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Strength of glass

Introduction

A materials is characterized by various kinds of strength. Yielding strength shows the elastic limit of a material (see yield). Ultimate strength shows the material's ability to resist fracture. And fatigue strength shows the material's property to resist cyclically environmental change. Generally, glass behaves like brittle materials (see brittleness) when it subjected to forces. That is, a glass material's yielding strength is close to its ultimate strength. In other words, one cannot see significant deformation after yielding before the failure of glass materials. This is the representation of the lack of toughness of brittle materials. In addition, brittle materials tend to fail under tension. The ultimate strength of glass might be seen as the tensile strength in this case (see ultimate tensile strength). In the following contents, if not specified, strength means the tensile strength. Ultimately, glass fails in repeatedly varying environment even if the loading is far from its tensile strength, and this is the topic of fatigue strength.

One feature distinguished the strength of glass is that, compared to typical ductile material, such as metals, glass exhibits larger variation of strength in terms of its thickness^[1].

The strength of materials is largely influenced by the flaws in the body ^[2]. Let’s first review the strength of a flawless material.

Theoretical strength

The theoretical strength represents the cohesive energy of the materials. That is, the energy needed to create two new surfaces in the material. This is first calculated by Orowan^[3] as following.

The minimum stress to overcome the interatomic forces in order to create two new surfaces is

$\sigma _{f}={\sqrt {\frac {\gamma _{s}E}{r_{0}}}}$

where $\gamma _{s}$ is the specific surface energy of the material, E the Young’s modulus, and r_0 the distance between two atomic planes. For typical values of $\gamma _{s}$ and r_0, the ideal strength can be represented in terms of E,

$\sigma _{f}\approx {\frac {E}{10}}\sim {\frac {E}{3}}$

Strength of flawed materials

The theoretical strength fails to describe the strength of glass in the real world. Take the theoretical strength of glass fibers for example, it is reported in excess of 14,000 MPa. However, the actual strength is only 7,000 MPa ^[4]. Most researchers suggest that this is due to the flaw in the glass. Thus, in the next section we will show the calculation of strength with flaws in the materials, that is, Griffith analysis.

Griffith analysis

One famous model to estimate the strength of brittle materials is presented by Griffith in 1921 ^[5]. Griffith modeled a infinite glass with a elliptical flaw in the middle under tension. As we already known, stress would be intensified at the tip of flaw. Griffith tells us that having stress at the flaw tip exceeded the ideal strength is not sufficient to rupture the material. Also, the flaw must reach its critical length to propagate the flaw then fracture. The estimated strength of flawed glass is

$\sigma _{f}={\sqrt {\frac {2\gamma _{s}E}{\pi c}}}$

where E is Young’s modulus, 2c the longer axis of the ellipse, $\gamma _{s}$ the specific surface energy of the material. If the environment is water rather than air, r will reduce, hence a lower strength can be expected. Generally, a glass may have serval flaws especially at the free surfaces, which are produced during manufacturing or handling process. But the fracture would most likely occur at the largest flaw because the largest c makes the lowest fracture strength. The largest flaw in the body is also known as Griffith’s flaw.

One should note that, E and $\gamma _{s}$ are affected by the composition of glass. However, the influences they make on the strength are less than the crack dimension c does^[6].

In addition, one should note that the dimension of the flaw in in the level of microns, which is difficult to observe even the utilization of optical microscope. Thus, the method to measure the geometry of the Griffith’s flaw is another important topic to estimate the strength of glass. ^[7] ^[8] ^[9] and others have proposed numerous practical methods to overcome the difficulty.

Measured strength

The measured strength^[10] listed below is the support of Griffith's analysis. One can observed that the strength of fibers are higher than of rods. The reason is, it's less possible to have larger Griffith's flaw in the thiner glass compared to the thick one.

Roughly strength level
Type of glass	MPa
Common glass products	14-70
Reshly drawn glass rods	70–140
Abraded glass rods	14-35
Wet, scored glass rods	3-7
Armored glass	350-500
Handled glass fibers	350-700
Freshly drawn glass fibers	700-2100

Fatigue strength

In the circumstances of cyclic loading, sometimes materials fail while the maximum and minimum values of the cyclic loading are much less than the ultimate strength. Researchers in Fracture mechanics explained this phenomenon of failure into three stages. That is, crack nucleation, crack propagation and fracture.

In 1967, Wiederhorn's research on the velocity of crack propagation versus stress intensity factor suggested that there are three stages of subcritical crack growth behavior ^[11]. In stage I, the velocity of crack propagation increases exponentially with applied force and humidity level. In stage II, the velocity of crack propagation is independent to the applied force but dependent to the humidity level. In stage III, the velocity of crack propagation increases exponentially with applied force again but this time does not respond to the humidity level.

In addition, in 1970, Wiederhorn's shows the composition of glass is also relative to the velocity of crack propagation^[12].

References

^ Varshneya, Arun K. (1994). Fundamentals of Inorganic Glasses, 3rd Edition. San Diego: Academic Press. ISBN 0-12-714970-8.
^ Pepi, John W. (2014). Strength Properties of Glass and Ceramics, 1st Edition. Washington: SPIE Press. ISBN 978-0-8194-9836-6.
^ E. Orowan, Z. Krist., A89, 327-343 (1934).
^ Pepi, John W. (2014). Strength Properties of Glass and Ceramics, 1st Edition. Washington: SPIE Press. ISBN 978-0-8194-9836-6.
^ A. A. Griffith, “The phenomenon of rupture and flow in solids,” Philosophical Transactions of the Royal Society of London Series A 221 pp. 163–198 (1921).
^ Varshneya, Arun K. (1994). Fundamentals of Inorganic Glasses, 3rd Edition. San Diego: Academic Press. ISBN 0-12-714970-8.
^ E. N. Andrade and L. C. Tsien, Proc. Roy. Soc. 159A, 346 (1937).
^ F. M. Ernsberger, Proc. Roy. Soc. A257, 213-223 (1960).
^ Y. Bando, S. Ito, and M. Tomozawa, J. Am. Ceram. Soc. 67(3), C36-37 (1984).
^ Varshneya, Arun K, 1994, Fundamentals of Inorganic Glasses, Academic Press, ISBN 0-12-714970-8.
^ Wiederhorn, S. M. "Influence of Water Vapor on Crack Propagation in Soda‐Lime Glass." Journal of the American Ceramic Society 50.8 (1967): 407-414.
^ Arora, A., et al. "Indentation deformation/fracture of normal and anomalous glasses." Journal of Non-Crystalline Solids 31.3 (1979): 415-428.

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