# Design & Form Optimization

September 1, 2011

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In recent years, efficiency improvements in many mechanical systems have become more important, and this situation has introduced new demands on certain mechanical components. Particularly for applications at high temperatures and corrosion and wear rates, the design of mechanical components using advanced ceramics has become very popular.

Advanced ceramics provide numerous advantages compared to other materials: durability, hardness, high mechanical strength at high temperature, stiffness, low density, electrical insulation and conductivity, thermal insulation and conductivity, etc. Most ceramic materials are almost entirely immune to corrosion.

Ceramics show a different fracture behavior than that of metals and polymers because of their brittle nature. Therefore, ceramic engineers should be aware of this behavior when designing reliable ceramic components. The design challenge for the use of ceramic materials in technical applications is the computation of their failure probability under prescribed boundary and loading conditions.

Since the strength of ceramics is not a constant value, design with ceramic materials cannot be done deterministically. Instead, a probabilistic approach should be used in which a ceramic component's failure probability is calculated. More than 70 years ago, Weibull derived a statistical theory of brittle fracture.3-4 Since that time, the Weibull distribution function has become the most widely used function in the mechanical design of ceramic components.

The simplest form of the so-called Weibull function5 is given in Equation 1 for a uniaxial homogeneous tensile stress state, where P

In general, the stress distribution within a ceramic component is not constant but takes different values at different positions. In addition, the true orientation of the cracks in ceramic components is randomly distributed. As a result, normal and shear stresses acting in the crack plane cause Mode-I, Mode-II and Mode-III loading of the cracks. A proper account of the spatial variation in the stress triaxiality (e.g., effect of shear stresses) must therefore be taken using a multiaxial failure criterion.

In this work, a reliable design of a displacing piston will be applied (taking into account the second requirement). The heat loss of the displacing piston, the friction between the body of the engine, and the displacing piston will not be considered.

Under these conditions, due to the low heat conductivity and high temperature stability, 3 mol % yttria-stabilized zirconia is particularly suitable for the production of the displacing piston.

The stress analysis was performed by ABAQUS, a commercial finite element program, and the failure probability of each model was calculated by STAU (STatistische AUswertung), a post-processor for ABAQUS. STAU was developed by the Probabilistic Group at the IZSM at Karlsruhe University, in cooperation with several partners.M

The loading and boundary conditions used in FEM analysis are shown in Figure 2(b). The temperature boundary conditions were T

As the ceramic material, 3 mol % yttria-stabilized zirconia was used due to its low heat conductivity, high-temperature strength and good Weibull parameters. The material parameters of 3 mol % yttria-stabilized zirconia used for the FEM and calculation of failure probability include: Young's Modulus E = 210 GPa, Poisson's ratio υ= 0.3, Weibull modulus

According to the results, it was observed that the parameters t

The region where the failure probability is P

Next, the temperature of the hot reservoir was increased from 900 to 1100 K, and the failure probability of the optimized form was calculated. Here, it is important to take into account the strength change of the 3 mol % yttria-stabilized zirconia with the temperature for the calculation of failure probability. Figure 5 illustrates the strength of the 3 mol % yttria- stabilized zirconia as a function of service temperature.

It can be seen that increasing the temperature from 900 to 1100 K did not influence the strength of the 3 mol % yttria-stabilized zirconia. Therefore, the same Weibull parameters can be used for this calculation. When the temperature of the hot reservoir is increased by 200 K, the maximum theoretical efficiency of the Stirling engine increases from 66.7% to 72.7%, which results in an approximately 9% increase in efficiency.

