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Procedia Structural (2017) 309–315 Structural IntegrityIntegrity Procedia600 (2016) 000–000
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XXVII International Conference “Mathematical and Computer Simulations in Mechanics of XXVII International Conference “Mathematical and Computer in Mechanics Solids and Structures”. Fundamentals of Static and DynamicSimulations Fracture (MCM 2017) of Solids and Structures”. Fundamentals of Static and Dynamic Fracture (MCM 2017)
Multiscale finite element modeling of nonlinear behavior of polycrystalline piezoceramics with account of tetragonal and polycrystalline piezoceramics with account of tetragonal and Thermo-mechanical modeling of a high pressure turbine blade of an rhombohedral phases rhombohedral phases airplane gas turbine engine Pudeleva Olgaa*, Semenov Artema, Melnikov Borisb
XV Portuguese Conference Fracture, modeling PCF 2016, 10-12 2016,behavior Paço de Arcos, Multiscale finite on element of February nonlinear of Portugal
a b Pudeleva Olgaa*, Semenov Artem a b , Melnikovc Boris Institute of Applied Mathematics and Mechanics, Peter the, Great St. Petersburg Polytechnic University, P. Brandão V. Infante , A.M. Deus * St. Petersburg 195251, Russia
a a
b Institute of AppliedofMathematics and Mechanics, St. Petersburg Polytechnic University, St. Petersburg 195251, Russia Institute Civil Engineering, Peter the Peter Great the St. Great Petersburg Polytechnic University, St. Petersburg 195251, Russia a b Department Engineering, Superior Técnico,Polytechnic Universidade de Lisboa, Rovisco 195251, Pais, 1, 1049-001 InstituteofofMechanical Civil Engineering, PeterInstituto the Great St. Petersburg University, St. Av. Petersburg Russia Lisboa, Portugal b IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal Abstract c CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Abstract Portugal
The nonlinear behavior of a polycrystalline piezoceramic materials based on the two-level finite element homogenization method The behavior of a polycrystalline based on the two-level finite element homogenization variants method was nonlinear investigated. Numerical experiments piezoceramic were carried materials out for rhombohedral and mixed (tetragonal-rhombohedral) was investigated. Numerical were carried for material. rhombohedral (tetragonal-rhombohedral) variants within a single crystal using experiments a micromechanical model out of the Basedand on mixed the results of numerical experiments, the Abstract within a single a micromechanical modelofofthe thesaturation material.strain Basedonon numerical experiments, the remanent part ofcrystal the freeusing energy accounting dependence thethe typeresults of theofmultiaxial deformed state was remanent the free energy accounting of the saturation strain on type of thedemanding multiaxial operating deformed state was constructed. During part theirofoperation, modern aircraft dependence engine components are subjected to the increasingly conditions, constructed. especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent © degradation, 2017 The Authors. Published by Elsevier B.V. which is creep. A using Copyright © 2017one TheofAuthors. Published bymodel Elsevier B.V. the finite element method (FEM) was developed, in order to be able to predict © the 2017creep The under Authors. Published by Elsevier B.V. Peer-review responsibility of the MCM 2017 organizers. behaviour of HPT blades. Flight data records (FDR) for a specific aircraft, provided by a commercial aviation Peer-review under responsibility of the MCM 2017 organizers. Peer-review responsibility the MCM organizers. company, under were used to obtainofthermal and2017 mechanical data for three different flight cycles. In order to create the 3D model Keywords: piezoceramic; strain was saturation;finite element constitutive behavior; phenomenological needed ferroelectrisity/ferroelasticity; for the FEM analysis, a HPT blade scrap scanned, and its homogenization; chemical composition and material properties were Keywords: ferroelectrisity/ferroelasticity; piezoceramic; strain element homogenization; constitutive phenomenological model obtained. The data that was gathered was fed into thesaturation;finite FEM model and different simulations were run,behavior; first with a simplified 3D model rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a model can be useful in the goal of predicting turbine blade life, given a set of FDR data. © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of PCF 2016. Keywords: High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation.
