J. N. Reddy Center for Innovations in Mechanics for Design and Manufacturing Texas A&M University, College Station, Texas jnreddy@tamu.edu; http://mechanics.tamu.edu City U Distinguished Lecture City University of Hong Kong 12 October, 2018 JN Reddy
Professional retrospectives: Primal-dual variational principles (PhD) Hypervelocity impact (Lockheed) Modeling of geological phenomena (OU) Modeling of bimodular materials (OU) Third-order structural theories (VPI) Penalty finite element models of flows of viscous incompressible fluids (VPI) Layerwise laminate theory (VPI & TAMU) Robust shell element (TAMU) Modeling of biological cells (TAMU) Least-squares FE models of fluid flow (TAMU) Strain gradient, non-local, and non- classical continuum models (TAMU) GraFEA (TAMU) Closing remarks 5 JN Reddy
Professor J. Tinsley Oden, Director, Institute for Computational Engineering and Sciences; Professor of Aerospace Engineering and Engineering Mechanics, University of Texas at Austin ( JN’s Ph.D. Thesis advisor and coauthor of papers and books ). 3 rd ed. to appear in 2017 JN Reddy
J.T. Oden and J.N. Reddy, “On dual-com plem entary variational principles in m athem atical physics,” Int. J. Engng Science , 12, 1-29 (1974). Supported by AFOSR T ET u * ( ) f 0 in and • Variational principles for + = Ω * S CS c ( ) 0 in h + = Ω • 14 Variational principles of elasticity: 7 primal and 7 dual; • Fluid mechanics, electrostatics, magnetostatics; and nonlinear operators All conventional as well as m ixed variational principles are derived. Several of these principles form ed the basis of the m ixed, hybrid, and assum ed strain finite elem ent m odels (they were not cited often because our work was a bit m athem atical and buried in the literature). 8 JN Reddy
CLPT FSDT J.N. Reddy, “A sim ple higher-order theory for z lam inated com posite plates,” J. of Ap p lied Mecha nics , 51, 745-752 (198 4). (over 20 0 0 citations) J.N. Reddy and C.F. Liu, “A higher-order shear deform ation theory for lam inated elastic shells,” Int. J. of Engng. Sci. , 23(3), 319-330 (198 5). (8 0 0 citations) Transverse shear stress s xz x y z ( , , ) ( , , ) x y z Q Q s g g = + xz 55 xz 45 yz TSDT u w v w ∂ ∂ ∂ ∂ ; g g = + = + xz yz z x z y ∂ ∂ ∂ ∂ Displacem ent field 2 3 u x y z ( , , ) u ( , ) x y z ( , ) x y z ( x y , ) z ( , ) x y f q l = + + + 0 x x x 2 3 v x y z ( , , ) v ( , ) x y z ( , ) x y z ( , ) x y z ( , ) x y f q l = + + + 0 y y y w ( x y z , , ) w ( x y , ) = JN Reddy 0 12
z Transverse shear strains u w w ∂ ∂ ∂ ( , ) x y 2 z ( , ) x y 3 z 2 ( , ) x y g f q l = + = + + + ∂ xz x x x z x x ∂ ∂ v w w ∂ ∂ ∂ 2 ( , ) x y 2 z ( x y , ) 3 z ( x , y ) g f q l = + = + + + y z y y y z y y ∂ ∂ ∂ s Vanishing of transverse shear stresses on the xz bounding planes ( , , x y h / 2) ( , , x y h / 2) 0 ( , ) x y ( , ) x y 0 s s q q ± = ± = = = xz yz x y TSDT 4 w 4 w ∂ ∂ ( , ) x y ( , ) x y , ( , ) x y ( , ) x y l f l f = − + = − + x x y y 3 h 2 x 3 h 2 y ∂ ∂ 3 4 z w ∂ φ x u x y z ( , , ) u ( , ) x y z f ( , ) x y f = + − + 0 x x 2 3 h x ∂ ( u , w ) ∂ w 0 3 4 z w ∂ x ∂ v x y z ( , , ) v ( , ) x y z ( , ) x y f f = + − + 0 y y 3 h 2 y ∂ ( u 0 w , ) 0 JN Reddy w x y z ( , , ) w ( , x y ) = 0
Bending of a symmetric cross-ply (0/90) s laminate (SS-1)under uniformly distributed load 0.020 3-D Elasticity 0.018 Solution CLPT 0.016 FSDT TSDT Deflection, w TSDT _ 0.014 0.012 3D E 2 = 10 6 psi (7 GPa) 0.010 E 1 =25 E 2 , G 12 = G 13 =0.5 E 2 0.008 G 23 =0.2 E 2 , n 12 =0.25 FSDT 0.006 CLPT 0.004 0 5 10 15 20 25 30 35 40 45 50 a/h JN Reddy Third-order laminate theory: 13
Bending of a symmetric cross-ply (0/90) s laminate (SS-1)under uniformly distributed load 0.50 TSDT (C) 0.30 TSDT CLPT (E) CLPT 0.10 FSDT (E) FSDT (C) TSDT (E) -0.10 TSDT (C) (E): equilibrium-derived (C): constitutively-derived FSDT (C) -0.30 FSDT (E) -0.50 0.00 0.04 0.08 0.12 0.16 0.20 _ Stress, σ yz ( a /2,0, z ) JN Reddy Third-Order Laminate Theory 16
