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Densification of window glass under very high pressure and its relevance to Vickers indentation
Scripta Materialia 55 (2006) 1159–1162 www.actamat-journals.com
Densification of window glass under very high pressure and its relevance to Vickers indentation Hui Ji,a Vincent Keryvin,a Tanguy Rouxela,* and Tahar Hammoudab a
LARMAUR, FRE-CNRS 2717, Baˆt. 10 B, Laboratoire de Recherche en Mecanique Appliquee, Universite´ de Rennes 1, campus de Beaulieu, 35042 Rennes cedex, France b Laboratoire Magmas et Volcans, CNRS-OPG, Universite´ Blaise Pascal, 5 rue Kessler, 63038 Clermont-Ferrand cedex, France Received 1 August 2006; revised 8 August 2006; accepted 9 August 2006 Available online 28 September 2006
Window glass experiences permanent densification under high hydrostatic pressure, like amorphous silica but to a lesser extent. Comparable hydrostatic stresses are reached beneath a sharp indenter during indentation loading. The behavior of glass under very high pressure is therefore a key factor in understanding indentation deformation. A careful modelling of a Vickers indentation process, accounting for a newly determined pressure-densification constitutive law, allows for a quantitative estimation of the contribution of densification to the indentation strain. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Very high pressure; Glass; Microindentation; Finite element analysis; Hardness
It is well known that the density of glasses with relatively low atomic packing density (Cg), defined as the ratio between the minimum theoretical volume occupied by the ions and the corresponding effective volume of glass, can be improved by very high pressure treatments. For instance the density of amorphous silica (a-SiO2) can be increased by about 20% and reaches that of quartz [1–4]. In the case of a-SiO2, the flow densification process is recognized to play a major role on the indentation behavior [5–8]. Whereas Vickers indentation results in median–radial cracks in most brittle materials, cone cracks show up in a-SiO2 with a concurrent collapsing of the matter under the indenter. a-SiO2 (Cg 0.45) was thus considered as an ‘‘anomalous’’ glass in contrast to ‘‘normal’’ window glass (WG) (Cg 0.52) for instance [9]. However, it was recently demonstrated that most glasses (metallic glasses may be the exception) partially behave ‘‘anomalously’’, depending on their atomic packing density or Poisson’s ratio (a low ratio favours densification) [10]. The present work focuses on a standard window glass (Planilux, Saint-Gobain Co., France) and, to the authors’ knowledge, the first pressure-densification data up to 25 GPa are reported. A three-dimensional (3D) finite element modelling of the Vickers * Corresponding author. Tel.: +33 223236718; fax: +33 223236359; e-mail: [email protected]
indentation process has been performed and shows the relevance of the densification mechanism to the permanent deformation observed after unloading and used to estimate hardness. Hardness is therefore found to significantly originate from the densification process. High pressure experiments were performed in an octahedral multi-anvil apparatus using a Walker [11] module and following the procedure described in Ref. [12]. Each run consisted of raising the load pressure of the main ram at the nominal rate of 0.5 MPa of oil per minute. After reaching the goal pressure, the sample was kept at high pressure for 1 h and then decompression was initiated. The resulting decompression time was between 12 and 14 h. Most specimens came out in one piece suggesting that the pressure device induced very little shear. Density was measured with a better than 0.001 g cm3 accuracy by means of a density gradient method using iodobenzene (q = 1.824 g cm3 (20 C)) and methylene iodide (q = 3.325 g cm3 (20 C)). The pristine glass and quartz were used to calibrate this float-sink method (see Ref. [13] for details). Vickers indentation experiments were conducted in air (20 C, 60% r.h.) using a dynamic hardness tester (Fischerscope H100C, Germany) on mirror-polished specimens (diamond suspensions up to 1 lm particle size). Specimens were annealed at Tg (823 K) for h prior to testing in order to limit the effect of the preparation history. The load was increased at a rate of 25 mN/s
1359-6462/$ - see front matter 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2006.08.038
H. Ji et al. / Scripta Materialia 55 (2006) 1159–1162
þ þ V d ¼ ðV o V a Þ þ ðV a V o Þ
ð1Þ
where () and (+) correspond to inside and piling-up volumes, respectively, and subscript (o) and (a) refer to the post-unloading and post-annealing volumes, respectively. Note that the piling-up volume was found þ to increase after annealing ðV þ o < V a Þ. This increase is likely to originate from the recovery of densification since the densification process extends to a region larger than the indentation print. Therefore, both the changes þ of the piling-up, ðV þ a V i Þ, and of the indentation volumes, ðV i V a Þ, are assumed to be constituents of the densified volume. A finite element analysis (FEA) of the indentation process was performed using Cast3M software (French Atomic Energy Agency [17]). The 3D simulations were conducted with eight-noded prismatic elements (except along the axis where four-noded pyramidal elements remain) and only 1/8th was meshed due to the symmetry of the Vickers indenter (see Ref. [18] for details of this meshing procedure). Simulations are displacement controlled and the maximum penetration is chosen to give the required load. At least 15 elements are in contact with the indenter at maximum load. The size of the mesh is chosen to be insensitive to the far-field boundary conditions. Note also that the large deformation formulation is used. For the diamond indenter, a linear isotropic elastic behavior is assumed with Young’s modulus and Pois-
