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A high order model for piezoelectric rods: An asymptotic approach
International Journal of Solids and Structures 81 (2016) 294–310
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A high order model for piezoelectric rods: An asymptotic approach J.M. Viaño a, C. Ribeiro b, J. Figueiredo b, Á. Rodríguez-Arós c,∗ a
Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, Spain Departamento de Matemática e Aplicações and Centro de Matemática, Universidade do Minho, Portugal c Departamento de Métodos Matemáticos e Representación, Universidade da Coruña, Spain b
a r t i c l e
i n f o
Article history: Received 21 March 2014 Revised 30 October 2015 Available online 11 December 2015 Keywords: Asymptotic analysis Elastic rod Beams Piezoelectric material Anisotropy
a b s t r a c t This paper is devoted to the study of an electromechanical model for a linear transversely inhomogeneous anisotropic rod made of piezoelectric symmetry class 2 materials. The materials in this class include, among many others, Barium Titanate (BT), Lead Zirconium Titanate (PZT) and Polyvinylidene Fluoride (PVDF). For that purpose, we use asymptotic analysis in order to derive from the 3D piezoelectricity problem a reduced model, without making any a priori assumptions of geometrical or mechanical nature. This process involves considering the diameter of the cross section h as small parameter, the assumption of existence of asymptotic expansions of the unknowns (displacements field and electric potential) and the characterization of their respective leading terms as the unique solutions of limit reduced coupled equations. Finally, the validity of the limit reduced model is supported by providing rigorous convergence results. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction The so-called “intelligent structures”, i.e. those that are able to adapt in an automatic way when submitted to external actions in order to provide an optimal behavior, play an increasing role in the framework of nowadays security and wellbeing tools, and their importance will certainly increase in the near future. The use of piezoelectric materials is a strategy that has been used with considerable success in the development of new materials and technologies for the production of these structures. The piezoelectric effect is an electromechanical property that is exhibited by many materials: when the material is deformed by the action of external forces or pressure, an electric field appears – direct piezoelectric effect – and conversely, when subject to an electric potential, the material suffers a deformation – inverse piezoelectric effect. This reciprocity provides both an efficient sensor capacity and the possibility of acting over large portions of structures without a considerable weight increase. For more complete information on the properties and use of piezoelectric materials see, for example, Banks et al. (1996); Ikeda (1990); Nye (1985); Royer and Dieulesaint (2000); Taylor et al. (1985). More specifically, information about traditional piezoelectric beam theories can be consulted in Bellis and Imperiale (2014); Bisegna (1998); Rovenski and Abramovich (2009). ∗
Corresponding author. Tel.: +34 981167000. E-mail addresses: [email protected] (J.M. Viaño), [email protected] (C. Ribeiro), jmfi[email protected] (J. Figueiredo), [email protected] (Á. Rodríguez-Arós). http://dx.doi.org/10.1016/j.ijsolstr.2015.12.005 0020-7683/© 2015 Elsevier Ltd. All rights reserved.
From the above considerations one easily concludes on the relevance of designing new multifunctional materials (e.g. piezoelectric materials) and, consequently, on the importance of accurately knowing the electromechanical behavior of structures in which these materials are applied. Some important results in this field were successfully obtained by applying two different modeling techniques: homogenization and asymptotic analysis (see Cagnol et al., 2008; Miara et al., 2007 for a review of the state of the art). See in particular, Castillero et al. (1998); Ghergu et al. (2007); Mechkour (2004); Miara et al. (2005) for homogenization of piezoelectric plates and shells and Collard and Miara (2002); Figueiredo and Leal (2006); Rahmoune et al. (1998); Sabu (2002); Sène (2001); Weller and Licht (2002); Weller and Licht (2004) for formal asymptotic analysis to justify lower dimensional constitutive laws for piezoelectric plates, shells and rods. The above cited works were based on relevant and pioneer work in applying homogenization and asymptotic analysis techniques in elasticity problems (linear and nonlinear). See in particular Ciarlet (1980); Collard and Miara (1997); Le Dret and Raoult ´ (1995); Lewinski and Telega (2000); Miara (1994); Sanchez-Hubert and Sanchez-Palencia (1997) for plates, Bermúdez and Viaño (1984a); Cimetière et al. (1988); Mascarenhas and Trabucho (1990); Trabucho and Viaño (1996); Tutek (1987); Tutek and Aganovic´ (1986) for rods, and Caillerie and Sanchez-Palencia (1995); Ciarlet (2000); Ciarlet et al. (1996); Pitkaranta and Sanchez-Palencia (1997) for shells. As it is well known, the classical theories of thin structures obtained from classical formulations are intuitive for understanding, but the a priori assumptions are often mechanically unjustified and can reveal inadequate for the analysis of real structures if those
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
a priori assumptions are not met. Although the amount of research in the asymptotic analysis field is quite large, only a small number of works exist (as far as we know) devoted to the derivation of models for piezoelectric rods (see Figueiredo and Leal, 2006; Weller and Licht, 2004). Despite the progress made in those papers in order to derive reduced models for piezoelectric beams, the subject is still open to further research. In this framework, our contribution consists in the obtention of a reduced model of a linear transversely inhomogeneous anisotropic rod made of piezoelectric symmetry class 2 materials, by using the asymptotic method introduced by Lions (see Lions, 1973), taking the diameter of the rod cross section h as small parameter, without any a priori hypothesis of geometrical or mechanical nature. The model obtained in this way allows to, simultaneously, justify and/or complement the traditional piezoelectric beam theories (such as those of Euler–Bernoulli, Timoshenko, and Vlasov). Additionally, in the reduced model, to our knowledge for the first time, the electric potential is coupled with first order terms of the asymptotic expansion of the displacements. The present work is organized in the following manner. In Section 2 we show the 3D piezoelectric problem posed in its domain of reference h (the details on the notation used will be specified in the corresponding sections and can be found in tables in the appendix as well) and formulate it in the form of the virtual work principle (see (2.13)): h Find (uh , ϕ h ) ∈ V h (h ) × H 1 (h ), such that ϕ h = ϕˆ h on eD , σihj (uh , ϕ h )ehij (vh )dxh + Dhk (uh , ϕ h )Ekh (ψ h )dxh h h = fih vhi dxh + ghi vhi d h ,
h
Nh
for all (v , ψ ) ∈ V ( ) × ( ), h
uh
h
(uhi )
h
h
h
h
ϕh
where = and are the displacements field and the electric potential, respectively, to be found in the corresponding admissible spaces Vh (h ) and H1 (h ). Also, σ h = (σihj ) and Dh =
(Dhi ) represent the stress tensor field and the electric displace-
ments field, respectively, while eh (vh ) = (ehij (vh )) = ( 21 (
∂vh ∂vhi + hj )) ∂ xhj ∂ xi
and E h = (Eih (ψ h )) = (− ∂ψh ) are the deformation operator and mih
∂ xi
nus the gradient operator. The constitutive equations are (see (2.3)):
σihj (uh , ϕ h ) = Cihjkl ehkl (uh ) − Pkih j Ekh (ϕ h ), h h ekl (uh ) + εihj E hj (ϕ h ), Dhi (uh , ϕ h ) = Pikl
in h , in h ,
where C h = (Cihjkl ), P h = (Pkih j ) and εh = (εihj ) are the stiffness, piezoelectric and dielectric tensors, respectively. In Section 3, we perform a change of variable and a scaling of the unknowns (to obtain the displacements field u(h) and the electric potential ϕ (h)) and data, so that the virtual work formulation is now posed in a domain independent on h, denoted by . In Section 4, we assume that the scaled unknowns can be written in the form of asymptotic developments, allowing to identify and characterize the corresponding leading terms; this procedure yields the new reduced model, which consists on a set of 1D variational problems for the displacements field and a variational problem for the electric potential and the first order term of the displacement, where some terms are not local. The strong convergence of the unknowns as the small parameter tends to zero is proved in Section 5. This is a key step in order to give a rigorous mathematical justification of the reduced models of Section 4. In Section 6 we undo the change of variable and de-scale the quantities appearing in the reduced model and constitutive equations derived in Section 4 in order to express both in the original reference
295
domain h and thus have a true physical meaning. We find that the asymptotic expansion of the true displacements field uh = (uhi ) is of the form:
uhα = ξαh + u1h α + O ( h ),
2 uh3 (xh ) = ξ3h − xhβ ∂3h ξβh + u1h 3 + O ( h ),
where the leading term u0h = (u0h ) is characterized by functions ξαh ∈ i 2 h 1 H0 (0, L ), ξ3 ∈ H0 (0, L ), satisfying the purely mechanical stretching and beam equations (see (6.5) and (6.6)):
ξαh ∈ H02 (0, L ), =
L 0
L 0
Fαh χαh dxh3 −
h h h Y Iαh ∂33 ξα ∂33 χαh dxh3
L 0
Mαh ∂3h χαh dxh3 ,
(no sum on α ),
χαh ∈ H02 (0, L ),
for all
ξ3h ∈ H01 (0, L ), =
L 0
L 0
YA(ωh )∂3h ξ3h ∂3h χ3h dxh3
F3h χ3h dxh3 , for all χ3h ∈ H01 (0, L ).
On the other hand, u1h = (u1h ), the following order term of uh , is of i the following form: h h h u1h α = zα + δα z ,
h h h h h h u1h 3 = −r − z3 − xα (zα ) − w (z ) ,
where wh ∈ R() is the warping function, depending only on the geh , zh ∈ H 1 (0, L ), zh ∈ L2 (0, L ), and ometry and stiffness tensor, while zα 0 3 h r ∈ R() is a function depending on the electric potential, thus giving the coupling between the mechanical and electrical quantities. More specifically, we find that the asymptotic expansion of the true electric potential ϕ h is
ϕ h = ϕˆ 0h + ϕ¯ 0h + O(h2 ), where ϕ¯ 0h ∈ l0 () and the coupled equations for zh , rh and ϕ¯ 0h are given by (see (6.8)–(6.10)):
∂αh wh − δαh ∂βh ϕ¯ 0h dxˆh ∂3h ζ h dx3 ωh 0 0 L h = Pβh3α ∂βh ϕˆ h ∂αh wh − δαh dxˆ ∂3h ζ h dx3 ∀ζ h ∈ H01 (0, L ), ωh 0 L h C3hα 3β ∂βh rh ∂αh ρh dxˆ ρ3h dx3 ωh 0 L h − Pβh3α ∂β ϕ¯ 0h ∂α ρh dxˆ ρ3h dx3 ωh 0 L h = Pβh3α ∂β ϕˆ h ∂α ρh dxˆ ρ3h dx3 ∀ρh ∈ Q h (ωh ),
L
Jh ∂3h zh ∂3h ζ dx3 −
L
Pβh3α
ωh
0
× ρ3h ∈ L2 (0, L ), L h εαβ ∂βh ϕ¯ 0h ∂αh ψh dxˆh ψ3h dx3 0
ωh L
ψ3h dx3 L h + Pβh3α ∂αh wh − δαh ∂βh ψh dxˆ ∂3h zh ψ3h dx3 ωh 0 L h =− εαβ ∂βh ϕˆ h ∂αh ψh dxˆh ψ3h dx3 ∀ψh ∈ S(ωh ), +
0
0
Pβh3α ∂αh rh ∂βh ψh dxˆ
h
ωh
ωh
× ψ3h ∈ L2 (0, L ). h and zh , these Notice that even though we know the existence of zα 3 functions are not characterized at this stage. Nevertheless, they are 0h not needed in order to obtain ϕ , which depends only on u1h through rh . We then briefly comment on some particular cases, specifically when the cross-section is square, which leads to some practical simplifications, and we finish by giving some conclusions.
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Table 1 List of relevant sets.
Table 3 List of scaled functions.
Notation
Description
ω
Domain in R2 with boundary γ , Reference configuration of the scaled beam/rod defined as = ω × (0, L ), ¯, A point in Cross section of the beam/rod defined as hω, with boundary γ h , Reference configuration of the beam/rod defined as h = ωh × (0, L ), Boundary of h , that is h = ∂ h = [0, L] × γ h , ¯ h, A point in Left end of the rod, that is 0h = ω ¯ h × { 0} Right end of the rod, that is Lh = ω ¯ h × {L} Part of h where the electric potential is given, h Part of h which is complementary of eD , Part of h with a condition of weak clampling. Dh = 0h ∪ Lh , Part of h where tractions are given.
x = ( xi )
ωh h h
xh = (xhi )
0h Lh h eD h eN Dh Nh
Table 2 List of functions. Description
nh = (nhi ) : h → R3 h ¯ h → R3 f = ( fih ) : gh = (ghi ) : ¯ Nh → R3 ¯ h → R3 uh = (uhi ) : ϕ h : ¯ h → R h ϕˆ h : eD →R
Unit outer normal vector to h , Density of volume forces, Density of tractions, Displacements field, Electric potential, h Electric potential given in eD and its trace lifting to H1 (h ), h Homogeneous part of ϕ , therefore given as ϕ h − ϕˆ h , Stress tensor,
h C¯ , Mh , Nh
δα : ω → R
¯ →R u(h ) = (ui (h )) :
Description 3
ϕ¯ (h ) : ¯ → R ϕˆ : eD → R ϕ (h ) : ¯ → R σ (h ) = (σi j (h )) : → S3 D(h ) = (Di (h )) : → R3
e(u(h )) = (ei j (h )) : → S3
Notation
ϕ¯ h : ¯ h → R σ h = (σihj ) : h → S3 Dh = (Dhi ) : h → R3 eh (uh ) = (ehij ) : h → S3 E (ϕ h ) = (Eih ) : h → R3 C h = (Cihjkl ) P h = (Pkih j ) εh = (εihj )
Notation
Electric displacement, Linearized strain tensor, Electric field, Stiffness tensor, Piezoelectric tensor, Dielectric tensor, Auxiliary matrices defined from components of Ch in (2.7)–(2.8), Auxiliary functions δ1 (x1 , x2 ) = x2 , δ2 (x1 , x2 ) = −x1 .
For the sake of readability we have also included Tables 1–5. where we list the main functions, constants, sets and functional spaces involved in this paper. 2. The three-dimensional problem Throughout this paper, Latin indices vary over the set {1, 2, 3} and Greek indices over the set {1, 2} for the components of both vectors and tensors. The summation convention on repeated indices will be also used. We denote by Sd the space of second order symmetric tensors on Rd , while “ · ” will represent the inner product on Sd and Rd (d = 2, 3, . . .). Let ω be an open, bounded, connected subset of R2 with Lipschitz continuous boundary γ = ∂ω, having area A(ω). We will study a prismatic rod that we identify with a solid occupying the reference configuration
¯h=ω ¯ h × [0, L], where h ∈ (0, 1] is very small with respect to the length of the rod L and ωh = hω so that A(ωh ) = h2 A(ω ) is the area of the rod cross section ωh , with boundary γ h = ∂ωh . Denoting the boundary of h by h = ∂ h , we consider two partitions for ∂ h : firstly, let us define a partition given by the h and h = γ h × (0, L ), where γ h ⊂ γ h and measurable parts eN eD eD eD h meas(γeD ) > 0; secondly, we consider a partition given by the measurable parts Nh and Dh = 0h ∪ Lh such that meas(Dh ) > 0, where 0h = ω¯ h × {0} and Lh = ω¯ h × {L} are the ends of the rod. An arbitrary
E (ϕ (h )) = (Ei (h )) : → R3
Scaled displacements, with uα (h )(x ) = huhα (xh ) and u3 (h )(x ) = uh3 (xh ); u(h) ∈ V(), Scaled homogeneous electric potential, with ϕ¯ (h )(x ) = h−1 ϕ¯ h (xh ); ϕ¯ (h ) ∈ (), Scaled electric potential in eD and its trace lifting to H1 (); Also, ϕˆ h (xh ) = hϕˆ (x ), Scaled electric potential, ϕ (h ) = ϕ¯ (h ) + ϕˆ , Scaled stress tensor, with h σαβ (xh ) = h2 σαβ (h )(x ), h σ3hα (xh ) = hσ3α (h )(x ), and σ33 (xh ) = σ33 (h )(x ), Scaled electric displacement, with Dhα (xh ) = Dα (h )(x ), and Dh3 (xh ) = h−1 D3 (h )(x ), Scaled linearized strain tensor, with ehαβ (xh ) = h−2 eαβ (h )(x ), eh3β (xh ) = h−1 e3β (h )(x ), and
eh33 (xh ) = e33 (h )(x ), Scaled electric field, with Eαh (ϕ h )(xh ) = Eα (ϕ (h ))(x ), and E3h (ψ h )(xh ) = hE3 (ψ (h ))(x ).
Table 4 List of functions and constants in the scaled domain . Notation
Description
¯ → R3 f = ( fi ) :
Density of scaled volume forces, with fαh (xh ) = h fα (x ), and f3h (xh ) = f3 (x ), Density of scaled tractions, with ghα (xh ) = h2 gα (x ), and gh3 (xh ) = hg3 (x ), Scaled stiffness tensor zero-th order term, Scaled stiffness tensor first order term, Scaled piezoelectric tensor, Scaled dielectric tensor, Auxiliary matrices defined from components of C and C , Terms of the asymptotic expansion of u(h), Terms of the asymptotic expansion of ϕ¯ (h ), Terms of the asymptotic expansion of σ (h), Terms of the asymptotic expansion of D(h), Components in the characterization of u0 ∈ VBN (), For a.e. x3 ∈ (0, L), first two components of u2 , Auxiliary matrices defined from components of C,
g = (gi ) : ¯ N → R3
C = (Ci jkl ) C = (Cijkl ) P = (Pki j ) ε = (εi j ) ¯ M, N, M , N C, u p : → R3 , p ≥ 0 ϕ¯ p : → R, p ≥ 0 σ p : → S3 , p ≥ −4 D p : → R3 , p ≥ −1 ξ = (ξi ) : (0, L ) → R3 uˆ (x3 ) = (uˆ2α (x3 )) : ω → R2 2
Mαβ
α : → R αβ : → R Iα Xαβ , Yα , Z : (0, L ) → R s, sα : (0, L ) → R
Y : (0, L ) → R Fi , Mα : (0, L ) → R P¯ : → R w: (0, L) → R() J : (0, L ) → R zα , z : (0, L ) → R
r : (0, L ) → Q (ω ), z3 : (0, L ) → R C¯ = (C¯i jkl ), P¯ = (P¯ki j ), ε¯ = (ε¯i j )
det M
Functions defined as α (x ) = − det Mαβ xβ , Functions defined using Mαβ , Inertia moment Iα = ω x2α dω, Functions Xαβ = ω αβ dω, Yα = ω αβ δβ dω, Z = ω α δα dω, 1 Functions sα = − X ξ , A(ω ) αβ β 1 Y ξ + Z ξ3 , s=− I1 + I2 β β ¯ C Function Y = detdet , M det N Resultants of forces and moments, det M αβ Function P¯ = P333 − P3αβ , det M
Warping function, Torsion function, Functions in H01 (0, L ) characterizing (u1α ), see (4.49), Functions which characterize u13 , see (4.56), Inverse constitutive tensors of C, P and ε.
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
where C h = (Cihjkl ), P h = (Pkih j ) and εh = (εihj ) are the stiffness, piezo-
Table 5 List of relevant functional spaces. Notation
electric and dielectric tensors, respectively. We assume that Cihjkl , Pkih j ,
Description
V h (h ) = {vh ∈ H 1 (h )
3 h : v = 0 on Dh }
h h (h ) = {ψ h ∈ H 1 (h ) : ψ h = 0 on eD }
V () = {v ∈ [H 1 ()]3 : v = 0 on D }
() = {ψ ∈ H () : ψ = 0 on eD } 1
ˆ () = ϕˆ + ()
V BN () = v ∈ V () : eαβ (v ) = e3β (v ) = 0
2 V 2m (ω ) = vˆ = (vˆ α ) ∈ H 1 (ω ) :
ˆ α dω = ω (x1 vˆ 2 − x2 vˆ 1 ) dω = 0 ωv
V 1 () = v ∈ V () : eαβ (v ) = 0
Subspace of transversal rigid displacements,
εihj ∈ L∞ (h ), implying the boundedness of these tensors, and also
Space of admissible displacements, Space of admissible electric potentials, Space of scaled admissible displacements, Space of scaled admissible electric potentials, Set consisting in {ψ ∈ () : ψ = ϕˆ on eD }, Space of Bernoulli-Navier displacements, Subspace where 2D Korn inequality holds,
Q = Q (ω ) = ρ ∈ H 1 (ω ) : ω ρ d ω = 0 , S = S (ω ) = {ψ ∈ H 1 (ω ) : ψ = 0 on γeD }, l () = {ψ ∈ L2 () : ∂α ψ ∈ L2 (ω )}
Space isomorph to L2 (0, L; H1 (ω)),
R() = L2 (0, L; Q (ω )), l0 () = L2 (0, L; S(ω )), T () = l0 () × R() × H01 (0, L ) X () = V () × (),
Product space, We also define the set ˆ , Xˆ () = V () × ()
ωh
xhα dωh = 0,
ωh
xh1 xh2 dωh = 0.
σ (u , ϕ ) = f , −div Dh (uh , ϕ h ) = 0, −div
h
h
∂ h /∂ xhi
in ,
(2.2)
in , h
= (σihj ) and the electric displacement vector Dh = related to the linear strain tensor (ehij (uh ) = 12 (∂ih uhj + the gradient of the electric potential (Ekh (ϕ h ) = −∂kh ϕ h )
σihj (uh , ϕ h ) = Cihjkl ehkl (uh ) − Pkih j Ekh (ϕ h ), h h ekl (uh ) + εihj E hj (ϕ h ), Dhi (uh , ϕ h ) = Pikl
h C1111
h C1211
h C1122
0
0
h C1133
h C1212
h C1222
0
0
h C1222
h C2222
0
0
0
0
h C3131
h C3132
0
0
h C3132
h C3232
0
h C1233
h C2233
0
0
h C3333
⎞
h ⎟ C1233 ⎟
⎟
h C2233 ⎟ ⎟, 0 ⎟
⎟ ⎠
⎛
h C1111
h C1211
h C1122
⎞
h M h = M h (xh ) := ⎝C1211
h C1212
h ⎠, C1222
h C1122
h C1222
h C2222
N h = N h (xh ) :=
h C3131
h C3132
h C3132
h C3232
.
(2.8)
From properties (2.4) it follows that these matrices are definite positive and therefore det C h > 0, det M h > 0 and det N h > 0. For the moment, and until not explicited otherwise, we will not assume that the conditions traduced by (2.6) hold. We suppose that the rod is weakly clamped at the extremities 0h ∪ Lh , which means that the displacement field uh is such that (see e.g. Trabucho and Viaño, 1996)
ah
uhi dωh = 0,
ah
xhj uhi − xhi uhj dωh = 0,
a = 0, L.
(2.9)
In the following we will represent this boundary condition by uh =
h
through the constitutive law (see e.g. Ikeda, 1990; Jackson, 1975):
ε3hθ = 0, in ¯ h .
(2.7)
the stress tensor σ h
(Dhi ) being ∂ hj uhi )) and
(2.5)
We will assume that the rod is made of piezoelectric materials of symmetry class 2, which comprises the following crystal systems (see e.g. Nye, 1985; Royer and Dieulesaint, 2000): monoclinic (except class ¯ ), m), orthorhombic, tetragonal, hexagonal (except classes 6¯ and 6m2 and cubic. For this class of materials, which includes among many other piezoelectric materials Barium Titanate (BT), Lead Zirconium Titanate (PZT) and Polyvinylidene Fluoride (PVDF), the following conditions hold:
h C1133
rium equations (see e.g. Ciarlet, 1988; Jackson, 1975): h
(2.4)
h h ¯ h. εihj = ε hji , Pmi in j = Pm ji ,
h h Cihjkl = Ckli j = C jikl ,
⎜C h ⎜ 1211 ⎜C h h h ⎜ h C¯ = C¯ (x ) := ⎜ 1122 ⎜ 0 ⎜ ⎝ 0
i
h
( ξi ) 2 ,
i=1
¯ h . Furthermore, the following symmetries hold: for every xh ∈
⎛
... derivative with respect to the (only) variable x3 (or xh3 ). ¯ h is a perfect dielectric body, so that there Now, assuming that is neither volume nor surface electric charge, and that volume forces of density f h = ( fih ) act in h , the resulting displacement field uh = ¯ h → R3 and electric potential ϕ h : ¯ h → R solve the equilib( uh ) : h
3
(2.6)
We define the following differential operators: = and ∂i = ∂ /∂ xi . For a function χ : [0, L] → R we note χ , χ , ... its first, second,
h
εihj (xh )ξi ξ j ≥ c
(Mi j )2 ,
i, j=1
(2.1)
∂ih
3
Cihjkl (xh )Mi j Mkl ≥ c
Accordingly, it is useful to define the following symmetric matrices:
¯ h will be denoted by xh = (xh ) and the unit outer norpoint of i mal vector to the boundary ∂ h by nh = (nhi ). The coordinate system Oxh1 xh2 xh3 will be assumed a principal system of inertia associated to ωh , which means that
the following coercive relations: there exists a constant c > 0 such that for all symmetric second order tensors M = (Mi j ) ∈ S3 and every vector ξ = (ξi ) ∈ R3 one has
h h C3hρ 33 = C3hθ αβ = 0, Pθhρσ = P33 α = Pβ 33 = 0,
M 1 () = τ = (τi j ) ∈ [L2 ()]3×3 : τi j = τ ji , M 2 () = [L2 ()]3 , M () = M 1 () × M 2 (), V RD () = {v ∈ V 1 () : v3 ∈ L2 (0, L; Q (ω ))}, V R () = {v ∈ V RD () : vα = δα s, s ∈ H01 (0, L )}, Subspace of admissible rotations, Subspace of admissible V T () = {v ∈ V RD () : vα = sα , translations, v3 = −xα sα , sα ∈ H01 (0, L ) , V˙ RD () := V RD ()/V T (), Quotient space, V H () = H 1 (0, L; V 2m (ω )).
297
in h , in h ,
(2.3)
0 on Dh . Remark 2.1. As we will see, the use of the “ average clamping” condition (2.9) instead of a strong clamping condition such as uh = 0 on Dh allows to avoid the “ boundary layer problem” (see Ref. Lions, 1973) that arises when a strong clamping condition is used - which turns out to be related to the fact that in general a strong clamping of the rod is physically impossible (see e.g. Trabucho and Viaño, 1996 and references therein).
Nh
We also assume that surface tractions of density gh = (ghi ) act on h . In the following and that an electric potential ϕˆ h is applied on eD
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h ) are we suppose that fih = L2 (h ), ghi ∈ L2 (Nh ), and ϕˆ h ∈ H 1/2 (eD given quantities. Thus, the boundary conditions satisfied by (uh , ϕ h ) are:
⎧ σ h (uh , ϕ h )nh = gh , ⎪ ⎪ h ⎪ ⎨ u = 0,
⎪ D (u , ϕ ) · n = 0, ⎪ ⎪ ⎩ ϕ h = ϕˆ h , h
h
h
h
on Nh , on Dh ,
(2.10)
h on eN , h on eD .
Let us now define the following closed subspaces of the Sobolev space of order one [H1 (h )]m , m ≥ 1:
h v ∈ [H 1 (h )]3 : vh = 0 on Dh ,
h h (h ) := ψ h ∈ H 1 (h ) : ψ h = 0 on eD .
V h (h ) :=
(2.11)
h ) > 0, it follows from Korn’s Since meas(Dh ) > 0 and meas(eD and Poincaré–Friedrichs’ inequalities that the following norms are equivalent to the standard Sobolev · 1,h norms on Vh and h , respectively,
vh ∈ V h (h ) → |eh (vh )|0,h , ψ h ∈ h (h ) → |E h (ψ h )|0,h , where | · |0,h stands for the usual norm in [L2 (h )]m , m ≥ 1.
ϕ¯ h
ϕh
− ϕˆ h ,
ϕˆ h
small parameter h as usually done in the elastic rod case (see e.g. Bermúdez and Viaño, 1984b; Trabucho and Viaño, 1996 and refer- ences therein). We will study the dependence of the solution uh , ϕ¯ h with respect to h. The technique of change of variable to a fixed domain and subsequent rescaling of the displacement and electric potential will allow us to derive variational problems equivalent to (2.12) and (2.13) where h shows up in an explicit way in the rescaled equations. To start, we perform a change of variable to the reference domain through the following transformation (see e.g. Trabucho and Viaño, 1996)
¯ h : x = (x1 , x2 , x3 ) ∈ h h h h h ¯ h. → x = (x ) = x1 , x2 , x3 = (hx1 , hx2 , x3 ) ∈
(3.1)
Moreover,
Nh = h (N ), Dh = h (D ), 0h = h (0 ), Lh = h (L ), h = h (eN ), n(x ) = nh (xh ), eN
h h γeD = h (γeD ), eD = h (eD ),
where n is the unit outer normal to the set ∂ = D ∪ N = eD ∪ eN and
eD = γeD × (0, L ).
