Composite Mechanics in the Last 50 Years - IIT Bombay · Composite Mechanics in the Last 50 Years Tarun Kant Indian Institute of Technology Bombay Powai, Mumbai -400 076 ... material

Composite Mechanics in the

Last 50 Years

Tarun Kant

Indian Institute of Technology Bombay

Powai, Mumbai - 400 076

Department of Civil Engineering

Introduction

FRC Lamina (or Ply)

is a single layer two-phase composite material in which high strength fibers are imbedded in a matrix material.

Composite LaminateComposite Laminate

2

FRC Laminate

consists of several laminae stacked together to achieve the required strength and stiffness to suit the needs of the designer.

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

Composite MechanicsComposite Mechanics

First paper

3

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


4

BasisTaylor’s Series Expansion

2-D approximate models/theories are deduced basedon different sets of assumptions:

� displacements� stresses� displacements and stresses

Kirchhoff (2D CLT)

� Plate is thin

� Transverse shear deformation is

neglected

� Tangential displacement vary

linearly through the thickness of

plate Before Deformation

0xz yz

γ γ= =

u u z v v zθ θ= + = +


5

� The thickness of laminate does not

change during deformation

� Transverse normal stress is very

small compared to other stresses

(neglected)

� State of stress is assumed as 2D

plane stress conditionAfter Deformation

o x o yu u z v v zθ θ= + = +

0z

σ ≅

0z

ε =

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.

Kirchhoff (2D CLT)


6

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.


Timoshenko, S.P. and Timoshenko, S.P. and WoinowskyWoinowsky--Krieger, S., Krieger, S.,

Theory of Plates and ShellsTheory of Plates and Shells, Second Edition, , Second Edition,

McGrawMcGraw--Hill, New York, 1959.Hill, New York, 1959.

for Isotropic Materialfor Isotropic Material

7

for Isotropic Materialfor Isotropic Material


Limitations

� Transverse shear deformability is not accountedin the formulation

� Transverse normal rotation/s of the cross section/sbecome/s first derivative/s of transverse displacement components

θθθθi = ±±±± ∂∂∂∂wo/ ∂∂∂∂xi

8

θθθθi = ±±±± ∂∂∂∂wo/ ∂∂∂∂xi

� Transverse displacement field turns C1 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.


Transverse Shear Energy

Need for Shear Deformation Theories

Laminated fibre-reinforced resin matrix composite structures

(beams, plates, shells etc.) exhibit more pronounced transverse

shear deformation than their conventional monolithic counterparts.

9

Need for Shear Deformation Theories

In contrast to in-plane properties,

� Transverse tensile &

� Interlaminar shear strengths of fibre-reinforced composite

laminates are quite low

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

� Transverse normal stress is very small

Before Deformation0z

ε =


10

� Transverse normal stress is very small

compared to other stresses (neglected)

After Deformation

0z

σ ≅

Comments

� Incorporates effect of transverse shear deformation� The transverse shear-angle is constant through

the thickness� A shear correction coefficient is necessarily used


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


11

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.

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

12

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.

3D domain subjected to the transverse loading

,z

w σ( , )p x y

z

Inplane Displacements : , ,

Inplane Stresses : , ,

Out-of-plane Stresses

Transverse Stresses : , ,

x y xy

zx zy z

u v w

σ σ τ

τ τ σ

/


13

,zx

u τ,

zyv τ

x

y

xσ

yσ

xyτ

yxτ

xzτ

yzτ

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.

3D Elasticity Solutions


14

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.

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

( ) ( )2 3

2 3

0 2 3

1 1, , , ,0

2! 3!

i i i

i i

u u uu x y z u x y z z z

z z zα

∂ ∂ ∂ = + + + + − − − −

∂ ∂ ∂


15

Contributions to modes of deformation

u = uo + z2 *

ou + …….

v = vo + z2*

ov + …….

w = zθz + z3 *

zθ + …….