son’s ratio of 1.1 GPa and 0.07, respectively. The glass deformation consists of a linear elastic part, with Young’s modulus and Poisson’s ratio of 71.5 GPa and of 0.23, respectively, and a permanent densification part. Because the driving force for the densification process (pressure and/or shear) as well as the pressure-densification constitutive law are not yet accurately known, the permanent deformation was brought into play as a consequence of the local stress level resulting from a pure linear elastic calculation. In each Gaussian point of all the meshes, density was computed according to the experimental pressure–density curve (see later). The local permanent volume change was integrated throughout the overall meshes. The sum of all the elementary permanent shrinkage volumes give the overall densified volume. The contact between the indenter and the glass is taken as frictionless. FEA and experimental investigations were carried out with a-SiO2 (E = 70 GPa, m = 0.15) as well for the sake of comparison. It is noteworthy that the instantaneous elastic response of the glass corresponds to an area of contact smaller than the one reached at the end of the indentation load plateau after the stresses relaxed through a flow process involving densification. Consequently, mean contact pressures higher than the conventional hardness values are reached during the loading stage (8.7 and 8.5 GPa for WG and a-SiO2, respectively). The increase of the density of the specimen with increasing pressure is plotted in Figure 1 together with the two data points obtained at 4 and 10 GPa by Cohen and Roy in the 1960s [2]. As for amorphous silica (aSiO2), the permanent densification increases rapidly above a pressure of 10 GPa and seems to saturate above 20 GPa. A maximum value of 2.672 g cm3 is achieved. This density is 6.3% higher than that of the as-melted glass (2.514 g cm3) and a bit higher than that of crystallized quartz (2.649 g cm3). The discrepancy between our data and those of Cohen and Roy probably arose from the high pressure testing configuration. Whereas these authors used powdered glass, glass cylinders were used in the present work. The use of powders favors some shear and may enhance the densification as suggested by Mackenzie and Laforce [19] and Uhlmann [20] who observed a similar disparity between their data on aSiO2 and those of Cohen and Roy. Note also in the latter case that the density changes for WG were estimated 8 7 6
ο
up to a maximum load between 100 and 500 mN. A load of 100 mN prevents microcracking. Experiments at 250 mN led to a more accurate indentation volumes estimation but eventually resulted in microcracking of the window glass. 500 mN was the upper limit beyond which both a-SiO2 and WG exhibited microcracks. The dwell time at maximum load was 5 s. The float process used to manufacture window glass consists of spreading the glass melt on molten tin and leaves some chemical gradients through the glass sheet thickness. The side in contact with molten tin was studied in the present work. The indentation volumes were measured using an atomic force microscope (AFM) (Veeco, Nanoscope III, DI 3000, USA) following the procedure detailed in Ref. [10]. Annealing was performed in air for all indented samples. The annealing temperature and time were set at 0.9 · Tg (K) and 2 h, respectively. As was previously reported by Neely and Mackenzie [14] and Yoshida et al. [15], these conditions allow for a nearly complete recovery of the densified region, while the viscous flow contribution remains negligible due to the much slower kinetics at this temperature. These authors argued that such a heat treatment is clearly unable to induce significant viscous relaxation and to relax the residual stresses occurring beneath the indentation print. Recently, Kese et al. [16] also showed that a 24 h long treatment at 0.97 · Tg is not sufficient to get a significant stress relaxation in a soda–lime glass. Consequently, it is assumed that the main effect of the present treatment is to favour the full recovery of the densified zone. Assuming that densification affects both the inside and the immediate neighbourhood of the indentation, the following expression holds for the densification volume:
Density change, Δ ρ/ρ (%)
1160
5 4 3 Linear approximation used for the simulation Experimental data Cohen and Roy [4] "Sigmoidal curve fitting"
2 1 0 0
5
10
15
20
25
30
Hydrostatic pressure, P (GPa)
Figure 1. Permanent change of the density (density gradient method) of a window glass (soda–lime–silica) after high pressure experiments conducted at room temperature with a 1 h plateau using an octahedral multi-anvil apparatus.
