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Published by mrityun.jgec, 2019-05-26 20:25:53

smart-structures-and-materials-2017

smart-structures-and-materials-2017

Computational Methods in Applied Sciences

Aurelio L. Araujo
Carlos A. Mota Soares Editors

Smart
Structures
and Materials

Selected Papers from the 7th ECCOMAS
Thematic Conference on Smart
Structures and Materials

Computational Methods in Applied Sciences

Volume 43

Series editor
E. Oñate
CIMNE
Edificio C-1, Campus Norte UPC
Gran Capitán, s/n
08034 Barcelona, Spain
[email protected]

More information about this series at http://www.springer.com/series/6899

Aurelio L. Araujo • Carlos A. Mota Soares

Editors

Smart Structures
and Materials

Selected Papers from the 7th ECCOMAS
Thematic Conference on Smart Structures
and Materials

123

Editors Carlos A. Mota Soares
Aurelio L. Araujo IDMEC, Instituto Superior Técnico
IDMEC, Instituto Superior Técnico University of Lisbon
University of Lisbon Lisbon
Lisbon Portugal
Portugal

ISSN 1871-3033

Computational Methods in Applied Sciences

ISBN 978-3-319-44505-2 ISBN 978-3-319-44507-6 (eBook)

DOI 10.1007/978-3-319-44507-6

Library of Congress Control Number: 2016954020

© Springer International Publishing Switzerland 2017
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
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or information storage and retrieval, electronic adaptation, computer software, or by similar or
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The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are exempt
from the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, express or implied, with respect to the material contained
herein or for any errors or omissions that may have been made.

Printed on acid-free paper

This Springer imprint is published by Springer Nature
The registered company is Springer International Publishing AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Smart Materials and Structures have been around, within the research community,
at least for the last three decades. The concept involves combined sensing, actuation
and control capabilities embedded within materials and structures at different scales,
from macro to micro scale. A variety of new research areas have emerged within
smart technologies, with impacts in the wider fields of engineering and science, and
more specifically in fields ranging from materials science to medicine. The disciplines
involved in smart technologies make this field one of the most interdisciplinary areas
of engineering science.

This book was compiled with expanded and reviewed contributions, originally
presented at the 7th ECCOMAS Thematic Conference on Smart Structures and
Materials (SMART2015), held at the University of the Azores, Ponta Delgada,
S. Miguel Island, from 3 to 6 June 2015. The aim of this thematic conference series
has been to gather the smart technologies community, providing a forum for the
discussion of the current state of the art in the field and generating inspiration for
future ideas on a multidisciplinary level.

Modeling aspects, design, fabrication and applications of smart materials and
structures along with structural control and structural health monitoring with
piezoelectric devices were the main topics of the conference. Application to mor-
phing wings, aircraft and aerospace vehicles are increasing, along with the incor-
poration of nanotechnologies in smart materials and structures. In the field of
modeling of smart structures, nonlinear aspects of material and structural response
have now a noticeable expression, including some damage mechanics studies
in smart structures and also modeling of electro-chemical and thermo-
electromechanical behavior. On the other hand, contributions in biomedical engi-
neering and biomimetic applications have started to bridge the gap between the
smart technologies and the biomedical engineering communities.

We hope that the different articles in this book help providing an insight into the
latest developments and future trends in smart structures and materials.

v

vi Preface

The editors would like to acknowledge all the contributing authors for their effort
in preparing and submitting extended articles that went through a peer-reviewing
process. The contribution of all the reviewers and the Springer editorial team is also
gratefully acknowledged. Last but not least, special thanks to Ms. Paula Jorge for
her tireless efforts and support in all the stages of the preparation of this book.

Lisbon, Portugal Aurelio L. Araujo
May 2016 Carlos A. Mota Soares

Contents

1 Role of the Structural Nonlinearity in Enhancing 1
the Performance of a Vibration Energy Harvester
Based on the Electrets Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Eugenio G.M. Brusa and Mircea Gh. Munteanu

2 Numerical Analysis of Fracture of Pre-stressed Ferroelectric
Actuator Taking into Account Cohesive Zone for Damage
Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Sergii Kozinov and Meinhard Kuna

3 Modelling the Constitutive Behaviour of Martensite
and Austenite in Shape Memory Alloys Using Closed-Form
Analytical Continuous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Arathi Pai, Thomas Niendorf, Phillip Krooss, Isabel Koke,
Ansgar Traechtler and Mirko Schaper

4 Experimental Investigations of Actuators
Based on Carbon Nanotube Architectures. . . . . . . . . . . . . . . . . . . . . 67
Sebastian Geier, Thorsten Mahrholz, Peter Wierach
and Michael Sinapius

5 Efficient Experimental Validation of Stochastic Sensitivity
Analyses of Smart Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Steffen Ochs, Sushan Li, Christian Adams and Tobias Melz

6 Design of Control Concepts for a Smart Beam Structure
with Sensitivity Analysis of the System . . . . . . . . . . . . . . . . . . . . . . . 115
Sushan Li, Steffen Ochs, Elena Slomski and Tobias Melz

7 Adaptive Inductor for Vibration Damping in Presence
of Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Bilal Mokrani, Ioan Burda and André Preumont

vii

viii Contents

8 Active Control of the Hinge of a Flapping Wing with Electrostatic
Sticking to Modify the Passive Pitching Motion . . . . . . . . . . . . . . . . 153
Hugo Peters, Qi Wang, Hans Goosen and Fred van Keulen

9 Control System Design for a Morphing Wing Trailing Edge. . . . . . 175
Ignazio Dimino, Monica Ciminello, Antonio Concilio, Andrè Gratias,
Martin Schueller and Rosario Pecora

10 Towards the Industrial Application of Morphing Aircraft
Wings—Development of the Actuation Kinematics
of a Droop Nose. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Stefan Storm and Johannes Kirn

11 Artificial Muscles Design Methodology Applied to Robotic
Fingers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
J.L. Ramírez, A. Rubiano, N. Jouandeau,
L. Gallimard and O. Polit

12 Methods for Assessment of Composite Aerospace Structures . . . . . 227
T. Wandowski, P. Malinowski, M. Radzienski, S. Opoka
and W. Ostachowicz

13 Design Optimization and Reliability Analysis
of Variable Stiffness Composite Structures . . . . . . . . . . . . . . . . . . . . 245
A. Sohouli, M. Yildiz and A. Suleman

14 Robust Multi-objective Evolutionary Optimization-Based Inverse
Identification of Three-Dimensional Elastic Behaviour
of Multilayer Unidirectional Fibre Composites . . . . . . . . . . . . . . . . . 267
Mohsen Hamdi and Ayech Benjeddou

Chapter 1

Role of the Structural Nonlinearity
in Enhancing the Performance
of a Vibration Energy Harvester
Based on the Electrets Materials

Eugenio G.M. Brusa and Mircea Gh. Munteanu

Abstract Films of the electrets material are currently proposed to design compact
vibration energy harvesters. They are used to cover the surface of electrodes of
some capacitive devices based on a deformable microbeam clamped at both its
ends. The performance of those energy harvesters is often predicted in the literature
by neglecting the effect of the geometric nonlinearity due to a mechanical coupling
occurring between the axial and flexural behaviors of the clamped-clamped
microbeam. This nonlinearity is herein investigated, by resorting to a distributed
model of the electromechanical coupling applied to the vibration energy harvester.
The analysis is performed by means of the finite element method. The performance
of energy conversion is then analyzed and some new configurations of the vibration
energy harvester are proposed.

⋅ ⋅Keywords Electrets material Vibration energy harvesting Finite element
⋅method Nonlinear dynamics

1.1 Introduction

The use of electrets materials in microsystems is currently proposed to improve the
performance of some smart devices and to increase the availability of autonomous
power supplies. An internal polarization of the electrets is exploited. It is due to a

E.G.M. Brusa (✉) 1

Department of Mechanical and Aerospace Engineering, Politecnico di Torino
Corso Duca degli Abruzzi 24, 10129 Turin, Italy
e-mail: [email protected]

M.Gh. Munteanu
Department of Electrical, Managerial, Mechanical Engineering, Università degli
Studi di Udine via delle Scienze 208, 33100 Udine, Italy
e-mail: [email protected]

© Springer International Publishing Switzerland 2017
A.L. Araujo and C.A. Mota Soares (eds.), Smart Structures and Materials,
Computational Methods in Applied Sciences 43,
DOI 10.1007/978-3-319-44507-6_1

2 E.G.M. Brusa and M.Gh. Munteanu

set of trapped electric charges or to some oriented molecular dipoles, but even to a
suitable combination of both those effects [1]. The behavior of electrets is very
close to that of some polar piezoelectric polymer like the polyvinylidene fluoride
(PVDF), in which the piezoelectric effect is due to a change of dipole density under
either mechanical stress or electric field [2]. The electrets materials exhibit a fairly
good electromechanical coupling, when suitably pre–charged, although the polar-
ization is sensitive to any increase of temperature. Pyroelectric phenomena are
negligible in these materials, if they are compared to the behavior of some typical
piezoelectric, and the electrical resonance usually occurs in the high range of kHz.
Those properties make the electrets suitable for application to microphones and
loudspeakers, since a fairly low distortion of sound can be assured. An increasing
application to MEMS is currently motivated by the structural properties of the
electrets. High resistance and thinness make the electrets materials suitable also for
manufacturing thin switches and digital keypads [3]. In addition, vibration energy
harvesters (or simply VEH) are nowadays used in several applications to convert
the energy associated to motion into electric charge, especially in wearable systems
aimed at monitoring the human health [4]. Very often those devices allow a
miniaturization of sensors and provide a local power supply to make autonomous
the monitoring system [5]. In case of a capacitive vibration energy harvester
(CVEH) an electric pre–charging is required to provide a bias voltage to operate [6].
This limit can be overcome by using a thin layer of the electrets material to supply a
constant voltage. It is directly positioned between the electrodes of the capacitor
and bonded upon either the fixed or the moveable one.

