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Finite element modelling of passive damping with resistively shunted piezocomposites
Computational Materials Science 19 (2000) 183±188
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Finite element modelling of passive damping with resistively shunted piezocomposites Christophe Poizat *, Matthias Sester Fraunhofer Institut f ur Werkstomechanik, W ohlerstraûe 11±13, D-79108 Freiburg, Germany
Abstract This work aims at modelling passive damping in structures made of composites with embedded resistively shunted piezoelectric ceramic ®bres. To this end, a 3D material model is implemented in the ®nite element (FE) code ABAQUS. Academical test cases as well as a structural case study are presented. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Passive damping; Piezocomposite; Material model; Finite element method
1. Introduction As shown by several authors [1±4], a promising way to increase the damping properties of ®bre reinforced composite materials consists in embedding resistively shunted piezoelectric ®bres [5,6]. The basic idea is illustrated in Fig. 1: due to the direct piezoelectric eect [7], the mechanical loading due to vibration induces electrical charges in the capacitive CpS piezoelectric element, so that a part of mechanical energy is converted in electrical energy. As a consequence, electrical charges ¯ow through the resistor (R) so that a given part of electrical energy is dissipated thermally through the Joule eect. The aim of this work consists in developing a tool allowing to design structures with optimal damping due to resistively shunted piezoelectric ®bres. Since closed form solutions are likely to be found only for simple systems, we use the ®nite element method. *
Corresponding author. Tel.: +49-761-51-42-230; fax: +49761-51-42-110. E-mail address: [email protected] (C. Poizat).
Two successive modelling levels have to be distinguished. First, the piezocomposites have to be homogenised. Several approaches may be used, whether analytical [8,9] or numerical [10,11]. We use the unit-cell method, as described in [11]. The resulting eective properties serve as input material data for the macromodelling part. In the following, all properties are eective properties. Then, passive damping is modelled on the macrolevel. The paper is organised as follows. First, basic knowledges on piezoelectricity are brie¯y recalled. Then, the model for passive damping and the principles of its implementation in the ®nite element (FE) code are described. Finally, numerical results are presented and discussed.
2. Piezoelectricity 2.1. Constitutive equations of piezoelectricity Coupled piezoelectric problems are those in which an electric potential gradient causes deformation (indirect eect), while stress causes an
0927-0256/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 0 ) 0 0 1 5 4 - 3
184
C. Poizat, M. Sester / Computational Materials Science 19 (2000) 183±188
2.2. Electromechanical coupling factors
Fig. 1. Resistively shunted piezoelectric ®bres composite (just one ®bre is schematically represented).
electric potential gradient in the material (direct eect). The coupling between mechanical and electrical ®eld is characterised by piezoelectric coecients (denoted d; e; g and h). In a compact matrix form and in terms of the stress coecients e, the constitutive equations of linear piezoelectricity are [7]: E S T c ÿet ; 1 E D e eS where D fD1 ; D2 ; D3 gt ;
E fE1 ; E2 ; E3 gt ;
S fS1 ; S2 ; S3 ; S4 ; S5 ; S6 gt ;
2
T fT1 ; T2 ; T3 ; T4 ; T5 ; T6 gt ; where Di and Ek , respectively, denote the components of the electric displacement and of the electric potential gradient and Ti and Sj are the components of the stress and strain. The third direction is associated with the direction of poling. The subscript t denotes the conventional matrix transpose. In the case of statistically distributed parallel ®bres made of a transversely isotropic piezoelectric solid (e.g., PZT-®bres poled in the axial direction) and isotropic polymer matrix, the resulting composite is a transversely isotropic piezoelectric material. In this case, the stiness matrix under short circuit conditions [cE ], the piezoelectric matrix [e] and the dielectric matrix under constant strains [eS ] simplify so that there remain 10 independent coecients [7].
In Section 4.1, we refer to the longitudinal coupling coecient k33 , associated with the so called d33 -eect or longitudinal eect. There exist several approaches to de®ne electromechanical coupling factors [7]. Physically, their squares give the percentage of mechanical strain energy which is converted in electrical energy and vice versa. That is the reason why they play an important role towards damping. In the example of interest in Section 4.1, the higher k33 , the higher the electrical energy induced piezoelectrically by longitudinal vibration.
