Composite Mechanics in the Last 50 Years

Composite Mechanics in the Last 50 Years Tarun Kant Department of Civil Engineering Indian Institute of Technology Bombay Powai, Mumbai - 400 076

Composite Laminate Introduction FRC Lamina (or Ply) is a single layer two-phase composite material in which high strength fibers are imbedded in a matrix material. FRC Laminate consists of several laminae stacked together to achieve the required strength and stiffness to suit the needs of the designer. 2

Composite Mechanics First paper Reissner E and Stavsky Y (1961), Bending and stretching of certain types of heterogeneous aelotropic elastic plates, ASME J. Appl. Mech. 28, 402-408. 3

Composite Mechanics Mathematical Modeling In reality, a Laminate is a 3-D Body But 3-D models/theories are: analytically difficult computationally expensive not feasible for practical problems 2-D approximate models/theories are deduced based on different sets of assumptions: displacements stresses displacements and stresses Basis Taylor s Series Expansion 4

Composite Mechanics Kirchhoff (2D CLT) Plate is thin Transverse shear deformation is neglected γ = γ = 0 x y Tangential displacement vary linearly through the thickness of plate u = u + θ v = v + θ o x o y Before Deformation The thickness of laminate does not change during deformation ε = 0 Transverse normal stress is very small compared to other stresses (neglected) σ 0 State of stress is assumed as 2D plane stress condition After Deformation 5

Composite Mechanics Kirchhoff (2D CLT) Reissner E and Stavsky Y (1961), Bending and stretching of certain types of heterogeneous aelotropic elastic plates, ASME J. Appl. Mech. 28, 402-408. Dong SB, Pister KS and Taylor RL (1962), On the theory of laminated anisotropic shells and plates, J. Aerospace Sciences. 29, 969-975. Ambartsumyan SA (1964), The Theory of Anisotropic Shells, NASA TT F-118. Stavsky Y (1963), Thermo-elasticity of heterogeneous aelotropic plates, ASCE J. Engg. Mech. 89, 89-105. Whitney JM (1969), Cylindrical bending of unsymmetrically laminated plates, J. Composite Materials 3, 715-719. 6

Composite Mechanics Timoshenko, S.P. and Woinowsky-Krieger, S., Theory of Plates and Shells,, Second Edition, McGraw-Hill, New York, 1959. for Isotropic Material 7

Limitations Composite Mechanics Transverse shear deformability is not accounted in the formulation Transverse normal rotation/s of the cross section/s become/s first derivative/s of transverse displacement components θ i = ± w o / x i Transverse displacement field turns C 1 continuous Comments These analyses indicated the inadequacy and also the range of applicability of the CLT. Significance of the transverse shear deformation effects have also come out clearly through these analyses. 8

Composite Mechanics Transverse Shear Energy Laminated fibre-reinforced resin matrix composite structures (beams, plates, shells etc.) exhibit more pronounced transverse shear deformation than their conventional monolithic counterparts. Need for Shear Deformation Theories In contrast to in-plane properties, Transverse tensile & Interlaminar shear strengths of fibre-reinforced composite laminates are quite low 9

Composite Mechanics Reissner-Mindlin (2D RM-FOST) Shear correction coefficient is necessarily used to correct the strain energy due to shear deformation The thickness of laminate does not change during deformation ε = 0 Transverse normal stress is very small compared to other stresses (neglected) σ 0 Before Deformation Comments Incorporates effect of transverse shear deformation The transverse shear-angle is constant through the thickness A shear correction coefficient is necessarily used After Deformation 10

Composite Mechanics Reissner-Mindlin (2D RM-FOST) Reissner (1945) and Mindlin (1951), type first order transverse shear deformation theory (FOST) is introduced. Whitney JM and Pagano, NJ (1970), Shear deformation in heterogeneous anisotropic plates, ASME J Appl. Mech. 31, 1031-1036. Dong SB and TSO FKW (1972), On a laminated orthotropic shell theory including transverse shear deformation, ASME J. Appl. Mech. 39, 1091-1097. Rolefs R and Rohwer K (1997), Improved transverse shear stresses in composite finite elements based on first order shear deformation theory, Int. J. Num. Meth. Engg. 70, 51-60. 11