Defining = where now represents a trace lifting h , the variational in H1 (h ) of the boundary potential acting on eD problem associated to (2.2), (2.3) and (2.10 ) is
Furthermore, condition (2.1) becomes now
Find (uh , ϕ¯ h ) ∈ V h (h ) × h (h ) such that × Cihjkl ehkl (uh )ehij (vh )dxh + εihj ∂ih ϕ¯ h ∂ hj ψ h dxh h h h h h h h + Pmi ¯ ei j (vh ) − ehij (uh )∂m ψ h )dxh j ( ∂m ϕ h = fih vhi dxh + ghi vhi d h − εihj ∂ih ϕˆ h ∂ hj ψ h dxh
In view of (2.9) we will represent by v = 0 on D the boundary condition
−
h
h N
h
h h h h Pmi ˆ ei j (vh )dxh , j ∂m ϕ
a
a
Remark 2.2. The derivation of (2.12) follows closely the steps presented in Bernadou and Haenel (1999) for the strong clamping case with minor changes to accommodate the weak clamping condition adopted here.
h Find (uh , ϕ h ) ∈ V h (h ) × H 1 (h ), such that ϕ h = ϕˆ h on eD , σihj (uh , ϕ h )ehij (vh )dxh + Dhk (uh , ϕ h )Ekh (ψ h )dxh × h h = fih vhi dxh + ghi vhi d h ,
Nh
× for all (v , ψ ) ∈ V h (h ) × h (h ), with the functions
σihj , Dhk
ω
x1 x2 dω = 0.
vi dω = 0,
a
vi dω = 0,
a
(3.2)
(x j vi − xi v j )dω = 0,
δα vα dω = 0,
a
a = 0, L,
xα v3 dω = 0,
a = 0, L,
where δ1 (x1 , x2 ) = x2 , δ2 (x1 , x2 ) = −x1 . We now define the following closed subspaces of [H1 ()]m , m ≥ 1 (cf. (2.11))
V () = {v ∈ [H 1 ()]3 : v = 0 on D },
() = {ψ ∈ H 1 () : ψ = 0 on eD }, endowed with the following norms equivalent to the usual Sobolev norms:
||v||V = |e(v )|0, , ||ψ|| = |∇ψ|0, , where | · |0, stands for the standard norm in [L2 ()]m , m ≥ 1, and
e(v ) = (ei j (v )),
Proposition 2.3. Problem (2.12) is formally equivalent to
h
(3.3)
for all (vh , ψ h ) ∈ V h (h )
It can be easily shown that the boundedness of Ch , Ph , εh , fh and gh together with properties (2.4) ensure the existence and uniqueness of solution of problem (2.12 ) as a result of Lax–Milgram Lemma.
h
xα dω = 0,
that is, taking (3.2) into account,
(2.12)
h
ω
h
× h (h ).
(2.13)
: → R defined as in (2.3).
The above formulation corresponds to the principle of virtual work. 3. Change of variable to the reference rod The major geometric feature of a three-dimensional rod is the fact that the largest cross sectional dimension is very small compared with its length (h L), cause of ill-conditioning of the threedimensional problem. We take advantage of this property to use an asymptotic expansion method (see Lions, 1973) with respect to the
ei j ( v ) =
1 (∂i v j + ∂ j vi ). 2
In order to obtain a problem in equivalent to (2.12) we proceed as follows (see e.g. Viaño et al., 2005; Viaño et al., 2015) . With the unknown and test displacement fields uh , vh in Vh (h ), we associate the (unknown and test) scaled displacement fields u(h ) = (ui (h )) and v(h ) = (vi (h )) in V() defined by the following scalings valid for all ¯: xh = h (x ), x ∈
uα (h )(x ) = huhα (xh ),
u3 (h )(x ) = uh3 (xh ),
vα (h )(x ) = hvα (x ), v3 (h )(x ) = v (x ). h
h
Similarly, the electric potential ϕ¯ h
h 3
h
and the test function ψ h
(3.4) (3.5) in h (h )
are associated with the scaled potential ϕ¯ (h ) and the scaled (test) function ψ (h) in () using the following scaling valid for all xh = ¯: h (x ), x ∈
ϕ¯ (h )(x ) = h−1 ϕ¯ h (xh ),
(3.6)
ψ (h )(x ) = h−1 ψ h (xh ).
(3.7)
Furthermore, we consider the following hypothesis on the magnitude of the data with respect to the diameter of the rod cross-section h:
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
1. There exist functions fi ∈ L2 () and gi ∈ L2 ( N ), independent of h, such that:
fαh (xh ) = h fα (x ), f3h (xh ) = f3 (x ), for all xh = h (x ) ∈ h , ghα (xh ) = h2 gα (x ), gh3 (xh ) = hg3 (x ), for all xh = h (x ) ∈ Nh .
(3.8) 2. There exists a function ϕˆ ∈ H 1/2 (eD ), independent of h, such that: h ϕˆ h (xh ) = hϕˆ (x ), for all xh = h (x ) ∈ eD .
We also denote by ϕˆ ∈
ϕ (h ) = ϕ¯ (h ) + ϕˆ .
H 1 ()
(3.9)
a trace lifting of ϕˆ and define
In addition to that, it is assumed that there exist functions Cijkl ∈
L∞ (0, L), and Cijkl , Pkij , ε ij ∈ L∞ (), independent of h, such that:
Cihjkl (xh1 , xh2 , xh3 ) = Ci jkl (x3 ) + hCijkl (x1 , x2 , x3 ), Pkih j (xh ) = Pki j (x ),
εihj (xh ) = εi j (x ),
(3.10)
¯ . We also assume that the for all xh = h (x ), x = (x1 , x2 , x3 ) ∈ properties (2.4)–(2.6) hold when we drop the superindex (take h = 1) and for Cijkl as well. Consequently, there exist tensor fields M, N, M and N such that
Mh (xh1 , xh2 , xh3 ) = M(x3 ) + hM (x1 , x2 , x3 ), Nh (xh ) = N(x3 ) + hN (x1 , x2 , x3 ), xh
h (x )
= where of h (see (2.8)).
(3.11)
¯ h, x ∈ ¯ , with M, N, M and N independent ∈
Remark 3.1. The Ansatz elastic tensor (3.10) includes the material’s inhomogeneity effects in the piezoelectric response predominantly given by the leading term. However, the elastic tensor also accounts 3D structural parameters (such as grain size for PZT’s, see e.g. Naceur et al., 2014; Su, 1997) effects on the piezoelectric response, through the perturbation introduced by the small parameter h. Note that other authors (e.g. Figueiredo and Leal, 2006) have assumed that there exist functions Cijkl independent of h, such that:
Cihjkl
(x ) = Ci jkl (x ) ∀x = (x ), h
h
h
¯. x∈
(3.12)
We shall further comment on the implications of these two different assumptions (see Remark 4.8 and Ref. Viaño et al., 2015). Combining transformation (3.1) with the notations (3.5) and (3.7) we have for all vh ∈ Vh (h ), ψ h ∈ h (h ) and xh = h (x ), x ∈ :
⎧ h h h −2 ⎨eαβ (v )(x ) = h eαβ (v(h ))(x ), eh (vh )(xh ) = h−1 e3β (v(h ))(x ), ⎩ 3hβ h h e33 (v )(x ) = e33 (v(h ))(x ),
(3.13)
Eαh (ψ h )(xh ) = −∂αh ψ h (xh ) = −∂α (ψ (h ))(x ) = Eα (ψ (h ))(x ),
E3h (ψ h )(xh ) = −∂3h ψ h (xh ) = −h∂3 (ψ (h ))(x ) = hE3 (ψ (h ))(x ). (3.14)
Using the scalings defined previously for the displacement vector and for the electric potential together with the above assumptions and results, we can reformulate the variational problem (2.12) into ¯ independent of h. Assuming another variational problem posed in from here on that we are dealing with piezoelectric materials of symmetry class 2 (i.e. properties (2.6) hold), we have the following result. Proposition 3.2. Let (u(h ), ϕ¯ (h )) ∈ V () × () be the functions obtained from (uh , ϕ¯ h ) ∈ V h (h ) × h (h ), the unique solution of problem (2.12), by transformation (3.1) and scalings (3.4) and (3.6). Bearing in mind (2.6) as well as scalings (3.5) and (3.7)–(3.13),
299
then (u(h ), ϕ¯ (h )) is the unique solution of the following variational problem:
(u(h ), ϕ¯ (h )) ∈ V () × (),
aˆ(h )((u(h ), ϕ¯ (h )), (v, ψ ))
= lˆ(h )(v, ψ ) ∀(v, ψ ) ∈ V () × (),
(3.15)
where the bilinear form aˆ(h ) and the linear functional lˆ(h ) are defined, respectively, by
h−4Cαβθ ρ + h−3Cαβθ aˆ(h )((u, ϕ¯ ), (v, ψ )) = ρ eθ ρ (u )eαβ (v )dx + 4h−2C3α 3β + 4h−1C3α 3β e3α (u )e3β (v )dx + h−2Cαβ 33 + h−1Cαβ e33 (u )eαβ (v ) + eαβ (u )e33 (v ) dx 33 −1 P3αβ ∂3 ϕ¯ eαβ (v ) − eαβ (u )∂3 ψ dx +h −1 +2h Pβ 3α ∂β ϕ¯ e3α (v ) − e3α (u )∂β ψ dx C3333 + hC3333 e33 (u )e33 (v )dx + + εαβ ∂α ϕ∂ ¯ β ψ dx + h P333 (∂3 ϕ¯ e33 (v ) − e33 (u )∂3 ψ )dx +h2 ε33 ∂3 ϕ∂ ¯ 3 ψ dx (3.16)
and
lˆ(h )((v, ψ )) = −2h−1 Pβ 3α ∂β ϕˆ e3α (v )dx + fi vi dx + gi vi d N − εαβ ∂α ϕ∂ ˆ β ψ dx − h P333 ∂3 ϕˆ e33 (v )dx − h2 ε33 ∂3 ϕ∂ ˆ 3 ψ dx. (3.17)
The existence and uniqueness of solution of problem (3.15)–(3.17) is consequence of Lax–Milgram Lemma, since aˆ(h ) is (V × )-elliptic and both aˆ(h ) and lˆ(h ) are (V × )-continuous, as the reader can easily check having in mind properties (2.4), the boundness of C, C , P, ε, f and g, and that 0 < h ≤ 1. We now consider the following scalings for the stress tensor σ (h ) = (σi j (h )) and the electric displacement ¯, vector D(h ) = (Di (h )), valid for all xh = h (x ), x ∈
⎧ h h 2 h h ⎪ ⎨σαβ (x ) = h σαβ (h )(x ), σ3α (x ) = hσ3α (h )(x ), h h σ33 (x ) = σ33 (h )(x ), ⎪ ⎩Dh (xh ) = D (h )(x ), Dh (xh ) = h−1 D (h )(x ). α 3 α 3
(3.18)
From transformation (3.1) and scalings (3.13)–(3.14) and (3.18), the variational problem (2.13) and the constitutive law (2.3) allow to deduce the following scaled principle of virtual work and scaled consti¯. tutive law posed in Proposition 3.3. Problem (3.15)–(3.17) is formally equivalent to
Find (u(h ), ϕ (h )) ∈ V () × H 1 (), such that ϕ (h ) = ϕˆ on eD , σi j (h )ei j (v )dx + Dk (h )Ek (ψ )dx = fi vi dx + gi vi d ,
for all (v, ψ ) ∈ V () × (),
N
(3.19)
300
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
where σ (h ) = σ (h )(u(h ), ϕ (h )) and D(h ) = D(h )(u(h ), ϕ (h )) are given by
⎧ σαβ (h ) = h−4Cαβθ ρ + h−3Cαβθ ⎪ ρ eθ ρ (u (h )) ⎪ ⎪ ⎪ ⎪ + h−2Cαβ 33 + h−1Cαβ e (u(h )) − h−1 P3αβ E3 (ϕ (h )), ⎪ 33 33 ⎪ ⎪ ⎪ −2 −1 −1 ⎪ ⎪ ⎨σ3α (h ) = 2 h C3α3β + h C3α3β e3β (u(h )) − h Pβ 3α Eβ (ϕ (h )), σ (h ) = h−2C33αβ + h−1C33 αβ eαβ (u(h )) ⎪ 33 ⎪ ⎪ ⎪ + C3333 + hC3333 e33 (u(h )) − hP333 E3 (ϕ (h )), ⎪ ⎪ ⎪ ⎪ ⎪ Dα (h ) = 2h−1 Pα 3β e3β (u(h )) + εαβ Eβ (ϕ (h )), ⎪ ⎪ ⎩ D3 (h ) = h−1 P3αβ eαβ (u(h )) + hP333 e33 (u(h )) + h2 ε33 E3 (ϕ (h )). (3.20)
−2 σ33 = C33αβ eαβ (u0 ), −1 0 σ33 = C33αβ eαβ (u1 ) + C33 αβ eαβ (u ),
(4.13)
0 1 0 σ33 = C33αβ eαβ (u2 ) + C33 αβ eαβ (u ) + C3333 e33 (u ), 0 D−1 α = 2Pα 3β e3β (u ),
D0α = 2Pα 3β e3β (u1 ) + εαβ Eβ (ϕ 0 ),
(4.15)
0 D−1 3 = P3αβ eαβ (u ),
D03 = P3αβ eαβ (u1 ),
(4.16)
D13 = P3αβ eαβ (u2 ) + P333 e33 (u0 ), D23 = P3αβ eαβ (u4 ) + P333 e33 (u3 ) + ε33 (ϕ 1 ).
4. Asymptotic method
(4.14)
(4.17)
We now assume that the solution of problem (3.15)–(3.17) can be expressed in the form
4.1. Characterization of u0 and u2
u(h ) = u0 + hu1 + h2 u2 + · · ·,
Let VBN () be the space of Bernoulli–Navier displacements defined by
ui ∈ V (),
(4.1)
ϕ¯ (h ) = ϕ¯ 0 + hϕ¯ 1 + h2 ϕ¯ 2 + · · ·, ϕ¯ i ∈ (),
(4.2)
where the successive coefficients of the powers of h are independent of h . Since ϕ¯ (h ) = ϕ (h ) − ϕˆ , then (4.2) leads to
ϕ (h ) = ϕ 0 + hϕ¯ 1 + h2 ϕ¯ 2 + · · · ,
(4.3)
with ϕ 0 = ϕ¯ 0 + ϕˆ . In view of (3.20), the asymptotic developments (4.1) and (4.3) induce the following formal expansions for tensors σ (h) and D(h)
σ (h ) = h σ + h σ + h σ + · · ·, D(h ) = h−1 D−1 + D0 + hD1 + · · · , −4
−4
−3
−3
−2
−2
(4.4)
with σ q = (σ i j ) and D p = (D j ) independent of h. Inserting developments (4.1)–(4.4) into problem (3.19) results in a set of variational equations that must be satisfied for all h > 0 and consequently the terms at the successive powers of h must be zero. This procedure yields the following problems at the successive powers of h for all (v, ψ ) ∈ V() × (): q
equipped with the norm |e(v)|0, . The space VBN () can be equivalently defined by (see e.g. Trabucho and Viaño, 1996)
V BN () =
{v = (vi ) ∈ V () : vα (x1 , x2 , x3 ) = χα (x3 ), χα ∈ H02 (0, L ), v3 (x1 , x2 , x3 ) = χ3 (x3 ) − xβ χβ (x3 ), χ3 ∈ H01 (0, L )}.
(4.18)
Theorem 4.1. Let u(h) be given by (4.1). The zeroth order term u0 is an element of the space of Bernoulli–Navier displacements:
u0α (x1 , x2 , x3 ) = ξα (x3 ), u03 (x1 , x2 , x3 ) = ξ3 (x3 ) − xβ ξβ (x3 ),
ξα ∈ H02 (0, L ), ξ3 ∈ H01 (0, L ).
p
Proof. In problem (4.5), we take v =
eαβ (u ) = 0, 0
σαβ = σ −4
−2 33
=
D−1 3
u0 ,
(4.19)
which yields:
= 0.
(4.20)
Applying similar arguments to variational Eq. (4.7), we take v = u0 , yielding:
−4 σαβ eαβ (v )dx = 0,
(4.5)
e3α (u0 ) = 0,
−3 σαβ eαβ (v )dx = 0,
(4.6)
Taking into account that eαβ (u0 ) = e3α (u0 ) = 0 will yield (4.19).
σi−2 ei j (v )dx = 0, j
(4.7)
Next, we go back to variational Eq. (4.6). Combining (4.6) with (4.10), using (4.20) and taking v = u1 we get
σ
−1 ei j ij
(v )dx +
σi0j ei j (v )dx +
D−1 Ek k
(ψ )dx = 0,
D0k Ek (ψ )dx =
(4.8)
fi vi dx +
N
gi vi d ,
(4.9)
(4.22)
Moreover, from (4.10), (4.13) and (4.16) we find that −3 −1 σαβ = σ33 = 0, D03 = 0.
−3 0 σαβ = Cαβθ ρ eθ ρ (u1 ) + Cαβθ ρ eθ ρ ( u ),
(4.10)
σαβ = Cαβθ ρ eθ ρ (u ) + Cαβ 33 e33 (u ) + Cαβθ ρ eθ ρ (u ), 2
0
1
(4.23)
(4.11)
V 2m (ω ) =
= 2C3α 3β e3β (u ), 0
= 2C3α 3β e3β (u ) +
2C3α 3β e3β
(u ) − Pβ 3α Eβ (ϕ ), 0
0
vˆ = (vˆ α ) ∈ [H 1 (ω )]2 :
(4.12)
(4.24)
(Cαβθ ρ eθ ρ (u2 ) + Cαβ 33 e33 (u0 ))eαβ (v )dx = 0, for all v ∈ V ().
Using the same arguments as in Viaño et al. (2015), we define
+ Cαβ e (u0 ) − P3αβ E3 (ϕ 0 ), 33 33
1
−2 σαβ eαβ (v )dx = 0, for all v ∈ V ().
In view of (4.11), (4.20) and (4.22) the previous equality becomes
−1 2 σαβ = Cαβθ ρ eθ ρ (u3 ) + Cαβ 33 e33 (u1 ) + Cαβθ ρ eθ ρ ( u )
−2 3α −1 3α
(4.21)
eαβ (u1 ) = 0.
−4 σαβ = Cαβθ ρ eθ ρ (u0 ),
−2
−1 σ3−2 α = Dα = 0.
From (4.20) and (4.21), Eq. (4.7) can be written
where
σ σ
V BN () = {v ∈ V () : eαβ (v ) = e3β (v ) = 0},
ω
ω
(x1 vˆ 2 − x2 vˆ 1 ) dω = 0 ,
vˆ α dω = 0,
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
which is a subspace of [H1 (ω)]2 where the bi-dimensional Korn’s ˆ ))|0, is a norm in the space inequality holds. Therefore |(eαβ (w L2 (0, L; V 2m (ω )). As a consequence, the problem
⎧ 2 2 ⎪ ⎪Find uˆ (x3 ) ∈ V m (ω ), ⎨
(4.25) has a unique solution for a.e. x3 ∈ (0, L). We now define some functions that will be useful in the following results. We consider
det M αβ det M
α = β ,
=−
where Mαβ (x3 ) are defined as follows:
C1133 C1233 C2233
M 11 =
M 12 = M 21
C1112 C1212 C1222
C1122 C1222 , M 22 = C2222
C1111 1 C1112 = 2 C1122
C1133 C1233 C2233
C1111 C1112 C1122
C1112 C1212 C1222
C1133 C1233 , C2233
C1122 C1222 . C2222
Iα =
ω
αβ dω, Yα =
ω
αβ δβ dω,
Z=
ω
α δα dω.
u ) + Cαβ 33 e33 (u0 ) = 0. Cαβθ ρ eθ ρ (! ! uα = sα + δα s + αβ ξβ + α ξ3 ,
(4.27)
Xαβ ξβ ,
1 s=− Y ξ + Z ξ3 I1 + I2 β β
2
u(x3 )) = −C˜θ ραβ (x3 )Cαβ 33 (x3 )e33 (u0 )(x3 ), in eθ ρ (!
ω
(4.29)
where (C˜θ ραβ ) is defined as follows:
C1111 C˜1211 C˜2211
det M
C˜1122 C˜1222 C˜2222
=
C1111 C1211 C2211
1 2
det M
1 1
3
(4.32) where sα are arbitrary functions depending only on x3 . Combining (4.31) with (4.32) leads to (4.27). Using (3.2) and the system (4.31) we have ω
det M αα A(ω )ξ3 , det M det M αα Iα ξα , (no sum on α ). = det M
∂α ! uα d ω = −
xα ∂α ! uα d ω
ω
(4.33)
uα d ω , ω x α ∂ α ! uα dω, ξα , ξ3 ∈ Since ! uα ∈ L2 (0, L; H 1 (ω )), then ω ∂α ! 2 L (0, L ). On the other hand, identity (4.27) allows to write
a.e. in (0, L ).
Proof. Let uˆ (x3 ) be a particular solution of the equation
2C˜1112 2C˜1212 2C˜2212
det M 11 det M 21 2x1 ξ1 − ξ3 . x1 ξ2 + det M det M
2 det M
(4.28)
˜
det M 22 det M 12 x2 ξ1 + 2x2 ξ2 − ξ3 , det M det M
⎧ ⎪ ⎨k1 (x2 , x3 ) = s1 + x2 s − det M 22 x22 ξ1 + det M 12 x22 ξ2 − x2 ξ3 ,
ω
where s, sα ∈ L2 (0, L) verify
A (ω )
∂2 k1 (x2 , x3 ) = s −
(4.26)
Moreover,
sα = −
Consequently, there exists a function s, depending only on x3 , such that
Theorem 4.2. Let u0 = (u0i ) ∈ V BN () be given by (4.19). Then, there exists ! u = (! uα ) ∈ L2 (0, L; V 2m (ω )) satisfying
1
det M 11 det M 21 x1 ξ2 + 2x1 ξ1 − ξ3 − ∂1 k2 (x1 , x3 ). det M det M
2 det M
and we define, on each cross-section ω × {x3 }, the following functions
Xαβ =
(4.31)
3
⎪ ⎩ k2 (x1 , x3 ) = s2 − x1 s − det M 11 x2 ξ + det M 21 x2 ξ − x1 ξ ,
xα d ω
1
Thus,
2
ω
2 2
∂1 k2 (x1 , x3 ) = −s −
We also introduce the following geometric constant (see e.g. Trabucho and Viaño, 1996, p. 701)
2 det M
det M 22 2 ⎪ ⎩! (x ξ + 2x1 x2 ξ − 2x2 ξ ) + k2 (x1 , x3 ). u2 =
det M 12 det M 22 x2 ξ1 − 2x2 ξ2 − ξ3 + ∂2 k1 (x2 , x3 ) det M det M
1 det M 11 x21 − det M 22 x22 , 2 det M
det M αθ x x , det M θ β
⎧ det M 11 2 ⎪ ⎨! (x1 ξ1 + 2x1 x2 ξ2 − 2x1 ξ3 ) + k1 (x2 , x3 ), u1 =
Substituting these expressions in the third equation of (4.30 ), we obtain
and
αβ (x ) =
(4.30)
and, taking into account that the matrix of coefficients (Cαβθ ρ ) depends only on x3 (see (3.10)), we have, by direct integration of the first two equations,
2 det M
xβ ,
11 (x ) = −22 (x ) =
⎧ det M 11 ⎪ xα ξα − ξ3 , u1 = ∂1 ! ⎪ ⎪ det M ⎪ ⎨ det M 22 ∂2 ! xα ξα − ξ3 , u2 = ⎪ det M ⎪ ⎪ ⎪ det M 12 + det M 21 ⎩∂ ! xα ξα − ξ3 , u1 = 1 u 2 + ∂2 ! det M
(Cαβθ ρ eθ ρ (uˆ2 (x3 )) + Cαβ 33 e33 (u0 )(x3 ))eαβ (wˆ ) dω = 0 ⎪ ⎪ ⎩ ω ∀wˆ ∈ V 2m (ω ),
α (x ) = −
301
ω
! uα dω = A(ω )sα + Xαβ ξβ
a.e. in (0, L ),
δβ ! uβ dω = (I1 + I2 )s + Yθ ξθ + Z ξ3 a.e. in (0, L ),
(4.34) (4.35)
from which the average conditions required for ! uα lead to (4.28). Corollary 4.3. A necessary and sufficient condition for problem (4.26) to have solutions u2 ∈ V() is that u0 ∈ VBN () be such that ξ α ∈ H3 (0, L), ξ 3 ∈ H2 (0, L) and that s, sα ∈ H1 (0, L). In addition to that, it is verified −2 = 0. that σαβ Proof. By using (4.26) in (4.25) we find that
2C1112 2C1212 2C2212
C1122 C1222 C2222
−1
.
It is easy to show that (4.29) is equivalent to the Equation (4.26). Fur2 ther, we verify that uˆ (x3 ) also solves (4.25). On the other hand, as u0 ∈ VBN (), Eq. (4.29) can be written in differential form
ω
(Cαβθ ρ eθ ρ (uˆ2 − u))eαβ (wˆ ) dω = 0 ∀wˆ ∈ V 2m (ω ),
2 and therefore (4.27) is in fact a charallowing to conclude that uˆ = u acterization of u2α , the transverse components of the second order displacement u2 . Now assume that u2 ∈ V(). Let u2α ∈ H 1 (). By relations (4.33)–(4.35) we deduce that ω ∂α u2α dω, ω xα ∂α u2α dω, s,
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sα ∈ H1 (0, L). Furthermore, the boundary condition for u2α (cf. (3.3)) leads to (4.28) for x3 = 0 and x3 = L. Conversely, taking u0 ∈ VBN () such that ξ α ∈ H3 (0, L), ξ 3 ∈ H2 (0, L), and s, sα ∈ H1 (0, L) now of the form (4.28) for all x3 ∈ (0, L), then (4.27) implies that u2α ∈ H 1 () verifies the boundary condition (3.3). Finally, from (4.11), (4.22) and −2 = 0. (4.26) we conclude that σαβ From (4.27) it follows that in general u2
Remark 4.4. α does not vanish at the rod ends, for that would require that both ξβ and ξ3 vanish there. Therefore, if we had chosen strong boundary conditions in the definition of V(), then we would not be able to ensure that u2 ∈ V() (the well-known “ boundary layer phenomenon” ) and consequently −2 = 0. that σαβ As a result of the previous discussion, we will be able to obtain variational problems that have ξ α and ξ 3 as unique solutions. Theorem 4.5. It is verified that u0 ∈ VBN () is the unique solution of the following variational problem:
0 σ33 e33 (v )dx =
fi vi dx +
¯ 33 (u0 ) = P¯ D13 = Pe
N
0 given by with σ33
ξ3 − xα ξα ∈ L2 (),
with
P¯ = P333 −
det M αβ det M
P3αβ .
Remark 4.8. In the case of taking assumption (3.12) instead of (3.10), we can still formulate the problem (4.25) (the coefficients Cαβθ ρ now depending also on x1 , x2 ) which has an unique solution in V 2m (ω ). The difference is that we cannot give explicitly u2 in terms of u0 as in (4.27). Following Theorem 3.1 in Girault and Raviart (1986), we have to introduce two auxiliary functions θ α ∈ H1 (ω) a.e. in (0, L) and 2 eθ ρ (uˆ ) = −C˜θ ραβ (Cαβ 33 e33 (u0 ) + θ ρ ) in
αβ =
∂2 θ 1 ∂2 θ 2
−∂1 θ1 , −∂1 θ2
det C¯ . det M det N
4.2. Characterization of u1 and ϕ 0 (4.38)
Notice that it is positive and bounded (cf. (2.7) and (2.8)). Proof. Problem (4.36) follows from variational Eq. (4.9) taking v ∈ VBN () and ψ = 0. Furthermore, after some calculations (4.14) becomes (4.37).
We start this section by introducing the space V1 () formed by all the infinitesimal rigid displacements of the set in the transversal directions:
V 1 () = {v ∈ V () : eαβ (v ) = 0}. In Trabucho and Viaño (1996) it is proved that V1 () coincides with the following space :
Corollary 4.6. The transverse components ξ α and the stretch component ξ 3 of the zeroth order displacement field solve respectively the following variational problems (no sum on α ):
V 1 () =
Find ξα ∈ H02 (0, L ) such that L L Y Iα ξα χα dx3 = Fα χα dx3
We also consider the spaces
−
(4.39)
0
L 0
Mα χα dx3 ,
for all
χα ∈ H02 (0, L ),
(4.39)
0
0
for all
χ3 ∈ H01 (0, L ),
(4.40)
with the resultant forces and moments given by:
Fi (x3 ) =
ω
f i ( x1 , x2 , x3 ) d ω +
∂ω
xα f 3 ( x1 , x2 , x3 ) d ω ω + xα g3 (x1 , x2 , x3 )dγ ∈ L2 (0, L ). ∂ω
Q = Q (ω ) =
ρ ∈ H 1 (ω ) :
ω
" ρ dω = 0 ,
S = S(ω ) = {ψ ∈ H 1 (ω ) : ψ = 0 on γeD },
(4.44)
equipped with the (semi-)norms
With the aid of Poincaré–Friedrichs’ inequality and Poincaré’s inequality in ω, it can be shown that these norms are equivalent to the usual H1 -norm. Also, let l () be the space
l () = {ψ ∈ L2 () : ∂α ψ ∈ L2 (ω )} ≡ L2 (0, L; H 1 (ω )),
gi (x1 , x2 , x3 )dγ ∈ L2 (0, L ), (4.41)
Mα (x3 ) =
{v ∈ V () : vα (x1 , x2 , x3 ) = ζα (x3 ) + δα ζ (x3 ), ζα , ζ ∈ H01 (0, L )}.
