Fig. 11. Evolution of the plastic zone under: (A) vertical loading, (B) rolling over a distance of 6 times the contact radius. Simulations were performed using sphere of radius R2 ¼0.2 mm at load P ¼45 N.

Fig. 12. Evolution of the maximum plastic strain under rolling conditions for the three materials.

4.2. Stress–strain curves It was reported that recorded load versus displacement curves for the studied materials deviate from linearity. This deviation was characterized by the identified values for the yield stress sy and hardening parameter K. However, the high values, greater than 20 GPa, indicate that Si3N4 behaviour is still dominated by the elastic part. The result agrees well with the conclusions of Lee et al. [22]. Concerning the plastic part of the mechanical behaviour, the decrease of the yield stress sy through the sequence F − Si3N4 → M − Si3N4 → C − Si3N4 is mainly attributed to the progressive increase of β phase composition and thus the grain sizes of the three materials. Values obtained for K do not seem to exhibit a clear trend, this makes us think that the bilinear law is only a first approximation and cannot thus catch all the damage mechanisms. Finally, the identified elastic moduli are in very good agreement with the experimental ones. The slight deviation observed in case of C-Si3N4 may be attributed to possible flattening of the indenter as C-Si3N4 specimens were the last to be tested. Grain boundaries play also an important role in the mechanical response of studied materials. Indeed, it is recognized that these last are susceptible to be weakened and their composition changed by the addition of a significant amount of a sintering additive

as it is the case for C-Si3N4 [59]. The weakening enhances the quasi-plastic deformation and partially suppresses cracking at the surface as it was experimentally demonstrated. The high values of the work hardening parameter K, typical for brittle materials, contradict the perfect plasticity assumption for silicon nitride ceramics under the Hertzian contact. This observation is confirmed by the discrepancy between the values of sy and that given by Tabor's relation, especially in the case of the finest microstructure. Nevertheless, the values obtained by inverse identification must be compared to other characterization techniques such as compression tests on micropillars before validation. For ductile materials, well known Hall–Petch [60] or Orowan equations describe an inverse relation between the yield stress and grain size in metals at room temperature. We noted the same trend for the materials of this study identified as brittle in the literature: Si3N4 ceramics with small grains have the highest yield strength. The latter property diminishes with increasing the microstructure coarseness. One can theorize that the mechanisms are similar for both material classes: the effect of the grain boundary density on the movement of dislocations for which the mean free paths depend on the grain size. Furthermore, it has been established in previous studies that the subsurface microfracture because of short cracks is raised for polycrystalline ceramics when increasing the grain size [39,59]. In other words, the damaged area extends more for coarse microstructures. This mechanism leads to material softening under high internal pressure and thus to a drop of the apparent yield stress sy. The profiles of the permanent indents on F-Si3N4, M-Si3N4 and C-Si3N4 were obtained using keyence VK-100 profilometer. In parallel, we have used the true stress–strain curves previously identified to numerically estimate the material response under same contact conditions. The results are shown in Fig. 13. One can notice that the numerical model reproduces well the experimental profile features for the three materials. On the other hand, it is found that the maximum depths were overestimated by FEM calculations in all cases with a maximum gap E20%. The mismatch is due to the unavoidable experimental and numerical uncertainties as well as to the limited number of parameters introduced to describe the non-linear behaviour of Si3N4 materials. More complex constitutive laws including other damage mechanisms (microcracks initiation and coalescence) may correct this discrepancy. We have also used the identified parameters to simulate rolling

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Fig. 13. Experimental versus simulated profiles for materials with coarse, medium and fine microstructures.

