# SPECIAL REPORT: REFRACTORIES: Predicting Refractory Failure

Most materials will fracture in severe thermal environments. When designing refractories for such environments, knowing the degree of damage that the material can sustain without complete fracture is crucial. Sometimes termed

*thermal stress damage resistance*, this is often measured by the degradation of mechanical properties based on the severity of the thermal shock.

It is common practice to assess damage resistance by measuring the material properties of the refractories that are contained in thermal stress damage resistance parameters such as Rst.^{1} However, thermal stress damage resistance is also dependent on the number of cracks that propagate during thermal shock, which is not taken into account in Rst. A better measure of damage resistance has been proposed^{2} and is given by:

_{c}= critical temperature difference, C

_{0}= initial crack length and N = crack density (number of cracks per unit cross-section area). The critical temperature difference, ΔT

_{c}, is the critical temperature difference at which fracture is initiated. R"

_{st}, is derived from the 2D Hasselman model

^{1}in which the body undergoing thermal shock is assumed to have N cracks per unit cross-sectional area of initial size C

_{0}.

Of course, C_{0} is difficult to measure directly. However, C_{o} and ΔT_{c}, and thus R"_{st}, can be determined from resonant frequency measurements by measuring the dynamic Young's modulus retained after a series of simple water quenching experiments and by comparing these results to a theoretical model.

## Resonant Frequency

A material will tend to vibrate at its natural resonant frequencies following a mechanical excitation. There are two methods to excite these frequencies:- Sonic resonance,
^{3}in which a driver attached to a frequency generator is used to vibrate the sample (forced vibration). The maximum amplitude is then measured on the sample as the applied frequency sweeps through the resonant frequency. - Impulse excitation,
^{4}in which a small hammer or impulse tool is used to strike the solid at specific locations.

In both cases, a well-defined standing wave is developed in the solid, and both methods yield the same results. For a bar with a uniform cross section, the fundamental flexural frequency of this wave is related to the size, shape, bulk density and elastic properties of the solid.^{5,6,7} All things being equal, the Young's modulus is proportional to the square of the frequency. Although the sonic resonance method was used for the purposes of this article (since it was based on some earlier work), the impulse excitation technique is now generally preferred because of its simplicity, speed and repeatability.^{8}

## Composite Model

A method to determine R"_{st}is presented here in which the observed Young's modulus after water quenching is used as a measure of damage resistance. A theoretical model is then fit to the observed values and R"

_{st}is determined. The theoretical model is described next.

_{d}) will be lower than that of the undamaged core (E

_{u}), as shown in Figure 1. Thus, when 2C

_{f}= W, the whole body is cracked and the Young's modulus of the composite (E) is equal to E

_{d}.

In this analysis(the original study used the Voigt assumption thus: E = E

_{d}.V

_{d}+ E

_{u}.V

_{u}, where V is the volume fraction of the damaged and undamaged regions as indicated by the subscripts d and u respectively), the flexural rigidity of a bar of square cross-section is considered and is given by the product of the Young's modulus (E) of the material and the second moment of area of the cross section (I) about the neutral axis (i.e., the stiffness due to the material multiplied by the stiffness due to shape). For the damaged refractory shown in Figure 1, the total rigidity of the bar is the sum of the flexural rigidities of each component, thus:

The subscripts d and u stand for the damaged and undamaged regions. Based on a theory advanced by Hasselman,

^{1}it has been shown

^{2}that if a uniform distribution of pre-existing cracks propagates simultaneously under the action of stresses induced by the quenching, the damaged layer will extend into the core as follows:

This leads to:

^{2}

The only unknowns in Equation 4 are C

_{0}and ΔT

_{c}; E

_{u}* and W can be measured directly. C

_{0}and ΔT

_{c}are determined simply from resonant frequency measurements. R"

_{st}can then be determined and the thermal stress damage resistance quantified.

*A minor correction was made for the lowering of E_{u}^{2} with soak temperature.

## Refractory Analyses

To demonstrate how thermal stress damage resistance can be quantified using resonant frequency measurements, some results obtained from the earlier study^{2}were reanalyzed. Two types of magnesia refractories were studied: one type was bonded with gels derived from ethyl silicate or with a gel derived from tetra-isopropyl titanate;

^{9}the other type was a commercial, pressed refractory. The gel bonding systems were used because they had been shown to enhance thermal stress damage resistance.

^{10}

Four different grain size distributions, based on Andreasen^{11} size gradings, were used for the ethyl silicate bonded refractories. From fine to coarse size gradings, the refractories were labeled EA1, EA2, EA3 and EA4. The same size grading for EA2 was used in the titanate gel-bonded refractory (T-T). The pressed refractory (PB) was used for comparison purposes, as this material is low density and is known to have good thermal stress damage resistance.

The refractories were thermally shocked by quenching them in water. Bars of 25 x 25 x 128 mm were suspended with platinum wire into the hot zone of an electrically heated vertical furnace. After the samples reached thermal equilibrium, they were dropped into a bath of water flowing at 2 l.min^{-1}. The bars were quenched from soak temperatures of 100-1300°C at 100°C intervals.

The temperature difference, ΔT, was taken to be the soak temperature. Five samples were quenched at each temperature. The extent of the damage caused by the thermal shock was evaluated by measuring the Young's modulus before and after quenching. Although the resonant frequencies were measured using the sonic resonance technique,^{12,13} similar results could have been obtained using impulse excitation.

C_{0} and ΔT_{c} were estimated simply by measuring the Young's modulus after quenching and fitting Equation 4 to the results. A best fit was obtained with Mathematica* 5.1 by using an iterative least squares method. Once C_{0} and ΔT_{c} were determined in this way, then the damage resistance of the refractories could be quantified through the parameter R"_{st} (ΔT_{c}/C_{0}^{3/2}). In addition, the failure temperature difference of the refractories, ΔT_{f}, could be predicted from Equation 3 (putting C_{f} = W/2).

The results of the quenching experiments are shown in Figure 2. As can be seen, the fit from Equation 4 is very good. R"_{st} and ΔT_{f} were determined for each refractory as shown and appear to give a reasonable measure of thermal stress damage resistance, as can be seen by comparing the left columns of the graphs in Figure 2. Generally, the damage resistance increased with the fines content and decreased with increasing mean pore size^{2} (see Figure 3), which is a similar result to that obtained by Carswell.^{14} As might be expected, the theoretical curves deviate most from the measured values when ΔT > ΔT_{f}.

*Wolfram Research, http://www.wolfram.com .

^{2}showed that this was because the ethyl silicate binder produced more second phase, causing the fines to agglomerate. Thus, EA2 had a mean pore size of 34µ vs. 21µ for T-T, even though both refractories had the same grain size distributions.

In this work, the damage resistance of the refractories could have been increased by increasing the fines content and by using the titanate gel binder in place of ethyl silicate.

## Improved Performance

Refractory manufacturers can use the resonant frequency techniques shown here to more easily and accurately measure the damage resistance of their refractories than by measuring traditional parameters such as R_{st}. Armed with this knowledge, manufacturers will be able to reduce development times and improve the quality of their products.

**For additional information regarding predicting refractory failure, contact BuzzMac International LLC at 620 North Sidney Place #104, Glendale, WI 53209; (414) 352-5419; fax (253) 540-9798; e-mail support@buzzmac.com ; or visit http://www.buzzmac.com .**

## Author Acknowledgements

**E. W. Roberts, Ph.D., (Leeds University) and H. G. Emblem, Ph.D., (Zirconal Processes Ltd.) are fondly remembered for their guidance and encouragement of the original work conducted at Leeds University, UK.**