Composite Mechanics Comments on CLT, FOST and 3D Both CLT and FOST are inadequate to model the distortion of the transverse normals due to transverse shear and transverse normal stresses. Further FOST requires the introduction of arbitrary shear correction coefficients which are problem dependent. The exact analysis of Pagano (1969, 1970) and others confirm that the distortion of transverse normal is dependent not only on the laminate thickness, but also on the orientation and degree of orthotropy of the individual layers. Therefore, the hypothesis of non-deformable normal while acceptable for isotropic plates/shells is often unacceptable for multilayered anisotropic plates/shells with very large ratio of E/G, even if they are relatively thin. Thus, a transverse shear-normal deformation theory which also accounts for the distortion of the deformed normals would be necessary. 12

3D domain subjected to the transverse loading p( x, y ) w, σ Composite Mechanics Inplane Displacements : u, v, w Inplane Stresses : σ, σ, τ Out-of-plane Stresses x y xy Transverse Stresses : τ, τ, σ / x y y u τ, x v, τ y τ y σ y τ x τ yx τ xy σ x x 13

Composite Mechanics 3D Elasticity Solutions Pagano NJ (1969), Exact solution for composite laminates in cylindrical bending, J. Compos. Mater. 3, 398-411. Pagano NJ (1970), Exact solution for rectangular bidirectional composites and sandwich plates, J. Compos. Mater. 4, 20-34. Pagano NJ (1970), Influence of shear coupling in cylindrical bending of anisotropic laminates, J. Compos. Mater. 4,330-343. Srinivas S and Rao AK (1970), Bending, vibration and buckling of simple supported thick orthotropic rectangular plates and laminates, Int. J. Solids and Structures 6,1463-1481. Srinivas S and Rao AK (1971), A three dimensional solution for plates and laminates, J. Franklin Institute 291, 469-481. 14

Composite Mechanics Displacement Formulation Hilderbrand et al. (1949) pioneered a systematic reduction procedure of a 3-D elasticity problem to a 2-D shell theory of any given finite order by use of Taylor s series in powers of the thickness coordinate. Taylor s Series u 1 i u i 1 u i u x, y, = u x, y,0 + + + + α i (,, ) (,,0 ) 2 3 2 3 i 0 2 3 0 2! 3! 0 0 Contributions to modes of deformation u = u o + 2 * u +. o v = v o + 2 * v o +. w = θ + 3 * θ +. Membrane mode θ + θ +... x 3 * x θ + θ +... w o y 3 * y + θ +... 2 * y Flexural mode 15

Composite Mechanics Reissner-Mindlin (2D RM-FOST) Plate is moderately thick Transverse shear deformation is considered Tangential displacement vary linearly through the thickness of plate u = u + θ v = v + θ o x o y Before Deformation Transverse shear angle is constant through the thickness of plate After Deformation 16

Composite Mechanics 2D HOSTs Displacement field in a form of polynomial in thickness () direction of a degree greater than one u = u + θ + u + θ +... + 2 * 3 * o x o x v = v + θ + v + θ +... + 2 * 3 * o y o y w = w + θ + w +... + 2 * o o Before Deformation Transverse shear deformation with distortion of normal is considered No shear correction coefficient is needed Generalied Hook s law is considered After Deformation 17

2D HOSTs Composite Mechanics Hildebrand FB, Reissner E and Thomas GB (1949), Notes on the foundations of small displacements of othotropic shells, NACA TN-1833. Whitney JM and Sun CT (1973), A higher-order theory for extensional motion of laminated composites, J. Sound Vibr. 30, 85-97. Whitney JM and Sun CT (1974), A refined theory for laminated anisotropic cylindrical shells, ASME J. Appl. Mech. 41, 471-476. Lo KH, Christensen RM and Wu EM (1977),A higher-order theory of plate deformation-parts 1 and 2, ASME J. Appl. Mech. 44, 663-676. Levinson M (1980), An accurate, simple theory of the statics and dynamics of elastic plates, Mech. Res. Comm. 7, 343-350. Murthy MVV (1981), An improved transverse shear deformation theory for laminated anisotropic plates, NASA TP-1903. 18

Composite Mechanics 2D HOSTs Kant T (1982), Numerical analysis of thick plates, Computer Meth. Appl. Mech. Engg., 31, 1-18. Kant T, Owen DRJ and Zienkiewic OC (1982), A refined higher-order C o plate bending element, Computers and Structures 15, 177-183. Reddy JN (1984), A simple higher-order theory for laminated composite plates, ASME J. Appl. Mech. 51, 745-752. Manjunatha BS and Kant T (2002), New theories for symmetric/ unsymmetric composite and sandwich beams with Co finite elements, Composite Structures 23, 61-73. Kant T and Swaminathan K (2002), Analytical solutions for static analysis of laminated composite and sandwich plates based on a higher order refined theory, Composite Structures 31, 1-18. 19

