1. Department of Aerospace Engineering
DYNAMIC STABILITY OF FUNCTIONALLY GRADED CYLINDRICAL SHELLS UNDER AXIAL FOLLOWER LOADS
By:
M. E. Torki, H. Haddadpour, J. N. Reddy
Department of Aerospace Engineering
Sharif University of Technology
Texas A&M University

2. Functionally Graded Materials (FGMs)
Constituents (Suresh and Mortensen, 1998)
Microscopic outlines of the nickel-stainless steel and stainless steel-alumina FGMs

3. Functionally Graded Materials (FGMs)
Advantages (Wu et al., 2006)(Suresh and Mortensen, 1998)
Controlling the places and amounts of temperature stresses
Delaying the yielding and fracture thresholds
Averting excessive stress concentration and discontinuity between layers
Increasing the Mechanical Properties of connecting layers
Maintaining structural integrity, esp. in high temperature settings
Reducing plastic deformations in locations with high strain rates, esp. in indented structures, e.g. missiles and turbine blades.

4. Functionally Graded Materials (FGMs)
Applications
Settings with high temperature gradients
Combustion engines
Furnaces (Smelting, Incinerators)
Rocket chambers
Heat exchangers
Contact-damage-resistant settings
Storage reservoirs
Fueling pipes
Radioactive settings
Implanted limbs
Magnetic devices
Cutting tools
Fire retardants
Optoelectric systems
Piezo-electric and ferro-electric materials, e.g. in turbine blades

5. Functionally Graded Materials (FGMs)
Applications
An example is the human bone which gradient is formed by its change in porosity and composition (Miyamoto et al., 2013).
Car engine cylinder
(EL-Wazery and EL-Desouky, 2015)
Turbine blade
(EL-Wazery and EL-Desouky, 2015)
Nuclear reactor inner wall
(EL-Wazery and EL-Desouky, 2015)

6. Functionally Graded Materials (FGMs)
Thermo-Mechanical Properties (Shen, 2009)
Volume fraction (V)
Mechanical Constants
Modulus of elasticity
Density
Poisson’s ratio (often considered identical for the two phases)
Thermal elongation coefficient
Thermal conductivity factor
Mixture Rule

7. Stability under Follower Forces (Flutter and Divergence)
A cylindrical shell under a follower load
Modeled as a follower load
The prevalent class of instability in small to moderate thicknesses is flutter.

8. Love’s Hypotheses for Thin to Moderately Thick Shells
The transverse normal is inextensible.
Normals to the reference surface of the shell before deformation remain straight after deformation (In case they remain straight and normal: Kirchhoff’s hypothesis).
Deflections and strains are infinitesimal.
The transverse normal stress is negligible (plane stress state is invoked).
First-Order Shear Deformation Theory (FSDT) (Reddy, 2004)
Only for thin (shallow) shells

9. Constitutive Relations
Kinematic Relations
Constitutive Relations
For a general cross-ply laminated or FGM shell
For a cylindrical shell

10. Generalized Hamilton’s Principle in Virtual Form
Effect of Axial Loading
Although we have considered strains to be infinitesimal, to introduce the effect of axial loads we must consider higher-order axial strain. Hence, using Sander’s theory (Reddy, 2007):

11. Effect of Beck’s Follower Loading
After rigorous mathematical operations, the work done by the Beck follower force is:
After integrating by parts:
Domain
Boundary

12. Solution Procedure
Orthogonality in terms of θ
Approximated Mode Functions
Galerkin Method: The functions φj must satisfy the essential boundary conditions. Thus, we need fewer terms for convergence.