3. Modelling passive damping with the ®nite element method The total damping of a given structure is strongly in¯uenced by the strain energy distribution in this structure [2]. Analytical models alone are appropriate to estimate the material damping properties but are not sucient for designing complex structures with embedded piezoelectric passive dampers. In order to allow analyses of arbitrarily shaped structures, we developed a User MATerial subroutine (UMAT) for the general purpose FE code ABAQUS [12]. Such subroutines allow the implementation of additional material laws for use with standard ®nite elements. In a displacement-based formulation as used by ABAQUS, the material model must be formulated in a stress±strain form. 3.1. The material law The method we use is basically the same as in Hagood and von Flotow [2]. We start from the mixed formulation in terms of the piezoelectric stress coecients eij , see Eq. (1). After some manipulations in the Laplace domain [13], the material law is found to have the form of a hereditary integral. In the case of a polarisation along direction 3 and a pair of electrodes shunted by a resistor as shown in Fig. 1, we obtain the 3D formulation (3) below.
C. Poizat, M. Sester / Computational Materials Science 19 (2000) 183±188
Ti t
Z t 0
S
cEij eÿ tÿs=RCp
e3i e3j _ Sj s ds: eS3
3
The domain of applicability of the model is as follows. The piezoelectric element is linear piezoelectric, transverse isotropic or orthotropic and non-conductive. In the case of a composite with piezo®bres embedded in an epoxy matrix, the internal damping (due to the viscoelastic matrix and the ®bre matrix interactions) is not taken into account. Detailed discussions may be found in [13]. 3.2. Implementation The hereditary integral formulation (3) allows the modelling of the damping behaviour of resistively shunted piezocomposites with embedded parallel ®bres. If the piezoelectric composite is transverse isotropic, the obtained material law has the mathematical form of transverse isotropic viscoelasticity with eective coecients evaluated with the help of micro±macro approaches as underlined before. The material library of ABAQUS [12] provides isotropic viscoelasticity, however, anisotropic viscoelasticity is not included. We hence implement the 3D form (3) of the model in the FE code ABAQUS. The numerical integration of Eq. (3) is based on the discrete spectrum approximation method, that uses an exponential form of the relaxation function [12±14]. The method as well as academic tests are described in details in [13].
185
isotropic composite with embedded resistively shunted piezoelectric ®bres. 4.1. Applications to bulk piezoelectric ceramics For an exact comparison with the analytical models [2,3], a one-dimensional formulation of the UMAT (called 1D-UMAT) was written to be used with one-dimensional truss elements. The material is a bulk ceramic with a coupling factor k33 of 0.46, referred as PZT-1. This material corresponds to the hard PZT ceramic of which the hard PZT®bres developed at the Fraunhofer-ISC, W urzburg, are made [6]. The set of material data is detailed in [13]. The results are shown in Fig. 2. The agreement is qualitatively and quantitatively good. Especially, there exists an optimal material damping: a frequency is optimally damped by choosing appropriately the shunt resistor and a maximal loss factor of about 0.124 is found in both cases. 4.2. Application to a 1±3 piezoelectric composite We consider a one-dimensional truss made of a 1±3 piezoelectric composite with three dierent volume fractions of PZT-1 ®bres. The matrix is an epoxy matrix and the eective properties are derived with the help of the unit-cell method described in [11]. The resulting normalised displacements u3 at the tip of the longitudinal free vibrating bar are
4. Numerical results We ®rst compare the loss factors (see the good review on damping [15]) obtained with a uniaxial version of our UMAT and those obtained with uniaxial analytical approaches [2,3]. The case study is a simple one-dimensional truss. The material is a bulk piezoelectric ceramic. To illustrate the interest of piezocomposites, we then study the damping behaviour of this truss in free longitudinal vibration; the material is a piezoelectric 1±3 composite [16] with varying ®bre volume fraction. We ®nally apply the 3D anisotropic model described above to a structure made of a transversely
Fig. 2. Loss factor as a function of the non-dimensional frequency (wimp imposed frequency, wopt optimal frequency).
