O. Sevecek2, M. Pletz2, R. Bermejo1, P. Supancic1

(1) Institut für Struktur- und Funktionskeramik(2) Materials Center Leoben Forschung GmbH

Analytical Stress-Strain analysis of the laminates with orthotropic (isotropic) layers using Classical Laminate Theory + Comparison with FEA

28.06.2011

Motivation

• Fast insight on the laminate behaviour without FEA

• Creation of the interactive Mathematica applet for the fast laminate design (tailoring)

Laminate theory

• Scheme of the laminate and used notationz

Geometric midplane

Layer 1

Layer 2

Layer k

Layer (n-1)

Layer n

h0

h1h2

hk-1hk

hn-2

hn-1hn

H h0=hn=H/2tk

zk

• Main idea of the Laminate Theory• Set up the global laminate stiffeness matrix („Hookes´ law“).

• Solve the global laminate behaviour.

• Return back to single layers and solve the desired quantities.

Laminate theory

• Properties of the single layer

2 2

2 2

2 2

cos sin sin cossin cos sin cos

2sin cos 2sin cos cos sin

φ φ φ φφ φ φ φ

φ φ φ φ φ φ

⎡ ⎤−⎢ ⎥= ⎢ ⎥⎢ ⎥− −⎣ ⎦

T

11 12

21 22

66

0 1/ / 00 / 1/ 0

0 0 0 0 1/

L LT L

TL T T

LT

C C E EC C E E

S G

νν

−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= = −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

C Compliance matrix in the material CS

Transformation matrix –rotation of material

compliance matrix into xy CS

T= ⋅ ⋅C T C T Compliance matrix in CS xy1−

=S C Stiffeness matrix in CS xy

el ⋅ε = C σT

φx

Ly

Laminate theory

• Properties of the single layer

Vector of CTEs in material CS

x

y

xy

ααα

⎡ ⎤⎢ ⎥= ⋅ = ⎢ ⎥⎢ ⎥⎣ ⎦

α αC T C

Vector of CTEs in CS xy

Rotation of the CTEs vector

0

L

T

αα

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

αC

T= ⋅ − ⋅ Δασ S ε S

tot T⋅ + ⋅ Δαε = C σ Cth T⋅ Δαε = C

= ⋅α αS S C „Stiffeness“ temperature vector in CS xy

Hooke´s law of one layer

xφ

x

Ly

Laminate theory

• Calculation of the whole laminate

( )

( )

( )

11

2 21

1

3 31

1

1213

n

ijij k kk k

n

ijij k kk k

n

ijij k kk k

A S h h

B S h h

D S h h

−=

−=

−=

⎡ ⎤= −⎣ ⎦

⎡ ⎤= −⎣ ⎦

⎡ ⎤= −⎣ ⎦

∑

∑

∑

/ 2

/ 2

x xH

y yH

xy xy

NN dzN

σσσ−

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∫/ 2

/ 2

x xH

y yH

xy xy

MM z dzM

σσσ−

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∫ Stress and moment equilibriumover laminate height

0 011 12 16

0 021 22 26

0 016 26 66

x x x x

y x x x

xy xy xy xy

S S S

S S S z T

S S S

σ ε κ ασ ε κ ασ ε κ α

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ = ⋅ + ⋅ − ⋅ Δ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦⎢ ⎥⎣ ⎦

th th

th th

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= − = −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

0 0

0 0

N NN A B ε εK

M MM B D κ κ

Extension Bending Temperature

Hooke´s law of one layer

th th

th th

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= − = −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

0 0

0 0

N NN A B ε εK

M MM B D κ κ

Laminate theory

• Stiffeness matrix of the whole laminate

011 12 13 11 12 13

021 22 23 21 22 23

031 32 33 31 32 33

011 12 13 11 12 13

021 22 23 21 22 23

031 32 33 31 32 33

x x

y y

xy xy

x x

y y

xy xy

N A A A B B BN A A A B B BN A A A B B BM B B B D D DM B B B D D DM B B B D D D

εεγκκκ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥

= ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢⎢ ⎥ ⎢ ⎥⎢⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦

