2019 Volume 60 Issue 8 Pages 1423-1427
In this study, micro-photoelastic measurements were performed to obtain three-dimensional stress maps of silica and soda-lime glasses during ball indentation. The stress components were calculated from retardations and azimuths, which were determined from photoelastic measurements with a spatial resolution of about 1 µm. During loading, it was observed that the tensile stress in the radial direction is generated near the surfaces of both glasses. During unloading, however, it was found that stress distributions of silica and soda-lime glasses are different from each other. It is concluded that the different stress distributions during indentation result in different crack geometries, ring and radial cracks.
Fig. 4 Principal stress maps of (a) silica and (b) soda-lime glasses under a ball indenter. The origin of each stress map corresponds to the contact point between the indenter and the glass. The positive values (red) and negative values (blue) indicate tensile and compressive stresses, respectively. Fine black lines denote directions of principal stresses.
Glass materials are widely used in various products, for example, bottles, windows and flat panel displays. In spite of the fact that glass is used for these many applications, glass is still one of the most brittle and fragile materials. This brittle behavior in glass results from stress concentration at a crack-tip on the glass surface. Due to very limited size of plastic zone at a crack tip, a cracked glass fails under a very low external stress. Therefore, in order to design high-strength and scratch-resistant glass, we need to understand the crack initiation behaviors in a pristine uncracked glass.
The indentation test is one solution to model cracking in glass, and the size of indentation-induced crack in glass is considered one measure of brittleness in glass.1) When the indentation test using a sharp diamond indenter is performed on glass, a permanent imprint, which is sometimes accompanied with some cracks, can be observed. This means that formation of the permanent imprint is a key mechanical response of glass prior to cracking. Actually, there are many kinds of researches which report that the crack initiation load during the indentation test depends on glass composition, on fictive temperature, and on indenter geometries.2–4)
Since the driving force of the indentation-induced crack is the stress field created by the indentation, the stress distribution should affect both crack geometries and threshold of crack nucleation. It has been reported that a ring/cone crack (Hertz crack) on silica glass is formed during a loading half cycle of Vickers indentation, and that a radial/median crack on soda-lime window glass is observed during an unloading half cycle of Vickers indentation.5) This experimental finding of differences in timing of crack nucleation and in crack geometries results from the fact that the indentation-induced stress field continuously changes during the indentation cycle, and that the indentation-induced stress field depends on glass composition. However, there is little quantitative information available on the stress field of glass during indentation. This is because the experimental determination of the stress field in glass during indentation is still a challenging task. In this study, we propose a micro-photoelastic evaluation technique to obtain in-situ three-dimensional stress maps around the contact point, and discuss a relation between the stress field and the crack initiation behavior.
The samples used were silica and soda-lime silicate glasses. The silica glass is an optical grade fused quartz (KV, Russia) and the soda-lime glass is a microscope slide glass (Matsunami 0050, Japan). Compositions and mechanical properties of the glasses used in this study are shown in Table 1. In order to perform micro-photoelastic measurements, the samples were shaped into square fibers with a cross-section of about 0.5 × 0.5 mm2. The geometries of the square fiber are suitable for micro-photoelastic observation from the side face of the sample during the indentation test on the top surface. In addition, the short optical path length (about 0.5 mm) enables us to obtain the photoelastic images with high signal/noise ratios. Silica glass fiber was made by drawing from a square rod at high temperature and soda-lime glass fiber was prepared by grinding and polishing of a square rod with the dimensions of about 1 × 1 × 35 mm3.
A cono-spherical diamond indenter with a tip radius of 0.05 mm and with a tip semi-angle of 60° was used for the indentation test. This indenter is called a ball indenter in this study. Indentation tests at a maximum load of 3.0 N were performed in an optical cell filled with an index matching oil which was the liquid mixture of methyl benzoate (refractive index: ∼1.515) and ethyl chloroacetate (refractive index: ∼1.422) to reduce light scattering and reflection during the photoelastic measurements. Details of the indentation test are explained in one previous study.8) Since the ball indenter used in this study was an axisymmetrical one, we can assume the indentation-induced stress field to be axisymmetric. This axisymmetrical stress field enables us to obtain the three-dimensional stress map of the glass around the indenter during the indentation cycle.