The failure probability of the optimized form with t

2. Andreasen, J. H., "Reliability-Based Design of Ceramics,"

3. Weibull, W., "A Statistical Theory of the Strength of Materials,"

4. Weibull, W., "A Statistical Representation of Fatigue Failures in Solids,"

5. Danzer, R., Supancic, P., Pascual, J. & Lube, T., "Fracture Statistics of Ceramics-Weibull Statistics and Deviations from Weibull Statistics,"

6. Danzer, R., Lube, T., Supancic, P. and Damani, R., "Fracture of Ceramics,"

7. Batdorf, S. B. & Crose, J. G., "A Statistical Theory for the Fracture of Brittle Structures Subjected to Nonuniform Polyaxial Stresses,"

8. Evans, A. G., "A General Approach for the Statistical Analysis of Multiaxial Fracture,"

9. Nohut, S., Usbeck, A., ™zcoban, H., Krause, D., and Schneider, G. A., "Determination of the Multiaxial Failure Criteria for Alumina Ceramics under Tension-Torsion Test,"

10. Ashby, M. F.,

11. Organ, A. J., Thermodynamics and Gas Dynamics of the Stirling Cycle Machine, University Press, Cambridge, 1992.

12. Brckner-Foit, A., Heger, A., Heiermann, K., Hlsmeier, P., Mahler, A., Mann, A., and Ziegler, C., STAU 5-User's Manual, Institut fr Materialforschung, Karlsruhe, 2005.

13. Ingel, R. P., Microfilms., Ph.D. Thesis, Catholic University, Washington D.C. University, Int. No. 8302474, 1982.

^{1}Advanced ceramics provide numerous advantages compared to other materials: durability, hardness, high mechanical strength at high temperature, stiffness, low density, electrical insulation and conductivity, thermal insulation and conductivity, etc. Most ceramic materials are almost entirely immune to corrosion.

Ceramics show a different fracture behavior than that of metals and polymers because of their brittle nature. Therefore, ceramic engineers should be aware of this behavior when designing reliable ceramic components. The design challenge for the use of ceramic materials in technical applications is the computation of their failure probability under prescribed boundary and loading conditions.

^{2}## The Weibull Theory

Ceramic materials exhibit strength scatter, and the failure of a ceramic begins with pre-existing defects in the material. The material's strength therefore depends on the size of the most critical flaw in the material, which varies from specimen to specimen. Essentially, the scatter of the ceramic strength is caused by the scatter of the flaw size.Since the strength of ceramics is not a constant value, design with ceramic materials cannot be done deterministically. Instead, a probabilistic approach should be used in which a ceramic component's failure probability is calculated. More than 70 years ago, Weibull derived a statistical theory of brittle fracture.3-4 Since that time, the Weibull distribution function has become the most widely used function in the mechanical design of ceramic components.

The simplest form of the so-called Weibull function5 is given in Equation 1 for a uniaxial homogeneous tensile stress state, where P

_{F,V}(σ, V) is the failure probability of a ceramic component due to the volume flaws; V is the volume of the component; V_{0}is the unit volume containing average number of cracks; σ is uniaxial applied stress; m is the Weibull modulus, which describes the scatter of the strength; and σ_{0}is the cumulative mean stress, at which the failure probability is 63.2% for a specimen with a volume V = V_{0}.In general, the stress distribution within a ceramic component is not constant but takes different values at different positions. In addition, the true orientation of the cracks in ceramic components is randomly distributed. As a result, normal and shear stresses acting in the crack plane cause Mode-I, Mode-II and Mode-III loading of the cracks. A proper account of the spatial variation in the stress triaxiality (e.g., effect of shear stresses) must therefore be taken using a multiaxial failure criterion.

^{6-8}In this article, the normal stress criterion will be used for the computation of failure probability.^{9}## Stirling Engines

The Stirling engine is a thermal engine in which the displacing piston separates the expansion area from the compression area. The displacing piston shifts the work gas from the hot area to the cold area through thin tubes. To be effective, the piston must fulfill two requirements: its contact surface with the cylinder should be gas-tight, since the flow of gas reduces the efficiency; and it should exhibit low heat conductivity, since a large temperature difference exists between the cold and hot areas.In this work, a reliable design of a displacing piston will be applied (taking into account the second requirement). The heat loss of the displacing piston, the friction between the body of the engine, and the displacing piston will not be considered.

Under these conditions, due to the low heat conductivity and high temperature stability, 3 mol % yttria-stabilized zirconia is particularly suitable for the production of the displacing piston.

