Lecture 6: Laminate theory for papermakers Sören Östlund

After Lecture 6 you should be able to ... • describe the basic assumptions in laminate theory • relate the features of laminate theory to basic concepts in solid mechanics • use laminate theory to predict stresses and strains in paper sheets subjected to mechanical loading and moisture changes

1

Literature • Pulp and Paper Chemistry and Technology Volume 4, Paper Products Physics and Technology, Chapter 11

With the help of laminate theory it is possible to … • Make sensitivity investigations of the causes of curl and twist • Calculate bending stiffness for different multi-ply structures • Calculate strains and stresses in paper sheets

2

Definition of a laminate • A laminate consists of a number of uniformly thick layers which do not move relative to each other other.

1 ϕ

x

• The laminate is ordered in a coordinate system 1-2, with the 3-axis, the Zdirection, directed downwards.

y

2

• Each layer is assumed to be orthotropic. • Each layer is ordered in a coordinate system x-y x y (on (on-axis) axis), which lies parallel with the 1-2 system (off-axis).

1 z0

• The positive angle is measured clockwise from the 1-system to the xsystem.

2

z1

z2 zk

zN

t/2

3

t t/2

k n

Z, 3

The thickness coordinates n

t = ∑ tk k =1

t z0 = − 2

1 z0

2

z1

z2 zk

zN

t/2

3 k

t t/2

n

z k = z k −1 + t k

Z, 3

tk = the thickness of layer k

3

Definition of positive stress and strain 1 N1

2

Stress N1 and strain

ε10 in the 1-direction

Stress N2 and strain

ε 20 in the 2-direction

N2 1

2 1

Shear stress N6 and shear strain

N6

ε 60

2

Curl of paper can be split into three curvature components 1 2

M1

κ1 = −

∂2w ∂x12

κ1 =

1 R1

⎡1⎤ ⎢⎣ m ⎥⎦

κ2 = −

∂2w ∂x 22

κ2 =

1 R2

⎡1⎤ ⎢⎣ m ⎥⎦

3 M2

1

o

o

M6 3

w

2

κ 6 = −2

∂2w ∂x1 x 2

(h )

κ 6 = −2 b l

⎡1⎤ ⎢m⎥ ⎣ ⎦

4

Twist

The twist, κ6, can be illustrated as the change in the inclination of the surface from point A to point B 1 l

h

B h

A 2 b

∂2w κ 6 = −2 ∂x1 x 2

κ6

h ) ( = −2 b

⎡1⎤ ⎢m⎥ ⎣ ⎦

l

Hooke’s law for plane stress ⎡ 1 ⎢ E ⎧εx ⎫ ⎢ x ⎪ ⎪ ⎢ ν yx ⎨ ε y ⎬ = ⎢− ⎪γ ⎪ ⎢ E y ⎩ xy ⎭ ⎢ ⎢ 0 ⎢⎣



ν xy Ex

1 Ey 0

⎡ 1 ⎤ 0 ⎥ ⎢ Ex ⎥ ⎧σ ⎫ ⎧εx ⎫ ⎢ x ⎢ ⎥⎪ ⎪ ν xy ⎪ ⎪ 0 ⎥ ⎨ σ y ⎬ or ⎨ ε y ⎬ = ⎢ − ⎥⎪ ⎪ ⎪ ⎪ ⎢ Ex ⎩γ xy ⎭ ⎢ ⎥ ⎩σ xy ⎭ 1 ⎥ ⎢ 0 ⎢⎣ Gxy ⎥⎦