1/2
ρ Q = ∂1 ρ 20,ω + ∂2 ρ 20,ω , for all ρ ∈ Q, 1/2
ψ S = ∂1 ψ 20,ω + ∂2 ψ 20,ω , for all ψ ∈ S.
and
Find ξ3 ∈ H01 (0, L ) such that L L YA(ω )ξ3 χ3 dx3 = F3 χ3 dx3 ,
∂1 θ1 = −∂2 θ2 .
(4.37)
where we can define the Young Modulus for the limit equations as
ω,
In these conditions, the limit problems cannot be fully determined (see Figueiredo and Leal, 2006).
0 σ33 = Ye33 (u0 ) = Y (ξ3 − xα ξα ),
0
(4.43)
Proof. This result follows straightforwardly from (4.17) taking into account that u2 verifies (4.26).
where
gi vi d, for all v ∈ V BN (), (4.36)
Y =
Corollary 4.7. The component D13 of D1 is given by
(4.42)
Proof. We consider (4.36) with the test function v ∈ VBN () of the form (4.18), implying e33 (v ) = χ3 (x3 ) − xθ χθ (x3 ). Then, taking successively χθ = 0 and χ3 = 0 in the resulting equation we obtain (4.39) and (4.40 ), respectively.
(4.45)
L2 (0, L; Q (ω ))
and l0 () = and let us define its subspaces R() = L2 (0, L; S(ω )). These spaces are equipped with the (semi-)norms
1/2
ψ l0 = ∂1 ψ 20, + ∂2 ψ 20, , for all ψ ∈ l0 (), 1/2
ρ R = ∂1 ρ 20, + |∂2 ρ 20, , for all ρ ∈ R().
Again, it can be shown that these norms are equivalent to the usual H1 -norm. We now define the warping function w ∈ R() as the unique solution of the variational problem (see e.g. Trabucho and Viaño, 1996, p. 700)
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
w(x3 ) ∈ Q (ω ) such that a.e. in(0, L ) : C3α 3β ∂β w∂α υ dω = C3α 3β δβ ∂α υ dω, ω
ω
Hence, if r is a solution of (4.48) then
for all υ ∈ H (ω ), 1
(4.46) as well as the torsion function J ∈ L2 (0, L), given by
J=
ω
=
ω
C3α 3β (δβ − ∂β w )(δα − ∂α w )dω C3α 3β (δβ − ∂β w )δα dω > 0.
(4.47)
(4.48)
Bearing in mind the previous definitions, we turn to the characterization of u1 and ϕ 0 . In (4.22), we obtained that eαβ (u1 ) = 0. Therefore, we can characterize the transverse components of u1 as follows (see e.g. Trabucho and Viaño, 1996)
u1α = zα + δα z,
zα , z ∈ H01 (0, L ).
(4.49)
Moreover, by taking v ∈ V1 () in (4.8) and v = 0 in (4.9), and considering (4.12), (4.15) and (4.23) we get, respectively,
2C3α 3β e3β (u1 ) + Pβ 3α ∂β ϕ 0 e3α (v )dx = 0,
for all v ∈ V 1 (), (4.50)
2Pβ 3α e3α (u1 ) − εαβ ∂α ϕ 0
which shows that ing result.
and
ϕ0
∂β ψ dx = 0, for all ψ ∈ (),
are coupled. In fact, we have the follow-
ˆ = (uˆi ) ∈ [L2 ()]3 be of the form (4.49) and ϕ 0 ∈ Theorem 4.9. Let u l () . Then uˆ is a solution of (4.50) if and only if
uˆ3 = −r − z3 − xα zα − wz ,
In view of the previous regularity result we conclude that although (4.49) implies that u1α ∈ H 1 (), we can only ensure that u13 ∈ L2 (0, L; H 1 (ω )). Furthermore, ω u13 dω vanishes at the rod ends as 1 long as z3 ∈ H0 (0, L ), but in general this is not the case for ω xα u13 dω. 1 Therefore, we cannot ensure that u ∈ V(), traducing a “ boundary layer phenomenon” . On the other hand, the regularity result obtained for u13 implies that U ∈ L2 (0, L; H1 (ω)) - cf. (4.54) - and therefore from (4.55) we can only guarantee that ϕ 0 is in l (), not in (). Hence, ϕ¯ 0 = ϕ 0 − ϕˆ ∈ l0 (). Nevertheless, even when u1 ∈ V(), the characterization of u1 done so far still holds, in particular as far as Eqs. (4.49) and (4.56) are concerned. This will allow to use Eqs. (4.50) and (4.51) to derive variational problems for ϕ¯ 0 .
(4.52)
Theorem 4.11. Element z ∈ H01 (0, L ) satisfies the variational problem
z ∈ H01 (0, L ),
0
L
Jz ζ dx3 =
Choosing v = (0, 0, χ (x3 )υ (x1 , x2 )), χ ∈ H01 (0, L ), υ ∈ Q (ω ) in (4.53) and taking (4.46) into account we get a.e. x3 ∈ (0, L)
C3α 3β ∂β uˆ3 (x3 )∂α υ dω ω = − C3α 3β ∂β (xθ zθ (x3 ) + wz (x3 ))∂α υ dω ω − Pβ 3α ∂β ϕ 0 (x3 )∂α υ dω, for all υ ∈ Q (ω ).
ω
(4.54)
for all υ ∈ Q (ω ).
(4.57)
(4.58)
Taking v = (χ (x3 )x2 , −χ (x3 )x1 , 0 ), χ ∈ H01 (0, L ) in the previous equation we get
(C3α3β (z (δβ − ∂β w ) − ∂β r ) + Pβ 3α ∂β ϕ 0 )δα χ dx = 0,
(4.59)
that is (cf. (4.47)),
d d (Jz ) = (C ∂ r − Pβ 3α ∂β ϕ 0 )δα dω, dx3 dx3 ω 3α 3β β
a.e. x3 ∈ (0, L ),
a.e. x3 ∈ (0, L ). (4.60)
is a solution a.e. x3 ∈ (0, L) of
C3α 3β ∂β U (x3 )∂α υ dω ω = − Pβ 3α ∂β ϕ 0 (x3 )∂α υ dω,
ζ dx3 ,
(C3α3β (z (δβ − ∂β w ) − ∂β r ) + Pβ 3α ∂β ϕ 0 )e3α (v )dx = 0,
d d Jz = P (∂α w − δα )∂β ϕ 0 dω, dx3 dx3 ω β 3α
ω
ω
or, in view of (4.46) and (4.48),
Therefore,
U = uˆ3 + xα zα + wz
(4.53)
0
Pβ 3α(∂α w − δα )∂β ϕ 0 dω
which plugged into (4.50) leads to
for all v ∈ V 1 ().
for all v ∈ V 1 ().
2e3α (u1 ) = −∂α r − z (∂α w − δα ),
ˆ = (uˆi ) is a solution of ( 4.50) and in view of Proof. Assuming that u (4.49) we conclude that uˆ3 verifies
L
Proof. We consider (4.49) and (4.56), yielding
(C3α3β (∂β uˆ3 + zβ + δβ z ) + Pβ 3α ∂β ϕ 0 )e3α (v )dx = 0,
∀ζ ∈ H01 (0, L ).
where r ∈ R() solves (4.48), w is the warping function and z3 ∈ L2 (0, L) is an arbitrary function of x3 .
(4.56)
4.2.1. Variational problem for ϕ¯ 0 , r and z
(4.51) u1
Corollary 4.10. Let u1α , the transverse component of u1 , be of the form (4.49) and let u1 verify (4.50). Then the axial component u13 is such that
D0α = −Pα 3β (∂β r + z (∂β w − δβ )) − εαβ ∂β ϕ 0 .
ω
ρ ∈ Q ( ω ).
where z3 is an arbitrary function of x3 , leading to (4.52 ). From (4.52) we conclude that since r ∈ L2 (0, L; H1 (ω)), zα , z ∈ H01 (0, L ) and w ∈ H1 (ω), then z3 ∈ L2 (0, L). The converse result is immediate.
where z3 ∈ L2 (0, L). Furthermore, (4.15) becomes
r (x3 ) ∈ Q (ω ) such that for a.e.x3 ∈ (0, L ) : C3α 3β ∂β r (x3 )∂α ρ dω = Pβ 3α ∂β ϕ 0 (x3 )∂α ρ dω, ω
a.e. x3 ∈ (0, L ),
U = −r − z3 ,
u13 = −r − z3 − xα zα − wz ,
Finally, let r ∈ R() satisfy the following variational problem,
for all
303
(4.55)
Finally, from (4.60) we also obtain that z satisfies (4.57). We note that if ϕ 0 is given, then the solution of (4.57) is unique since J is bounded and positive. Remark 4.12. From the previous results we conclude that in general it is not possible to obtain the elasticity result z = 0 (see e.g. ÁlvarezDios and Viaño, 1993a; 1993b; Trabucho and Viaño, 1996). However, we can prove that this evidence happens when we take ϕ 0 = ϕ 0 (x3 ).
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Note that this implies ϕ 0 = ϕˆ and is only possible if ϕˆ = ϕˆ (x3 ). In this case, Eq. (4.59) takes the form
L
ω
0
C3α 3β [z (δβ − ∂β w ) − ∂β r]δα dω
χ dx3 = 0, ∀χ ∈ H01 (0, L ),
and, by a density argument, we deduce
z
ω
C3α 3β (δβ − ∂β w )δα dω −
ω
C3α 3β ∂β rδα dω = 0.
C3α 3β ∂β r∂α wdω =
and, therefore, we deduce that Jz = 0 with J > 0. Thus, z = 0 and since z ∈ H01 (0, L ) one gets z = 0 a.e in (0, L).
(4.61)
(ϕ¯ 0 , r, z )
Theorem 4.13. The element the following variational problem
0
J (ς )2 dx3 +
C3α 3β ∂β ρ∂α ρ dx
J (ς )2 dx3 > c1 |ς |20,(0,L) ,
C3α 3β ∂β ρ∂α ρ dx ≥ c2
∈ T () is the unique solution of
((∂1 ρ )2 + (∂2 ρ )2 )dx
= c2 (|∂1 ρ|0, + |∂2 ρ|0, ), ∀ ρ ∈ R(), εαβ ∂β ψ∂α ψ dx ≥ c3 ((∂1 ψ )2 + (∂2 ψ )2 )dx 2
(4.62)
for all ς ∈ H01 (0, L ).
Bearing in mind properties (2.4), there exists constants c2 , c3 > 0 such that
(ψ , ρ , ς ) 2T () = |∂1 ψ|20, + |∂2 ψ|20, + |∂1 ρ|20,
0
2
equipped with the norm
Find
L
Let us define the space
+ |∂2 ρ|20, + |ς |20,(0,L) .
L
Pβ 3α ∂α w∂β ϕ 0 dω = 0,
T () = l0 () × R() × H01 (0, L ),
! a((ψ , ρ , ς ), (ψ , ρ , ς )) = + εαβ ∂α ψ∂β ψ dx,
for all (ψ , ρ , ς ) ∈ T(). On the other hand, since J is positive and bounded, we conclude that there exists a constant c1 > 0 such that
From (4.46)–(4.48), the previous relation becomes
Jz =
Since () is dense in l0 (), we obtain (4.63)–(4.65). One can easily show that both ! a and ! l are T -continuous. The coercivity of ! a is shown as follows. From (4.64) one has
= c3 (|∂1 ψ|0, + |∂2 ψ|0, ), 2
2
∀ ψ ∈ l0 ().
Thus, there exists a constant c4 > 0 such that (cf. (4.62)) 2 ! a((ψ , ρ , ς ), (ψ , ρ , ς )) ≥ c4 (ψ , ρ , ς ) T ,
ϕ¯ 0 , r, z ∈ T () such that
!, ρ !, ρ !, ρ ! !, ς )) = ! !, ς ), for all (ψ !, ς ) ∈ T (), a((ϕ¯ 0 , r, z ), (ψ l (ψ
that is, ! a is T-elliptic. Hence, the uniqueness of solution of the variational problem (4.63)–(4.65) follows from Lax–Milgram Lemma.
(4.63) 5. Convergence analysis
where the bilinear form ! a and the linear functional ! l are given by
L
! Jz ς dx3 + C3α 3β ∂β r∂α ρ dx a((ϕ¯ 0 , r, z ), (ψ , ρ , ς )) = 0 + εαβ ∂α ϕ¯ 0 ∂β ψ dx + Pβ 3α (∂α r∂β ψ − ∂β ϕ¯ 0 ∂α ρ )dx + Pβ 3α (∂α w − δα )(z ∂β ψ − ς ∂β ϕ¯ 0 )dx (4.64)
and
! l (ψ , ρ , ς ) = Pβ 3α ∂β ϕˆ (∂α ρ + ζ (∂α w − δα ))dx εαβ ∂β ϕ∂ ˆ α ψ dx, −
(4.65)
Pβ 3α (∂α r + z (∂α w − δα )) + εαβ ∂α ϕ¯ 0
=−
∂β ψ dx
εαβ ∂α ϕ∂ ˆ β ψ dx ∀ ψ ∈ ().
L 0
(4.66)
for all ψ ∈ (),
ρ ∈ R(), ζ ∈ H01 (0, L ).
∀(v, ψ ) ∈ X (),
(5.2)
where
X () = V () × (), M 1 () =
Jz ζ dx3 + C3α 3β ∂β r∂α ρ dx + εαβ ∂β ϕ¯ 0 ∂α ψ dx + Pβ 3α (∂α r∂β ψ − ∂β ϕ¯ 0 ∂α ρ )dx + Pβ 3α (∂α w − δα )(z ∂β ψ − ∂β ϕ¯ 0 ζ )dx = Pβ 3α ∂β ϕˆ (∂α ρ + ζ (∂α w − δα ))dx − εαβ ∂β ϕ∂ ˆ α ψ dx,
(5.1)
ˆ Xˆ () = V () × () ,
M () = M 1 () × M 2 (),
Combining (4.66) with (4.48) and (4.57),and taking into account that ϕ¯ 0 = ϕ 0 − ϕˆ , we conclude that ϕ¯ 0 , r, z ∈ T () verifies:
∀(τ , d ) ∈ M (), bH ((σ (h ), D(h )), (v, ψ )) = lH (v, ψ ),
Proof. We consider the variational problem (4.51) and write it as (cf. (4.58))
We start with the scaled variational problem associated to the primal variational problem (3.15)–(3.17): Find ((σ (h ), D(h )), (u(h ), ϕ (h ))) ∈ M () × Xˆ () such that
aH ((σ (h ), D(h )), (τ , d )) + bH ((τ , d ), (u(h ), ϕ (h ))) = 0,
respectively.
In this section we will provide a strong convergence result. Since we will need a mixed formulation to prove some results, in this section we have to invert the constitutive laws (2.3). Therefore, in order to simplify the exposition we assume throughout this section that in (3.10) we have Cijkl = 0.
τ = (τi j ) ∈ [L2 ()]9 : τi j = τ ji , M 2 () = [L2 ()]3 ,
ˆ () = ϕˆ + () = {ψ ∈ () : ψ = ϕˆ on eD }, and the forms aH , lH and bH are defined as follows:
aH ((σ (h ), D(h )), (τ , d )) = h−2 a−2 ((σ (h ), D(h )), (τ , d )) + h−1 a−1 ((σ (h ), D(h )), (τ , d )) + a0 ((σ (h ), D(h )), (τ , d )) + ha1 ((σ (h ), D(h )), (τ , d )) + h2 a2 ((σ (h ), D(h )), (τ , d )) + h4 a4 ((σ (h ), D(h )), (τ , d )), lH (v, ψ ) = − fi vi dx − gi vi d ,
bH ((σ , D ), (v, ψ )) = −
dN
σi j ei j (v )dx −
Dk Ek (ψ )dx,
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
with
exists a subsequence, still denoted by (u(h ), ϕ¯ (h ))h>0 which satisfies
a−2 ((σ , D ), (τ , d )) =
u ( h ) → u0
ε¯33 D3 d3 dx,
a−1 ((σ , D ), (τ , d )) = P¯333 D3 τ33 dx − P¯333 σ33 d3 dx, a0 ((σ , D ), (τ , d )) = ε¯αβ Dβ dα dx, C¯3333 σ33 τ33 dx + a1 ((σ , D ), (τ , d )) = P¯3αβ D3 ταβ dx − P¯3αβ σαβ d3 dx −2 P¯α 3θ σ3θ dα dx + 2 P¯β 3α Dβ τ3α dx, a2 ((σ , D ), (τ , d )) = C¯33αβ σαβ τ33 dx + 4 C¯3α 3β σ3β τ3α dx
Furthermore
u1 ∈ V 1 () :
e˜ =
e˜33 (h ) := e33 (u(h ))(x ), E˜α (h ) := Eα (ϕ (h ))(x ),
E˜3 (h ) := hE3 (ϕ (h ))(x ),
verifying the constitutive equations (cf. (3.20))
Sαβ (h ) := h2 σαβ (h ) = Cαβθ ρ e˜θ ρ (h ) + Cαβ 33 e˜33 (h ) − P3αβ E˜3 (h ), (5.4)
S33 (h ) :=
(5.5)
Let us now define
u (h ) := h
(u(h ) − u ) = u + h.o.t., 0
1
uˆ2α (h ) := h−2 uα (h ) − u0α − hu1α = u2α + h.o.t. In order to have strong convergence we need the additional hypotheses:
u1 (h ) ∈ V RD (),
2
ˆ (h ) ∈ L 0, L; V 2m (ω ) , u 2
u13 = −r − z3 − xα zα − wz ,
eαβ (u )
e3α (u1 )
e3α (u1 )
e33 (u0 )
2
, E˜ = −(∂1 ϕ 0 , ∂2 ϕ 0 , 0 ),
Proof. The proof is divided into 7 steps, numbered (i) to (vii). Step (i): We first show that there exists a constant c (ϕˆ ) > 0, depending only the applied potential ϕˆ , such that, for all h ∈ (0, 1):
u(h ) 21, + |e˜(h )|20, + |E¯ (h )|20, ≤ c(ϕˆ ),
(5.7)
ϕ¯ (h ) 0, ≤ c(ϕˆ ),
(5.8)
¯ (h )) = −(∂1 ϕ¯ (h ), ∂2 ϕ¯ (h ), h∂3 ϕ¯ (h )). where E¯ (h ) = E˜ (ϕ The proof of this step is straightforward and can be found in Ref. Figueiredo and Leal (2006). Step (ii): There exists a subsequence of (u(h ), e˜(h ), ϕ (h ), E (h ))0
u(h ) u in[H 1 ()]3 ,
S3α (h ) := hσ3α (h ) = 2Cα 33θ e˜3θ (h ) − Pθ α 3 E˜θ (h ),
−1
ξα ∈ H02 (0, L ), u03 (x1 , x2 , x3 ) = ξ3 (x3 ) − xρ ξρ (x3 ), ξ3 ∈ H01 (0, L ), 1 uα = zα + δα z, zα , z ∈ H01 (0, L )
where z3 ∈ L2 (0, L), functions ξ α , ξ 3 , z solve the variational problems (4.39), (4.40) and (4.57), respectively, and w, r ∈ Q(ω) solve the variational problems (4.46) and (4.48).
e˜3β (h ) := h−1 e3β (u(h ))(x ),
σ33 (h ) = C33θ ρ e˜θ ρ (h ) + C3333 e˜33 (h ) − P333 E˜3 (h ), Tθ (h ) := Dθ (h ) = 2Pθ 3α e˜3α (h ) + εθ α E˜α (h ), T3 (h ) := h−1 D3 (h ) = P3αβ e˜αβ (h ) + P333 e˜33 (h ) + ε33 E˜3 (h ).
in M 2 ().
u0α (x1 , x2 , x3 ) = ξα (x3 ),
u ∈ V BN () :
where C¯ = (C¯i jkl ), P¯ = (P¯ki j ), ε¯ = (ε¯i j ) are inverse constitutive tensors of C = (Ci jkl ), P = (Pki j ) and ε = (εi j ) and inherit their properties of ellipticity and boundedness (see e.g. Viaño et al., 2015). We also define the following auxiliary tensor (e˜i j (h )) and vector (E˜i (h )):
e˜αβ (h ) := h−2 eαβ (u(h ))(x ),
ˆ ϕ¯ (h ) → ϕ 0 − ϕˆ in () ,
0
(5.3)
in V (),
e˜(h ) → e˜ in M 1 () E˜ (h ) → E˜
+ C¯αβ 33 σ33 ταβ dx, a4 ((σ , D ), (τ , d )) = C¯αβθ ρ σθ ρ ταβ dx,
1
305
(5.6)
where
V RD () = {v ∈ V 1 () : v3 ∈ L2 (0, L; Q (ω ))},
(5.9)
e˜(h ) e˜ in M 1 (),
(5.10)
ϕ (h ) ϕ in L2 (),
(5.11)
E˜ (h ) E˜
(5.12)
in M 2 ().
From step (i) it is easy to show that there exists subsequences (u(h))h > 0 , (e˜(h ))h>0 , (ϕ¯ (h ))h>0 , (E¯ (h ))h>0 and functions u, e˜ , ϕ¯ and E¯ satisfying (5.9 ) and (5.10),
ϕ¯ (h ) ϕ¯ in L2 (), and E¯ (h ) E¯ in [L2 ()]3 ,
(5.13)
respectively. The weak convergence results (5.11) and (5.12) follow straightforwardly from (5.13) taking into account that ϕ (h ) = ϕ¯ (h ) + ϕˆ , E˜ (h ) = E¯ (h ) + Eˆ (h ) and Eˆ (h ) = (−∂1 ϕˆ , −∂2 ϕˆ , −h∂3 ϕˆ ) →
(−∂1 ϕˆ , −∂2 ϕˆ , 0 ) as h → 0.
is the space of admissible rigid displacements. We also define
Step (iii): There exist subsequences of (u(h))h > 0 and (ϕ (h))h > 0 still denoted the same way, and e˜ ∈ M 1 (), E˜ ∈ M 2 () such that
V R () = {v ∈ V RD () : vα = δα s, s ∈ H01 (0, L )},
e33 (u(h )) e˜33 , h−1 eα 3 (u(h )) e˜α 3 , h−2 eαβ (u(h )) e˜αβ ,
V T () = {v ∈ V RD () : vα = sα ,
v3 = −xα sα , sα ∈ H01 (0, L )},
which are the the subspaces of admissible rotations and translations, respectively. We note that |e3β ( · )|0, is a norm for VR () but not for VRD (). In order to overcome this difficulty, we can think of using the quotient space V˙ RD () := V RD ()/V T (), where the induced application
˙ |||v||| = |e3β (v )|0, , ∀ v˙ = v + V T () ∈ V˙ RD (), is a norm. Theorem 5.1. Let (u(h ), ϕ¯ (h )) ∈ V () × () be the solution of the scaled problem (3.15)–(3.17) and assume that (5.6) holds. Then, there
eα 3 (u(h )) 0,
eαβ (u(h )) 0,
E˜α (ϕ (h )) E˜α ,
E˜3 (ϕ (h )) E˜3 ,
hEα (ϕ (h )) 0,
(5.14)
in L2 (). Moreover,
e˜33 = e33 (u ), E˜α = −∂α ϕ ,
e3α (u ) = eαβ (u ) = 0 (i.e., u ∈ V BN ()), E˜3 = 0.
(5.15) (5.16)
The obtention of (5.14) is straightforward. Indeed, from (5.10) and (5.12) we deduce that there exist (e˜αβ , e˜3α , e˜33 , E˜α , E˜3 ) ∈ M (), which is the weak limit of a subsequence of
(h−2 eαβ (u(h )), h−1 e3α (u(h )), e33 (u(h )), Eα (ϕ (h )), hE3 (ϕ (h ))),
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still denoted by the same way. Also, from (5.9) and the uniqueness of the limit as h → 0, we deduce that (5.15) holds. Similarly, from (5.11) we obtain (5.16). Step (iv): Let ((σ (h), D(h)), (u(h), ϕ (h))) be the solution of the mixed variational problem (5.1)–(5.3). Then, there exists a constant C = C (ϕˆ ) > 0, independent of h, such that, for all 0 < h ≤ 1:
S33 (h ) 0, ≤ C,
Sα3 (h ) 0, ≤ C,
Sαβ (h ) 0, ≤ C,
Tα (h ) 0, ≤ C, T3 (h ) 0, ≤ C.
(5.17) (5.18)
Moreover, there exists a subsequence, still parameterized by h, and there exist ∈ M1 () and T ∈ M2 (), such that:
S33 (h ) 33 ,
Sα 3 ( h ) α 3 ,
Sαβ (h ) αβ ,
in L2 () (5.19)
Tα (h ) Tα ,
T3 (h ) T3 ,
in L () 2
(5.20)
when h goes to zero. Furthermore,
Multiplying Eq. (5.2) by h, taking the test function v ∈ VRD () and passing to the limit as h → 0, we find
3α e3α (v¯ )dx = 0 ∀v¯ ∈ V RD ().
(5.30)
Therefore, from (5.28)–(5.30) we get (5.26). The relations (5.27) follow from the inversion of (5.21)–(5.25). Step (vi): The following characterizations hold:
e˜θ ρ = eθ ρ (u2 ), e˜33 = e33 (u0 ), e˜3β = e3β (u1 ), E˜β = Eβ (ϕ 0 ). (5.31) Moreover, −2 αβ = σαβ ,
T3 = D13 ,
0 33 = σ33 ,
Tα = D0α ,
3α = σ3−1 α,
in L2 (),
in L2 ().
(5.32) (5.33)
ˆ ∈ L2 (0, L; V 2m (ω )) From (5.7) and (5.6) we find that there exist u and u¯˙ 1 ∈ V˙ RD () such that the following weak convergences hold: 2
e˜αβ = C¯θ αβρ θ ρ + C¯33αβ 33 + P¯3αβ T3 ,
(5.21)
uˆ (h ) uˆ ,
e˜3α = 2C¯3α 3θ 3θ + P¯θ 3α Tθ ,
(5.22)
Moreover, since eαβ : V 2m (ω ) → [L2 (ω )]2×2 and e3β : V˙ RD () → s [L2 ()]2 are weakly continuous applications, by using the closed graph theorem (see, for example Ref. Brézis, 1983) we find that
e˜33 = C¯3333 33 + P¯333 T3 + C¯33θ ρ θ ρ ,
(5.23)
E˜α = −2P¯θ 3α 3θ − ε¯θ α Tθ ,
(5.24)
E˜3 = −P¯3θ ρ θ ρ − P¯333 33 + ε¯33 T3 .
(5.25)
αβ eαβ (vˆ )dx +
+ 33 e33 (v )dx + = fi vi dx + gi vi d ,
N
∀(vˆ , v¯ , v, ψ¯ ) ∈ V H ()
∀(v, ψ¯ ) ∈ V BN () × ().
(5.27)
−2 ¯ )dx αβ − σαβ eαβ (vˆ )dx + 3α − σ3−1 α e3α (v
+
0 33 − σ33 e33 (v )dx + Tα − D0α Eα (ψ¯ )dx = 0,
∀(vˆ , v¯ , v, ψ¯ ) ∈ V H () × V RD () × V BN () × l0 (), which can be equivalently written as follows, taking into account the constitutive Eqs. (4.11), (4.12), (4.14), (4.15) and (5.27),
T
eαβ (vˆ ) e33 (v )
Cαβ 33 C3333
C3α 3β Pα 3β
−Pβ 3α
T
e3α (v¯ ) Eα (ψ¯ )
Cαβθ ρ C33θ ρ
εαβ
e˜θ ρ − eθ ρ (u2 ) dx = 0, e˜33 − e33 (u0 )
fi vi dx +
N
αβ eαβ (vˆ )dx = 0, ∀vˆ ∈ V H ().
(5.35)
e˜3β − e3β (u1 ) dx = 0, ˜ Eβ − Eβ (ϕ¯ 0 + ϕˆ ) (5.36)
for all (vˆ , v¯ , v, ψ¯ ) ∈ V H () × V RD () × V BN () × l0 (). By using density arguments, the integral Eq. (5.35) is also valid when we replace VH () by L2 (0, L; V 2m (ω )). Therefore, we can take as test functions
vˆ = uˆ 2 − u2 , v¯ = u¯ 1 − u1 , v = u − u0 , ψ¯ = ϕ − (ϕ¯ 0 + ϕˆ ).
gi vi d , (5.28)
Since () is dense in l0 (), the previous equation is also valid for all (v, ψ¯ ) ∈ V BN () × l0 (). Now, we multiply Eq. (5.2) by h2 . Let us now define the space of horizontal displacements V H () = H 1 (0, L; V 2m (ω )). By choosing v3 = 0 and (vα ) ∈ VH (), and passing to the limit as h → 0, we obtain
Taking v ∈ VBN () in (5.2) and passing to the limit when h → 0, we obtain
Tα Eα (ψ¯ )dx =
(5.34)
(5.26)
⎧ ⎨αβ = Cαβθ ρ e˜θ ρ + Cαβ 33 e˜33 , 3α = 2C3α3θ e˜3θ − Pθ 3α E˜θ , 33 = C33θ ρ e˜θ ρ + C3333 e˜33 , ⎩ Tα = 2Pα 3β e˜3β + εαβ E˜β , T3 = P3αβ e˜αβ + P333 e˜33 .