3 * .......x xz zθ θ+ +

3 * .......y y

z zθ θ+ +

2 * .......o yw z θ+ +

Membrane mode Flexural mode

( ) ( )0 2 3

0 0 0

, , , ,02! 3!

i iu x y z u x y z z z

z z zα= + + + + − − − −

∂ ∂ ∂

� Plate is moderately thick

� Transverse shear deformation is

considered

� Tangential displacement vary linearly

through the thickness of plate

Before Deformation



16


� Transverse shear angle is constant


After Deformation

o x o y

u u z v v zθ θ= + = +

Before Deformation

� Displacement field in a form of

polynomial in thickness (z) direction

of a degree greater than one

2D HOSTs

2 * 3 *

2 * 3 *

2 *

..........

..........

..........

o x o x

o y o y

u u z z u z

v v z z v z

w w z z w

θ θ

θ θ

θ

= + + + + + ∞

= + + + + + ∞

= + + + + ∞


17

After Deformation

� Transverse shear deformation with

distortion of normal is considered

� No shear correction coefficient is

needed

� Generalized Hook’s law is considered

2 * ..........o z ow w z z wθ= + + + + ∞

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.

2D HOSTs

Whitney JM and Sun CT (1974), A refined theory for laminated anisotropic cylindrical shells, ASME J. Appl. Mech. 41, 471-476.


18

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.

2D HOSTs

Kant T (1982), Numerical analysis of thick plates, Computer Meth. Appl. Mech. Engg., 31, 1-18.

Kant T, Owen DRJ and Zienkiewicz OC (1982), A refined higher-order Co 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.


19

plates, ASME J. Appl. Mech. 51, 745-752.

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.

Manjunatha BS and Kant T (2002), New theories for symmetric/ unsymmetric composite and sandwich beams with Co finite elements , Composite Structures 23, 61-73.

Summary


Before DeformationCLT

20

CLT

FOST HOST


PRESENT HIGHER ORDER THEORY

REISSNER-MINDLIN THEORY

CLASSICAL PLATE THEORY

THREE-DIMENSIONAL ELASTICITY

HIGHER-ORDER SHEAR DEFORMATION THEORY

FIRST-ORDER SHEAR DEFORMATION THEORY

0.8

Simply supported ( 00/900/00 ) cross-ply

square laminate under sinusoidal transverse

load.

0.7

21

(σx

×× ××m

2)

a/h

0.6

0.5

0.4

0.3

0

0 25 50 75 100

[Kant T and Pandya BN (1988), Comput. Meth. Appl. Mech. Engng. 66, 173-198]


22

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.]


23

Percentage error in a two-layer (00/900) cross-ply laminate [Kant T and

Swaminathan K (2002) , Comp. Struct. 56, 329-344.]

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

Accuracy in prediction of inplane stresses and displacements


24

� 2D HOSTs

� 2D HOSNTs

� 2D Layer-wise Theories

� 3D Theories

stresses and displacements

“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*.”


25

* 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.

3D domain subjected to the transverse loading

,z

w σ( , )p x y

z

Inplane Displacements : , ,

Inplane Stresses : , ,

Out-of-plane Stresses

Transverse Stresses : , ,

x y xy

zx zy z

u v w

σ σ τ

τ τ σ

/


26

,zx

u τ,

zyv τ

x

y

xσ

yσ

xyτ

yxτ

xzτ

yzτ

( )1i

zxτ

+

( )1i

zyτ

+

( )1i

σ+

( 1) thi Layer+

y

Interlaminar Transverse Stresses


27

thi Layer

( )zyτ ( )zσ

( )i

zyτ

( )i

zσ

( )i

zxτ

zy x

Evaluation of Interlaminar Transverse Stresses

Use of Constitutive Relations

Interface

At an interface

DISPLACEMENT-BASED APPROACHES


28

At an interface

� Displacements :- Continuous

� All strains components :- Continuous

� All stress components :- discontinuous ( through material

constitutive relations)


while the actual situation is like

CONTINUOUS DISCONTINUOUS

Inplane Displacements

( , , ) u v w


29

( , , ) u v w

x y xy

Inplane S trains

( , , ) ε ε γ

x y xy

Inplane Stresses

( , , ) σ σ τzx zy z

Transverse Stresses

( , , ) τ τ σ

zx zy z

Transverse Strains

( , , ) γ γ ε


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 and transverse stresses .zx zy z( , , )γ γ ε zx zy z( , , ) τ τ σ