y k k 1 + z s s xx xx s s ≠ yy yy s s xy xy x ( k + 1)th layer k th layer Equilibrium of k + 1 ( k+ 1) σ zx interlaminar ( k+ 1) ( k+ 1) stresses σ zy σ zz ( k+ 1) ( k ) σ zx = σ zx k ( k ) ( k ) σ zy σ zz ( k+ 1) ( k ) σ zy = σ zy ( k ) σ zx ( k+ 1) ( k ) σ zz = σ zz JN Reddy 9
Equilibrium Requirements k k 1 k k 1 + + s s e e zz zz zz zz s s e e ( ) k ( k 1) because Q Q = ≠ + ≠ yz yz yz yz ij ij s s e e xz xz xz xz y Single-Layer Theories z k k 1 + k k 1 + e e e e xx xx zz zz , e e e e = = x yy yy yz yz e e e e ( k + 1)th layer k th layer xy xy xz xz JN Reddy 10
z U N N I th layer I+1 U I + 1 z I U I I − 1 U I − 1 U I + 1 I+1 x U 4 4 N � 3 U I U I ( x, y, t ) Φ I ( z ) U 3 u ( x, y, z, t ) = 2 I I =1 2 U 2 1 N � V I ( x, y, t ) Φ I ( z ) v ( x, y, z, t ) = u 1 I − 1 U 1 U I − 1 U I Φ I ( z ) I =1 M � W I ( x, y, t ) Ψ I ( z ) w ( x, y, z, t ) = I =1 J.N. Reddy 11
Conventional 3D Layerwise 2D + 1D Cubic Linear serendipity Lagrange element element (in-plane) (through (1a) thickness) (1b) Quadratic Quadratic serendipity Lagrange element element (in-plane) (through (2a) thickness) J.N. Reddy (2b) 12
Table: Comparison of the number of operations needed to form the element sti ff ness matrices for equivalent el- ements in the conventional 3-D format and the lay- erwise 2-D format. Full quadrature is used in all. Element Type † Multipli. Addition Assignments 1a (3-D) 1,116,000 677,000 511,000 1b (LWPT) 423,000 370,000 106,000 2a (3-D) 1,182,000 819,000 374,000 2b (LWPT) 284,000 270,000 69,000 † Element 1 a : 72 degrees of freedom, 24-node 3-D isopara- metric hexahedron with cubic in-plane interpolation and linear transverse interpolation. Element 1 b : 72 degrees of freedom, E12—L1 layerwise element. Element 2 a : 81 degrees of freedom, 27-node 3-D isopara- metric hexahedron with quadratic interpolation in all three directions. Element 2 b : 81 degrees of freedom, E9—Q1 layerwise el- ement. JN Reddy 13
Layerwise Kinematic Model 3D modeling with 2D & 1D elements z z y y a 2 a 2 h 2-D quadratic x Lagrangian element a 2 a 2 x three quadratic layers through the thickness E 1 = 25 × 10 6 psi , E 2 = E 3 = 10 6 psi G 12 = 0 . 5 × 10 6 psi , G 13 = G 23 = 0 . 2 × 10 6 psi , ν 12 = ν 13 = ν 23 = 0 . 25 u ( x, a/ 2 , z ) = u ( a/ 2 , y, z ) = 0 v ( a/ 2 , y, z ) = u ( x, a/ 2 , z ) = 0 w ( x, a, z ) = u ( a, y, z ) = 0 22 J.N. Reddy
In-plane Stresses predicted by the Layerwise Theory 1.0 0.8 Exact 3-D Elasticity Layerwise Mesh 1 Layerwise Mesh 2 z/h =0.667 CLPT 0.6 FSDT z/h 90° 0.4 z/h =0.333 0° 0.2 0.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 _ Inplane normal stress, σ xx JN Reddy 23
Transverse shear stresses predicted by the Layerwise Theory 1.0 0.8 Exact 3-D Elasticity Layerwise Mesh 1 Layerwise Mesh 2 z/h =0.667 CLPT (equilibrium) 0.6 FSDT (equilibrium) z/h 90° FSDT 0.4 z/h =0.333 0.2 0° 0.0 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 _ Transverse shear stress, σ JN Reddy 24 yz
j1 Variable Kinematic Model for Global-Local Analysis Composite displacement field: u i ( x, y, z ) = u ESL ( x, y, z ) + u LWT ( x, y, z ) i i ESL Displacement field: u ESL ( x, y, z ) = u 0 ( x, y ) + z φ x ( x, y ) 1 u ESL ( x, y, z ) = v 0 ( x, y ) + z φ y ( x, y ) 2 u ESL ( x, y, z ) = w 0 ( x, y ) 3 LWT Displacement field: N � u LWT U I ( x, y ) Φ I ( z ) ( x, y, z ) = 1 I =1 N � u LWT V I ( x, y ) Φ I ( z ) ( x, y, z ) = 2 I =1 M � u LWT W I ( x, y ) Ψ I ( z ) ( x, y, z ) = 3 JN Reddy 25 I =1
Slide 17 j1 jnreddy, 6/30/2014
An Efficient Shell Finite Element Objective : Develop a robust shell element for the linear and nonlinear analysis of shell structures made of multilayered composites and functionally graded materials that is computationally efficient (i.e., accurate and computationally inexpensive). Thickness stretch 7-parameter displacement field is included h h ( ) 2 ˆ u X ( , , ) u ( , ) ( , ) ( , ) n ξ η ζ = ξ η + ζ ϕ ξ η + ζ Ψ ξ η 2 2 F.E. approximation of displacement field h h n k k 2 k ˆ k u ( , ) u n k k = ψ ξ η + ζ ϕ + ζ Ψ k 2 2 k 1 = JN Reddy 32
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