Issue 29

Volume VIII Issue 29 July 2014

Rivista Internazionale Ufficiale del Gruppo Italiano Frattura

Editor-in-chief:

Francesco Iacoviello

ISSN 1971-8993

Associate Editors:

Alfredo Navarro Thierry Palin-Luc

Luca Susmel John Yates Elio Sacco Sonia Marfia

Guest Editor:

Editorial Advisory Board:

Harm Askes Alberto Carpinteri Andrea Carpinteri Donato Firrao M. Neil James Gary Marquis Robert O. Ritchie Ashok Saxena Darrell F. Socie Cetin Morris Sonsino Ramesh Talreja

David Taylor Shouwen Yu Frattura ed integrità strutturale The International Journal of the Italian Group of Fracture

www.gruppofrattura.it

Frattura ed Integrità Strutturale, 29 (2014); International Journal of the Italian Group of Fracture

Table of Contents

A. Bacigalupo, L. Gambarotta A micropolar model for the analysis of dispersive waves in chiral mass-in-mass lattices ………...….. 1 L. Cabras, M. Brun Effective properties of a new auxetic triangular lattice: an analytical approach ……………...……. 9 A. Caporale, R. Luciano A micromechanical four-phase model to predict the compressive failure surface of cement concrete ...….. 19 G. Carta, M. Brun, A.B. Movchan Elastic wave propagation and stop-band generation in strongly damaged solids ……………............. 28 M.L. De Bellis, D. Addessi A micromechanical approach for the micropolar modeling of heterogeneous periodic media ……...…... 37 C. Maruccio, L. De Lorenzis A multilevel finite element approach for piezoelectric textiles made of polymeric nanofibers ………… 49 S. Terravecchia, T. Panzeca, C. Polizzotto Strain gradient elasticity within the symmetric BEM formulation ………………...……………. 61 A. Fortini, M. Merlin, R. Rizzoni, S. Marfia TWSME of a NiTi strip in free bending conditions: experimental and theoretical approach …...….. 74 V. Sepe, F. Auricchio, S. Marfia Response of porous SMA: a micromechanical study ……………………...…………………... 85 M. Marino An ideal model for stress-induced martensitic transformations in shape-memory alloys …………...... 96 N.A. Nodargi, E. Artioli, F. Caselli, P. Bisegna State update algorithm for associative elastic-plastic pressure-insensitive materials by incremental energy minimization …………………………………………………………………………… 111 A. Castellano, P. Foti, A. Fraddosio, S. Marzano, M. D. Piccioni Geometric numerical integrators based on the magnus expansion in bifurcation problems for non-linear elastic solids …………………………………………………………………………….. 128 L. Facchini, M. Betti An efficient Bouc & Wen approach for seismic analysis of masonry tower ………………………. 139 G. Giambanco, E. La Malfa Ribolla, A. Spada CH of masonry materials via meshless meso-modeling ………………………………………... 150

I

Frattura ed Integrità Strutturale, 29(2014); ISSN 1971-9883

J. Toti, V. Gattulli, E. Sacco Damage propagation in a masonry arch subjected to slow cyclic and dynamic loadings …………….. 166 D. Addessi, P. Di Re A 3D mixed frame element with multi-axial coupling for thin-walled structures with damage …....... 178 L. Contrafatto, R. Cosenza Prediction of the pull-out strength of chemical anchors in natural stone ………………………….. 196 D. De Domenico, A.A. Pisano, P. Fuschi Limit analysis on FRP-strengthened RC members …………………………………………... 209 N. Cefis, C. Comi Damage modelling in concrete subject to sulfate attack ………………………………………... 222 R. Massabò Influence of boundary conditions on the response of multilayered plates with cohesive interfaces and delaminations using a homogenized approach ………………………………………………... 230 M. Marino, F. Nerilli, G. Vairo A finite-element approach for the analysis of pin-bearing failure of composite laminates …………… 241 F. Tornabene, N. Fantuzzi, M. Bacciocchi The strong formulation finite element method: stability and accuracy ……………………………. 251 R. Dimitri, M. Trullo, G. Zavarise, L. De Lorenzis A consistency assessment of coupled cohesive zone models for mixed-mode debonding problems ……... 266 R. Serpieri, L. Varricchio, E. Sacco, G. Alfano Bond-slip analysis via a cohesive-zone model simulating damage, friction and interlocking …………. 284 S. de Miranda, A. Palermo, F. Ubertini A simple beam model to analyse the durability of adhesively bonded tile floorings in presence of shrinkage ………………………………………………………………………………. 293 A. Infuso, M. Paggi Flaw-tolerance of nonlocal discrete systems and interpretation according to network theory …………. 302 P. Casini, O. Giannini, F. Vestroni Crack detection in beam-like structures by nonlinear harmonic identification …………………….. 313 E. Grande, M. Imbimbo A data fusion based approach for damage detection in linear systems …………………………… 325 V. Zega, C. Comi, A. Corigliano, C. Valzasina Integrated structure for a resonant micro-gyroscope and accelerometer ……………………………. 334 A. De Rosis, S. de Miranda, C. Burrafato A numerical framework for simulating fluid-structure interaction phenomena ……………………. 343 G. Maurelli, N. Maini, P. Venini Mixed methods for viscoelastodynamics and topology optimization ……………………………... 351 L. Petrini, W. Wu, D. Gastaldi, L. Altomare, S. Farè, F. Migliavacca, A. Gökhan Demir, B. Previtali, M. Vedani Development of biodegradable magnesium alloy stents with coating ……………………………... 364 W. Guodong The research for mechanics stimulation method of nonlinear random vibration based on statistical linear theory …………...…………………………………………………...…………… 376