30

The evaluation of transverse stresses 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.

zx zy z( , , ) τ τ σ


0y xx z x

x y z

τσ τ∂∂ ∂+ + =

∂ ∂ ∂

In order to avoid the above discrepancy, the 3D equilibrium equations of elasticity are

integrated through the thickness after knowing inplane stresses


31

0

0

x y y z y

y zx z z

x y z

x y z

τ σ τ

ττ σ

∂ ∂ ∂+ + =

∂ ∂ ∂

∂∂ ∂+ + =

∂ ∂ ∂

3D Equations of Equilibrium


y xz x x

z y x y y

y zx zz

z x y

z x y

ττ σ

τ τ σ

ττσ

∂ ∂ ∂= − +

∂ ∂ ∂

∂ ∂ ∂ = − +

∂ ∂ ∂

∂ ∂∂= − +


32

DIRECT INTEGRATION METHOD

y zx zz

z x y

ττσ ∂ ∂∂= − +

∂ ∂ ∂

2 222

2 22

y zx zz

y x yxz

z z z x y

z x y x y

ττσ

σ τσσ

∂ ∂∂∂ ∂ = − +

∂ ∂ ∂ ∂ ∂

∂ ∂∂∂= + +

∂ ∂ ∂ ∂ ∂


( )( )

( )( )

( 1 )

1

1

1

1

i

L

i

i

hLL xyx

zx z h

i h

hLL y xy

dz Cx y

dz C

τστ

σ ττ

+

+

==

+

∂ ∂= − + +

∂ ∂

∂ ∂ = − + +

∑ ∫

∑ ∫


33

( )( 1 ) 2

1L

i

L y xy

zy z h

i h

dz Cy x

σ ττ

+==

∂ ∂ = − + +

∂ ∂ ∑ ∫

values obtained may not satisfy both boundary conditions at

as only one constant of integration is present2

hz = ±

( )( )

( 1)

1 2 22

3 42 21

2i

L

i

hLL y xyx

z z h

i h z

dz dz zC Cx y x y

σ τσσ

+

+

==

∂ ∂∂= + + + + ∂ ∂ ∂ ∂ ∑ ∫ ∫

and from last equation of equilibrium


34

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).

Problems/Difficulties

�The mathematical model for the integration of the transverse

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.


35

�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.

Motivation for, what we describe now, comes from a desire to

have an:

�effective,

�efficient and

Motivation


36

�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.

3D Plate

( , , ) , ( , , )z

w x y z x y zσ( , , ), ( , , )

zyv x y z x y zτ

( , )p x y

3D rectangular domain under transverse loading

37

( , , ), ( , , )zx

u x y z x y zτ

Dependent variable on a plane z = a constant = , , , , and zx zy z

u v w τ τ σ

� Each layer in the plate, is considered

to be in a 3D state of stress

� Bottom surface is free of any

3D Plate

Laminate mid

plane

y

z

h

38

Bottom surface is free of any

stresses and top surface is loaded

with transverse loading system

a

1Lz +

Lz

b

x

L = NL

L = 1L = 2

Constitutive Relations

with reference to the lamina coordinates

(before transformation)

i i i

1 11 12 13 1

2 22 23 2

3 33 3

12 44 12

13 55 13

23 66 23

C C C 0 0 0

C C 0 0 0

C 0 0 0

C 0 0

Sym. C 0

C

σ ε

σ ε

σ ε

τ γ

τ γ

τ γ

=

2 2

2 2

c s 0 2cs 0 0

s c 0 2cs 0 0

−

Basic Elasticity Relations in 3D

3D Plate

13, z2

α

39

( )[ ]

2 2

s c 0 2cs 0 0

0 0 1 0 0 0T

cs cs 0 c s 0 0

0 0 0 0 s c

0 0 0 0 c s

=

− −

−

x x11 12 13 14

y y22 23 24

z z33 34

xy xy44

55 56zx zx

66zy zy

Q Q Q Q 0 0

Q Q Q 0 0

Q Q 0 0

Q 0 0

Sym Q Q

Q

σ ε

σ ε

σ ε

τ γ

τ γ

τ γ

=

.