In the literature some configurations of CVEH based on the electrets material
were analyzed. The electrets layer was connected in parallel with two variable
comb-drive capacitors operating in anti–phase in [7]. In this case a transfer of
electric charge was activated between two variable capacitors by the motion of a
proof mass. A more efficient layout was tested in [8], where only one capacitor was
used, thus focusing the interest of the literature on this simpler configuration.
Moreover, a comparison among the piezoceramics, electrets and electromagnetic
materials was proposed in [9]. It shows that the efficiency of energy conversion
strictly depends on the size and the properties of each system analyzed and the
prediction of performance significantly depends upon the accuracy of the mathe-
matical models implemented.

According to the above mentioned investigation, some benefits were found when
the electrets are surface bonded on a moveable plate of a CVEH, within a slightly
variable gap. The electric charging is fairly high, the system is quite small and
the relative speed is fairly low, thus assuring a low dissipation. This screening sug-
gests that refining the modeling approaches applied to the electrets–based VEH is a
goal of the current research activity, thus motivating a deeper analysis of the per-
formance in presence of structural nonlinearity as in some other microsystems [10].

1 Role of the Structural Nonlinearity in Enhancing … 3

1.2 Investigation

1.2.1 State-of-the-Art of the Design of a Electrets–Based
Vibration Energy Harvester

A capacitive microsystem based on the electrets materials was very often modeled
in the literature through some discrete mathematical models [6] by resorting to a
simple plane capacitor with multiple layers, like in gas sensors [11] and in very
small actuators [12]. In case of a capacitor with embedded electrets, where the gap
is variable, the above mentioned approach is effective only when each electrode
behaves like a rigid body. An equivalent voltage generator is used to represent the
electrets effect [13]. When flexible electrodes are exploited to convert the energy
associated to vibration, as in a beam clamped at both its ends, the electromechanical
coupling is distributed upon the surface of electrodes. It is affected by some phe-
nomena like the axial–flexural coupling of the deformable electrode [14]. The
electromechanical stability, the pull–in phenomenon and the charge distribution are
considerably depending on the structural behavior of the flexible structure, when
the electrets layer is fixed to the wafer. Modeling becomes difficult when the
electrets layer is itself a part of the deformable structure [15].

This paper proposes a detailed analysis of the distributed voltage and the loading
condition upon a VEH consisting of a clamped-clamped beam with embedded
electrets. Its performance will be compared to that predicted by some discrete
models [15]. The nonlinearities associated to the electromechanical coupling will be
then analyzed and compared to some previous papers [16], where the dynamic
behavior of the VEH was analyzed in the frequency domain. Moreover, the non-
linear dynamic response of the system will be compared to that of the test case
analyzed in [17], where some nonlinear spring elements were introduced to suspend
the moveable electrode. The numerical investigation herein performed will be
finally used to improve the configuration of the VEH to reach the highest perfor-
mance, by suitably exploiting its nonlinear dynamic behavior and flexibility.

1.2.2 Configurations of the Electrets–Based Vibration
Energy Harvester

The electrets–based VEH looks like in Fig. 1.1, when a unique compliant element
is applied to the moveable electrode and allows both the in–plane and out–of–plane
oscillations, respectively [18]. If vibration excites the in–plane oscillation the area
of electrodes actually interfaced varies over time, while the gap between electrodes
is variable, when vibration excites a vertical displacement. In both those cases the
value of capacitance changes. A source of nonlinearity in this configuration is the
electromechanical force applied between electrodes, since it nonlinearly depends on
the voltage, the charge and the gap [19]. Moveable electrode is usually assumed to

4 E.G.M. Brusa and M.Gh. Munteanu

Fig. 1.1 Discrete model of In plane I, current
the electrets–based vibration oscillaƟon
energy harvester
Moveable
electrode

Out of plane Q1 R, resistance
oscillaƟon V, voltage

Generic compliant C1 g(t), gap
link to fixed frame ε1 d

electret C2 ε2
Fixed Q2

electrode

Area A, side l,

width w

be rigid and often the relation between force and displacement in lateral springs of
Fig. 1.1 is assumed to be linear. To decrease the structural stiffness and calibrate the
resonance of the VEH a cantilever–based configuration is applied (Fig. 1.2).

The cantilever beam behaves as a compliant element and vibration is usually
applied along the vertical direction, through the clamp [15]. Both the ends are often
clamped, as in Fig. 1.3, to limit the rotation of the tip mass when the beam is
bended and to control better the relative position of electrodes [16]. It is worthy
noticing that constraints of Figs. 1.2 and 1.3 are very different. The beam of
Fig. 1.2 is statically and kinematically determinate, therefore the number of degrees
of freedom of the structure and those inhibited by the clamp are just equal and no
motion of the tip mass is allowed without deforming the beam. The system in
Fig. 1.3 is kinematically determinate and statically indeterminate, since the number
of degrees of freedom inhibited by the clamps is larger. The benefit of this con-
figuration is a lack of rotation of the proof mass. Its lower face remains almost

Out of I, current

Flexible plane Tip mass
electrode oscillaƟon

Deformed Q1 C1 g(t), R, resistance
shape ε1 gap V, voltage
Area
inter-faced C2 ε2 d

electret Q2
Fixed

electrode

Area A, side l,

width w

Fig. 1.2 Electrets–based vibration energy harvester with cantilever flexible electrode

1 Role of the Structural Nonlinearity in Enhancing … 5

Flexible I, current

electrode Proof mass Out of
plane

oscillaƟon

Q1 C1 g(t), R, resistance
ε1 gap V, voltage
Area
inter-faced C2 ε2 d Deformed
shape
electret
Fixed Q2

electrode

Area A, side l,

width w

Fig. 1.3 Electrets–based vibration energy harvester with clamped–clamped beamlike electrode

horizontal, being better interfaced to the lower electrode and the surface distribution
of electric charge is regular. Nevertheless, when the displacement of the proof mass
is large the beam is stretched by the clamps, because of a mechanical coupling
between the flexural and axial behavior, respectively. As the proof mass induces a
large displacement of the beam cross section, the reactions of clamps grow up and
an axial loading is applied, thus stiffening the structure.

1.2.3 Goals of This Study

As it was above mentioned three configurations are mainly proposed in the liter-
ature to evaluate the effectiveness of the electrets–based VEH. It is known that the
electrets materials exhibit an electromechanical coupling weaker than some other
smart materials like piezoelectrics, but the possibility of embedding the electrets
layers into a CVEH could motivate their use in some application. Therefore, some
issues are herein investigated:

• the effects of nonlinear dynamics in harvesting
• the influence of the mechanical coupling on the power generation associated to

the maximum displacement of the proof mass
• the frequency range in which the device could effectively operate
• some criteria which might be applied in design activity to eventually exploit the

nonlinear behavior to enhance the system performance.

As a main test case the layout described in Fig. 1.3 will be analyzed, as it looks
in the graphical impression of Fig. 1.4.

6 E.G.M. Brusa and M.Gh. Munteanu

Upper electrode

Tip mass

Gap
Clamp

Support Layer of electret material
Lower electrode

Fig. 1.4 Investigated configuration of the electrets–based vibration energy harvester

1.3 Analysis

1.3.1 A Basic Model of the Electrets–Based Vibration
Energy Harvester

A basic model of the electromechanical coupling occurring in the electrets-based
VEH is often used in the literature (Fig. 1.3). Considering that coordinate ξ is the
instantaneous distance between the lower surface of the flexible electrode and the
upper surface of the electrets layer, in a simple model with a single mechanical
degree of freedom, the dynamic equilibrium of system is [15]:

mξ̈ + bξ̇ + kξ = − mη̈ + Fe Fe = 1 Q21ðtÞ ð1:1Þ
2 Aε0ε1

being m the proof mass, b the damping, k the structural stiffness and coordinate η
describes the excitation applied to the clamps in the fixed reference frame. The
electromechanical force Fe includes the relative permittivity of dielectric material ε1
and of vacuum ε0, the area of electrode interfaced A, and the electric charge Q1,
while t is the time.

The equivalent electric circuit is depicted in Fig. 1.5. The electrets material

supplies a constant bias voltage V2. The electric charge on the lower electrode is Q2,

Variable I V

capacitance V1 C ResisƟve
load
Equivalent +R
generator for the

electrets layer V2

Fig. 1.5 Equivalent circuit and simplified sketch of the configuration of Fig. 1.1

1 Role of the Structural Nonlinearity in Enhancing … 7

while on the upper one is Q1, because of the embedded layer of electrets. Capac-
itance C2 and voltage V2 of the electrets material are almost constant, although their
polarization suffers a degradation induced by temperature and aging [3]. The power
converted P can be measured through a resistive load, R, as a product of voltage, V,
and current I. Those two variables can be related to the energy harvester parameters
as follows:

V2 = V1 + V; V = V2 − V1; V1 = Q1 ; V2 = const.
CðtÞ

CðtÞ = C1C2 ; C1 = Aε0ε1 ; C2 = const.
C1 + C2 ξðtÞ
2 2
R CðtÞ dV R dQ1 ; ð1:2Þ
PðtÞ = RI 2 ðtÞ = =

dt dt
32
2 Aε0 ε1 + C2 5
PðtÞ = R4V22 − Q1 ξðtÞ

R Aε0 ε1 C2
ξðtÞ

Total capacitance C is a result of the series of two capacitances C1 and C2. The
first one depends on the relative position of electrodes ξ, as well as the elec-

tromechanical force Fe depends on the electric charge Q1. The electromechanical
coupled system is therefore described by the following equations:

mξ̈ + bξ̇ + kξ = − mη̈ 1 Q21ðtÞ
+ Aε0ε1

2 ð1:3Þ
Aε0 ε1
IðtÞ = dQ1 V2 − Q1 ξðtÞ + C2
dt R
= Aε0 ε1 C2R
ξðtÞ

Several simplifications are often applied when the identification of parameters of
Eq. (1.3) is performed. Mass m is usually assumed to be corresponding only to the
proof mass, although a contribution is given by the beam, being variable with the
excited vibration mode. Damping can be related to the loss factor of material as well
as to the air damping or the squeeze film effect. Beam stiffness is simply calculated
by referring to the linear static deflection under a concentrated load applied to the
proof mass [16]. This assumption does not consider the real contribution of each
vibration mode to the dynamic response of the VEH. Moreover, the nonlinearities
due to the beam stretching and to the large displacement of its cross sections are
completely neglected.