II

Frattura ed Integrità Strutturale, 29 (2014); International Journal of the Italian Group of Fracture

M. Romano, C. J. J. Hoinkes, I. Ehrlich Experimental investigation of fibre reinforced plastics with hybrid layups under high-velocity impact loads …………………………………………………………………………………... 385 M. Scafè, G. Raiteri, A. Brentari, R. Dlacic, E. Troiani, M. P. Falaschetti, E. Besseghini Estimate of compressive strength of an unidirectional composite lamina using cross-ply and angle-ply laminates …………………………………………………………...…………………... 399 L. Zhao, H. Xue, F. Yang, Y. Suo Numerical investigation on stress corrosion cracking behavior of dissimilar weld joints in pressurized water reactor plants ……………………………………………………………………… 410 S. K. Kudari, K. G. Kodancha A new formulation for estimating maximum stress intensity factor at the mid plane of a SENB specimen: Study based on 3D FEA ……………………………………………...………... 419

III

Frattura ed Integrità Strutturale, 29(2014); ISSN 1971-9883

Editor-in-Chief Francesco Iacoviello Associate Editors Alfredo Navarro

(Università di Cassino e del Lazio Meridionale, Italy)

(Escuela Superior de Ingenieros, Universidad de Sevilla, Spain) (Ecole Nationale Supérieure d'Arts et Métiers, Paris, France)

Thierry Palin-Luc

Luca Susmel John Yates

(University of Sheffield, UK) (University of Manchester, UK)

Guest Editor Elio Sacco

(Università di Cassino e del Lazio Meridionale, Italy) (Università di Cassino e del Lazio Meridionale, Italy)

Sonia Marfia

Advisory Editorial Board Harm Askes

(University of Sheffield, Italy) (Politecnico di Torino, Italy) (Università di Parma, Italy) (Politecnico di Torino, Italy) (University of Plymouth, UK)

Alberto Carpinteri Andrea Carpinteri Donato Firrao M. Neil James Gary Marquis Ashok Saxena Darrell F. Socie Shouwen Yu Ramesh Talreja David Taylor Robert O. Ritchie Cetin Morris Sonsino Editorial Board Stefano Beretta Nicola Bonora Roberto Citarella Claudio Dalle Donne Manuel de Freitas Vittorio Di Cocco Giuseppe Ferro Tommaso Ghidini Paolo Leonetti Carmine Maletta Liviu Marsavina Daniele Dini

(Helsinki University of Technology, Finland)

(University of California, USA)

(Galgotias University, Greater Noida, UP, India; University of Arkansas, USA)

(University of Illinois at Urbana-Champaign, USA)

(Tsinghua University, China) (Fraunhofer LBF, Germany) (Texas A&M University, USA) (University of Dublin, Ireland)

(Politecnico di Milano, Italy)

(Università di Cassino e del Lazio Meridionale, Italy)

(Università di Salerno, Italy) (EADS, Munich, Germany) (EDAM MIT, Portugal)

(Università di Cassino e del Lazio Meridionale, Italy)

(Imperial College, UK)

(Politecnico di Torino, Italy)

(European Space Agency - ESA-ESRIN) (Università della Calabria, Italy) (Università della Calabria, Italy) (University of Timisoara, Romania) (University of Porto, Portugal)

Lucas Filipe Martins da Silva

Hisao Matsunaga

(Kyushu University, Japan) (University of Sheffield, UK) (Politecnico di Torino, Italy) (Università di Parma, Italy) (Università di Messina, Italy) (Università di Brescia, Italy) (Università di Bologna, Italy) (Università di Parma, Italy)

Mahmoud Mostafavi

Marco Paggi Oleg Plekhov

(Russian Academy of Sciences, Ural Section, Moscow Russian Federation)

Alessandro Pirondi Giacomo Risitano Roberto Roberti

Marco Savoia

Andrea Spagnoli Charles V. White

(Kettering University, Michigan,USA)

IV

Frattura ed Integrità Strutturale, 29 (2014); International Journal of the Italian Group of Fracture

Journal description and aims Frattura ed Integrità Strutturale (Fracture and Structural Integrity) is the official Journal of the Italian Group of Fracture. It is an open-access Journal published on-line every three months (July, October, January, April). Frattura ed Integrità Strutturale encompasses the broad topic of structural integrity, which is based on the mechanics of fatigue and fracture, and is concerned with the reliability and effectiveness of structural components. The aim of the Journal is to promote works and researches on fracture phenomena, as well as the development of new materials and new standards for structural integrity assessment. The Journal is interdisciplinary and accepts contributions from engineers, metallurgists, materials scientists, physicists, chemists, and mathematicians. Contributions Frattura ed Integrità Strutturale is a medium for rapid dissemination of original analytical, numerical and experimental contributions on fracture mechanics and structural integrity. Research works which provide improved understanding of the fracture behaviour of conventional and innovative engineering material systems are welcome. Technical notes, letters and review papers may also be accepted depending on their quality. Special issues containing full-length papers presented during selected conferences or symposia are also solicited by the Editorial Board. Manuscript submission Manuscripts have to be written using a standard word file without any specific format and submitted via e-mail to iacoviello@unicas.it. The paper may be written in English or Italian (with an English 1000 words abstract). A confirmation of reception will be sent within 48 hours. The review and the on-line publication process will be concluded within three months from the date of submission. Peer review process Frattura ed Integrità Strutturale adopts a single blind reviewing procedure. The Editor in Chief receives the manuscript and, considering the paper’s main topics, the paper is remitted to a panel of referees involved in those research areas. They can be either external or members of the Editorial Board. Each paper is reviewed by two referees. After evaluation, the referees produce reports about the paper, by which the paper can be: a) accepted without modifications; the Editor in Chief forwards to the corresponding author the result of the reviewing process and the paper is directly submitted to the publishing procedure; b) accepted with minor modifications or corrections (a second review process of the modified paper is not mandatory); the Editor in Chief returns the manuscript to the corresponding author, together with the referees’ reports and all the suggestions, recommendations and comments therein. c) accepted with major modifications or corrections (a second review process of the modified paper is mandatory); the Editor in Chief returns the manuscript to the corresponding author, together with the referees’ reports and all the suggestions, recommendations and comments therein. d) rejected. The final decision concerning the papers publication belongs to the Editor in Chief and to the Associate Editors. The reviewing process is completed within three months. The paper is published in the first issue that is available after the end of the reviewing process.

