NOTICE: this is the author’s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering 310 (2016) 802-839, DOI: 10.1016/j.cma.2016.07.042
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[en] The aim of this work is to study the thermo-elastic quality factor (Q) of micro-resonators with a stochastic multi-scale approach. In the design of high-Q micro-resonators, thermo-elastic damping is one of the major dissipation mechanisms, which may have detrimental effects on the quality factor, and has to be predicted accurately. Since material uncertainties are inherent to and unavoidable in micro-electromechanical systems (MEMS), the effects of those variations have to be considered in the modeling in order to ensure the required MEMS performance. To this end, a coupled thermo-mechanical stochastic multi-scale approach is developed in this paper. Thermo-mechanical micro-models of polycrystalline materials are used to represent micro-structure realizations. A computational homogenization procedure is then applied on these statistical
volume elements to obtain the stochastic characterizations of the elasticity tensor, thermal expansion, and conductivity tensors at the meso-scale. Spatially correlated meso-scale random fields can thus be generated to represent the stochastic behavior of the homogenized material properties. Finally, the distribution of the thermo-elastic quality factor of MEMS resonators is studied through a stochastic finite element method using as input the generated stochastic random field.
3SMVIB: The research has been funded by the Walloon Region under the agreement no 1117477 (CT-INT 2011-11-14) in the context of the ERA-NET MNT framework.
Funders :
Service public de Wallonie : Direction générale opérationnelle de l'économie, de l'emploi et de la recherche - DG06
[1] Yasumura, K., Stowe, T., Chow, E., Pfafman, T., Kenny, T., Stipe, B., Rugar, D., Quality factors in micron- and submicron-thick cantilevers. J. Microelectromech. Syst. 9:1 (2000), 117–125.
[2] Duwel, A., Gorman, J., Weinstein, M., Borenstein, J., Ward, P., Experimental study of thermoelastic damping in MEMS gyros. Sensors Actuators A 103:1–2 (2003), 70–75 Micromechanics section of Sensors and Actuators, based on contributions revised from the Technical Digest of the 15th IEEE International conference on Micro Electro mechanical Systems (MEMS 2002). URL http://www.sciencedirect.com/science/article/pii/S0924424702003187.
[3] Houston, B., Photiadis, D., Vignola, J., Marcus, M., Liu, X., Czaplewski, D., Sekaric, L., Butler, J., Pehrsson, P., Bucaro, J., Loss due to transverse thermoelastic currents in microscale resonators. Mater. Sci. Eng. A 370:1–2 (2004), 407–411 13th International Conference on Internal Friction and Ultrasonic Attenuation in Solids. URL http://www.sciencedirect.com/science/article/pii/S0921509303009055.
[4] Zener, C., Elasticity and Anelasticity of Metals. 1948, University of Chicago Press.
[5] Lifshitz, R., Roukes, M.L., Thermoelastic damping in micro- and nanomechanical systems. Phys. Rev. B 61 (2000), 5600–5609 http://link.aps.org/doi/10.1103/PhysRevB.61.5600.
[6] S. Lepage, Stochastic finite element method for the modeling of thermoelastic damping in micro-resonators. Leloup, 2007.
[7] Serra, E., Bonaldi, M., A finite element formulation for thermoelastic damping analysis. Internat. J. Numer. Methods Engrg. 78:6 (2009), 671–691 URL http://dx.doi.org/10.1002/nme.2502.
[8] Mariani, S., Martini, R., Ghisi, A., Corigliano, A., Simoni, B., Monte carlo simulation of micro-cracking in polysilicon MEMS exposed to shocks. Int. J. Fract. 167:1 (2011), 83–101 URL http://dx.doi.org/10.1007/s10704-010-9531-4.
[9] Harada, T., Shinozuka, M., The Scale of Correlation for Stochastic Fields–Technical Report., 1986, Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY.
[10] Baxter, S., Graham, L., Characterization of random composites using moving-window technique. J. Eng. Mech. 126:4 (2000), 389–397.