1 e˜3β = e3β (u¯ ).
Now, taking into account Eq. (4.8) for test functions v ∈ VRD () and ψ = 0, Eq. (4.9) with v = 0 and ψ ∈ (), dense in l0 (), and rela-
where
33 e33 (v )dx +
2
Tα Eα (ψ¯ )dx
× V 1 () × V BN () × l0 (),
u1 (h ) u¯˙ 1 .
e˜αβ = eαβ (uˆ ),
3α e3α (v¯ )dx
2
tions (4.24), (4.36) and (5.26), we obtain
Indeed, (5.17) and (5.18) follow from (5.7) and (5.8) and (5.4) and (5.5). Moreover, (5.19) and (5.20) is an immediate consequence of (5.17) and (5.18). Furthermore, by choosing τ33 = τ3α = d3 = dα = 0 as test functions in (5.1), multiplying by h−2 and passing to the limit, we deduce (5.21) in L2 (). Proceeding similarly, we obtain (5.22)– (5.25). Step (v): The limit model is given by:
2
(5.29)
Having in mind (5.15), (5.16) and (5.34), the ellipticity in (5.35)–(5.36) gives (5.31). Consequently, from (4.11), (4.12), (4.14), (4.15), (4.17) and (5.27), we can deduce (5.32)–(5.33). Step (vii): The following strong convergences hold:
e˜(h ) → e˜ in M 1 (),
E˜ (h ) → E˜
u ( h ) → u0
ˆ ϕ (h ) → ϕ 0 in () .
in V (),
in [L2 ()]3 ,
Let X(h) be the norm of (e˜(h ), −E˜ (h )) in M():
X (h ) =
+
Ci jkl e˜kl (h )e˜i j (h )dx +
ε33 E˜3 (h )E˜3 (h )dx
εαβ E˜α (h )E˜β (h )dx
"1/2
.
(5.37)
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
Taking (v, ψ ) = (u(h ), ϕ (h )) in (3.15)–(3.17) we find that (u(h), ϕ (h))h > 0 verifies
Ci jkl e˜kl (h )e˜i j (h )dx + εαβ E˜α (h )E˜β (h )dx + ε33 E˜3 (h )E˜3 (h )dx fi ui (h )dx + gi ui (h )d . (5.38) =
N
Combining (5.37) and (5.38) we have
(X (h ))2 =
fi ui (h )dx +
gi ui (h )d .
N
From the weak convergences (5.9) we get
lim (X (h ))2 =
h→0
fi ui dx +
N
gi ui d .
Since we have already proved that (e˜(h ), ∂1 ϕ (h ), ∂2 ϕ (h ), h∂3 ϕ (h )) converges weakly, it suffices to show that the limit of its norm tends to the norm of its weak limit to obtain strong convergence. We denote by
X=
Ci jkl e˜kl e˜i j dx +
εαβ E˜α E˜β dx
1/2
˜ E˜1 , E˜2 , E˜3 ). So, we want to show that the L2 norm of (e,
X 2 = lim (X (h ))2 , h→0
or, equivalently,
Ci jkl e˜kl e˜i j dx +
εαβ E˜α E˜β dx =
fi ui dx +
N
gi ui d .
⎧ 1 2 1 2 ⎨uα (h ) = ξα + huα + O(h ), u3 (h ) = ξ3 − xβ ξβ + hu3 + O(h ), 1 1 u = zα + δα z, u3 = −r − z3 − xα zα − wz , ⎩ α ϕ¯ (h ) = ϕ¯ 0 + O(h ), (6.1) and
⎧ −1 + O(h0 ), σαβ (h ) = h−1 σαβ ⎪ ⎪ ⎪ −1 ⎪ σαβ (h ) = h−1 σαβ + O(h0 ), ⎪ ⎪ ⎪ ⎪ −1 0 ⎪ ⎨σ3α (h ) = h [Pβ 3α ∂β (ϕ¯ + ϕˆ ) −C3α 3β (∂β r + z (∂β w − δβ ))] + O(h0 ), ⎪ ⎪ ⎪ ⎪σ33 (h ) = Y (ξ3 − xα ξα ) + O(h ), ⎪ ⎪ ⎪ ⎪Dα (h ) = −Pα3β (∂β r + z ∂β w − δβ ) − εαβ ∂β (ϕ¯ 0 + ϕˆ ) + O(h ), ⎪ ⎩ D3 (h ) = hP¯ (ξ3 − xα ξα ) + O(h2 ), (6.2) where Y and P¯ are given by (4.38) and (4.43 ), respectively, and the (Bernoulli–Navier) displacement components ξα ∈ H02 (0, L ) and ξ3 ∈ H01 (0, L ) are, respectively, the unique solutions of the 1D variational (limit) problems ( 4.39) and (4.40), just as in the elastic case. This result holds under the assumption (3.10). Regarding the electric potential, we have showed that even if no conditions are imposed onthe dependence of C, P and ε with respect to x, the element ϕ¯ 0 , r, z ∈ T () is the unique solution of the 3D variational (limit) problem (4.63)–(4.65). 6.1. The limit model on the original rod h
It easy to check the following relation
Ci jkl e˜kl e˜i j dx + εαβ E˜α E˜β dx = (Cαβθ ρ e˜θ ρ + Cαβ 33 e˜33 )e˜αβ dx + 4 C3α 3θ e˜3θ e˜3α dx + (C33θ ρ e˜θ ρ + C3333 e˜33 )e˜33 dx + εαβ E˜α E˜β dx.
We now return to the original rod h and define the following spaces (cf. (4.44), (4.45) and (4.61)):
lh (h ) = L2 (0, L; H 1 (ωh )), h l0h (h ) = {ψ h ∈ L2 (0, L; H 1 (ωh )) : ψ h = 0 on eD }
≡ L2 (0, L; Sh (ωh )),
R ( ) = L2 (0, L; Q h (ωh )), Q h (ωh ) = {qh ∈ H 1 (ωh ) : qh dωh = 0}, h
and, having in mind the relations (5.27), one deduce
Ci jkl e˜kl e˜i j dx + εαβ E˜α E˜β dx = (Cαβθ ρ e˜θ ρ + Cαβ 33 e˜33 )e˜αβ dx +
%$Given the scalings (3.4), (3.6) and ( 3.18), the developments (4.1 ), (4.2) and (4.4) induce formal developments on uh , ϕ¯ h , σ h and Dh , respectively, whose leading terms we will identify and characterize in the following. For that we will undo the change of variable xh = h (x ) and accordingly define the de-scaled quantities:
(C33θ ρ e˜θ ρ + C3333 e˜33 )e˜33 dx $% &
#
33
+2
h (h ) × Rh (h ) × H01 (0, L ). T h (h ) = l0
&
αβ
2C3α 3θ e˜3θ e˜3α dx +
# $% & 3α +Pβ 3α E˜β
ξαh := ξαh xh3 = h−1 ξα (x3 ), ξ3h := ξ3h xh3 = ξ3 (x3 ),
εαβ E˜α E˜β dx. # $% &
ϕ¯ 0h := ϕ¯ 0h (xh ) = hϕ¯ 0 (x ), rh := rh (xh ) = hr (x ),
Tβ −2Pα 3β e˜3β
2 By using (5.26) with test functions v = u0 , v¯ = u1 , vˆ = uˆ and ψ¯ = ϕ 0 and the equality relations (5.31) and (5.32), we find
Ci jkl e˜kl e˜i j dx + εαβ E˜α E˜β dx = αβ e˜αβ dx + 2 3α e˜3α dx 0 0 ˜ + 33 e˜33 dx + Tβ Eβ dx = fi ui dx + gi ui d ,
h
ωh
#
307
N
as required. 6. Leading terms and limits models for the reference rod The results obtained in the previous sections concerning the leading terms of the developments for the scaled displacement vector (ui (h)), electric potential ϕ¯ (h ), stress tensor (σ ij (h)) and electric displacement vector (Di (h)), for the reference rod , can be summarized as follows:
zαh := zα , zh := h−1 z, as well as the warping function
wh = wh xh1 , xh2 = h2 w(x1 , x2 ), which is the unique solution of (cf. (4.46))
wh ∈ Q h (ωh ) such that C3α 3β ∂βh wh ∂αh υ h dωh = ωh
ωh
C3α 3β δβh ∂αh υ h dωh ,
for all υ h ∈ H 1 (ωh ), the torsion constant (cf. (4.47))
Jh = h4 J =
ωh
=
ωh
C3α 3β (δβh − ∂βh wh )(δαh − ∂αh wh )dωh C3α 3β (δβh − ∂βh wh )δαh dωh ,
(6.3)
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and the second moment of area of the cross section with respect to axis Oxhβ (α = β )
Iαh =
ωh
(xhα )2 dωh ,
where δ1h (xh1 , xh2 ) = xh2 , δ2h (xh1 , xh2 ) = −xh1 . We also define the resultant of the applied loads and moments in each cross section (cf. (4.41) and (4.42)):
Fαh (xh3 ) = h3 Fα (x3 ) = fαh (xh1 , xh2 , xh3 )dωh ωh + ghα (xh1 , xh2 , xh3 )dγ h ∈ L2 (0, L ), γNh
F3h
( ) = h F3 (x3 ) = xh3
2
+
γNh
ωh
f3h
(
xh1 , xh2 , xh3
)d ω
h − h εαβ ∂βh ϕˆ h ∂αh ψ h dxh ,
(6.7)
for all (ψ , ρ , ς ) ∈ T ( ), h
h
Proposition 6.1. The leading terms of the formal developments induced by (4.1), (4.2) and (4.4) on the displacement vector (uhi ), electric potential ϕ¯ h , stress tensor (σihj ) and electric displacement vector (Dhi ) are such that:
⎧h h h h h h h 1h 2 u = ξαh + u1h ⎪ α + O ( h ) , u 3 ( x ) = ξ 3 − xβ ∂ 3 ξ β + u 3 + O ( h ) , ⎪ ⎪α ⎪ ⎪ h h h 1h h h h h h h ⎪ u1h ⎪ α = zα + δα z , u3 = −r − z3 − xα (zα ) − w (z ) , ⎪ ⎪ ⎪ ⎪ ⎪ ϕ¯ (h ) = ϕ¯ 0 + O(h ), ⎪ ⎪ ⎪ ⎪ h ⎪ 0h 2 ⎪ ⎪ ⎪ϕ¯ = ϕ¯ + O(h ), ⎨ h σαβ = O ( h ), ⎪ ⎪ ⎪ h ⎪ σ3α = Pβ 3α ∂βh (ϕ¯ 0h + ϕˆ ) − C3α3β (∂βh rh + ∂3h zh (∂βh wh − δβh )) + O(h ), ⎪ ⎪ ⎪ ⎪ ⎪ h h h ⎪ σ33 = Y (∂3h ξ3h − xhα ∂33 ξα ) + O ( h ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dhα = −Pα 3β (∂βh rh + ∂3h zh (∂βh wh − δβh )) − εαβ ∂βh (ϕ¯ 0h + ϕˆ ) + O(h ), ⎪ ⎪ ⎪ ⎪ ⎩ h ¯ h h h h h D3 = P (∂3 ξ3 − xα ∂33 ξα ) + O(h ), (6.4) (we note that α ∼ ∼ O ( h ), α ∼ ∼ and z3h ∼ O(h )) where ξαh ∈ H02 (0, L ), ξ3h ∈ H01 (0, L ) and ϕ¯ 0h , rh , zh ∈ T h are the unique solutions of the following variational problems,
ξαh ∈ H02 (0, L ),
L 0
h h h Y h Iαh ∂33 ξα ∂33 χαh dxh3 =
L
0
Fαh χαh dxh3 −
O(h0 ), zh
(no sum on α ), for all χα ∈ h
H02
L 0
for all χ ∈
H01
Mαh ∂3h χαh dxh3 , (6.5)
σ3hα = Pβh3α ∂βh (ϕ¯ 0h + ϕˆ ) − C3hα3β (∂βh rh + ∂3h zh (∂βh wh − δβh )), h h h σ33 = Y h (∂3h ξ3h − xhα ∂33 ξα ) , h h h h h ∂βh (ϕ¯ 0h + ϕˆ ), Dα = −Pα 3β (∂β r + ∂3h zh (∂βh wh − δβh )) − εαβ h h ξα ) . Dh3 = P¯ h (∂3h ξ3h − xhα ∂33
From (6.7), by taking appropriate choices of test functions we obtain the following set of independent equations (note that xˆ = (xh1 , xh2 )):
L 0
Jh ∂3h zh ∂3h ζ dx3 −
=
L
L
=
L
L
ωh
0
Pβh3α ∂βh ϕˆ h (∂αh wh − δαh ) dxˆ
ρ3h dx3
Pβh3α ∂β ϕ¯ 0h ∂α ρh dxˆ
h
ωh
Pβh3α ∂β ϕˆ h ∂α ρh dxˆ
h
ωh
(0, L ),
(ϕ¯ 0h , rh , zh ) ∈ T h (h ),
0
L
F3h χ3h dxh3 ,
(6.6)
∂3h ζ h dx3
∂3h ζ h dx3 ∀ζ h ∈ H01 (0, L ),
ρ3h dx3
ρ3h dx3
∀ρh ∈ Q h (ωh ), ρ3h ∈ L2 (0, L ),
(6.9)
h εαβ ∂βh ϕ¯ 0h ∂αh ψh dxˆh ψ3h dx3 ωh 0 L h + Pβh3α ∂αh rh ∂βh ψh dxˆ ψ3h dx3 ωh 0 L h + Pβh3α (∂αh wh − δαh )∂βh ψh dxˆ ∂3h zh ψ3h dx3 ωh 0 L h =− εαβ ∂βh ϕˆ h ∂αh ψh dxˆh ψ3h dx3
L
∀ψ
(6.8)
0
0
Pβh3α (∂αh wh − δαh )∂βh ϕ¯ 0h dxˆ
h
h
−
C3hα 3β ∂βh rh ∂αh ρh dxˆ
ωh
0
L
h
ωh
0
0 h ∈
ωh
S(ωh ), ψ3h ∈ L2 (0, L ).
(6.10)
ωh
Y h A(ωh )∂3h ξ3h ∂3h χ3h dxh3 = h 3
L
O(h−1 )
(0, L ),
ξ3h ∈ H01 (0, L ),
0
h
h σαβ = 0,
Bearing in mind these definitions and in view of (6.1) and (6.2), one has the following result for the leading terms of the induced formal developments for uh , ϕ¯ h , σ h and Dh .
zh
h
h
The variational problems obtained for the transverse displacement ξαh , the axial displacement ξ3h and the electric potential ϕ¯ 0h constitute the limit model for the original rod. On the other hand, the limit constitutive equations are (cf. ( 6.4))
γNh
ϕ¯ 0h
h
whereas wh ∈ Qh (ωh ) is the unique solution of (6.3).
Mαh (xh3 ) = h3 Mα (x3 ) = xhα f3h (xh1 , xh2 , xh3 )dωh ωh + xhα gh3 (xh1 , xh2 , xh3 )dγ h ∈ L2 (0, L ).
O(h−1 ),
L
h
gh3 (xh1 , xh2 , xh3 )dγ h ∈ L2 (0, L ),
ξh
Jh ∂3h zh ∂3h ς h dxh3 + C3hα 3β ∂βh rh ∂αh ρ h dxh h 0 h + εαβ ∂αh ϕ¯ 0h ∂βh ψ h dxh + Pβh3α (∂α rh ∂β ψ h − ∂βh ϕ¯ 0h ∂αh ρ h )dxh h h + Pβh3α (∂αh wh − δαh )(∂3h zh ∂βh ψ h − ∂3h ς h ∂βh ϕ¯ 0h )dxh h = Pβh3α ∂βh ϕˆ h (∂αh ρ h + ∂3h ς h (∂αh wh − δαh ))dxh
[−ah , ah ] ×
= Now, let us assume the particular case where h = (−ah , ah ) × {−bh }. In this case we can take as [−bh , bh ] and γeD transversal test functions ρh (xh1 , xh2 ) = xh1 xh2 and ψh (xh1 , xh2 ) = xh1 (xh2 + bh ). In addition to that, we define the following quantities:
θ1h = xh2 , θ2h = xh1 , θˆ1h = xh2 + bh , θˆ2h = xh1 , Cβh = C3hα3β θαh , h ˆh Pβh = Pβh3α θαh , Pα†h = Pβh3α θβh , εβh = εαβ θα .
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
Therefore, the latter two variational equations can be reformulated as follows:
L h ρ3h dx3 − Pβh ∂βh ϕ¯ 0h dxˆ ρ3h dx3 ωh ωh 0 0 L h = Pβh ∂βh ϕˆ h dxˆ ρ3h dx3 ∀ρ3h ∈ L2 (0, L ), ωh 0 L L h εβh ∂βh ϕ¯ 0h dxˆh ψ3h dx3 + Pα†h ∂αh rh dxˆ ψ3h dx3 ωh ωh 0 0 L h + Pα†h (∂αh wh − δαh ) dxˆ ∂3h zh ψ3h dx3 ωh 0 L =− εβh ∂βh ϕˆ h dxˆh ψ3h dx3 ∀ψ3h ∈ L2 (0, L ).
L
Cβh ∂βh rh dxˆ
h
0
ωh
Moreover, the strong formulation of our problem can be formulated as follows:
⎧ h ⎪ Pβ 3α (∂αh wh − δαh )∂βh ϕ¯ 0h dxˆ ∂3h Jh ∂3h zh − ∂3h ⎪ ⎪ ωh ⎪ ⎪ ⎪ ⎪ h ⎪ h h h h h h ⎪ ˆ = ∂ P ∂ ϕ ˆ ( ∂ w − δ ) d x , h α α ⎪ ω β 3α β 3 ⎪ ⎪ ⎪ ⎪ ⎪ zh (a ) = 0, (Jh ∂3h zh )(a ) = 0, Pβ 3α (∂αh wh ⎪ ⎪ ωh ×{a} ⎪ ⎨ h −δαh )∂βh (ϕ¯ 0h + ϕˆ h ) dxˆ = 0, for a ∈ {0, L} ⎪ ⎪ h h ⎪ C h ∂ h rh dxˆh − Pβh ∂βh ϕ¯ 0h dxˆ = Pβh ∂βh ϕˆ h dxˆ , ⎪ β ⎪ β h h h ⎪ ω ω ω ⎪ ⎪ ⎪ ⎪ h ⎪ εβh ∂βh ϕ¯ 0h dxˆh + Pα†h ∂αh rh dxˆ + Pα†h (∂αh wh ⎪ ⎪ h h h ⎪ ω ω ω ⎪ ⎪ −δ h ) dxˆh ∂ h zh = − ε h ∂ h ϕˆ h dxˆh , ⎪ ⎪ α ωh β β 3 ⎩ a.e. in (0, L )
(6.11)
ωh
rh
ϕ¯ 0h
ϕˆ h ,
from which we deduce that = = 0 and = i.e., the first order component of the displacements u1h is of the Bernoulli–Navier kind and the zeroth order component of the electric potential ϕ 0h coincides with the applied potential ϕˆ h . Remark 6.3. For materials having an isotropic mechanical behavior, such as PVDF, the limit model and constitutive equations are somewhat simplified since in this case the stiffness tensor C has the form
Ci jkl =
h
with, from (4.46) and (4.47)
Jh =
Y (∂1h wh )2 − (∂2h wh )2 ) (I1h + I2h − 2 (1 + ν ) ωh ωh
and
wh ∈ Q h ( ω h ), ∂αh wh ∂αh ψ h dωh = (xh2 ∂1h ψ h − xh1 ∂2h ψ h )dωh , ωh
ωh
for all ψ ∈ H (ω ). 1
h
7. Conclusions
⎧ h h h h ∂3 J ∂3 z = 0 a.e. in (0, L ), zh (a ) = ∂3h (Jh zh )(a ) = 0, ⎪ ⎪ ⎪ ⎨ h h h h Cβ ∂β r dxˆ = 0 a.e. in (0, L ), ωh ⎪ ⎪ ⎪ h ⎩ P†h ∂ h rh dxˆh + Pα†h (∂αh wh − δαh ) dxˆ ∂3h zh = 0 a.e. in (0, L ), α α zh
L ! Jh ∂3h zh ∂3h ς h dxh3 ahiso ((zh , rh , ϕ¯ 0h ), (ς h , ρ h , ψ h )) = 0 Y + ∂αh rh ∂αh ρ h dxh 2(1 + ν ) h + εαβ ∂αh ϕ¯ 0h ∂βh ψ h dxh + Pβ 3α (∂α rh ∂β ψ h − ∂βh ϕ¯ 0h ∂αh ρ h )dxh h h + Pβ 3α (∂αh wh − δαh )(∂3h zh ∂βh ψ h − ∂3h ς h ∂βh ϕ¯ 0h )dxh ,
h
Remark 6.2. Assume that ∂βh ϕˆ h = ∂βh ϕ¯ 0h = 0, i.e., both the applied potential and the first order component of the electric potential ϕ h distributed on the beam depend only on x3 . In this case, the previous equations give
ωh
309
Y νY δi j δkl + δik δ jl + δil δ jk , 1 + ν 1 − 2 ν 2 1 + ν ( )( ) ( )
where Y is the Young’s modulus, ν the Poisson’s ratio and δ the Kronecker’s delta. As a consequence, Y defined by (4.38) is indeed the Young’s modulus and
Y C3α 3β = δ . 2(1 + ν ) αβ Therefore, the variational problems (6.5) and (6.6) remain unchanged, while in the variational problem for ϕ¯ 0h the bilinear form becomes
The main result in this paper is the model shown in Eqs. (6.5)– (6.7). From (6.5)–(6.6) we observe that the electric potential has no effect on the zeroth order components of the bendings ξαh and the stretching ξ3h . This effect only appears on higher order terms as we can see from (6.7). Unfortunately, unless a specific dependence of ϕ h on (xh1 , xh2 ) is established, we cannot reduce (6.7) to a set of onedimensional expressions as we did for the zeroth order displacements. Still, by taking particular cases of transversal section ωh we can give a strong formulation of (6.7) as a set of two-dimensional expressions, as shown in (6.11). There we see that some of the terms are still global since they involve an integration over the whole transversal section. This is a remarkable difference with our previous work Viaño et al. (2015) (the electric potential was applied at the beam ends), where in the limit model the stretching was coupled with the electric potential, even for the zeroth order terms, and that the set of equations was reduced to one-dimensional expressions (despite having some global terms acting as coefficients). We ended this work by showing two interesting particular cases. In Remark 6.2 we show that a trivial case is obtained when the potential depends only on the axial coordinate, and in Remark 6.3 we show the equations for the homogeneous isotropic case. Acknowledgments This research was partially financed by FEDER Funds through “Programa Operacional Factores de Competitividade - COMPETE”, by Portuguese Funds through FCT - “Fundação para a Ciência e a Tecnologia”, within the Project PEst-OE/MAT/UI0013/2014, and by the project “Modelización y simulación numérica de sólidos y fluidos en dominios con pequeñas dimensiones. Aplicaciones en estructuras, biomecánica y aguas someras, MTM2012-36452-C02-01 financed by the Spanish Ministry of Economía y Competitividad with the participation of FEDER. For the sake of readability we give a list of the functional spaces and more important functions and notations involved in this paper. References Álvarez-Dios, J.A., Viaño, J.M., 1993a. An asymptotic general model for linear elastic homogeneous anisotropic rods. Int. J. Numer. Meth. Eng 36 (18), 3067–3095. doi:10.1002/nme.1620361804. Álvarez-Dios, J.A., Viaño, J.M., 1993b. On a bending and torsion asymptotic theory for linear nonhomogeneous anisotropic elastic rods. Asympt. Anal 7 (2), 129–158. Banks, H., Smith, R., Wang, Y., 1996. Smart Material Structures. Modeling, Estimation and Control. Chichester: Wiley; Paris: Masson. Bellis, C., Imperiale, S., 2014. Dynamical 1D models of passive piezoelectric sensors . Math. Mech. Solids 19 (5), 451–476. doi:10.1177/1081286512469980.
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Sène, A., 2001. Modelling of piezoelectric static thin plates. Asymptot. Anal 25 (1), 1–20. Su, B., 1997. Novel Fabrication Processing for Improved Lead Zirconate Titanate (PZT) Ferroelectric Ceramic Materials Ph.D. thesis. Faculty of Engineering, University of Birmingham. Taylor, G.W., Gagnepain, J.J., Meeker, T.R., Nakamura, T., Shuvalov, L.A., 1985. Piezoelectricity. Pure and Applied Mathematics (Boca Raton), 82. Gordon and Breach Science Publishers New York. Trabucho, L., Viaño, J.M., 1996. Mathematical modelling of rods. In: Ciarlet, P.G., Lions, J.L. (Eds.), Handbook of Numerical Analysis, IV. North-Holland, Amsterdam, pp. 487–974. Tutek, Z., 1987. A homogenized model of rod in linear elasticity,. In: Ciarlet, P.G., Sanchez-Palencia, E. (Eds.), Applications of Multiple Scaling in Mechanics, IV. Masson, Paris, pp. 316–332. ´ I., 1986. A justification of the one-dimensional linear model of elasTutek, Z., Aganovic, tic beam. Math. Methods Appl. Sci 8 (4), 502–515. doi:10.1002/mma.1670080133. Viaño, J.M., Ribeiro, C., Figueiredo, J., 2005. Asymptotic modelling of a piezoelectric beam. In: Proceedings of the II ECCOMAS Thematic Conference on Smart Structures and Materials. Viaño, J.M., Figueiredo, J., Ribeiro, C., Rodríguez-Arós, Á., 2015. A model for bending and stretching of piezoelectric rods obtained by asymptotic analysis. Z. Angew. Math. Phys 66 (3), 1207–1232. doi:10.1007/s00033-014-0438-1. Weller, T., Licht, C., 2002. Analyse asymptotique de plaques minces linéairement piézoélectriques. C. R., Math., Acad. Sci. Paris 335 (3), 309–314. doi:10.1016/ S1631-073X(02)02457-3. Weller, T., Licht, C., 2004. Asymptotic modeling of linearly piezoelectric slender rods. C. R., Méc., Acad. Sci. Paris 13, 57–63. Juan Viaño, born in 1955, got a Ph.D. in Mathematics at the University of Santiago de Compostela and a Ph.D. in Numerical Analysis at the University Pierre et Marie Curie (Paris 6). He is Professor of Applied Mathematics at the University of Santiago de Compostela since 1988, member of the Promoter Group of the Node in Galicia of National Spanish Centre of Mathematics (IEMath-Galicia) since 2008 and member of the Promoter Group of Consortium of Technological Institute of Industrial Mathematics since 2012. He has been Head of the Department of Applied Mathematics ath the University of Santiago de Compostela for 6 years, Dean of the Faculty of Mathematics at the University of Santiago de Compostela for 8 years and Vice-rector of Teaching and Academic Organisation of the University of Santiago de Compostela for 2 years. He is and has been assesor and member of a number of committes of national and international level. He has published more than 70 papers in international journals, several books and directed more than 10 doctoral theses. He has also directed more than 10 national and international research projects. Jorge Figueiredo, born in 1966, has a Bachelor degree in Physics/Applied Mathematics (Porto University, 1988) and a M.Sc. in Astrophysics and Spatial Techniques (Paris University, 1989). He got his Ph.D. in Applied Mathematical Sciences form Porto University (1996), under the theme A new numerical code to model pre-main-sequence lowmass stars. He joined the Dep. of Mathematics and Applications of Minho University as a Junior Teaching Assistant in 1993 and became Assistant Professor in 1996. The main research interests concern PDEs with particular emphasis on deriving reduced beam models from 3D piezoelectricity and developing high-order finite volume schemes for the shallow-water equations. Carolina Ribeiro received her Bachelor degree in Mathematics from University of Porto and her Ph.D. degree in Mathematical Science, under the theme Asymptotic derivation of models for anisotropic piezoelectric beams and shallow arches, from University of Minho in 2009. She joined the Department of Mathematics and Applications of the University of Minho, as a Junior Teaching Assistant in 2000 and was promoted to Assistant Professor in 2009. Carolina’s research interests focus in the area of partial differential equations with a particular emphasis on deriving beam models from 3D piezoelectricity and developing high-order finite volume schemes for the shallow water equations. Ángel Rodríguez-Arós, born in 1976, has a Ph.D. in Applied Mathematics from the University of Santiago de Compostela, Spain, with European special mention (2005) and spent a post-doctoral year at the ONERA-site Toulouse, France (2008). Currently, he is Lecturer at the University of A Coruña. Over the years he has published over 20 papers in international journals, given around 20 talks in national and international meetings and has been member of nearly 10 national and international research projects. He has also published a text book in spherical trigonometry with applications to navigation (2012).