Composite Mechanics Summary Before Deformation CLT FOST HOST 20

Composite Mechanics Simply supported ( 0 0 /90 0 /0 0 ) cross-ply square laminate under sinusoidal transverse load. 0.8 0.7 PRESENT HIGHER ORDER THEORY REISSNER-MINDLIN THEORY CLASSICAL PLATE THEORY THREE-DIMENSIONAL ELASTICITY HIGHER-ORDER SHEAR DEFORMATION THEORY FIRST-ORDER SHEAR DEFORMATION THEORY (σ x m 2 ) 0.6 0.5 0.4 0.3 0 0 25 50 75 100 a/h [Kant T and Pandya BN (1988), Comput. Meth. Appl. Mech. Engng. 66, 173-198] 21

Composite Mechanics Variation of nondimensional in-plane displacement through thickness of a three layer simply supported plate under sinusoidal transverse load. [Kant T and Swaminathan K (2002), Comp. Struct. 56, 329-344.] 22

Composite Mechanics Percentage error in a two-layer (0 0 /90 0 ) cross-ply laminate [Kant T and Swaminathan K (2002), Comp. Struct. 56, 329-344.] 23

Composite Mechanics It is established that inplane stresses and displacements can be evaluated reliably and reasonably accurately by the following analytical models (in ascending order of accuracy) 2D CLT 2D RM-FOST 2D HOSTs 2D HOSNTs 2D Layer-wise Theories 3D Theories Accuracy in prediction of inplane stresses and displacements 24

Composite Mechanics The main aim of entire investigation is to bring out clearly the accuracy of various shear deformation theories in predicting the inplane stresses so that the claims made by various investigators regarding the superemacy of their models are put to rest *. * Kant, T. and Swaminathan, K., Analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher order refined theory, Composite Structures, 56, 329-344, 2002. 25

3D domain subjected to the transverse loading p( x, y ) w, σ Composite Mechanics Inplane Displacements : u, v, w Inplane Stresses : σ, σ, τ Out-of-plane Stresses x y xy Transverse Stresses : τ, τ, σ / x y y u τ, x v, τ y τ y σ y τ x τ yx τ xy σ x x 26

Composite Mechanics Interlaminar Transverse Stresses ( ) i + 1 τ x ( i + 1) th Layer y x i + ( τ 1 y ) ( ) i σ ( σ ) i+ 1 σ ( τ ) i y i th Layer ( ) i τ x 27

DISPLACEMENT-BASED APPROACHES Evaluation of Interlaminar Transverse Stresses Use of Constitutive Relations Composite Mechanics Interface At an interface Displacements :- Continuous All strains components :- Continuous All stress components :- discontinuous ( through material constitutive relations) 28

Evaluation of Interlaminar Transverse Stresses while the actual situation is like Composite Mechanics CONTINUOUS DISCONTINUOUS Inplane Displacements ( u, v, w ) Inplane S trains ( ε, ε, γ ) x y xy Transverse Stresses ( τ, τ, σ ) x y Transverse Strains ( γ, γ, ε ) x y Inplane Stresses ( σ, σ, τ ) x y xy 29

Evaluation of Interlaminar Transverse Stresses Composite Mechanics Therefore, this path (displacement strains stresses through constitutive relations) for evaluation of these stresses is not suitable for layered systems. Completely wrong predictions are made concerning transverse strains ( γ x, γ y, ε ) and transverse stresses ( τ x, τ y, σ ). The evaluation of transverse stresses ( τ x, τ y, σ ) from the stress-strain constitutive relations lead to discontinuity at the interface of two adjacent layers (laminae) of a laminate and thus violates the Newton s third law- to every action there is an equal and opposite reaction. 30

Evaluation of Interlaminar Transverse Stresses Composite Mechanics In order to avoid the above discrepancy, the 3D equilibrium equations of elasticity are integrated through the thickness after knowing inplane stresses σ x τ τ y x x + + = x y τ x y σ y τ y + + = x y τ x τ σ x y y + + = 0 0 0 3D Equations of Equilibrium 31

Evaluation of Interlaminar Transverse Stresses Composite Mechanics τ x σ τ x = + x y τ τ σ = + x y σ τ τ x = + x y y x y x y y y σ τ τ x y = + x y 2 2 2 2 σ σ σ x y τ x y = + + 2 2 2 x y x y DIRECT INTEGRATION METHOD 32