H. Ji et al. / Scripta Materialia 55 (2006) 1159–1162
1161
from changes in the refractive index with possible additional error. The densification data could be fitted with a sigmoid consisting of three tunable parameters (a, b and P0): Dq 1 1 ð2Þ ¼a q0 1 þ b expðP =P 0 Þ 1 þ b where a = 6.3% (maximum relative density change at saturation), b = 700 and P0 = 2.22 GPa. The conversion to relative volume changes is straightforward from the mass conservation law: DV =V o ¼ Dq=q0 =ð1 þ Dq=q0 Þ
ð3Þ
The density at saturation corresponds to a maximum volume change of 5.9%, much smaller than the one observed for a-SiO2 (14%). Eq. (3) was used to estimate the permanent volume change in every mesh from its original volume Vo and by calculating the relative density change of the mesh from the mean pressure in the mesh. For the sake of simplicity and to ease the FEA calculation it was considered that no densification occurs below a pressure onset (taken as 8 GPa) and that densification is complete for P higher than a saturation value (20 GPa). A linear increase was considered between both extreme regimes, so that the sigmoid (Eq. (2)) was replaced by a linear curve consisting of three straight segments. A typical set of FEA patterns is illustrated by Figure 2 for a 100 mN Vickers indentation. It is obvious from the top view (Fig. 2(b)) that densification extends along the edges of the indentation print due to singular stresses in this region. However, the side view in an edge-containing plane (Fig. 2(c)) reveals that densification along the edges is rather localized near the surface in comparison with what is observed in other vertical planes (Fig. 2(d)). Comparison between WG and a-SiO2 shows that although densification is less significant in WG than in a-SiO2 (the saturation level is 3 times smaller in WG), it extends further (20% deeper, for instance) in WG. The corresponding overall densified volume is DVd = 0.31 lm3. It is worth noting that a FEA simulation performed with an equivalent cone indenter leading to the same contact area as the Vickers’ one for a given load, i.e. with a 140.6 cone angle, led to a larger amount of densification, by about 20%. This is presumably because the stress singularity along the edges of the Vickers indentation gives a minor contribution to the overall densification, mainly governed by the stresses on the inverse pyramid faces, which are smaller (on average) than the stresses produced by the equivalent cone. A direct measurement of the densified volume by means of indentation topometry using AFM and annealing treatments (Fig. 3) results in this case in DVd = 0.89 lm3 and represents up to 68% of the permanent indentation volume measured just after unloading. Although both experiments and simulations bring to light the importance of the densification process to the irreversible indentation deformation of window glass, the discrepancy between the FEA and the experimental results suggests that the shear part of the indentation stress tensor plays a significant role. Let us consider that shear favours densification but that hydrostatic pressure is a necessary condition, then the effect of shear may be introduced as follows:
Figure 2. 3D FEA simulation of the Vickers indentation process: (a) AFM image of a 100 mN Vickers indentation and schematic drawing of the different views; (b) WG, top view of the elementary unit of symmetry (one eighth of the indent) showing the singular intensity of the densification along the edge; (c) WG, densification gradient in an edge containing vertical (loading axis) plane; (d) WG, face view of the densification gradient; (e) a-SiO2, edge view of the densification gradient; (f) WG, edge view of the densification gradient accounting for a shear contribution. Colours from dark to light correspond to 0 to maximum densification (saturation).
a
b μm
μm 0.2
0.2
0
0 6
-0.2
6
-0.2 4
2 4
2 6 μm
4
2 4
2 6 μm
Figure 3. 3-D AFM images of Vickers indentations performed on float glass with a load of 100 mN: (a) before annealing; (b) after annealing at 0.9 Tg for 2 h.
P e ¼ ð1 þ s=s0 ÞP
ð4Þ
where Pe is an equivalent pressure, s is a value of the shear stress intensity (the Tresca maximum shear stress was considered here) and s0 is a normalization stress which allows the shear effect to be tuned. For a-SiO2, the FEA simulations were conducted assuming an onset pressure of 10 GPa and a saturation plateau at 20 GPa corresponding to a 20% relative density change, as was previously reported [3,21]. Results are summarized in Table 1. The ratio Vr between the densification volume and the post-unloading indentation volume reaches maximum values of 68% and 92%, respectively for
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H. Ji et al. / Scripta Materialia 55 (2006) 1159–1162
Table 1. Estimations of the densified volume resulting from Vickers indentation by means of indentation topometry (AFM) and FEA simulations Volumes (lm3) ± 0.01 lm3
WG, 100 mN WG, 250 mN WG, 500 mN [10] a-SiO2, 100 mN [10] a-SiO2, 500 mN
Indentation topometry
FEA (Vickers 3D)
FEA (cone)
V o
Vþ o
V a
Vþ a
Vd
Vr (%)
Vd
s0
Vd
1.30 5.22 16.7 1.03 8.53
0.18 0.91 2.42 0.02 0.95
0.45 1.96 6.57 0.2 1.13
0.22 0.96 2.52 0.14 1.05
0.89 3.30 10.26 0.95 7.50
68 63 61 92 88
0.31 1.2 3.5 0.66 5.1
2.9 3.1 2.9 24 22
0.39 1.46 3.88 0.80 8.3
þ þ V d ¼ ðV o V a Þ þ ðV a V o Þ, V r ¼ V d =V o ; annealing treatment: 0.9 Tg for 2 h; n.d.: non determined.