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International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
A high order model for piezoelectric rods: An asymptotic approach J.M. Viaño a, C. Ribeiro b, J. Figueiredo b, Á. Rodríguez-Arós c,∗ a
Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, Spain Departamento de Matemática e Aplicações and Centro de Matemática, Universidade do Minho, Portugal c Departamento de Métodos Matemáticos e Representación, Universidade da Coruña, Spain b
a r t i c l e
i n f o
Article history: Received 21 March 2014 Revised 30 October 2015 Available online 11 December 2015 Keywords: Asymptotic analysis Elastic rod Beams Piezoelectric material Anisotropy
a b s t r a c t This paper is devoted to the study of an electromechanical model for a linear transversely inhomogeneous anisotropic rod made of piezoelectric symmetry class 2 materials. The materials in this class include, among many others, Barium Titanate (BT), Lead Zirconium Titanate (PZT) and Polyvinylidene Fluoride (PVDF). For that purpose, we use asymptotic analysis in order to derive from the 3D piezoelectricity problem a reduced model, without making any a priori assumptions of geometrical or mechanical nature. This process involves considering the diameter of the cross section h as small parameter, the assumption of existence of asymptotic expansions of the unknowns (displacements field and electric potential) and the characterization of their respective leading terms as the unique solutions of limit reduced coupled equations. Finally, the validity of the limit reduced model is supported by providing rigorous convergence results. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction The so-called “intelligent structures”, i.e. those that are able to adapt in an automatic way when submitted to external actions in order to provide an optimal behavior, play an increasing role in the framework of nowadays security and wellbeing tools, and their importance will certainly increase in the near future. The use of piezoelectric materials is a strategy that has been used with considerable success in the development of new materials and technologies for the production of these structures. The piezoelectric effect is an electromechanical property that is exhibited by many materials: when the material is deformed by the action of external forces or pressure, an electric field appears – direct piezoelectric effect – and conversely, when subject to an electric potential, the material suffers a deformation – inverse piezoelectric effect. This reciprocity provides both an efficient sensor capacity and the possibility of acting over large portions of structures without a considerable weight increase. For more complete information on the properties and use of piezoelectric materials see, for example, Banks et al. (1996); Ikeda (1990); Nye (1985); Royer and Dieulesaint (2000); Taylor et al. (1985). More specifically, information about traditional piezoelectric beam theories can be consulted in Bellis and Imperiale (2014); Bisegna (1998); Rovenski and Abramovich (2009). ∗
Corresponding author. Tel.: +34 981167000. E-mail addresses: [email protected] (J.M. Viaño), [email protected] (C. Ribeiro), jmfi[email protected] (J. Figueiredo), [email protected] (Á. Rodríguez-Arós). http://dx.doi.org/10.1016/j.ijsolstr.2015.12.005 0020-7683/© 2015 Elsevier Ltd. All rights reserved.
From the above considerations one easily concludes on the relevance of designing new multifunctional materials (e.g. piezoelectric materials) and, consequently, on the importance of accurately knowing the electromechanical behavior of structures in which these materials are applied. Some important results in this field were successfully obtained by applying two different modeling techniques: homogenization and asymptotic analysis (see Cagnol et al., 2008; Miara et al., 2007 for a review of the state of the art). See in particular, Castillero et al. (1998); Ghergu et al. (2007); Mechkour (2004); Miara et al. (2005) for homogenization of piezoelectric plates and shells and Collard and Miara (2002); Figueiredo and Leal (2006); Rahmoune et al. (1998); Sabu (2002); Sène (2001); Weller and Licht (2002); Weller and Licht (2004) for formal asymptotic analysis to justify lower dimensional constitutive laws for piezoelectric plates, shells and rods. The above cited works were based on relevant and pioneer work in applying homogenization and asymptotic analysis techniques in elasticity problems (linear and nonlinear). See in particular Ciarlet (1980); Collard and Miara (1997); Le Dret and Raoult ´ (1995); Lewinski and Telega (2000); Miara (1994); Sanchez-Hubert and Sanchez-Palencia (1997) for plates, Bermúdez and Viaño (1984a); Cimetière et al. (1988); Mascarenhas and Trabucho (1990); Trabucho and Viaño (1996); Tutek (1987); Tutek and Aganovic´ (1986) for rods, and Caillerie and Sanchez-Palencia (1995); Ciarlet (2000); Ciarlet et al. (1996); Pitkaranta and Sanchez-Palencia (1997) for shells. As it is well known, the classical theories of thin structures obtained from classical formulations are intuitive for understanding, but the a priori assumptions are often mechanically unjustified and can reveal inadequate for the analysis of real structures if those
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
a priori assumptions are not met. Although the amount of research in the asymptotic analysis field is quite large, only a small number of works exist (as far as we know) devoted to the derivation of models for piezoelectric rods (see Figueiredo and Leal, 2006; Weller and Licht, 2004). Despite the progress made in those papers in order to derive reduced models for piezoelectric beams, the subject is still open to further research. In this framework, our contribution consists in the obtention of a reduced model of a linear transversely inhomogeneous anisotropic rod made of piezoelectric symmetry class 2 materials, by using the asymptotic method introduced by Lions (see Lions, 1973), taking the diameter of the rod cross section h as small parameter, without any a priori hypothesis of geometrical or mechanical nature. The model obtained in this way allows to, simultaneously, justify and/or complement the traditional piezoelectric beam theories (such as those of Euler–Bernoulli, Timoshenko, and Vlasov). Additionally, in the reduced model, to our knowledge for the first time, the electric potential is coupled with first order terms of the asymptotic expansion of the displacements. The present work is organized in the following manner. In Section 2 we show the 3D piezoelectric problem posed in its domain of reference h (the details on the notation used will be specified in the corresponding sections and can be found in tables in the appendix as well) and formulate it in the form of the virtual work principle (see (2.13)): h Find (uh , ϕ h ) ∈ V h (h ) × H 1 (h ), such that ϕ h = ϕˆ h on eD , σihj (uh , ϕ h )ehij (vh )dxh + Dhk (uh , ϕ h )Ekh (ψ h )dxh h h = fih vhi dxh + ghi vhi d h ,
h
Nh
for all (v , ψ ) ∈ V ( ) × ( ), h
uh
h
(uhi )
h
h
h
h
ϕh
where = and are the displacements field and the electric potential, respectively, to be found in the corresponding admissible spaces Vh (h ) and H1 (h ). Also, σ h = (σihj ) and Dh =
(Dhi ) represent the stress tensor field and the electric displace-
ments field, respectively, while eh (vh ) = (ehij (vh )) = ( 21 (
∂vh ∂vhi + hj )) ∂ xhj ∂ xi
and E h = (Eih (ψ h )) = (− ∂ψh ) are the deformation operator and mih
∂ xi
nus the gradient operator. The constitutive equations are (see (2.3)):
σihj (uh , ϕ h ) = Cihjkl ehkl (uh ) − Pkih j Ekh (ϕ h ), h h ekl (uh ) + εihj E hj (ϕ h ), Dhi (uh , ϕ h ) = Pikl
in h , in h ,
where C h = (Cihjkl ), P h = (Pkih j ) and εh = (εihj ) are the stiffness, piezoelectric and dielectric tensors, respectively. In Section 3, we perform a change of variable and a scaling of the unknowns (to obtain the displacements field u(h) and the electric potential ϕ (h)) and data, so that the virtual work formulation is now posed in a domain independent on h, denoted by . In Section 4, we assume that the scaled unknowns can be written in the form of asymptotic developments, allowing to identify and characterize the corresponding leading terms; this procedure yields the new reduced model, which consists on a set of 1D variational problems for the displacements field and a variational problem for the electric potential and the first order term of the displacement, where some terms are not local. The strong convergence of the unknowns as the small parameter tends to zero is proved in Section 5. This is a key step in order to give a rigorous mathematical justification of the reduced models of Section 4. In Section 6 we undo the change of variable and de-scale the quantities appearing in the reduced model and constitutive equations derived in Section 4 in order to express both in the original reference
295
domain h and thus have a true physical meaning. We find that the asymptotic expansion of the true displacements field uh = (uhi ) is of the form:
uhα = ξαh + u1h α + O ( h ),
2 uh3 (xh ) = ξ3h − xhβ ∂3h ξβh + u1h 3 + O ( h ),
where the leading term u0h = (u0h ) is characterized by functions ξαh ∈ i 2 h 1 H0 (0, L ), ξ3 ∈ H0 (0, L ), satisfying the purely mechanical stretching and beam equations (see (6.5) and (6.6)):
ξαh ∈ H02 (0, L ), =
L 0
L 0
Fαh χαh dxh3 −
h h h Y Iαh ∂33 ξα ∂33 χαh dxh3
L 0
Mαh ∂3h χαh dxh3 ,
(no sum on α ),
χαh ∈ H02 (0, L ),
for all
ξ3h ∈ H01 (0, L ), =
L 0
L 0
YA(ωh )∂3h ξ3h ∂3h χ3h dxh3
F3h χ3h dxh3 , for all χ3h ∈ H01 (0, L ).
On the other hand, u1h = (u1h ), the following order term of uh , is of i the following form: h h h u1h α = zα + δα z ,
h h h h h h u1h 3 = −r − z3 − xα (zα ) − w (z ) ,
where wh ∈ R() is the warping function, depending only on the geh , zh ∈ H 1 (0, L ), zh ∈ L2 (0, L ), and ometry and stiffness tensor, while zα 0 3 h r ∈ R() is a function depending on the electric potential, thus giving the coupling between the mechanical and electrical quantities. More specifically, we find that the asymptotic expansion of the true electric potential ϕ h is
ϕ h = ϕˆ 0h + ϕ¯ 0h + O(h2 ), where ϕ¯ 0h ∈ l0 () and the coupled equations for zh , rh and ϕ¯ 0h are given by (see (6.8)–(6.10)):
∂αh wh − δαh ∂βh ϕ¯ 0h dxˆh ∂3h ζ h dx3 ωh 0 0 L h = Pβh3α ∂βh ϕˆ h ∂αh wh − δαh dxˆ ∂3h ζ h dx3 ∀ζ h ∈ H01 (0, L ), ωh 0 L h C3hα 3β ∂βh rh ∂αh ρh dxˆ ρ3h dx3 ωh 0 L h − Pβh3α ∂β ϕ¯ 0h ∂α ρh dxˆ ρ3h dx3 ωh 0 L h = Pβh3α ∂β ϕˆ h ∂α ρh dxˆ ρ3h dx3 ∀ρh ∈ Q h (ωh ),
L
Jh ∂3h zh ∂3h ζ dx3 −
L
Pβh3α
ωh
0
× ρ3h ∈ L2 (0, L ), L h εαβ ∂βh ϕ¯ 0h ∂αh ψh dxˆh ψ3h dx3 0
ωh L
ψ3h dx3 L h + Pβh3α ∂αh wh − δαh ∂βh ψh dxˆ ∂3h zh ψ3h dx3 ωh 0 L h =− εαβ ∂βh ϕˆ h ∂αh ψh dxˆh ψ3h dx3 ∀ψh ∈ S(ωh ), +
0
0
Pβh3α ∂αh rh ∂βh ψh dxˆ
h
ωh
ωh
× ψ3h ∈ L2 (0, L ). h and zh , these Notice that even though we know the existence of zα 3 functions are not characterized at this stage. Nevertheless, they are 0h not needed in order to obtain ϕ , which depends only on u1h through rh . We then briefly comment on some particular cases, specifically when the cross-section is square, which leads to some practical simplifications, and we finish by giving some conclusions.
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Table 1 List of relevant sets.
Table 3 List of scaled functions.
Notation
Description
ω
Domain in R2 with boundary γ , Reference configuration of the scaled beam/rod defined as = ω × (0, L ), ¯, A point in Cross section of the beam/rod defined as hω, with boundary γ h , Reference configuration of the beam/rod defined as h = ωh × (0, L ), Boundary of h , that is h = ∂ h = [0, L] × γ h , ¯ h, A point in Left end of the rod, that is 0h = ω ¯ h × { 0} Right end of the rod, that is Lh = ω ¯ h × {L} Part of h where the electric potential is given, h Part of h which is complementary of eD , Part of h with a condition of weak clampling. Dh = 0h ∪ Lh , Part of h where tractions are given.
x = ( xi )
ωh h h
xh = (xhi )
0h Lh h eD h eN Dh Nh
Table 2 List of functions. Description
nh = (nhi ) : h → R3 h ¯ h → R3 f = ( fih ) : gh = (ghi ) : ¯ Nh → R3 ¯ h → R3 uh = (uhi ) : ϕ h : ¯ h → R h ϕˆ h : eD →R
Unit outer normal vector to h , Density of volume forces, Density of tractions, Displacements field, Electric potential, h Electric potential given in eD and its trace lifting to H1 (h ), h Homogeneous part of ϕ , therefore given as ϕ h − ϕˆ h , Stress tensor,
h C¯ , Mh , Nh
δα : ω → R
¯ →R u(h ) = (ui (h )) :
Description 3
ϕ¯ (h ) : ¯ → R ϕˆ : eD → R ϕ (h ) : ¯ → R σ (h ) = (σi j (h )) : → S3 D(h ) = (Di (h )) : → R3
e(u(h )) = (ei j (h )) : → S3
Notation
ϕ¯ h : ¯ h → R σ h = (σihj ) : h → S3 Dh = (Dhi ) : h → R3 eh (uh ) = (ehij ) : h → S3 E (ϕ h ) = (Eih ) : h → R3 C h = (Cihjkl ) P h = (Pkih j ) εh = (εihj )
Notation
Electric displacement, Linearized strain tensor, Electric field, Stiffness tensor, Piezoelectric tensor, Dielectric tensor, Auxiliary matrices defined from components of Ch in (2.7)–(2.8), Auxiliary functions δ1 (x1 , x2 ) = x2 , δ2 (x1 , x2 ) = −x1 .
For the sake of readability we have also included Tables 1–5. where we list the main functions, constants, sets and functional spaces involved in this paper. 2. The three-dimensional problem Throughout this paper, Latin indices vary over the set {1, 2, 3} and Greek indices over the set {1, 2} for the components of both vectors and tensors. The summation convention on repeated indices will be also used. We denote by Sd the space of second order symmetric tensors on Rd , while “ · ” will represent the inner product on Sd and Rd (d = 2, 3, . . .). Let ω be an open, bounded, connected subset of R2 with Lipschitz continuous boundary γ = ∂ω, having area A(ω). We will study a prismatic rod that we identify with a solid occupying the reference configuration
¯h=ω ¯ h × [0, L], where h ∈ (0, 1] is very small with respect to the length of the rod L and ωh = hω so that A(ωh ) = h2 A(ω ) is the area of the rod cross section ωh , with boundary γ h = ∂ωh . Denoting the boundary of h by h = ∂ h , we consider two partitions for ∂ h : firstly, let us define a partition given by the h and h = γ h × (0, L ), where γ h ⊂ γ h and measurable parts eN eD eD eD h meas(γeD ) > 0; secondly, we consider a partition given by the measurable parts Nh and Dh = 0h ∪ Lh such that meas(Dh ) > 0, where 0h = ω¯ h × {0} and Lh = ω¯ h × {L} are the ends of the rod. An arbitrary
E (ϕ (h )) = (Ei (h )) : → R3
Scaled displacements, with uα (h )(x ) = huhα (xh ) and u3 (h )(x ) = uh3 (xh ); u(h) ∈ V(), Scaled homogeneous electric potential, with ϕ¯ (h )(x ) = h−1 ϕ¯ h (xh ); ϕ¯ (h ) ∈ (), Scaled electric potential in eD and its trace lifting to H1 (); Also, ϕˆ h (xh ) = hϕˆ (x ), Scaled electric potential, ϕ (h ) = ϕ¯ (h ) + ϕˆ , Scaled stress tensor, with h σαβ (xh ) = h2 σαβ (h )(x ), h σ3hα (xh ) = hσ3α (h )(x ), and σ33 (xh ) = σ33 (h )(x ), Scaled electric displacement, with Dhα (xh ) = Dα (h )(x ), and Dh3 (xh ) = h−1 D3 (h )(x ), Scaled linearized strain tensor, with ehαβ (xh ) = h−2 eαβ (h )(x ), eh3β (xh ) = h−1 e3β (h )(x ), and
eh33 (xh ) = e33 (h )(x ), Scaled electric field, with Eαh (ϕ h )(xh ) = Eα (ϕ (h ))(x ), and E3h (ψ h )(xh ) = hE3 (ψ (h ))(x ).
Table 4 List of functions and constants in the scaled domain . Notation
Description
¯ → R3 f = ( fi ) :
Density of scaled volume forces, with fαh (xh ) = h fα (x ), and f3h (xh ) = f3 (x ), Density of scaled tractions, with ghα (xh ) = h2 gα (x ), and gh3 (xh ) = hg3 (x ), Scaled stiffness tensor zero-th order term, Scaled stiffness tensor first order term, Scaled piezoelectric tensor, Scaled dielectric tensor, Auxiliary matrices defined from components of C and C , Terms of the asymptotic expansion of u(h), Terms of the asymptotic expansion of ϕ¯ (h ), Terms of the asymptotic expansion of σ (h), Terms of the asymptotic expansion of D(h), Components in the characterization of u0 ∈ VBN (), For a.e. x3 ∈ (0, L), first two components of u2 , Auxiliary matrices defined from components of C,
g = (gi ) : ¯ N → R3
C = (Ci jkl ) C = (Cijkl ) P = (Pki j ) ε = (εi j ) ¯ M, N, M , N C, u p : → R3 , p ≥ 0 ϕ¯ p : → R, p ≥ 0 σ p : → S3 , p ≥ −4 D p : → R3 , p ≥ −1 ξ = (ξi ) : (0, L ) → R3 uˆ (x3 ) = (uˆ2α (x3 )) : ω → R2 2
Mαβ
α : → R αβ : → R Iα Xαβ , Yα , Z : (0, L ) → R s, sα : (0, L ) → R
Y : (0, L ) → R Fi , Mα : (0, L ) → R P¯ : → R w: (0, L) → R() J : (0, L ) → R zα , z : (0, L ) → R
r : (0, L ) → Q (ω ), z3 : (0, L ) → R C¯ = (C¯i jkl ), P¯ = (P¯ki j ), ε¯ = (ε¯i j )
det M
Functions defined as α (x ) = − det Mαβ xβ , Functions defined using Mαβ , Inertia moment Iα = ω x2α dω, Functions Xαβ = ω αβ dω, Yα = ω αβ δβ dω, Z = ω α δα dω, 1 Functions sα = − X ξ , A(ω ) αβ β 1 Y ξ + Z ξ3 , s=− I1 + I2 β β ¯ C Function Y = detdet , M det N Resultants of forces and moments, det M αβ Function P¯ = P333 − P3αβ , det M
Warping function, Torsion function, Functions in H01 (0, L ) characterizing (u1α ), see (4.49), Functions which characterize u13 , see (4.56), Inverse constitutive tensors of C, P and ε.
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
where C h = (Cihjkl ), P h = (Pkih j ) and εh = (εihj ) are the stiffness, piezo-
Table 5 List of relevant functional spaces. Notation
electric and dielectric tensors, respectively. We assume that Cihjkl , Pkih j ,
Description
V h (h ) = {vh ∈ H 1 (h )
3 h : v = 0 on Dh }
h h (h ) = {ψ h ∈ H 1 (h ) : ψ h = 0 on eD }
V () = {v ∈ [H 1 ()]3 : v = 0 on D }
() = {ψ ∈ H () : ψ = 0 on eD } 1
ˆ () = ϕˆ + ()
V BN () = v ∈ V () : eαβ (v ) = e3β (v ) = 0
2 V 2m (ω ) = vˆ = (vˆ α ) ∈ H 1 (ω ) :
ˆ α dω = ω (x1 vˆ 2 − x2 vˆ 1 ) dω = 0 ωv
V 1 () = v ∈ V () : eαβ (v ) = 0
Subspace of transversal rigid displacements,
εihj ∈ L∞ (h ), implying the boundedness of these tensors, and also
Space of admissible displacements, Space of admissible electric potentials, Space of scaled admissible displacements, Space of scaled admissible electric potentials, Set consisting in {ψ ∈ () : ψ = ϕˆ on eD }, Space of Bernoulli-Navier displacements, Subspace where 2D Korn inequality holds,
Q = Q (ω ) = ρ ∈ H 1 (ω ) : ω ρ d ω = 0 , S = S (ω ) = {ψ ∈ H 1 (ω ) : ψ = 0 on γeD }, l () = {ψ ∈ L2 () : ∂α ψ ∈ L2 (ω )}
Space isomorph to L2 (0, L; H1 (ω)),
R() = L2 (0, L; Q (ω )), l0 () = L2 (0, L; S(ω )), T () = l0 () × R() × H01 (0, L ) X () = V () × (),
Product space, We also define the set ˆ , Xˆ () = V () × ()
ωh
xhα dωh = 0,
ωh
xh1 xh2 dωh = 0.
σ (u , ϕ ) = f , −div Dh (uh , ϕ h ) = 0, −div
h
h
∂ h /∂ xhi
in ,
(2.2)
in , h
= (σihj ) and the electric displacement vector Dh = related to the linear strain tensor (ehij (uh ) = 12 (∂ih uhj + the gradient of the electric potential (Ekh (ϕ h ) = −∂kh ϕ h )
σihj (uh , ϕ h ) = Cihjkl ehkl (uh ) − Pkih j Ekh (ϕ h ), h h ekl (uh ) + εihj E hj (ϕ h ), Dhi (uh , ϕ h ) = Pikl
h C1111
h C1211
h C1122
0
0
h C1133
h C1212
h C1222
0
0
h C1222
h C2222
0
0
0
0
h C3131
h C3132
0
0
h C3132
h C3232
0
h C1233
h C2233
0
0
h C3333
⎞
h ⎟ C1233 ⎟
⎟
h C2233 ⎟ ⎟, 0 ⎟
⎟ ⎠
⎛
h C1111
h C1211
h C1122
⎞
h M h = M h (xh ) := ⎝C1211
h C1212
h ⎠, C1222
h C1122
h C1222
h C2222
N h = N h (xh ) :=
h C3131
h C3132
h C3132
h C3232
.
(2.8)
From properties (2.4) it follows that these matrices are definite positive and therefore det C h > 0, det M h > 0 and det N h > 0. For the moment, and until not explicited otherwise, we will not assume that the conditions traduced by (2.6) hold. We suppose that the rod is weakly clamped at the extremities 0h ∪ Lh , which means that the displacement field uh is such that (see e.g. Trabucho and Viaño, 1996)
ah
uhi dωh = 0,
ah
xhj uhi − xhi uhj dωh = 0,
a = 0, L.
(2.9)
In the following we will represent this boundary condition by uh =
h
through the constitutive law (see e.g. Ikeda, 1990; Jackson, 1975):
ε3hθ = 0, in ¯ h .
(2.7)
the stress tensor σ h
(Dhi ) being ∂ hj uhi )) and
(2.5)
We will assume that the rod is made of piezoelectric materials of symmetry class 2, which comprises the following crystal systems (see e.g. Nye, 1985; Royer and Dieulesaint, 2000): monoclinic (except class ¯ ), m), orthorhombic, tetragonal, hexagonal (except classes 6¯ and 6m2 and cubic. For this class of materials, which includes among many other piezoelectric materials Barium Titanate (BT), Lead Zirconium Titanate (PZT) and Polyvinylidene Fluoride (PVDF), the following conditions hold:
h C1133
rium equations (see e.g. Ciarlet, 1988; Jackson, 1975): h
(2.4)
h h ¯ h. εihj = ε hji , Pmi in j = Pm ji ,
h h Cihjkl = Ckli j = C jikl ,
⎜C h ⎜ 1211 ⎜C h h h ⎜ h C¯ = C¯ (x ) := ⎜ 1122 ⎜ 0 ⎜ ⎝ 0
i
h
( ξi ) 2 ,
i=1
¯ h . Furthermore, the following symmetries hold: for every xh ∈
⎛
... derivative with respect to the (only) variable x3 (or xh3 ). ¯ h is a perfect dielectric body, so that there Now, assuming that is neither volume nor surface electric charge, and that volume forces of density f h = ( fih ) act in h , the resulting displacement field uh = ¯ h → R3 and electric potential ϕ h : ¯ h → R solve the equilib( uh ) : h
3
(2.6)
We define the following differential operators: = and ∂i = ∂ /∂ xi . For a function χ : [0, L] → R we note χ , χ , ... its first, second,
h
εihj (xh )ξi ξ j ≥ c
(Mi j )2 ,
i, j=1
(2.1)
∂ih
3
Cihjkl (xh )Mi j Mkl ≥ c
Accordingly, it is useful to define the following symmetric matrices:
¯ h will be denoted by xh = (xh ) and the unit outer norpoint of i mal vector to the boundary ∂ h by nh = (nhi ). The coordinate system Oxh1 xh2 xh3 will be assumed a principal system of inertia associated to ωh , which means that
the following coercive relations: there exists a constant c > 0 such that for all symmetric second order tensors M = (Mi j ) ∈ S3 and every vector ξ = (ξi ) ∈ R3 one has
h h C3hρ 33 = C3hθ αβ = 0, Pθhρσ = P33 α = Pβ 33 = 0,
M 1 () = τ = (τi j ) ∈ [L2 ()]3×3 : τi j = τ ji , M 2 () = [L2 ()]3 , M () = M 1 () × M 2 (), V RD () = {v ∈ V 1 () : v3 ∈ L2 (0, L; Q (ω ))}, V R () = {v ∈ V RD () : vα = δα s, s ∈ H01 (0, L )}, Subspace of admissible rotations, Subspace of admissible V T () = {v ∈ V RD () : vα = sα , translations, v3 = −xα sα , sα ∈ H01 (0, L ) , V˙ RD () := V RD ()/V T (), Quotient space, V H () = H 1 (0, L; V 2m (ω )).
297
in h , in h ,
(2.3)
0 on Dh . Remark 2.1. As we will see, the use of the “ average clamping” condition (2.9) instead of a strong clamping condition such as uh = 0 on Dh allows to avoid the “ boundary layer problem” (see Ref. Lions, 1973) that arises when a strong clamping condition is used - which turns out to be related to the fact that in general a strong clamping of the rod is physically impossible (see e.g. Trabucho and Viaño, 1996 and references therein).
Nh
We also assume that surface tractions of density gh = (ghi ) act on h . In the following and that an electric potential ϕˆ h is applied on eD
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h ) are we suppose that fih = L2 (h ), ghi ∈ L2 (Nh ), and ϕˆ h ∈ H 1/2 (eD given quantities. Thus, the boundary conditions satisfied by (uh , ϕ h ) are:
⎧ σ h (uh , ϕ h )nh = gh , ⎪ ⎪ h ⎪ ⎨ u = 0,
⎪ D (u , ϕ ) · n = 0, ⎪ ⎪ ⎩ ϕ h = ϕˆ h , h
h
h
h
on Nh , on Dh ,
(2.10)
h on eN , h on eD .
Let us now define the following closed subspaces of the Sobolev space of order one [H1 (h )]m , m ≥ 1:
h v ∈ [H 1 (h )]3 : vh = 0 on Dh ,
h h (h ) := ψ h ∈ H 1 (h ) : ψ h = 0 on eD .
V h (h ) :=
(2.11)
h ) > 0, it follows from Korn’s Since meas(Dh ) > 0 and meas(eD and Poincaré–Friedrichs’ inequalities that the following norms are equivalent to the standard Sobolev · 1,h norms on Vh and h , respectively,
vh ∈ V h (h ) → |eh (vh )|0,h , ψ h ∈ h (h ) → |E h (ψ h )|0,h , where | · |0,h stands for the usual norm in [L2 (h )]m , m ≥ 1.
ϕ¯ h
ϕh
− ϕˆ h ,
ϕˆ h
small parameter h as usually done in the elastic rod case (see e.g. Bermúdez and Viaño, 1984b; Trabucho and Viaño, 1996 and refer- ences therein). We will study the dependence of the solution uh , ϕ¯ h with respect to h. The technique of change of variable to a fixed domain and subsequent rescaling of the displacement and electric potential will allow us to derive variational problems equivalent to (2.12) and (2.13) where h shows up in an explicit way in the rescaled equations. To start, we perform a change of variable to the reference domain through the following transformation (see e.g. Trabucho and Viaño, 1996)
¯ h : x = (x1 , x2 , x3 ) ∈ h h h h h ¯ h. → x = (x ) = x1 , x2 , x3 = (hx1 , hx2 , x3 ) ∈
(3.1)
Moreover,
Nh = h (N ), Dh = h (D ), 0h = h (0 ), Lh = h (L ), h = h (eN ), n(x ) = nh (xh ), eN
h h γeD = h (γeD ), eD = h (eD ),
where n is the unit outer normal to the set ∂ = D ∪ N = eD ∪ eN and
eD = γeD × (0, L ).