186
C. Poizat, M. Sester / Computational Materials Science 19 (2000) 183±188
given in Fig. 3. In all cases, the resistance is set to its optimum value (since the ®bre volume fraction is dierent, the frequencies are dierent as well. The shunt resistor hence has to be tuned to its optimal value in order to get optimal damping). The ®bre volume fraction has a strong in¯uence. A slight dierence is observed between res Vf 100% (gres max 0:124) and Vf 30% (gmax 0:112), whereas the damping capacity is much smaller at Vf 4% (gres max 0:055). The dependence on the volume fraction is clearly to be understood with the Fig. 4(a), that shows the coupling factor k33 as a function of the ®bres volume fraction Vf . The resulting coupling factor (Fig. 4(b)), that is determinant in the maximum damping capacity, varies non-linearly with the volume fraction. For light weight structure
Fig. 3. Longitudinal free vibration of a bar, and in¯uence of the PZT-1 ®bre volume fraction.
applications and from a material point of view, a volume fraction between 20% and 40% gives the best compromise between weight and damping capacity. 4.3. Cantilever beam with three layers We consider a cantilever beam with three layers in free vibration bending (Fig. 5). The layers are made of a piezoelectric composite, called PZT-2. The eective properties are listed in Appendix A (see [13] for details). The poling direction in the third one. The middle layer, called substrate, is not damped. Two cases are treated. In the ®rst case, the substrate is fully covered with passively damped piezocomposite. In the second case, just one half of the substrate is covered (Fig. 6). It should be pointed out that for regions strained in the opposite direction, dierent shunt resistors have to be used since otherwise, the electrical charges with dierent signs would cancel each other. The chosen resistor is the optimal resistor in both cases. As seen in Fig. 6, the vibration frequency is almost the same (64 and 66 Hz). As a consequence, the optimal resistors are almost equal. The results are shown in Fig. 6 in terms of the vertical beam tip displacement over time. Clearly, the damping eect is equivalent. In both cases, the logarithmic decrement [15] is about 0.11. For longitudinal vibration, the logarithmic decrement is about 1.3. For bending vibration, the damping eect is much smaller due to the presence of a neutral layer and the strain energy concentration in the clamped area. The proposed FE approach
Fig. 4. Eective coupling factor k33 (a) and maximal loss factor (b) as a function of the volume fraction Vf .
C. Poizat, M. Sester / Computational Materials Science 19 (2000) 183±188
187
Fig. 5. Cantilever beam with three layers (l 131:9 mm, b 19 mm, h1 h3 0:1; h2 0:685 mm).
Hagood and von Flotow [2] in an suitable 3D stress±strain form. The model allows for the estimation of the added damping due to resistive shunting of linear, non-conductive piezoelectric materials. The analogy with viscoelasticity has the advantage to reduce the degrees of freedom to 3, instead of 4 in the case of piezoelectric elements. As shown with a simple cantilever beam, structural analyses will allow to give more realistic estimates on damping eects and useful informations for an optimal placement of damping materials. In a short future, a plane stress formulation of the model will be written that is suitable with the use of shell-elements. Fig. 6. Free bending vibration of a cantilever beam: vertical tip displacement over time.
Acknowledgements
allows to take inhomogeneous energy distribution into account.
Fruitful discussions with Prof. W. Kreher (TUDresden, Germany) are gratefully acknowledged.
5. Conclusion
Appendix A
For a numerical implementation in the displacement-based FE code ABAQUS, we have reformulated the constitutive law developed by
The elastic properties, piezoelectric constants and permittivities are, respectively, given in GPa, in C/m2 and in 10ÿ9 F/m (see Table 1).
