Thermo-Mechanical Beam Element for Analyzing Stresses in Functionally Graded Materials

121 Pages·2015·2.62 MB·English

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USF Tampa Graduate Theses and Dissertations USF Graduate Theses and Dissertations
2011
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Simon Caraballo
University of South Florida, [email protected]
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Caraballo, Simon, "Thermo-Mechanical Beam Element for Analyzing Stresses in Functionally Graded
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Thermo-Mechanical Beam Element for Analyzing

Stresses in Functionally Graded Materials

by

Simón A. Caraballo

A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Department of Mechanical Engineering
College of Engineering
University of South Florida

Major Professor: Autar K. Kaw, Ph.D.
Kandethody M. Ramachandran, Ph.D.
Muhammad M. Rahman, Ph.D.
Glen H. Besterfield, Ph.D.
Ali Yalcin, Ph.D.

Date of Approval:
March 29, 2011

Keywords: Multilayer Beams, Thermal Stresses, Nonhomogeneous Material Properties,
Temperature-Dependent Properties, Finite Element Formulation, Numerical Simulation

Copyright © 2011, Simón A. Caraballo
Dedication

First, I give praise and honor to my Lord and savior Jesus Christ for allowing me
to accomplish this goal.
I dedicate this dissertation to my beloved wife Marisolin and my daughter
Marielsy; without their love and emotional support this work would have not been
possible. I would like to thank them for their sacrifice, patience and love, and for all that
they have given to me in my life.
I would like to express my sincere thanks to my advisor, Professor Autar K. Kaw,
for his constructive guidance, financial support, understanding and patience throughout
the study, especially when I was in down moments. My appreciation extends as well to
the members of my supervisory committee; Dr. Muhammad M. Rahman and Dr. Glen H.
Besterfield from the Department of Mechanical Engineering, Dr. Kandethody M.
Ramachandran from the Department of Mathematics, and Dr. Ali Yalcin from the
Department of Industrial and Management Systems.
I also would like to express my sincere thanks to Ms. Marsha Brett, Ms. Kate
Johnson and Dr. Rajiv Dubey from the College of Engineering for their financial support.
I wish to express my sincere gratitude to my parents for their unconditional love
and confidence in me all the time.
Last but not least, the financial support of the Experimental Polytechnic National
University “Antonio Jose de Sucre” (UNEXPO) is deeply appreciated.
Table of Contents

List of Tables ..................................................................................................................... iii
List of Figures .................................................................................................................... iv
Nomenclature .................................................................................................................... vii
Abstract .............................................................................................................................. ix
Chapter 1 Introduction ........................................................................................................ 1
Motivation ....................................................................................................................... 1
Research Goals................................................................................................................ 2
Outcomes ........................................................................................................................ 3
Dissertation Organization ............................................................................................... 4
Chapter 2 Review of Relevant Literature ........................................................................... 6
Introduction ..................................................................................................................... 6
Chapter 3 Theoretical Background ..................................................................................... 9
Introduction ..................................................................................................................... 9
FGM Theoretical Fundamentals ..................................................................................... 9
Conceptual Idea of FGMs ........................................................................................... 9
Effective Properties of FGMs ................................................................................... 14
Consideration of Temperature Dependence of Material Properties for FGMs ......... 19
FGMs Applications ................................................................................................... 30
Chapter 4 Formulation of Governing Equations ............................................................... 32
Introduction ................................................................................................................... 32
Beam Theory for FGM Structures ................................................................................ 32
Finite Element Formulations......................................................................................... 36
Two-node Element Formulation ............................................................................... 36
Three-node FOSD Element....................................................................................... 43
Temperature Profile Modeling ...................................................................................... 48
One-dimensional Heat Conduction Steady-State Exact Solution for a 3-layer FG
Beam ......................................................................................................................... 48
Two-dimensional Heat Conduction Steady-State Numerical Solution for a
3-Layer FG Beam with Temperature Dependency of the Material Properties ......... 53
Chapter 5 Analyses and Results ........................................................................................ 57
i
Introduction ................................................................................................................... 57
Comparisons of the Element Formulation Simulations with Related Literature .......... 57
Comparison with Suresh and Mortensen’s Model .................................................... 58
Comparison with Chakraborty et al. Models ............................................................ 62
Simulations with Generic Temperature Distributions and Temperature Independence
of the Material Properties.............................................................................................. 68
Simulations with Actual Temperature Distributions with and without Temperature
Dependence of the Material Properties ......................................................................... 77
Influence of the Interlayer Thickness on the Factor of Safety in Composite Beams.... 92
Determination of the Baseline Thickness of the Metallic Layer for Studying the
Influence of the FGM Interlayer in the Factor of Safety .............................................. 93
Effect of Thickness of the Graded Interlayer in the Factor of Safety for the Tri-layer
Model ............................................................................................................................ 95
Chapter 6 Conclusions and Future Work ........................................................................ 100
Introduction ................................................................................................................. 100
Conclusions ................................................................................................................. 100
Recommendations and Future Work........................................................................... 103
References ....................................................................................................................... 104
About the Author ............................................................................................................ End

ii
List of Tables

Table 1. Effective property formulas of FGMs [24]. ....................................................... 18
Table 2. Thermal properties of steel [31, 35, 39]............................................................. 26
Table 3. Thermal properties of alumina [31, 38]. ............................................................ 26
Table 4. Material and geometrical parameters of a tri-layered beam .............................. 51
Table 5. Thermo-elastic properties of nickel and alumina at 300 K ................................ 60
Table 6. Thermo-elastic properties of steel and alumina at 300 K .................................. 63
Table 7. Loading cases for models from literature paper [1]. .......................................... 63
Table 8. Temperature distributions. ................................................................................. 71
Table 9. Factor of safety for the different models............................................................ 91
Table 10. Layer thickness variation for the bi-material model. ........................................ 93
Table 11. Layer thickness variation for the 3-layer model. .............................................. 96