^{10}In addition to low heat conductivity, properties such as better wear resistance and lower density compared to conventionally used material (e.g., chromium-nickel stainless steel), enable the use of 3 mol % yttria-stabilized zirconia to offer many advantages in this application. The geometry and the boundary conditions of the displacing piston have been realized according to a beta series Stirling engine,^{11}as represented in Figure 1, where T_{h}is the temperature of the hot part, T_{c}is the temperature of the cold part, P_{h}is the pressure of the hot part, and P_{h}is the pressure of the cold part.## FEM Model

In general, T_{h}= 873 K, T_{c}= 333 K, P_{h}= 17.67 bar, and P_{c}= 3.2 bar. The diameter of the displacing piston is D_{dp}= 55 mm and H_{dp}= 80 mm. The technical drawing and the geometric variables of the displacing piston are given in Figure 2(a), where t_{1}= 12.5 mm, t_{2}= 20 mm, t_{3}= 10 mm, d_{1}= 24 mm, d_{2}= 36 mm and d_{3}= 24 mm. Using these variables, a parametric study was performed, and the form was optimized related to minimum failure probability.The stress analysis was performed by ABAQUS, a commercial finite element program, and the failure probability of each model was calculated by STAU (STatistische AUswertung), a post-processor for ABAQUS. STAU was developed by the Probabilistic Group at the IZSM at Karlsruhe University, in cooperation with several partners.M

^{12}The loading and boundary conditions used in FEM analysis are shown in Figure 2(b). The temperature boundary conditions were T

_{h}= 900 K and T_{c}= 300 K. The mechanical boundary conditions were P_{h}= 18 bar, P_{c}= 3 bar and P_{ps}= 1100 bar, which is the pre-stressing due to the used screw. In the calculations, an M10 screw was selected, which produces a pre-stressing force of 26 kN according to the strength class of 8.8.As the ceramic material, 3 mol % yttria-stabilized zirconia was used due to its low heat conductivity, high-temperature strength and good Weibull parameters. The material parameters of 3 mol % yttria-stabilized zirconia used for the FEM and calculation of failure probability include: Young's Modulus E = 210 GPa, Poisson's ratio υ= 0.3, Weibull modulus

*m*= 20, mean strength σ0 = 600 MPa, heat conductivity λ = 1.5 W/mK and thermal expansion coefficient α_{30-1000}= 12.5 x 10^{-6}K^{-1}.## Results and Discussion

The stress distribution in the displacing piston with the boundary and loading conditions given in Figure 2(b) is shown in Figure 3. In order to find the optimized form, all geometrical variables given in Figure 2(a) were changed one by one, and the effects of the variables on the failure probability were investigated.According to the results, it was observed that the parameters t

_{1}and t_{2}have the most significant effects on the failure probability. Thus, the failure probability of the ceramic displacing piston was calculated for t_{1}intervals of 2.5-15 mm and t_{2}intervals of 10-65 mm. The failure probability values for different combinations of t_{1}and t_{2}are shown in Figure 4.The region where the failure probability is P

_{F,V}> 10^{-4}is shown with a red color, since it is not acceptable for practical applications. According to these results, a displacing piston with t_{1}= 2.5 mm and t_{2}= 55 mm gives the lowest failure probability, which is P_{F,V}= 2.7 x 10^{-7}. As previously stated, the main aim of this work was to find the optimum shape of a zirconia displacing piston according to the calculation of the probability of failure, which occurs due to the stresses that are mainly caused by the temperature difference between the hot and cold part of the displacing piston. From a fracture mechanics point of view, it is possible to use 3 mol % yttria-stabilized zirconia for the displacing piston in a Stirling engine; by changing the shape, the failure probability can be reduced to an order of 10^{-6}.A Stirling engine uses the temperature difference between its hot end and cold end to establish a cycle of a fixed mass of gas-heated-expanded and cooled-compressed-to convert thermal energy into mechanical energy. The greater the temperatures difference between the hot and cold sources, the greater the thermal efficiency. The maximum theoretical efficiency is equivalent to the Carnot cycle. According to the Carnot cycle, the theoretical efficiency can be calculated as n