Table 1 PZT-2 composite material constant for the 3D-UMAT PZT-2
cE11
cE12
cE22
cE13
cE23
cE33
cE44
cE66
e33
e31
eS3
9.88
3.42
9.88
3.99
3.99
25.2
1.71
2.5
14.9
)0.28
7.9
188
C. Poizat, M. Sester / Computational Materials Science 19 (2000) 183±188
References [1] O. Lacour, In¯uence de la piezoe lectricite sur l'absorption du son dans les milieux composites, Ph.D. Thesis, Grenoble I, THOMSON-SINTRA, France, 1989. [2] N.W. Hagood, A. von Flotow, J. Sound Vib. 146 (2) (1991) 243±268. [3] W. Kreher, Internal Report MW 280.331, TU-Dresden, Germany, 1998. [4] G.A. Lesieutre et al., J. Mater. Eng. Performance 2 (6) (1993) 887±892. [5] S. Yoshikawa et al., J. Intell. Mater. Syst. Struct. 6 (2) (1995) 152±158. [6] D. Sporn et al., in: P. Vincenzini (Ed.), Proceedings of the Ninth CIMTEC Conference on Advances in Science and Technology, vol. 25, 14±19 June 1998, Florence, 1999, pp. 279±286. [7] T. Ikeda, Fundamentals of Piezoelectricity, Oxford University Press, Oxford, 1996.
[8] M.L. Dunn, M. Taya, Int. J. Solids Struct. 30 (2) (1993) 161±175. [9] V.M. Levin, M.I. Rakovskaya, W.S. Kreher, Int. J. Solids Struct. 36 (1999) 2683±2705. [10] J. Pastor, Mech. Res. Commun. 24 (2) (1997) 145±150. [11] C. Poizat, M. Sester, Comput. Mater. Sci. 16 (1±4) (1999) 89±97. [12] ABAQUS Standard and Theory Manuals, Version 5.8, HKS, 1998. [13] C. Poizat, Modelisation nume rique de materiaux et structures composites a ®bres piezoelectriques, Ph.D. Thesis 00TROY0004, Universite de Technologie de Troyes, France, 2000. [14] M.A. Puso, J.A. Weiss, Trans. ASME 120 (1998) 62±70. [15] C.W. Bert, J. Sound Vib. 29 (2) (1973) 129±153. [16] R.E. Newnham, et al., Mater. Res. Bull. 13 (1978) 525±536.
www.elsevier.com/locate/commatsci
Finite element modelling of passive damping with resistively shunted piezocomposites Christophe Poizat *, Matthias Sester Fraunhofer Institut f ur Werkstomechanik, W ohlerstraûe 11±13, D-79108 Freiburg, Germany
Abstract This work aims at modelling passive damping in structures made of composites with embedded resistively shunted piezoelectric ceramic ®bres. To this end, a 3D material model is implemented in the ®nite element (FE) code ABAQUS. Academical test cases as well as a structural case study are presented. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Passive damping; Piezocomposite; Material model; Finite element method
1. Introduction As shown by several authors [1±4], a promising way to increase the damping properties of ®bre reinforced composite materials consists in embedding resistively shunted piezoelectric ®bres [5,6]. The basic idea is illustrated in Fig. 1: due to the direct piezoelectric eect [7], the mechanical loading due to vibration induces electrical charges in the capacitive CpS piezoelectric element, so that a part of mechanical energy is converted in electrical energy. As a consequence, electrical charges ¯ow through the resistor (R) so that a given part of electrical energy is dissipated thermally through the Joule eect. The aim of this work consists in developing a tool allowing to design structures with optimal damping due to resistively shunted piezoelectric ®bres. Since closed form solutions are likely to be found only for simple systems, we use the ®nite element method. *
Corresponding author. Tel.: +49-761-51-42-230; fax: +49761-51-42-110. E-mail address: [email protected] (C. Poizat).
Two successive modelling levels have to be distinguished. First, the piezocomposites have to be homogenised. Several approaches may be used, whether analytical [8,9] or numerical [10,11]. We use the unit-cell method, as described in [11]. The resulting eective properties serve as input material data for the macromodelling part. In the following, all properties are eective properties. Then, passive damping is modelled on the macrolevel. The paper is organised as follows. First, basic knowledges on piezoelectricity are brie¯y recalled. Then, the model for passive damping and the principles of its implementation in the ®nite element (FE) code are described. Finally, numerical results are presented and discussed.