Defining = where now represents a trace lifting h , the variational in H1 (h ) of the boundary potential acting on eD problem associated to (2.2), (2.3) and (2.10 ) is
Furthermore, condition (2.1) becomes now
Find (uh , ϕ¯ h ) ∈ V h (h ) × h (h ) such that × Cihjkl ehkl (uh )ehij (vh )dxh + εihj ∂ih ϕ¯ h ∂ hj ψ h dxh h h h h h h h + Pmi ¯ ei j (vh ) − ehij (uh )∂m ψ h )dxh j ( ∂m ϕ h = fih vhi dxh + ghi vhi d h − εihj ∂ih ϕˆ h ∂ hj ψ h dxh
In view of (2.9) we will represent by v = 0 on D the boundary condition
−
h
h N
h
h h h h Pmi ˆ ei j (vh )dxh , j ∂m ϕ
a
a
Remark 2.2. The derivation of (2.12) follows closely the steps presented in Bernadou and Haenel (1999) for the strong clamping case with minor changes to accommodate the weak clamping condition adopted here.
h Find (uh , ϕ h ) ∈ V h (h ) × H 1 (h ), such that ϕ h = ϕˆ h on eD , σihj (uh , ϕ h )ehij (vh )dxh + Dhk (uh , ϕ h )Ekh (ψ h )dxh × h h = fih vhi dxh + ghi vhi d h ,
Nh
× for all (v , ψ ) ∈ V h (h ) × h (h ), with the functions
σihj , Dhk
ω
x1 x2 dω = 0.
vi dω = 0,
a
vi dω = 0,
a
(3.2)
(x j vi − xi v j )dω = 0,
δα vα dω = 0,
a
a = 0, L,
xα v3 dω = 0,
a = 0, L,
where δ1 (x1 , x2 ) = x2 , δ2 (x1 , x2 ) = −x1 . We now define the following closed subspaces of [H1 ()]m , m ≥ 1 (cf. (2.11))
V () = {v ∈ [H 1 ()]3 : v = 0 on D },
() = {ψ ∈ H 1 () : ψ = 0 on eD }, endowed with the following norms equivalent to the usual Sobolev norms:
||v||V = |e(v )|0, , ||ψ|| = |∇ψ|0, , where | · |0, stands for the standard norm in [L2 ()]m , m ≥ 1, and
e(v ) = (ei j (v )),
Proposition 2.3. Problem (2.12) is formally equivalent to
h
(3.3)
for all (vh , ψ h ) ∈ V h (h )
It can be easily shown that the boundedness of Ch , Ph , εh , fh and gh together with properties (2.4) ensure the existence and uniqueness of solution of problem (2.12 ) as a result of Lax–Milgram Lemma.
h
xα dω = 0,
that is, taking (3.2) into account,
(2.12)
h
ω
h
× h (h ).
(2.13)
: → R defined as in (2.3).
The above formulation corresponds to the principle of virtual work. 3. Change of variable to the reference rod The major geometric feature of a three-dimensional rod is the fact that the largest cross sectional dimension is very small compared with its length (h L), cause of ill-conditioning of the threedimensional problem. We take advantage of this property to use an asymptotic expansion method (see Lions, 1973) with respect to the
ei j ( v ) =
1 (∂i v j + ∂ j vi ). 2
In order to obtain a problem in equivalent to (2.12) we proceed as follows (see e.g. Viaño et al., 2005; Viaño et al., 2015) . With the unknown and test displacement fields uh , vh in Vh (h ), we associate the (unknown and test) scaled displacement fields u(h ) = (ui (h )) and v(h ) = (vi (h )) in V() defined by the following scalings valid for all ¯: xh = h (x ), x ∈
uα (h )(x ) = huhα (xh ),
u3 (h )(x ) = uh3 (xh ),
vα (h )(x ) = hvα (x ), v3 (h )(x ) = v (x ). h
h
Similarly, the electric potential ϕ¯ h
h 3
h
and the test function ψ h
(3.4) (3.5) in h (h )
are associated with the scaled potential ϕ¯ (h ) and the scaled (test) function ψ (h) in () using the following scaling valid for all xh = ¯: h (x ), x ∈
ϕ¯ (h )(x ) = h−1 ϕ¯ h (xh ),
(3.6)
ψ (h )(x ) = h−1 ψ h (xh ).
(3.7)
Furthermore, we consider the following hypothesis on the magnitude of the data with respect to the diameter of the rod cross-section h:
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
1. There exist functions fi ∈ L2 () and gi ∈ L2 ( N ), independent of h, such that:
fαh (xh ) = h fα (x ), f3h (xh ) = f3 (x ), for all xh = h (x ) ∈ h , ghα (xh ) = h2 gα (x ), gh3 (xh ) = hg3 (x ), for all xh = h (x ) ∈ Nh .
(3.8) 2. There exists a function ϕˆ ∈ H 1/2 (eD ), independent of h, such that: h ϕˆ h (xh ) = hϕˆ (x ), for all xh = h (x ) ∈ eD .
We also denote by ϕˆ ∈
ϕ (h ) = ϕ¯ (h ) + ϕˆ .
H 1 ()
(3.9)
a trace lifting of ϕˆ and define
In addition to that, it is assumed that there exist functions Cijkl ∈
L∞ (0, L), and Cijkl , Pkij , ε ij ∈ L∞ (), independent of h, such that:
Cihjkl (xh1 , xh2 , xh3 ) = Ci jkl (x3 ) + hCijkl (x1 , x2 , x3 ), Pkih j (xh ) = Pki j (x ),
εihj (xh ) = εi j (x ),
(3.10)
¯ . We also assume that the for all xh = h (x ), x = (x1 , x2 , x3 ) ∈ properties (2.4)–(2.6) hold when we drop the superindex (take h = 1) and for Cijkl as well. Consequently, there exist tensor fields M, N, M and N such that
Mh (xh1 , xh2 , xh3 ) = M(x3 ) + hM (x1 , x2 , x3 ), Nh (xh ) = N(x3 ) + hN (x1 , x2 , x3 ), xh
h (x )
= where of h (see (2.8)).
(3.11)
¯ h, x ∈ ¯ , with M, N, M and N independent ∈
Remark 3.1. The Ansatz elastic tensor (3.10) includes the material’s inhomogeneity effects in the piezoelectric response predominantly given by the leading term. However, the elastic tensor also accounts 3D structural parameters (such as grain size for PZT’s, see e.g. Naceur et al., 2014; Su, 1997) effects on the piezoelectric response, through the perturbation introduced by the small parameter h. Note that other authors (e.g. Figueiredo and Leal, 2006) have assumed that there exist functions Cijkl independent of h, such that:
Cihjkl
(x ) = Ci jkl (x ) ∀x = (x ), h
h
h
¯. x∈
(3.12)
We shall further comment on the implications of these two different assumptions (see Remark 4.8 and Ref. Viaño et al., 2015). Combining transformation (3.1) with the notations (3.5) and (3.7) we have for all vh ∈ Vh (h ), ψ h ∈ h (h ) and xh = h (x ), x ∈ :
⎧ h h h −2 ⎨eαβ (v )(x ) = h eαβ (v(h ))(x ), eh (vh )(xh ) = h−1 e3β (v(h ))(x ), ⎩ 3hβ h h e33 (v )(x ) = e33 (v(h ))(x ),
(3.13)
Eαh (ψ h )(xh ) = −∂αh ψ h (xh ) = −∂α (ψ (h ))(x ) = Eα (ψ (h ))(x ),
E3h (ψ h )(xh ) = −∂3h ψ h (xh ) = −h∂3 (ψ (h ))(x ) = hE3 (ψ (h ))(x ). (3.14)
Using the scalings defined previously for the displacement vector and for the electric potential together with the above assumptions and results, we can reformulate the variational problem (2.12) into ¯ independent of h. Assuming another variational problem posed in from here on that we are dealing with piezoelectric materials of symmetry class 2 (i.e. properties (2.6) hold), we have the following result. Proposition 3.2. Let (u(h ), ϕ¯ (h )) ∈ V () × () be the functions obtained from (uh , ϕ¯ h ) ∈ V h (h ) × h (h ), the unique solution of problem (2.12), by transformation (3.1) and scalings (3.4) and (3.6). Bearing in mind (2.6) as well as scalings (3.5) and (3.7)–(3.13),
299
then (u(h ), ϕ¯ (h )) is the unique solution of the following variational problem:
(u(h ), ϕ¯ (h )) ∈ V () × (),
aˆ(h )((u(h ), ϕ¯ (h )), (v, ψ ))
= lˆ(h )(v, ψ ) ∀(v, ψ ) ∈ V () × (),
(3.15)
where the bilinear form aˆ(h ) and the linear functional lˆ(h ) are defined, respectively, by
h−4Cαβθ ρ + h−3Cαβθ aˆ(h )((u, ϕ¯ ), (v, ψ )) = ρ eθ ρ (u )eαβ (v )dx + 4h−2C3α 3β + 4h−1C3α 3β e3α (u )e3β (v )dx + h−2Cαβ 33 + h−1Cαβ e33 (u )eαβ (v ) + eαβ (u )e33 (v ) dx 33 −1 P3αβ ∂3 ϕ¯ eαβ (v ) − eαβ (u )∂3 ψ dx +h −1 +2h Pβ 3α ∂β ϕ¯ e3α (v ) − e3α (u )∂β ψ dx C3333 + hC3333 e33 (u )e33 (v )dx + + εαβ ∂α ϕ∂ ¯ β ψ dx + h P333 (∂3 ϕ¯ e33 (v ) − e33 (u )∂3 ψ )dx +h2 ε33 ∂3 ϕ∂ ¯ 3 ψ dx (3.16)
and
lˆ(h )((v, ψ )) = −2h−1 Pβ 3α ∂β ϕˆ e3α (v )dx + fi vi dx + gi vi d N − εαβ ∂α ϕ∂ ˆ β ψ dx − h P333 ∂3 ϕˆ e33 (v )dx − h2 ε33 ∂3 ϕ∂ ˆ 3 ψ dx. (3.17)
The existence and uniqueness of solution of problem (3.15)–(3.17) is consequence of Lax–Milgram Lemma, since aˆ(h ) is (V × )-elliptic and both aˆ(h ) and lˆ(h ) are (V × )-continuous, as the reader can easily check having in mind properties (2.4), the boundness of C, C , P, ε, f and g, and that 0 < h ≤ 1. We now consider the following scalings for the stress tensor σ (h ) = (σi j (h )) and the electric displacement ¯, vector D(h ) = (Di (h )), valid for all xh = h (x ), x ∈
⎧ h h 2 h h ⎪ ⎨σαβ (x ) = h σαβ (h )(x ), σ3α (x ) = hσ3α (h )(x ), h h σ33 (x ) = σ33 (h )(x ), ⎪ ⎩Dh (xh ) = D (h )(x ), Dh (xh ) = h−1 D (h )(x ). α 3 α 3
(3.18)
From transformation (3.1) and scalings (3.13)–(3.14) and (3.18), the variational problem (2.13) and the constitutive law (2.3) allow to deduce the following scaled principle of virtual work and scaled consti¯. tutive law posed in Proposition 3.3. Problem (3.15)–(3.17) is formally equivalent to
Find (u(h ), ϕ (h )) ∈ V () × H 1 (), such that ϕ (h ) = ϕˆ on eD , σi j (h )ei j (v )dx + Dk (h )Ek (ψ )dx = fi vi dx + gi vi d ,
for all (v, ψ ) ∈ V () × (),
N
(3.19)
300
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where σ (h ) = σ (h )(u(h ), ϕ (h )) and D(h ) = D(h )(u(h ), ϕ (h )) are given by
⎧ σαβ (h ) = h−4Cαβθ ρ + h−3Cαβθ ⎪ ρ eθ ρ (u (h )) ⎪ ⎪ ⎪ ⎪ + h−2Cαβ 33 + h−1Cαβ e (u(h )) − h−1 P3αβ E3 (ϕ (h )), ⎪ 33 33 ⎪ ⎪ ⎪ −2 −1 −1 ⎪ ⎪ ⎨σ3α (h ) = 2 h C3α3β + h C3α3β e3β (u(h )) − h Pβ 3α Eβ (ϕ (h )), σ (h ) = h−2C33αβ + h−1C33 αβ eαβ (u(h )) ⎪ 33 ⎪ ⎪ ⎪ + C3333 + hC3333 e33 (u(h )) − hP333 E3 (ϕ (h )), ⎪ ⎪ ⎪ ⎪ ⎪ Dα (h ) = 2h−1 Pα 3β e3β (u(h )) + εαβ Eβ (ϕ (h )), ⎪ ⎪ ⎩ D3 (h ) = h−1 P3αβ eαβ (u(h )) + hP333 e33 (u(h )) + h2 ε33 E3 (ϕ (h )). (3.20)
−2 σ33 = C33αβ eαβ (u0 ), −1 0 σ33 = C33αβ eαβ (u1 ) + C33 αβ eαβ (u ),
(4.13)
0 1 0 σ33 = C33αβ eαβ (u2 ) + C33 αβ eαβ (u ) + C3333 e33 (u ), 0 D−1 α = 2Pα 3β e3β (u ),
D0α = 2Pα 3β e3β (u1 ) + εαβ Eβ (ϕ 0 ),
(4.15)
0 D−1 3 = P3αβ eαβ (u ),
D03 = P3αβ eαβ (u1 ),
(4.16)
D13 = P3αβ eαβ (u2 ) + P333 e33 (u0 ), D23 = P3αβ eαβ (u4 ) + P333 e33 (u3 ) + ε33 (ϕ 1 ).
4. Asymptotic method
(4.14)
(4.17)
We now assume that the solution of problem (3.15)–(3.17) can be expressed in the form
4.1. Characterization of u0 and u2
u(h ) = u0 + hu1 + h2 u2 + · · ·,
Let VBN () be the space of Bernoulli–Navier displacements defined by
ui ∈ V (),
(4.1)
ϕ¯ (h ) = ϕ¯ 0 + hϕ¯ 1 + h2 ϕ¯ 2 + · · ·, ϕ¯ i ∈ (),
(4.2)
where the successive coefficients of the powers of h are independent of h . Since ϕ¯ (h ) = ϕ (h ) − ϕˆ , then (4.2) leads to
ϕ (h ) = ϕ 0 + hϕ¯ 1 + h2 ϕ¯ 2 + · · · ,
(4.3)
with ϕ 0 = ϕ¯ 0 + ϕˆ . In view of (3.20), the asymptotic developments (4.1) and (4.3) induce the following formal expansions for tensors σ (h) and D(h)
σ (h ) = h σ + h σ + h σ + · · ·, D(h ) = h−1 D−1 + D0 + hD1 + · · · , −4
−4
−3
−3
−2
−2
(4.4)
with σ q = (σ i j ) and D p = (D j ) independent of h. Inserting developments (4.1)–(4.4) into problem (3.19) results in a set of variational equations that must be satisfied for all h > 0 and consequently the terms at the successive powers of h must be zero. This procedure yields the following problems at the successive powers of h for all (v, ψ ) ∈ V() × (): q
equipped with the norm |e(v)|0, . The space VBN () can be equivalently defined by (see e.g. Trabucho and Viaño, 1996)
V BN () =
{v = (vi ) ∈ V () : vα (x1 , x2 , x3 ) = χα (x3 ), χα ∈ H02 (0, L ), v3 (x1 , x2 , x3 ) = χ3 (x3 ) − xβ χβ (x3 ), χ3 ∈ H01 (0, L )}.
(4.18)
Theorem 4.1. Let u(h) be given by (4.1). The zeroth order term u0 is an element of the space of Bernoulli–Navier displacements:
u0α (x1 , x2 , x3 ) = ξα (x3 ), u03 (x1 , x2 , x3 ) = ξ3 (x3 ) − xβ ξβ (x3 ),
ξα ∈ H02 (0, L ), ξ3 ∈ H01 (0, L ).
p
Proof. In problem (4.5), we take v =
eαβ (u ) = 0, 0
σαβ = σ −4
−2 33
=
D−1 3
u0 ,
(4.19)
which yields:
= 0.
(4.20)
Applying similar arguments to variational Eq. (4.7), we take v = u0 , yielding:
−4 σαβ eαβ (v )dx = 0,
(4.5)
e3α (u0 ) = 0,
−3 σαβ eαβ (v )dx = 0,
(4.6)
Taking into account that eαβ (u0 ) = e3α (u0 ) = 0 will yield (4.19).
σi−2 ei j (v )dx = 0, j
(4.7)
Next, we go back to variational Eq. (4.6). Combining (4.6) with (4.10), using (4.20) and taking v = u1 we get
σ
−1 ei j ij
(v )dx +
σi0j ei j (v )dx +
D−1 Ek k
(ψ )dx = 0,
D0k Ek (ψ )dx =
(4.8)
fi vi dx +
N
gi vi d ,
(4.9)
(4.22)
Moreover, from (4.10), (4.13) and (4.16) we find that −3 −1 σαβ = σ33 = 0, D03 = 0.
−3 0 σαβ = Cαβθ ρ eθ ρ (u1 ) + Cαβθ ρ eθ ρ ( u ),
(4.10)
σαβ = Cαβθ ρ eθ ρ (u ) + Cαβ 33 e33 (u ) + Cαβθ ρ eθ ρ (u ), 2
0
1
(4.23)
(4.11)
V 2m (ω ) =
= 2C3α 3β e3β (u ), 0
= 2C3α 3β e3β (u ) +
2C3α 3β e3β
(u ) − Pβ 3α Eβ (ϕ ), 0
0
vˆ = (vˆ α ) ∈ [H 1 (ω )]2 :
(4.12)
(4.24)
(Cαβθ ρ eθ ρ (u2 ) + Cαβ 33 e33 (u0 ))eαβ (v )dx = 0, for all v ∈ V ().
Using the same arguments as in Viaño et al. (2015), we define
+ Cαβ e (u0 ) − P3αβ E3 (ϕ 0 ), 33 33
1
−2 σαβ eαβ (v )dx = 0, for all v ∈ V ().
In view of (4.11), (4.20) and (4.22) the previous equality becomes
−1 2 σαβ = Cαβθ ρ eθ ρ (u3 ) + Cαβ 33 e33 (u1 ) + Cαβθ ρ eθ ρ ( u )
−2 3α −1 3α
(4.21)
eαβ (u1 ) = 0.
−4 σαβ = Cαβθ ρ eθ ρ (u0 ),
−2
−1 σ3−2 α = Dα = 0.
From (4.20) and (4.21), Eq. (4.7) can be written
where
σ σ
V BN () = {v ∈ V () : eαβ (v ) = e3β (v ) = 0},
ω
ω
(x1 vˆ 2 − x2 vˆ 1 ) dω = 0 ,
vˆ α dω = 0,
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
which is a subspace of [H1 (ω)]2 where the bi-dimensional Korn’s ˆ ))|0, is a norm in the space inequality holds. Therefore |(eαβ (w L2 (0, L; V 2m (ω )). As a consequence, the problem
⎧ 2 2 ⎪ ⎪Find uˆ (x3 ) ∈ V m (ω ), ⎨
(4.25) has a unique solution for a.e. x3 ∈ (0, L). We now define some functions that will be useful in the following results. We consider
det M αβ det M
α = β ,
=−
where Mαβ (x3 ) are defined as follows:
C1133 C1233 C2233
M 11 =
M 12 = M 21
C1112 C1212 C1222
C1122 C1222 , M 22 = C2222
C1111 1 C1112 = 2 C1122
C1133 C1233 C2233
C1111 C1112 C1122
C1112 C1212 C1222
C1133 C1233 , C2233
C1122 C1222 . C2222
Iα =
ω
αβ dω, Yα =
ω
αβ δβ dω,
Z=
ω
α δα dω.
u ) + Cαβ 33 e33 (u0 ) = 0. Cαβθ ρ eθ ρ (! ! uα = sα + δα s + αβ ξβ + α ξ3 ,
(4.27)
Xαβ ξβ ,
1 s=− Y ξ + Z ξ3 I1 + I2 β β
2
u(x3 )) = −C˜θ ραβ (x3 )Cαβ 33 (x3 )e33 (u0 )(x3 ), in eθ ρ (!
ω
(4.29)
where (C˜θ ραβ ) is defined as follows:
C1111 C˜1211 C˜2211
det M
C˜1122 C˜1222 C˜2222
=
C1111 C1211 C2211
1 2
det M
1 1
3
(4.32) where sα are arbitrary functions depending only on x3 . Combining (4.31) with (4.32) leads to (4.27). Using (3.2) and the system (4.31) we have ω
det M αα A(ω )ξ3 , det M det M αα Iα ξα , (no sum on α ). = det M
∂α ! uα d ω = −
xα ∂α ! uα d ω
ω
(4.33)
uα d ω , ω x α ∂ α ! uα dω, ξα , ξ3 ∈ Since ! uα ∈ L2 (0, L; H 1 (ω )), then ω ∂α ! 2 L (0, L ). On the other hand, identity (4.27) allows to write
a.e. in (0, L ).
Proof. Let uˆ (x3 ) be a particular solution of the equation
2C˜1112 2C˜1212 2C˜2212
det M 11 det M 21 2x1 ξ1 − ξ3 . x1 ξ2 + det M det M
2 det M
(4.28)
˜
det M 22 det M 12 x2 ξ1 + 2x2 ξ2 − ξ3 , det M det M
⎧ ⎪ ⎨k1 (x2 , x3 ) = s1 + x2 s − det M 22 x22 ξ1 + det M 12 x22 ξ2 − x2 ξ3 ,
ω
where s, sα ∈ L2 (0, L) verify
A (ω )
∂2 k1 (x2 , x3 ) = s −
(4.26)
Moreover,
sα = −
Consequently, there exists a function s, depending only on x3 , such that
Theorem 4.2. Let u0 = (u0i ) ∈ V BN () be given by (4.19). Then, there exists ! u = (! uα ) ∈ L2 (0, L; V 2m (ω )) satisfying
1
det M 11 det M 21 x1 ξ2 + 2x1 ξ1 − ξ3 − ∂1 k2 (x1 , x3 ). det M det M
2 det M
and we define, on each cross-section ω × {x3 }, the following functions
Xαβ =
(4.31)
3
⎪ ⎩ k2 (x1 , x3 ) = s2 − x1 s − det M 11 x2 ξ + det M 21 x2 ξ − x1 ξ ,
xα d ω
1
Thus,
2
ω
2 2
∂1 k2 (x1 , x3 ) = −s −
We also introduce the following geometric constant (see e.g. Trabucho and Viaño, 1996, p. 701)
2 det M
det M 22 2 ⎪ ⎩! (x ξ + 2x1 x2 ξ − 2x2 ξ ) + k2 (x1 , x3 ). u2 =
det M 12 det M 22 x2 ξ1 − 2x2 ξ2 − ξ3 + ∂2 k1 (x2 , x3 ) det M det M
1 det M 11 x21 − det M 22 x22 , 2 det M
det M αθ x x , det M θ β
⎧ det M 11 2 ⎪ ⎨! (x1 ξ1 + 2x1 x2 ξ2 − 2x1 ξ3 ) + k1 (x2 , x3 ), u1 =
Substituting these expressions in the third equation of (4.30 ), we obtain
and
αβ (x ) =
(4.30)
and, taking into account that the matrix of coefficients (Cαβθ ρ ) depends only on x3 (see (3.10)), we have, by direct integration of the first two equations,
2 det M
xβ ,
11 (x ) = −22 (x ) =
⎧ det M 11 ⎪ xα ξα − ξ3 , u1 = ∂1 ! ⎪ ⎪ det M ⎪ ⎨ det M 22 ∂2 ! xα ξα − ξ3 , u2 = ⎪ det M ⎪ ⎪ ⎪ det M 12 + det M 21 ⎩∂ ! xα ξα − ξ3 , u1 = 1 u 2 + ∂2 ! det M
(Cαβθ ρ eθ ρ (uˆ2 (x3 )) + Cαβ 33 e33 (u0 )(x3 ))eαβ (wˆ ) dω = 0 ⎪ ⎪ ⎩ ω ∀wˆ ∈ V 2m (ω ),
α (x ) = −
301
ω
! uα dω = A(ω )sα + Xαβ ξβ
a.e. in (0, L ),
δβ ! uβ dω = (I1 + I2 )s + Yθ ξθ + Z ξ3 a.e. in (0, L ),
(4.34) (4.35)
from which the average conditions required for ! uα lead to (4.28). Corollary 4.3. A necessary and sufficient condition for problem (4.26) to have solutions u2 ∈ V() is that u0 ∈ VBN () be such that ξ α ∈ H3 (0, L), ξ 3 ∈ H2 (0, L) and that s, sα ∈ H1 (0, L). In addition to that, it is verified −2 = 0. that σαβ Proof. By using (4.26) in (4.25) we find that
2C1112 2C1212 2C2212
C1122 C1222 C2222
−1
.
It is easy to show that (4.29) is equivalent to the Equation (4.26). Fur2 ther, we verify that uˆ (x3 ) also solves (4.25). On the other hand, as u0 ∈ VBN (), Eq. (4.29) can be written in differential form
ω
(Cαβθ ρ eθ ρ (uˆ2 − u))eαβ (wˆ ) dω = 0 ∀wˆ ∈ V 2m (ω ),
2 and therefore (4.27) is in fact a charallowing to conclude that uˆ = u acterization of u2α , the transverse components of the second order displacement u2 . Now assume that u2 ∈ V(). Let u2α ∈ H 1 (). By relations (4.33)–(4.35) we deduce that ω ∂α u2α dω, ω xα ∂α u2α dω, s,
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J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
sα ∈ H1 (0, L). Furthermore, the boundary condition for u2α (cf. (3.3)) leads to (4.28) for x3 = 0 and x3 = L. Conversely, taking u0 ∈ VBN () such that ξ α ∈ H3 (0, L), ξ 3 ∈ H2 (0, L), and s, sα ∈ H1 (0, L) now of the form (4.28) for all x3 ∈ (0, L), then (4.27) implies that u2α ∈ H 1 () verifies the boundary condition (3.3). Finally, from (4.11), (4.22) and −2 = 0. (4.26) we conclude that σαβ From (4.27) it follows that in general u2
Remark 4.4. α does not vanish at the rod ends, for that would require that both ξβ and ξ3 vanish there. Therefore, if we had chosen strong boundary conditions in the definition of V(), then we would not be able to ensure that u2 ∈ V() (the well-known “ boundary layer phenomenon” ) and consequently −2 = 0. that σαβ As a result of the previous discussion, we will be able to obtain variational problems that have ξ α and ξ 3 as unique solutions. Theorem 4.5. It is verified that u0 ∈ VBN () is the unique solution of the following variational problem:
0 σ33 e33 (v )dx =
fi vi dx +
¯ 33 (u0 ) = P¯ D13 = Pe
N
0 given by with σ33
ξ3 − xα ξα ∈ L2 (),
with
P¯ = P333 −
det M αβ det M
P3αβ .
Remark 4.8. In the case of taking assumption (3.12) instead of (3.10), we can still formulate the problem (4.25) (the coefficients Cαβθ ρ now depending also on x1 , x2 ) which has an unique solution in V 2m (ω ). The difference is that we cannot give explicitly u2 in terms of u0 as in (4.27). Following Theorem 3.1 in Girault and Raviart (1986), we have to introduce two auxiliary functions θ α ∈ H1 (ω) a.e. in (0, L) and 2 eθ ρ (uˆ ) = −C˜θ ραβ (Cαβ 33 e33 (u0 ) + θ ρ ) in
αβ =
∂2 θ 1 ∂2 θ 2
−∂1 θ1 , −∂1 θ2
det C¯ . det M det N
4.2. Characterization of u1 and ϕ 0 (4.38)
Notice that it is positive and bounded (cf. (2.7) and (2.8)). Proof. Problem (4.36) follows from variational Eq. (4.9) taking v ∈ VBN () and ψ = 0. Furthermore, after some calculations (4.14) becomes (4.37).
We start this section by introducing the space V1 () formed by all the infinitesimal rigid displacements of the set in the transversal directions:
V 1 () = {v ∈ V () : eαβ (v ) = 0}. In Trabucho and Viaño (1996) it is proved that V1 () coincides with the following space :
Corollary 4.6. The transverse components ξ α and the stretch component ξ 3 of the zeroth order displacement field solve respectively the following variational problems (no sum on α ):
V 1 () =
Find ξα ∈ H02 (0, L ) such that L L Y Iα ξα χα dx3 = Fα χα dx3
We also consider the spaces
−
(4.39)
0
L 0
Mα χα dx3 ,
for all
χα ∈ H02 (0, L ),
(4.39)
0
0
for all
χ3 ∈ H01 (0, L ),
(4.40)
with the resultant forces and moments given by:
Fi (x3 ) =
ω
f i ( x1 , x2 , x3 ) d ω +
∂ω
xα f 3 ( x1 , x2 , x3 ) d ω ω + xα g3 (x1 , x2 , x3 )dγ ∈ L2 (0, L ). ∂ω
Q = Q (ω ) =
ρ ∈ H 1 (ω ) :
ω
" ρ dω = 0 ,
S = S(ω ) = {ψ ∈ H 1 (ω ) : ψ = 0 on γeD },
(4.44)
equipped with the (semi-)norms
With the aid of Poincaré–Friedrichs’ inequality and Poincaré’s inequality in ω, it can be shown that these norms are equivalent to the usual H1 -norm. Also, let l () be the space
l () = {ψ ∈ L2 () : ∂α ψ ∈ L2 (ω )} ≡ L2 (0, L; H 1 (ω )),
gi (x1 , x2 , x3 )dγ ∈ L2 (0, L ), (4.41)
Mα (x3 ) =
{v ∈ V () : vα (x1 , x2 , x3 ) = ζα (x3 ) + δα ζ (x3 ), ζα , ζ ∈ H01 (0, L )}.