,

,

,

,

,

,

x th

y th

xy th

x th

y th

xy th

NNNMMM

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

− ⎢ ⎥⎢ ⎥

⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎣ ⎦

External Stress and Moment resultants Midplane

deformationsStiffeness matrix of the

laminate

„Hooke´s law“ forthe whole laminate

Thermal Stress and Moment resultants

1 1

x xn n

kk ky k y kk k

xy xyk k

T t T t zα αα αα α= =

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= ⋅ Δ ⋅ = ⋅ Δ ⋅ ⋅⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∑ ∑th thN S M S

knownunknown

Laminate theory

• Calculation of strain and stresses

,

,

,

x th x

y th y

xy th xyk k

Tε αε αε α

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= ⋅ Δ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

0 0,

0 0,

0 0,

x tot x x

y tot y y

xy tot xy xy

zε ε κε ε κε ε κ

⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ = + ⋅⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

el tot thε ε ε= −

[ ]0 011 12 16

0 021 22 26

0 016 26 66

x x x x

y x x x zz

xy xy xy xyz zz

S S S

S S S z T S

S S S

σ ε κ ασ ε κ ασ ε κ α

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎡ ⎤= ⋅ + ⋅ − ⋅ Δ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎣ ⎦⎣ ⎦

elε

1 th

th

− ⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤= +⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

0

0

NNεK

MMκDeformation and curvatures of the midplane

Strains over laminate height

Stresses over laminate height

Laminate theory

• Apparent properties of the whole laminate

0,

0 1,

0,

1x x th

x x th

xy xy th

NN

TN

ααα

−

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= ⋅ ⋅⎢ ⎥ ⎢ ⎥ Δ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

K

• Apparent CTEs

• Apparent E-modul, Poissons´ ratios ν, G-modul

,11

,22

1

1

x app

y app

EH A

EH A

=⋅

=⋅

1; − ⎡ ⎤⎡ ⎤= = ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦

A B A BK K

B D B D

12 12,

22 11

12 12,

11 22

xy app

yx app

A AA AA AA A

ν

ν

= = −

= = −, , 66

66

1xy app yx appG G A

H A= = =

⋅

H – total height of the laminate

Laminate theory

• Deformation of the laminate (plate curvatures)2 2

2

2 2

2

2

( )2

( )2

2 ( , )2

x x x

y y y

xy xy xy

w xw x dx dxx

w yw y dy dyx

w x yw x y dy dxx y

κ κ κ

κ κ κ

κ κ κ

∂− = = − = − ⋅

∂

∂− = = − = − ⋅

∂

∂ ⋅− = = − = − ⋅

∂ ∂

∫ ∫

∫ ∫

∫ ∫

( ) ( ) ( )2 2

( , )2 2 2tot x y xyx y x yw x y κ κ κ ⋅

= − ⋅ + − ⋅ + − ⋅

( , ) ( ) ( ) ( , )totw x y w x w y w x y= + +

deformation along x axis

deformation along y axis

twisting deformation

yx

Examples(for pure thermal loading)

Examples1) Symmetric laminate with isotropic layers:

Property Units M1 M2Young´s modulus E MPa 390000 280000Poissons ratio ν - 0.22 0.22CTE α (20-1200°C) 10-6K-1 9.8 8.0

M1

M2

M1THM1=1mmTHM2=1mm

ΔT=-1230°y

x

z

Examples

Stresses:

Deformation:

1) Symmetric laminate with isotropic layers:

σx σy

y x

yx

Examples

Strains:

1) Symmetric laminate with isotropic layers:

εx,el

εy,el

εx,th

εy,th

εx,tot

εy,tot

y x

Examples2) Non-symmetric laminate with isotropic layers:

