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.
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