Publisher Gruppo Italiano Frattura (IGF) http://www.gruppofrattura.it ISSN 1971-8993 Reg. Trib. di Cassino n. 729/07, 30/07/2007

V

Frattura ed Integrità Strutturale, 29(2014); ISSN 1971-9883

Some news about our Journal …

D

ear Friend, this is the 29 th issue of the IGF Journal … and we have other news for you!!

First of all, some details about this issue. As you can see, 37 papers! 32 papers are dedicated to the Computational Mechanics and Mechanics of Materials activities in Italy. Sonia Marfia and Elio Sacco (Università di Cassino e del Lazio Meridionale) are the guest editors and we are grateful for all their efforts that allowed to obtain and important state of art of the research activities in Italy on the Computational Mechanics and Mechanics of Materials. Secondly, we started a cooperation with CrossRef, an association of scholarly publishers that develops shared infrastructure to support more effective scholarly communications. CrossRef citation-linking network today covers over 67 million journal articles and other content items (books chapters, data, theses, technical reports) from thousands of scholarly and professional publishers around the globe. Last, but not least, the new address (and the new server) for our journal: www.fracturae.com. Starting from July 4 th the "Fracture and Structural Integrity" journal is hosted on a cloud server. Cloud servers mean virtual servers which run on cloud computing environment. The key benefits of using cloud servers are: - Flexibility and scalability; extra resource can be accessed as and when required; - Cost-effectiveness; whilst being available when needed, clients only pay for what they are using at a particular time; - Reliability; due to the number of available servers, if there are problems with some, the resource will be shifted so that clients are unaffected. Thanks to the new server it will be possible to activate in the future new and (we hope!) useful services. At www.fracturae.com you can find both the IGF journal Frattura ed Integrità Strutturale and the other two journals we publish on line in our OJS server with the permission of AIM ( La Metallurgia Italiana ) and of Teksid ( Metallurgical Science and Technology ), respectively. We hope you will appreciate the possibility to read all these journals using the same portal.

Francesco Iacoviello Direttore F&IS

VI

Frattura ed Integrità Strutturale, 29 (2014); International Journal of the Italian Group of Fracture

Computational Mechanics and Mechanics of Materials in Italy

T

he special issue of Fracture and Structural Integrity is dedicated to the novelties in fields of Computational Mechanics and on the Mechanics of Materials in Italy, as the invited authors are very active Italian researchers. The works in the special issue cover a wide range of topics ranging from Mechanics, Micromechanics, Homogenization, Fracture Mechanics, Finite Element Method, Boundary Method, Structural Dynamics, Nonlinear Mechanics, Constitutive Modeling. The common aim is to propose innovative theoretical, numerical and experimental approaches in order to study mechanical problems inherent to the Computational Mechanics and Mechanics of Materials. In this framework, several models and interesting applications are proposed in order to simulate processes and materials in the engineering and industrial fields. The set of the whole paper presented in this special issue can be grouped in the following macro-topics: - derivation of the response of innovative heterogeneous materials adopting homogenization techniques; - analysis of enhanced continua; - experimental investigation and development of constitutive laws for shape memory alloys; - derivation and implementation of new algorithms for time integration of nonlinear evolutive constitutive equations; - modeling and analysis of masonry structure in static as well as in dynamic framework; - new modeling approaches and analysis of composite laminates;

- development of interface models for the analysis of adhesion and detachment problems; - identification methods for the detection of damage and fracture in structural element; - numerical techniques for the simulation of fluid-structure interaction; - studies concerning the topology optimization; - simulation of the behavior of prosthesis in biomechanics.

Finally, the papers published in this special issue represent a non-exhaustive overview of the state-of-art in Italy in Mechanics of Materials and Computational Mechanics, which are two aspect of the same problem of solving engineering problems concerning the development of new analytical, numerical and experimental procedures for the analysis of existing structural systems or of new innovative devices.

Sonia Marfia & Elio Sacco Università di Cassino e del Lazio Meridionale (Guest Editors)

VII

A. Bacigalupo et alii, Frattura ed Integrità Strutturale, 29 (2014) 1-8; DOI: 10.3221/IGF-ESIS.29.01

Focussed on: Computational Mechanics and Mechanics of Materials in Italy

A micropolar model for the analysis of dispersive waves in chiral mass-in-mass lattices

A. Bacigalupo University of Trento andrea.bacigalupo@unitn.it L. Gambarotta University of Genoa luigi.gambarotta@unige.it