[11] Matthies, H.G., Brenner, C.E., Bucher, C.G., Guedes~Soares, C., Uncertainties in probabilistic numerical analysis of structures and solids-stochastic finite elements. Struct. Saf. 19:3 (1997), 283–336 Devoted to the work of the Joint Committee on Structural Safety.
[12] Yamazaki, F., Member, A., Shinozuka, M., Dasgupta, G., Neumann expansion for stochastic finite element analysis. J. Eng. Mech. 114:8 (1988), 1335–1354.
[13] Shinokuza, M., Astill, C.J., Random eigenvalue problems in structural analysis. AIAA J. 10:4 (1972), 456–462.
[15] Geers, M., Kouznetsova, V., Brekelmans, W., Multi-scale computational homogenization: Trends and challenges. J. Comput. Appl. Math. 234:7 (2010), 2175–2182.
[16] Hill, R., Elastic properties of reinforced solids: Some theoretical principles. J. Mech. Phys. Solids 11:5 (1963), 357–372.
[17] Kanit, T., Forest, S., Galliet, I., Mounoury, V., Jeulin, D., Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Struct. 40:13–14 (2003), 3647–3679.
[18] Hoang, T., Guerich, M., Yvonnet, J., Determining the size of RVE for nonlinear random composites in an incremental computational homogenization framework. J. Eng. Mech., 142(5), 2016, 04016018 American Society of Civil Engineers, URL http://dx.doi.org/10.1061/(ASCE)EM.1943-7889.0001057.
[19] Ostoja-Starzewski, M., Wang, X., Stochastic finite elements as a bridge between random material microstructure and global response. Comput. Methods Appl. Mech. Engrg. 168:1–4 (1999), 35–49.
[20] Gusella, V., Cluni, F., Random field and homogenization for masonry with nonperiodic microstructure. J. Mech. Mater. Struct. 1:2 (2006), 357–386.
[21] Trovalusci, P., Ostoja-Starzewski, M., De~Bellis, M.L., Murrali, A., Scale-dependent homogenization of random composites as micropolar continua. Eur. J. Mech. A Solids 49:0 (2015), 396–407.
[22] Mariani, S., Martini, R., Ghisi, A., Corigliano, A., Beghi, M., Overall elastic properties of polysilicon films: a statistical investigation of the effects of polycrystal morphology. Int. J. Multiscale Comput. Eng. 9:3 (2011), 327–346.
[23] Stefanou, G., Savvas, D., Papadrakakis, M., Stochastic finite element analysis of composite structures based on material microstructure. Compos. Struct. 132 (2015), 384–392 URL http://www.sciencedirect.com/science/article/pii/S0263822315004183.
[24] Yin, X., Chen, W., To, A., McVeigh, C., Liu, W., Statistical volume element method for predicting microstructure-constitutive property relations. Comput. Methods Appl. Mech. Engrg. 197:43–44 (2008), 3516–3529 Stochastic Modeling of Multiscale and Multiphysics Problems.
[25] Clément, A., Soize, C., Yvonnet, J., Computational nonlinear stochastic homogenization using a nonconcurrent multiscale approach for hyperelastic heterogeneous microstructures analysis. Internat. J. Numer. Methods Engrg. 91:8 (2012), 799–824.
[26] Ma, J., Sahraee, S., Wriggers, P., De~Lorenzis, L., Stochastic multiscale homogenization analysis of heterogeneous materials under finite deformations with full uncertainty in the microstructure. Comput. Mech. 55:5 (2015), 819–835 URL http://dx.doi.org/10.1007/s00466-015-1136-3.
[27] V. Lucas, J.-C. Golinval, R. Voicu, M. Danila, R. Müller, M. Danila, A. Dinescu, L. Noels, L. Wu, Propagation of material and surface profile uncertainties on MEMS micro-resonators using a stochastic second-order computational multi-scale 3 approach, Internat. J. Numer. Methods Engrg., submitted for publication.
[28] Wright, T., The Physics and Mathematics of Adiabatic Shear Bands Cambridge Monographs on Mechanics, 2002, Cambridge University Press URL https://books.google.com/books?id=FGilEcC2tg0C.