Although glass is an optically isotropic material, it is known that the glass specimen under a stress shows birefringence. When linearly polarized light is incident on the stressed glass, for example, the incident light is transformed into elliptically polarized one. This phenomenon is caused by the stress-induced birefringence, which is also called photoelasticity. In order to evaluate the indentation-induced stress field, photoelastic measurements can be performed around the contact point between the glass and the indenter during the indentation test. Photoelastic parameters, which are retardation and azimuth (the orientation of the slow axis), were measured by using a birefringence imaging system, LC-PolScope (CRi, Inc., USA). The schematic illustration of the self-made micro-photoelastic measurement system used in this study is shown in Fig. 1. This system consists of three parts, which are a light source, a sample stage, and the LC-PolScope system. Details of this micro-photoelastic measurement system are explained elsewhere.8,9) Circularly polarized light from the light source part is incident on the sample during the indentation test. The wavelength of the incident light was 546 nm. The elliptically polarized light passed through the birefringent sample was analyzed by the LC-PolScope system consisting of a liquid crystal compensator and an analyzer.
Schematic illustration of an experimental set-up for micro-photoelastic measurement system.
From the photoelastic measurements performed during the indentation, retardations and azimuths were obtained with a spatial resolution of about 1 µm (Fig. 2). Using the values of retardation and azimuth, indentation-induced stresses can be calculated by photoelastic tomography.10) In the case of the axisymmetric stress field, four kinds of stress components (shear, axial, radial, and hoop stresses) are represented in cylindrical coordinates. These stress components are schematically shown in Fig. 3. The numerical algorithm to determine the stress components from photoelastic parameters were described elsewhere.8–10) Although the indentation maximum load was 3.0 N, photoelastic measurements were performed at 0.5 N both during loading and during unloading. This is because the numerical algorithm is valid only if the retardation is less than one-quarter of the wavelength used.10) Under a load of more than 0.5 N, some retardation values become larger than the limit.
Schematic illustration of photoelastic measurements of retardation and azimuth during indentation.
Axisymmetric stress states.
From the stress values of the four kinds of stress components, the maximum principal stress (σ1) was calculated by using the following equation.11)
| \begin{equation} \sigma_{1} = (\sigma_{r} + \sigma_{z})/2 + \{[(\sigma_{r} - \sigma_{z})/2]^{2} + \tau_{zr}{}^{2}\}^{1/2} \end{equation} | (1) |
| \begin{equation} \tan\theta p = -(\sigma_{r} - \sigma_{z})/2\tau_{zr} + \{[(\sigma_{r}-\sigma_{z})/2\tau_{zr}]^{2} + 1\}^{1/2} \end{equation} | (2) |
The maximum principal stress distributions of silica and soda-lime silicate glasses are shown in Fig. 4. The origin of each stress map corresponds to the contact point between the indenter and glass surface. The red and blue colored regions in the maps represent tensile and compressive stressed regions, respectively. Fine black lines denote trajectories of the principal directions. In the stress map of silica glass (Fig. 4(a)) on loading, the compressive stress is generated under the indenter, and the tensile stress is located near the contact edge on glass surface.
Principal stress maps of (a) silica and (b) soda-lime glasses under a ball indenter. The origin of each stress map corresponds to the contact point between the indenter and the glass. The positive values (red) and negative values (blue) indicate tensile and compressive stresses, respectively. Fine black lines denote directions of principal stresses.
