* l W C y d e S
m IOAS cycles A 5.1 0% Cycles X 10% Cycles
*
Io'cyctesI
Figure 9. Flexural strength as a h c t i o n of Hertzian contact stress.
Figure 10. Fracture Surfhce of Flexwe Bar of Si3N4 subjected to Long- Term Spherical Indenter Cycling Test (10,000,000 cycles; 5 GPa). The vertical arrows indicate the width of the crack and the horizontal arrow indicates the depth of the crack fiom the surface.
The samples in the high failure stress range failed away fiom the Hertzian damaged area. The stress level and number of cycles were not large enough to create a critical flaw larger than the pre-existing cracks due to surfke preparation at that location. Therefore, the samples simply failed fiom the flaws that were already present in the material. An example of the k t u r e surfaces showing the failure origins for bars that have been subjected to long term cyclic loading (10,000,000 cycles at 5 GPa) are shown in Figure 10. Note that the arrows provide a guide to the fiacture initiating crack size. The vertical arrows are the width (2 b) of the crack and the horizontal arrow estimates the depth (a) of the crack. These fiacture surfaces are representative of those bars that hiled fiom
"~tural" and indentation cracks. However, the combination of contact stress above approximately 12 GPa and above approximately 105 cycles was such that a critical flaw was created larger than the pre-existing flaws. Thus, the cracks fiom the cyclic loading caused the low hilure stress. Therefore, the Hertzian damaged area became the site of the critical flaw and was the source of Wure. The size of the failure-inducing flaw depended upon both Hertzian contact stress as well as the number of cycles. There is a threshold value of contact stress of 10-12 GPa that exists for surfaces that are finished to bearing ball quality based on the present results as well as research results in the literature.
To determine a critical flaw size, the measured flaw size was compared with a calculated flaw size. The calculated flaw size was obtained using a fracture mechanics equation l0:
where Y is the crack and loading geometry constant, and c is the equivalent semi- elliptical flaw size measured on the fiacture surfke and is given by:
where a and b are the depth and half-width of the equivalent semi-elliptical flaw.
The value of Y is the geometric and load factor, which has a value of 1.65 for cracks with local residual stress (fiom indentation) and 1.24 for cracks without local residual stress. The average toughness values of Si3N4 were estimated by measuring the crack sizes on the fiacture surhce of pre-indented bars and bars fiactured in four-point flexure. The average value was determined to be about 7.2 In order to try and establish a relationship between the ring crack diameter and the critical crack size, both were measured on the bars that hiled after cyclic loading. An example of failure fi-om a ring crack is shown in Figure 11. The MPam?
crack that caused failure is similar to those shown in Figure 10. However, the origin of the crack is at the edge of the ring crack.
The above equations are based on fiacture mechanics and have been adapted to predict the size of the critical crack for damage in cyclic loading. By combining experimental results of damage accumulation with finite element modeling of cracks in a bearing load simulation, we have been able to determine the size of the critical crack. Finite element analysis (FEA) allows us to change the angle and size of the crack in realistic geometric loading conditions.
Combining the experimental and analytical approaches provide more reassurance that the size determined is reasonable for bearing applications.
Figure 11. SEM Micrograph of the contact surface (upper micrograph) and hcture s h e (lower portion of micrograph) of NBD 200. The arrow points to the fiacture origin at the edge of the ring crack.
Notice the Hertzian-type crack to the right on the fiacture surface (black line fiom the surface at an angle of about 22 degrees fiom the horizontal.
The fiacture 6 e s of all of the bars fractured in the above cyclic loading test were examined and the crack sizes at the origin measured. The actual flaw sizes are graphed versus the calculated flaw sizes in Figure 13. The open circles in Figure 13 indicate the specimens that were subjected to Hertzian cyclic loading (as indicated in Figure 9). However, only the three with crack sizes larger than 150 microns failed fkom the Hertzian damage. The others fkiled fiom damage as a result of the surface finish. Note that bars with no contact damage (solid triangles) had the same flaw size as the open circles at
-
80 microns. The solid diamonds show the crack sizes fiom the strength indentation tests forcomparison, and the verification of the technique. Thus, the indentation technique results in cracks of similar magnitude as the Hertzian damage for the load levels used in the present study. The results of these tests imply that cracks smaller than about 150 microns will not result in damage due to contact fatigue.
