HIGHER-ORDER THEORIES

HIGHER-ORDER THEORIES Third-order Shear Deformation Plate Theory Displacement and strain fields Equations of motion Navier s solution for bending Layerwise Laminate Theory Interlaminar stress and strain continuity Equations of motion Numerical results 1

TSDT Displacement Field (continued) Reduction of the Displacement Field : Require that the top and bottom faces of the plate are free of shear stress h h xz( x, y,,) t yz( x, y,,) t 0 h h xz( x, y,,) t yz( x, y,,) t 0 w xz x zx 3z x x w 3h w 3h x hx x 0, x hx x 0 x 4 x 4 4 w x, 0 x x 3h x Third-Order Laminate Plate Theory

Third-Order Shear Deformation Plate Theory (TSDT) Displacement Field φ x (u,w) w 0 x ( u 0, w 0 ) Assumed Displacement Field 1 3 3 x x x u ( x, y, z,) t u( x, y,) t z ( x, y,) t z ( x, y,) t z ( x, y,) t 3 y y y u ( x, y, z,) t v( x, y,) t z ( x, y,) t z ( x, y,) t z ( x, y,) t u( xyzt,,,) wxyt (,,) Third-Order Laminate Plate Theory 3

Displacement Field of the Reddy Third-Order Laminate Plate Theory (RLPT) Displacement Field 3 4 w u1 ( x, y, z,) t u( x, y,) t zx( x, y,) t z x 3h x 3 4 w u( x, y, z,) t v( x, y,) t zy( x, y,) t z y 3h y u( xyzt,,,) wxyt (,,) 3 Strain Field (0) (1) (3) xx xx xx xx 3 yy yy z yy z yy xy xy xy xy (0) () xz xz xz z yz yz yz 4

Strain Field of the Reddy Third-Order Laminate Plate Theory Strain Field (continued) u 1 w x (0) x x (1) x xx xx v 1 w y yy, yy 4 y y, c1 xy y 3h xy u v w w x y y x x y y x x w (3) x x w xx y w (0) () xz yy c 1 x 1, xz x y y yz 3 1 c yz w xy y x y w y y x xy Third-Order Laminate Plate Theory 5

RLPT Equations of Motion @N xx @x @N xy @x + @N xy + @N yy @ ¹ Q x @x + @ ¹ Q y + @ @x @ P xx + c 1 @x " + c 1 I 0 Äu 0 + J 1 Ä Áx c 1 I 3 @ Äw 0 @x I 0 Äv 0 + J 1Áy Ä @ Äw 0 c 1 I 3 @w 0 Nxx @x + N @w 0 xy + @ @w 0 Nxy @x + N @w 0 yy + @ P xy @x + @ P yy + q I0 Äw 0 c 1I 6 @ Äw 0 @x + @ Äw 0 @Äu 0 I 3 @x + @Äv 0 @ Á Ä x + J4 @x + @ Á Ä y # Omit subscript 0 from u, v, and w @ M ¹ xx @x @ M ¹ xy @x + @ ¹ M xy + @ ¹ M yy ¹ Q x J 1 Äu 0 + K Ä Áx c 1 J 4 @ Äw 0 @x ¹ Q y J 1 Äv 0 + K Ä Áy c 1 J 4 @ Äw 0 ¹M M c 1 P ¹ Q Q c R Third-Order Laminate Plate Theory 6

RLPT Definition of Stress Resultants Conventional Stress Resultants < : N xx N yy N xy Z h h < : ¾ xx ¾ yy ¾ xy dz < : M xx M yy M xy Z h h < : ¾ xx ¾ yy ¾ xy z dz Higher-Order Stress Resultants < < : P xx P yy P xy Z h h : ¾ xx ¾ yy ¾ xy z3 dz ½ Rx R y ¾ Z h h ½ ¾yz ¾ z dz ¾ xz Mass Inertias I i Z h h ½ (z) i dz (i 0 1 6) J i I i c 1 I i+ K I c 1 I 4 + c 1I 6

RLPT Boundary Conditions Primary Variables : u n u s w 0 @w 0 @n Á n Á s Secondary Variables : N nn N ns ¹ Vn P nn ¹ Mnn ¹ Mns ¹V n c 1 µ @Pxx @x + @P xy n x + c 1 " µ I 3 Äu 0 + J 4 Ä Áx c 1 I 6 @ Äw 0 @x µ @Pxy @x n x " + @P yy n y µ # + I 3 Äv 0 + J 4Áy Ä @ Äw 0 c 1 I 6 n y + @P ns ¹Q x n x + ¹Q y n y + P(w0 ) + c 1 @s µ µ @w 0 P(w 0 ) N xx @x + N @w 0 @w 0 xy n x + N xy @x + N @w 0 yy n y Third-Order Laminate Plate Theory