with reference to the laminate coordinates

(after transformation)

x

α

0

0

0

yxx zxx

xy y zy

y

yzxz zz

Bx y z

Bx y z

Bx y z

τσ τ

τ σ τ

ττ σ

∂∂ ∂+ + + =

∂ ∂ ∂

∂ ∂ ∂+ + + =

∂ ∂ ∂

∂∂ ∂+ + + =

∂ ∂ ∂

Equations of Equilibrium

Basic Elasticity Relations in 3D

3D Plate

40

; ; ;x y z

u v w

x y zε ε ε

∂ ∂ ∂= = =

∂ ∂ ∂

; ;xy xz yz

u v u w v w

y x z x z yγ γ γ

∂ ∂ ∂ ∂ ∂ ∂= + = + = +

∂ ∂ ∂ ∂ ∂ ∂

Strain-Displacement Relationship

fifteen unknowns in fifteen equations

, , , , , , , , , , , , , and x y z xy xz yz x y z xy xz yzu w v ε ε ε γ γ γ σ σ σ τ τ τ

Partial Differential Equations

( ) ( )65 66 55 56

55 66 56 65 55 66 56 65

31 34 32 34

33

1 1

1

zy zx zy zx

z

u w v wQ Q Q Q

z Q Q Q Q x z Q Q Q Q Y

w u u v vQ Q Q Q

z Q x y y x

τ τ τ τ

σσ

∂ ∂ ∂ ∂ = − + − = − − ∂ − ∂ ∂ − ∂

∂∂ ∂ ∂ ∂ ∂= − − − − ∂ ∂ ∂ ∂ ∂

zyzxzz

Bz x y

ττ ∂∂= − − −

∂ ∂ ∂

Primary Variables

, , , , &xz yz zu v w τ τ σ

3D Plate

41

2 2 2 2

13 31 13 3411 142 2

33 33

2 2 2 2

43 31 43 3441 442 2

33 33

2

13 3212

3

zxQ Q Q Qu u u u

Q Qz x Q x Q x y x y

Q Q Q Qu u u uQ Q

x y Q x y Q y y

Q QvQ

x y Q

τ ∂ ∂ ∂ ∂ ∂= − + + −

∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂− + + −

∂ ∂ ∂ ∂ ∂ ∂

∂− +

∂ ∂

2 2 2

13 34142 2

3 33

2 2 2 2

43 32 43 3442 442 2

33 33

13 43

33 33

z zx

Q Qv v vQ

x y Q x x

Q Q Q Qv v v vQ Q

y Q y Q x y x y

Q QB

Q x Q y

σ σ

∂ ∂ ∂+ −

∂ ∂ ∂ ∂

∂ ∂ ∂ ∂− + + −

∂ ∂ ∂ ∂ ∂ ∂

∂ ∂− − −

∂ ∂

2 2 2 2

23 31 23 3421 242 2

33 33

2 2 2 2

43 31 43 3441 442 2

33 33

2 2

23 32 23 3422 2 2

33 3

zy Q Q Q Qu u u uQ Q

z x y Q x y Q y y

Q Q Q Qu u u uQ Q

x Q x Q x y x y

Q Q Q Qv vQ

y Q y Q

τ∂ ∂ ∂ ∂ ∂= − + + −

∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂− + + −

∂ ∂ ∂ ∂ ∂ ∂

∂ ∂− + +

∂ ∂

2 2

24

3

2 2 2 2

43 32 43 3442 442 2

33 33

43 23

33 33

z zy

v vQ

x y x y

Q Q Q Qv v v vQ Q

x y Q x y Q x x

Q QB

Q x Q y

σ σ

∂ ∂−

∂ ∂ ∂ ∂

∂ ∂ ∂ ∂− + + −

∂ ∂ ∂ ∂ ∂ ∂

∂ ∂− − −

∂ ∂

3D Plate

Plate with simply (diaphragm) supported end conditions on all four edges

0yu w σ= = =

0x

v w σ= = =

y

z

42

0x

v w σ= = =

0yu w σ= = =

x

For a plate simply (diaphragm) supported on all four edges,

0

1,3,..... 1,3,.....