8 E.G.M. Brusa and M.Gh. Munteanu

1.3.2 Modeling of Continuous Beam Configuration: Linear
Approach

To investigate the nonlinear effects above mentioned the test case of [12] was first
considered. It looks like in Fig. 1.2. Its main properties are resumed in Table 1.1. It
is worthy noticing that as the beam bends, gap g(t) is not constant along the electrets
layer. Its local value can be written as:

gðx, tÞ = g0 − vðx, tÞ ð1:4Þ

where g0 is the equilibrium condition about which vibration of electrode occurs, v is
the displacement of moveable electrode from its initial shape, being measured along
the portion of the line axis interfaced with the electrets layer. Coordinate x runs
along the line axis of the beam, from the clamp to the free end. To suitably describe
the variable capacitance of the device, it is required to integrate the effects of the
variable gap as follows:

CðtÞ = 1 R 1 ⇔CðtÞ = 1 ð1:5Þ

d + 1 d + 11
ε2 ε0 A ε1 ε0 ε2 ε0 A n
w tÞdz ε1 ε0 Ai
gðx, ∑
λ gðxi, tÞ
i=1

where λ is the length of the electrets layer along the x–axis. When the system is
discretized through the finite element method the expression of capacitance
becomes the second one in Eq. (1.5). Discretization includes n electrical degrees of
freedom, found through a regular subdivision of the electrode area in elementary

Table 1.1 Test case analyzed in the numerical simulation 5 × 10−3
1.6 × 1011
Mass (kg) 1.3 × 10−2
Young’s modulus, E (MPa) 3 × 10−4
Beam width (m) 3 × 10−2
Beam thickness (m)
Beam length (m) 1400
Voltage (V) 2.18 × 109
Resistance (Ω) 1.27 × 10−4
Electret thickness, d (m) 5.93 × 10−4
Initial gap, g (m) 9.6 × 10−3
Interfaced length of electrode, λ (m)
Imposed acceleration, ÿ (m/s2) 4
Frequency of vibration (Hz)
Dielectric permittivity of vacuum, ε0 (pF/μm) 51.32
Relative dielectric permittivity, ε1 8.854 × 10−6
Relative dielectric permittivity, ε2
Length of tip mass (m) 1.00059

2.0
4 × 10−3

1 Role of the Structural Nonlinearity in Enhancing … 9

subareas, Ai, whose middle point along the line axis is detected by the xi coordinate.
A possible refinement of that discretization might be performed by resorting to
n segments not exactly equal.

The continuous structure of the VEH can be modeled by means of the finite
element method (FEM) as follows:

89
 Ã><> 1 =>>
ÂÃ ÂÃ + ÂÃ = ÈÉ − y0̈ M >:> ⋮ >>; − d Q2
M fvg̈ + C fvġ K fvg F 1 dfvg 2CðtÞ

ð1:6Þ

yðtÞ = y0 sinðω ⋅ tÞ dQðtÞ = V − Q
dt R CðtÞR

where {v} describes the vertical displacement of the n electrical degrees of freedom
of the moveable electrode with respect of the fixed counter–electrode, while vector
{F} includes all of mechanical actions. The goal of the analysis is investigating the
electromechanical coupling occurring between the compliant structure and the
capacitor in correspondence of the interfaced area of the electrodes. This result is
reached through two steps. The beam is discretized with two–dimensional beam
finite elements, with two nodes and three degrees of freedom per each node (u,
v and rotation θ) (Fig. 1.6). All relevant matrices are derived, namely the mass [M],
the damping [C], the stiffness [K] and the mechanical actions vector {F}. The
capacitor electrodes are even discretized in several capacitive elements, to allow the
prediction of the local effect induced by bending on the electric charge distribution.
Each element corresponds to a defined node of the beam along the line axis. To
describe the dynamic response of the VEH it is mainly required of investigating the
degrees of freedom corresponding to the nodes distributed along the electrodes.
Therefore, all the above mentioned matrices are reduced, by selecting the nodes of
the beam corresponding to the electrical elements, as master nodes and considering
as slave nodes all the other ones. In Eq. (1.6) over lined symbols mean that a

Fig. 1.6 Example of I
discretization through the
finite element method of Slave nodes Master mass
system of Fig. 1.2 v nodes

u R
V
Beam Q1 Q1,i C1,i gi(t)
nodes Vi

electret C2,i
Q2
Fixed
electrode

Electric
nodes

10 E.G.M. Brusa and M.Gh. Munteanu

reduction of degrees of freedom was applied. Mass was considered to be concen-
trated in those nodes, thus applying a reduction somehow similar to the so-called
static condensation [17].

The mass matrix looks like:

23
m1 0 0
ÂÃ = 40 ⋱ 05 ð1:7Þ
M

|fflfflfflffl0fflfflfflfflfflfflfflffl{0zfflfflfflfflfflmfflfflfflfflnfflfflffl}

nxn

The sum of all of partial masses mi is equal to the total mass. Damping matrix is
usually defined by resorting to the assumption of proportional damping, while
acceleration of all the nodes of the discretized system was assumed to be equal, i.e.
only a translational motion along the vertical direction was considered, although a
rotation about the clamp may be even added. Unit vector appearing in Eq. (1.6)
includes n elements.

The electromechanical action appears on the right hand of Eq. (1.6) as a column
of n elements:

fFemg = d 2CQð21tÞ ⇔ d Q12 = − Q21 ε0!ε12Ai ð1:8Þ
dfvg dyi 2CðtÞ
∑n ε0ε1Aj ðg0 − viðtÞÞ2

j = 1 g0 − vjðtÞ

where g0 is the initial constant gap between two electrodes. The above numerical
system can be written to be solved by means of the Runge–Kutta solution method

as:

8
<> Q̇1 V − Q1
= R CðtÞR

>: fv̇g = Âfzgà À Âà  à Á ð1:9Þ
fzġ = M C fzg K fvg ðvÞg
− 1 − − + fFem − fy0̈ g

being composed by 2n + 1 differential equations of first order. Solution can be
found iteratively by updating alternately the vector of electromechanical forces Fem
and the value of capacitance C for each increment of displacement v and finding the
new displaced configuration of the beam tip region.

Numerical results obtained by using the model of Eq. (1.9) and those described
in [12] were compared, for the test case with optimized configuration. It was found
a good agreement (Fig. 1.7), after that the damping coefficient was set at the same
value. In that case was found a maximum displacement of the tip mass of about 2%

of the beam length (geometric linearity).

1 Role of the Structural Nonlinearity in Enhancing … 11

1000 1000

800 800
Output Voltage [V]
Output Voltage [V]600600

400 400

200 200

0 0 0.5 1 1.5 2
- 200 0 0.5 1 1.5 2 2.5 3 - 200 0

- 400 - 400

- 600 - 600

- 800 - 800

- 1000 - 1000

Time [s] Time [s]

Fig. 1.7 Comparison between results for the test case presented in [12] and results of the
numerical simulation based on the linear model

1.3.3 Modeling of Continuous Beam Configuration:
Nonlinear Approach

The model above described is valid in case of small displacements and linear
behavior. Stiffness matrix is accordingly written [20]. When conditions for the so–
called geometrical nonlinearity occur (improperly called large displacement non-
linearity) it is required resorting to the second order theory of beam [14]. Difference
between the first order beam theory and the second order can be shortly shown in
following equations:

& d2u ( d4v
dx2 dx4
= 0 EI = p ð1:10Þ

N = EA du M = EI d2v
dx dx2

The above set of Eq. (1.10) describes the linear first order theory. It can be
basically appreciated that for a linear distribution of vertical load, p, in linearity
flexural and axial behaviors are uncoupled, thus allowing to compute the axial effort
N separately from the bending moment M. Those actions are calculated by means of
corresponding strains, which are directly expressed as a function of the axial dis-
placement u and the vertical displacement v, but appear separately in the two above
mentioned equations. Other symbols are Young’s modulus of elasticity, E, cross
section area of beam, A, transversal moment of area of the second order (improperly
flexural inertia), I.

When the second order theory of beam is considered the above equations
become:

8 d2u dv d2v
<>>>> dx2 + dx dx2 =0
>>>:>
NEI=ddx4E4v A=hpddux++EA21 ÀdddduxxvddÁx222vi ð1:11Þ

M = EI d2v
dx2

12 E.G.M. Brusa and M.Gh. Munteanu

Table 1.2 Components of the stiffness matrix of the beam structure
½KŠ = ½K0Š + ½KNLŠ
½K0Š ⇒ I order theory; linear, small displacements, axial and flexural behaviors uncoupled ,
½KNLŠ ⇒ Second order approximation theory, larger displacements, axial and flexural behaviors
coupled by a variable normal effort
N ðclamped−clamped beam like Fig. 3Þ

As it looks clear a coupling effect between axial and flexural behavior, respec-

tively, occurs. The load distribution affects the axial strain of beam, while the axial

effort N is coupled with the vertical displacement, v. This coupling can be due in

cantilever configuration to a rotation of the tip sufficiently large to apply a load

component along the line axis (Fig. 1.2). Moreover, in case of a clamped–clamped

configuration, the two constraints apply an axial force to the beam under bending

even when vertical displacements are fairly small (Fig. 1.3).

To introduce the second order theory inside the model of Eq. (1.6) it is sufficient

to formulate the stiffness matrix by including all the elements describing its

dependence on the increasing normal effort N, as a function of displacement v. More

details are described in [14]. Procedure is sketched in Table 1.2.

