By changing the limit of integration within the zone where fracture occurs in brittle glassy materials a modification of the equation for theoretical strength can be achieved

**Introduction**

In a previous article on float glass technology in Kanch[1], apart from various chemical aspects, the problem of thickness variations was discussed. However, the mechanical properties in terms of fracture of such glasses are also of great importance, since these glasses are mostly used in the building industry and façade decoration purposes, wherein fracture in any form can cause damage to the installation, which will result in a loss to the investment made. Although the present article deals with the problem of fracture in brittle materials such as glasses, importance has to be given to float glasses, as these glasses have aesthetic appeal with the highest quality requirements as far as the glass industry is concerned.

A tremendous amount of work has been carried out in the field of fracture mechanics in brittle materials and glasses, particularly on their theoretical strength over long periods of time[2-8]. This is mainly carried out using the Griffith equation based on the formation of an elliptical crack, and its main plank is the utilization of material parameters or constants, which are measurable, in designing suitable materials for various important applications[2]. An insight should be obtained on the nature of theoretical strength based on a sinusoidal approximation in the stress versus spatial elongation curve (see Figure 1 in the next section for clarity). In the context of this approximation, in the present article, our main focus is on the ...

**Introduction**

In a previous article on float glass technology in Kanch[1], apart from various chemical aspects, the problem of thickness variations was discussed. However, the mechanical properties in terms of fracture of such glasses are also of great importance, since these glasses are mostly used in the building industry and façade decoration purposes, wherein fracture in any form can cause damage to the installation, which will result in a loss to the investment made. Although the present article deals with the problem of fracture in brittle materials such as glasses, importance has to be given to float glasses, as these glasses have aesthetic appeal with the highest quality requirements as far as the glass industry is concerned.

A tremendous amount of work has been carried out in the field of fracture mechanics in brittle materials and glasses, particularly on their theoretical strength over long periods of time[2-8]. This is mainly carried out using the Griffith equation based on the formation of an elliptical crack, and its main plank is the utilization of material parameters or constants, which are measurable, in designing suitable materials for various important applications[2]. An insight should be obtained on the nature of theoretical strength based on a sinusoidal approximation in the stress versus spatial elongation curve (see Figure 1 in the next section for clarity). In the context of this approximation, in the present article, our main focus is on the “fluctuation” of the spatial elongation value at or near fracture so that a modified equation can be developed to better predict the theoretical strength of glasses.

Although the subject of much research over the past decades, the fracture of brittle glassy materials remains in many ways not understood. Of particular interest is the mechanism by which energy in the system is dissipated. Experimental measurements of the flow of energy into the tip of a running crack have indicated that the fracture energy (i.e., the energy needed to create a unit extension of a crack) is a strong function of the crack’s velocity and that the majority of the energy stored in the system prior to the onset of fracture ends up as heat. An example of the fracture of soda-lime-silica glass has been taken into consideration. Residual stress profiles were introduced in sodium aluminosilicate glass disks using an ion-exchange process, i.e. after chemical strengthening. They were fractured in two loading conditions: indentation and biaxial flexure. The fractal dimension of the macroscopic crack branching pattern called the crack branching coefficient (CBC), as well as the number of fragments (NOF) were used to quantify the crack patterns. The fracture surfaces were analyzed to determine the stresses responsible for the crack branching patterns. The total strain energy in the body was calculated. The CBC was a good measure of the NOF. They are directly related to the tensile strain energy due to the residual stress profile for fractures due to indentation loading. However, in general for materials with residual stresses, CBC (or NOF) is not related to the strength or the stress at fracture, or even to the total stored tensile strain energy. A study was carried out to determine the geometric characteristics associated with the critical crack caused by cyclic loading in baria silicate glass[9] (see the references therein for other useful references). Next, we show the theoretical side of the story.

**Theoretical development**

The theoretical strength of a ‘body’ is the stress required to separate it into “two parts”, with the separation taking place simultaneously across the cross-section. To estimate sTh, let us consider ‘pulling’ on a cylindrical bar of unit cross section.

The “force of cohesion” between the two planes of atoms varies with their separation, after the interatomic spacing.

A part of the curve is approximated by the sinusoidal relation [3]. This equation represents the so-called governing equation of stress (σs) against the spatial elongation (X). The work per unit area to separate the two planes of atoms is then calculated by the integral of the curve between X = 0 and X = ll/2.

This work or energy is then equated with the surface energy (2gϒ) of the two newly created surfaces.