13. Verification with Very Long Shells (Simitses and Hodges, 2006)
Equivalent Eigen-Value Problem
Beam theory results
Verification with Very Long Shells (Simitses and Hodges, 2006)

14. Dynamic stability of the Nickel-Stainless Steel FGM
Effect of N on the Flutter Load
Hardening FGM
Softening FGM

15. Dynamic stability of the Nickel-Stainless Steel FGM
Effect of N on the Flutter Load
Hardening FGM
Softening FGM

16. Dynamic stability of the Nickel-Stainless Steel FGM
Effect of N on the Flutter Load
Hardening FGM
Softening FGM

17. Dynamic stability of the Nickel-Stainless Steel FGM
Effect of N on the Flutter Load
Hardening FGM
Softening FGM

18. Dynamic stability of the Nickel-Stainless Steel FGM
Effect of N on the Flutter Load
Hardening FGM
Softening FGM

19. Dynamic stability of the Nickel-Stainless Steel FGM
Effect of N on the Flutter Load
Hardening FGM
Softening FGM

20. Concluding Remarks
The effect of N on flutter load is most considerable in long, thick shells, and it decreases significantly in moderately long and very long shells.
The flutter load remains almost constant with N in thin shells.
At a constant N, the flutter load always gets increased with the thickness ratio.
When the shell is not very long, in a constant thickness ratio, the utmost increase (decrease) in the flutter load occurs due to 𝑵=𝟏.
In long shells, the curve of flutter load approaches an envelope curve, usu. named as beam-like region.

21. Thank You!

22. Thermomechanical Properties of FGMs
Modulus of elasticity for ceramics and metals (Pa) (Shen, 2009) P3 P2 P1
P-1 P0
Materials
-3.681×10-10
1.214×10-6
-1.371×10-3
244.27×109
Zirconia
-1.673×10-10
4.027×10-7
-3.853×10-4
349.55×109
Aluminum oxide
-8.946×10-11
2.160×10-7
-3.070×10-4
348.43×109
Silicon nitride
-4.586×10-4
122.56×109
-6.534×10-7
3.079×10-4
201.04×109
Stainless steel
-3.998×10-9
-2.794×10-4
223.95×109
Nickel

23. Thermal conductivity for ceramics and metals (W.m.K-1) (Shen, 2009)
Thermal expansion coefficient for ceramics and metals (K-1) (Shen, 2009) P3 P2 P1
P-1 P0
Materials
-6.778×10-11
1.006×10-6
-1.491×10-3
12.766×10-6
Zirconia
1.838×10-4
6.827×10-6
Aluminum oxide
9.095×10-4
5.872×10-6
Silicon nitride
-3.147×10-6
6.638×10-4
7.579×10-6
8.076×10-4
12.330×10-6
Stainless steel
8.705×10-4
9.921×10-6
Nickel
Thermal conductivity for ceramics and metals (W.m.K-1) (Shen, 2009) P3 P2 P1
P-1 P0
Materials
6.648×10-8
1.276×10-4
1.700
Zirconia
-6.227×10-3
Aluminum oxide
-7.876×10-11
5.466×10-7
-1.032×10-3
13.723
Silicon nitride
1.704×10-2
1.000
-7.223×10-10
2.092×10-6
-1.264×10-3
15.379
Stainless steel
-1.983×10-9
4.005×10-6
-2.869×10-3
Nickel (when )
-1.523×10-10
6.670×10-7
-4.614×10-4
58.754
Nickel (when )
Poisson’s ratio for ceramics and metals (Shen, 2009) P3 P2 P1
P-1 P0
Materials
1.133×10-4
0.288
Zirconia
0.260
Aluminum oxide
0.240
Silicon nitride
1.121×10-4
3.797×10-7
-2.002×10-4
0.326
Stainless steel
0.310
Nickel

24. Basic Modes of Instability
Beam-like modes:
First bending mode
Second bending mode
Shell modes:
Rayleigh mode
Love mode
Combined modes:
First bending mode
Basic Modes of instability (Park and Kim, 2000)

25. Generalized Hamilton’s Principle

26. Strong Form
Using integration by parts, we will obtain the strong form:
And the new boundary conditions will be:

27. Effect of N on the Critical Circumferential Wave Number (ncr)
L/R 10 20 40 N
0.5 1 5 30
100 ∞
h/R=0.01 4 3
h/R=0.03 2
h/R=0.05
h/R=0.075
h/R=0.1
h/R=0.125
h/R=0.15
h/R=0.175
h/R=0.2 60 80
100
0.5 1 5 30 ∞ 3 2
The effect of N on ncr is more discernible in long (20<=L/R<=40) and thick (h/R>=0.01) shells. For moderately-long (L/R<=10) and very long (L/R>=80) shells, it is not so considerable.