1/2
ρ Q = ∂1 ρ 20,ω + ∂2 ρ 20,ω , for all ρ ∈ Q, 1/2
ψ S = ∂1 ψ 20,ω + ∂2 ψ 20,ω , for all ψ ∈ S.
and
Find ξ3 ∈ H01 (0, L ) such that L L YA(ω )ξ3 χ3 dx3 = F3 χ3 dx3 ,
∂1 θ1 = −∂2 θ2 .
(4.37)
where we can define the Young Modulus for the limit equations as
ω,
In these conditions, the limit problems cannot be fully determined (see Figueiredo and Leal, 2006).
0 σ33 = Ye33 (u0 ) = Y (ξ3 − xα ξα ),
0
(4.43)
Proof. This result follows straightforwardly from (4.17) taking into account that u2 verifies (4.26).
where
gi vi d, for all v ∈ V BN (), (4.36)
Y =
Corollary 4.7. The component D13 of D1 is given by
(4.42)
Proof. We consider (4.36) with the test function v ∈ VBN () of the form (4.18), implying e33 (v ) = χ3 (x3 ) − xθ χθ (x3 ). Then, taking successively χθ = 0 and χ3 = 0 in the resulting equation we obtain (4.39) and (4.40 ), respectively.
(4.45)
L2 (0, L; Q (ω ))
and l0 () = and let us define its subspaces R() = L2 (0, L; S(ω )). These spaces are equipped with the (semi-)norms
1/2
ψ l0 = ∂1 ψ 20, + ∂2 ψ 20, , for all ψ ∈ l0 (), 1/2
ρ R = ∂1 ρ 20, + |∂2 ρ 20, , for all ρ ∈ R().
Again, it can be shown that these norms are equivalent to the usual H1 -norm. We now define the warping function w ∈ R() as the unique solution of the variational problem (see e.g. Trabucho and Viaño, 1996, p. 700)
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
w(x3 ) ∈ Q (ω ) such that a.e. in(0, L ) : C3α 3β ∂β w∂α υ dω = C3α 3β δβ ∂α υ dω, ω
ω
Hence, if r is a solution of (4.48) then
for all υ ∈ H (ω ), 1
(4.46) as well as the torsion function J ∈ L2 (0, L), given by
J=
ω
=
ω
C3α 3β (δβ − ∂β w )(δα − ∂α w )dω C3α 3β (δβ − ∂β w )δα dω > 0.
(4.47)
(4.48)
Bearing in mind the previous definitions, we turn to the characterization of u1 and ϕ 0 . In (4.22), we obtained that eαβ (u1 ) = 0. Therefore, we can characterize the transverse components of u1 as follows (see e.g. Trabucho and Viaño, 1996)
u1α = zα + δα z,
zα , z ∈ H01 (0, L ).
(4.49)
Moreover, by taking v ∈ V1 () in (4.8) and v = 0 in (4.9), and considering (4.12), (4.15) and (4.23) we get, respectively,
2C3α 3β e3β (u1 ) + Pβ 3α ∂β ϕ 0 e3α (v )dx = 0,
for all v ∈ V 1 (), (4.50)
2Pβ 3α e3α (u1 ) − εαβ ∂α ϕ 0
which shows that ing result.
and
ϕ0
∂β ψ dx = 0, for all ψ ∈ (),
are coupled. In fact, we have the follow-
ˆ = (uˆi ) ∈ [L2 ()]3 be of the form (4.49) and ϕ 0 ∈ Theorem 4.9. Let u l () . Then uˆ is a solution of (4.50) if and only if
uˆ3 = −r − z3 − xα zα − wz ,
In view of the previous regularity result we conclude that although (4.49) implies that u1α ∈ H 1 (), we can only ensure that u13 ∈ L2 (0, L; H 1 (ω )). Furthermore, ω u13 dω vanishes at the rod ends as 1 long as z3 ∈ H0 (0, L ), but in general this is not the case for ω xα u13 dω. 1 Therefore, we cannot ensure that u ∈ V(), traducing a “ boundary layer phenomenon” . On the other hand, the regularity result obtained for u13 implies that U ∈ L2 (0, L; H1 (ω)) - cf. (4.54) - and therefore from (4.55) we can only guarantee that ϕ 0 is in l (), not in (). Hence, ϕ¯ 0 = ϕ 0 − ϕˆ ∈ l0 (). Nevertheless, even when u1 ∈ V(), the characterization of u1 done so far still holds, in particular as far as Eqs. (4.49) and (4.56) are concerned. This will allow to use Eqs. (4.50) and (4.51) to derive variational problems for ϕ¯ 0 .
(4.52)
Theorem 4.11. Element z ∈ H01 (0, L ) satisfies the variational problem
z ∈ H01 (0, L ),
0
L
Jz ζ dx3 =
Choosing v = (0, 0, χ (x3 )υ (x1 , x2 )), χ ∈ H01 (0, L ), υ ∈ Q (ω ) in (4.53) and taking (4.46) into account we get a.e. x3 ∈ (0, L)
C3α 3β ∂β uˆ3 (x3 )∂α υ dω ω = − C3α 3β ∂β (xθ zθ (x3 ) + wz (x3 ))∂α υ dω ω − Pβ 3α ∂β ϕ 0 (x3 )∂α υ dω, for all υ ∈ Q (ω ).
ω
(4.54)
for all υ ∈ Q (ω ).
(4.57)
(4.58)
Taking v = (χ (x3 )x2 , −χ (x3 )x1 , 0 ), χ ∈ H01 (0, L ) in the previous equation we get
(C3α3β (z (δβ − ∂β w ) − ∂β r ) + Pβ 3α ∂β ϕ 0 )δα χ dx = 0,
(4.59)
that is (cf. (4.47)),
d d (Jz ) = (C ∂ r − Pβ 3α ∂β ϕ 0 )δα dω, dx3 dx3 ω 3α 3β β
a.e. x3 ∈ (0, L ),
a.e. x3 ∈ (0, L ). (4.60)
is a solution a.e. x3 ∈ (0, L) of
C3α 3β ∂β U (x3 )∂α υ dω ω = − Pβ 3α ∂β ϕ 0 (x3 )∂α υ dω,
ζ dx3 ,
(C3α3β (z (δβ − ∂β w ) − ∂β r ) + Pβ 3α ∂β ϕ 0 )e3α (v )dx = 0,
d d Jz = P (∂α w − δα )∂β ϕ 0 dω, dx3 dx3 ω β 3α
ω
ω
or, in view of (4.46) and (4.48),
Therefore,
U = uˆ3 + xα zα + wz
(4.53)
0
Pβ 3α(∂α w − δα )∂β ϕ 0 dω
which plugged into (4.50) leads to
for all v ∈ V 1 ().
for all v ∈ V 1 ().
2e3α (u1 ) = −∂α r − z (∂α w − δα ),
ˆ = (uˆi ) is a solution of ( 4.50) and in view of Proof. Assuming that u (4.49) we conclude that uˆ3 verifies
L
Proof. We consider (4.49) and (4.56), yielding
(C3α3β (∂β uˆ3 + zβ + δβ z ) + Pβ 3α ∂β ϕ 0 )e3α (v )dx = 0,
∀ζ ∈ H01 (0, L ).
where r ∈ R() solves (4.48), w is the warping function and z3 ∈ L2 (0, L) is an arbitrary function of x3 .
(4.56)
4.2.1. Variational problem for ϕ¯ 0 , r and z
(4.51) u1
Corollary 4.10. Let u1α , the transverse component of u1 , be of the form (4.49) and let u1 verify (4.50). Then the axial component u13 is such that
D0α = −Pα 3β (∂β r + z (∂β w − δβ )) − εαβ ∂β ϕ 0 .
ω
ρ ∈ Q ( ω ).
where z3 is an arbitrary function of x3 , leading to (4.52 ). From (4.52) we conclude that since r ∈ L2 (0, L; H1 (ω)), zα , z ∈ H01 (0, L ) and w ∈ H1 (ω), then z3 ∈ L2 (0, L). The converse result is immediate.
where z3 ∈ L2 (0, L). Furthermore, (4.15) becomes
r (x3 ) ∈ Q (ω ) such that for a.e.x3 ∈ (0, L ) : C3α 3β ∂β r (x3 )∂α ρ dω = Pβ 3α ∂β ϕ 0 (x3 )∂α ρ dω, ω
a.e. x3 ∈ (0, L ),
U = −r − z3 ,
u13 = −r − z3 − xα zα − wz ,
Finally, let r ∈ R() satisfy the following variational problem,
for all
303
(4.55)
Finally, from (4.60) we also obtain that z satisfies (4.57). We note that if ϕ 0 is given, then the solution of (4.57) is unique since J is bounded and positive. Remark 4.12. From the previous results we conclude that in general it is not possible to obtain the elasticity result z = 0 (see e.g. ÁlvarezDios and Viaño, 1993a; 1993b; Trabucho and Viaño, 1996). However, we can prove that this evidence happens when we take ϕ 0 = ϕ 0 (x3 ).
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Note that this implies ϕ 0 = ϕˆ and is only possible if ϕˆ = ϕˆ (x3 ). In this case, Eq. (4.59) takes the form
L
ω
0
C3α 3β [z (δβ − ∂β w ) − ∂β r]δα dω
χ dx3 = 0, ∀χ ∈ H01 (0, L ),
and, by a density argument, we deduce
z
ω
C3α 3β (δβ − ∂β w )δα dω −
ω
C3α 3β ∂β rδα dω = 0.
C3α 3β ∂β r∂α wdω =
and, therefore, we deduce that Jz = 0 with J > 0. Thus, z = 0 and since z ∈ H01 (0, L ) one gets z = 0 a.e in (0, L).
(4.61)
(ϕ¯ 0 , r, z )
Theorem 4.13. The element the following variational problem
0
J (ς )2 dx3 +
C3α 3β ∂β ρ∂α ρ dx
J (ς )2 dx3 > c1 |ς |20,(0,L) ,
C3α 3β ∂β ρ∂α ρ dx ≥ c2
∈ T () is the unique solution of
((∂1 ρ )2 + (∂2 ρ )2 )dx
= c2 (|∂1 ρ|0, + |∂2 ρ|0, ), ∀ ρ ∈ R(), εαβ ∂β ψ∂α ψ dx ≥ c3 ((∂1 ψ )2 + (∂2 ψ )2 )dx 2
(4.62)
for all ς ∈ H01 (0, L ).
Bearing in mind properties (2.4), there exists constants c2 , c3 > 0 such that
(ψ , ρ , ς ) 2T () = |∂1 ψ|20, + |∂2 ψ|20, + |∂1 ρ|20,
0
2
equipped with the norm
Find
L
Let us define the space
+ |∂2 ρ|20, + |ς |20,(0,L) .
L
Pβ 3α ∂α w∂β ϕ 0 dω = 0,
T () = l0 () × R() × H01 (0, L ),
! a((ψ , ρ , ς ), (ψ , ρ , ς )) = + εαβ ∂α ψ∂β ψ dx,
for all (ψ , ρ , ς ) ∈ T(). On the other hand, since J is positive and bounded, we conclude that there exists a constant c1 > 0 such that
From (4.46)–(4.48), the previous relation becomes
Jz =
Since () is dense in l0 (), we obtain (4.63)–(4.65). One can easily show that both ! a and ! l are T -continuous. The coercivity of ! a is shown as follows. From (4.64) one has
= c3 (|∂1 ψ|0, + |∂2 ψ|0, ), 2
2
∀ ψ ∈ l0 ().
Thus, there exists a constant c4 > 0 such that (cf. (4.62)) 2 ! a((ψ , ρ , ς ), (ψ , ρ , ς )) ≥ c4 (ψ , ρ , ς ) T ,
ϕ¯ 0 , r, z ∈ T () such that
!, ρ !, ρ !, ρ ! !, ς )) = ! !, ς ), for all (ψ !, ς ) ∈ T (), a((ϕ¯ 0 , r, z ), (ψ l (ψ
that is, ! a is T-elliptic. Hence, the uniqueness of solution of the variational problem (4.63)–(4.65) follows from Lax–Milgram Lemma.
(4.63) 5. Convergence analysis
where the bilinear form ! a and the linear functional ! l are given by
L
! Jz ς dx3 + C3α 3β ∂β r∂α ρ dx a((ϕ¯ 0 , r, z ), (ψ , ρ , ς )) = 0 + εαβ ∂α ϕ¯ 0 ∂β ψ dx + Pβ 3α (∂α r∂β ψ − ∂β ϕ¯ 0 ∂α ρ )dx + Pβ 3α (∂α w − δα )(z ∂β ψ − ς ∂β ϕ¯ 0 )dx (4.64)
and
! l (ψ , ρ , ς ) = Pβ 3α ∂β ϕˆ (∂α ρ + ζ (∂α w − δα ))dx εαβ ∂β ϕ∂ ˆ α ψ dx, −
(4.65)
Pβ 3α (∂α r + z (∂α w − δα )) + εαβ ∂α ϕ¯ 0
=−
∂β ψ dx
εαβ ∂α ϕ∂ ˆ β ψ dx ∀ ψ ∈ ().
L 0
(4.66)
for all ψ ∈ (),
ρ ∈ R(), ζ ∈ H01 (0, L ).
∀(v, ψ ) ∈ X (),
(5.2)
where
X () = V () × (), M 1 () =
Jz ζ dx3 + C3α 3β ∂β r∂α ρ dx + εαβ ∂β ϕ¯ 0 ∂α ψ dx + Pβ 3α (∂α r∂β ψ − ∂β ϕ¯ 0 ∂α ρ )dx + Pβ 3α (∂α w − δα )(z ∂β ψ − ∂β ϕ¯ 0 ζ )dx = Pβ 3α ∂β ϕˆ (∂α ρ + ζ (∂α w − δα ))dx − εαβ ∂β ϕ∂ ˆ α ψ dx,
(5.1)
ˆ Xˆ () = V () × () ,
M () = M 1 () × M 2 (),
Combining (4.66) with (4.48) and (4.57),and taking into account that ϕ¯ 0 = ϕ 0 − ϕˆ , we conclude that ϕ¯ 0 , r, z ∈ T () verifies:
∀(τ , d ) ∈ M (), bH ((σ (h ), D(h )), (v, ψ )) = lH (v, ψ ),
Proof. We consider the variational problem (4.51) and write it as (cf. (4.58))
We start with the scaled variational problem associated to the primal variational problem (3.15)–(3.17): Find ((σ (h ), D(h )), (u(h ), ϕ (h ))) ∈ M () × Xˆ () such that
aH ((σ (h ), D(h )), (τ , d )) + bH ((τ , d ), (u(h ), ϕ (h ))) = 0,
respectively.
In this section we will provide a strong convergence result. Since we will need a mixed formulation to prove some results, in this section we have to invert the constitutive laws (2.3). Therefore, in order to simplify the exposition we assume throughout this section that in (3.10) we have Cijkl = 0.
τ = (τi j ) ∈ [L2 ()]9 : τi j = τ ji , M 2 () = [L2 ()]3 ,
ˆ () = ϕˆ + () = {ψ ∈ () : ψ = ϕˆ on eD }, and the forms aH , lH and bH are defined as follows:
aH ((σ (h ), D(h )), (τ , d )) = h−2 a−2 ((σ (h ), D(h )), (τ , d )) + h−1 a−1 ((σ (h ), D(h )), (τ , d )) + a0 ((σ (h ), D(h )), (τ , d )) + ha1 ((σ (h ), D(h )), (τ , d )) + h2 a2 ((σ (h ), D(h )), (τ , d )) + h4 a4 ((σ (h ), D(h )), (τ , d )), lH (v, ψ ) = − fi vi dx − gi vi d ,
bH ((σ , D ), (v, ψ )) = −
dN
σi j ei j (v )dx −
Dk Ek (ψ )dx,
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
with
exists a subsequence, still denoted by (u(h ), ϕ¯ (h ))h>0 which satisfies
a−2 ((σ , D ), (τ , d )) =
u ( h ) → u0
ε¯33 D3 d3 dx,
a−1 ((σ , D ), (τ , d )) = P¯333 D3 τ33 dx − P¯333 σ33 d3 dx, a0 ((σ , D ), (τ , d )) = ε¯αβ Dβ dα dx, C¯3333 σ33 τ33 dx + a1 ((σ , D ), (τ , d )) = P¯3αβ D3 ταβ dx − P¯3αβ σαβ d3 dx −2 P¯α 3θ σ3θ dα dx + 2 P¯β 3α Dβ τ3α dx, a2 ((σ , D ), (τ , d )) = C¯33αβ σαβ τ33 dx + 4 C¯3α 3β σ3β τ3α dx
Furthermore
u1 ∈ V 1 () :
e˜ =
e˜33 (h ) := e33 (u(h ))(x ), E˜α (h ) := Eα (ϕ (h ))(x ),
E˜3 (h ) := hE3 (ϕ (h ))(x ),
verifying the constitutive equations (cf. (3.20))
Sαβ (h ) := h2 σαβ (h ) = Cαβθ ρ e˜θ ρ (h ) + Cαβ 33 e˜33 (h ) − P3αβ E˜3 (h ), (5.4)
S33 (h ) :=
(5.5)
Let us now define
u (h ) := h
(u(h ) − u ) = u + h.o.t., 0
1
uˆ2α (h ) := h−2 uα (h ) − u0α − hu1α = u2α + h.o.t. In order to have strong convergence we need the additional hypotheses:
u1 (h ) ∈ V RD (),
2
ˆ (h ) ∈ L 0, L; V 2m (ω ) , u 2
u13 = −r − z3 − xα zα − wz ,
eαβ (u )
e3α (u1 )
e3α (u1 )
e33 (u0 )
2
, E˜ = −(∂1 ϕ 0 , ∂2 ϕ 0 , 0 ),
Proof. The proof is divided into 7 steps, numbered (i) to (vii). Step (i): We first show that there exists a constant c (ϕˆ ) > 0, depending only the applied potential ϕˆ , such that, for all h ∈ (0, 1):
u(h ) 21, + |e˜(h )|20, + |E¯ (h )|20, ≤ c(ϕˆ ),
(5.7)
ϕ¯ (h ) 0, ≤ c(ϕˆ ),
(5.8)
¯ (h )) = −(∂1 ϕ¯ (h ), ∂2 ϕ¯ (h ), h∂3 ϕ¯ (h )). where E¯ (h ) = E˜ (ϕ The proof of this step is straightforward and can be found in Ref. Figueiredo and Leal (2006). Step (ii): There exists a subsequence of (u(h ), e˜(h ), ϕ (h ), E (h ))0
u(h ) u in[H 1 ()]3 ,
S3α (h ) := hσ3α (h ) = 2Cα 33θ e˜3θ (h ) − Pθ α 3 E˜θ (h ),
−1
ξα ∈ H02 (0, L ), u03 (x1 , x2 , x3 ) = ξ3 (x3 ) − xρ ξρ (x3 ), ξ3 ∈ H01 (0, L ), 1 uα = zα + δα z, zα , z ∈ H01 (0, L )
where z3 ∈ L2 (0, L), functions ξ α , ξ 3 , z solve the variational problems (4.39), (4.40) and (4.57), respectively, and w, r ∈ Q(ω) solve the variational problems (4.46) and (4.48).
e˜3β (h ) := h−1 e3β (u(h ))(x ),
σ33 (h ) = C33θ ρ e˜θ ρ (h ) + C3333 e˜33 (h ) − P333 E˜3 (h ), Tθ (h ) := Dθ (h ) = 2Pθ 3α e˜3α (h ) + εθ α E˜α (h ), T3 (h ) := h−1 D3 (h ) = P3αβ e˜αβ (h ) + P333 e˜33 (h ) + ε33 E˜3 (h ).
in M 2 ().
u0α (x1 , x2 , x3 ) = ξα (x3 ),
u ∈ V BN () :
where C¯ = (C¯i jkl ), P¯ = (P¯ki j ), ε¯ = (ε¯i j ) are inverse constitutive tensors of C = (Ci jkl ), P = (Pki j ) and ε = (εi j ) and inherit their properties of ellipticity and boundedness (see e.g. Viaño et al., 2015). We also define the following auxiliary tensor (e˜i j (h )) and vector (E˜i (h )):
e˜αβ (h ) := h−2 eαβ (u(h ))(x ),
ˆ ϕ¯ (h ) → ϕ 0 − ϕˆ in () ,
0
(5.3)
in V (),
e˜(h ) → e˜ in M 1 () E˜ (h ) → E˜
+ C¯αβ 33 σ33 ταβ dx, a4 ((σ , D ), (τ , d )) = C¯αβθ ρ σθ ρ ταβ dx,
1
305
(5.6)
where
V RD () = {v ∈ V 1 () : v3 ∈ L2 (0, L; Q (ω ))},
(5.9)
e˜(h ) e˜ in M 1 (),
(5.10)
ϕ (h ) ϕ in L2 (),
(5.11)
E˜ (h ) E˜
(5.12)
in M 2 ().
From step (i) it is easy to show that there exists subsequences (u(h))h > 0 , (e˜(h ))h>0 , (ϕ¯ (h ))h>0 , (E¯ (h ))h>0 and functions u, e˜ , ϕ¯ and E¯ satisfying (5.9 ) and (5.10),
ϕ¯ (h ) ϕ¯ in L2 (), and E¯ (h ) E¯ in [L2 ()]3 ,
(5.13)
respectively. The weak convergence results (5.11) and (5.12) follow straightforwardly from (5.13) taking into account that ϕ (h ) = ϕ¯ (h ) + ϕˆ , E˜ (h ) = E¯ (h ) + Eˆ (h ) and Eˆ (h ) = (−∂1 ϕˆ , −∂2 ϕˆ , −h∂3 ϕˆ ) →
(−∂1 ϕˆ , −∂2 ϕˆ , 0 ) as h → 0.
is the space of admissible rigid displacements. We also define
Step (iii): There exist subsequences of (u(h))h > 0 and (ϕ (h))h > 0 still denoted the same way, and e˜ ∈ M 1 (), E˜ ∈ M 2 () such that
V R () = {v ∈ V RD () : vα = δα s, s ∈ H01 (0, L )},
e33 (u(h )) e˜33 , h−1 eα 3 (u(h )) e˜α 3 , h−2 eαβ (u(h )) e˜αβ ,
V T () = {v ∈ V RD () : vα = sα ,
v3 = −xα sα , sα ∈ H01 (0, L )},
which are the the subspaces of admissible rotations and translations, respectively. We note that |e3β ( · )|0, is a norm for VR () but not for VRD (). In order to overcome this difficulty, we can think of using the quotient space V˙ RD () := V RD ()/V T (), where the induced application
˙ |||v||| = |e3β (v )|0, , ∀ v˙ = v + V T () ∈ V˙ RD (), is a norm. Theorem 5.1. Let (u(h ), ϕ¯ (h )) ∈ V () × () be the solution of the scaled problem (3.15)–(3.17) and assume that (5.6) holds. Then, there
eα 3 (u(h )) 0,
eαβ (u(h )) 0,
E˜α (ϕ (h )) E˜α ,
E˜3 (ϕ (h )) E˜3 ,
hEα (ϕ (h )) 0,
(5.14)
in L2 (). Moreover,
e˜33 = e33 (u ), E˜α = −∂α ϕ ,
e3α (u ) = eαβ (u ) = 0 (i.e., u ∈ V BN ()), E˜3 = 0.
(5.15) (5.16)
The obtention of (5.14) is straightforward. Indeed, from (5.10) and (5.12) we deduce that there exist (e˜αβ , e˜3α , e˜33 , E˜α , E˜3 ) ∈ M (), which is the weak limit of a subsequence of
(h−2 eαβ (u(h )), h−1 e3α (u(h )), e33 (u(h )), Eα (ϕ (h )), hE3 (ϕ (h ))),
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J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
still denoted by the same way. Also, from (5.9) and the uniqueness of the limit as h → 0, we deduce that (5.15) holds. Similarly, from (5.11) we obtain (5.16). Step (iv): Let ((σ (h), D(h)), (u(h), ϕ (h))) be the solution of the mixed variational problem (5.1)–(5.3). Then, there exists a constant C = C (ϕˆ ) > 0, independent of h, such that, for all 0 < h ≤ 1:
S33 (h ) 0, ≤ C,
Sα3 (h ) 0, ≤ C,
Sαβ (h ) 0, ≤ C,
Tα (h ) 0, ≤ C, T3 (h ) 0, ≤ C.
(5.17) (5.18)
Moreover, there exists a subsequence, still parameterized by h, and there exist ∈ M1 () and T ∈ M2 (), such that:
S33 (h ) 33 ,
Sα 3 ( h ) α 3 ,
Sαβ (h ) αβ ,
in L2 () (5.19)
Tα (h ) Tα ,
T3 (h ) T3 ,
in L () 2
(5.20)
when h goes to zero. Furthermore,
Multiplying Eq. (5.2) by h, taking the test function v ∈ VRD () and passing to the limit as h → 0, we find
3α e3α (v¯ )dx = 0 ∀v¯ ∈ V RD ().
(5.30)
Therefore, from (5.28)–(5.30) we get (5.26). The relations (5.27) follow from the inversion of (5.21)–(5.25). Step (vi): The following characterizations hold:
e˜θ ρ = eθ ρ (u2 ), e˜33 = e33 (u0 ), e˜3β = e3β (u1 ), E˜β = Eβ (ϕ 0 ). (5.31) Moreover, −2 αβ = σαβ ,
T3 = D13 ,
0 33 = σ33 ,
Tα = D0α ,
3α = σ3−1 α,
in L2 (),
in L2 ().
(5.32) (5.33)
ˆ ∈ L2 (0, L; V 2m (ω )) From (5.7) and (5.6) we find that there exist u and u¯˙ 1 ∈ V˙ RD () such that the following weak convergences hold: 2
e˜αβ = C¯θ αβρ θ ρ + C¯33αβ 33 + P¯3αβ T3 ,
(5.21)
uˆ (h ) uˆ ,
e˜3α = 2C¯3α 3θ 3θ + P¯θ 3α Tθ ,
(5.22)
Moreover, since eαβ : V 2m (ω ) → [L2 (ω )]2×2 and e3β : V˙ RD () → s [L2 ()]2 are weakly continuous applications, by using the closed graph theorem (see, for example Ref. Brézis, 1983) we find that
e˜33 = C¯3333 33 + P¯333 T3 + C¯33θ ρ θ ρ ,
(5.23)
E˜α = −2P¯θ 3α 3θ − ε¯θ α Tθ ,
(5.24)
E˜3 = −P¯3θ ρ θ ρ − P¯333 33 + ε¯33 T3 .
(5.25)
αβ eαβ (vˆ )dx +
+ 33 e33 (v )dx + = fi vi dx + gi vi d ,
N
∀(vˆ , v¯ , v, ψ¯ ) ∈ V H ()
∀(v, ψ¯ ) ∈ V BN () × ().
(5.27)
−2 ¯ )dx αβ − σαβ eαβ (vˆ )dx + 3α − σ3−1 α e3α (v
+
0 33 − σ33 e33 (v )dx + Tα − D0α Eα (ψ¯ )dx = 0,
∀(vˆ , v¯ , v, ψ¯ ) ∈ V H () × V RD () × V BN () × l0 (), which can be equivalently written as follows, taking into account the constitutive Eqs. (4.11), (4.12), (4.14), (4.15) and (5.27),
T
eαβ (vˆ ) e33 (v )
Cαβ 33 C3333
C3α 3β Pα 3β
−Pβ 3α
T
e3α (v¯ ) Eα (ψ¯ )
Cαβθ ρ C33θ ρ
εαβ
e˜θ ρ − eθ ρ (u2 ) dx = 0, e˜33 − e33 (u0 )
fi vi dx +
N
αβ eαβ (vˆ )dx = 0, ∀vˆ ∈ V H ().
(5.35)
e˜3β − e3β (u1 ) dx = 0, ˜ Eβ − Eβ (ϕ¯ 0 + ϕˆ ) (5.36)
for all (vˆ , v¯ , v, ψ¯ ) ∈ V H () × V RD () × V BN () × l0 (). By using density arguments, the integral Eq. (5.35) is also valid when we replace VH () by L2 (0, L; V 2m (ω )). Therefore, we can take as test functions
vˆ = uˆ 2 − u2 , v¯ = u¯ 1 − u1 , v = u − u0 , ψ¯ = ϕ − (ϕ¯ 0 + ϕˆ ).
gi vi d , (5.28)
Since () is dense in l0 (), the previous equation is also valid for all (v, ψ¯ ) ∈ V BN () × l0 (). Now, we multiply Eq. (5.2) by h2 . Let us now define the space of horizontal displacements V H () = H 1 (0, L; V 2m (ω )). By choosing v3 = 0 and (vα ) ∈ VH (), and passing to the limit as h → 0, we obtain
Taking v ∈ VBN () in (5.2) and passing to the limit when h → 0, we obtain
Tα Eα (ψ¯ )dx =
(5.34)
(5.26)
⎧ ⎨αβ = Cαβθ ρ e˜θ ρ + Cαβ 33 e˜33 , 3α = 2C3α3θ e˜3θ − Pθ 3α E˜θ , 33 = C33θ ρ e˜θ ρ + C3333 e˜33 , ⎩ Tα = 2Pα 3β e˜3β + εαβ E˜β , T3 = P3αβ e˜αβ + P333 e˜33 .