* Corresponding author. Tel.: +7-921-383-66-71. * E-mail Corresponding Tel.: +7-921-383-66-71. address:author.
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[email protected] 2452-3216 © 2017 The Authors. Published by Elsevier B.V. 2452-3216 © 2017 Authors. Published Elsevier B.V. Peer-review underThe responsibility of theby MCM 2017 organizers. * Corresponding Tel.: +351of218419991. Peer-review underauthor. responsibility the MCM 2017 organizers. E-mail address:
[email protected] 2452-3216 © 2016 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of the Scientific Committee of PCF 2016. 2452-3216 Copyright 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the MCM 2017 organizers. 10.1016/j.prostr.2017.11.047
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Pudeleva et al. Integrity / Procedia Structural 6 (2017) 309–315 Author name /Olga Structural Procedia 00 Integrity (2017) 000–000
Nomenclature strain tensor ε electric displacement vector D 4 tensor of elastic modulus S 3 tensor of piezoelectric modulus d tensor of dielectric permittivity k stress tensor σ electric field intensity vector E εr remanent part of strain tensor Pr remanent part of electric displacement vector μ Schmid orientation tensor s unit vector in the direction of the change in remanent polarization remanent part of the free energy associated only with the internal state of material Ψr stored elastic energy Ψs H 0m , c parameters of the material external stress tensor σ external electric displacement vector D external electric field intensity vector E external strain tensor ε 1. Introduction Ferroelectroelastic materials are widely used in modern engineering and research investigations as shown in Zhukov (2009) and Ivashov (2014). One of such materials is polycrystalline piezoelectric ceramics. Ferroelastic materials are used as elements of sensors and actuators (fuel injection valves, vibration dampers, micromotors, nanopositioners, sensors for monitoring the integrity of structures, etc.). The main purpose of this work is to simulate the nonlinear behavior of the polycrystalline ferroelectroelastic materials by means of homogenization using a finite element method with the tetragonal and rhombohedral phases. A new phenomenological model of ferroelectroelastic materials, which is based on determination of the effective properties of polycrystals is also presented in the paper. The constitutive response of the ferroelectroelastic solid can be written as
4 S 3d T εr (1) 3 , k E Pr D d 4 3 where ε is the strain tensor, D is the electric displacement vector, S is the tensor of elastic modulus, d is the tensor of piezoelectric modulus, k is the dielectric permittivity, σ is the stress field, E is the electric field intensity vector, εr and Pr are the remanent parts of strain and electric displacement, respectively. Domains with different orientations and different values of
4
S , 3d , k exist inside the single crystal.
2. Ferroelectric/ferroelastic switching Two differently oriented domains can be combined into a ferroelectroelastic switching system. In a tetragonal variant within the single crystal M = 6 orientations of the spontaneous polarization are realized (along the positive and negative directions of the three crystallographic axes) corresponding to the N = M (M-1) = 30 switching systems. In a rhombohedral single crystal, M = 8 spontaneous polarization orientations are realized along the directions of the four main diagonals of the crystal cell corresponding to N = 56 switching systems. Ferroelectric switching occurs as a result of the movement of the walls of domains, which leads to a change in the concentration of domains in the crystal.
Author name / Structural Integrity Procedia 00 (2017) 000–000 Pudeleva Olga et al. / Procedia Structural Integrity 6 (2017) 309–315
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Fig.1. Domain orientations in a) tetragonal variant, b) rhombohedral variant, c) mixed tetragonal/rhombohedral variant.