because



ν yx Ey

1 Ey 0

⎤ 0 ⎥ ⎥ ⎧σ ⎫ ⎥⎪ x ⎪ 0 ⎥ ⎨σ y ⎬ ⎥⎪ ⎪ ⎥ ⎩σ xy ⎭ 1 ⎥ Gxy ⎥⎦

E x ν xy = E y ν yx

5

Stiffness relations ⎡σ x ⎤ ⎡Qxx ⎢ ⎥ ⎢ ⎢σ y ⎥ = ⎢Qyx ⎢τ xy ⎥ ⎣⎢ 0 ⎣ ⎦ Qxx =

where

Qyx =

Ex

1 −ν xyν yx

ν yx E x 1 −ν xyν yx

Qxy Qyy 0

0 ⎤ ⎡εx ⎤ ⎢ ⎥ 0 ⎥⎥ ⎢ ε y ⎥ Qss ⎦⎥ ⎢⎣γ xy ⎥⎦

Qxy = Qyy =

ν xy E y 1 −ν xyν yx Ey 1 −ν xyν yx Qss = Gxy

Comparison between isotropic and orthotropic elastic materials Isotropic

G=

E 2(1+ ν )

Orthotropic paper materials (Baum)

Gxy =

Ex E y 2 ⎡⎣1 + ν xyν yx ⎤⎦

ν xyν yx = 0, 0 293 Gxy = 0.387 E x E y

E x ν xy = E y ν yx

ν xy = 0.293

Ex Ey

6

Strain in the plane due to moisture absorption βH = hygroexpansion coefficient

ε H = β H ΔH

ΔH = the change in the moisture content of the paper

ε H = β RH ΔRH

RH = relative humidity

ε H = β mc Δmc

mc = moisture content

Total T t l strain t i in i a layer l d due tto b both th moisture i t sorption and mechanical forces ε =εM +εH

Strain due to bending

M

R

M

ε B = zκ =

z R

Compression

Total strain

ε = ε 0 + εB

Tension Neutral

where the strain of the mean surface is

ε0

7

Total strain in a given layer due to moisture changes, mechanical influences and bending

strain due to change in moisture

ε M = ε 0 + zκ − β H ΔH bending strain

membrane strain

Forces and bending moments Not σ because 2D-structure

(

) ΔH ) dz

N = ∫ Q ε 0 + zκ − β H ΔH dz

(

M = ∫ Qz ε 0 + zk − β H

Integrate over sheet thickness n 1 ⎛ ⎞ N = ∑ ⎜ Qε 0 ( zi − zi −1 ) + Qk zi2 − zi2−1 − Q β H ΔH ( zi − zi −1 ) ⎟ 2 ⎠ i =1 ⎝

(

)

n 1 1 ⎛1 ⎞ M = ∑ ⎜ Qε 0 zi2 − zi2−1 + Qκ zi3 − zi3−1 − Q β H ΔH zi2 − zi2−1 ⎟ 3 2 ⎠ i =1 ⎝ 2

(

)

(

)

(

)

8

Compact notation ⎡ N ⎤ ⎡ A B ⎤ ⎡ε 0 ⎤ ⎡ N H ⎤ ⎢ M ⎥ = ⎢ B D ⎥ ⎢ ⎥ − ⎢ H ⎥ or ⎣ ⎦ ⎣ ⎦ ⎣⎢ κ ⎥⎦ ⎣⎢ M ⎦⎥

⎡ N ⎤ ⎡ N H ⎤ ⎡ A B ⎤ ⎡ε 0 ⎤ ⎢M ⎥ + ⎢ H ⎥ = ⎢B D⎥ ⎢ ⎥ ⎣ ⎦ ⎣⎢ M ⎦⎥ ⎣ ⎦ ⎣⎢ κ ⎥⎦

where n

A = ∑ Qi [ zi − zi −1 ] i =1

B=

[

1 n Qi z i2 − z i2−1 ∑ 2 i =1

[

1 n D = ∑ Qi z i3 − 3i −1 3 i =1

]

]

Note! Coupling between tension and bending!

