A stochastic 3-scale method to predict the thermo-elastic behaviors of polycrystalline structures

The purpose of this work is to upscale material uncertainties in the context of thermo-elastic response of polycrystalline structures.
The probabilistic behavior of micro-resonators made of polycrystalline materials is evaluated using a stochastic multi-scale approach defined using the following methodology.
1. Stochastic volume elements (SVEs) [1] are defined from Voronoi tessellations using experimental measurements of the grain size, orientation, and surface roughness [2];
2. Mesoscopic apparent thermo-elastic properties such as elasticity tensor, thermal conductivity tensor, and thermal dilatation tensor are extracted using a coupled homogenization theory [3, 4] applied on the SVE realizations;
3. A stochastic model of the homogenized properties extracted from Voronoi tessellations using a moving window technique is then constructed in order to be able to generate spatially correlated meso-scale random fields;
4. These meso-scale random fields are then used as input for stochastic finite element
simulations.
As a result, the probabilistic distribution of micro-resonator properties can be extracted.
The applications are two-fold:
1. A stochastic thermo-elastic homogenization, see Fig. 1(a), is coupled to thermoelastic 3D models of the micro-resonator in order to extract the probabilistic distribution of the quality factor of micro-resonators [5];
2. A stochastic second-order mechanical homogenization, see Fig. 1(b), is coupled to a plate model of the micro-resonator in order to extract the effect of the uncertainties related to the surface roughness of the polycrystalline structures [2].
References
[1] Ostoja-Starzewski, M., Wang, X. Stochastic finite elements as a bridge between random
material microstructure and global response. Comput. Meth. in Appl. Mech. and Eng. (1999) 168: 35-49.
[2] Lucas, V., Golinval, J.-C., Voicu, R., Danila, M., Gravila, R., Muller, R., Dinescu, A., Noels, L., Wu, L. Propagation of material and surface profile uncertainties on MEMS micro-resonators using a stochastic second-order computational multi-scale approach. Int. J. for Num. Meth. in Eng. (2017).
[3] Temizer, I., Wriggers, P. Homogenization in finite thermoelasticity.J. of the Mech. and Phys. of Sol. (2011) 59, 344-372.
[4] Nguyen, V. D., Wu, L., Noels, L. Unified treatment of boundary conditions and efficient algorithms for estimating tangent operators of the homogenized behavior in the computational homogenization method. Computat. Mech. (2017) 59, 483-505.
[5] Wu, L., Lucas, V., Nguyen, V. D., Golinval, J.-C., Paquay, S., Noels, L. A Stochastic Multi-Scale Approach for the Modeling of Thermo-Elastic Damping in Micro-Resonators. Comput. Meth. in Appl. Mech. and Eng. (2016) 310, 802-839.