1 e˜3β = e3β (u¯ ).
Now, taking into account Eq. (4.8) for test functions v ∈ VRD () and ψ = 0, Eq. (4.9) with v = 0 and ψ ∈ (), dense in l0 (), and rela-
where
33 e33 (v )dx +
2
Tα Eα (ψ¯ )dx
× V 1 () × V BN () × l0 (),
u1 (h ) u¯˙ 1 .
e˜αβ = eαβ (uˆ ),
3α e3α (v¯ )dx
2
tions (4.24), (4.36) and (5.26), we obtain
Indeed, (5.17) and (5.18) follow from (5.7) and (5.8) and (5.4) and (5.5). Moreover, (5.19) and (5.20) is an immediate consequence of (5.17) and (5.18). Furthermore, by choosing τ33 = τ3α = d3 = dα = 0 as test functions in (5.1), multiplying by h−2 and passing to the limit, we deduce (5.21) in L2 (). Proceeding similarly, we obtain (5.22)– (5.25). Step (v): The limit model is given by:
2
(5.29)
Having in mind (5.15), (5.16) and (5.34), the ellipticity in (5.35)–(5.36) gives (5.31). Consequently, from (4.11), (4.12), (4.14), (4.15), (4.17) and (5.27), we can deduce (5.32)–(5.33). Step (vii): The following strong convergences hold:
e˜(h ) → e˜ in M 1 (),
E˜ (h ) → E˜
u ( h ) → u0
ˆ ϕ (h ) → ϕ 0 in () .
in V (),
in [L2 ()]3 ,
Let X(h) be the norm of (e˜(h ), −E˜ (h )) in M():
X (h ) =
+
Ci jkl e˜kl (h )e˜i j (h )dx +
ε33 E˜3 (h )E˜3 (h )dx
εαβ E˜α (h )E˜β (h )dx
"1/2
.
(5.37)
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
Taking (v, ψ ) = (u(h ), ϕ (h )) in (3.15)–(3.17) we find that (u(h), ϕ (h))h > 0 verifies
Ci jkl e˜kl (h )e˜i j (h )dx + εαβ E˜α (h )E˜β (h )dx + ε33 E˜3 (h )E˜3 (h )dx fi ui (h )dx + gi ui (h )d . (5.38) =
N
Combining (5.37) and (5.38) we have
(X (h ))2 =
fi ui (h )dx +
gi ui (h )d .
N
From the weak convergences (5.9) we get
lim (X (h ))2 =
h→0
fi ui dx +
N
gi ui d .
Since we have already proved that (e˜(h ), ∂1 ϕ (h ), ∂2 ϕ (h ), h∂3 ϕ (h )) converges weakly, it suffices to show that the limit of its norm tends to the norm of its weak limit to obtain strong convergence. We denote by
X=
Ci jkl e˜kl e˜i j dx +
εαβ E˜α E˜β dx
1/2
˜ E˜1 , E˜2 , E˜3 ). So, we want to show that the L2 norm of (e,
X 2 = lim (X (h ))2 , h→0
or, equivalently,
Ci jkl e˜kl e˜i j dx +
εαβ E˜α E˜β dx =
fi ui dx +
N
gi ui d .
⎧ 1 2 1 2 ⎨uα (h ) = ξα + huα + O(h ), u3 (h ) = ξ3 − xβ ξβ + hu3 + O(h ), 1 1 u = zα + δα z, u3 = −r − z3 − xα zα − wz , ⎩ α ϕ¯ (h ) = ϕ¯ 0 + O(h ), (6.1) and
⎧ −1 + O(h0 ), σαβ (h ) = h−1 σαβ ⎪ ⎪ ⎪ −1 ⎪ σαβ (h ) = h−1 σαβ + O(h0 ), ⎪ ⎪ ⎪ ⎪ −1 0 ⎪ ⎨σ3α (h ) = h [Pβ 3α ∂β (ϕ¯ + ϕˆ ) −C3α 3β (∂β r + z (∂β w − δβ ))] + O(h0 ), ⎪ ⎪ ⎪ ⎪σ33 (h ) = Y (ξ3 − xα ξα ) + O(h ), ⎪ ⎪ ⎪ ⎪Dα (h ) = −Pα3β (∂β r + z ∂β w − δβ ) − εαβ ∂β (ϕ¯ 0 + ϕˆ ) + O(h ), ⎪ ⎩ D3 (h ) = hP¯ (ξ3 − xα ξα ) + O(h2 ), (6.2) where Y and P¯ are given by (4.38) and (4.43 ), respectively, and the (Bernoulli–Navier) displacement components ξα ∈ H02 (0, L ) and ξ3 ∈ H01 (0, L ) are, respectively, the unique solutions of the 1D variational (limit) problems ( 4.39) and (4.40), just as in the elastic case. This result holds under the assumption (3.10). Regarding the electric potential, we have showed that even if no conditions are imposed onthe dependence of C, P and ε with respect to x, the element ϕ¯ 0 , r, z ∈ T () is the unique solution of the 3D variational (limit) problem (4.63)–(4.65). 6.1. The limit model on the original rod h
It easy to check the following relation
Ci jkl e˜kl e˜i j dx + εαβ E˜α E˜β dx = (Cαβθ ρ e˜θ ρ + Cαβ 33 e˜33 )e˜αβ dx + 4 C3α 3θ e˜3θ e˜3α dx + (C33θ ρ e˜θ ρ + C3333 e˜33 )e˜33 dx + εαβ E˜α E˜β dx.
We now return to the original rod h and define the following spaces (cf. (4.44), (4.45) and (4.61)):
lh (h ) = L2 (0, L; H 1 (ωh )), h l0h (h ) = {ψ h ∈ L2 (0, L; H 1 (ωh )) : ψ h = 0 on eD }
≡ L2 (0, L; Sh (ωh )),
R ( ) = L2 (0, L; Q h (ωh )), Q h (ωh ) = {qh ∈ H 1 (ωh ) : qh dωh = 0}, h
and, having in mind the relations (5.27), one deduce
Ci jkl e˜kl e˜i j dx + εαβ E˜α E˜β dx = (Cαβθ ρ e˜θ ρ + Cαβ 33 e˜33 )e˜αβ dx +
%$Given the scalings (3.4), (3.6) and ( 3.18), the developments (4.1 ), (4.2) and (4.4) induce formal developments on uh , ϕ¯ h , σ h and Dh , respectively, whose leading terms we will identify and characterize in the following. For that we will undo the change of variable xh = h (x ) and accordingly define the de-scaled quantities:
(C33θ ρ e˜θ ρ + C3333 e˜33 )e˜33 dx $% &
#
33
+2
h (h ) × Rh (h ) × H01 (0, L ). T h (h ) = l0
&
αβ
2C3α 3θ e˜3θ e˜3α dx +
# $% & 3α +Pβ 3α E˜β
ξαh := ξαh xh3 = h−1 ξα (x3 ), ξ3h := ξ3h xh3 = ξ3 (x3 ),
εαβ E˜α E˜β dx. # $% &
ϕ¯ 0h := ϕ¯ 0h (xh ) = hϕ¯ 0 (x ), rh := rh (xh ) = hr (x ),
Tβ −2Pα 3β e˜3β
2 By using (5.26) with test functions v = u0 , v¯ = u1 , vˆ = uˆ and ψ¯ = ϕ 0 and the equality relations (5.31) and (5.32), we find
Ci jkl e˜kl e˜i j dx + εαβ E˜α E˜β dx = αβ e˜αβ dx + 2 3α e˜3α dx 0 0 ˜ + 33 e˜33 dx + Tβ Eβ dx = fi ui dx + gi ui d ,
h
ωh
#
307
N
as required. 6. Leading terms and limits models for the reference rod The results obtained in the previous sections concerning the leading terms of the developments for the scaled displacement vector (ui (h)), electric potential ϕ¯ (h ), stress tensor (σ ij (h)) and electric displacement vector (Di (h)), for the reference rod , can be summarized as follows:
zαh := zα , zh := h−1 z, as well as the warping function
wh = wh xh1 , xh2 = h2 w(x1 , x2 ), which is the unique solution of (cf. (4.46))
wh ∈ Q h (ωh ) such that C3α 3β ∂βh wh ∂αh υ h dωh = ωh
ωh
C3α 3β δβh ∂αh υ h dωh ,
for all υ h ∈ H 1 (ωh ), the torsion constant (cf. (4.47))
Jh = h4 J =
ωh
=
ωh
C3α 3β (δβh − ∂βh wh )(δαh − ∂αh wh )dωh C3α 3β (δβh − ∂βh wh )δαh dωh ,
(6.3)
308
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
and the second moment of area of the cross section with respect to axis Oxhβ (α = β )
Iαh =
ωh
(xhα )2 dωh ,
where δ1h (xh1 , xh2 ) = xh2 , δ2h (xh1 , xh2 ) = −xh1 . We also define the resultant of the applied loads and moments in each cross section (cf. (4.41) and (4.42)):
Fαh (xh3 ) = h3 Fα (x3 ) = fαh (xh1 , xh2 , xh3 )dωh ωh + ghα (xh1 , xh2 , xh3 )dγ h ∈ L2 (0, L ), γNh
F3h
( ) = h F3 (x3 ) = xh3
2
+
γNh
ωh
f3h
(
xh1 , xh2 , xh3
)d ω
h − h εαβ ∂βh ϕˆ h ∂αh ψ h dxh ,
(6.7)
for all (ψ , ρ , ς ) ∈ T ( ), h
h
Proposition 6.1. The leading terms of the formal developments induced by (4.1), (4.2) and (4.4) on the displacement vector (uhi ), electric potential ϕ¯ h , stress tensor (σihj ) and electric displacement vector (Dhi ) are such that:
⎧h h h h h h h 1h 2 u = ξαh + u1h ⎪ α + O ( h ) , u 3 ( x ) = ξ 3 − xβ ∂ 3 ξ β + u 3 + O ( h ) , ⎪ ⎪α ⎪ ⎪ h h h 1h h h h h h h ⎪ u1h ⎪ α = zα + δα z , u3 = −r − z3 − xα (zα ) − w (z ) , ⎪ ⎪ ⎪ ⎪ ⎪ ϕ¯ (h ) = ϕ¯ 0 + O(h ), ⎪ ⎪ ⎪ ⎪ h ⎪ 0h 2 ⎪ ⎪ ⎪ϕ¯ = ϕ¯ + O(h ), ⎨ h σαβ = O ( h ), ⎪ ⎪ ⎪ h ⎪ σ3α = Pβ 3α ∂βh (ϕ¯ 0h + ϕˆ ) − C3α3β (∂βh rh + ∂3h zh (∂βh wh − δβh )) + O(h ), ⎪ ⎪ ⎪ ⎪ ⎪ h h h ⎪ σ33 = Y (∂3h ξ3h − xhα ∂33 ξα ) + O ( h ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dhα = −Pα 3β (∂βh rh + ∂3h zh (∂βh wh − δβh )) − εαβ ∂βh (ϕ¯ 0h + ϕˆ ) + O(h ), ⎪ ⎪ ⎪ ⎪ ⎩ h ¯ h h h h h D3 = P (∂3 ξ3 − xα ∂33 ξα ) + O(h ), (6.4) (we note that α ∼ ∼ O ( h ), α ∼ ∼ and z3h ∼ O(h )) where ξαh ∈ H02 (0, L ), ξ3h ∈ H01 (0, L ) and ϕ¯ 0h , rh , zh ∈ T h are the unique solutions of the following variational problems,
ξαh ∈ H02 (0, L ),
L 0
h h h Y h Iαh ∂33 ξα ∂33 χαh dxh3 =
L
0
Fαh χαh dxh3 −
O(h0 ), zh
(no sum on α ), for all χα ∈ h
H02
L 0
for all χ ∈
H01
Mαh ∂3h χαh dxh3 , (6.5)
σ3hα = Pβh3α ∂βh (ϕ¯ 0h + ϕˆ ) − C3hα3β (∂βh rh + ∂3h zh (∂βh wh − δβh )), h h h σ33 = Y h (∂3h ξ3h − xhα ∂33 ξα ) , h h h h h ∂βh (ϕ¯ 0h + ϕˆ ), Dα = −Pα 3β (∂β r + ∂3h zh (∂βh wh − δβh )) − εαβ h h ξα ) . Dh3 = P¯ h (∂3h ξ3h − xhα ∂33
From (6.7), by taking appropriate choices of test functions we obtain the following set of independent equations (note that xˆ = (xh1 , xh2 )):
L 0
Jh ∂3h zh ∂3h ζ dx3 −
=
L
L
=
L
L
ωh
0
Pβh3α ∂βh ϕˆ h (∂αh wh − δαh ) dxˆ
ρ3h dx3
Pβh3α ∂β ϕ¯ 0h ∂α ρh dxˆ
h
ωh
Pβh3α ∂β ϕˆ h ∂α ρh dxˆ
h
ωh
(0, L ),
(ϕ¯ 0h , rh , zh ) ∈ T h (h ),
0
L
F3h χ3h dxh3 ,
(6.6)
∂3h ζ h dx3
∂3h ζ h dx3 ∀ζ h ∈ H01 (0, L ),
ρ3h dx3
ρ3h dx3
∀ρh ∈ Q h (ωh ), ρ3h ∈ L2 (0, L ),
(6.9)
h εαβ ∂βh ϕ¯ 0h ∂αh ψh dxˆh ψ3h dx3 ωh 0 L h + Pβh3α ∂αh rh ∂βh ψh dxˆ ψ3h dx3 ωh 0 L h + Pβh3α (∂αh wh − δαh )∂βh ψh dxˆ ∂3h zh ψ3h dx3 ωh 0 L h =− εαβ ∂βh ϕˆ h ∂αh ψh dxˆh ψ3h dx3
L
∀ψ
(6.8)
0
0
Pβh3α (∂αh wh − δαh )∂βh ϕ¯ 0h dxˆ
h
h
−
C3hα 3β ∂βh rh ∂αh ρh dxˆ
ωh
0
L
h
ωh
0
0 h ∈
ωh
S(ωh ), ψ3h ∈ L2 (0, L ).
(6.10)
ωh
Y h A(ωh )∂3h ξ3h ∂3h χ3h dxh3 = h 3
L
O(h−1 )
(0, L ),
ξ3h ∈ H01 (0, L ),
0
h
h σαβ = 0,
Bearing in mind these definitions and in view of (6.1) and (6.2), one has the following result for the leading terms of the induced formal developments for uh , ϕ¯ h , σ h and Dh .
zh
h
h
The variational problems obtained for the transverse displacement ξαh , the axial displacement ξ3h and the electric potential ϕ¯ 0h constitute the limit model for the original rod. On the other hand, the limit constitutive equations are (cf. ( 6.4))
γNh
ϕ¯ 0h
h
whereas wh ∈ Qh (ωh ) is the unique solution of (6.3).
Mαh (xh3 ) = h3 Mα (x3 ) = xhα f3h (xh1 , xh2 , xh3 )dωh ωh + xhα gh3 (xh1 , xh2 , xh3 )dγ h ∈ L2 (0, L ).
O(h−1 ),
L
h
gh3 (xh1 , xh2 , xh3 )dγ h ∈ L2 (0, L ),
ξh
Jh ∂3h zh ∂3h ς h dxh3 + C3hα 3β ∂βh rh ∂αh ρ h dxh h 0 h + εαβ ∂αh ϕ¯ 0h ∂βh ψ h dxh + Pβh3α (∂α rh ∂β ψ h − ∂βh ϕ¯ 0h ∂αh ρ h )dxh h h + Pβh3α (∂αh wh − δαh )(∂3h zh ∂βh ψ h − ∂3h ς h ∂βh ϕ¯ 0h )dxh h = Pβh3α ∂βh ϕˆ h (∂αh ρ h + ∂3h ς h (∂αh wh − δαh ))dxh
[−ah , ah ] ×
= Now, let us assume the particular case where h = (−ah , ah ) × {−bh }. In this case we can take as [−bh , bh ] and γeD transversal test functions ρh (xh1 , xh2 ) = xh1 xh2 and ψh (xh1 , xh2 ) = xh1 (xh2 + bh ). In addition to that, we define the following quantities:
θ1h = xh2 , θ2h = xh1 , θˆ1h = xh2 + bh , θˆ2h = xh1 , Cβh = C3hα3β θαh , h ˆh Pβh = Pβh3α θαh , Pα†h = Pβh3α θβh , εβh = εαβ θα .
J.M. Viaño et al. / International Journal of Solids and Structures 81 (2016) 294–310
Therefore, the latter two variational equations can be reformulated as follows:
L h ρ3h dx3 − Pβh ∂βh ϕ¯ 0h dxˆ ρ3h dx3 ωh ωh 0 0 L h = Pβh ∂βh ϕˆ h dxˆ ρ3h dx3 ∀ρ3h ∈ L2 (0, L ), ωh 0 L L h εβh ∂βh ϕ¯ 0h dxˆh ψ3h dx3 + Pα†h ∂αh rh dxˆ ψ3h dx3 ωh ωh 0 0 L h + Pα†h (∂αh wh − δαh ) dxˆ ∂3h zh ψ3h dx3 ωh 0 L =− εβh ∂βh ϕˆ h dxˆh ψ3h dx3 ∀ψ3h ∈ L2 (0, L ).
L
Cβh ∂βh rh dxˆ
h
0
ωh
Moreover, the strong formulation of our problem can be formulated as follows:
⎧ h ⎪ Pβ 3α (∂αh wh − δαh )∂βh ϕ¯ 0h dxˆ ∂3h Jh ∂3h zh − ∂3h ⎪ ⎪ ωh ⎪ ⎪ ⎪ ⎪ h ⎪ h h h h h h ⎪ ˆ = ∂ P ∂ ϕ ˆ ( ∂ w − δ ) d x , h α α ⎪ ω β 3α β 3 ⎪ ⎪ ⎪ ⎪ ⎪ zh (a ) = 0, (Jh ∂3h zh )(a ) = 0, Pβ 3α (∂αh wh ⎪ ⎪ ωh ×{a} ⎪ ⎨ h −δαh )∂βh (ϕ¯ 0h + ϕˆ h ) dxˆ = 0, for a ∈ {0, L} ⎪ ⎪ h h ⎪ C h ∂ h rh dxˆh − Pβh ∂βh ϕ¯ 0h dxˆ = Pβh ∂βh ϕˆ h dxˆ , ⎪ β ⎪ β h h h ⎪ ω ω ω ⎪ ⎪ ⎪ ⎪ h ⎪ εβh ∂βh ϕ¯ 0h dxˆh + Pα†h ∂αh rh dxˆ + Pα†h (∂αh wh ⎪ ⎪ h h h ⎪ ω ω ω ⎪ ⎪ −δ h ) dxˆh ∂ h zh = − ε h ∂ h ϕˆ h dxˆh , ⎪ ⎪ α ωh β β 3 ⎩ a.e. in (0, L )
(6.11)
ωh
rh
ϕ¯ 0h
ϕˆ h ,
from which we deduce that = = 0 and = i.e., the first order component of the displacements u1h is of the Bernoulli–Navier kind and the zeroth order component of the electric potential ϕ 0h coincides with the applied potential ϕˆ h . Remark 6.3. For materials having an isotropic mechanical behavior, such as PVDF, the limit model and constitutive equations are somewhat simplified since in this case the stiffness tensor C has the form
Ci jkl =
h
with, from (4.46) and (4.47)
Jh =
Y (∂1h wh )2 − (∂2h wh )2 ) (I1h + I2h − 2 (1 + ν ) ωh ωh
and
wh ∈ Q h ( ω h ), ∂αh wh ∂αh ψ h dωh = (xh2 ∂1h ψ h − xh1 ∂2h ψ h )dωh , ωh
ωh
for all ψ ∈ H (ω ). 1
h
7. Conclusions
⎧ h h h h ∂3 J ∂3 z = 0 a.e. in (0, L ), zh (a ) = ∂3h (Jh zh )(a ) = 0, ⎪ ⎪ ⎪ ⎨ h h h h Cβ ∂β r dxˆ = 0 a.e. in (0, L ), ωh ⎪ ⎪ ⎪ h ⎩ P†h ∂ h rh dxˆh + Pα†h (∂αh wh − δαh ) dxˆ ∂3h zh = 0 a.e. in (0, L ), α α zh
L ! Jh ∂3h zh ∂3h ς h dxh3 ahiso ((zh , rh , ϕ¯ 0h ), (ς h , ρ h , ψ h )) = 0 Y + ∂αh rh ∂αh ρ h dxh 2(1 + ν ) h + εαβ ∂αh ϕ¯ 0h ∂βh ψ h dxh + Pβ 3α (∂α rh ∂β ψ h − ∂βh ϕ¯ 0h ∂αh ρ h )dxh h h + Pβ 3α (∂αh wh − δαh )(∂3h zh ∂βh ψ h − ∂3h ς h ∂βh ϕ¯ 0h )dxh ,
h
Remark 6.2. Assume that ∂βh ϕˆ h = ∂βh ϕ¯ 0h = 0, i.e., both the applied potential and the first order component of the electric potential ϕ h distributed on the beam depend only on x3 . In this case, the previous equations give
ωh
309
Y νY δi j δkl + δik δ jl + δil δ jk , 1 + ν 1 − 2 ν 2 1 + ν ( )( ) ( )
where Y is the Young’s modulus, ν the Poisson’s ratio and δ the Kronecker’s delta. As a consequence, Y defined by (4.38) is indeed the Young’s modulus and
Y C3α 3β = δ . 2(1 + ν ) αβ Therefore, the variational problems (6.5) and (6.6) remain unchanged, while in the variational problem for ϕ¯ 0h the bilinear form becomes
The main result in this paper is the model shown in Eqs. (6.5)– (6.7). From (6.5)–(6.6) we observe that the electric potential has no effect on the zeroth order components of the bendings ξαh and the stretching ξ3h . This effect only appears on higher order terms as we can see from (6.7). Unfortunately, unless a specific dependence of ϕ h on (xh1 , xh2 ) is established, we cannot reduce (6.7) to a set of onedimensional expressions as we did for the zeroth order displacements. Still, by taking particular cases of transversal section ωh we can give a strong formulation of (6.7) as a set of two-dimensional expressions, as shown in (6.11). There we see that some of the terms are still global since they involve an integration over the whole transversal section. This is a remarkable difference with our previous work Viaño et al. (2015) (the electric potential was applied at the beam ends), where in the limit model the stretching was coupled with the electric potential, even for the zeroth order terms, and that the set of equations was reduced to one-dimensional expressions (despite having some global terms acting as coefficients). We ended this work by showing two interesting particular cases. In Remark 6.2 we show that a trivial case is obtained when the potential depends only on the axial coordinate, and in Remark 6.3 we show the equations for the homogeneous isotropic case. Acknowledgments This research was partially financed by FEDER Funds through “Programa Operacional Factores de Competitividade - COMPETE”, by Portuguese Funds through FCT - “Fundação para a Ciência e a Tecnologia”, within the Project PEst-OE/MAT/UI0013/2014, and by the project “Modelización y simulación numérica de sólidos y fluidos en dominios con pequeñas dimensiones. Aplicaciones en estructuras, biomecánica y aguas someras, MTM2012-36452-C02-01 financed by the Spanish Ministry of Economía y Competitividad with the participation of FEDER. For the sake of readability we give a list of the functional spaces and more important functions and notations involved in this paper. References Álvarez-Dios, J.A., Viaño, J.M., 1993a. An asymptotic general model for linear elastic homogeneous anisotropic rods. Int. J. Numer. Meth. Eng 36 (18), 3067–3095. doi:10.1002/nme.1620361804. Álvarez-Dios, J.A., Viaño, J.M., 1993b. On a bending and torsion asymptotic theory for linear nonhomogeneous anisotropic elastic rods. Asympt. Anal 7 (2), 129–158. Banks, H., Smith, R., Wang, Y., 1996. Smart Material Structures. Modeling, Estimation and Control. Chichester: Wiley; Paris: Masson. Bellis, C., Imperiale, S., 2014. Dynamical 1D models of passive piezoelectric sensors . Math. Mech. Solids 19 (5), 451–476. doi:10.1177/1081286512469980.
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Sène, A., 2001. Modelling of piezoelectric static thin plates. Asymptot. Anal 25 (1), 1–20. Su, B., 1997. Novel Fabrication Processing for Improved Lead Zirconate Titanate (PZT) Ferroelectric Ceramic Materials Ph.D. thesis. Faculty of Engineering, University of Birmingham. Taylor, G.W., Gagnepain, J.J., Meeker, T.R., Nakamura, T., Shuvalov, L.A., 1985. Piezoelectricity. Pure and Applied Mathematics (Boca Raton), 82. Gordon and Breach Science Publishers New York. Trabucho, L., Viaño, J.M., 1996. Mathematical modelling of rods. In: Ciarlet, P.G., Lions, J.L. (Eds.), Handbook of Numerical Analysis, IV. North-Holland, Amsterdam, pp. 487–974. Tutek, Z., 1987. A homogenized model of rod in linear elasticity,. In: Ciarlet, P.G., Sanchez-Palencia, E. (Eds.), Applications of Multiple Scaling in Mechanics, IV. Masson, Paris, pp. 316–332. ´ I., 1986. A justification of the one-dimensional linear model of elasTutek, Z., Aganovic, tic beam. Math. Methods Appl. Sci 8 (4), 502–515. doi:10.1002/mma.1670080133. Viaño, J.M., Ribeiro, C., Figueiredo, J., 2005. Asymptotic modelling of a piezoelectric beam. In: Proceedings of the II ECCOMAS Thematic Conference on Smart Structures and Materials. Viaño, J.M., Figueiredo, J., Ribeiro, C., Rodríguez-Arós, Á., 2015. A model for bending and stretching of piezoelectric rods obtained by asymptotic analysis. Z. Angew. Math. Phys 66 (3), 1207–1232. doi:10.1007/s00033-014-0438-1. Weller, T., Licht, C., 2002. Analyse asymptotique de plaques minces linéairement piézoélectriques. C. R., Math., Acad. Sci. Paris 335 (3), 309–314. doi:10.1016/ S1631-073X(02)02457-3. Weller, T., Licht, C., 2004. Asymptotic modeling of linearly piezoelectric slender rods. C. R., Méc., Acad. Sci. Paris 13, 57–63. Juan Viaño, born in 1955, got a Ph.D. in Mathematics at the University of Santiago de Compostela and a Ph.D. in Numerical Analysis at the University Pierre et Marie Curie (Paris 6). He is Professor of Applied Mathematics at the University of Santiago de Compostela since 1988, member of the Promoter Group of the Node in Galicia of National Spanish Centre of Mathematics (IEMath-Galicia) since 2008 and member of the Promoter Group of Consortium of Technological Institute of Industrial Mathematics since 2012. He has been Head of the Department of Applied Mathematics ath the University of Santiago de Compostela for 6 years, Dean of the Faculty of Mathematics at the University of Santiago de Compostela for 8 years and Vice-rector of Teaching and Academic Organisation of the University of Santiago de Compostela for 2 years. He is and has been assesor and member of a number of committes of national and international level. He has published more than 70 papers in international journals, several books and directed more than 10 doctoral theses. He has also directed more than 10 national and international research projects. Jorge Figueiredo, born in 1966, has a Bachelor degree in Physics/Applied Mathematics (Porto University, 1988) and a M.Sc. in Astrophysics and Spatial Techniques (Paris University, 1989). He got his Ph.D. in Applied Mathematical Sciences form Porto University (1996), under the theme A new numerical code to model pre-main-sequence lowmass stars. He joined the Dep. of Mathematics and Applications of Minho University as a Junior Teaching Assistant in 1993 and became Assistant Professor in 1996. The main research interests concern PDEs with particular emphasis on deriving reduced beam models from 3D piezoelectricity and developing high-order finite volume schemes for the shallow-water equations. Carolina Ribeiro received her Bachelor degree in Mathematics from University of Porto and her Ph.D. degree in Mathematical Science, under the theme Asymptotic derivation of models for anisotropic piezoelectric beams and shallow arches, from University of Minho in 2009. She joined the Department of Mathematics and Applications of the University of Minho, as a Junior Teaching Assistant in 2000 and was promoted to Assistant Professor in 2009. Carolina’s research interests focus in the area of partial differential equations with a particular emphasis on deriving beam models from 3D piezoelectricity and developing high-order finite volume schemes for the shallow water equations. Ángel Rodríguez-Arós, born in 1976, has a Ph.D. in Applied Mathematics from the University of Santiago de Compostela, Spain, with European special mention (2005) and spent a post-doctoral year at the ONERA-site Toulouse, France (2008). Currently, he is Lecturer at the University of A Coruña. Over the years he has published over 20 papers in international journals, given around 20 talks in national and international meetings and has been member of nearly 10 national and international research projects. He has also published a text book in spherical trigonometry with applications to navigation (2012).