Internal forces and moments caused by hygroexpansion n

N H = ∑ ΔH i β iH Qi [ zi − zi −1 ] i=1

MH =

1 n ∑ ΔH i βiH Qi ⎡⎣ zi2 − zi2−1 ⎤⎦ 2 i =1

External forces, bending moments, strains and curvatures have three components p each: ⎡ N1 ⎤ ⎢ ⎥ ⎢ N2 ⎥ ⎢ N6 ⎥ ⎣ ⎦

⎡ N1H ⎤ ⎢ H⎥ ⎢ N2 ⎥ ⎢ N 6H ⎥ ⎣ ⎦

⎡M1 ⎤ ⎢ ⎥ ⎢M 2 ⎥ ⎢M 6 ⎥ ⎣ ⎦

⎡ M 1H ⎤ ⎢ H⎥ ⎢M 2 ⎥ ⎢ M 6H ⎥ ⎣ ⎦

⎡ε1 ⎤ ⎢ ⎥ ⎢ε 2 ⎥ ⎢ε 6 ⎥ ⎣ ⎦

⎡κ1 ⎤ ⎢ ⎥ ⎢κ 2 ⎥ ⎢κ 6 ⎥ ⎣ ⎦

9

Transformation of stresses and strains ⎡ ⎤ ⎢ ε1 ⎥ ⎢ ⎥ −1 ⎢ ε 2 ⎥ = [T ] ⎢1 ⎥ ⎢ γ 12 ⎥ ⎣2 ⎦

⎡ ⎤ ⎢ εx ⎥ ⎢ ⎥ ⎢ ε y ⎥ where ⎢1 ⎥ ⎢ γ xy ⎥ ⎣2 ⎦

[T ]

−1

⎡c 2 ⎢ = ⎢s 2 ⎢ cs ⎣

− 2cs ⎤ ⎥ c 2cs ⎥ − cs c 2 − s 2 ⎥⎦ s2

2

⎧c = cos ϕ ⎨ ⎩ s = sin ϕ

⎡σ x ⎤ ⎡σ 1 ⎤ ⎢σ ⎥ = [T ]−1 ⎢σ ⎥ ⎢ y⎥ ⎢ 2⎥ ⎢τ xy ⎥ ⎢⎣τ 6 ⎥⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ ⎢ β1H ⎥ ⎢ β xH ⎥ ⎢ H ⎥ −1 ⎢ H ⎥ ⎢ β 2 ⎥ = [T ] ⎢ β y ⎥ ⎢1 H ⎥ ⎢1 H ⎥ ⎢ β12 ⎥ ⎢ β12 ⎥ ⎣2 ⎦ ⎣2 ⎦

Vanishes in general!

Transformation of the stiffness matrix, Q ⎡σ 1 ⎤ ⎡Q11 ⎢σ ⎥ = ⎢Q ⎢ 2 ⎥ ⎢ 12 ⎣⎢τ 12 ⎦⎥ ⎢⎣Q16 ⎡ Q11 ⎤ ⎡ c 4 ⎢Q ⎥ ⎢ 4 ⎢ 22 ⎥ ⎢ s ⎢Q12 ⎥ ⎢c 2 s 2 ⎢ ⎥=⎢ 2 2 ⎢Q66 ⎥ ⎢c s ⎢Q16 ⎥ ⎢ c 3 s ⎢ ⎥ ⎢ 3 ⎢⎣Q26 ⎥⎦ ⎢⎣ cs

Q12 Q22 Q26

Q16 ⎤ ⎡ ε 1 ⎤ Q26 ⎥⎥ ⎢⎢ ε 2 ⎥⎥ Q66 ⎥⎦ ⎣⎢γ 12 ⎥⎦

⎤ ⎥ ⎥ ⎡Q xx ⎤ ⎥ ⎢Q yy ⎥ ⎥⎢ ⎥ ⎥ ⎢Q xy ⎥ ⎢ ⎥ − cs 3 cs 3 − c 3 s 2 cs 3 − c 3 s ⎥ ⎣ Qss ⎦ ⎥ − c 3 s c 3 s − cs 3 2 c 3 s − cs 3 ⎥⎦ s4 c4 c2s2 c2s2