WG and a-SiO2 for a load of 100 mN. These values are relatively load-insensitive for loads between 100 and 500 mN (values of 61% and 88% were obtained for a 500 mN indentation load). In order to have the results of the simulations match the experimental data, values of 2.9 and 24 GPa were derived for s0 for WG and aSiO2, respectively for a load of 100 mN. The shear effect on the densification process appears to be composition dependent but additional FEA investigations with different load values and various glasses are clearly needed to go further on interpretation. Poisson’s ratio seems to play a key role in the indentation–densification phenomena. On the one hand, this ratio reflects the glass network atomic packing density (see Ref. [22]) so that a low Poisson’s ratio, such as the one of a-SiO2, corresponds to a low packing density and to a high sensitivity to pressure densification. On the other hand, the Poisson’s ratio governs the size of the densified zone beneath the indenter. For instance, the pressure along the loading axis at a distance r from the loading surface is expressed as [23]: P ¼ ð1 þ mÞF =ð3pr2 Þ
ð5Þ
where F is the contact force. This means that the depth of the densified zone (saturation level) for a given indentation load is proportional to [(1+m)/P]1/2. As a consequence, the area affected by the densification process increases with m. So that the overall densification volume is the result of the combination of the local intensity of the process and the domain over which it extends. In contrast to the common idea that WG (and glasses with higher packing density as well) exhibits little indentation–densification deformations, FEA investigations and AFM observations lead to the conclusion that both a-SiO2 and WG experience densification contributions to rather comparable amounts. The densified volume stemming from an indentation load of 500 mN is even larger in WG than in a-SiO2 (Table 1). In conclusion: (i) The permanent change of the density of a standard window glass was studied for pressure ranging up to 25 GPa. The density is found to increase by about 6% starting at an onset pressure of 8 GPa and saturating for pressures above 20 GPa. (ii) Quantitative investigations of the densification beneath a Vickers indenter by means of AFM topometry result in a densified volume representing as much as 68% of the post-unloading indentation print (92% for a-SiO2). (iii) 3D FEA simulations were conducted and show that the densified area extends further in WG than in a-SiO2, with a strong densification along the indentation edges. The predicted
densified volumes are smaller than the experimental ones suggesting a shear-contribution to densification. (iv) Poisson’s ratio seems to be a controlling parameter and even silicate glasses with high m values, e.g. those containing relatively large amounts of compensating and modifying cations, experience indentation–densification. The authors are very grateful to Bernard Truffin, Jean-Christophe Sangleboeuf and Ronan Tartivel, (LARMAUR, CNRS-University of Rennes 1, France), for their experimental assistance with the AFM observation and the density measurements, and to Satoshi Yoshida (Prefectural University of Shiga, Japan) for stimulating discussions. [1] P.W. Bridgman, I. Simon, J. Appl. Phys. 24 (4) (1953) 405. [2] H.M. Cohen, R. Roy, J. Am. Ceram. Soc., Discussions and Notes 44 (1961) 523. [3] J.D. Mackenzie, J. Am. Ceram. Soc. 46 (1963) 461. [4] H.M. Cohen, R. Roy, Phys. Chem. Glasses 6 (5) (1965) 149. [5] T. Rouxel, S. Yoshida, H. Shang, J.C. Sangleboeuf, MRS Proceedings [904E], Mechanisms of Mechanical Deformation in Brittle Materials, 0904-BB02-03 (2005). [6] F.M. Ernsberger, J. Am. Ceram. Soc. 51 (1968) 545. [7] K.W. Peter, J. Non-Cryst. Sol. 5 (1970) 103. [8] C.R. Kurkjian, G.W. Kammlott, M.M. Chaudhri, J. Am. Ceram. Soc. 78 (1995) 737. [9] A. Arora, D.B. Marshall, B.R. Lawn, M.V. Swain, J. Non-Cryst. Sol. 31 (1979) 415. [10] S. Yoshida, J.C. Sangleboeuf, T. Rouxel, J. Mater. Res. 20 (2005) 3404. [11] D. Walker, M.A. Carpenter, C.M. Hitch, Am. Mineral. 75 (1990) 1020. [12] T. Hammouda, Earth Planet Sci. 214 (2003) 357. [13] F. Hubard Horn, Phys. Rev. 97 (6) (1955) 1521. [14] J.E. Neely, J.D. Mackenzie, J. Mater. Sci. 3 (1968) 603. [15] S. Yoshida, S. Isono, J. Matsuoka, N. Soga, J. Am. Ceram. Soc. 84 (2001) 2141. [16] Kwadwo Kese, Matilda Tehler, Bill Bergman, J. Eur. Ceram. Soc. 26 (6) (2006) 1003. [17]. [18] A.E. Giannakopoulos, P.L. Larsson, Mech. Mater. 25 (1997) 1. [19] J.D. Mackenzie, R.P. Laforce, Nature 197 (1963) 480. [20] D.R. Uhlmann, J. Non-Cryst. Sol. 13 (1973/74) 89. [21] M. Grimsditch, Phys. Rev. Lett. 52 (1984) 2379. [22] A. Makishima, J.D. Mackenzie, J. Non-Cryst. Sol. 17 (1975) 147. [23] J. Boussinesq, Applications des potentiels a` l’e´tude de l’e´quilibre et du mouvement des solides e´lastiques, Gauthiers-Villars, 1885.