contact of an elastic ball over the sintered ceramics. It is found that the plastic strain strongly increases for coarser microstructures. Obviously, it is confirmed that the resistance of the Si3N4 to rolling contact fatigue significantly different from one microstructure to another. 4.3. Contact damage The contact damage was investigated using spheres of radii 0.2 and 1.0 mm. The surface and subsurface observations revealed different responses of Si3N4 with coarse microstructure under Hertzian contact conditions. It is admitted that elongated β grains improve the resistance of coarse microstructures to the propagation of long cracks leading to materials with R-curve behaviour. However, the resistance to short cracks is found to be deteriorated [55]. Concerning the contact damage, the fracture mode (classical ring cracks) is dominant at the large scale for indentation with 1 mm sphere. The ring cracks initiated slightly outside of the contact zone despite the fact that the tensile stress value is reached at r ¼a [61]. At smaller scale, the material undergone a permanent deformation without evidence of ring cracks at the surface. The deformation is accompanied by high density of short cracks. Their size is about few micrometers, they initiated from the weak grain boundaries and propagate along and across the adjacent grains. In addition, traces of slip system activation along large β grains were clearly stated. They were activated because of the high shear stress, several GPa, imposed to the specimen by contact with the tip and necessary to overcome the predominantly covalent atomic bonds. The exact characteristics of the lattice defect has not been established yet, further work is in progress to find well oriented grains in TEM. According to Evans and Sharp, the dislocations in β-Si3N4 are expected to have 〈0 0 0 1〉 Burgers vector and {1 0 1 0} slip system [62]. All these observations confirm a transition in the damage mode from brittle tensile-driven mode to quasi-plastic mode. This latter one causes strength degradation and is detrimental for rolling bearing applications because of subsurface localized plasticity as previously demonstrated. For a better estimation of the lifetime of the rolling elements, damage modes which occur under contact loading should be better quantified. In this context, Rhee et al. [55] assumed that shear and tensile stresses lead to quasi-plastic and fracture modes in brittle materials, respectively. The authors then experimentally

measured the critical loads for fracture Pc and localized deformation Py and finally introduced a transition radius r ⁎ for various brittle ceramic materials. Quasi-plastic mode occurs if the indenter radius R is smaller than r ⁎, i.e. Py ≤ Pc while fracture occurs when R is larger than r ⁎, i.e. Py ≥ Pc . The authors calculated a value r ⁎ of 12 mm for coarse Si3N4 having approximatively the same mechanical properties. We have experimentally demonstrated that the transition radius for the sintered material with heterogeneous grain size ranges between 0.2 and 1 mm. This value is significantly lower than that predicted by the previous model. The deviation from the obtained results of Rhee et al. is probably due in one hand to the hypothesis of perfect plasticity used to build the described model. In fact, we have demonstrated that the constraint factor H/Y depends on the grain sizes (no longer a material invariant), and on the other hand to the microstructural properties especially the β grains and the glassy boundary phase which nature has a significant effect on the degree of contact damage and the crack initiation critical load [63].

5. Conclusion Silicon nitride ceramics with different grain sizes were manufactured using SPS technique at temperature of 1700 °C with an addition of 1%, 3% and 5% of Y2O3. A gradual evolution from materials with fine equiaxed α grains to microstructures containing large elongated β grains was observed. Significant effect of the processing conditions on the mechanical properties was also noticed. The mechanical behaviour at small scale was investigated using indentation testing technique with spherical diamond tips. Obtained load–displacement curves were coupled with identification algorithm to take into account the non-linear part of the mechanical behaviour. A good agreement is found between the measured elastic modulus and the calculated one. The yield stress, sy, depends significantly on the grain size while high values were obtained for the hardening parameter K. The identified parameters were used to simulate the mechanical response of materials under a vertical or rolling loadings. It was numerically demonstrated that these latter conditions enlarge the damage zone independently of the material microstructure. Finally, Hertzian contact tests at macroscopic and mesoscopic scales were performed to investigate damage modes at the surface and beneath the indents. Experimental observations revealed a

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remarkable size effect in the damage mode for coarse microstructure: fracture at large scale and quasi-plastic deformation at smaller scale. In addition, the transition radius for a coarse Si3N4 microstructure was experimentally obtained.

Acknowledgements This research was supported in part by a mobility grant from Région Rhône-Alpes in the framework of Avenir Lyon-Saint-Etienne program and Elyt Laboratory (CNRS LIA Laboratory). The authors are grateful to Dr. Y. Kodama for FIB machining and Mr. Y. Hayasaka for TEM observations.

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