2c 2 s 2 2c 2 s 2 c4 + s4 − 2c 2 s 2

4c 2 s 2 4c 2 s 2 − 4c 2 s 2 c2 − s2

( ( (

)

) )

10

Forces and moments Expanded equations H ⎡ N 1 ⎤ ⎡ N 1 ⎤ ⎡ A11 ⎢ ⎥ ⎢ H⎥ ⎢ ⎢ N 2 ⎥ + ⎢ N 2 ⎥ = ⎢ A12 ⎢⎣ N 6 ⎥⎦ ⎢⎣ N 6H ⎦⎥ ⎢⎣ A16 H ⎡ M 1 ⎤ ⎡ M 1 ⎤ ⎡ B11 ⎢M ⎥ + ⎢M H ⎥ = ⎢ B ⎢ 2 ⎥ ⎢ 2 ⎥ ⎢ 12 ⎢⎣ M 6 ⎥⎦ ⎢⎣ M 6H ⎥⎦ ⎢⎣ B16

A12 A22 A26 B12 B22 B26

A16 ⎤ ⎡ε 10 ⎤ ⎡ B11 ⎢ ⎥ A26 ⎥⎥ ⎢ε 20 ⎥ + ⎢⎢ B12 A66 ⎦⎥ ⎢⎣ε 60 ⎥⎦ ⎢⎣ B16 B16 ⎤ ⎡ε 10 ⎤ ⎡ D11 ⎢ ⎥ B26 ⎥⎥ ⎢ε 20 ⎥ + ⎢⎢ D12 B66 ⎦⎥ ⎢⎣ε 60 ⎥⎦ ⎣⎢ D16

B12 B22 B26 D12 D22 D26

B16 ⎤ ⎡κ 1 ⎤ B26 ⎥⎥ ⎢⎢κ 2 ⎥⎥ B66 ⎦⎥ ⎣⎢κ 6 ⎥⎦ D16 ⎤ ⎡κ 1 ⎤ D26 ⎥⎥ ⎢⎢κ 2 ⎥⎥ D66 ⎦⎥ ⎢⎣κ 6 ⎦⎥

⎡ N ⎤ ⎡ N H ⎤ ⎡ A B ⎤ ⎡ε 0 ⎤ ⎢M ⎥ + ⎢ H ⎥ = ⎢ B D⎥ ⎢ ⎥ ⎣ ⎦ ⎣⎢ M ⎦⎥ ⎣ ⎦ ⎣⎢ κ ⎥⎦

Bending stiffness and tensile stiffness NH = M H = 0 Strains and curvatures give external forces and external moments

⎡ N ⎤ ⎡ A B ⎤ ⎡ε °⎤ ⎢M ⎥ = ⎢ B D⎥ ⎢ κ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ Forces and curvatures give external strains and external moments

⎡ε 0 ⎤ ⎡ A∗ ⎢ ⎥=⎢ ∗ ⎣⎢ M ⎦⎥ ⎣⎢C

B∗ ⎤ ⎡ N ⎤ ⎥⎢ ⎥ D∗ ⎦⎥ ⎣ κ ⎦

Forces and moments give external strains and external curvatures

⎡ε 0 ⎤ ⎡ A′ B′ ⎤ ⎡ N ⎤ ⎢ ⎥=⎢ ′ ⎥⎢ ⎥ ⎢⎣ κ ⎥⎦ ⎣C D′⎦ ⎣ M ⎦

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Expanded equations ⎡ε 10 ⎤ ⎡ A11′ ⎢ 0⎥ ⎢ ⎢ε 2 ⎥ = ⎢ A12′ ⎢ 0⎥ ⎣ε 6 ⎦ ⎢⎣ A16′