In the nonlinear case Eq. (1.9) becomes:

8
>< Q1̇ = V − Q1
R CðtÞR

>: fv̇g = Âfzgà À Âà Á ð1:12Þ
fzġ = M C fzg ðvÞg
− 1 − − fFel g + fFem − fy0̈ g

where {Fel} represents the vector of elastic forces that in the linear case are
fFelg = ½K̄Šfvg. When the structural behavior is no longer linear, the elastic forces are
computed by means of an iterative approach. In the frame of the Runge–Kutta method,
at each time step, displacements {v} are known and elastic forces and electrome-

chanical forces are accordingly computed. The elastic forces are computed iteratively
by following the Newton–Raphson method. The Runge–Kutta method is used to
solve the differential system of Eq. (1.12). It requires very small time steps, therefore a
couple of iterations is needed to apply the Newton–Raphson method. This approach
was deeply developed within the theory of the nonlinear finite elements in [20].

1.4 Advantages of the Structural Nonlinearity
in the Design of the Electrets-Based Energy Harvester

1.4.1 Stiffening Effect on a Clamped–Clamped Structure

A practical comparison between the real behavior of a clamped–clamped beam in
linear and nonlinear operating conditions can be performed on a second test case.

1 Role of the Structural Nonlinearity in Enhancing … 13

Fig. 1.8 Comparison between the linear and the geometrically nonlinear behaviors of a
clamped-clamped beam

A beam with length L = 60 mm, width w = 26 mm, thickness h = 0.3 mm,
Young’s modulus E = 160000 MPa and Poisson’s coefficient ν = 0.3 was ana-
lyzed. Actually it exhibits the same stiffness of the cantilever beam described in 1.1
and frequency of the first vibration mode is 51.33 Hz.

The analysis of the static behavior of this test case is shown in Fig. 1.8. A first
solution was found without considering the electromechanical coupling, but only
the deflection of beam under a mechanical concentrated load applied to the middle
span. In case of two clamps an axial–flexural coupling occurs, thus requiring to
resort to the second order theory. By converse in case of one end simply guided
along the line axis, without constraining the axial displacement, the linear theory is
sufficient to describe the characteristic curve of force–versus–flexural displacement.
As Fig. 1.8 shows the difference is evident, even for fairly low values of force and
under the assumption of perfect clamps.

1.4.2 Role of the Constraint Compliance on the Stiffening
Effect

As Fig. 1.8 points out the two extreme conditions are corresponding to the
clamped–free and clamped–clamped configuration, respectively. When the clamps
exhibit an intrinsic compliance, the axial displacement of the beam is inhibited, but

14 E.G.M. Brusa and M.Gh. Munteanu

Fig. 1.9 Role of the constraint compliance on the geometric nonlinearity of beam: a infinitely
rigid constraint, b k0 = 1.53e + 005 N/m—high stiffness of clamp, c k0 = 4.94e + 004 N/m—
moderate stiffness of clamp

under the effect of a local axial stiffness of the constraint, k0. The characteristic
curve described in Fig. 1.8 changes as in Fig. 1.9.

The stiffening effect introduces in the frequency response of the system a clear
nonlinearity. It is evidenced by the so–called jump of the curve close to the apparent
resonance of the dynamic system (to nonlinear system resonance concept is not
applicable) in Fig. 1.10. This effect has two relevant consequences for the energy
harvesting purpose. The amplitude of the dynamic response is fairly high not only
in correspondence of a narrow range of frequency values, like in the linear systems
happens just close to the resonance. Moreover, above a defined value of frequency
the system response is somehow damped. This effect might be useful to prevent an
unforeseen failure of the device. In addition, the slope of the amplitude curve is
regulated by the constraint compliance and the dependence on the value of fre-
quency is almost linear. Obviously the stiffening effect increases the frequency at
which the amplitude reaches its maximum, before jumping down (Fig. 1.10).

1 Role of the Structural Nonlinearity in Enhancing … 15

(a) (b)

Amplitude [μm]
Amplitude [μm]

Frequency [Hz] Frequency [Hz]

Fig. 1.10 Role of constraint compliance on the nonlinear dynamic response of beam: a infinitely
rigid constraint, b moderate stiffness of clamp

1.5 Some Design Criteria for the Electrets–Based Energy
Harvester

1.5.1 Clamped-Sliding Configuration

To identify some practical criteria for the design, the solution for the electrome-

chanical coupled system was analyzed. As a matter of facts, if the gap is fairly large

the power conversion tends to be lower.
Nevertheless, a configuration like in Fig. 1.8 with only a clamp and the other one

allowing the axial displacement (to be referred to as sliding) can improve the
efficiency of the energy conversion. If the simulation is run in case of the first test
case and the cantilever–based configuration is compared to the clamped–sliding
layout it can be appreciated that a slight improvement is found. If the numerical
inputs are the same for both the configurations, i.e. V = 1400 V, d = 127 mm,
R = 300 MΩ, but the gap is increased up to g = 1 mm, results are those of
Fig. 1.11. Damping ratio was set at ζ = 0.025, the exciting frequency was
ω = 50 Hz and the acceleration amplitude ÿ0 = 4.5 m/s2.

Results point out in Fig. 1.11 that the dynamic excitation basically acts in the

same way on the dynamic response of the beam, in terms of maximum displace-

ment of proof mass. However, the simulation shows that a slight rotation of the tip

mass in case of a cantilever allows to have a slightly less effective coupling. Output

voltage is larger in case of a clamp and a sliding device, because the proof mass is

kept with its surface aligned with the lower electrode, thus exploiting better the gap
between electrodes. In this case it is worthy noticing that first order theory was
sufficient for the second configuration, since one clamp allows the axial
displacement.

16 E.G.M. Brusa and M.Gh. Munteanu

Fig. 1.11 Comparison of performance between (a, b) cantilever and (c, d) clamped–sliding
configurations

Stiffening effect with perfect clamps requires to resort to the second order theory.
For the same inputs of the above mentioned cases, it can be immediately appre-
ciated how much the dynamic response is lower, because of the beam stretching
(Fig. 1.12).

Fig. 1.12 Performance of the clamped–clamped beam with same inputs of cases in Fig. 1.11

1 Role of the Structural Nonlinearity in Enhancing … 17

1.5.2 Clamped-Clamped Configuration

To fit the need of providing a device fairly sensitive to a wide range of frequency
actually the geometric nonlinearity may help. If the clamped–sliding and the
clamped–clamped beams are compared in terms of dynamic response in the fre-
quency domain, it can be clearly appreciated a significant difference in Fig. 1.13,
where results were plotted within the range of variation of voltage described in
above Fig. 1.11. In Fig. 1.13c, d it can be appreciated how the nonlinearity can be
exploited to have an amplitude variable with frequency, almost linearly. This effect
is due to the curved backbone of the path in Fig. 1.13c, d. In linear system the
dynamic response curve is almost symmetric with respect to the so–called back-
bone, i.e. a symmetry axis which could be plotted along the vertical direction at
resonance. In a nonlinear system with stiffening effect as the amplitude grows up,
the frequency increases because of the higher stiffness. Therefore the peak of
resonance moves towards the right side of the diagram, thus creating a superpo-
sition of numerical solution with the lower path of the curve. The nonlinear system
naturally tends to reach the solution with associated the lowest energy and

Fig. 1.13 Frequency response of (a, b) clamped–sliding and (c, d) clamped–clamped beam
configurations

Force [N]18 E.G.M. Brusa and M.Gh. Munteanu

apparently it suddenly jumps down from the peak to a very low level of amplitude.
According to Figs. 1.10b and 1.12 a key issue of design might be assessing a
suitable value for the constraint compliance to find a compromise between the
amplitude of the dynamic response and the narrow frequency range in which it can
be exploited in the linear system behavior.

1.5.3 Application of Additional Constraints

The configurations based on clamp–sliding and clamped–clamped constraints show
some weakness. A linear system provides a narrow range of frequency to usefully
operate the energy harvester, the nonlinear system with larger range unfortunately
provides a weaker dynamic response in terms of amplitude.

A possible solution could be resorting, like in case of some RF–MEMS [21], to a
variable constrained configuration as depicted in Fig. 1.14. In practice, some sup-
ports are positioned below the deformable electrode at a certain distance from it.
When the bended beam touches the supports its behavior looks like that of a more
compliant structure, because of the portion of beam supported between the clamp
and the middle part of the structure. The performance of this configuration is
described in Fig. 1.15. The amplitude of the dynamic response is kept almost
maximum within a wider range of frequency, while the deflection is fairly large in a
first step of bending.

Maximum vertical displacement [μm]
Fig. 1.14 Characteristic curve of the clamped–clamped beam configuration with supports

1 Role of the Structural Nonlinearity in Enhancing … 19

Fig. 1.15 Frequency response of the clamped–clamped beam configuration with supports

This layout may overcome some problems evidenced in [18], where the authors
introduced some nonlinear springs expressively microfabricated to exploit some
benefits of the nonlinear behavior of structural elements for the energy harvesting.

1.6 Conclusion

The literature claim that a main benefit of the electrets materials applied to
microtechnology and energy harvesters is making possible to provide a local and
autonomous power supply for miniaturized devices, based on capacitive systems,
where a bias voltage is required to operate the energy conversion. Nevertheless,
performance of those materials is still considered somehow insufficient to have a
relevant technological impact.

