zbMATH — the first resource for mathematics

Minimal failure probability for ceramic design via shape control. (English) Zbl 1322.49070
Summary: We consider the probability of failure for components made of brittle materials under one time application of a load, as introduced by Weibull and Batdorf-Crosse. These models have been applied to the design of ceramic heat shields of space shuttles and to ceramic components of the combustion chamber in gas turbines, for example. In this paper, we introduce the probability of failure as an objective functional in shape optimization. We study the convexity and the lower semi-continuity properties of such objective functionals and prove the existence of optimal shapes in the class of shapes with a uniform cone property. We also shortly comment on shape derivatives and optimality conditions.

49Q10 Optimization of shapes other than minimal surfaces
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
Full Text: DOI arXiv
[1] Weibull, EW, A statistical theory of the strength of materials, Ingeniors Vetenskaps Akad. Handl., 151, 1-45, (1939)
[2] Allaire, G.: Numerical Analysis and Optimization. Oxford University Press, Oxford (2007) · Zbl 1120.65001
[3] Delfour, M.C., Zolesio, J.P.: Shapes and Geometries. Advances in Design and Control, 2nd edn. SIAM, Philadelphia (2011) · Zbl 1251.49001
[4] Eppler, K.: Efficient Shape Optimization Algorithms for Elliptic Boundary Value Problems. Habilitationsschrift Technische Universität Chemnitz, Chemnitz (1997)
[5] Haslinger, J., Mäkinen, R.A.E.: Introduction to Shape Optimization—Theory, Approximation and Computation. SIAM, Philadelphia (2003) · Zbl 1020.74001
[6] Sokolowski, J., Zolesio, J.P.: Introduction to Shape Optimization—Shape Sensitivity Analysis. Springer, Berlin (1992) · Zbl 0761.73003
[7] Batdorf, SB; Crosse, JG, A statistical theory for the fracture of brittle structures subject to nonuniform polyaxial stress, J. Appl. Mech., 41, 459-465, (1974) · Zbl 0288.73074
[8] Heger, A.: Bewertung der Zuverlässigkeit mehrachsig belasteter keramischer Bauteile. No. 132 in Fortschrittberichte VDI/18. VDI-Verlag, Düsseldorf (1993)
[9] Nemeth, N.N., Manderscheid, J., Gyekenyeshi, J.: Ceramic analysis and reliability evaluation of structures (cares). Report TP-2916, NASA (1990)
[10] Riesch-Oppermann, H., Brückner-Foit, A., Ziegler, C.: Stau—a general purpose tool for probabilistic reliability assessment of ceramic components under multi axial loading. In: Proceedings of the 13th International Conference on ECF 13, San Sebastian (2000)
[11] Riesch-Oppermann, H; Scherrer-Rudiya, S; Erbacher, T; Kraft, O, Uncertainty analysis of reliability predictions for brittle fracture, Eng. Fract. Mech., 74, 2933-2942, (2007)
[12] Weil, NA; Daniel, IM, Analysis of fracture probabilities in nonuniformly stressed brittle materials, J. Am. Ceram. Soc., 47, 268-274, (1964)
[13] Ziegler, C.: Bewertung der Zuverlässigkeit keramischer Komponenten bei zeitlich veränderlichen Spannungen unter Hochtemperaturbelastung. No. 238 in Fortschritt-Berichte des VDI, Series 18. VDI-Verlag (1998) · Zbl 0629.49006
[14] Munz, D., Fett, D.: Mechanische Eigenschaften von Keramik. Springer, Berlin (1989)
[15] Brückner-Foit, A., Hülsmeier, P., Diegele, E., Rettig, U., Hohmann, C.: Simulating the failure behaviour of ceramic components under gas turbine conditions. In: Proceedings of ASME TURBO EXPO 2002 June 3-6, 2002, Amsterdam (2002)
[16] Hülsmeier, P.: Lebensdauervorhersage für keramische Bauteile. Dissertation, Universität Karlsruhe (2004)
[17] Brückner-Foit, A; Fett, T; Munz, D; Schirmer, K, Discrimination of multiaxiality criteria with the Brasilian disk test, J. Eur. Ceram. Soc., 17, 689-696, (1997)
[18] Gambarotta, L; Lagomarsino, S, A microcrack damage model for brittle materials, Int. J. Solids Struct., 30, 177-198, (1993) · Zbl 0775.73196
[19] Fuji, N, Lower semicontinuity in domain optimization problems, J. Optim. Theory Appl., 59, 407-422, (1988) · Zbl 0629.49006
[20] Ciarlet, P.: Mathematical Elasticity—Volume I: Three-Dimensional Elasticity. Studies in Mathematics and Its Applications, vol. 20. North-Holland, Amsterdam (1988) · Zbl 0648.73014
[21] Gottschalk, H., Schmitz, S.: Optimal reliability in design for fatigue life. SIAM J. Control Optim. (to appear) · Zbl 1307.49041
[22] Allaire, G; Bonneter, E; Francfort, G; Jouve, G, Shape optimization by the homogenization method, Numer. Math., 76, 27-68, (1997) · Zbl 0889.73051
[23] Chenais, D, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl., 52, 189-219, (1975) · Zbl 0317.49005
[24] Evans, AG, A general approach for the statistical analysis of multiaxial fracture, J. Am. Ceram. Soc., 61, 302-308, (1978)
[25] Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, New York (2003) · Zbl 1098.46001
[26] Gross, D., Seelig, T.: Bruchmechanik, 4th edn. Springer, Berlin (2007)
[27] Kallenberg, O.: Random Measures. Akademie Verlag, Berlin (1975) · Zbl 0345.60031
[28] Schmitz, S.: A Local and Probabilistic Model for Low Cycle Fatigue—New Aspects of Structural Analysis. Dissertation Thesis, Universitá della Svizzera Italiana, Lugano (2014)
[29] Escobar, L.A., Meeker, W.Q.: Statistical Methods for Reliability Data. Wiley, New York (1998) · Zbl 0949.62086
[30] Nitsche, JA, On korn’s second inequality, RAIRO Anal. Numer., 15, 237-248, (1981) · Zbl 0467.35019
[31] Schulz, V.: A Riemannian view on shape optimization. arXiv:1203.1493 (2012)
[32] Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Kluwer Academic Publishers, Dordrecht (1994) · Zbl 0932.53003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.