⎡κ 1 ⎤ ⎡C11′ ⎢κ ⎥ = ⎢C ′ ⎢ 2 ⎥ ⎢ 12 ⎣⎢κ 6 ⎦⎥ ⎢⎣C16′

′ A26

A16′ ⎤ ⎡ N 1 ⎤ ⎡ B11′ ′ ⎥ ⎢ N 2 ⎥ + ⎢ B12′ A26 ⎥⎢ ⎥ ⎢ ′ ⎦⎥ ⎣⎢ N 6 ⎥⎦ ⎢⎣ B16′ A66

B12′ ′ B22 ′ B26

C12′ ′ C 22 ′ C 26

C16′ ⎤ ⎡ N 1 ⎤ ⎡ D11′ ′ ⎥ ⎢ N 2 ⎥ + ⎢ D12′ C 26 ⎥⎢ ⎥ ⎢ ′ ⎦⎥ ⎢⎣ N 6 ⎥⎦ ⎢⎣ D16′ C 66

D12′ ′ D22 ′ D26

A12′ ′ A22

Tensile stiffness N

N 2 = N6 = 0

D16′ ⎤ ⎡ M 1 ⎤ ′ ⎥ ⎢M 2 ⎥ D26 ⎥⎢ ⎥ ′ ⎦⎥ ⎣⎢ M 6 ⎥⎦ D66

Eb =

x3 x2

B16′ ⎤ ⎡ M 1 ⎤ ′ ⎥ ⎢M 2 ⎥ B26 ⎥⎢ ⎥ ′ ⎦⎥ ⎣⎢ M 6 ⎥⎦ B66

N ε°

x1

⎫ ⎬ ⇒ M1 = M 2 = M 6 = 0⎭

N

N1

1 E = o = ε1 A11′ b 1

E 2b = E 6b =

N2

ε

o 2

N 61

ε

o 16

=

1 ′ A22

=

1 ′ A66

δ = ε 0L

shear stiffness

12

Bending stiffness is important for the stacking strength of boxes ⎛1⎞ M = EI ⎜ ⎟ = EI κ ⎝R⎠

S ≡ EI =

E 3 bt 12

The bending stiffness of the beam is the slope of the moment vs. curvature relation. Bending stiffness per unit width

Sb ≡

S E 3 = t b 12

Bending stiffness of a single-ply sheet Sb ≡

S E 3 Eb 2 = t = t b 12 12

where

Eb E= t

introduce

ρ=

yields

w t

E b w2 S = 12 ρ 2 b

13

Bending stiffness of a single-ply sheet Tensile stiffness index is related to density according to (experimental)

E = kρ w

Eb E = w w

a

The bending stiffness per unit width then becomes

S b = kw3 ρ a − 2 /12 Define bending stiffness index (independent of grammage) as

Sw =

S b 1 ⎛ Ew ⎞ = ⎜ ⎟ w3 12 ⎝ ρ 2 ⎠

NOTE! Bending stiffness index depends on density

Bending stiffness of three-ply sheet

Introduce

E surface = n surface Eref E core = n core Eref

14

Bending stiffness of three-ply sheet The bending stiffness is then given by

S = Eref I ref = Eref ∑ ni I i i

Calculation of the surface moment of inertia yields

I ref = ∑ n I i = n i

surface

i

2 3 ⎛ btS3 ⎛ tC tS ⎞ ⎞ core btC + btC ⎜ + ⎟ ⎟ 2 + n ⎜⎜ 12 12 ⎝ 2 2 ⎠ ⎟⎠ ⎝

c.f. Steiner’s theorem from basic courses

Bending stiffness of three-ply sheet

Assume

I ref bt 3

⎧n surface = 2 ⎪ core ⎪n = 1 ⎨ ⎪tS = t / 3 ⎪t = t / 3 ⎩C =

13 1 53 + = 81 324 324

surface plies

middle ply

15

Bending stiffness

M k=1/R

N1 = N 2 = N 6 = 0 ⎫ ⎬ ⇒ κ1 = D11′ ⋅ M1 M2 = M6 = 0 ⎭ κ2 ≠ 0 Note!