3. Micromechanical and phenomenological models The motion of the domain walls with the velocity of f generates in the single crystal an increment of remanent strain and remanent polarization. The total remanent strain and polarization are summation over all switching systems as proposed in Huber (1999) and Huber (2001):
˙ M ˙ ε N ˙ μ εr cI I f , ˙ PI 1 s P Pr I 1
2
μ is identical to the Schmid orientation tensor, while s is a unit vector in the direction of the change in remanent polarization. The scalars and P define the maximum shear strain and polarisation change due to the transformation of system .
where
Let introduce the Helmholtz free energy per unit volume of material:
Ψ Ψs ε, ε r , D, P r Ψr ε r , P r , s
3
r
where Ψ is the stored elastic energy, Ψ – part of the free energy associated only with the internal state of material. Landis (2003) proposed the formulation of mechanical part of remanent free energy in the following way: 2
e m J m 1 , Ψ r H 0m c 2 exp 2 c 1 c r ijr where H , c , m are parameters of the material, e ij m 0
m
kkr 3
ij ,
(4)
is the dependence between remanent
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strain and the type of strain rate. It is assumed that
is the function of the form:
J 3e . e J2
J 2e f
(5)
4. Formulation of the problem and boundary conditions In this paper a micromechanical finite element model for the representative volume element of polycrystalline piezoceramics is considered in the same way with Liskowsky (2005), Pathak (2008), Semenov (2011) and Neumeister (2011). As the representative volume element in form of a cube with a regular partition on finite elements was taken. Each Gaussian point was considered as a single crystal. The orientation of single crystals was generated randomly. In this paper ferroelectric models with only rhombohedral variants, only tetragonal variants and mixed tetragonal/rhombohedral variants were considered. The piezoceramic material PZT-5H was considered in the computations. The formulation of coupled electro-mechanical boundary value problem in the volume of the representative volume element is defined by equation:
σ 0, E ,
D 0, ε us .
S The boundary conditions for these equations on the outer surface n σ S n σ,
(6)
S S SD are: Su
n D S n D,
(7)
D
uS ε r, φS E r, u
φ
where σ is the external stress tensor, D is the electric displacement vector, ε is the external strain tensor, E is the external electric field intensity vector. The vector potential finite-element formulation (see for details Landis (2002), Semenov (2006), Semenov (2010-1)) is used in simulations. The effective methods of integration of constitutive equations are used in a similar way with Semenov (2010-1), Kitaeva (2007) and Semenov (2010-2). The numerical results are obtained with help of finite-element program PANTOCRATOR described in Semenov (2003). 5. Results Computations have been made for different values of the ratio
J 3e / J 2e to obtain the function (5).
For the rhombohedral and mixed tetragonal-rhombohedral variants within the single crystal calculations were carried out for three orientations. After these calculations an approximating function with the least squares method. For the rhombohedral variant function
f J 3e / J 2e .was yielded
f J 3e / J 2e : 3
6
Je Je Je Je f 3e 0.0972 3e 0.0071 3e 0.8444, 3e 0 J2 J2 J2 J2 rh
(8)
Pudeleva Olga et al. / Procedia Structural Integrity 6 (2017) 309–315 Author name / Structural Integrity Procedia 00 (2017) 000–000 3
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313 5
21
Je Je Je Je Je f 3e 0.1518 3e 0.0650 3e 0.1222 3e 0.8419, 3e 0. J2 J2 J2 J2 J2 rh
For the tetragonal variant function
Je f 3e J2
: 3
f
tet
6
J 3e J 3e J 3e J 3e 0.112 0.0052 0.899, 0 e e e J 2e J2 J2 J2 3
f
tet
6
(9)
21
J 3e J 3e J 3e J 3e J 3e 0.1206 0.0047 0.0373 0.899, 0. e e e e J 2e J2 J2 J2 J2
J 3e For the mixed tetragonal-rhombohedral variant function f e : J2 3
f
mix
3
f
mix
6
J 3e J 3e J 3e J 3e 0.1452 e 0.0449 e 0.8699, e 0 e J2 J2 J2 J2 6
(10)
21
J 3e J 3e J 3e J 3e J 3e 0.1127 0.0098 0.0604 0.8684, 0. e e e e e J J J J J 2 2 2 2 2
The dependence of the saturation strain on the form of the stressed state is described by a certain function
g J / J 2e , which is related to the function f J 3e / J 2e by the following relation: e 3
g J 3e / J 2e where
c
c 1 , e 0 f J 3 / J 2e
(11)
is the deformation of compression saturation.