Densification of window glass under very high pressure and its relevance to Vickers indentation Hui Ji,a Vincent Keryvin,a Tanguy Rouxela,* and Tahar Hammoudab a
LARMAUR, FRE-CNRS 2717, Baˆt. 10 B, Laboratoire de Recherche en Mecanique Appliquee, Universite´ de Rennes 1, campus de Beaulieu, 35042 Rennes cedex, France b Laboratoire Magmas et Volcans, CNRS-OPG, Universite´ Blaise Pascal, 5 rue Kessler, 63038 Clermont-Ferrand cedex, France Received 1 August 2006; revised 8 August 2006; accepted 9 August 2006 Available online 28 September 2006
Window glass experiences permanent densification under high hydrostatic pressure, like amorphous silica but to a lesser extent. Comparable hydrostatic stresses are reached beneath a sharp indenter during indentation loading. The behavior of glass under very high pressure is therefore a key factor in understanding indentation deformation. A careful modelling of a Vickers indentation process, accounting for a newly determined pressure-densification constitutive law, allows for a quantitative estimation of the contribution of densification to the indentation strain. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Very high pressure; Glass; Microindentation; Finite element analysis; Hardness
It is well known that the density of glasses with relatively low atomic packing density (Cg), defined as the ratio between the minimum theoretical volume occupied by the ions and the corresponding effective volume of glass, can be improved by very high pressure treatments. For instance the density of amorphous silica (a-SiO2) can be increased by about 20% and reaches that of quartz [1–4]. In the case of a-SiO2, the flow densification process is recognized to play a major role on the indentation behavior [5–8]. Whereas Vickers indentation results in median–radial cracks in most brittle materials, cone cracks show up in a-SiO2 with a concurrent collapsing of the matter under the indenter. a-SiO2 (Cg 0.45) was thus considered as an ‘‘anomalous’’ glass in contrast to ‘‘normal’’ window glass (WG) (Cg 0.52) for instance [9]. However, it was recently demonstrated that most glasses (metallic glasses may be the exception) partially behave ‘‘anomalously’’, depending on their atomic packing density or Poisson’s ratio (a low ratio favours densification) [10]. The present work focuses on a standard window glass (Planilux, Saint-Gobain Co., France) and, to the authors’ knowledge, the first pressure-densification data up to 25 GPa are reported. A three-dimensional (3D) finite element modelling of the Vickers * Corresponding author. Tel.: +33 223236718; fax: +33 223236359; e-mail: [email protected]
indentation process has been performed and shows the relevance of the densification mechanism to the permanent deformation observed after unloading and used to estimate hardness. Hardness is therefore found to significantly originate from the densification process. High pressure experiments were performed in an octahedral multi-anvil apparatus using a Walker [11] module and following the procedure described in Ref. [12]. Each run consisted of raising the load pressure of the main ram at the nominal rate of 0.5 MPa of oil per minute. After reaching the goal pressure, the sample was kept at high pressure for 1 h and then decompression was initiated. The resulting decompression time was between 12 and 14 h. Most specimens came out in one piece suggesting that the pressure device induced very little shear. Density was measured with a better than 0.001 g cm3 accuracy by means of a density gradient method using iodobenzene (q = 1.824 g cm3 (20 C)) and methylene iodide (q = 3.325 g cm3 (20 C)). The pristine glass and quartz were used to calibrate this float-sink method (see Ref. [13] for details). Vickers indentation experiments were conducted in air (20 C, 60% r.h.) using a dynamic hardness tester (Fischerscope H100C, Germany) on mirror-polished specimens (diamond suspensions up to 1 lm particle size). Specimens were annealed at Tg (823 K) for h prior to testing in order to limit the effect of the preparation history. The load was increased at a rate of 25 mN/s
1359-6462/$ - see front matter 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2006.08.038
H. Ji et al. / Scripta Materialia 55 (2006) 1159–1162
þ þ V d ¼ ðV o V a Þ þ ðV a V o Þ
ð1Þ
where () and (+) correspond to inside and piling-up volumes, respectively, and subscript (o) and (a) refer to the post-unloading and post-annealing volumes, respectively. Note that the piling-up volume was found þ to increase after annealing ðV þ o < V a Þ. This increase is likely to originate from the recovery of densification since the densification process extends to a region larger than the indentation print. Therefore, both the changes þ of the piling-up, ðV þ a V i Þ, and of the indentation volumes, ðV i V a Þ, are assumed to be constituents of the densified volume. A finite element analysis (FEA) of the indentation process was performed using Cast3M software (French Atomic Energy Agency [17]). The 3D simulations were conducted with eight-noded prismatic elements (except along the axis where four-noded pyramidal elements remain) and only 1/8th was meshed due to the symmetry of the Vickers indenter (see Ref. [18] for details of this meshing procedure). Simulations are displacement controlled and the maximum penetration is chosen to give the required load. At least 15 elements are in contact with the indenter at maximum load. The size of the mesh is chosen to be insensitive to the far-field boundary conditions. Note also that the large deformation formulation is used. For the diamond indenter, a linear isotropic elastic behavior is assumed with Young’s modulus and Pois-