As a preliminary test, the indentation test at a maximum load of 0.5 N was performed on silica glass using the same ball indenter. After the test, no permanent imprint was left on the glass surface. This means that silica glass responds elastically under this loading condition. In order to compare this experimental result with Hertzian analytical solution,11) Hertzian principal stresses and their trajectories were calculated for silica glass using the values of Young’s modulus (1141 GPa) and Poisson’s ratio (0.07) of the diamond indenter. These elastic parameters of the indenter are widely used in the analysis of indentation test data,12) and these values are originally from the experimental data.13) The Hertzian stress map is shown in Fig. 5. In Fig. 5, the tensile stress is observed on the top of sample surface near the contact edge about r = 6 µm. This surface-localized tensile stress is also observed in the experimental result (Fig. 4(a)). Therefore, it can be said that the photoelastic stress field in an elastic regime is in reasonable agreement with Hertzian analytical solution.
Principal stress map of Hertzian stress of silica glass (Young’s modulus and Poisson’s ratio are 73 GPa and 0.17, respectively.) Young’s modulus and Poisson’s ratio of the diamond indenter are 1141 GPa and 0.07, respectively. The radius of ball indenter is 0.05 mm and the indentation load is 0.5 N.
In Fig. 4(b) for soda-lime glass, however, the tensile stress region on loading extends to sub-surface. This enlarged tensile region in soda-lime glass on loading suggests faint plastic deformation of this glass. Using the Hertzian analytical solution, the maximum contact pressure, σmax can be determined from the equation below:11)
| \begin{equation} \sigma_{\text{max}} = (3P^{1/3}/2\pi)(3R/4E^{*})^{-2/3} \end{equation} | (3) |
It is considered that the compositional variation of the stress field during unloading comes from the difference in the mechanism of the plastic deformation. It has been known that plastic deformation of glass at room temperature can be divided into two, densification and shear flow.15–17) The plastic deformation of silica glass is primarily dominated by densification, or volume shrinkage, because silica glass has larger free volume as compared with other glass compositions. On the other hand, soda-lime glass under the indenter prefers shear flow which is a volume conservative deformation process. This is because soda-lime glass has less free volume and more non-bridging oxygens. Under the indenter, shear flow of soda-lime glass would occur in a way of breakage and recombination of bonds in glass. According to the expanding cavity model, volume conservative flow of glass under the indenter causes volume strain around the plastic zone.5,18–20) Therefore, it is considered that such a difference in deformation mechanism leads to a difference in photoelastic stress field under the indenter.
It can be expected that these different stress fields result in different crack patterns between silica and soda-lime glasses. In order to evaluate the crack pattern of these glasses, the indentation tests at a larger load were performed. Figure 6 shows optical microscope images of the imprints on these glasses. The indentation maximum load was 5.0 N, which was larger than the load for micro-photoelastic observations in this study. Silica glass shows only ring cracks (Fig. 6(a)), whereas soda-lime glass shows radial cracks accompanied with ring cracks (Fig. 6(b)). Both glasses show ring cracks, because the stress maps of both glasses on loading show the tensile stress regions near the surface (Fig. 4(a) and (b)), and because the tensile stress acts in radial direction. During unloading, on the other hand, the tensile stress at the bottom of the plastic zone in soda-lime glass assists the opening of median/radial crack. This is the reasons why only soda-lime glass shows radial cracks around the imprint.
Optical microscope images of indentation imprints on (a) silica and (b) soda-lime glasses after the ball indentation at 5.0 N.
The indentation-induced stress fields of both silica and soda-lime glass were quantitatively determined by using a micro-photoelastic measurement technique. The obtained photoelastic stress fields in an elastic regime are in reasonable agreement with Hertzian analytical solutions. It is also found that the indentation-induced stress filed of soda-lime glass during unloading was different from that of silica glass. This results from different deformation mechanisms of these glasses under the indenter, and results in different crack patterns after the indentation.
The authors would like to thank Dr. S. Semjonov at Fiber Optics Research Center, RAS, Russia for drawing silica fiber. The authors are also grateful to Hinds Instruments, Inc, U.S. for donating the equipment, LC-PolScope, and to Nippon Electric Glass Co., Ltd., Japan for financial supports. One of the authors (SY) acknowledges the JSPS KAKENHI grant Number 16K06730.