[29] Der~Kiureghian, A., Ke, J., The stochastic finite element method in structural reliability. Probab. Eng. Mech. 3:2 (1988), 83–91.
[30] Shinozuka, M., Deodatis, G., Response variability of stochastic finite element systems. J. Eng. Mech. 114:3 (1988), 499–519.
[31] Lanczos, C., Dan iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Natl. Bur. Stand., 45(4), 1950, 255.
[32] Kouznetsova, V., Brekelmans, W.A.M., Baaijens, F.P.T., An approach to micro-macro modeling of heterogeneous materials. Comput. Mech. 27:1 (2001), 37–48.
[33] Miehe, C., Koch, A., Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Arch. Appl. Mech. 72:4–5 (2002), 300–317.
[34] Özdemir, I., Brekelmans, W.A.M., Geers, M.G.D., Computational homogenization for heat conduction in heterogeneous solids. Internat. J. Numer. Methods Engrg. 73:2 (2008), 185–204 URL http://dx.doi.org/10.1002/nme.2068.
[35] Temizer, I., Wriggers, P., Homogenization in finite thermoelasticity. J. Mech. Phys. Solids 59:2 (2011), 344–372 URL http://www.sciencedirect.com/science/article/pii/S002250961000219X.
[36] Ainsworth, M., Essential boundary conditions and multi-point constraints in finite element analysis. Comput. Methods Appl. Mech. Engrg. 190:48 (2001), 6323–6339 URL http://www.sciencedirect.com/science/article/pii/S0045782501002365.
[37] Larsson, F., Runesson, K., Saroukhani, S., Vafadari, R., Computational homogenization based on a weak format of micro-periodicity for rve-problems. Comput. Methods Appl. Mech. Engrg. 200:1–4 (2011), 11–26 URL http://www.sciencedirect.com/science/article/B6V29-50J9GSY-1/2/50ad840717a7cc397ab171da344df990.
[38] Terada, K., Hori, M., Kyoya, T., Kikuchi, N., Simulation of the multi-scale convergence in computational homogenization approaches. Int. J. Solids Struct. 37:16 (2000), 2285–2311 URL http://www.sciencedirect.com/science/article/B6VJS-3YDFYCH-4/2/788d085d620c6c5ae804d8c4e4f8ae9b.
[39] Nguyen, V.-D., Béchet, E., Geuzaine, C., Noels, L., Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation. Comput. Mater. Sci. 55 (2012), 390–406.
[40] Shinozuka, M., Jan, C.-M., Digital simulation of random processes and its applications. J. Sound Vib. 25:1 (1972), 111–128.
[41] Yamazaki, F., Shinozuka, M., Digital generation of non-gaussian stochastic fields. J. Eng. Mech. 114:7 (1988), 1183–1197.
[43] Deodatis, G., Micaletti, R., Simulation of highly skewed non-gaussian stochastic processes. J. Eng. Mech. 127:12 (2001), 1284–1295.
[44] Hopcroft, M., Nix, W., Kenny, T., What is the young's modulus of silicon?. J. Microelectromech. Syst. 19:2 (2010), 229–238.
[45] McConnell, A., Uma, S., Goodson, K.E., Thermal conductivity of doped polysilicon layers. J. Microelectromech. Syst. 10:3 (2001), 360–369.
[46] Asheghi, M., Leung, Y.K., Wong, S.S., Goodson, K.E., Phonon-boundary scattering in thin silicon layers. Appl. Phys. Lett. 71:13 (1997), 1798–1800 URL http://scitation.aip.org/content/aip/journal/apl/71/13/10.1063/1.119402.
[47] Devanarayanan, S., Krishnan, R.S., Srinivasan, R., Thermal expansion of crystals. International Series in the Science of the Solid State International Series in the Science of the Solid State, vol.~12, 1979, Pergamon URL http://www.sciencedirect.com/science/article/pii/B9780080214054500019.
[48] Maugin, G.A., The Thermomechanics of Nonlinear Irreversible Behaviors World Scientific Series on Nonlinear Science Series A, vol.~27, 1999, World Scientific URL http://www.worldscientific.com/worldscibooks/10.1142/3700.
