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

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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

Â 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 = ÂfzgÃ À ÂÃ Â Ã 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

½K = ½K0 + ½KNL

½K0 ⇒ I order theory; linear, small displacements, axial and flexural behaviors uncoupled ,

½KNL ⇒ 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 = ÂfzgÃ À ÂÃ 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 = ½K̄fvg. 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

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

Heinemann

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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

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induced crack initiation at electrode edges in piezoelectric ceramics. Acta Mater 49(14):2751–

2759

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trode tip in multilayer ferroelectric actuators of two designs and their optimizations. Int J Plast

26(4):533–548

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tric polycrystals. Eng Fract Mech 76(6):742–760

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18. Hao TH, Shen ZY (1994) A new electric boundary condition of electric fracture mechanics

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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