Some explanation of what is meant by crack sizes is in order. The crack sizes above are the equivalent semi-circular cracks, which cause failure in flexure (Fig. 10). They are approximately perpendicular to the surfhce into the depth of the specimen, whether a plate or a ball. Note that for Hertzian type cracks, the cracks propagate into the depth at an angle of about 22” fiom the vertical, if they are loaded perpendicular to the surfkce. This angle change will most likely affect the Y parameter in Equation 5. The cracks, which can be inspected, are approximately in the plane of the surface extending down into the surface.
Typically the Hertzian type cracks are circular or circular arcs on the surface.
Thus, it is necessary to develop a relationship between the “C-crack” or circular crack that can easily be observed and the in-depth cracks that lead to damage.
The arrow indicates the location of the ring crack and failure origin on the fiacture surfhce. The results for the comparison of ring crack size and the size of the crack on the Eracture surface are presented in Figure 12. The measured crack sizes of the silicon nitride bars that were fiactured in four-point flexure are shown in Figure 13 as a function of the calculated crack sizes. The cracks were a result of several treatments. The treatment designated as “Hertzian” means that the bars were subjected to cyclic loading at various stress levels before testing. They all failed fiom cracks associated with the surface finish of the bars. The bars designated with “Vickers” means that the bars were indented with a Vickers hardness diamond to produce controlled semi-elliptical cracks. The samples designated as “No Indent” are representative of the cracks produced by the surface treatment. All of the measured crack sizes below 150 microns failed fiom the surfhce finish and not fiom the other surfhce treatments. The cracks below 150 microns are all about the same size indicating the distribution of random flaws associated with the surface finish. Since the semi-circular crack is a radius, this means that the surface trace would be expected to be at least twice that size, i.e., 300 microns. An equation ( Equation 5) was adapted to be used to determine the critical crack size for resistance to cyclic contact loading. In using this equation, it was determined that a crack size of about 300 microns on the surface, i.e., 150 microns in depth can be tolerated without failure fiom contact stresses for (pressure) loads less than 10 GPa. There is a threshold contact stress level below which failure fiom contact damage will not dominate the fhilure. The value of this threshold depends on the surface finish on the material. For surface finishes suitable for bearing applications the threshold contact pressure is about 10 GPa.
The information obtained here along with the finite element analysis and rig tests
should be sufficient to provide a critical crack size methodology for the non- destructive evaluation of silicon nitride balls.
Figure 12. Crack size as a h c t i o n of ring crack diameter.
This last example of the development of a flaw size methodology for the hybrid bearing demonstrates one of the many different processes that must be addressed before a material and structural design is implemented. After the size of the crack is determined, then the appropriate NDE techniques can be implemented. In addition, pilot scale testing, in bearing systems, must be performed. After these pilot scale tests are successful with their inherent in situ monitoring of the stresses and strains, then large-scale tests of bearing systems are implemented.
This
process of concept, testing, selection, development and insertion may take 10 years, if there are no major complications. In a for-profit company, this is a major investment. Thus, the input of the materials engineer at the onset is very important. In addition, it is the team of engineers, economists, business consultants that are necessary for the final decision to proceed with a new material and structural design.Figure 13. Measured vs. calculated crack sizes for fracture of flexure bars.
These two examples were selected to demonstrate that a combination of stress analysis, fiacture mechanics and material selection principles to form a material design methodology. They also show that successful structural designs are possible using toughened ceramics. However, this success is dependent on a team effort. The materials engineer is just one, albeit a critical one, of the team members.
ACKNOWLEDGEMENTS
The author thanks the many colleagues, co-workers and students who have contributed to the work discussed in this paper. I especially want to acknowledge the former students who participated in the hybrid bearing program:
Dr.
Zheng Chen, Ms. Tammy Simpson and Gary Ross. The help of Dr. Ken Anusavice,Dr.
Alvaro Della BOM and Dr. Thomas J. Hill with dental biomaterials information and slides is gratefilly acknowledged. The support of DARPA and AFOSR on part of this work is also gratefilly acknowledged. Dr. Herb Chin has been inspirational in his guidance in the design methodology for a high performance
component. I also thank Pratt and Whitney and St. Gobain Industries for the use of testing facilities and materials, respectively.
REFERENCES
‘M. F. Ashby, Engineering Materials Design, Cambridge University Press, Cambridge, UK 1997.
2Mecholsky, Jr., J. J. 1996 Fractography, Fracture mechanics and fiactal geometry: An Integration; in Fractography of Glasses and Ceramics 111, Varner, Frechette and Quinn, eds., American Ceramic Society, Westerville, OH pp. 385- 93.
