Fundamentals

Good Reference for Fracture Mechanics of Ceramics

Ceramics are an important material for precision engineers.  Ceramics are often used for their low coefficients of thermal expansion, high Young’s modulus, as well as other properties.  The text Ceramics: Mechanical Properties, Failure Behaviour, Materials Selection by Dietrich Munz and Theo Fett (google books link, amazon.com link) is a very good reference for ceramics and for their failure in particular.  The text derives formulas for lifetime (time to failure) under constant and cyclic loads.  The text also provides extensive information on materials testing, statistical methods (Weibull distributions), and probability of fracture for ceramics.

One topic that is particularly interesting and well written is the section on subcritical crack growth.  Subcritical crack growth (also called stress corrosion) is the growth or extension of a crack over time with stress intensity factor less than the critical stress intensity factor K_{Ic}.  This is interesting because in classic linear elastic fracture mechanics (LEFM) cracks in brittle materials are typically viewed as stable (no growth under constant load) if the the stress intensity factor K is less than the the critical value K_c.  The text only briefly mentions the underlying cause of subcritical crack growth in terms of chemical bonds breaking in the neighborhood of the crack tip.

The crack growth is given by

    \[ v=\frac{da}{dt}=A^{\ast} \left(\frac{K_I}{K_{Ic}} \right)^n \]

where v is the crack growth velocity with respect to time, a is the length of the crack, t is time, K_I is the stress intensity factor, K_{Ic} is the critical stress intensity factor, and A^\ast and n are constants based on the particular material, temperature and environmental conditions.  Another common parameter is B which is given by

    \[ B=\frac{2 K_{Ic}^2}{A^{\ast} Y^2 (n-2)} \]

where Y is a constant related to the geometry, typically of order 1, with value \sqrt{\pi} for and infinite plate.

Zerodur (manufacturer Schott’s website) is one of the most commonly used ceramics or glass ceramics  by precision engineers.  Subcritical crack growth is an important consideration for Zerodur in the presence of water or water vapor.  The term stress corrosion is often used in this case as many like to think of the water molecules attacking the bonds at the root of the crack.

“Fracture Toughness and Crack Growth of Zerodur”  (NASA Technical Memorandum 4185) by Michael J. Viens, April 1990 (link to document)  gives the stress corrosion constants to be n=30.7 and \mbox{ln} (B)=-6.837 with 100% humidity.  The Properties of Optical Glass edited by Hans Bach and Norbert Neuroth (google books link, amazon.com link) gives the value of n=33 for Zerodur.  They also give the values of n for Duran and soda-lime glass as 22.4 and 18.1, respectively.  The Zerodur technical document by Schott TIE-33 (link) has an extensive discussion, examples, and list of fracture and related properties including stress corrosion.