Actually, it can be noticed that the effectiveness of the electrets–based system
does not depend uniquely upon the coupling coefficients of such smart material, but
even on the dynamic behavior exploited for converting the energy. In this paper the
case of energy harvesters based on flexible beams was analyzed. A first contribution
consists of focusing on the nonlinear behavior exhibited by the clamped–clamped
configuration, because of the mechanical coupling between axial and flexural
behavior, respectively. Stiffening associated to the beam stretching actually might
reduce the peak of amplitude reached in dynamic behavior, but the range of fre-
quency assuring a high amplitude of the vibration mode of the beam is larger. As a
consequence a key issue of the design are the compliance and the layout of sup-
ports. If the beam is supported by compliant clamps, with a defined stiffness, the
best configuration should be set up by looking at the maximum peak achievable by
the dynamic response and the frequency bandwidth in which it is reached. The
nonlinear behavior might help in assuring the best conversion over a suitable range
of uncertainty concerning the frequency of excitation. Obviously the design activity
must assure a suitable reliability against the structural damage. A improved

20 E.G.M. Brusa and M.Gh. Munteanu

constructive solution was proposed, it consists of adding some supports to excite
the strongly nonlinear response of the system, above a certain amplitude of
vibration. A future work will investigate some practical issues about the proposed
solution on small prototypes of vibration energy harvester. However, a slight
benefit in efficiency was demonstrated, despite of the low coupling effect provided
by some electrets material.

References

1. Wilson S (2007) New materials for microscale sensors and actuators—an engineering review.
Mater Sci Eng R56:1–129

2. Uchino K (2010) Advanced piezoelectric materials. Woodhead Publishing Ltd., Cambridge
3. Evreinov G, Raisamo R (2005) One–directional position–sensitive force transducer based on

EMFi. Sens Actuators A 123–124:204–209
4. Pozzi M, Zhu M (2011) Plucked piezoelectric bimorphs for knee-joint energy harvesting:

modeling and experimental validation. Smart Mater Struct 20
5. Mitcheson P, Yeatman E, Rao GK, Holmes AS, Green TC (2008) Energy Harvesting from

human and machine motion for wireless electronic devices. Proc IEEE 96(9):1457–1486
6. Boisseau S, Despesse G, Seddik BA (2012) Electrostatic conversion for vibration energy

harvesting. In: Small–scale energy harvesting. Intech, 2012
7. Sterken T, Fiorini P, Baert K, Puers K, Borghs G (2003) An electrets–based electrostatic–

generator. In: Proceedings of the 12th International conference on solid state sensors,
actuators microsystems (Transducers), Boston, MA, pp 1291–1294
8. Okamoto H, Suzuki T, Mori K, Cao Z, Onuki T, Kuwano H (2009) The advantages and
potential of electrets–based vibration–driven micro energy harvesters. Int J Energy Res
33:1180–1190
9. Mizuno M, Chetwynd DG (2003) Investigation of a resonance microgenerator. J Micromech
Microeng 13:209–216
10. Brusa E, Munteanu M (2009) Role of nonlinearity and chaos on RF-MEMS structural
dynamics. In: Proceedings of the IEEE, design test integration packaging of MEMS and
MOEMS—DTIP 2009, 1–3 April 2009, Roma, Italy, IEEE Catalog N. CFP09DTI
11. Paajanen M, Vaklimakki H, Lekkala J (2000) Modelling the electromechanical film (EMFi).
J Electrostat 48:193–204
12. Paajanen M, Lekkala J, Kirjavainen K (2000) ElectroMechanical Film EMFi—a new
multipurpose electret material. Sens Actuators 84:95–102
13. Deng Q, Liu L, Sharma P (2014) Electrets in soft materials: nonlinearity, size effects, and
giant electromechanical coupling. Phys Rev E 90:0126031–0126037
14. Bettini P , Brusa E, Munteanu M, Specogna R, Trevisan F (2008) Innovative numerical
methods for nonlinear MEMS: the non incremental FEM vs. the discrete geometric approach.
Comput Modell Eng Sci (CMES), 33(3):215–242
15. Boisseau S, Despesse G, Ricart T, Defay E, Sylvestre A (2011) Cantilever–based energy
harvesters. Smart Mater Struct 20:105013
16. Chiu Y, Lee Y-C (2013) Flat and robust out–of–plane vibrational electret energy harvester.
J Micromech Microeng 23:015012–8 pp
17. Miki D, Honzumi M, Suzuki Y, Kasagi N (2013) Large amplitude MEMS electret generator
with nonlinear spring. Proc IEEE:176–179. ISBN-978-1-4244-5764-9
18. Suzuki Y, Miki D, Edamoto M, Honzumi M (2010) A MEMS electret generator with
electrostatic levitation for vibration-driven energy-harvesting applications. J Micromech
Microeng 20:104002–8 pp

1 Role of the Structural Nonlinearity in Enhancing … 21

19. Brusa E (2006) Dynamics of mechatronic systems at microscale. In: Microsystem mechanical
design, CISM Lectures Series, 478, Springer Verlag, Wien, 2006, pp 57–80

20. Zienlkiewicz OC, Taylor RL (2004) The finite element method, 5th edn. Butterworth-
Heinemann

21. Lucyszyn S (2010) Advanced RF MEMS. Cambridge University Press, Cambridge

Chapter 2

Numerical Analysis of Fracture
of Pre-stressed Ferroelectric Actuator
Taking into Account Cohesive Zone
for Damage Accumulation

Sergii Kozinov and Meinhard Kuna

Abstract Operational safety of smart-structures as well as ferroelectric multilayer
actuators (MLA) is essentially reduced by crack formation. Such failure processes
are numerically simulated in this work by the finite element method (FEM) employ-
ing coupled electro-mechanical analyses. First step of the simulation is the poling
process during manufacturing of the actuator. During this step, the pre-stress due to
the presence of an external frame which protects the MLA against tensile forces is
obtained. After the end of the poling process, an alternating electric loading with
constant amplitude is applied. In order to model the bulk material behavior, ferro-
electric user elements are implemented into the commercial software ABAQUS, thus
allowing to simulate the poling process of the actuator as a result of the microme-
chanical domain switching. Material damage is accumulated in accordance with the
traction-separation law (TSL) of an electro-mechanical cyclic cohesive zone model
(EMCCZM). The cohesive zone technique allows to capture initiation and accumu-
lation of damage, while domain switching modeling ensures a realistic simulation of
the non-linear processes occurring in the ferroelectric material. In the cohesive zone,
a finite electric permittivity is assumed, which degrades with damage accumulation.
Another important feature is that applied cyclic loading of a constant amplitude leads
to increasing damage which can not be modeled with a monotonous TSL. The results
of the numerical simulation qualitatively coincide with the experimentally observed
patterns of crack initiation. It was found that the poling process of ceramics may
induce cracking at an electrode surface, which further developes under purely cyclic
electric loading. Damage evolution is observed due to mechanical and electrical field
concentrations near the electrode tip. The methodology for the analysis dealing with
coupled ferroelectromechanical modeling combined with damage accumulation in

S. Kozinov (✉) ⋅ M. Kuna 23

Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg,
Lampadiusstraße 4, 09596 Freiberg, Germany
e-mail: [email protected]

M. Kuna
e-mail: [email protected]

© Springer International Publishing Switzerland 2017
A.L. Araujo and C.A. Mota Soares (eds.), Smart Structures and Materials,
Computational Methods in Applied Sciences 43,
DOI 10.1007/978-3-319-44507-6_2

24 S. Kozinov and M. Kuna

smart structures was recently developed by the authors and published in [1, 2]. The
new problem of the current paper is the analysis of the effect of the pre-stress in
the actuator, which is an important step towards future optimizations of the actuator
design.

2.1 Introduction

Piezo-(ferro-)electric multilayer actuators (MLA) are widely used in the field of
high-precision positioners due to their accuracy, high stiffness, large generative
forces and fast response [3]. A present-day multilayer actuator consists of hundreds
of ceramic layers aggregated with internal electrodes, which terminate inside the
ceramics. High electrical fields or local stress concentrations in the vicinity of the
electrode edge, originating during actuators exploitation, make it necessary to cor-
rectly predict fracture of such smart devices. Profound reviews about cracking in
ferroelectric ceramics can be found in [4, 5].

Piezoelectric actuators show the non-linear electromechanical behavior of ferro-
electric bulk ceramics due to the domain-switching phenomenon caused by the high
electric field concentration. Early finite element method (FEM) simulations of actu-
ators mostly focused on the modeling of linear piezoelectric materials, which is a
rough approximation of inherent non-linear electromechanical properties of ferro-
electrics. Moreover, in most of these analyses, electromechanical fields are not cou-
pled and are only treated in a sequential way. In the present investigation a micro-
mechanical material model, based on tetragonal volumetric domain switching [6], is
used.

The crack propagation mechanisms in the actuators are still not clear due to the
complex non-linear interactions between electromechanical fields and microstruc-
ture near the electrode edge. Lack of accurate experimental data is due to several
reasons. The strains found from strain gages are only average values in the local area
due to the large size of gages in comparison with the thickness of ceramic layers.
Therefore deformations near the electrode edge can not be precisely measured. On
the contrary, interferometry only provides limited pattern information in the small
region near the electrode edge.

In general, experimental observations indicate that the electric-field induced
cracks initiate substantially from the electrode edge and they can propagate along
the ceramic-electrode interface [7–9]. The concept of cohesive zone models is quite
efficient to simulate initiation and evolution of damage and cracking, provided that
one or several possible damage paths can be introduced a priori with embedded cohe-
sive elements, as, for example, along interfaces in polycrystalline ceramics. A first
adaptation of the classical exponential cohesive zone model to ferroelectric materi-
als for simulating electric fatigue was done by Arias et al. [10], though some physical
simplifications were made. Other simulations with cohesive zone elements but with
piezoelectric bulk behavior were performed by Utzinger et al. [11] and Verhoosel
and Gutiérrez [12].

2 Numerical Analysis of Fracture of Pre-stressed Ferroelectric . . . 25

The coupled electromechanical cyclic cohesive zone model (EMCCZM) was sug-
gested by Kozinov et al. [2] as an extention of the pure mechanical cyclic cohe-
sive zone elements developed by Roth et al. [13]. Both mechanical and dielectric
properties of the material during its ongoing degradation are taken into account in
EMCCZM. It allows to track initiation and evolution of the interface cracks dur-
ing mechanical and/or electrical loading, applied to simple structures and actuators.
Damage accumulation is captured resulting from the electromechanical response of
the actuators during operational loading.