Typical values of E = 3 x 1011 dynes/cm2, gg= 103 ergs/cm2 and a = 3 x 10-8cm, and sσTh= 1011 dynes/cm2 as per equation (4). If ll≈a, then we can show sσTh varies from E/5 to E/10. For window glass, the strength is 104 psi, σsTh = E/1000, and for alumina ceramics, the strength is 5 x 104 psi, σsTh = E/1000.

It has to be noted here that alumina is an important component in the window glass composition, as suitable mechanical strength is desired by the consumers.

Therefore, between the theoretical predictions and the actual experimental values, there is a discrepancy, which needs to be solved. It should be mentioned that involving material’s constants (E and gg) and half of the elliptical crack length (c), Griffith’s criterion of the maximum strength at which the material fails on cracking is based on the above equation. Hence, this equation certainly merits careful attention. Moreover, in line with Griffith’s concept of micro-flaw formation, the reduction of theoretical strength also merits further attention. Flaws in this kind of brittle glassy materials may not have the nature of classical Griffith micro cracks, but may rather take the form of embryonic defects with intensely concentrated residual stress fields[9,10].

In the above mathematical formulation, the limits of integration have been taken between 0 and ll/2, i.e. the work or energy is calculated up to a limiting point (i.e. l/2) before which the material has already cracked, whereas linear Hook’s law has been applied to the other end, i.e. at X = 0, when rrσ = 0. It should be clearly mentioned that the value of s is maximum, which is the value of sσTh, at X = ll/4.

In this theoretical development, we are inclined to take a ‘small variation’ around this value of X around ll/4. Let us assume that this variation is dd, i.e. the maximum strength can be assumed to be arrived at (ll/4). A brittle material cracks at or just after sσTh, we can do the integration of equation (1) up to a limit of (ll/4 +dd) for the energy formulation in order to be able to be equated with the surface energy (2gg), instead of extending it up to ll/2, when the material or glass has already cracked or fractured. However, for linear Hook’s law, it can be easily applied at (ll/4 −dd), when it is perfectly possible to differentiate s at X = (ll/4 −dd), which was carried out at X = 0 as in equation (3). Therefore, the basic tenets of these two approaches are clear from the above. Under the above assumption, we find the total work carried out and equate with the surface energy of the newly created surfaces due to fracture. After some differentiation and mathematical work-outs, we arrive at the modified Griffith equation.

Therefore, it is clearly seen that our above equation will put an incremental effect on the theoretical estimate of maximum strength with respect to a simple Griffith criterion (Elg/a) 1/2 involving the material parameters, with “a” replaced by half of the elliptical crack length. It is known for a long time that Griffith criterion of predicting and eventually designing the right materials, only through measurable material properties like surface energy (g) and elastic modulus (E), has been very popular, since the equilibrium interatomic distance (a) is approximately known for glasses.

**Results and discussion**

It should be pointed out that if we put d/l to be much less than 1/4, then it would be possible for us to predict the correct theoretical strength of brittle glassy materials. Therefore, this equation (5) can be used to precisely do this prediction by adjusting the value of d/l. For example, for three different values of d/l = 0.1, 0.01, 0.001, we have to multiply Griffith value (under the square root sign) with 1.40, 5.47 and 17.10 respectively. In the literature, very often, there is a factor of √2 in the Griffith’s value. In the first case, our assumption of taking the value d/l = 0.1 gives rise to a multiplication factor of 1.40 (close to 1.414 = √2).

The above treatment will help us analyze a variety of materials with different values of the ratio of d/l (non-dimensional value) to fit the experimental value with that of the theoretical estimate. Since both the values of d and l are not measurable, it is always better to take a ratio to estimate the strength as per equation (5).

Let us take the example of a common glass, where the value of sσTh is 14 GPa as per equation (4), but as the experimental values are always lower, Griffith[2] put forward a new equation of sσ = (2Egg/PL) 1/2, where L = length of the micro flaws, which were considered to reduce the strength, as in many other brittle materials. As per this revised equation, Griffith[2] postulated that even micron (10-6 m) sized flaw could reduce the observed strength of the glass by a factor of 100. Thus, the ratio d/l is to be still lower, and the multiplication factor is higher. Actually, this ratio clearly dictates the presence of micro-flaws.