2. Piezoelectricity 2.1. Constitutive equations of piezoelectricity Coupled piezoelectric problems are those in which an electric potential gradient causes deformation (indirect eect), while stress causes an
0927-0256/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 0 ) 0 0 1 5 4 - 3
184
C. Poizat, M. Sester / Computational Materials Science 19 (2000) 183±188
2.2. Electromechanical coupling factors
Fig. 1. Resistively shunted piezoelectric ®bres composite (just one ®bre is schematically represented).
electric potential gradient in the material (direct eect). The coupling between mechanical and electrical ®eld is characterised by piezoelectric coecients (denoted d; e; g and h). In a compact matrix form and in terms of the stress coecients e, the constitutive equations of linear piezoelectricity are [7]: E S T c ÿet ; 1 E D e eS where D fD1 ; D2 ; D3 gt ;
E fE1 ; E2 ; E3 gt ;
S fS1 ; S2 ; S3 ; S4 ; S5 ; S6 gt ;
2
T fT1 ; T2 ; T3 ; T4 ; T5 ; T6 gt ; where Di and Ek , respectively, denote the components of the electric displacement and of the electric potential gradient and Ti and Sj are the components of the stress and strain. The third direction is associated with the direction of poling. The subscript t denotes the conventional matrix transpose. In the case of statistically distributed parallel ®bres made of a transversely isotropic piezoelectric solid (e.g., PZT-®bres poled in the axial direction) and isotropic polymer matrix, the resulting composite is a transversely isotropic piezoelectric material. In this case, the stiness matrix under short circuit conditions [cE ], the piezoelectric matrix [e] and the dielectric matrix under constant strains [eS ] simplify so that there remain 10 independent coecients [7].
In Section 4.1, we refer to the longitudinal coupling coecient k33 , associated with the so called d33 -eect or longitudinal eect. There exist several approaches to de®ne electromechanical coupling factors [7]. Physically, their squares give the percentage of mechanical strain energy which is converted in electrical energy and vice versa. That is the reason why they play an important role towards damping. In the example of interest in Section 4.1, the higher k33 , the higher the electrical energy induced piezoelectrically by longitudinal vibration.
3. Modelling passive damping with the ®nite element method The total damping of a given structure is strongly in¯uenced by the strain energy distribution in this structure [2]. Analytical models alone are appropriate to estimate the material damping properties but are not sucient for designing complex structures with embedded piezoelectric passive dampers. In order to allow analyses of arbitrarily shaped structures, we developed a User MATerial subroutine (UMAT) for the general purpose FE code ABAQUS [12]. Such subroutines allow the implementation of additional material laws for use with standard ®nite elements. In a displacement-based formulation as used by ABAQUS, the material model must be formulated in a stress±strain form. 3.1. The material law The method we use is basically the same as in Hagood and von Flotow [2]. We start from the mixed formulation in terms of the piezoelectric stress coecients eij , see Eq. (1). After some manipulations in the Laplace domain [13], the material law is found to have the form of a hereditary integral. In the case of a polarisation along direction 3 and a pair of electrodes shunted by a resistor as shown in Fig. 1, we obtain the 3D formulation (3) below.
C. Poizat, M. Sester / Computational Materials Science 19 (2000) 183±188
Ti t
Z t 0
S
cEij eÿ tÿs=RCp
e3i e3j _ Sj s ds: eS3
3
The domain of applicability of the model is as follows. The piezoelectric element is linear piezoelectric, transverse isotropic or orthotropic and non-conductive. In the case of a composite with piezo®bres embedded in an epoxy matrix, the internal damping (due to the viscoelastic matrix and the ®bre matrix interactions) is not taken into account. Detailed discussions may be found in [13]. 3.2. Implementation The hereditary integral formulation (3) allows the modelling of the damping behaviour of resistively shunted piezocomposites with embedded parallel ®bres. If the piezoelectric composite is transverse isotropic, the obtained material law has the mathematical form of transverse isotropic viscoelasticity with eective coecients evaluated with the help of micro±macro approaches as underlined before. The material library of ABAQUS [12] provides isotropic viscoelasticity, however, anisotropic viscoelasticity is not included. We hence implement the 3D form (3) of the model in the FE code ABAQUS. The numerical integration of Eq. (3) is based on the discrete spectrum approximation method, that uses an exponential form of the relaxation function [12±14]. The method as well as academic tests are described in details in [13].
