טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentBen-Atia Ariel
SubjectMorphology Geometry and Loading Interaction in
Stochastically Heterogeneous Structures
DepartmentDepartment of Mechanical Engineering
Supervisors Professor Emeritus Eli Altus
Professor Josef Givli
Full Thesis textFull thesis text - English Version


Abstract

In small scale structures the heterogeneity is dominant, and "macro-engineering" concepts are not always applicable. Even current micro/nano structures are commonly heterogeneous, with stochastic material properties (elastic, thermal, optical etc.), characterized by a microstructure which is similar in size to the structure itself. For illustration, consider an ensemble of cantilever beams used as sensors, cut from the same polycrystalline wafer. Commonly, each micro-beam is constructed from a finite number of grains, in some cases even less than 20. The beams are statistically similar, yet not identical. Applying the same force at the end of each beam will produce a different deflection, where the statistical properties of the ensemble are directly related to material morphology, such as elastic properties of the single grain, relative grain size, orientation etc. In this research the effect of heterogeneity on the mechanical response of the structure is studied. The analysis is essentially analytical, and is based on the Functional Perturbation Method, developed and generalized in the past few years. Analytical results have been compared with numerical simulations (Monte-Carlo, Rain Drops Method) and found to be in good agreement. The main goal of this research is to investigate the contribution of three different components on the mechanical response of the structure: morphology, geometry and loading. Specifically we wish to find the interactions between morphological (moduli) characteristics, mechanical properties (reaction forces, stability, failure, displacement, etc.) and structural geometry. First, we expand the 2-Point Correlation concept and its morphological meaning, i.e., correlation length, from 1D to 2D fields. Correlation lengths can be defined in many ways, and attention is devoted to the necessary conditions for proper tensorial definitions of such properties. 1D and 2D stochastic fields are created using the Cholesky method and results are validated by Monte-Carlo simulations. Next, these concepts are applied to the case of a beam with a single indeterminacy (clamped-simply supported). By referring to analytical approximations, the variance of the reaction force is studied. This helps in understanding the inter-relations between geometry, loading, and morphology. Further, these analytical relations may enable better exploitation of the manufacturing techniques in order to control the mechanical behavior of nano-structures in a well defined systematic way. In order to study the effect of random heterogeneity on the overall behavior of the structure, we chose to focus on moduli heterogeneity. Nevertheless, other types of non-uniformity, such as Poisson's ratio, grain size, and orientation, can be examined similarly.