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Sandwich Panels with Functionally Graded Core SANDWICH PANELS WITH FUNCTIONALLY GRADED CORE By NICOLETA ALINA APETRE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005 Copyright 2005 by Nicoleta Alina Apetre This document is dedicated to my loving family: my mother, Elisabeta Apetre, my sister, Dana and my brother, Nicu and to my devoted boyfriend Laurence as they have offered their support and love throughout this endeavor. No words said can express how indebted I am to them for what I am today and for what I shall accomplish in the future. ACKNOWLEDGMENTS I am profoundly grateful to my advisor, Dr. Bhavani Sankar, for scientific guidance, moral support and endless patience. I would like to thank Dr. Peter Ifju for his help and ideas throughout this enterprise. I would also like to thank my friend, mentor and committee member, Dr. Oana Cazacu, for her guidance and support on my research and my life. Her comments and advice made this dissertation a much more useful document. At the same time, I wish to thank Dr. Raphael T. Haftka and Dr. Bjorn Birgisson for serving on my committee and for their assistance and guidance. I express sincere appreciation to a friend and mentor, Dr. Satchi Venkataraman, for his continuous scientific guidance and moral support during the years. I would like to thank all my professors from whom I learned so much, especially the late Dr. E. Soos, who showed me the path I am walking now. Special thanks go to all my friends, colleagues and family for their continual love, support, and encouragement throughout my time in graduate school and my life. iv TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................................................................................. iv LIST OF TABLES............................................................................................................ vii LIST OF FIGURES ......................................................................................................... viii ABSTRACT...................................................................................................................... xii CHAPTER 1 INTRODUCTION AND BACKGROUND .................................................................1 Introduction...................................................................................................................1 Literature Review .........................................................................................................7 Sandwich Plate Theories .......................................................................................7 Contact on Composite Materials .........................................................................14 Cellular Solids and Functionally Graded Materials ............................................28 Objectives and Approaches ........................................................................................36 2 ANALYTICAL MODELING OF SANDWICH BEAMS WITH FUNCTIONALLY GRADED CORE ........................................................................38 Exact Method for Sandwich Structures with Functionally Graded Core ...................38 Fourier – Galerkin Method for Sandwich Structures with Functionally Graded Core.........................................................................................................................42 Analytical Models for Sandwich Structures with Functionally Graded Core ............47 An Equivalent Single-Layer First-Order Shear Deformation Theory.................47 An Equivalent Single-Layer Third-Order Shear Deformation Theory ...............50 A Higher-Order Shear Deformation Theory .......................................................52 Results for Sandwich Beam with Functionally Graded Core under a Distributed Load ........................................................................................................................57 Summary and Conclusions .........................................................................................67 3 CONTACT AND LOW-VELOCITY IMPACT Of SANDWICH BEAMS WITH FUNCTIONALLY GRADED CORE ........................................................................69 Assumed Stress Distribution Method .........................................................................71 Method of Point Matching..........................................................................................74 v Quasi-Static Impact of a Sandwich Beam with FG Core ...........................................76 Results.........................................................................................................................78 Summary and Conclusions .......................................................................................103 4 MASS OPTIMIZATION FOR A SANDWICH BEAM WITH FUNCTIONALLY GRADED CORE......................................................................................................105 Linear Model ............................................................................................................105 Case 1: Same Flexural Rigidity.........................................................................109 Case 2: Same Mass and Flexural Rigidity.........................................................112 Quadratic Model .......................................................................................................118 Case 1: Same Flexural Rigidity.........................................................................121 Case 2: Same Total Mass ..................................................................................130 Summary and Conclusions .......................................................................................138 5 SUMMARY AND CONCLUSIONS.......................................................................140 LIST OF REFERENCES.................................................................................................144 BIOGRAPHICAL SKETCH ...........................................................................................153 vi LIST OF TABLES Table page 3-1 Maximum normal and shear strains for a given impact energy ...............................94 3-2 Core thicknesses for different materials with same flexural stiffness, D11 and same global stiffness ................................................................................................97 4-1 Variations of Young’s modulus, density, mass and flexural rigidity.....................107 4-2 Total mass of sandwich structures with different density variations and different core thicknesses......................................................................................................111 4-3 Maximum normal and shear strains for a given impact energy .............................117 4-4 Variations of Young’s modulus, density, mass and flexural rigidity.....................120 4-5 Total mass of sandwich structures with same geometry and different density variations ................................................................................................................123 4-6 Maximum normal and shear strains normal and interfacial shear stresses for a given impact energy ...............................................................................................127 4-7 Flexural rigidity of sandwich structures with same geometry and different density variations....................................................................................................132 4-8 Maximum normal and shear strains and stresses for a given impact energy .........134 vii LIST OF FIGURES Figure page 1.1 The behavior of (a) flexible face sheet and (b) rigid face sheet.................................4 2.1 Sandwich beam with functionally graded core with schematic of the analysis elements....................................................................................................................39 2.2 Traction forces and displacements at the interfaces of each element in the FGM sandwich beam .........................................................................................................40 2.3 The beam geometry..................................................................................................48 2.4 The beam geometry..................................................................................................53 2.5 Through-the-thickness variations of core modulus..................................................58 2.6 Variation of transverse shear stresses in the sandwich structure with FGM core....59 2.7 Variation of σzz stresses in the sandwich structure with FGM core .........................60 2.8 Variation of bending stresses through the thickness of the FGM core ....................61 2.9 Variation of u displacements through the thickness of the sandwich structure with FGM core .........................................................................................................61 2.10 Non-dimensional core modulus ...............................................................................62 2.11 Comparison of deflections .......................................................................................64 2.12 Comparison of longitudinal displacement................................................................64 2.13 Comparison of axial stress in the core. ....................................................................65 2.14 Comparison of shear strain in the core.....................................................................66
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