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 2010-11

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

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.

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

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)

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

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.

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

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

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):

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

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.

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)

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

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

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

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

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

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

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.

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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

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 -1123.600 -14.087 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 -2.869×10-3 187.660 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

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)

Generalized Hamilton’s Principle

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

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.