M k

Sb =

κ6 ≠ 0

Bending or torsional stiffness in all directions:

M

κ

⎧ b M1 1 = ⎪ S1 = κ1 D11′ ⎪ ⎪ b M2 1 = ⎨ S2 = ′ κ 2 D22 ⎪ ⎪ b M6 1 = ⎪ S6 = κ 6 D66′ ⎩

Simplified expression for bending stiffness Uniaxial loading

⎡ N ⎤ ⎡ N H ⎤ ⎡ A B ⎤ ⎡ε 0 ⎤ ⎢M ⎥ + ⎢ H ⎥ = ⎢ B D⎥ ⎢ ⎥ ⎣ ⎦ ⎢⎣ M ⎥⎦ ⎣ ⎦ ⎢⎣ κ ⎥⎦ N =M H

H

0 ⎪⎧ N = Aε + Bκ =0 ⇒ ⎨ 0 ⎪⎩M = Bε + Dκ

N =0 ⇒

⎡ B2 ⎤ M = ⎢D − ⎥ κ A⎦ ⎣ Sb = D −

B2 A

16

Example Calculation of bending stiffness for 4-ply sheet w

Ply k

Ew =

E

g/m2

E = Ew ⋅ ρ

ρ

ρ MNm/kg

mm

80

7.5

800

6000

0.1

2

80

2.5

400

1000

0.2

3

50

4.0

500

2000

0.1

4

70

7.14

700

5000

0.1

z k2 mm2

n

A = ∑ Qi [ zi − zi −1 ] i =1

-0,25

0,0625

-15,625 · 10-3

1

-0,15

0,0225

-3,375 · 10-3

2

0,05

0,0025

0,125 · 10-3

3

0,15

0,0225

3,374 · 10-3

4

0,25

0,0625

15,625 · 10-3

Qk (zk − zk−1 ) ≈ ≈ E k (zk − zk−1 )

[mm]

z k3 mm3

0

Layer k

ρ

N/mm2 ((MPa))

1

zk mm

w

kg/m3

t = ∑ tk = 0.1 + 0.2 + 0.1 + 0.1 = 0.5⎫ ⎪ 05 0.5 ⎪ z0 = − = − 0.25 ⎪ 2 ⎪ z1 = z0 + t1 = −0.25 + 0.1 = −0.15 ⎬ ⎪ z2 = 0.05 ⎪ z3 = 0.15 ⎪ ⎪ z4 = 0.25 ⎭

Layer k

t =

2 Qk (z k2 − z k−1 )≈

3 Qk (z k3 − z k−1 )≈

≈ E k (z − z

3 ≈ E k (z k3 − z k−1 )

2 k

2 k−1

)

1

6000 [-0,15-(-0,25)]= = 600

6000 (0,0225-0,0625)= = -240

6000 [-3.375-(15,625)]·10-3= = 73,5

2

1000 [0,05-(-0,15)]= = 200

1000 (0,0025-0,0225)= = -20

1000 [0.125-(-3,375)]·10-3= = 3,5

3

2000 [0,15-0,05]= = 200

2000 (0,0225-0,0025)= = 40

2000 [3,375-0.125]·10-3= = 6,5

4

5000 [0,25-0,15]= = 500

5000 (0,0625-0,0225)= = 200

5000 [15,625-3.375]·10-3= = 61,25

A = ∑ = 1500

B=

1 ∑ = − 10 2

D=

[

B=

1 n ∑ Qi z i2 − z i2−1 2 i =1

D=

1 n ∑ Qi z i3 − 3i −1 3 i =1

Sb = D −

[

]

B2 A



]

(−10) 2 = 1500 48.18 mNm

S b = 48.25 −

1 ∑ = 48, 25 3

17

Curl, twist, strain and shear, as a result of a change in moisture N1 = N 2 = N 6 = 0 ⎫ ⎡ε 10 ⎤ ⎡ A11′ ⎬ ⇒ ⎢ 0⎥ ⎢ M1 = M 2 = M 6 = 0 ⎭ ⎢ε 2 ⎥ = ⎢ A12′ ⎢ 0⎥ ⎣ε 6 ⎦ ⎢⎣ A16′

