THEORETICAL ASPECTS

4.1. THEORETICAL ASPECTS

There has been a tremendous amount of work done in the field of fracture mechanics in brittle ceramic materials and glasses, particularly on their theoretical strength for a long time [1 - 7]. This is mainly based on 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 vs. spatial elongation curve.

The theoretical strength (σ Th ) 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 σ Th , let us con- sider ‘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 inter-atomic spacing (a). As shown in Figure - 4.1, a part of the curve is approximated by the sinusoidal relation as [3] :

This equation represents the so-called governing equation of stress ( σ) against the spatial elonga- tion (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 = λ/2 as :

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σ Th

a 0 λ/2

Figure 4.1: Applied stress vs. spatial elongation curve.

⎟ d X = σ Th ⎜⎟ ∫ (4.2) 0

This work or energy is then equated with the surface energy (2 γ) of the two newly created sur- faces, giving :

For the initial part of the curve near the equilibrium spacing (a), as in Figure 4.1, the Hook’s law has been expressed as :

where, E = Young’s modulus. For this small part of X of this curve, from equation (4.1), the following relation has been deduced :

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⎟ (4.5) ⎣ d X

⎦ X0 =

This is done at X = 0. By equating this with obtained from equation (4.4), the following

relation has been obtained as :

. σ Th =

and substituting equation (4.6) into equation (4.3), we get :

Typical values of E = 3 × 10 11 dynes/cm 2 , γ = 10 3 ergs/cm 2 and a = 3 × 10 –8 cm, and σ Th = 10 11 dynes/cm 2 as per equation (4.7). If λ ≈ a, then σ Th = E/5 to E/10 as per equation (4.6). For window glass, the strength is 10 4 psi, σ Th = E/1000, and for alumina ceramics, the strength is 5 × 10 4 psi, σ Th = E/1000.

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 γ) 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 mico-flaw formation, the reduction of theoretical strength also merits further attention.

Therefore, changing the ‘limit of integration, and a very small ‘fluctuation’ of δ around the λ value, we arrive at the revised value of the theoretical strength after which the material will crack. This is found out as [8] :

Therefore, it is clearly seen that the above equation will put an incremental effect on the theoreti- cal estimate of maximum strength with respect to a simple Griffith criterion (E γ/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 measur- able material properties like surface energy ( γ) and elastic modulus (E), has been very popular, since the equilibrium inter-atomic distance (a) is more or less known for all the crystalline brittle ceramic materi- als, and even approximately for glasses.

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4.1.1. Data Analysis of Theoretical Strength

It should be pointed out that if we put δ/λ to be much less than 1/4, then it would be possible for us to predict the correct theoretical strength of brittle materials. Therefore, this equation (4.8) can be used to precisely do this prediction by adjusting the value of δ/λ . For example, for three different values of δ/λ = 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 δ/λ = 0.1 gives rise to a multiplication factor of 1.40 (close

to 1.414 ≈ 2 ). The above analysis will help us analyse a variety of materials with different values of the ratio of

δ/λ (non-dimensional value) to fit the experimental value with that of the theoretical estimate. Since both the values of δ and λ are not measurable, it is always better to take a ratio to estimate the strength as per equation (4.8).

Let us take the example of a common glass, where the value of σ Th is 14 GPa as per equation (4.7), but as the experimental values are always lower, Griffith [2] put forward a new equation of σ= (2E γ/πL) 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 can reduce the observed strength of the glass by a factor of 100. In such situations, the ratio of δ/λ has to be still lower so that the multiplication factor is higher. Actually, this ratio clearly indicates the presence of micro-flaws.

Finally, an example of fused silica, whose γ = 1.75 J/m 2 and E = 72 GPa and taking a = 1.6 × 10 –10 m, we find a theoretical value of strength as per equation (4.7) 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. δ/λ = 0.1), we estimate the strength value as per equation (4.8) 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]. However, an analysis can be based on the estimated value of strength as 28.1 GPa.

Table 4.1 : Theoretical strength at different values of flaw size of fused silica [ σσ σσ σ Th = 28.1 GPa as per equation (4.7)]

Multiplication (flaw size in

a=L σ Th estimated

Level of

Optimum Ratio of

Griffith Rev. Equ.

Crack

δ/λ to get 28.1 GPa

Factor needed to

10 –10 m)

(E γ/πL) 1/2

[eqn. (4.8)]

get σ Th

1.6 15.84 Quantum

16 5.01 Nano

1.58 Nano

0.50 > Nano

0.16 Granular

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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 (4.7), the theoretical strength goes down by almost half to 15.84 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, as shown in Table 4.1, as also mentioned above. This necessi- tates the inclusion of the ratio δ/λ in the calculation of theoretical strength, which should also be in consonance with the data on fused silica on the probable flaw size, as shown in Table 4.1.

The last two columns clearly shows the need of a multiplication factor, when the concept of micro flaws needs to be introduced, which is calculated from our equation (4.8) taking smaller values of δ/λ ratio. It is seen that as the level of micro flaws goes to an “usual” granular level, the value of δ/λ ratio becomes still smaller, and the need for a higher multiplication factor arises, as shown in the last column of the above table.

It is pertinent to mention here that although the data for fused silica are fitted here, the informa- tion shown in the above table 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.

In summary, it can be said that the modification of the basic equation on theoretical strength can

be achieved, within the context of a sinusoidal approximation in the applied stress vs. spatial elongation curve, 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 multi- plication 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.

Finally, it is pertinent to mention here that the above modified Griffith equation can be also utilized to determine the theoretical strength, based on certain assumptions of crack length, of sintered compacts of nano-sized silicon carbide and alumina particles. However, the analysis of these materials are done in the usual line of interpretation of fracture strength with the hope that the above formalism might be used by other researchers in the field of nano-materials to derive more information on this emerging field with some degree of refinement, if needed.