Bending of a symmetric cross-ply (0/0/0/0) laminate under uniformly distributed load Deflection, w _ 0.00 0.01 0.016 0.014 0.01 0.010 0.00 0.006 3-D Elasticity Solution CLPT FSDT TSDT E 10 6 psi (7 Gpa) E 1 5E, G 1 G 13 0.5E G 3 0.E, ν 1 0.5 at x0 and xa v 0 w 0 0 _ φ y N xx M xx 0 y a SS 1 at y0 and yb u 0 w 0 0 _ φ x N yy M yy 0 b x 0.004 0 5 10 15 0 5 30 35 40 45 50 a/h SS-1 Figure Boundary 11.5 3 Conditions Third-Order Laminate Plate Theory

Bending of a symmetric cross-ply (0/0/0/0) laminate under uniformly distributed load 0.50 0.30 c ess coo d ate, 0.10-0.10-0.30 CLPT (E) FSDT (E) FSDT (C) TSDT (E) TSDT (C) (E): equilibrium-derived (C): constitutively-derived -0.50 0.00 0.10 0.0 0.30 0.40 0.50 0.60 Stress, σ _ xz (0,b/,z) Figure 11.5 6 Third-Order Laminate Plate Theory 10

Bending of a symmetric cross-ply (0/0/0/0) laminate under uniformly distributed load 0.50 0.30 c ess coo d ate, 0.10-0.10-0.30 CLPT (E) FSDT (E) FSDT (C) TSDT (E) TSDT (C) (E): equilibrium-derived (C): constitutively-derived -0.50 0.00 0.04 0.0 0.1 0.16 0.0 Stress, σ _ yz (a/,0,z) Third-Order Laminate Plate Theory 11

LAYERWISE LAMINATE THEORY y z Equilibrium of Interlaminar Stresses kth layer x (k+1)th layer z (k+1) σ zx (k+1) σ zy (k+1) σ zz (k) σ zx (k) σ zy (k) σ zz (k+1) σ zy (k+1) σ zx (k) σ zz (k+1) σ zz (k) σ zy (k) σ zx k+1 k x Layerwise Laminate Theory 1

INTERLAMINAR STRESS AND STRAIN CONTINUITY Equilibrium Requirements < ¾ xx ¾ : yy ¾ xy (k) < ¾ xx 6 ¾ : yy ¾ xy (k+1) < ¾ xz ¾ : yz ¾ zz (k) < ¾ xz ¾ : yz ¾ zz (k+1) < ¾ xz ¾ : yz ¾ zz (k) < ¾ xz ¾ : yz ¾ zz (k+1)! < xz : yz " zz (k) < xz 6 : yz " zz (k+1) Single-Layer Theories < ¾ xx ¾ : yy ¾ xy (k) < ¾ xx 6 ¾ : yy ¾ xy (k+1) < ¾ xz ¾ : yz ¾ zz (k) < ¾ xz 6 ¾ : yz ¾ zz (k+1) < " xx " : yy xy (k) < : " xx " yy xy (k+1) < xz : yz " zz (k) < xz yz : " zz (k+1) Layerwise Laminate Theory 13

LAYERWISE KINEMATIC MODEL NX u(x y z t) U I (x y t) I (z) I1 NX v(x y z t) V I (x y t) I (z) I1 MX w(x y z t) W I (x y t)ª I (z) I1 z Ith layer x z N I+1 I I 1 4 U N U I+1 U I U I 1 U 4 I+1 U I+1 U 3 U 3 1 I U I U 1 1 u I 1 U I 1 U I Φ I (z) Layerwise Laminate Theory 14

u(x y z t) v(x y z t) w(x y z t) Layerwise Displacement Field, Governing Equations, and FEM Approximation NX U I (x y t) I (z) I1 NX V I (x y t) I (z) I1 MX W I (x y t)ª I (z) I1 @N I xx @x @N I xy @x + @N I xy QI x N X + @N I yy QI y J1 NX J1 I IJ @ U J @t I IJ @ V J @t @ ~ Q I x @x + @ ~ Q I y ~ Q I z + ~ N I + q b ± I1 + q t ± IM MX J1 ~I IJ @ W J @t Finite element approximation p I3 I I1 U I (x y t) V I (x y t) W I (x y t) px U j I (t)ã j(x y) j1 px j1 V j I (t)ã j(x y) qx W j I (t)' j(x y) j1 15

Layerwise Kinematic Model Conventional 3D Layerwise D + 1D Cubic serendipity element Linear Lagrange element (1a) (in-plane) (through thickness) (1b) Quadratic serendipity element Quadratic Lagrange element (a) (in-plane) (b) (through thickness)