Kamlah and Böhle [14] did pioneering studies of multilayer actuator response
incorporating the simulation of a ferroelectric behavior. Subsequent papers by Zhao
et al. [9] and Abdollahi and Arias [15] offered improvements and modifications, as
well as simplifications.

The first fully coupled electromechanical simulation of the MLA, considering
ferroelectric bulk material behavior together with the cohesive zone implementation,
was recently presented by the authors [1].

In order to achieve a longer life-time and better performance, the MLAs are usu-
ally pre-stressed by an outer clamping frame. Numerical simulations are needed for
a better understanding and optimization of the effect of pre-stressing on the opera-
tional performance of the actuator. To the authors’ knowledge a fully coupled electro-
mechanical simulation of the pre-stressed actuator with a cohesive zone implemen-
tation for damage accumulation is done for the first time. It enables to study electro-
mechanical fields in the critical regions of a MLA and to investigate possible gradual
failure.

2.2 Ferroelectric Materials Constitutive Behavior and
Electromechanical Cyclic Cohesive Zone Model

For a detailed explanation about EMCCZM as well as the constitutive behavior of fer-

roelectric materials the reader is referred to [2]. In this section only a brief overview

is provided.
Constitutive behavior of a piezoelectric material with remanent strain lrs and

remanent polarization Pir of the polycrystal in a Cartesian coordinate system xk(k =
1, 2, 3) is formulated by the following equations:

ij = Cijls( ls − lrs) − esijEs, Di = eils( ls − lrs) + isEs + Pir. (2.1)

where uk, ls, , ij, Di, Esi are mechanical displacements, strains, electric potential,
mechanical stresses, electric displacements and electric field, respectively. The coef-

ficients Cijls, esij, is are elastic, piezoelectric and dielectric moduli, correspondingly.

Domains are subregions of a grain, in which all dipole moments of the neigh-
lssp and
boring unit cells are aligned and have identical spontaneous strain electric sponta-
neous polarization Psip (Fig. 2.1, left). During application of a strong loading

26 S. Kozinov and M. Kuna

Fig. 2.1 Reorientation of the domains during poling process ( strain in a representative poly-
crystalline volume element, r—remanent strain, homogenized over the polycrystal)
Fig. 2.2 Strain hysteresis
loop

(Fig. 2.1, center) domains orient along the direction of the electric field. After ter-
mination of the external electric loading a remanent polarization Pr as well as a
remanent strain r remain (Fig. 2.1, right).

According to the nonlinear evolution law defined by Eq. (2.1), the strain hystere-

sis (Fig. 2.2) as well as the polarization hysteresis (Fig. 2.3) can be reproduced as

consequence of domain switching.

Ferroelectric domain switching happens when the critical work barrier [5]

√ (2.2)
c±90◦ = 2PspEc, 1c80◦ = 2PspEc

is overcome by the energy supply [16]

ij ij + ij isjp + EiDi + Ei Pisp ≥ c , ∈ {−90◦, 90◦, 180◦}. (2.3)

For tetragonal domains two types of switching are possible: 90◦ switching, alter-
ing both spontaneous strain and polarization, and 180◦ switching, leading to spon-

taneous polarization change with no influence on the spontaneous strain value. This

2 Numerical Analysis of Fracture of Pre-stressed Ferroelectric . . . 27

Fig. 2.3 Polarization
hysteresis loop

micromechanical ferroelectric model was implemented as user routine for the finite
element code Abaqus©, see [17].

In the numerical modelling the specific data of a lead zirconate titanate PZT-5H
are used. The linear material properties are presented in Table 2.1, while the non-
linear ferroelectric quantities are shown in Table 2.2.

One of the most commonly used approaches to model the crack growth is an
irreversible cohesive law with loading-unloading hysteresis. According to the cohe-
sive zone model (CZM), the material gradually loses its load-bearing capacity. Thus
the whole damage process, starting from the crack formation until complete failure,
can be modelled. Information about the developed electromechanical cyclic cohe-
sive zone model (EMCCZM) can be found in [2] and is schematically presented in
Fig. 2.4, right. Orange dots show different positions on the traction-separation law
(TSL) t( ) during the course of material damage. TSL connects normalized effective
traction t and separation , introduced as following:

√√ (2.4)
t = tn2 + tr2 + ts2∕t0, = ⟨ n⟩2 + r2 + s2∕ 0.

Here ⟨ n⟩ = ( n + | n|)∕2; t0 denotes maximum cohesive traction, 0—critical sep-
aration at maximum cohesive traction; indices n, r and s stand for normal and two

tangential components.

Electrically, cohesive zone element (CZE) has the behavior of a medium with
limited dielectric permittivity [18]. With the accumulation of damage D, dielectric
permittivity of the cohesive elements degrades

c = (1 − D) a + D 0 (2.5)

28 S. Kozinov and M. Kuna

Table 2.1 Properties of the PZT-5H ceramics (polarized along x1 axis) [5]
Elastic moduli (MPa)

C1111 C2222, C3333 C1122, C1133 C2233 C1212, C1313 C2323
117000 126000 53000 55000 35300 35500

Dielectric constants ( F/m) Piezoelectric constants (C/m2)

11 22, 33 e111 e122, e133 e212, e313
0.0151 0.0130 23.3 −6.5 17.0

Table 2.2 Nonlinear ferroelectric constants of the PZT-5H ceramics [1]

Spontaneous Psp 0.3 C/m2

polarization

Coercive field strength Ec 0.8 kV/mm

Spontaneous strain sp 0.3 %

Fig. 2.4 Parallel plate capacitor model (left) and cohesive zone approach (right)

and the dielectric displacement field Dn inside the CZE is defined as [2]:

Dn = − [ − D) a + ] + − − . (2.6)
(1 D 0 0 + n

The CCZM captures damage accumulation during cycling with constant amplitude

and distinguishes between active separation and endurance threshold e using an
evolution law of the following form [13]:

Ḋ = (1 − D) 1 − − D) ⟨ ̇ ⟩ H( − e), (2.7)
log(1

where H is the Heaviside step function. Formula (2.7) is appropriate for modeling
piezoelectric ceramics degradation during exploitation loading.

2 Numerical Analysis of Fracture of Pre-stressed Ferroelectric . . . 29

An example of the CZE behavior during monotonic loading from an initially
undamaged state and subsequent unloading-reloading cycles is presented in Fig. 2.5.

The electromechanical cohesive element has eight nodes and four integration
points located in a midplane (see Fig. 2.6) and each node has three mechanical and
one electrical DOF. In ferroelectric and cohesive elements full integration scheme
is used. During electromechanical loading the upper and lower faces of CZE split in
normal and/or tangential directions.

The properties of the EMCCZM are specified in Table 2.3.

Fig. 2.5 Response of CZE
to initial monotonic loading
followed by two
unloading-reloading cycles
according to the TSL

Fig. 2.6 Illustration of
EMCCZE; midplane values
are marked in red

30 S. Kozinov and M. Kuna

Table 2.3 Material constants of the cohesive zone [1] N/m
MPa
Fracture energy 0 2.34 nm
t0 100 nm
Maximum cohesive F/m
traction
F/m
Critical separation 0 = 0∕(e t0) 8.608
0 5
Thickness of a grain
boundary

Grain boundary 0 8.854 ×10−6

permittivity at failure

(air)

Initial grain boundary a = 0 r 0.006
permittivity

2.3 Numerical Simulation

The main aim of the present study is to investigate damage initiation and accumu-
lation in the MLA. The presence of an external frame results in the pre-compressed
state in the MLA after the poling process.

In the numerical calculations, domain switching in the bulk material is accounted
for by means of the non-linear ferroelectric model and damage is captured using the
electromechanical cyclic CZE. Some simple examples as well as model validation
can be found in [2, 19]. The proposed approach adequately describes the processes
occurring in the microstructure of smart ceramics.

The geometry of the MLA, the finite element mesh and the applied boundary con-
ditions are presented in Fig. 2.7. In x2 direction plane strain conditions are applied;
the cohesive elements have zero initial thickness. The profile of the outer frame is
schematically presented in Fig. 2.7. In practical realizations, the frame is a steel cage
acting as an elastic spring. The lower faces of the actuator and the external frame
are fixed while their upper faces move jointly. The cross-sectional area of the frame,
made of steel with Young’s modulus 2 × 1011 Pa, is 2.5 times smaller than that of
the actuator. It is assumed that the frame and the actuator are assembled stress-free
before poling. Pre-stress of the actuator arises as a result of the poling process and
remains during the electrical cyclic loading.

Fig. 2.7 Finite element model of the actuator

2 Numerical Analysis of Fracture of Pre-stressed Ferroelectric . . . 31

During poling, the electric potential is increased from 0 to 100 V and then
reduced to zero (see Fig. 2.8). This leads to an electric potential gradient growth
up to 2 kV/mm (2.5 Ec) between the electrodes, which is enough to polarize the
ceramics (Fig. 2.9).

The polarization of the actuator at maximum electric voltage is illustrated in
Fig. 2.9. The PZT-5H ceramics is initially unpoled. Large number of vectors emerge
along the electrode due to the high mesh density. Three different regions are easily
observed:

∙ an active zone where after the poling process the ceramics is polarized along the
x3 axis (left),

∙ an inactive zone with low electric field and minor (no) polarization (right),
∙ a transition zone in-between (center).

The electric potential along the electrode plane is plotted in Fig. 2.10 for the
maximum voltage (red, dotted line) and after poling (red, solid line). The electrode
(0...125 µm), where the electric potential = 100 V is prescribed as the boundary
condition, is followed by a gradual decay towards zero at the free edge of the actuator
(125...250 µm). After poling, the residual electric potential of about −5 V is notice-
able ahead of the electrode tip. Such electric behavior was first reported in [14].

The electric potential distribution during subsequent operational cyclic electric
loading (Fig. 2.8) is shown in Fig. 2.10 with the black dotted and solid lines for max-
imum and minimum applied voltage, respectively.