A BSTRACT . The possibility of obtaining band gap structures in chiral auxetic lattices is here considered and applied to the case of inertial locally resonant structures. These periodic materials are modelled as beam-lattices made up of a periodic array of rigid rings, each one connected to the others through elastic slender ligaments. To obtain low-frequency stop bands, elastic circular resonating inclusions made up of masses located inside the rings and connected to them through an elastic surrounding interface are considered and modeled. The equations of motion are obtained for an equivalent homogenized micropolar continuum and the overall elastic moduli and the inertia terms are given for both the hexachiral and the tetrachiral lattice. The constitutive equation of the beam lattice given by the Authors [15] are then applied and a system of six equations of motion is obtained. The propagation of plane waves travelling along the direction of the lines connecting the ring centres of the lattice is analysed and the secular equation is derived, from which the dispersive functions may be obtained. K EYWORDS . Auxetic materials; Chirality; Cellular materials; Mass-in-mass dynamic systems; Dispersive waves. Tee et al. [7] have recently obtained a band gap structure in periodic tetrachiral materials. Spadoni et al. [8] have obtained analogous results with reference to hexachiral lattices. These kind of materials are auxetics [9] and their structure is I I NTRODUCTION n recent years a considerable interest in acoustic metamaterials was witnessed by several research (see for reference [1]), most of them focused to the design of artificial materials having periodic microstructure conceived to get complete sound attenuation for a certain frequency range, namely acoustic wave spectral gap. Sonic crystals with spectral gaps [2] have been developed on the realization that composites with locally resonant structural units may exhibit effective negative elastic constants at certain frequency ranges, as shown by Liu et al. [3] and by Huang et al. [4,5]. Recently, Bigoni et al. [6] proposed a periodic metamaterial with internal locally resonant structures that supports tunable low frequency stop bands. This effect is associated with localized rotational modes obtained from a chiral microstructure of the periodic cell.

1

A. Bacigalupo et alii, Frattura ed Integrità Strutturale, 29 (2014) 1-8; DOI: 10.3221/IGF-ESIS.29.01

characterized by a periodic array of the rings connected by four or six elastic ligaments. Lakes [10], with reference to the planar isotropic case of hexachiral lattice, proposed this microstructure geometry firstly. Later Prall and Lakes [11] showed for this material a Poisson’s ratio of -1 under the hypothesis of ignoring the axial strain of the ligaments. Further studies on the homogenization of hexachiral auxetic materials have been carried out by Spadoni and Ruzzene [12] and by Liu et al. [13]. Liu et al. [14] proposed a hexachiral metacomposite by integrating a two-dimensional hexachiral lattice with elastic resonating inclusions to obtain low-frequency band gap. This metacomposite has been analysed through a numerical model where the ligaments are modelled as multi-beam elements and the inclusions as a two dimensional FEM model. In this paper, hexa- and tetra-chiral beam lattices are considered having local resonators at the nodes of the periodic array. The model is developed in closed-form and is based on a micropolar homogenization of the lattice. This approach partially relies on the results by Bacigalupo and Gambarotta [15], which developed and compared the results from the micropolar and a second displacement gradient homogenization for both the hexachiral and the tetrachiral periodic cells in order to evaluate the validity limits of the beam lattice model. The possibility of obtaining band gap structures in chiral auxetic lattices is here considered and applied to the case of inertial locally resonant structures. These structures are obtained following Huang et al. [4] and Liu et al. [14], by the insertion of a circular mass connected through an elastic surrounding interface to each ring of the microstructure. The equations of motion are given within a micropolar continuum model and the overall elastic moduli and the inertia terms are obtained for both the hexachiral and the tetrachiral lattice. The constitutive equation of the beam lattice given in [15] are then applied and a system of six equations of motion is obtained. The propagation of plane waves travelling along the direction of the lines connecting the ring centres of the lattice is analysed and the secular equation is derived, from which the dispersive functions may be obtained.

(a)

(b)

Figure 1 : (a) Hexachiral lattice; (b) tetrachiral lattice.

C HIRAL MASS - IN - MASS PERIODIC MATERIAL : M ICROPOLAR HOMOGENIZATION

T

he periodic materials shown in Figure 1 are considered as beam-lattices made up of a periodic array of rigid rings, each one connected to the others through n elastic slender ligaments rigidly connected to the rings. In Figure 2.a the periodic hexachiral cell ( n =6) is shown, while in Figure 1.b is shown the tetrachiral cell ( n =4). Each ligament is tangent to the joined rigid rings and has length l measured between the connection points, with section width t, thickness d and Young’s modulus s E . Both the ligaments and the rings have mass density s  , so that the mass and the rotation inertia of the rings are 1 M 2 s rt   and 2 1 1 J M r  , respectively. The two dimensional composite materials are auxetic. The hexachiral lattice is isotropic and its Poisson’s ratio becomes negative when increasing the chirality angle  . On the other hand, the tetrachiral material is strongly anisotropic and presents a directional dependency of the chirality on the direction of the applied stress, as shown in [15]. To obtain low-frequency stop bands, a metamaterial inclusion consisting of a softly heavy disk is located inside the ring as shown in Figure 3.a. This inclusion plays the role of low-frequency resonator with mass and rotation inertia denoted with J , respectively. The motion of the rigid ring of the beam lattice is denoted by the displacement vector u and the rotation  , respectively (see Figure 3.b), while the motion of the mass of the resonator is denoted by the displacement vector v and the rotation  (see Figure 3.c). 2 M and 2

2

A. Bacigalupo et alii, Frattura ed Integrità Strutturale, 29 (2014) 1-8; DOI: 10.3221/IGF-ESIS.29.01

(a) (b) Figure 2 : Periodic cell of the (a) hexachiral lattice; (b) tetrachiral lattice.

(a) (b) (c)

Figure 3 : (a) Internal mass with external elastic thick layer; (b) rigid ring and related dofs; (c) internal mass and related dofs. A soft elastic interface connects the internal mass to the rigid ring. To get a simplified formulation, the constitutive equation of the interface is assumed in the form     , c = d k k         f v u (1) f being the force exerted by the rigid ring on the internal mass and c the corresponding couple (see Figure 4). The parameters d k and k  are the isotropic translational stiffness and the rotational stiffness, respectively.