Property Units M1 M2Young´s modulus E MPa 390000 280000Poissons ratio ν - 0.22 0.22CTE α (20-1200°C) 10-6K-1 9.8 8.0

M1

M2

THM1=1.5mmTHM2=1.5mm

ΔT=-1230°y

x

z

Examples2) Non-symmetric laminate with isotropic layers:

Stresses:

Deformation:

σx σy

Cut: y=0Cut: x=0

y x

yx

Examples2) Non-symmetric laminate with isotropic layers:

Strains

εx,el

εy,el

εx,th

εy,th

εx,tot

εy,tot

y x

Examples3a) Non-symmetric laminate with orthotropic layers:

0.238-Poissons ratio νLT(LZ)

8010-6K-1CTE αT (20°C)-410-6K-1CTE αL (20°C)

76000MPaYoung´s modulus EL

Property Units Value *

Young´s modulus ET MPa 5500

Shear modulus GLT(LZ) MPa 2300

L1 (90°)

L2 (0°)

THM1=1.5mmTHM2=1.5mm

ΔT=500°90°

θLX (L1)=90°

θLX (L1)=0°

* Epoxy Matrix Composite reinforced by 50% Kevlar fibers

y

x

z

Examples

Stresses:

Deformation:

3a) Non-symmetric laminate with orthotropic layers:

σx σy

Cut: y=0Cut: x=0

y x

yx

Examples

Strains

3a) Non-symmetric laminate with orthotropic layers:

εx,el

εy,el

εx,th

εy,th

εx,tot

εy,tot

y x

Examples

0.238-Poissons ratio νZL(TL)

8010-6K-1CTE αT (20°C)-410-6K-1CTE αL (20°C)

76000MPaYoung´s modulus EL

Property Units Value *

Young´s modulus ET MPa 5500

Shear modulus GZL(TL) MPa 2300

L1 (45°)

L2 (0°)THM1=1.5mmTHM2=1.5mm

ΔT=500°45°

θLX (L1)=45°

θLX (L1)=0°

* Epoxy Matrix Composite reinforced by 50% Kevlar fibers

3b) Non-symmetric laminate with orthotropic layers:

y

x

z

Examples

Stresses:

Deformation:

3b) Non-symmetric laminate with orthotropic layers:

σx σy

Cut: y=0Cut: x=0

y x

yx

Examples

Strains:

3b) Non-symmetric laminate with orthotropic layers:

εx,el

εy,el

εx,th

εy,th

εx,tot

εy,tot

y x

Summary

CODE INPUT *):

• Number of layers + layer thicknesses.

• Material properties of each layer (isotropic or orthotropic with general fiber orientation).

• BC - External force, moment or ΔT applied on the laminate(also combination of them).

CODE OUTPUT *):

• Stress and strain (total, thermal) distribution over all layers – in XY or LT coordinate system.

• Deformation of the laminate midplane (bending and also twisting)

• Apparent material properties of the whole laminate (homogenization).

• Position of the neutral plane.

• And another... (e.g. ply failure analysis, ...).

Model possibilities

*) Processed in Mathematica 8 or Matlab 2010

Conclusions

• Laminate theory and FE calculations are a in good agreement.

• Laminate theory is thus suitable for the fast laminate design (tailoring) – without need of FEM!

Application

• Design of the layer number, thicknesses or their material properties to:

• reach some maximal (residual) stresses in each layer.

• meet the requirements on the global laminate behaviour (total deformation).

• Determination of the critical laminate loading.

• ...

Literature

• Verbundwerkstoffe – Vorlesungsbehelf zu den Vorlesungen, Inst. für Konstruieren in Kunst- und Verbundstoffen, MU Leoben.

• A.T.Nettles, Basic Mechanics of Laminated Composite plates, NASA Reference Publication, MSFC, Alabama, 1994.

• R.M. Jones, Mechanics of Composite Materials – 2. edition, Taylor & Francis, Philadelphia, 1999.