Evaluation of Interlaminar Transverse Stresses τ ( ) L x = h ( L + 1 ) i = 1 ( h + 1) L i x xy = + d + h i ( h + 1) σ x τ y σ τ = + d + i = 1 y x h L i L y xy Composite Mechanics ( ) y = h C ( 1 ) 2 τ L + i C 1 values obtained may not satisfy both boundary conditions at as only one constant of integration is present = ± h 2 33

Composite Mechanics and from last equation of equilibrium ( ) σ = h( L+ 1) ( h + ) = + + d d + C + C i L i 1 2 2 2 L σ σ y τ x xy 2 2 2 i= 1 x y x y h 3 4 Two constants of integration are presents. The above equation is solved as a two-point boundary value problem (BVP) instead of initial value problem (IVP). 34

Composite Mechanics Problems/Difficulties There are serious limitations even in the approach just described. The estimates are not only inaccurate but the method is unreliable and the methodology lacks robustness. The mathematical model for the integration of the transverse shear stresses is an improperly posed BVP. Error accumulation due to the numerical evaluation of the higher derivatives of the displacements. 35

Composite Mechanics Motivation Motivation for, what we describe now, comes from a desire to have an: effective, efficient and accurate technique for evaluation /estimation of transverse interlaminar stresses in general laminated composites starting from the governing 3D partial differential equation (PDE) system of laminated composites. 36

3D Plate 3D rectangular domain under transverse loading p ( x, y ) w ( x, y, ), σ ( x, y, ) v( x, y, ), τ ( x, y, ) y u( x, y, ), τ ( x, y, ) x Dependent variable on a plane = a constant = u, v, w, τ x, τ y and σ 37

3D Plate y Laminate mid plane h Each layer in the plate, is considered to be in a 3D state of stress L = NL b Bottom surface is free of any stresses and top surface is loaded with transverse loading system L + 1 L x L = 2 L = 1 a 38

3D Plate Constitutive Relations i i i σ 1 C11 C12 C13 0 0 0 ε1 σ 2 C22 C23 0 0 0 ε 2 σ 3 C33 0 0 0 ε3 = τ 12 C44 0 0 γ 12 τ 13 Sym. C55 0 γ 13 τ 23 C66 γ 23 Basic Elasticity Relations in 3D with reference to the lamina coordinates (before transformation) 2 3, 1 2 2 c s 0 2cs 0 0 2 2 s c 0 2cs 0 0 0 0 1 0 0 0 [ T ] = 2 2 cs cs 0 ( c s ) 0 0 0 0 0 0 s c 0 0 0 0 c s α x with reference to the laminate coordinates (after transformation) σ x σ y 22 23 24 y σ Q33 Q34 0 0 ε = τ xy Q44 0 0 γ xy τ x τ y Q11 Q12 Q13 Q14 0 0 Q Q Q 0 0 Sym. Q Q 55 56 Q 66 ε x ε γ x γ y 39

3D Plate Basic Elasticity Relations in 3D Equations of Equilibrium σ τ x yx τ x + + + Bx x y = 0 τ xy σ y τ y + + + By x y = 0 τ τ x y σ + + + B x y = 0 u v w ε x = ; ε y = ; ε = ; x y u v xy ; u w x ; v γ = + γ = + γ w y = + y x x y Strain-Displacement Relationship fifteen unknowns in fifteen equations u, w, v, ε x, ε y, ε, γ xy, γ x, γ y, σ x, σ y, σ, τ xy, τ x and τ y 40

3D Plate Partial Differential Equations Primary Variables u, v, w, τ x, τ y & σ u 1 w v 1 w = Q τ + Q τ = Q τ Q τ Q Q Q Q x Q Q Q Q Y ( ) ( ) 65 y 66 x 55 y 56 x 55 66 56 65 55 66 56 65 w 1 u u v v σ τ τ x y = σ Q31 Q34 Q32 Q34 Q 33 x y y x = x y B 2 2 2 2 τ x u Q13Q 31 u Q13Q 34 u u = Q11 + 2 + 2 Q14 x Q33 x Q33 x y x y 2 2 2 2 u Q43Q 31 u Q43Q 34 u u Q41 + + Q 2 44 2 x y Q33 x y Q33 y y 2 2 2 2 v Q13Q 32 v Q13Q 34 v v Q12 + + Q14 x y Q 2 2 33 x y Q33 x x 2 2 2 2 v Q43Q 32 v Q43Q 34 v v Q42 + Q 2 + 2 44 y Q33 y Q33 x y x y Q σ Q σ Bx Q x Q y 13 43 33 33 2 2 2 2 τ y u Q23Q 31 u Q23Q 34 u u = Q21 + + Q 2 24 2 x y Q33 x y Q33 y y 2 2 2 2 u Q43Q 31 u Q43Q 34 u u Q41 + Q 2 + 2 44 x Q33 x Q33 x y x y 2 2 2 2 v Q23Q 32 v Q23Q 34 v v Q22 + + y 2 Q 2 Q24 33 y Q 33 x y x y 2 2 2 2 v Q43Q 32 v Q43Q 34 v v Q42 + + Q 2 44 2 x y Q33 x y Q33 x x Q43 σ Q23 σ By Q x Q y 33 33 41