Figure 4 : Contact force and couple between the rigid ring and the internal mass. The lattice model is here approximated as a micropolar continuum model resulting from a homogenization process based on the macro-homogeneity criterion, involving both the total potential energy and the kinetic energy through the Hamilton principle (see [13, 16] for reference). The equations of motion for the homogenized beam-lattice are

3

A. Bacigalupo et alii, Frattura ed Integrità Strutturale, 29 (2014) 1-8; DOI: 10.3221/IGF-ESIS.29.01

ˆ





                  11,1 12,2 1 1 21,1 22,2 2 2 ˆ d d k v u k v u

1 1 u 

 ˆ k

1 2  u







          

                   1,1 m m k u v k u v 2,2 21 1 1 2 2 2 ˆ ˆ ˆ d d k I       

I



12

1

(2)

2 1  v

2 2  v

where the stiffnesses ˆ d

d cell k k A  and ˆ

cell A being the area of the periodic cell, together with

k k A   

are introduced,

cell

M

M

J

J

 

 

1 A  and



and

, and the micro-inertia terms

, respectively.

the mass densities

I

I

1

2

2

A

A

A

1

2

1

2

cell

cell

cell

cell

i m are the overall stress components, namely the asymmetric stress components and the micro

ij  and

In Eq. (2)

cell A 

2 2 2 3 cos l

 and the following inertial parameters

couples, respectively. In case of hexachiral lattice, one obtains

are obtained

        3 sin 2

s

1

24

2       2 3 tan sin l 2       2 3 sin 24 i A 1 24 M s I 2

cell

2     3 J

2

2 sin tan

2  

I

l

192 i

2

A

cell

  24 s       , 1 sin 2

cell A 

 and the resulting inertia parameters are

For the tetrachiral lattice one obtains

2

2 cos

l

48 s    

2       , M

2     J

2

2

1   . To obtain the displacement formulation of the equations of motion, the compatibility equations involving the macrostrain components 11 1,1 u   , 22 2,2 u   , 12 1,2 u     ,     21 2,1 u and the curvatures 1 ,1    and 2 ,2    have to be considered together with the constitutive equation. sin I l  , 2 2 sin 48 i cell A 2 2 sin tan 2 2 192 i cell I l A

M ICROPOLAR CONSTITUTIVE EQUATION FOR HEXA - AND TETRA - CHIRAL LATTICES

T

he constitutive equation of hexachiral honeycomb corresponds to that obtained in [13, 15] and is written as follows

   

    

11                            22 12 21 1 2      

A A A A

2

0 0 0 0 S 0

0 0 0 0 0 S

11             22

2      A A k A A k 



       

k k

(3)

12



21       1 m m  

   

0 0

0 0

0 0

0 0

2   

in which the five elastic moduli

4

A. Bacigalupo et alii, Frattura ed Integrità Strutturale, 29 (2014) 1-8; DOI: 10.3221/IGF-ESIS.29.01

3





2

 

1   

E

s

4

3 1      E







2

2

2    sin

cos

s

4 3 sin





2       2 2 cos k E

(4)

s

2

3





2

2 3sin 4 cos

2     2

 

S

E a

s

12

 3 1 sin cos 2 s         2 A E

depend on the Young’s modulus, the slenderness ratio

t l   and the angle  of inclination of the ligaments. The

constitutive Eq. (3) show the coupling between the extensional strains 11

 and 22 

and the asymmetric strains 12

 and

21  through the elastic constant A . The elastic moduli, with the exception of parameter  , depend on the parameter of chirality  , but only the constant A is an odd function of this parameter, i.e. it reverses its sign when the handedness of the material pattern is flipped over. In case of symmetric macro-strain fields, the fourth order elastic tensor for the hexagonal system corresponds to that of the transversely isotropic system whose elastic moduli in the plane of the lattice are:        2 4 2 2 3 2 hom 2 2 2 2 3 cos sin 2 3 1 cos sin 1 s E E                

hom  

(5)

2       4 2 2 cos sin 3

 3= 1 s E



G

2

  

hom

4

For the tetrachiral lattice, the constitutive equation is written [15]:

11              22 21            1 2 m m            12

2 0 0 

11                  22 12 21    2                1

B

0 0

 

B

0 2

0 0 0 0 0 0

0 0   B

0

(6)



B

0 0

S

0 0 0 0

0 S

0 0 0 0 0

where four elastic moduli are related to the lattice parameters as follows         2 2 2 2 2 2 2 2 2 2 2 cos sin 2 sin cos 1 sin cos 1 3sin 4 cos 12 s s s s E k E B E S E a                          

(7)

Similarly to the hexachiral honeycomb, a coupling is obtained between the extensional strains and the asymmetric strains through the elastic modulus B which is an odd function of the parameter of chirality  , while the other elastic moduli are even functions. In case of symmetric macro-strain fields, the resulting classical fourth order elasticity tensor has the elastic

5

A. Bacigalupo et alii, Frattura ed Integrità Strutturale, 29 (2014) 1-8; DOI: 10.3221/IGF-ESIS.29.01

moduli of the tetragonal system. The elasticity tensor depends on the chirality parameter, but unlike the hexachiral honeycomb, some elasticities are odd functions of  [15].