Figure 2 shows a comparison of the results obtained for the rhombohedral, tetragonal and mixed variants.
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Author name / Structural Integrity Procedia 00 (2017) 000–000 Pudeleva Olga et al. / Procedia Structural Integrity 6 (2017) 309–315
Fig.2. Effective saturation strain level as a function of the remanent strain invariant ratio for tetragonal, rhombohedral and mixed variants.
6. Conclusions
Multivariant computational studies were carried out with taking into account the presence of tetragonal and rhombohedral phases within the single crystals in order to determine the free energy parameters of the phenomenological model of ferroelectroelastic material by the method of finite element homogenization. For each type of lattice, the dependences of the saturation strain on the form of the stress state for various distributions of the orientations of single crystals in a polycrystal were established, and analytical dependences of the saturation strain on the form of the stress state were obtained. The mail result of this study is that an increase of portion of rhombohedral single crystals in a polycrystal leads to an increase of the value of the residual strain under different regimes of multiaxial loading. References Zhukov, S.A., 2009. O piezokeramike i perspektivah ee primenenija. The World of Technics and Technologics №5, pp.56-60. Ivashov, I.V., Semenov, A.S., 2014. Vlijanie granichnyh uslovij na beregah treshhiny na process razrushenija polikristallicheskoj p'ezokeramiki. Magazine of Civil Engineering, №7(51), pp.5-15 Huber, J.E., Fleck, N.A., Landis, C.M., McMeeking, R.M., 1999. A constitutive model for ferroelectric polycrystals. J. Mech. Phys. Solids, Vol. 47, pp.1663–1697. Huber JE, Fleck NA. Multi-axial electrical switching of a ferroelectric: theory versus experiment. J. Mech. Phys. Solids 2001; 49:785–811. Landis, C. M., 2003. On the Strain Saturation Conditions for Polycrystalline Ferroelastic Materials. J. of App. Mechanics, 70(4), pp.470-478. Liskowsky, A.C., Semenov, A.S., Balke, H., McMeeking, R.M., 2005. Finite element modeling of the ferroelectroelastic material behavior in consideration of domain wall motions. In: R.M. McMeeking, M. Kamlah, S. Seelecke and D. Viehland (Ed.). Coupled Nonlinear Phenomena – Modeling and Simulation for Smart, Ferroic and Multiferroic Materials, Proceedings of the 2005 MRS Spring Meeting, Vol. 881E. Pathak, A., McMeeking, R.M., 2008. Three-dimensional finite element simulations of ferroelectric polycrystals under electrical and mechanical loading. J. Mech. Phys. Solids, Vol. 56, pp.663-683. Semenov, A.S., Balke, H., Melnikov, B.E., 2011. Modelirovanie polikristallicheskoj piezokeramiki metodom konechno-jelementnoj gomogenizacii. Marine Intellectual Technologies №3, pp.109-115. Neumeister, P., Balke, H., 2011. Micromechanical modelling of the remanent properties of morphotropic PZT // J. of the Mech. and Physics of Solids, Vol. 59, pp.1794-1807. Landis, C.M., 2002. A new finite-element formulation for electromechanical boundary value problems. International Journal for Numerical Methods in Engineering, 55, pp.613-628. Semenov, A.S., Liskowsky, A.C., Balke, H., 2010. Return mapping algorithms and consistent tangent operators in ferroelectroelasticity. International Journal for Numerical Methods in Engineering, 81, pp.1298–1340.
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Kitaeva, D.A., Pazylov, Sh.T., Rudaev, Ya.I., 2007. On applications of nonlinear dynamics methods for mechanics of materials. Vestnik Permskogo gosudarstvennogo tekhnicheskogo universiteta. Matematicheskoe modelirovanie system i protcesov 15, pp.46–70. Semenov, A.S., 2003. PANTOCRATOR – the finite element program specialized on the nonlinear problem solution. In B.E Melnikov (ed.), Proc. V Int. Conf. "Scientific and engineering problems of predicting the reliability and service life of structures and methods of their solution", pp.466–480.