( , ) sin sinmn

m n

m x m yp x y p

a b

π π∞ ∞

= =

= ∑ ∑

Intensity of transverse loading can be expressed in the form of a Fourier series,

3D Plate

43

0 0

00

, for bi-directional sinusoidal load

corresponding to 1 harmonic

16 =

mn

st

mn

where p p

pp

mn

=

2 for uniformly distributed load

corresponding to harmonicthmn

π

Semi-analytical Approach

( , , ) ( ) cos sin

( , , ) ( ) sin cos

( , , ) ( )sin sin

mn

mn

mn

mn

mn

mn

m x n yu x y z u z

a b

m x n yv x y z v z

a b

m x n yw x y z w z

a b

π π

π π

π π

=

=

=

∑

∑

∑

Assumed Variation of Primary displacements Variables(Kantorovich method of transforming PDEs to ODEs)

satisfying the simple

(diaphragm) support end

conditions exactly on the

3D Plate

44

mn a b∑

and basic elasticity relations, it can be shown

conditions exactly on the

all four edges of plate

( , , ) ( ) cos sin

( , , ) ( ) sin cos

( , , ) ( )sin sin

zx zxmn

mn

zy zymn

mn

z zmn

mn

m x n yx y z z

a b

m x n yx y z z

a b

m x n yx y z z

a b

π πτ τ

π πτ τ

π πσ σ

=

=

=

∑

∑

∑Semi-analytical Approach

66

55 66 56 65

55

55 66 56 65

31 32

( )( ) ( )

( )( ) ( )

( ) 1( ) ( ) ( )

mnmn zxmn

mnmn zymn

mn

du z Qmw z z

dz a Q Q Q Q

dv z Qnw z z

dz b Q Q Q Q

dw z Q Qm nu z v z z

πτ

πτ

π πσ

= − +

−

= − +

−

= + +

2 2 2 2

13 31 43 3411 442 2

33 33

2

13 32 43 3412 44

33 33

13

33

( )( )

+ ( )

zxmn

mn

mn

d z Q Q Q Qm nQ Q u z

dz Q a Q b

Q Q Q Q mnQ Q v z

Q Q ab

Q m

Q a

τ π π

π

π

= − + −

− − +

−

( ) ( , )zmn xz B x zσ −

First-order Ordinary Differential Equations

3D Plate

45

31 32

33 33 33

( ) 1( ) ( ) ( )

( )( )

mnmn mn zmn

zmnzxmn

dw z Q Qm nu z v z z

dz Q a Q b Q

d z m nz

dz a b

π πσ

σ π πτ

= + +

= +

( ) ( , )zymn z

z B x zτ

−

2

31 23 43 3421 44

33 33

2 2 2 2

23 32 43 3422 442 2

33 33

23

33

( )( )

+ ( )

zymn

mn

mn

d z Q Q Q Q mnQ Q u z

dz Q Q ab

Q Q Q Qn mQ Q v z

Q b Q a

Q n

Q b

τ π

π π

πσ

= − − +

− + −

−

( ) ( , )zmn y

z B x z−

OR

( ) ( ) ( ) ( )d

z z z zdz

= +Cy y f

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

3D Plate

46

We propose to carryout partial discretization (finite element

discritization in x-y plane only) – results in a system of coupled

discrete first-order ordinary differential equations connecting all

finite element nodes.

To remove the above limitations

Idea of Generalization

47

Semi Discrete Approach

y

z

3D Plate

48

x

e yl e xl element ( )i

h∆

Concept of partial discretization

2 2( ), ( )z

w z zσ

4 4( ), ( )zyv z zτ

1 1( ), ( )zw z zσ

4 4( ), ( )z

w z zσ 3 3( ), ( )z

w z zσ

( ), ( )v z zτ

4 4( ), ( )zxu z zτ 3 3( ), ( )zxu z zτ

z

y

34

3D Plate

49

Bi-linear Plate Element

2 2z1 1( ), ( )zw z zσ

1 1( ), ( )zy

v z zτ

2 2( ), ( )zy

v z zτ

3 3( ), ( )zyv z zτ

x1 1( ), ( )