Fig. 2.8 Diagram of the applied electric potential
Fig. 2.9 Vector plot showing polarization at maximum voltage during the poling process

32 S. Kozinov and M. Kuna
Fig. 2.10 Electric potential
along the electrode plane
over the poling procedure
and during exploitation

Fig. 2.11 Change in normal
stress along the electrode
plane during poling and
exploitation

The red-coloured curves in Fig. 2.11 illustrate the normal stress distribution
33(x1) in the cohesive layer during poling. The dash-dot red line shows stresses at
maximum voltage, while the solid red line corresponds to the residual stresses after
completion of the poling process. The analyses show distinguished regions of com-
pression, tension and transition, which agree well with the previous simulations for
the actuator without an external shell [1, 14]. The presence of the frame expectedly
offsets the stresses in the MLA by a few tens of MPa downwards compared to the
standard MLA, thus enhancing lifetime of the actuator and giving it the possibility
to work in more severe operating conditions. The external frame experiences tensile
loading at all times.

After finishing the poling phase, cyclic electric voltage with constant amplitude is
applied, which alternates from 0 to 33 V (see Fig. 2.8). Damage accumulates merely
due to the electric cyclic loading. A contour plot of the damage value, accumu-
lated from 90 cycles, is shown in Fig. 2.12. For better visualisation, the deformation
near the electrode tip is scaled 150 times. It is remarkable that the simulated open-

2 Numerical Analysis of Fracture of Pre-stressed Ferroelectric . . . 33
Fig. 2.12 Damage
accumulated after 90 electric
cycles

Fig. 2.13 Speckle image of
the crack caused by electric
loading (optical microscopy,
taken from [9])

ing of cohesive elements predicts the experimental observations [9], namely, the
biggest separation and primarily damaged region lies ahead of the electrode edge
(see Fig. 2.13, where the dark segment ahead of the electrode is an interlayer gap).

Four characteristic points are selected for further illustration. Point A is located
1 µm ahead of the electrode tip, points B and C—at 4 µm and 18 µm, respectively.
Point D belongs to the right vertical face of the actuator, namely, x1D = 250 µm.

Damage initiation and accumulation at the characteristic points A, B, C and D
is shown in Fig. 2.14 over 1000 cycles. According to the TSL, damage accumu-
lates when cohesive traction is high enough to overcome endurance locus [13].
Below endurance limit, unloading/reloading curves coincide for an infinite number
of cycles. Points A and B belong to the region where endurance locus is overcome
during the poling process. During cyclic voltage this region continues to accumu-
late damage and extends with each cycle. At the point C damage initiates after 500
cycles (Fig. 2.14, blue dotted line) which is seen as a propagation of the crack from
the actuator tip. Further crack growth is restricted according to the stress distribution
along the electrode, where a stress drop is observed (see Fig. 2.11, red curves). At
the point D there is no damage accumulation since the applied loading is insufficient
to overcome the endurance limit in contrast to the MLA which is not pre-stressed
[1].

34 S. Kozinov and M. Kuna
Fig. 2.14 Damage
accumulation shown at
characteristic points along
the electrode

Fig. 2.15 Change in
normalized effective opening
during cyclic electrical
loading

In Fig. 2.15 normalized effective opening variation is presented. At the points A
and B opening is increasing as a consequence of diminishing resistance of the cohe-
sive elements. Point B possesses bigger magnitude after 100 cycles compared to
the point A, since point A is closer to the compressed area at the left of the elec-
trode tip (see Fig. 2.12). After 300 cycles effective opening growth in the cohesive
elements slows down with complete damage at the points A and B. Opening at the
point C is gradually increasing with damage accumulation as it is seen from Fig. 2.15.
Since the rightmost region of the electrode is separated from the developing crack
by a compressed area and no damage accumulation is observed, there is no change
in normalized effective opening at the point D during electric cycling.

Figures 2.16 and 2.17 illustrate oscillations of the normal tractions tn as a result
of electric cycling with constant amplitude. While Fig. 2.16 displays general view
of the normal traction variation, Fig. 2.17 magnifies plots in selected cycling inter-
vals, since cycling loops at Fig. 2.16 are too condensed. Therefore, two characteristic
ranges are chosen and a break in the x-axis is introduced. One can readily see that

2 Numerical Analysis of Fracture of Pre-stressed Ferroelectric . . . 35

Fig. 2.16 Normal traction
variation during cyclic
electric loading of constant
amplitude: general view

Fig. 2.17 Normal traction
variation during cyclic
electric loading of constant
amplitude: detailed view
with a break in x-axis

during the poling process CZEs A and B experience an effective traction, which
is lower than the maximum cohesive traction, but higher than the endurance limit.
In this cohesive elements cyclic electric loading as well leads to an effective trac-
tion, which exceeds the endurance limit. From the beginning, the peak value of the
normal tractions and its amplitude are observed at the point A. This leads to ear-
lier damage initiation than at the point B (see Fig. 2.14). With continuing cycling,
the damage zone is developing and peak stresses are shifting to the right, thus after
500 electric cycles the point C becomes the focus of the damage area. By this time
the region containing points A and B is completely fractured (see Figs. 2.17, inter-
val (II) and 2.14). The peak tractions at the cohesive zone element C correspond to
the damage origination at this point. It is also worth to observe the distribution of
stresses along the electrode plane after 1000 cycles (Fig. 2.11): The peak of stresses
is only slightly shifted to the right from the point C since the zone of compression is
reached. No crack propagation is expected in the further course, unless overloading
happens, which may raise the stresses near the free edge of the actuator beyond the

36 S. Kozinov and M. Kuna

endurance limit. At the point D the mean value and the amplitude of the effective
tractions remain constant, since stresses during poling and exploitation never exceed
the endurance threshold and damage does not accumulate. It should be noticed, that
for case of a fairly low endurance threshold, damage would accumulate at the point
D as well.

According to the traction-separation law, cyclic electric loading leads to
unloading-reloading behavior in the cohesive elements (see Fig. 2.5). Figure 2.18
presents actual traction-separation behavior at the characteristic points A, B, C and
D over 1000 electric cycles. At the points A and B, peak stresses during poling of the
MLA are easily observed. On the contrary, poling process has no direct influence on
the behavior of the point C, which is governed only by the initiation and expansion
of the damaged region. At the point D opening according to the TSL is limited to the
initial curve, since no damage is observed during cycling.

Figure 2.19 illustrates cohesive zone opening, as well as specimen fracture, after
1000 electric cycles.

Figure 2.20 represents the formation and expansion of the damaged zone during
the operational cyclic electrical loading. After 50 cycles damage appears merely
ahead of the electrode tip in the region containing CZE A. With continuing cycling,
damage magnitude is gradually increasing and after 400 cycles complete failure is

Fig. 2.18 Traction-separation behavior in the cohesive layer as a result of electric loading

Fig. 2.19 Damage of the actuator after 1000 electric cycles (magnified 200 times)

2 Numerical Analysis of Fracture of Pre-stressed Ferroelectric . . . 37

Fig. 2.20 Damage initiation
and accumulation along the
electrode plane

reached at the points A and B. During this period, damaged zone is progressing
away from the electrode edge towards the right face of the actuator. This process
continues up to the 700th cycle, when the compression region, generated during the
poling process, is reached (see Fig. 2.11). After that, damage accumulation is almost
negligible and after 1000 cycles crack propagation is arrested.

Thus, two different damage stages can be distinguished in the MLA:

∙ The first one is associated with the poling process of the actuator. Domain reorien-
tation due to the strong electric field between the electrodes leads to the formation
of active and inactive (in terms of piezoelectric response) zones (Fig. 2.9). These
two regions and an external frame have to be balanced during deformation. This
results in the occurrence of compressed and stretched regions inside the MLA
(Fig. 2.11) with the stretched outer frame. During the first step the endurance limit
is surpassed in the region close to the actuator tip leading to the damage initiation.

∙ In the second step, electric cycling is applied as under exploitation conditions.
During this stage, damage accumulates in the region close to the electrode tip
(Figs. 2.14 and 2.20) resulting in complete failure in the zone, depicted in Fig. 2.19.

2.4 Conclusions

In the present paper, failure of a pre-stressed multi-layer actuator is studied by
means of the finite element method. A coupled electromechanical simulation, com-
bined with a cohesive zone model for the damage accumulation, is carried out. An
advanced cyclic CZM is used together with a micromechanical domain switching
approach. The cohesive zone technique allows to capture initiation and accumula-
tion of damage, while domain switching modeling provides realistic simulation of
the non-linear processes occurring in ferroelectric smart materials. Electrically, the
cohesive zone is being treated as a capacitor with finite electric permittivity, which

38 S. Kozinov and M. Kuna

degrades during damage accumulation. An important feature of the employed TSL
is that electric cyclic loading of a constant amplitude leads to increase of damage,
which can not be modeled with a monotonous TSL.

By means of the simulation, the poling process of the ceramics in the MLA is stud-
ied and electric potential distribution, polarization vectors, mechanical stresses are
obtained and analysed. Piezoelectrically “active” and “inactive” zones are observed,
with an external frame leading to the pre-stressed state of the actuator before exploita-
tion. The formation and evolution of damage in the MLA is analyzed as a con-
sequence of a purely cyclic electric loading. Results of our numerical simulations
qualitatively coincide with the experimentally observed crack patterns.

It should be emphasized, that the mode of the crack initiation and growth is cer-
tainly dependent on the specific material, cohesive properties of the interface, stiff-
ness of the external frame, type and magnitude of the electric in-service loading and
poling technology.

Based on the current and forthcoming analyses, design suggestions can be pro-
posed regarding the geometry of the actuators as well as their electromechanical
properties in order to reduce the failure probability of MLAs.

Acknowledgements The research was funded by DFG under grant KU 929/20.