P LANE WAVES PROPAGATION

T

he free wave motion in the hexachiral material is written in terms of the components of the generalized displacement field     1 2 1 2 T u u v v    U x in the following form



 





 k u





       

1,11 u    

  



k u      

2,12 2 A k k v u k A k v u                      (2 ) 2 d Au u 2,22 ,1 ,2 1 1 2 2

Au

Au

1 1 u 

(2 )

2

1,12

1,22

2,11









 k u

     

  



Au

1,12 k u Au

Au

1 2  u

2

d

1,11

1,22

2,11

2,12

2,22

,1

,2

2

2









   



1,2 2   ku

4 k k         I

,11 S S k u v k u v 1 1 d

Au

Au

ku

2

2

2



,22

1,1

2,2

2

,1

1

(8)

ˆ ˆ ˆ

  



      

2 1  v

, ,

2 2  v

          2 2 2 d k I 

.

x in an infinite planar micropolar medium is admitted, the generalized

If a harmonic plane wave propagating along axis 1

 1 ˆ exp T i qx t        U U , where q and  denote the 

displacement field at a point is assumed in the following form





ˆ 

ˆ 

ˆ U

ˆ ˆ u u

1 v v ˆ ˆ

T



is the vector of the amplitudes

wave number and the circular frequency, respectively, and

1

2

2

  1 i   . Substituting the assumed generalized displacement field in the equation of motion (8) one obtains the secular equation system for the equivalent continuum model   2 2 q        

ˆ k

2





    

Aq

iAq

2

0

0

  

  

ˆ k

d

2

    

d

1

                

  k q     2

  

ˆ k

2



Aq

ikq

2

0

0

  

ˆ k

d

1                                   0 2 ˆ u u ˆ ˆ

2

    

d

1

  

4   

2 Sq k

 

ˆ k





iAq

ikq

2

2

0

0

  

ˆ



2

   

k I 

1

(9)

ˆ

v

ˆ k

ˆ k



2      2 

1



0

0

0

0

d

d

ˆ

v

2

ˆ

ˆ k

ˆ k



2      2 



0

0

0

0

d

d

     

ˆ k

ˆ

 

 

2



 

k I 

0

0

0

0



2



The solution of the eigenproblem (9), characterized by a hermitian matrix, provides dispersion functions   q  defined in the domain , q      , a being the characteristic size of the cell as shown in Figure 2. Since the model has six degrees of freedom, it follows that in the general case six plane waves may propagate, each characterized by a dispersion function. It follows that this model exhibits plane waves more complex than those obtained and discussed by Parfitt and Eringen [17] for isotropic achiral two dimensional micropolar solids and more recently by Khurana and Tomar [18] for 3D chiral a a     

6

A. Bacigalupo et alii, Frattura ed Integrità Strutturale, 29 (2014) 1-8; DOI: 10.3221/IGF-ESIS.29.01

micropolar media. Since the matrix of the coefficient in the eigenproblem (9) is hermitian, the eigenvalues are real, but not necessarily positive. Therefore, for some values of the model parameters is expected a reduced number of dispersive functions. In case of a tetrachiral lattice with internal resonant masses the free wave equation of motion takes the form:           1,11 1,22 2,11 2,22 ,1 ,2 1 1 1 1 1,11 1,22 2,11 2,22 ,1 ,2 2 2 1 2 ,11 ,22 1,1 2,2 1,2 2,1 1 1 1 2 1 2 2 2 2 2 2 2 d d d u ku Bu Bu B k k v u u Bu Bu ku u k B k v u u S S Bu Bu ku ku k k I k u v v k u v v                                                       (10)

ˆ ˆ ˆ                  d   2 k I 

In this case the secular equation system is written as follows:

ˆ

                    

           

   

2 q k    

2

ˆ k

d

2



Bq

iBq

0

0

  

d

2

  

1

ˆ

  

  

2 kq k

ˆ k

d

2



Bq

ikq

0

0

  

d

1           2 ˆ u u ˆ ˆ

2

  

1

  

2   

2 Sq k

ˆ k







iBq

ikq

0

0

  

ˆ



2

   

k I 

1

ˆ                  v

0 (11)

ˆ k

ˆ k



2      2 

1



0

0

0

0

d

d

ˆ

v

2

ˆ

ˆ k

ˆ k

 

 

2



  

0

0

0

0

     

d

d

2

ˆ k

ˆ

 

 

2



 

k I 

0

0

0

0



2

Again the solution of the eigenproblem (11) with hermitian matrix provides dispersion functions   q  , defined in the domain , q a a          . The analytical formulation here derived, which is based on a micropolar continuum model enriched with additional degrees of freedom of the resonating masses located inside the rings, will be developed in future research to analyze the wave propagation and to appreciate the influence of the geometrical and mechanical parameters of the microstructure and of the resonant devices on the frequency spectra and on the conditions of wave attenuation. The reliability of the results has to be evaluated from the comparison with the rigorous solutions obtained from a Floquet-Bloch analysis of the generalized beam lattice proposed in this paper. Finally, it should be noted that the validity of this model relies on the rigidity assumption of the rings [15], a condition that may be easily implemented in the physical model.

A CKNOWLEDGEMENTS

he Authors acknowledge financial support of the (MURST) Italian Department for University and Scientific and Technological Research in the framework of the research MIUR Prin09 project XWLFKW, Multi-scale modelling of materials and structures, coordinated by prof. A. Corigliano. Andrea Bacigalupo gratefully thanks financial T

7

A. Bacigalupo et alii, Frattura ed Integrità Strutturale, 29 (2014) 1-8; DOI: 10.3221/IGF-ESIS.29.01

support of the Italian Ministry of Education, University and Research in the framework of the FIRB project 2010, “Structural mechanics models for renewable energy applications" .