iii
List of Figures

Figure 1. Examples of material grading in functionally graded materials. ..................... 10
Figure 2. Graphical FGM representation of gradual transition in the direction of the
temperature gradient. ...................................................................................... 12
Figure 3. Ceramic volume fraction throughout the FGM layer ...................................... 14
Figure 4. Effect of the grading parameter n on the volume fraction V .......................... 15
c
Figure 5. Material properties throughout the FGM layer................................................ 16
Figure 6. Temperature dependence of elastic modulus and thermal conductivity for
aluminum titanate ceramics. ........................................................................... 20
Figure 7. Temperature dependence of thermal conductivity for several engineering
materials. ......................................................................................................... 21
Figure 8. Temperature dependence of the linear thermal expansion for several
engineering materials. ..................................................................................... 23
Figure 9. Temperature dependence of the Young’s modulus for several ceramic
materials. ......................................................................................................... 24
Figure 10. Temperature dependence of the flexural strength for several engineering
materials. ......................................................................................................... 25
Figure 11. Temperature dependence of the thermoelastic properties of steel. ................. 28
Figure 12. Temperature dependence of the thermoelastic properties of alumina. ............ 29
Figure 13. FGM application for a turbine blade design. ................................................... 31
Figure 14. FGM application for relaxation of stress concentration in lathe bits. ............. 31
Figure 15. Beam coordinate system. ................................................................................. 34
Figure 16. Nodes and degrees of freedom for the 2-node element ................................... 40
iv
Figure 17. Nodes and degrees of freedom for the 3-node element ................................... 44
Figure 18. Three-layer beam with perfect thermal contact at the interface. ..................... 49
Figure 19. Depth-wise exact temperature distribution obtained from the solution of
the heat conduction differential equation. ....................................................... 52
Figure 20. Three-layer beam geometry and boundary conditions .................................... 53
Figure 21. Geometry and nomenclature for a tri-layered composed beam model from
literature reference [1]..................................................................................... 59
Figure 22. Axial thermal stress distribution in a Ni-Graded Layer-Al O trilayer beam
2 3
subject to a T = -100 oC. ............................................................................... 61
Figure 23. Geometry and loading cases for models from literature paper [1]. ................. 63
Figure 24. Axial stress through the thickness for case 1................................................... 65
Figure 25. Transverse shear stress through the thickness for case 1................................. 65
Figure 26. Axial stress through the thickness for case 2................................................... 66
Figure 27. Axial stress through the thickness for case 3................................................... 66
Figure 28. Transverse shear stress through the thickness for case 3................................. 67
Figure 29. Beam configurations. ....................................................................................... 68
Figure 30. Beam geometry and boundary conditions. ...................................................... 70
Figure 31. Normalized axial stress through the thickness for case 1, T=100. ................ 72
Figure 32. Normalized axial stress through the thickness for case 2, T(z) =
100exp230(zh/2). .................................................................................. 73
Figure 33. Normalized axial stress through the thickness for case 3, T(x) =x/BL....... 74
Figure 34. Normalized transverse shear stress through the thickness for case 3,
T(x) =x/BL. ................................................................................................ 74
Figure 35. Normalized axial stress through the thickness for case 4, T(x,z) =
100exp230(zh/2)x/BL......................................................................... 76
Figure 36. Normalized transverse shear stress through the thickness for case 4,
T(x,z)100exp230(zh/2)x/BL. ...................................................... 76
v
Figure 37. Beam geometry and boundary conditions (Bimaterial)................................... 78
Figure 38. Thermal conductivity k distribution with and without temperature
dependence (Bimaterial case). ........................................................................ 79
Figure 39. Temperature profile with and without temperature dependence
(Bimaterial case). ............................................................................................ 79
Figure 40. Beam geometry and boundary conditions (Average interlayer)...................... 81
Figure 41. Thermal conductivity k distribution with and without temperature
dependence (Average interlayer case). ........................................................... 82
Figure 42. Temperature profile with and without temperature dependence (Average
interlayer case). ............................................................................................... 82
Figure 43. Beam geometry and boundary conditions (FGM interlayer)........................... 83
Figure 44. Thermal conductivity k distribution with and without temperature
dependence (FGM interlayer case). ................................................................ 85
Figure 45. Temperature profile with and without temperature dependence (FGM
interlayer case). ............................................................................................... 85
Figure 46. Normalized axial stress through the thickness for actual temperature
distribution. ..................................................................................................... 87
Figure 47. Normalized transverse shear stress through the thickness for actual
temperature distribution. ................................................................................. 89
Figure 48. Factor of safety for the different models ......................................................... 90
Figure 49. Beam models for studying the effect of the FGM interlayer thickness in
the factor of safety........................................................................................... 92
Figure 50. Effect of thickness of FGM interlayer in the factor of safety for the
tri-layer model ................................................................................................. 97
Figure 51. Effect of thickness of FGM interlayer in the specific factor of safety for
the tri-layer model ........................................................................................... 98

vi
Nomenclature

The following symbols are used in this dissertation:

A ,A ,B , D stiffness coefficients
11 55 11 11
AT , BT thermal stiffness coefficients
11 11
BL beam length
E(z) Young’s modulus
G(z) shear modulus
L element length
S strain energy
t beam thickness
u axial displacement
w transverse displacement
 axial strain
xx
(z) coefficient of thermal expansion
0 rotation of reference axis about y-axis
 shear strain
xz
 axial stress
xx
vii