_{c}= 1-(T_{c}/T_{h}) where T_{c}is the temperature of the cold reservoir and T_{h}is the temperature of the hot reservoir. According to these conditions, the maximum theoretical efficiency of the Stirling engine is equal to 66.7%. When the temperature of the hot reservoir is increased, the efficiency of the Stirling engine can also be increased.Next, the temperature of the hot reservoir was increased from 900 to 1100 K, and the failure probability of the optimized form was calculated. Here, it is important to take into account the strength change of the 3 mol % yttria-stabilized zirconia with the temperature for the calculation of failure probability. Figure 5 illustrates the strength of the 3 mol % yttria- stabilized zirconia as a function of service temperature.

^{13}It can be seen that increasing the temperature from 900 to 1100 K did not influence the strength of the 3 mol % yttria-stabilized zirconia. Therefore, the same Weibull parameters can be used for this calculation. When the temperature of the hot reservoir is increased by 200 K, the maximum theoretical efficiency of the Stirling engine increases from 66.7% to 72.7%, which results in an approximately 9% increase in efficiency.

The failure probability of the optimized form with t

_{1}= 2.5 mm and t_{2}= 55 mm is equal to P_{F,V}= 1.4 x 10-5. This means that increasing the temperature by 200 K increases the efficiency while keeping the displacing piston in the reliable region. In the future, a more comprehensive reliability analysis should be done by taking into account the heat loss of the displacing piston and the friction between the body and the displacing piston.*For more information, contact the author at (90) 0342-2116789; fax (90) 0342-2116677; or email serkan.nohut@zirve.edu.tr.*## References

1. Nohut S., and Schneider, G. A., "Failure Probability of Ceramic Coil Springs,"*Journal of the European Ceramic Society*, 2009, 29, 1013-1019.2. Andreasen, J. H., "Reliability-Based Design of Ceramics,"

*Materials and Design*, 1994, 15[1], 3-13.3. Weibull, W., "A Statistical Theory of the Strength of Materials,"

*Ingeni”rsvetenskapsakademiens Handlingar*, Stockholm, 1939, 151, 1-45.4. Weibull, W., "A Statistical Representation of Fatigue Failures in Solids,"

*Transactions of the Royal Institute of Technology*, Stockholm, Sweden, 1949, 27, 5-50.5. Danzer, R., Supancic, P., Pascual, J. & Lube, T., "Fracture Statistics of Ceramics-Weibull Statistics and Deviations from Weibull Statistics,"

*Engineering Fracture Mechanics*, 2007, 74, 2919-2932.6. Danzer, R., Lube, T., Supancic, P. and Damani, R., "Fracture of Ceramics,"

*Advanced Engineering Materials*, 2008, 10[4], 275-298.7. Batdorf, S. B. & Crose, J. G., "A Statistical Theory for the Fracture of Brittle Structures Subjected to Nonuniform Polyaxial Stresses,"

*Journal of the Applied Mechanics*, 1974, 41, 267-272.8. Evans, A. G., "A General Approach for the Statistical Analysis of Multiaxial Fracture,"

*Journal of the American Ceramic Society*, 1978, 61, 302-308.9. Nohut, S., Usbeck, A., ™zcoban, H., Krause, D., and Schneider, G. A., "Determination of the Multiaxial Failure Criteria for Alumina Ceramics under Tension-Torsion Test,"

*Journal of the European Ceramic Society*, 2010, 30[16], 3339-3349.10. Ashby, M. F.,

*Materials Selection in Mechanical Design*, 3rd Ed. Elsevier, 2005.11. Organ, A. J., Thermodynamics and Gas Dynamics of the Stirling Cycle Machine, University Press, Cambridge, 1992.

12. Brckner-Foit, A., Heger, A., Heiermann, K., Hlsmeier, P., Mahler, A., Mann, A., and Ziegler, C., STAU 5-User's Manual, Institut fr Materialforschung, Karlsruhe, 2005.

13. Ingel, R. P., Microfilms., Ph.D. Thesis, Catholic University, Washington D.C. University, Int. No. 8302474, 1982.

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