son’s ratio of 1.1 GPa and 0.07, respectively. The glass deformation consists of a linear elastic part, with Young’s modulus and Poisson’s ratio of 71.5 GPa and of 0.23, respectively, and a permanent densification part. Because the driving force for the densification process (pressure and/or shear) as well as the pressure-densification constitutive law are not yet accurately known, the permanent deformation was brought into play as a consequence of the local stress level resulting from a pure linear elastic calculation. In each Gaussian point of all the meshes, density was computed according to the experimental pressure–density curve (see later). The local permanent volume change was integrated throughout the overall meshes. The sum of all the elementary permanent shrinkage volumes give the overall densified volume. The contact between the indenter and the glass is taken as frictionless. FEA and experimental investigations were carried out with a-SiO2 (E = 70 GPa, m = 0.15) as well for the sake of comparison. It is noteworthy that the instantaneous elastic response of the glass corresponds to an area of contact smaller than the one reached at the end of the indentation load plateau after the stresses relaxed through a flow process involving densification. Consequently, mean contact pressures higher than the conventional hardness values are reached during the loading stage (8.7 and 8.5 GPa for WG and a-SiO2, respectively). The increase of the density of the specimen with increasing pressure is plotted in Figure 1 together with the two data points obtained at 4 and 10 GPa by Cohen and Roy in the 1960s [2]. As for amorphous silica (aSiO2), the permanent densification increases rapidly above a pressure of 10 GPa and seems to saturate above 20 GPa. A maximum value of 2.672 g cm3 is achieved. This density is 6.3% higher than that of the as-melted glass (2.514 g cm3) and a bit higher than that of crystallized quartz (2.649 g cm3). The discrepancy between our data and those of Cohen and Roy probably arose from the high pressure testing configuration. Whereas these authors used powdered glass, glass cylinders were used in the present work. The use of powders favors some shear and may enhance the densification as suggested by Mackenzie and Laforce [19] and Uhlmann [20] who observed a similar disparity between their data on aSiO2 and those of Cohen and Roy. Note also in the latter case that the density changes for WG were estimated 8 7 6
ο
up to a maximum load between 100 and 500 mN. A load of 100 mN prevents microcracking. Experiments at 250 mN led to a more accurate indentation volumes estimation but eventually resulted in microcracking of the window glass. 500 mN was the upper limit beyond which both a-SiO2 and WG exhibited microcracks. The dwell time at maximum load was 5 s. The float process used to manufacture window glass consists of spreading the glass melt on molten tin and leaves some chemical gradients through the glass sheet thickness. The side in contact with molten tin was studied in the present work. The indentation volumes were measured using an atomic force microscope (AFM) (Veeco, Nanoscope III, DI 3000, USA) following the procedure detailed in Ref. [10]. Annealing was performed in air for all indented samples. The annealing temperature and time were set at 0.9 · Tg (K) and 2 h, respectively. As was previously reported by Neely and Mackenzie [14] and Yoshida et al. [15], these conditions allow for a nearly complete recovery of the densified region, while the viscous flow contribution remains negligible due to the much slower kinetics at this temperature. These authors argued that such a heat treatment is clearly unable to induce significant viscous relaxation and to relax the residual stresses occurring beneath the indentation print. Recently, Kese et al. [16] also showed that a 24 h long treatment at 0.97 · Tg is not sufficient to get a significant stress relaxation in a soda–lime glass. Consequently, it is assumed that the main effect of the present treatment is to favour the full recovery of the densified zone. Assuming that densification affects both the inside and the immediate neighbourhood of the indentation, the following expression holds for the densification volume:
Density change, Δ ρ/ρ (%)
1160
5 4 3 Linear approximation used for the simulation Experimental data Cohen and Roy [4] "Sigmoidal curve fitting"
2 1 0 0
5
10
15
20
25
30
Hydrostatic pressure, P (GPa)
Figure 1. Permanent change of the density (density gradient method) of a window glass (soda–lime–silica) after high pressure experiments conducted at room temperature with a 1 h plateau using an octahedral multi-anvil apparatus.