Finally, an example of fused silica is given here, as we normally try to understand its behaviour with that of float or other glasses. The parameters are: gg= 1.75 J/m2 and E = 72 GPa and taking a = 1.6 x 10-10 m, we find a theoretical value of strength as per equation (4) as 28.1 GPa, whereas the experimental value is 24.1 GPa. The close similarity of these values clearly indicates that it does not take the ‘micro-flaws’ into account. The theoretical value should be much higher. By multiplying the Griffith value with 1.40 (i.e. d/l = 0.1), we estimate the strength value as per equation (5) as 39.34 GPa.

This discrepancy (or even more discrepancy) will actually justify the presence of the ‘micro-flaws’ in fused silica, which is a known fact[7, 10]. However, an analysis can be based on the estimated value of strength as 28.1 GPa.

As per the revised equation of Griffith involving micro-flaws, if we take the size (L) of the flaws at the quantum level, i.e. the value of “a” in equation (4), the theoretical strength goes down to 22.39 Gpa.

As the size of the flaw increases to a level normally considered in the micron level, the value goes down by a factor of 100, i.e it becomes 0.2239, as also mentioned above. This necessitates the inclusion of the ratio d/l in the calculation of theoretical strength, which should also be in consonance with the data on fused silica on the probable flaw size.

The concept of micro flaws needs to be introduced, which is calculated from our equation (5) taking smaller values of d/l ratio. It is seen that as the level of micro flaws goes to a ‘usual’ granular level, the value of d/l ratio becomes still smaller, and the need for a higher multiplication factor.

It is pertinent to mention that although the data for fused silica are fitted here, the information given above can be obtained on a variety of other ceramic brittle materials in order to be able to explain the discrepancy between theoretical and experimental values of strength for effective design.

Here, we have calculated theoretical strength of different materials at the flaw size of 1.6×10-10 meter using the equation (4) and are trying to show that how theoretical strength of materials (σTh) changes with fluctuation as per Equation (5).

It is always true to the fact that different materials have different theoretical strength (sσTh). It is clear from the equation (5) that with an increase in the ratio, there is always decrease in the theoretical strength of the material. Thus, the characteristics of different materials are shown in Figure 2. Except NaCl, all the materials are used in the glass industry and their comparison with the data of fused silica makes sense in understanding the overall mechanical behaviour of commercial glasses.

It is clear that the different materials of the same ratio have different value of theoretical strength.

The theoretical strength of magnesia (MgO) is at the highest level and its value is 64.23 GPa. The next material is alumina (Al2O3), then glass, fused silica and sodium chloride (NaCl) respectively. The material having the lowest theoretical strength is sodium chlo-ride (NaCl) whose theoretical strength value is 18.36 GPa.

**Conclusion**

The modification of the basic equation on theoretical strength has been achieved, within the context of a sinusoidal approximation in the applied stress vs. spatial elongation curve[3], by assuming a small spatial variation and by changing the limit of integration in the energy formulation for crack formation. This modification yields a ratio of this variation, giving rise to a multiplication factor, which can correctly predict the theoretical strength of brittle ceramic materials. The available data on fused silica has been fitted with this new model and found to be effective in explaining a lower observed strength due to the presence of micro-flaws. Many such data on other brittle ceramic materials can be fitted in future to give it a comprehensive shape.

Acknowledgements

Acknowledgements

The authors would like to thank Prof. Loc Vu-Quoc of Department of Mechanical Engineering at University of Florida (US) for many stimulating discussions during the preparation of this manuscript.

**References**

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2. E. Orowan, Inst. of Engg. & Shipbuilders in Scotland, 42 (1945-6) 165

3. A. A. Griffith, Phil. Trans. of Royal Society, Series A, 221 (1920) 163

4. W. D. Kingery, H. K. Bowen and D. R. Uhlmann, Introduction to Ceramics, John Wiley & Sons, New York (1976) pp. 783

5. S. M. Wiederhorn, J. Am. Ceram. Soc., 50 (1967) 407

6. G. R. Irwin, Encyclopedia of Physics, Vol. IV, Springer-Verlag, Berlin (1958) pp. 551

7. W. B. Hillig and R. J. Charles, High Strength Materials, Ed. V. F. Zackay, John Wiley, New York (1965) pp. 682

8. A. K. Varshneya, Strength of Glass in “Chemistry of Glasses”, Ed. A. Paul, Chapman & Hall, London (1982) pp. 140

9. K. Supputtamongkol, K. A. Anusavice and J. J. Mecholsky, “Fracture mechanics analysis of crack shapes by cyclic loading in Ba-silicate glass” J. Mater. Res. (2007)

10. A. K. Bandyopadhyay, “Nano Materials (Chap 5)”, New Age Sci., (Tunbridge Wells, Kent, UK, 2010)

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