R EFERENCES

[1] Lu, M.H., Feng, L., Chen, Y.-F., Phononic crystals and acoustic metamaterials, MaterialsToday, 12 (2009) 34-42. [2] Pennec, Y., Vasseur, J.O., Djafari-Rouhani, B., Dobrzyński, L., Deymier, P.A., Two-dimensional phononic crystals: Examples and applications, Surface Science Reports, 65 (2010) 229–291. [3] Liu, Z., Zhang, X., Mao, Y., Zhu, Y., Yang, Z., Chang, C.T., Sheng, P., Locally Resonant Sonic Materials, Science, 289 (2000) 1734-1736. [4] Huang, H.H., Sun, C.T., Huang, G.L., On the negative effective mass density in acoustic metamaterials, Int. J. of Engineering Science, 47 (2009) 610-617. [5] Huang, H.H., Sun, C.T., Wave attenuation mechanism in an acoustic metamaterial with negative effective mass density, New Journal of Physiscs, 11 (2009) 013003. [6] Bigoni, D., Guenneau, S., Movchan, A. B., Brun M., Elastic metamaterials with inertial locally resonant structures: Application to lensing and localization, Physical Review, B 87 (2013) 174303. [7] Tee, K.F., Spadoni, A., Scarpa, F., Ruzzene, M., Wave propagation in auxetic tetrachiral honeycombs, J. of Vibration and Acoustics ASME, 132 (2010) 031007-1/8. [8] Spadoni, A., Ruzzene, M., Gonella, S., Scarpa, F., Phononic properties of hexagonal chiral lattices, Wave Motion, 46 (2009) 435-450. [9] Prawoto, Y., Seeeing auxetic materials from the mechanics point of view: A structural review on the negative Poisson's ratio, Computational Material Science, 58 (2012) 140-153. [10] Lakes, R. S.,Foam structures with a negative Poisson’s ratio, Science, 235 (1987) 1038-1040. [11] Prall, D., Lakes, R.S., Properties of chiral honeycomb with a Poisson ratio of -1, Int. J. Mechanical Sciences, 39 (1997) 305-314. [12] Spadoni, A., Ruzzene, M., Elasto-static micropolar behavior of a chiral auxetic lattice, J. Mechanics and Physics of Solids, 60 (2012) 156-171. [13] Liu, X.N., Huang, G.L., Hu, G.K., Chiral effect in plane isotropic micropolar elasticity and it’s application to chiral lattices, J. Mechanics and Physics of Solids, 60 (2012) 1907-1921. [14] Liu, X.N., Hu, G.K., Sun, C.T., Huang, G.L., Wave propagation characterization and design of two-dimensional elastic chiral metacomposite, J. of Sound and Vibration, 330 (2011) 2536-2553. [15] Bacigalupo, A., Gambarotta, L., Homogenization of periodic hexa- and tetrachiral cellular solids, Composite Structures, DOI 10.1016/j.compstruct.2014.05.033, 2014.to appear. [16] Stefanou, I., Sulem, J., Vardoulakis, I., Three-dimensional Cosserat homogenization of masonry structures: elasticity, Acta Geotechnica, 3 (2008) 71–83. [17] Parfitt, V.R., Eringen, A.C., Reflection of Plane Waves from the Flat Boundary of a Micropolar Elastic Half-Space, J. Acoustical Society of America, 45 (1969) 1258-1272. [18] Khurana, A., Tomar, S.K., Longitudinal wave response of a chiral slab interposed between micropolar solid half spaces, Int. J. of Solids and Structures, 46 (2009) 135–150.

8

L. Cabras et alii, Frattura ed Integrità Strutturale, 29 (2014) 9-18; DOI: 10.3221/IGF-ESIS.29.02

Focussed on: Computational Mechanics and Mechanics of Materials in Italy

Effective properties of a new auxetic triangular lattice: an analytical approach L. Cabras Università degli Studi di Cagliari, Dipartimento di Ingegneria Civile, Ambientale e Architettura luigi.cabras@unica.it M. Brun Università degli Studi di Cagliari, Dipartimento di Ingegneria Meccanica, Chimica e dei Materiali; University of Liverpool, Department of Mathematical Sciences mbrun @unica.it

A BSTRACT . In this article we propose a new auxetic periodic lattice with negative Poisson's ratio which tends to the limit ν=-1 under particular conditions. We have studied its generation and kinematic, and we give a full description of the mechanical properties of this innovative model. Calibrating the geometrical configuration of the lattice and the mechanical properties of the constituent material we are able to have a Poisson's ratio which is arbitrarily close to -1. K EYWORDS . Auxetic lattice; Negative Poisson’s ratio; Mechanical properties.

I NTRODUCTION

oisson's Ratio, usually represented by ν, is defined as the ratio of transverse contraction strain to longitudinal extension strain with respect to the direction of stretching force applied. Since tensile deformation is considered positive and compressive deformation is considered negative in the definition of Poisson's ratio is introduced a minus sign, so that common materials have a positive ratio. However, there are particular materials that expand laterally when stretched longitudinally with a negative Poisson's ratio, they were named for the first time auxetic materials by Ken Evans in an article in Nature (1991). For isotropic materials it may be shown that Poisson's ratio is between -1≤ν≤ ½ in 3D and -1≤ν≤1 in 2D, for anisotropic materials ν is not restricted by the above limits. The value of the Poisson's ratio has also important consequences for other aspects of the behavior of materials, in fact the most materials resist a change in volume as determined by the bulk modulus K more than they resist a change in shape, as determined by the shear modulus μ, the values of K are typically larger than the values of μ. By changing the microstructure of a material in such a way that the Poisson's ratio ν is lower, the values of K and μ can be altered. Decreasing the value of ν to negative value, it would result into a material with a higher shear modulus μ than the bulk modulus K. Different geometrical structures and models are created trying to reproduce some observed feature in auxetic materials, ranging from the macroscopic to microscopic and to the molecular levels. A simple classification can be based on mechanical considerations. Almost all of these models are based on a simple mechanism that is treated as a unit cell leading to a global stiffening effect. One of the earliest models used to describe these special materials was that with re-entrant structure, firstly suggested in [1]. Over the years, many more sophisticated models have been proposed. Another model is based on chiral structure, the researchers in this area use the adjective "chiral" to mean a physical property of spinning. In this type of structures, basic chiral units P