H. Ji et al. / Scripta Materialia 55 (2006) 1159–1162
1161
from changes in the refractive index with possible additional error. The densification data could be fitted with a sigmoid consisting of three tunable parameters (a, b and P0): Dq 1 1 ð2Þ ¼a q0 1 þ b expðP =P 0 Þ 1 þ b where a = 6.3% (maximum relative density change at saturation), b = 700 and P0 = 2.22 GPa. The conversion to relative volume changes is straightforward from the mass conservation law: DV =V o ¼ Dq=q0 =ð1 þ Dq=q0 Þ
ð3Þ
The density at saturation corresponds to a maximum volume change of 5.9%, much smaller than the one observed for a-SiO2 (14%). Eq. (3) was used to estimate the permanent volume change in every mesh from its original volume Vo and by calculating the relative density change of the mesh from the mean pressure in the mesh. For the sake of simplicity and to ease the FEA calculation it was considered that no densification occurs below a pressure onset (taken as 8 GPa) and that densification is complete for P higher than a saturation value (20 GPa). A linear increase was considered between both extreme regimes, so that the sigmoid (Eq. (2)) was replaced by a linear curve consisting of three straight segments. A typical set of FEA patterns is illustrated by Figure 2 for a 100 mN Vickers indentation. It is obvious from the top view (Fig. 2(b)) that densification extends along the edges of the indentation print due to singular stresses in this region. However, the side view in an edge-containing plane (Fig. 2(c)) reveals that densification along the edges is rather localized near the surface in comparison with what is observed in other vertical planes (Fig. 2(d)). Comparison between WG and a-SiO2 shows that although densification is less significant in WG than in a-SiO2 (the saturation level is 3 times smaller in WG), it extends further (20% deeper, for instance) in WG. The corresponding overall densified volume is DVd = 0.31 lm3. It is worth noting that a FEA simulation performed with an equivalent cone indenter leading to the same contact area as the Vickers’ one for a given load, i.e. with a 140.6 cone angle, led to a larger amount of densification, by about 20%. This is presumably because the stress singularity along the edges of the Vickers indentation gives a minor contribution to the overall densification, mainly governed by the stresses on the inverse pyramid faces, which are smaller (on average) than the stresses produced by the equivalent cone. A direct measurement of the densified volume by means of indentation topometry using AFM and annealing treatments (Fig. 3) results in this case in DVd = 0.89 lm3 and represents up to 68% of the permanent indentation volume measured just after unloading. Although both experiments and simulations bring to light the importance of the densification process to the irreversible indentation deformation of window glass, the discrepancy between the FEA and the experimental results suggests that the shear part of the indentation stress tensor plays a significant role. Let us consider that shear favours densification but that hydrostatic pressure is a necessary condition, then the effect of shear may be introduced as follows:
Figure 2. 3D FEA simulation of the Vickers indentation process: (a) AFM image of a 100 mN Vickers indentation and schematic drawing of the different views; (b) WG, top view of the elementary unit of symmetry (one eighth of the indent) showing the singular intensity of the densification along the edge; (c) WG, densification gradient in an edge containing vertical (loading axis) plane; (d) WG, face view of the densification gradient; (e) a-SiO2, edge view of the densification gradient; (f) WG, edge view of the densification gradient accounting for a shear contribution. Colours from dark to light correspond to 0 to maximum densification (saturation).
a
b μm
μm 0.2
0.2
0
0 6
-0.2
6
-0.2 4
2 4
2 6 μm
4
2 4
2 6 μm
Figure 3. 3-D AFM images of Vickers indentations performed on float glass with a load of 100 mN: (a) before annealing; (b) after annealing at 0.9 Tg for 2 h.
P e ¼ ð1 þ s=s0 ÞP
ð4Þ
where Pe is an equivalent pressure, s is a value of the shear stress intensity (the Tresca maximum shear stress was considered here) and s0 is a normalization stress which allows the shear effect to be tuned. For a-SiO2, the FEA simulations were conducted assuming an onset pressure of 10 GPa and a saturation plateau at 20 GPa corresponding to a 20% relative density change, as was previously reported [3,21]. Results are summarized in Table 1. The ratio Vr between the densification volume and the post-unloading indentation volume reaches maximum values of 68% and 92%, respectively for
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Table 1. Estimations of the densified volume resulting from Vickers indentation by means of indentation topometry (AFM) and FEA simulations Volumes (lm3) ± 0.01 lm3
WG, 100 mN WG, 250 mN WG, 500 mN [10] a-SiO2, 100 mN [10] a-SiO2, 500 mN
Indentation topometry
FEA (Vickers 3D)
FEA (cone)
V o
Vþ o
V a
Vþ a
Vd
Vr (%)
Vd
s0
Vd
1.30 5.22 16.7 1.03 8.53
0.18 0.91 2.42 0.02 0.95
0.45 1.96 6.57 0.2 1.13
0.22 0.96 2.52 0.14 1.05
0.89 3.30 10.26 0.95 7.50
68 63 61 92 88
0.31 1.2 3.5 0.66 5.1
2.9 3.1 2.9 24 22
0.39 1.46 3.88 0.80 8.3
þ þ V d ¼ ðV o V a Þ þ ðV a V o Þ, V r ¼ V d =V o ; annealing treatment: 0.9 Tg for 2 h; n.d.: non determined.