Table: Comparison of the number of operations needed to form the element sti ness matrices for equivalent elements in the conventional 3-D format and the layerwise -D format. Full quadrature is used in all. Element Type y Multipli. Addition Assignments 1a (3-D) 1,116,000 677,000 511,000 1b (LWPT) 43,000 370,000 106,000 a (3-D) 1,1,000 1,000 374,000 b (LWPT) 4,000 70,000 6,000 y Element 1a: 7 degrees of freedom, 4-node 3-D isoparametric hexahedron with cubic in-plane interpolation and linear transverse interpolation. Element 1b: 7 degrees of freedom, E1{L1 layerwise element. Element a: 1 degrees of freedom, 7-node 3-D isoparametric hexahedron with quadratic interpolation in all three directions. Element b: 1 degrees of freedom, E{Q1 layerwise element. Layerwise Laminate Theory 17

Layerwise Kinematic Model 3D modeling with D & 1D elements z z y y a h x a a a x -D quadratic Lagrangian element three quadratic layers through the thickness E 1 5 10 6 psi E E 3 10 6 psi G 1 0:5 10 6 psi G 13 G 3 0: 10 6 psi º 1 º 13 º 3 0:5 u(x a z) u(a y z) 0 v(a y z) u(x a z) 0 w(x a z) u(a y z) 0 Layerwise Laminate Theory 1

Validation of the Layerwise Theory Layerwise Laminate Theory 1

Verification of the Layerwise Theory Layerwise Laminate Theory 0

Variable Kinematic Model for Global-Local Analysis Composite displacement field: ESL Displacement field: u ESL 1 (x y z) u 0 (x y) + zá x (x y) u ESL (x y z) v 0 (x y) + zá y (x y) u ESL 3 (x y z) w 0 (x y) LWT Displacement field: u i (x y z) u ESL i (x y z) + ui LW T (x y z) u LW T 1 (x y z) u LW T (x y z) u LW T 3 (x y z) NX U I (x y) I (z) I1 NX V I (x y) I (z) I1 MX W I (x y)ª I (z) Layerwise Laminate Theory 1 I1

Laywerwise Kinematic Model (continued) z z 5 z 4 B FSDT Rotation U 5 0 U 4 Layerwise z 3 Translation U 3 U 1 z U z 1 z 1 A U 1 0 Layerwise Laminate Theory

y Sub-region Continuity of the Solution y LWT LWT LWT LWT LWT LWT LWT1 FSDT LWT1 LWT1 FSDT LWT1 FSDT FSDT FSDT FSDT FSDT FSDT At nodes At nodes, set U j V j 0, j1,,..,n, set W 0, j1,,..,n j x At nodes, set U j V j 0, j1,,..,n x (a) Enforcing strict subregion compatibility (b) Enforcing relaxed subregion compatibility Layerwise Laminate Theory 3

Free-Edge Problem E 1 0 10 6 psi. E.1 10 6 psi. E 3.1 10 6 psi. G 1 0.5 10 6 psi. G 13 G 3 G 1 ν 13 ν 3 ν 1 0.1 y a 10b b 4h h k Local Region Global Region (LWT) (FSDT) b x b a a 4 Layerwise Laminate Theory 4

Free-Edge Problem (continued) z h k 45 45 b b y Local Region LWT elements h k h k h k h k y Layerwise Laminate Theory 5

Free-Edge Problem (continued) Table: Description of global{local meshes for the (45/{45) s laminate under axial extension. Global-local meshes Remarks Mesh 1 Mesh Mesh 3 Mesh 4 Mesh 5 3D mesh Number of Elements in Local LWT Region 3 4 3 5 3 6 3 7 5 11 Width of Local Region 1 h k h k h k 3h k 16h k 6 Length of Local Region 5 a 6 5 a 6 5 a 6 5 a a Total Number of Active D.O.F. in VKFE Mesh 1,6,400,14 3,,116 (Strict Compatibility) Total Number of Active D.O.F. in VKFE Mesh,354,00 3,46 3,60,116 (Relaxed Compatibility) h k thickness of a single material ply. All ve VKFE meshes have the exact same in-plane discretization (5 11). Layerwise Laminate Theory 6

Free-Edge Problem (continued) z/hk σ zz Layerwise Laminate Theory 7

Free-Edge Problem (continued) z/hk σ xz Layerwise Laminate Theory

Free-Edge Problem (continued) z/hk σ xz σ zz Layerwise Laminate Theory

Summary In this lecture, we have discussed the following topics: Third-order Shear Deformation Plate Theory Development of governing equations Numerical results Layerwise Laminate Theory Development of governing equations Global-local analysis Numerical results Layerwise Laminate Theory 30