9

L. Cabras et alii, Frattura ed Integrità Strutturale, 29 (2014) 9-18; DOI: 10.3221/IGF-ESIS.29.02

are firstly formed by connecting straight ligaments to central nodes which may be circles or rectangles or other geometrical forms. The auxetic effects are achieved through wrapping or unwrapping of the ligaments around the nodes in response to an applied force, as shown in [2], the Poisson's ratio ν of a chiral structure for in-plane deformations, with flexible ribs and rigid node, can be tailored to be around -1. Later Ruzzene and Spadoni, in [3], have considered the behavior of structures by introducing the flexibility of the nodes. Other models, as in [4], derive the auxetic behavior by the rotation of rigid or semi-rigid shapes (triangle, squares, rectangles and tetrahedron) when loaded, this type of structures has been developed to reproduce the behavior of foams and hypothetical nanostructure networked polymers. A different approach is followed by Bathurst and Rothenburg in [5], they formulate the incremental response of an assembly of elastic spheres, considering an isotropic distribution of contacts around a particle. Since the negative Poisson's ratio is a scale independent property the auxetic behavior can be achieved at a macroscopic or microstructural level, or even at the mesoscopic and molecular levels, many models were developed to simulate polymeric structure or anisotropic fibrous composites. The first auxetic microporous polymeric material was investigated in [6]. It was an expanded foam of PTFE which has a highly anisotropic negative ν =-12. Several cases of negative Poisson's ratios have been discovered in the analysis of anisotropic fibrous composites. In these composites there is a high degree of anisotropy and the negative Poisson's ratio only occurs in some directions; in some cases only over a narrow range of orientation angle between the applied load and the fibers. In a recent advance, laminate structures have been presented which give rise to intentional negative Poisson's ratios combined with mechanical isotropy in two dimensions or in three dimensions. These laminates have structure on several levels of scale; they are hierarchical. By appropriate choice of constituent properties one can achieve Poisson's ratios approaching the lower limit of -1. Auxetic systems perform better than classical material in a number of applications, due to their superior properties. They have been shown to provide better indentation resistance [7, 8 and 9] for their property of “densification” in the vicinity of an impact. The auxetic materials form dome shaped structures [10, 11] when they are subjected to out of plane bending moments instead the saddle shape adopted by the common materials. Also, they can be useful when we need better acoustic and vibration properties than the conventional materials [12, 13 and 14].

M ODEL OF PERIODIC LATTICE WITH AUXETIC MACROSCOPIC BEHAVIOR

W

e consider radially foldable structure formed by two angulated elements ABC and DBE, shown in Fig. 1, connected together through a hinge in B.

Figure 1 : Pair of linkages movable with a single degree of freedom. The two rigid linkages ABC and EBD are shown in grey and black, respectively. They are constraint at the “coupler” point B to have the same displacement components. Points A and D and E and C can only move along straight lines. The coupled angulated elements ABC and DBE can roto-translate with a single degree of freedom and the end point A, C, D and E can only translate along to Ox-axis and the axis inclined by the angle α with respect to the Ox-axis, respectively.

10

L. Cabras et alii, Frattura ed Integrità Strutturale, 29 (2014) 9-18; DOI: 10.3221/IGF-ESIS.29.02

In analyzing the trajectory of the central point B, we also follow the more general formulation given in [15]. In Fig. 1, p is the length of the arms, θ the internal angle between them and α is the angle between the two straight lines along which the points A, D and C, E are constrained to move. B is the “coupler” point of the linkage. The equation for the one parameter trajectory followed by the point B is obtained fixing the values of the geometric variables p, θ, α; then, the position of B is determined by the angle γ. When we couple the movement of the linkage ABC with the linkage EBD, we obtain a relation between angles θ and α. The common point B follow the radial line OB:

sinθ



y

x

(1)



cosθ 1

The two linkages are assembled in order to create a radially foldable structure, as depicted in Fig. 2 and to avoid crossover with other pairs in a polar arrangement of the fully radially foldable structure the angle γ has to satisfy the bound η π γ ηα  , (2) where EDBCAB      . Different configurations are shown in Fig. 2b; the point B for each pair of linkages moves radially and the corresponding Poisson's ratios is equal to -1. We consider the triangular geometries with α=2π/3.

(a) (b) (c) Figure 2 : (a) Radially foldable structure with geometric parameters. (b) Configurations of the single degree of freedom lattices at different values of the geometrical parameter. (c) The radial distance OB as a function of γ is also given for p=1. Construction of periodic lattice The kinematically compatible periodic structures shown in Fig. 3 is obtained by a periodic distribution of the single cell elements shown in Fig. 2 as in [16].

Figure 3 : Periodic microstructure. Three different configurations, for different values of α are shown. The grey dashed region is the unit cell of the Bravais lattice where t 1 and t 2 are the primitive vectors. The microstructure is composed of shaped elements with 12 arms of the same length. A system of cross couple is built where two elements are disposed in two different planes. Each cross couple is mutually constrained to have the same displacement at the central point where a hinge is introduced. Different couples of crosses are then constrained each other by internal hinges at the external end of each arm. The periodic structures have a Bravais periodic lattice [17] consisting of points: R = n 1 t 1 +n 2 t 2 (3)

11