3D Plate Plate with simply (diaphragm) supported end conditions on all four edges v = w = σ x = 0 u = w = σ y = 0 y u = w = σ y = 0 x v = w = σ x = 0 42

3D Plate For a plate simply (diaphragm) supported on all four edges, Intensity of transverse loading can be expressed in the form of a Fourier series, mπ x mπ y p( x, y) = p0mn sin sin a b m = 1,3,... n= 1,3,... where, p = p for bi-directional sinusoidal load p 0mn 0 0mn st corresponding to 1 harmonic 16p0 = for uniformly distributed load 2 mnπ th corresponding to mn harmonic Semi-analytical Approach 43

3D Plate Assumed Variation of Primary displacements Variables (Kantorovich method of transforming PDEs to ODEs) mπ x nπ y u( x, y, ) = umn ( )cos sin a b mn mπ x nπ y v( x, y, ) = vmn ( )sin cos a b mn mπ x nπ y w( x, y, ) = wmn ( )sin sin mn a b and basic elasticity relations, it can be shown mπ x nπ y τ x ( x, y, ) = τ x mn( )cos sin a b mn mπ x nπ y τ y ( x, y, ) = τ y mn( )sin cos a b mn mπ x nπ y σ ( x, y, ) = σ mn ( )sin sin a b mn satisfying the simple (diaphragm) support end conditions exactly on the all four edges of plate Semi-analytical Approach 44

3D Plate First-order Ordinary Differential Equations dumn( ) mπ Q = w + τ d a Q Q Q Q 66 mn ( ) xmn( ) 55 66 56 65 dvmn ( ) nπ Q = w + τ d b Q Q Q Q 55 mn( ) ymn ( ) 55 66 56 65 dw mn ( ) Q m π Q n π 1 u ( ) v ( ) σ ( ) d Q a Q b Q 31 32 = mn + mn + mn 33 33 33 dσ mn ( ) mπ nπ = τ xmn ( ) + τ d a b ymn d d ( ) B ( x, ) OR 2 2 2 2 dτ xmn( ) Q13Q 31 m π Q43Q 34 n π = Q11 + Q 2 44 u ( ) 2 mn d Q33 a Q33 b 2 Q13Q 32 Q43Q 34 mnπ + Q12 + Q44 vmn ( ) Q33 Q33 ab Q Q 13 33 23 33 mπ σ a mn ( ) B ( x, ) 2 dτ ymn ( ) Q31Q 23 Q43Q 34 mnπ = Q21 + Q44 umn ( ) d Q33 Q33 ab 2 2 2 2 Q23Q 32 n π Q43Q 34 m π + Q22 + Q 2 44 v ( ) 2 mn Q33 b Q33 a Q Q nπ σ b y( ) = C( ) y( ) + f ( ) mn ( ) B ( x, ) y x 45

3D Plate Limitations of Semi-Analytical Approach restricted to only simple support end conditions not capable to handle general angle-ply laminates To remove the above limitations We propose to carryout partial discretiation (finite element discritiation in x-y plane only) results in a system of coupled discrete first-order ordinary differential equations connecting all finite element nodes. 46

Idea of Generaliation 47

3D Plate Semi Discrete Approach y l e y l e x element ( i ) x h Concept of partial discretiation 48

3D Plate y w ( ), ( ) w ( ), ( ) 4 σ 4 3 σ 3 4 4 ( ), τ x4 ( ) 3 3 τ x3 u u ( ), ( ) w ( ), σ ( ) 1 1 v ( ), τ ( ) 4 y 4 w ( ), σ ( ) 2 2 v ( ), τ ( ) 3 y3 l ey v ( ), τ ( ) 1 y1 1 u ( ), τ ( ) 1 x1 l ex v ( ), τ ( ) 2 y 2 2 u ( ), τ ( ) 2 x2 x Bi-linear Plate Element 49