References

1. Kozinov S, Kuna M (2015) Simulation of damage in ferroelectric actuators by means of cohe-
sive zone model. Sensor Actuat A-Phys 233:176–183

2. Kozinov S, Kuna M, Roth S (2014) A cohesive zone model for the electromechanical damage
of piezoelectric/ferroelectric materials. Smart Mater Struct 23(5):055024

3. Uchino K, Takahashi S (1996) Multilayer ceramic actuators. Curr Opin Solid State Mater Sci
1:698–705

4. Kuna M (2010) Fracture mechanics of piezoelectric materials—where are we right now? Eng
Fract Mech 77(2):309–326

5. Schneider GA (2007) Influence of electric field and mechanical stresses on the fracture of
ferroelectrics. Annu Rev Mater Res 37:491–538

6. Huber Je, Fleck Na, Landis CM, McMeeking RM (1999) A constitutive model for ferroelectric
polycrystals. J Mech Phys Solids 47(8):1663–1697

7. Furuta A, Uchino K (1993) Dynamic observation of crack propagation in piezoelectric multi-
layer actuators. J Am Ceram Soc 76(8):1615–1617

8. dos Santos e Lucato SL, Lupascu DC, Kamlah M, Rödel J, Lynch CS (2001) Constraint-
induced crack initiation at electrode edges in piezoelectric ceramics. Acta Mater 49(14):2751–
2759

9. Zhao XJ, Liu B, Fang DN (2010) Study on electroelastic field concentration around the elec-
trode tip in multilayer ferroelectric actuators of two designs and their optimizations. Int J Plast
26(4):533–548

10. Arias I, Serebrinsky S, Ortiz M (2006) A phenomenological cohesive model of ferroelectric
fatigue. Acta Mater 54(4):975–984

11. Utzinger J, Steinmann P, Menzel A (2008) On the simulation of cohesive fatigue effects in
grain boundaries of a piezoelectric mesostructure. Int J Solids Struct 45(17):4687–4708

12. Verhoosel CV, Gutiérrez MA (2009) Modelling inter- and transgranular fracture in piezoelec-
tric polycrystals. Eng Fract Mech 76(6):742–760

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13. Roth S, Hütter G, Kuna M (2014) Simulation of fatigue crack growth with a cyclic cohesive
zone model. Int J Fract 188(1):23–45

14. Kamlah M, Böhle U (2001) Finite element analysis of piezoceramic components taking into
account ferroelectric hysteresis behavior. Int J Solids Struct 38(4):605–633

15. Abdollahi A, Arias I (2012) Crack initiation patterns at electrode edges in multilayer ferroelec-
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Solids Struct 36(10):1541–1556

17. Li Q, Kuna M (2012) Inhomogeneity and material configurational forces in three dimensional
ferroelectric polycrystals. Eur J Mech A-Solid 31(1):77–89

18. Hao TH, Shen ZY (1994) A new electric boundary condition of electric fracture mechanics
and its applications. Eng Fract Mech 47(6):793–802

19. Kozinov S, Roth S, Kuna M (2014) Development of a cohesive model for damage simulation
in ferroelectric materials subjected to electromechanical loading. In: Oñate E, Oliver J, Huerta
A (eds) 11th world congress on computational mechanics 4320–4331

Chapter 3

Modelling the Constitutive Behaviour
of Martensite and Austenite in Shape
Memory Alloys Using Closed-Form
Analytical Continuous Equations

Arathi Pai, Thomas Niendorf, Phillip Krooss, Isabel Koke,
Ansgar Traechtler and Mirko Schaper

Abstract Shape Memory Alloy (SMA) actuators capable of precise position control
are faced with numerous challenges attributed mostly to the extreme non-linearities
of such alloys. The development of control strategies for such actuators is alleviated
by the use models incorporating these non-linearities. Such models should, however,
among other characteristics, be real-time capable in order to bring reasonable bene-
fits. This work presents a novel constitutive phenomenological model for martensite
and austenite. The model is based entirely on continuous differentiable analytical
equations and these closed-form equations are capable of depicting the smooth cur-
vatures observed in the SMA stress-strain characteristic with few and easy to identify
physical parameters. They can describe shape changes in both SMA phases (marten-
site or austenite) when subjected to monotonic as well as cyclic loading, includ-
ing minor loop behaviour. The model is validated by stress-strain experiments and
the results show outstanding correlation with experimental data. Since the model is

A. Pai (✉) ⋅ I. Koke ⋅ A. Traechtler 41

Fraunhofer Institute for Production Technology,
Zukunftsmeile 1, 33102 Paderborn, Germany
e-mail: [email protected]

I. Koke
e-mail: [email protected]

A. Traechtler
e-mail: [email protected]

T. Niendorf ⋅ P. Krooss ⋅ M. Schaper
Department of Materials Science, University of Paderborn, Warburger Strasse 100,
33098 Paderborn, Germany
e-mail: [email protected]

P. Krooss
e-mail: [email protected]

M. Schaper
e-mail: [email protected]

© Springer International Publishing Switzerland 2017
A.L. Araujo and C.A. Mota Soares (eds.), Smart Structures and Materials,
Computational Methods in Applied Sciences 43,
DOI 10.1007/978-3-319-44507-6_3

42 A. Pai et al.

based on simple closed-form equations, it is extremely computational efficient and
can build the foundation for the development of real-time capable SMA models for
control algorithms.

3.1 Introduction

Shape Memory Alloys (SMAs) have been investigated numerously in the last decades
due to their ability to display two kinds of shape recovery effects, namely, the
one-way-shape-memory-effect and pseudoelasticity [1–4]. These effects arise due
to the fact that microscopically, SMAs feature two phases: a parent high tempera-
ture austenitic phase and a low-temperature martensitic phase. Characterization of
the stability of both phases is done on the basis of the phase transformation tem-
peratures: Ms, Mf (martensite start and finish) and As, Af (austenite start and fin-
ish), where, generally, Mf < Ms < As < Af . The one-way-shape-memory-effect is
observed when the transformation between phases is thermally activated: at tem-
peratures below Mf , where the alloy is initially fully martensitic, mechanical load-
ing causes a macroscopic shape change. This shape is stable until the martensite is
heated above Af , triggering a transformation to austenite and a macroscopic shape
recovery. Pseudoelasticity is observed during a stress-driven transformation, where
austenite transforms to martensite above certain critical stresses causing a macro-
scopic shape change. Removal of the driving stress, triggers an (almost) complete
reverse transformation from martensite to austenite, and a respective shape recov-
ery. Both of these effects can be exploited to work on a load, and SMAs can hereby
be used as actuators. Such SMA actuators have attractive properties such as high
energy density, smooth and silent actuation, reduced part counts compared to tra-
ditional alternatives, scalability etc. [5, 6]. However, due to the inherent extremely
nonlinear and hysteretic behaviour of SMAs, the design of actuators for position
control, for example, is non-trivial. One of the approaches used to develop control
algorithms for SMA actuators is model-based design, where models that attempt to
predict the non-linearities are employed. In the past 20 years, models to describe
SMA behaviour have been developed from various perspectives: Thermodynamics
[7–13], phenomenological and thermomechanical [14–22], micromechanical [23–
25], Finite element [26, 27], constitutive [28, 29]. etc. Existing SMA models, nev-
ertheless, show various deficiencies: the models usually have several, in some cases
difficult to identify, parameters [28], the model equations are either extremely com-
plicated and computationally expensive or they are too elementary, in consequence
showing poor correlation to observed behaviour. This is primarily the case when
constitutive models are considered. Constitutive models considering the stress-strain
relation are usually either approximated based on linear piecewise models, imple-
mented using a series of conditional statements, that do not fully predict observed
behaviour, or with complex equations. Although, some researchers (including [28])

3 Modelling the Constitutive Behaviour of Martensite and Austenite . . . 43

have published numerical solutions to such complicated equations, they are useful
only for numerical simulations e.g. finite element, but are not sufficient enough to be
used in real-time control algorithms. The consequence of this is that although one
of major advantages of SMA actuators are their miniaturization (made possible due
to their high energy density) [6], the control units that are commonly employed are
large, since they need the computational capacity dictated by the complexity of the
models and the control algorithms.

This paper concentrates on reducing the complexity in modelling and presents a
novel SMA constitutive phenomenological model based entirely on closed-form con-
tinuous differentiable analytical equations that are capable of depicting the smooth
curvatures typical in the SMA stress-strain characteristics. The model can be used for
monotonic and arbitrary loading-unloading cycles irrespective of the phases present,
i.e. martensite or austenite. Stress-strain tensile experiments are subsequently used
for validation. The main advantages of this model are the excellent agreement with
experimental data, the few and easy to identify model parameters and since the model
is based on simple closed-form continuous equations, it is extremely computation-
ally efficient and, thus, can be used as the basis for the development of real-time
capable SMA models.

It is important to note that the developed model focuses on modelling macroscopic
phenomena observed in SMAs, as these are most relevant for actuator development.
In this regard, although the model equations can be justified with hypotheses from
material science, the motivation of the authors was not to describe material spe-
cific processes with the model, but to predict macroscopic SMA behaviour relevant
for control and industrial applications. This modelling method has been used very
successfully in the famous Pacejka Magic Formula for tire dynamics [30] which is
widely used as the industrial standard for vehicle dynamic simulations [31].

3.2 SMA Model Base Equation

Figure 3.1 shows typical stress-strain curves for monotonic loading and unloading
in martensite and austenite (plots (a) and (b)) and cyclical loading of martensite and
austenite (plots (c) and (d)). Inspection of the data shows a ubiquitous ‘s-shaped’
curve (shown as red dashed curves in Fig. 3.1. Note that in plot (b), two s-shaped
curves, one for loading and the other for unloading behaviour are present). This
curve, plotted by the blue solid line on the stress-strain ( − ) plane in Fig. 3.2,
is characterised by three slopes connected by two ‘knees’ and describes, physically,
the evolution of phases in the SMA during an iso-thermal tensile test. Depending
on the initial unloaded phase (austenite or martensite), the three segments can be
related to the following [4]: For a test done below Mf (100 % martensite), twinned
martensite is elastically deformed in the first segment a − b. Detwinning starts at 1
and proceeds until complete conversion to detwinned martensite at 2, after which


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