185
isotropic composite with embedded resistively shunted piezoelectric ®bres. 4.1. Applications to bulk piezoelectric ceramics For an exact comparison with the analytical models [2,3], a one-dimensional formulation of the UMAT (called 1D-UMAT) was written to be used with one-dimensional truss elements. The material is a bulk ceramic with a coupling factor k33 of 0.46, referred as PZT-1. This material corresponds to the hard PZT ceramic of which the hard PZT®bres developed at the Fraunhofer-ISC, W urzburg, are made [6]. The set of material data is detailed in [13]. The results are shown in Fig. 2. The agreement is qualitatively and quantitatively good. Especially, there exists an optimal material damping: a frequency is optimally damped by choosing appropriately the shunt resistor and a maximal loss factor of about 0.124 is found in both cases. 4.2. Application to a 1±3 piezoelectric composite We consider a one-dimensional truss made of a 1±3 piezoelectric composite with three dierent volume fractions of PZT-1 ®bres. The matrix is an epoxy matrix and the eective properties are derived with the help of the unit-cell method described in [11]. The resulting normalised displacements u3 at the tip of the longitudinal free vibrating bar are
4. Numerical results We ®rst compare the loss factors (see the good review on damping [15]) obtained with a uniaxial version of our UMAT and those obtained with uniaxial analytical approaches [2,3]. The case study is a simple one-dimensional truss. The material is a bulk piezoelectric ceramic. To illustrate the interest of piezocomposites, we then study the damping behaviour of this truss in free longitudinal vibration; the material is a piezoelectric 1±3 composite [16] with varying ®bre volume fraction. We ®nally apply the 3D anisotropic model described above to a structure made of a transversely
Fig. 2. Loss factor as a function of the non-dimensional frequency (wimp imposed frequency, wopt optimal frequency).
186
C. Poizat, M. Sester / Computational Materials Science 19 (2000) 183±188
given in Fig. 3. In all cases, the resistance is set to its optimum value (since the ®bre volume fraction is dierent, the frequencies are dierent as well. The shunt resistor hence has to be tuned to its optimal value in order to get optimal damping). The ®bre volume fraction has a strong in¯uence. A slight dierence is observed between res Vf 100% (gres max 0:124) and Vf 30% (gmax 0:112), whereas the damping capacity is much smaller at Vf 4% (gres max 0:055). The dependence on the volume fraction is clearly to be understood with the Fig. 4(a), that shows the coupling factor k33 as a function of the ®bres volume fraction Vf . The resulting coupling factor (Fig. 4(b)), that is determinant in the maximum damping capacity, varies non-linearly with the volume fraction. For light weight structure
Fig. 3. Longitudinal free vibration of a bar, and in¯uence of the PZT-1 ®bre volume fraction.
applications and from a material point of view, a volume fraction between 20% and 40% gives the best compromise between weight and damping capacity. 4.3. Cantilever beam with three layers We consider a cantilever beam with three layers in free vibration bending (Fig. 5). The layers are made of a piezoelectric composite, called PZT-2. The eective properties are listed in Appendix A (see [13] for details). The poling direction in the third one. The middle layer, called substrate, is not damped. Two cases are treated. In the ®rst case, the substrate is fully covered with passively damped piezocomposite. In the second case, just one half of the substrate is covered (Fig. 6). It should be pointed out that for regions strained in the opposite direction, dierent shunt resistors have to be used since otherwise, the electrical charges with dierent signs would cancel each other. The chosen resistor is the optimal resistor in both cases. As seen in Fig. 6, the vibration frequency is almost the same (64 and 66 Hz). As a consequence, the optimal resistors are almost equal. The results are shown in Fig. 6 in terms of the vertical beam tip displacement over time. Clearly, the damping eect is equivalent. In both cases, the logarithmic decrement [15] is about 0.11. For longitudinal vibration, the logarithmic decrement is about 1.3. For bending vibration, the damping eect is much smaller due to the presence of a neutral layer and the strain energy concentration in the clamped area. The proposed FE approach
Fig. 4. Eective coupling factor k33 (a) and maximal loss factor (b) as a function of the volume fraction Vf .