Assumed Variations of Displacements in x-y Plane u uˆ( x, y, ) N ( x, y ) u ( ) = 4 i = 1 v vˆ( x, y, ) N ( x, y) v ( ) = 4 i= 1 w wˆ ( x, y, ) N ( x, y ) w ( ) = 4 i =1 and through basic elasticity relations, it can be shown 4 τ ˆ τ ( x, y, ) N ( x, y ) τ ( ) = x x i x i i = 1 τ ˆ τ ( x, y, ) N ( x, y ) τ ( ) = y y i y i i = 1 σ σˆ ( x, y, ) N ( x, y ) σ ( ) = i i i = 1 4 4 i i i (Kantorovich method of transforming PDEs to ODEs) i i i where, N N N N 1 2 3 4 x y x y ( x, y) = 1 l l l l ( x, y ) ( x, y) x = l = ex x y l l ex ey y ( x, y) = l ey ex ey ex ey x y l l ex ey x y l l ex ey 3D Plate 50

Strong Bubnov-Galerkin Weighted Residual Statements (with the help of governing Partial Differential Equations) 3D Plate A uˆ ( x, y, ) 1 wˆ ( x, y, ) N ˆ ˆ i ( x, y ) Q 6 5τ y ( x, y, ) Q 6 6τ x ( x, y, ) + + d A = 0 ( Q 5 5Q 6 6 Q 5 6Q 6 5 ) x vˆ ( x, y, ) 1 wˆ ( x, y, ) N ( x, y ˆ ˆ i ) 55 (,, ) 56 (,, ) 0 A y x ( 55 66 56 65 ) Q τ x y Q τ Q Q Q Q x y + + + y da = A uˆ ( x, y, ) uˆ ( x, y, ) σˆ ( x, y, ) Q 31 Q 34 wˆ ( x, y, ) 1 x y N i ( x, y ) da = 0 Q ˆ ˆ 33 v( x, y, ) v( x, y, ) Q 32 Q 34 y x 51

A N i 2 2 ˆ τ ˆ ˆ x ( x, y, ) Q13Q 31 u ( x, y, ) Q 43Q 34 u ( x, y, ) + Q11 + 2 Q 44 2 Q 33 x Q 3 3 y 2 2 Q ˆ ˆ 43Q 3 1 Q1 3Q 3 4 u ( x, y, ) Q13Q 34 v ( x, y, ) + Q 4 1 + Q14 + Q14 2 Q 3 3 Q 33 x y Q 3 3 x ( x, y ) 2 2 Q 43Q 32 vˆ( x, y, ) Q13Q 32 Q 4 3Q 34 vˆ ( x, y, ) + Q 4 2 + 2 Q1 2 + Q 44 Q 3 3 y Q 3 3 Q 33 x y Q ˆ ˆ 13 σ ( x, y, ) Q 43 σ ( x, y, ) + + + Bˆ x ( x, y, ) Q 3 3 x Q 33 y d A = 0 A N i 2 2 ˆ τ y ( x, y, ) Q 43Q ˆ ˆ 31 u ( x, y, ) Q 23Q 34 u ( x, y, ) + Q 41 + 2 Q 24 2 Q 33 x Q 33 y 2 2 Q 23Q 31 Q 43Q 34 uˆ ( x, y, ) Q 43Q 34 v ˆ ( x, y, ) + Q 21 + Q 44 + Q 44 2 Q 33 Q 33 x y Q 33 x ( x, y ) 2 2 Q 23Q 32 vˆ ( x, y, ) Q 43Q 32 Q 23Q 34 vˆ ( x, y, ) + Q 22 + 2 Q 24 + Q 42 Q 33 y Q 33 Q 33 x y Q ˆ ˆ 43 σ ( x, y, ) Q 23 σ ( x, y, ) + + + Bˆ y ( x, y, ) Q 33 x Q 33 y both equations contain second order derivatives of & û ˆv da = 0 and ˆ σ (,, ) ˆ (,, ) ˆ (,, ) x y τ x x y τ y x y N (, ) ˆ i x y + + + B ( x, y, ) da = 0 A x y 52

