Geometrically Nonlinear Composite Shells with … · RHEINISCH-WESTFÄLISCHE TECHNISCHE HOCHSCHULE AACHEN Geometrically Nonlinear Composite Shells with Integrated Piezoelectric Layers

RHEINISCH-WESTFÄLISCHETECHNISCHEHOCHSCHULEAACHEN


Geometrically Nonlinear Composite Shells

with Integrated Piezoelectric Layers

S. Lentzen and R. Schmidt

Institute of General Mechanics, RWTH Aachen University



Contents:

• Nonlinear FOSD theory of shells

• Total Lagrangian formulation

• FE implementation

• Numerical examples

• Summary



Nomenclature and Definition

Θ2

Θ3

Ω0

a1~a2~

Θ1

0

V21V10

1

V11

1

1

V21

0

V311V

C

C0

1

C2

V∆

a3~ ~

~



Nonlinear FOSD theory of shells for small strains and

moderate rotation

Kinematic hypothesis: mV∼ = mV0

∼ + Θ3 mV1

∼

Assumptions:

• small strains εij = O(θ2) with θ2 1

• small rotations about the normal ωαβ = O(θ2)

• moderate rotations of the normal ωα3 = O(θ)

Green-Lagrange strain tensor:

εαβ =0εαβ +Θ3 1

εαβ +(

Θ3)

2 2εαβ

εα3 =0εα3 +Θ3 1

εα3

ε33 = 0



Strain-Displacement Relations

Tangential strains:

0εαβ=

0θαβ +

1

2

0ϕα

0ϕβ

1εαβ=

1

2

(

1vα|β +

1vβ|α −bλ

α0ϕλβ −bλ

β0ϕλα +

0ϕα bλ

β1vλ +

0ϕβ bλ

α1vλ

)

2εαβ=

1

2

(

bλαbκ

β1vλ

1vκ − bλ

α1vλ|β −bλ

β1vλ|α

)

Transverse shear strains:0εα3=

1

2

(

0ϕα +

1vα +

1vλ 0

ϕλα

)

and1εα3=

1

2

1vλ 1

vλ|α

Linearised tangential strains:0θαβ=

12

(

0vα|β +

0vβ|α

)

− bαβ0v3

Linearised rotations:0ϕαβ=

0vα|β −bαβ

0v3 and

0ϕα=

0v3,α + bλ

α0vλ,



Constitutive Relations

0D = [e] 0ε + [δ] 0E

0S = [c] 0ε − [e]T 0E

[e] = [d] [c]

[e]T = [c] [d]T

0S =

σ11

σ22

τ12

τ23

τ13

, 0ε =

ε11ε222ε122ε232ε13

, 0D =

D1

D2

D3

, 0E =

E1E2E3



[c] =

c11 c12 c13 0 0c12 c22 c23 0 0c13 c23 c33 0 00 0 0 c44 c450 0 0 c45 c55

Elasticity Matrix

[d]T =

0 0 d310 0 d310 0 00 d15 0

d15 0 0

Piezoelectric Constant Matrix

[δ] =

δ11 0 00 δ22 00 0 δ33

Permittivity Matrix



Total Lagrangian Formulation

Electric field:

tE∼ = −grad (φ) 0E∼ = −GRAD(φ) = FTtE∼ ⇒ 0Ei = tEi

C0

Ct

0dA

t dA

N~

n~

Electric displacement:

tq = limt∆A→0

∆Qt∆A

= tD∼ ·n∼ 0q = 0D∼ ·N∼ =tdA0dA

tq = J tD∼ ·n∼ = JF−1tD∼ ·N∼

0D∼ = JF−1tD∼ ⇒ 0Di = J tD

i



State of Equilibrium

δWi = δWe

δWi =∫

0V

(

0δεT 0S − 0δET 0D)

0dV

δWe =∫

VρF iδVi dV +

∫

Aσ

σiδVi dAσ −∫

AQ

Diniδφ dAQ

Further Assumptions

• Electric field is only existent in transverse direction

⇒ 0E∼ = 0 0 E3

• Electric field is homogeneous between one electrode pair



Incremental FE Formulation

δWi =

δq

δφ

T

·

[

qq

] [

qφ

]

[

qφ

]T [

φφ

]

·

∆q

∆φ

+

i

i

[

qq

]

= Incremental stiffness matrix[

qφ

]

= Electromechanical coupling matrix[

φφ

]

= Piezoelectric capacity matrix



Bimorph Cantilever Beam

0 00.025 0.0250.05 0.050.075 0.0750.10

0.10

0.1100

0.2200

0.3

300

Deflection [ µm]

x [m]

AnalysisLab −TzouFEM −TzouMRT

Sensor Position [m]

Sensor Voltage [V]

Tzou et al. Piefort linear [1cm/0.025058N] MRT [1cm/0.025914N] MRT [0.96895cm/0.025058N]