C. Poizat, M. Sester / Computational Materials Science 19 (2000) 183±188
187
Fig. 5. Cantilever beam with three layers (l 131:9 mm, b 19 mm, h1 h3 0:1; h2 0:685 mm).
Hagood and von Flotow [2] in an suitable 3D stress±strain form. The model allows for the estimation of the added damping due to resistive shunting of linear, non-conductive piezoelectric materials. The analogy with viscoelasticity has the advantage to reduce the degrees of freedom to 3, instead of 4 in the case of piezoelectric elements. As shown with a simple cantilever beam, structural analyses will allow to give more realistic estimates on damping eects and useful informations for an optimal placement of damping materials. In a short future, a plane stress formulation of the model will be written that is suitable with the use of shell-elements. Fig. 6. Free bending vibration of a cantilever beam: vertical tip displacement over time.
Acknowledgements
allows to take inhomogeneous energy distribution into account.
Fruitful discussions with Prof. W. Kreher (TUDresden, Germany) are gratefully acknowledged.
5. Conclusion
Appendix A
For a numerical implementation in the displacement-based FE code ABAQUS, we have reformulated the constitutive law developed by
The elastic properties, piezoelectric constants and permittivities are, respectively, given in GPa, in C/m2 and in 10ÿ9 F/m (see Table 1).
Table 1 PZT-2 composite material constant for the 3D-UMAT PZT-2
cE11
cE12
cE22
cE13
cE23
cE33
cE44
cE66
e33
e31
eS3
9.88
3.42
9.88
3.99
3.99
25.2
1.71
2.5
14.9
)0.28
7.9
188
C. Poizat, M. Sester / Computational Materials Science 19 (2000) 183±188
References [1] O. Lacour, In¯uence de la piezoe lectricite sur l'absorption du son dans les milieux composites, Ph.D. Thesis, Grenoble I, THOMSON-SINTRA, France, 1989. [2] N.W. Hagood, A. von Flotow, J. Sound Vib. 146 (2) (1991) 243±268. [3] W. Kreher, Internal Report MW 280.331, TU-Dresden, Germany, 1998. [4] G.A. Lesieutre et al., J. Mater. Eng. Performance 2 (6) (1993) 887±892. [5] S. Yoshikawa et al., J. Intell. Mater. Syst. Struct. 6 (2) (1995) 152±158. [6] D. Sporn et al., in: P. Vincenzini (Ed.), Proceedings of the Ninth CIMTEC Conference on Advances in Science and Technology, vol. 25, 14±19 June 1998, Florence, 1999, pp. 279±286. [7] T. Ikeda, Fundamentals of Piezoelectricity, Oxford University Press, Oxford, 1996.
[8] M.L. Dunn, M. Taya, Int. J. Solids Struct. 30 (2) (1993) 161±175. [9] V.M. Levin, M.I. Rakovskaya, W.S. Kreher, Int. J. Solids Struct. 36 (1999) 2683±2705. [10] J. Pastor, Mech. Res. Commun. 24 (2) (1997) 145±150. [11] C. Poizat, M. Sester, Comput. Mater. Sci. 16 (1±4) (1999) 89±97. [12] ABAQUS Standard and Theory Manuals, Version 5.8, HKS, 1998. [13] C. Poizat, Modelisation nume rique de materiaux et structures composites a ®bres piezoelectriques, Ph.D. Thesis 00TROY0004, Universite de Technologie de Troyes, France, 2000. [14] M.A. Puso, J.A. Weiss, Trans. ASME 120 (1998) 62±70. [15] C.W. Bert, J. Sound Vib. 29 (2) (1973) 129±153. [16] R.E. Newnham, et al., Mater. Res. Bull. 13 (1978) 525±536.