3Anusavice, K. J., (1996) Phillips’ Science of Dental Biomaterials, lO* ed.
Philadelphia: W.B
.
Sauders.4Tsai, Y. L., Petsche, P.E., Anusavice, K. J., Yang, C. (1998), Influence of Glass Ceramic thickness on Hertzian and Bulk Fracture mechanisms, Int. J.
Prosthodont., 1 1 :27-32.
’Thompson J. Y., Anusavice, KJ, Naman, A, Morris, HF (1994). Fracture S u r h e Characterization of Clinically Failed All-Ceramic Crowns. J. Dental Res.
6Hucklenbroich I, Stein G, Chin H, Trojahn W, Streit E, (1999 )High nitrogen mmtensitic steel for critical components in aviation
,
Materials Science 7Z. Chen, J. Cuneo, J. J. Mecholsky, Jr. and Hu, S.F., “Damage Process of*B. Lawn, “Fracture of Brittle Solids”, Second Edition, University Press, 9H. Hertz, “Miscellaneous Papers by H. Hertz”, London, Macmillan,
‘OK. Johnson, “Contact Mechanics”, Cambridge University Press, A. Higdon, E. Ohlsen, W. Stiles, J. Weese, W. Riley, Mechanics of 12Gundepudi, MK, Sankar, BV, Mecholsky, JJ and Clupper, DC (1997) 13Hu, SF, Chen, 2, Mecholsky, JJ (1996), On the Mertzian Fatigue Cone 14D. Broek, “Elementary Engineering Fracture Mechanics”, Sijthoff and 73 : 1824-32.
F oV. 318- pp. ~ 161-166
SIN4 Bearing Material Under Contact Loading”, Wear 198, 197-207 (1 996).
Cambridge, (1 995).
(1 986).
Cambridge (1985).
Materials, Fourth Edition, John Wiley and Sons, New York (1 985).
Powder Technology 94(2) 153-161.
Crack Propagation in Ceramics, International J. of Fracture 79 (3) 295-307.
Noordhohh, Alphen aan den Rijin, Netherlands (1978).
11
PROBLEMS
1. Calculate the expected crack size for a material to be used for dental crown veneer if the fiacture toughness, &, is 0.7MPam” and the expected applied stress was determined fiom finite element analysis to be 21 MPa.
2. Determine the maximum contact stress in an elastic plate of silicon nitride, which is in contact with a 2.5 cm diameter spherical indentation of WC-Co loaded at 1OON. [ Si3N4 Properties: KC = 6 MPam’’, E = 600 GPa, v = 0.24; WC-CO Properties: & = 12 MPamln, E = 700 GPa, v = 0.3.1
3. There are generally considered to be three stages of the wear process in well- polished materials. Stage I is the development of surhce relief, Stage I1 is the production of the Hertzian cone crack and Stage I11 is the removal of material.
Why would a rough surface make a difference in the order of the stages, i.e., Stage I1 would occur before Stage I?
4. (a) Explain the shape of the graph in Figure 9 in terms of damage to the surfhce of the material experiencing the contact.
the graph?
(b) How would the quality of the surhce finish on the contacted surface affect 5 . The graph shown in Figure 13 has three different types of cracks. Explain the source of these cracks, i.e., what caused the cracks? If the surfhce finish was improved (less rough) how would this change affect the graph shown in Figure 13?
6. How does the mesh size in a finite element analysis affect the calculation of stresses?
7. In a finite element analysis grid, should there be more or fewer mesh units near the corners and surfaces? Why?
8. There are two critical regions associated with a crown contacted by hard material as shown in Figure 3. What material properties and geometric changes will affect the value of stress in these regions? Will the stresses increase or decrease in each region with the changes? Why?
ROLE OF THERMAL EXPANSION AND CONDUCTIVITY IN DESIGN D. P. H. Hasselman and K. Y. Donaldson
Department of Materials Science and Engineering Virginia Polytechnic Institute & State University Blacksburg, Virginia 24061 USA
ABSTRACT
A discussion, illustrated by many numerical examples, is presented on the role the thermal properties of ceramics play in a variety of engineering applications, including the accommodation of thermal expansion, thermal insulation, heat storage, and thermal management.