zxu z zτ

2 2( ), ( )zxu z zτ1

2

exl

eyl

Assumed Variations of Displacements in x-y Plane

4

1

ˆ ( , , ) ( , ) ( )i i

i

u u x y z N x y u z=

≈ = ∑

4

1

ˆ( , , ) ( , ) ( )i i

i

v v x y z N x y v z=

≈ = ∑4

1

ˆ ( , , ) ( , ) ( )i i

i

w w x y z N x y w z=

≈ = ∑

1

2

where,

( , ) 1

( , )

ex ey ex ey

x y x yN x y

l l l l

x x yN x y

l l l

= − − −

= −

(Kantorovich method of transforming PDEs to ODEs)

3D Plate

50

and through basic elasticity relations, it can be

shown

1i =

∑

4

1

ˆ ( , , ) ( , ) ( )zx xz i zx i

i

x y z N x y zτ τ τ=

≈ = ∑4

1

ˆ ( , , ) ( , ) ( )zy yz i zy i

i

x y z N x y zτ τ τ=

≈ = ∑4

1

ˆ ( , , ) ( , ) ( )z z i z i

i

x y z N x y zσ σ σ=

≈ = ∑

2

3

4

( , )

( , )

( , )

ex ex ey

ex ey

ey ex ey

N x yl l l

x yN x y

l l

y x yN x y

l l l

= −

=

= −

Strong Bubnov-Galerkin Weighted Residual Statements(with the help of governing Partial Differential Equations)

( ) 6 5 6 6

5 5 6 6 5 6 6 5

ˆ ˆ( , , ) 1 ( , , )ˆ ˆ( , ) ( , , ) ( , , ) 0

i zy zxA

u x y z w x y zN x y Q x y z Q x y z d A

z Q Q Q Q xτ τ

∂ ∂ + − + = ∂ − ∂

∫∫

( ) 55 56

ˆ ˆ( , , ) 1 ( , , )ˆ ˆ( , ) ( , , ) ( , , ) 0

i zy zx

v x y z w x y zN x y Q x y z Q x y z dAτ τ

∂ ∂ + − + + = ∫∫

3D Plate

51

( ) 55 56

55 66 56 65

ˆ ˆ( , ) ( , , ) ( , , ) 0i zy zx

AN x y Q x y z Q x y z dA

z Q Q Q Q yτ τ + − + + = ∂ − ∂

∫∫

31 34

3332 34

ˆ ˆ( , , ) ( , , )ˆ ( , , )

ˆ ( , , ) 1( , ) 0

ˆ ˆ( , , ) ( , , )

z

iA

u x y z u x y zx y z Q Q

x yw x y zN x y dA

v x y z v x y zz QQ Q

y x

σ ∂ ∂

− − ∂ ∂∂ − = ∂ ∂∂ − − ∂ ∂

∫∫

2 2

13 31 43 3411 442 2

33 3 3

2 2

43 3 1 1 3 3 4 13 344 1 14 14 2

3 3 33 3 3

2

43 324 2

3 3

ˆ ˆ ˆ( , , ) ( , , ) ( , , )

ˆ ˆ( , , ) ( , , )+

( , )ˆ ( ,

zx

i

x y z Q Q Q Qu x y z u x y zQ Q

z Q x Q y

Q Q Q Q Q Qu x y z v x y zQ Q Q

Q Q x y Q xN x y

Q Q v x yQ

Q

τ ∂ ∂ ∂+ − + −

∂ ∂ ∂

∂ ∂+ − − + −

∂ ∂ ∂

∂+ −

2

13 32 4 3 341 2 442

3 3 33

13 43

3 3 33

0ˆ, ) ( , , )

+

ˆ ˆ( , , ) ( , , ) ˆ+ ( , , )

A

z zx

d A

Q Q Q Qz v x y zQ Q

y Q Q x y

Q Qx y z x y zB x y z

Q x Q y

σ σ

= ∂

+ − − ∂ ∂ ∂ ∂ ∂

+ + ∂ ∂

∫∫

2 2

43 31 23 3441 242 2

33 33

2 2

23 31 43 34 43 34

ˆ ( , , ) ˆ ˆ( , , ) ( , , )