WG and a-SiO2 for a load of 100 mN. These values are relatively load-insensitive for loads between 100 and 500 mN (values of 61% and 88% were obtained for a 500 mN indentation load). In order to have the results of the simulations match the experimental data, values of 2.9 and 24 GPa were derived for s0 for WG and aSiO2, respectively for a load of 100 mN. The shear effect on the densification process appears to be composition dependent but additional FEA investigations with different load values and various glasses are clearly needed to go further on interpretation. Poisson’s ratio seems to play a key role in the indentation–densification phenomena. On the one hand, this ratio reflects the glass network atomic packing density (see Ref. [22]) so that a low Poisson’s ratio, such as the one of a-SiO2, corresponds to a low packing density and to a high sensitivity to pressure densification. On the other hand, the Poisson’s ratio governs the size of the densified zone beneath the indenter. For instance, the pressure along the loading axis at a distance r from the loading surface is expressed as [23]: P ¼ ð1 þ mÞF =ð3pr2 Þ
ð5Þ
where F is the contact force. This means that the depth of the densified zone (saturation level) for a given indentation load is proportional to [(1+m)/P]1/2. As a consequence, the area affected by the densification process increases with m. So that the overall densification volume is the result of the combination of the local intensity of the process and the domain over which it extends. In contrast to the common idea that WG (and glasses with higher packing density as well) exhibits little indentation–densification deformations, FEA investigations and AFM observations lead to the conclusion that both a-SiO2 and WG experience densification contributions to rather comparable amounts. The densified volume stemming from an indentation load of 500 mN is even larger in WG than in a-SiO2 (Table 1). In conclusion: (i) The permanent change of the density of a standard window glass was studied for pressure ranging up to 25 GPa. The density is found to increase by about 6% starting at an onset pressure of 8 GPa and saturating for pressures above 20 GPa. (ii) Quantitative investigations of the densification beneath a Vickers indenter by means of AFM topometry result in a densified volume representing as much as 68% of the post-unloading indentation print (92% for a-SiO2). (iii) 3D FEA simulations were conducted and show that the densified area extends further in WG than in a-SiO2, with a strong densification along the indentation edges. The predicted
densified volumes are smaller than the experimental ones suggesting a shear-contribution to densification. (iv) Poisson’s ratio seems to be a controlling parameter and even silicate glasses with high m values, e.g. those containing relatively large amounts of compensating and modifying cations, experience indentation–densification. The authors are very grateful to Bernard Truffin, Jean-Christophe Sangleboeuf and Ronan Tartivel, (LARMAUR, CNRS-University of Rennes 1, France), for their experimental assistance with the AFM observation and the density measurements, and to Satoshi Yoshida (Prefectural University of Shiga, Japan) for stimulating discussions. [1] P.W. Bridgman, I. Simon, J. Appl. Phys. 24 (4) (1953) 405. [2] H.M. Cohen, R. Roy, J. Am. Ceram. Soc., Discussions and Notes 44 (1961) 523. [3] J.D. Mackenzie, J. Am. Ceram. Soc. 46 (1963) 461. [4] H.M. Cohen, R. Roy, Phys. Chem. Glasses 6 (5) (1965) 149. [5] T. Rouxel, S. Yoshida, H. Shang, J.C. Sangleboeuf, MRS Proceedings [904E], Mechanisms of Mechanical Deformation in Brittle Materials, 0904-BB02-03 (2005). [6] F.M. Ernsberger, J. Am. Ceram. Soc. 51 (1968) 545. [7] K.W. Peter, J. Non-Cryst. Sol. 5 (1970) 103. [8] C.R. Kurkjian, G.W. Kammlott, M.M. Chaudhri, J. Am. Ceram. Soc. 78 (1995) 737. [9] A. Arora, D.B. Marshall, B.R. Lawn, M.V. Swain, J. Non-Cryst. Sol. 31 (1979) 415. [10] S. Yoshida, J.C. Sangleboeuf, T. Rouxel, J. Mater. Res. 20 (2005) 3404. [11] D. Walker, M.A. Carpenter, C.M. Hitch, Am. Mineral. 75 (1990) 1020. [12] T. Hammouda, Earth Planet Sci. 214 (2003) 357. [13] F. Hubard Horn, Phys. Rev. 97 (6) (1955) 1521. [14] J.E. Neely, J.D. Mackenzie, J. Mater. Sci. 3 (1968) 603. [15] S. Yoshida, S. Isono, J. Matsuoka, N. Soga, J. Am. Ceram. Soc. 84 (2001) 2141. [16] Kwadwo Kese, Matilda Tehler, Bill Bergman, J. Eur. Ceram. Soc. 26 (6) (2006) 1003. [17]