INTRODUCTION
Many applications of ceramic materials involve high temperatures andor significant levels of steady-state or transient heat flow, which can present significant challenges in selecting appropriate materials for a given design. For example, a material may be required to optimize heat-flow, a criterion critical for heat exchangers. Other applications may require that heat losses be minimized, as in the design of buildings, furnaces, ovens and kilns. Some applications may require that prescribed temperatures not be exceeded, such as in the design of solid-state electronics and propulsion systems such as internal combustion or turbine engines.
Thermal expansions may need to be limited or at least matched for multi-component systems made up of a number of different materials with differing thermal expansion behavior.
It is the purpose of this paper to present a number of specific examples of materials selection and design for a range of thermal environments and performance criteria. At the risk of over-simplification, each example will be discussed in the briefest manner. The sum total of these examples, however, will serve to demonstrate the complexity of the selection of ceramic materials for thermal applications. No attempt is made to present an exhaustive review of the literature on heat transfer. A
listing of appropriate text books for further information is included at the end of this chapter.
ACCOMMODATION OF THERMAL EXPANSION
Structures designed for operation at elevated temperatures, such as kilns, fiunaces and ovens, generally consist of individual components with different compositions and associated property values. Since such structures are usually assembled at room temperature, bringing them to the required operating temperatures will involve dimensional changes due to thermal expansion, which will differ from component to component. In order to avoid structural failure due to imposed tensile or compressive stresses, thermal buckling or other causes, the thermal expansions of the individual components need to be accommodated freely, without being constrained by neighboring components.
A well-known example of this problem is the use of silicon carbide heating elements in the construction of electrically heated laboratory fiunaces. Generally, such heating elements are quite long and extend through the furnace walls to allow attachment to an electric power supply outside of the furnace. The strength of such elements is not particularly high, and furthermore silicon carbide is quite brittle.
Accordingly, the strains-at-hcture of these elements is quite low, implying that even minor constraints to their free thermal expansion may well lead to premature failure.
Also, their resistance to creep is quite high so that any stresses induced by constraints to thermal expansion are not easily accommodated by creep. Accordingly, these furnaces are generally constructed to allow adequate free space between the heating elements and the associated openings through the furnace walls so the heating elements can slide freely through the openings over the total range of operating temperatures. Any heat losses fiom the b a c e through the gaps between the heating elements and the furnace wall can be reduced by packing the gaps loosely with a thermally insulating refractory wool.
The design of a fuel rod in an uranium oxide-based nuclear power reactor is a unique example illustrating the need to accommodate unequal thermal expansions.
Uranium oxide in the form of right circular cylinders with a diameter of some ?4 inch by some ?4 inch high, referred to as pellets, is the usual fuel source. These pellets are inserted into a vertical zircalloy metal tube of ten or more feet in length. During operation the interior core of the fuel pellet may reach temperatures as high as 1600
"C or greater while the surface temperature reaches about 800 "C. For simplicity let's assume the mean temperature of the pellet is about 1000 "C. The zircalloy tube is surrounded by circulating pressurized cooling water which removes the heat generated within the nuclear pellet. During full-power operation the zircalloy tubing operates at a temperature near 600 "C. Not only do the pellets and surrounding tubing
operate at a different mean temperature, but they also exhibit different thermal expansion characteristics. Insertion of the fuel pellets into the tubing takes place at the same (i. e., room) temperature. The differences in thermal expansions between the fuel pellet and surrounding zircalloy tubing upon reaching the operating temperature need to be accommodated by careful control of the outer dimension of the fuel pellet and the inside diameter of the zircalloy tubing.
As a simple numerical example, let us assume that at full-power operation of the reactor the fuel pellet and the zircalloy tubing are in direct contact. The diameter of the pellet in this case is identical to the value of the inner diameter of the zircalloy tube. We wish to calculate the maximum diameter of the pellet and minimum inside diameter of the zircalloy tube at room temperature prior to assembly so that they are just in contact at operating temperature.
The coefficient of linear thermal expansion, a , of a solid is defined by:
where (AL) is the change in length, L, is the original length and (A T) is the change in temperature. In effect, equation 1 defines the coefficient of linear thermal expansion as the thermal strain per unit temperature difference.
Equation 1 can be rewritten in terms of the change in length:
( A L ) = a L o ( A T )
Writing
L
= L,+
(A L), yields:L = L o ( l + a ( A T ) ) (3) For the current example, L = ?4 inch. Let us assume that the insertion of the pellet into the tube occurs at an ambient temperature of 25 "C.
With an assumed value for the mean operating temperature of 1000 "C, (A