After replacing the above two equations in their weak forms and substitution of assumed variations in x-y plane, twenty-four first-order coupled ordinary differential equations are obtained e e e e e e e e 01 02 03 02 e A A A A 01 02 1 ( ) B B y 03 04 e e B B 1 ( ) y p 1 e e e e e e e e e e A 02 A01 A02 A03 d y ( ) 2 e B 05 B 06 B 07 B 08 y ( ) 2 p 2 e e e e e = e + e d ( ) A03 A02 A01 A02 y e e e e ( ) B 09 B 10 B 11 B 12 y 3 p 3 e e e e e e A02 A03 A02 A 01 ( ) 4 4 B13 B14 B15 B y 16 p 3 e e e e e y ( ) 4 in which ( ) = u ( ), v ( ), w ( ), τ ( ), τ ( ), σ ( ) t e e e e e e e e e e e yi i i i xi yi i and i i4 i5 i6 A OR t 3D Plate p ( x, y, ) = 0,0,0, p, p, p for, i = 1-4 d ( x, y ) y ( ) = B ( x, y, ) y ( ) + p ( x, y, ) d e e e e e Standard semi-discrete system of equations 53

3D Plate After contributions of all the elements are taken into account d ( x, y) y ( ) = B ( x, y, ) y ( ) + p ( x, y, ) d n n n e e e e e A k = 1 k = 1 k = 1 where OR d A( x, y) y( ) = B( x, y, ) y( ) + p( x, y, ) d On multiplication by [ A ( x, y )] 1 d y ( ) = C ( x, y, ) y ( ) + f ( x, y, ) d [ ] f [ ] 1 1 C ( x, y, ) = A ( x, y ) B ( x, y, ) and ( x, y, ) = A ( x, y ) p ( x, y, ) 54

Numerical Investigation Static analysis of simply supported three-layered cross-ply symmetric plate under bi-directional sinusoidal loading a/h Source a b h σ x, ; ± 2 2 2 h τ xy 0,0, ± 2 b τ x 0,,0 2 a τ y,0,0 2 a b w,,0 2 2 Semi-analytical 0.8010-0.7550 (.0000) (.0000) -0..0510 0.0505 (.0000) (.0000) 0.2560 (.0000) 0.2170 (.0000) 2.0060 4 Partial FEM 0.7556-0.7128 (-5.668) (-5.589) -0.0464 0.0458 (-9.019) (-8.400) 0.2583 (.8980) 0.2231 (2.811) Pagano (1970) 0.8010-0.7550-0.0510 0.0500 0.2560 0.2170 -- 2.0046 Ramtekkar et al. (2002) 0.8080-0.7600-0.0510 0.0500 0.2570 0.2210 2.0070 Kant et al. (2002) 0.7670 -- - 0.0500 -- -- 1.9260 Semi-analytical 0.5900-0.5900 (.0000) (.0000) -0.0290 0.0290 (.0000) (.0000) 0.3570 (.0000) 0.1230 (.0000) 0.7530 10 Partial FEM 0.5750-0.5750 (-2.542) (-2.542) -0.0268 0.0268 (-7.586) (-7.586) 0.3550 (-.5600) 0.1200 (-2.439) Pagano (1970) 0.5900-0.5900-0.0290 0.0290 0.3570 0.1230 -- 0.7471 Ramtekkar et al. (2002) 0.5940-0.5940-0.0290 0.0290 0.3580 0.1240 0.8560 Kant et al. (2002) 0.5850 -- - 0.0281 -- -- 0.7176 55

Numerical Investigation Static analysis of simply supported three-layered sandwich plate under bi-directional sinusoidal loading a/h=4 0.50 0.50 a/h=4 Semi-analytical Partial FEM 0.25 Pagano (1970) Ramtekkar et al. (2002) σ x (a/2,b/2,) 0.00-1.5-1.0-0.5 0.0 0.5 1.0 1.5 0.25 u (0,b/2,) 0.00-0.02-0.01 0.00 0.01 0.02-0.25-0.25 Semi-analytical Partial FEM Ramtekkar et al. (2003) -0.50-0.50 56

Numerical Investigation Static analysis of simply supported two-layered angle-ply composite plate under bi-directional sinusoidal loading 0.50 0.50 a/h=4 a/h=4 0.25 0.25 τ xy (0,0,) 0.00-0.10-0.05 0.00 0.05 0.10 0.15 τ y (a/2,0,) 0.00 0.00 0.05 0.10-0.25 Partial FEM Savoia and Reddy (1992) -0.25 Partial FEM Savoia and Reddy (1992) -0.50-0.50 ICCMS06, IIT Guwahati,, 8-10 December 2006 57