ˆ ˆ( , , ) ( , , )+ +

zyx y z Q Q Q Qu x y z u x y z

Q Qz Q x Q y


τ∂ ∂ ∂+ − + −

∂ ∂ ∂

∂ ∂+ − − −

52

23 31 43 34 43 3421 44 44 2

33 33 33

23 3222

33

ˆ ˆ( , , ) ( , , )+ +

( , )i


Q Q x y Q xN x y

Q QQ

Q

∂ ∂+ − − −

∂ ∂ ∂

∂+ −

2 2

43 32 23 3424 422

33 33

43 23

33 33

0ˆ ˆ( , , ) ( , , )

+

ˆ ˆ( , , ) ( , , ) ˆ + ( , , )

A

z zy

dA

Q Q Q Qv x y z v x y zQ Q

y Q Q x y

Q Qx y z x y zB x y z

Q x Q y

σ σ

= ∂

+ − − ∂ ∂ ∂ ∂ ∂

+ + ∂ ∂

∫∫

both equations contain second order derivatives of &u v

ˆ ( , , )ˆˆ ( , , )( , , ) ˆ( , ) ( , , ) 0zyzxz

i zA

x y zx y zx y zN x y B x y z dA

z x y

ττσ ∂ ∂∂+ + + =

∂ ∂ ∂ ∫∫

and

After replacing the above two equations in their weak forms andsubstitution of assumed variations in x-y plane, twenty-four first-ordercoupled ordinary differential equations are obtained

2

3

4

01 02 03 02 01 021

02 01 02 03

03 02 01 02

( )

( )

( )

( )

e e e e e ee

e e e e e

ee e e e

ee e e e

z

zd

zdz

z

=

A A A A B B

A A A A

A A A A

A A A A

y

y

y

y

1

2 2

3 3

4 4

03 041

05 06 07 08

09 10 11 12

( )

( )

( )

( )

e eee

e e e e e e

e ee e e e

e ee e e e

z

z

z

z

+

B B

B B B B

B B B B

B B B B

py

y p

y p

y p

3D Plate

53

402 03 02 01

( )e e e e z A A A Ay

4 413 14 15 16

( )e e e e z B B B By p

( ) ( ), ( ), ( ), ( ), ( ), ( )t

e e e e e e e

i i i i zxi zyi ziz u z v z w z z z zτ τ σ = y 4 5 6( , , ) 0,0,0, , , for, 1- 4

te e e e

i i i ix y z p p p i = = pand

in which

OR

( , ) ( ) ( , , ) ( ) ( , , )e e e e edx y z x y z z x y z

d z= +A By y p

Standard semi-discrete system of equations

After contributions of all the elements are taken into account

1 1 1

( , ) ( ) ( , , ) ( ) ( , , )n n n

e e e e e

k k k

dx y z x y z z x y z

dz= = =

= +∑ ∑ ∑A By y p

OR

( , ) ( ) ( , , ) ( ) ( , , )d

x y z x y z z x y zdz

= +A By y p

3D Plate

54

dz

On multiplication by [ ]1

( , )x y−

A

( ) ( , , ) ( ) ( , , )d

z x y z z x y zdz

= +Cy y f

[ ] [ ]1 1

( , , ) ( , ) ( , , ) and ( , , ) ( , ) ( , , )x y z x y x y z x y z x y x y z− −

= =C A B Af p

where

Static analysis of simply supported three-layered cross-ply symmetric plate under bi-directional

sinusoidal loading

, ;2 2 2

x

a b hσ

±

0,0,

2xy

hτ

±

0, ,0

2xz

bτ

,0,02

yz

aτ

, , 02 2

a bw a/h Source

4

Semi-analytical0.8010 -0.7550

(.0000) (.0000)

-0..0510 0.0505

(.0000) (.0000)

0.2560

(.0000)

0.2170

(.0000)2.0060

Partial FEM0.7556 -0.7128

(-5.668) (-5.589)

-0.0464 0.0458

(-9.019) (-8.400)

0.2583

(.8980)