A12′ ′ A22 ′ A26

A16′ ⎤ ⎡ N 1H ⎤ ⎡ B11′ ⎢ ⎥ ′ ⎥ ⎢ N 2H ⎥ + ⎢ B12′ A26 ⎥ ⎢ ′ ⎦⎥ ⎢⎣ N 6H ⎥⎦ ⎢⎣ B16′ A66

B12′ ′ B22 ′ B26

B16′ ⎤ ⎡ M 1H ⎤ ⎢ ⎥ ′ ⎥ ⎢ M 2H ⎥ B26 ⎥ ′ ⎦⎥ ⎢⎣ M 6H ⎥⎦ B66

⎡κ 1 ⎤ ⎡C11′ ⎢κ ⎥ = ⎢C ′ ⎢ 2 ⎥ ⎢ 12 ⎢⎣κ 6 ⎥⎦ ⎢⎣C16′

C12′ ′ C 22 ′ C 26

C16′ ⎤ ⎡ N 1H ⎤ ⎡ D11′ ⎢ ⎥ ′ ⎥ ⎢ N 2H ⎥ + ⎢ D12′ C 26 ⎥ ⎢ H ′ ⎥⎦ ⎢⎣ N 6 ⎥⎦ ⎢⎣ D16′ C 66

D12′ ′ D22 ′ D26

D16′ ⎤ ⎡ M 1H ⎤ ⎢ ⎥ ′ ⎥ ⎢ M 2H ⎥ D26 ⎥ H ′ ⎥⎦ ⎢⎣ M 6 ⎥⎦ D66

⎫ ⎪ ⎪ i =1 ⎬⇒ n 1 2 2 H ⎪ = ∑ ΔH i β i Qi ⎡⎣ zi − zi −1 ⎤⎦ ⎪⎭ 2 i =1 n

N H = ∑ ΔH i β iH Qi [ zi − zi −1 ] MH

κ1 = [C11′ ] ⎣⎡ N1H ⎦⎤ + [C12′ ] ⎣⎡ N 2H ⎦⎤ + [C16′ ] ⎣⎡ N6H ⎦⎤ +

[ D11′ ] ⎡⎣ M1H ⎤⎦ + [ D12′ ] ⎡⎣ M 2H ⎤⎦ + [ D16′ ] ⎡⎣ M 6H ⎤⎦ κ 2 = ... κ 6 = ...

Stress and strains in the thickness direction of the laminate due to hygroexpansion

σ

z

18

Laminate theory program Excel Christer Fellers Innventia/KTH

1 z0

2

z1

z2 zk

zN

t/2

3 k

1 t

ϕ

t/2

x

N Z, 3

y

2

19

G xy = 0 , 3 8 7

ExEy

εy

x

εx

x

εx

y

ν x yν E E

ν

x y

xy

=

yx

ν ν

= 0, 293

εy

xy yx

= 0, 293

y

Ex Ey

20

Change in moisture

2004-09-02 / 41

Results ε κ

κ1 1 2

M1

κ2

M2

κ6 M6

Twist

1

Tensile stiffness index

2 Shear ε 6

21

Calculate strains and curvatures

1 2

N2

κ1

M1

N1

⎡ε 0 ⎤ ⎡ A′ B′ ⎤ ⎡ N ⎤ ⎢ ⎥=⎢ ′ ⎥⎢ ⎥ ⎢⎣ κ ⎥⎦ ⎣C D′⎦ ⎣ M ⎦

M2

1 κ2 N6

M6

κ6

2

Shear

Home assignments

22

23

After Lecture 6 you should be able to ... • describe the basic assumptions in laminate theory • relate the features of laminate theory to basic concepts in solid mechanics • use laminate theory to predict stresses and strains in paper sheets subjected to mechanical loading and moisture

24