Concluding Remarks Displacement Based 3D Finite Element Model Displacements are the degree of freedoms Involved assumptions in all three directions Equations form is algebraic Mixed 3D Finite Element Model Displacements and corresponding stresses are the degree of freedoms Involved assumptions in all three directions Equations form is algebraic Mixed Partial Finite Element Model Displacements and corresponding stresses are the degree of freedoms No assumption along the thickness direction Equations form is ODE system 58

Concluding Remarks Motivation for this presentation came from a desire to have an effective, an efficient and an accurate technique for evaluation/estimation of transverse interlaminar stresses. In the available approaches, the inplane lamina stresses are first computed in the first phase of any general laminate analysis. The transverse interlaminar stresses are then estimated by integrating the 3D elasticity equilibrium equations in the second postprocessing phase. The post processing phase is unfortunately beset with both analytical and numerical problems/difficulties. Kantorovich method of transforming PDEs into a set of ODEs is generalied here by introducing FEM discretiation in place of assumed global functions for prismatic domain defined by all but one independent coordinates. d A y( ) = By( ) + p d We can call this technique as a partial disretiation procedure for BVPs defined by elliptic equations. One can contrast this with usual partial discretiation for time dependent IVPs defined by parabolic and hyperbolic equations.... M d + C d + K d = F 59

Concluding Remarks This technique occupies an intermediate position between exact (?) and fully discrete solutions. The advantage of this technique, apart from its great accuracy, consists in that only part of the expression giving the solution is chosen a priori (global or discrete), part of the functions being determined in accordance with the character of the physics of the problem. The technique is applicable to general BVPs, i.e., homogeneous equations with nonhomogeneous BCs, non-homogeneous equations with homogeneous and/or nonhomogeneous BCs. Both displacements and corresponding stresses are evaluated simultaneously with same degree of accuracy. 60

Standard form of semi-discrete equation, A d ( x) y ( ) = B ( x, ) y ( ) + p ( x, ) d Concluding Remarks is always obtained for any problem wherein global properties are obtained by the summation of the elemental properties as follows in the usual manner, A B p = = = n k = 1 n k = 1 n k = 1 A B p e e e ( x) ( x, ) ( x, ) 61

Recent References Tarun Kant, Yogesh Desai and Sandeep Pendhari, 2008, Stress analyses of laminates under cylindrical bending. Communication in Numerical Methods in Engineering, 24(1), pp. 15-32. Tarun Kant, Sandeep S. Pendhari and Yogesh M. Desai, 2007, A new partial finite element model for statics of sandwich plates. Journal of Sandwich Structures and Materials,, 9(5), pp. 487-520. Tarun Kant, Sandeep S. Pendhari and Yogesh M. Desai, 2007, A novel finite element numerical integration model for composite laminates supported on two opposite edges. ASME Journal of Applied Mechancis, 74(6), pp. 1114-1124. Tarun Kant, Sandeep S. Pendhari and Yogesh M. Desai, 2007, A general discretiation methodology for interlaminar stress computation in composite laminates. Computer Modeling in Engineering and Science, 17(2), pp. 135-161. 62

Recent References Tarun Kant, Sandeep S. Pendhari and Yogesh M. Desai, 2007, On accurate stress analysis of composite and sandwich narrow beams. International Journal for Computational Methods in Engineering Sciences and Mechanics, 8(3), pp. 165-177. Tarun Kant, Avani B. Gupta, Sandeep S. Pendhari and Yogesh M. Desai, 2008, Elasticity solution of cross ply composite and sandwich plates. Composite Structures, 83, pp. 13-24. 63

IIT Bombay Aeronautics R&D Board, Ministry of Defence Board of Research in Nuclear Sciences, DAE Acknowledgements NP Sahani (MTech 1984) MG Kollegal (MTech 1992) AS Bookwala (MTech 1985) JR Kommineni (PhD 1993) S Sharma (MTech 1986) HS Patil (PhD 1993) BN Pandya (PhD 1987) Vijay Rode (PhD 1996) JH Varaiya (PhD 1988) RK Khare (PhD 1996) CP Arora (MTech 1988) SR Bhate (PhD 1999) Mallikarjuna (PhD 1989) Shrish Kale (PhD 2000) AB Gupta (MTech 1990) K Swaminathan (PhD 2000) BS Manjunatha (PhD 1991) C Sarath Babu (PhD 2001) TS Reddy (MTech 1991) VPV Ramana (PhD 2003) MP Menon (PhD 1992) C. V. Subbaiah (MTech 2005) SS Pendhari (PhD 2007) 64

Dedication I wish to dedicate this lecture of mine to the memory of the following: to my late uncle to late Professor C.K. Ramesh 65

for your kind attention 66