F

1V



Bimorph Clamped Beam

0 00.025 0.025

Deflection [

0.05

µm]

0.050.075 0.0750.1 0.1−500

−300 1

−1002

100

3

300

4

5

x [m]

Theory MRT no clampingMRT

Sensor Position [m]

Sensor Voltage [V]

Theory Linear [2mm/0.32133N]MRT [2mm/1.1843N] MRT [1.09mm/0.32133N]

F

200V



Actuation and Sensing of a Hinged Beam

0 0.2 0.40

250

500

750

1000

non−dimensional deflection and stress [−]

actu

ator

vol

tage

[V]

linear mid−point deflectionMRT mid−point deflection

MRT mid−point deflection

σ

σ

xx

xx

|z=0

0.045

PVDF (0.028 mm)

Steel (0.2 mm)

PVDF (0.028 mm)10 mm

200 mm

vz0

/ H

0.04

0.035

0.03

0.025

0.02

0.015

0.01

0.005

0

12

34

56

78

910

Sen

sor

volta

ge [V

]

Sensor number

LinearMRT

.A

(EA)steel + (EA)

total x 105

piezo



Shape Control of a Hinged Beam

00 0.050.05 0.10.1 0.150.15 0.2 0.2−0.04−0.02

00.02

0.04

0.06

0.08

0.1

0.14

coordinate x [m]

coor

dina

te y

[mm

]

0V 100V

coordinate x [m]

200V

coor

dina

te y

[mm

]

300V

400V

0V

500V

100V

200V

600V 300V

700V

400V

800V 500V

900V

600V

1000V

700V

800V

900V

1000V

MRTlinear

RH

EIN

ISC

H-

WE

ST

FÄ

LIS

CH

ET

EC

HN

ISC

HE

HO

CH

SC

HU

LE

AA

CH

EN

RH

EIN

ISC

H-

WE

ST

FÄ

LIS

CH

ET

EC

HN

ISC

HE

HO

CH

SC

HU

LE

AA

CH

EN

C-Blo

ck

Actuator

Ele

ctro

de; h

= 6

.5

m; E

= 7

00 M

Pa

17.5mm22mm

mm20

R =

14.

32 m

m

mm17.522mm

R =

13.

6 m

m

17.5 mm22 mm

R =

14.

9 m

m

Su

bstr

ate;

h =

25

m; E

= 1

900

MPa

µ

Subs

trat

e; h

= 2

5 m

; E =

650

0 M

Paµ

Piez

o; h

= 5

2 m

; E =

290

0 M

Paµ

Polin

g D

irec

tion;

d

= 2

.3E

−11

m/V

31µM

oska

lik&

Bre

i

The

ory

Exp

erim

ent

Pief

ort

Lin

ear

MR

T

Displacement (mm)

Displacement (mm)

Voltage (V)

Voltage (V)

Voltage (V)

Displacement (mm)

b)

c)

a)



Static Stability Control

−2 0 2

2

4 6 8 10 12 14 160

5

10

Midpoint displacement [mm]

Increasing Applied Voltage

Load

p [1

000

N/m

]

15

20

25

Piezoelectric Layers

p



Cantilever Piezolaminated Shell

−1 −0.75 −0.5 −0.25 00

250

80

500

90

750

100

1000

[302 S/0]T300/976

PZT G1195

PZT G1195

[30T300/976 /0]

PZT G1195

PZT G1195

Def

lect

ion

(w2/

a), T

wis

t (w

3−w 1)

/a

Kioua et alLee et alPresent

Radius of Curvature (R/a)10 706050403020

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Twist

Deflection

Pres

sure

[N

/m2 ]

TwistDeflection

300 N/m2

Deflection (w2/a), Twist (w3−w1) x 100/a

Linear

MRT

pressure

100 V1

23

1

2 3

a

a



Sensor Voltages of a Cantilever Piezolaminated Shell

at 300N/m2

Linear 300 N/m 2 Nonlinear 300 N/m2

250

200

150

100

50

0

180

140

100

60

20



Shape Control of a Cantilever Piezolaminated Shell

at 300N/m2

−0.2 −0.4−0.15 −0.2−0.1 0−0.05 0.20 0.40.05 0.60

0.8

250

1

500

0

750

250

1000

500

750

1000

Deflection (w2/a) Twist (w3−w1) x 100/a

LinearMRT

LinearMRT

Max

imum

app

lied

volta

ge

Max

imum

app

lied

volta

ge



Summary

• A geometrically nonlinear finite piezolaminated shell element is

presented

• Special attention is given to the definition of the electric field

variables due to total Lagragian formulation

• Several numerical examples are shown and discussed in which the

geometrical nonlinear effect is profound

Documents

Geometrically Nonlinear Composite Shells with … · RHEINISCH-WESTFÄLISCHE TECHNISCHE HOCHSCHULE AACHEN Geometrically Nonlinear Composite Shells with Integrated Piezoelectric Layers