0.2231

(2.811)2.0046

Pagano (1970) 0.8010 -0.7550 -0.0510 0.0500 0.2560 0.2170 --

Numerical Investigation

55

Pagano (1970) 0.8010 -0.7550 -0.0510 0.0500 0.2560 0.2170 --

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

10

Semi-analytical0.5900 -0.5900

(.0000) (.0000)

-0.0290 0.0290

(.0000) (.0000)

0.3570

(.0000)

0.1230

(.0000)0.7530

Partial FEM0.5750 -0.5750

(-2.542) (-2.542)

-0.0268 0.0268

(-7.586) (-7.586)

0.3550

(-.5600)

0.1200

(-2.439)0.7471

Pagano (1970) 0.5900 -0.5900 -0.0290 0.0290 0.3570 0.1230 --

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

Static analysis of simply supported three-layered sandwich plate under bi-directional sinusoidal

loading

0.25

0.50a/h=4

z

Semi-analytical

Partial FEM

Pagano (1970)

Ramtekkar

et al. (2002)

0.25

0.50

a/h=4

z


56

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-0.50

-0.25

0.00σ

x (a/2,b/2,z)

et al. (2002)

-0.02 -0.01 0.00 0.01 0.02

-0.50

-0.25

0.00

u (0,b/2,z)

Semi-analytical

Partial FEM

Ramtekkar

et al. (2003)

Static analysis of simply supported two-layered angle-ply composite plate under bi-directional

sinusoidal loading

0.25

0.50

a/h=4

z

0.25

0.50a/h=4


57

-0.10 -0.05 0.00 0.05 0.10 0.15

-0.50

-0.25

0.00τ

xy(0,0,z)

Partial FEM

Savoia and

Reddy (1992)

0.00 0.05 0.10

-0.50

-0.25

0.00

τyz

(a/2,0,z)z

Partial FEM

Savoia and

Reddy (1992)

ICCMS06, IIT Guwahati,, 8-10 December 2006

Concluding Remarks

Displacement Based 3D Finite Element Model

Mixed 3D Finite Element Model

• Displacements are the degree of freedoms

• Involved assumptions in all three directions

• Equations form is algebraic

• Displacements and corresponding stresses

are the degree of freedoms

58

Mixed Partial Finite Element Model


• Involved assumptions in all three directions

• Equations form is algebraic

• Displacements and corresponding stresses


• No assumption along the thickness direction

• Equations form is ODE system

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 post-

processing phase. The post processing phase is unfortunately beset with both

analytical and numerical problems/difficulties.

59

� Kantorovich method of transforming PDEs into a set of ODEs is generalized here by

introducing FEM discretization in place of assumed global functions for prismatic

domain defined by all but one independent coordinates.

� We can call this technique as a partial disretization procedure for BVPs defined by

elliptic equations. One can contrast this with usual partial discretization for time

dependent IVPs defined by parabolic and hyperbolic equations. . . .

d d d =M + C + K F

( ) ( )d

z zdz

= +A By y p

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 non-

60

� The technique is applicable to general BVPs, i.e., homogeneous equations with non-

homogeneous BCs, non-homogeneous equations with homogeneous and/or non-

homogeneous BCs.

� Both displacements and corresponding stresses are evaluated simultaneously with

same degree of accuracy.

Concluding Remarks

�Standard form of semi-discrete equation,

is always obtained for any problem wherein global properties are obtained by the

summation of the elemental properties as follows in the usual manner,

( ) ( ) ( , ) ( ) ( , )d

x z x z z x zdz

= +A By y p

( )n

ex= ∑A A

61

1

1

1

( )

( , )

( , )

e

k

ne

k

ne

k

x

x z

x z

=

=

=

=

=

=

∑

∑

∑

A A

B B

p p

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 .

62

� 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 discretization methodology for interlaminar stress computation in

composite laminates.” Computer Modeling in Engineering and Science, 17(2), pp. 135-161.

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

Composite Structures, 83, pp. 13-24 .

Acknowledgements

IIT BombayAeronautics R&D Board, Ministry of DefenceBoard of Research in Nuclear Sciences, DAE

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)

64

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)

Dedication

I wish to dedicate this lecture of mine to the

memory of the following:

� to my late uncle

65

� to late Professor C.K. Ramesh

66

for your kind attention

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