• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    Simple Efficient Smart Finite Elements for the Analysis of Smart Composite Beams

    2015-12-13 01:54:16RayDongandAtluri
    Computers Materials&Continua 2015年9期

    M.C.Ray,L.Dongand S.N.Atluri

    Simple Efficient Smart Finite Elements for the Analysis of Smart Composite Beams

    M.C.Ray1,L.Dong2and S.N.Atluri3

    This paper is concerned with the development of new simple 4-noded locking-alleviated smart finite elements for modeling the smart composite beams.The exact solutions for the static responses of the overall smart composite beams are also derived for authenticating the new smart finite elements.The overall smart composite beam is composed of a laminated substrate conventional composite beam,and a piezoelectric layer attached at the top surface of the substrate beam.The piezoelectric layer acts as the actuator layer of the smart beam.Alternate finite element models of the beams,based on an“equivalent single layer high order shear deformation theory”,and a“l(fā)ayer-wise high order shear deformation theory”,are also derived for the purpose of investigating the required number of elements across the thickness of the overall smart composite beams.Several cross-ply substrate beams are considered for presenting the results.The responses computed by the present new “smart finite element model”excellently match with those obtained by the exact solutions.The new smart finite elements developed here reveal that the development of finite element models of smart composite beams does not require the use of conventional first order or high order or layer-wise shear deformation theories of beams.Instead,the use of the presently developed locking-free 4-node elements based on conventional linear piezo-elasticity is sufficient.

    Piezoelectricity,exact solutions,smart finite element,smart structures.

    1 Introduction

    In the quest for developing very light weight high performance flexible structures,a concept has emerged for developing structures with self-controlling and/or selfmonitoring capabilities.Expediently,utilizing the piezoelectric effects Foreward(1981)first attempted to demonstrate the feasibility of using the piezoelectric actuators to damp out the vibrations of a cylindrical fiber glass mast.Subsequently,Bailey and Hubbard(1985),Bruke and Hubbard(1987),Crawley and Leuis(1987),Im and Atluri(1989),Shi and Atluri(1990)successfully reported that the patches of piezoelectric actuators being bonded with the host beams efficiently perform as the distributed actuators of the host beams.Miller and Hubbard(1987)first demonstrated that a layer of the piezoelectric material being integrated with a cantilever beam can act as the distributed sensor of the host cantilever beam.When these distributed sensors and actuators are the elements of the control systems such that the distributed piezoelectric actuators can be activated with a proper control voltage,the host structure attains the self-controlling and self-sensing capabilities.Such flexible host structures possessing built-in mechanism for achieving self-controlling and self-sensing capabilities are being customarily called as smart structures.Since its inception,tremendous research on smart structures has been going on for developing very light weight smart flexible structures.Needless to say that the finite element method has been established as the most widely accepted analytical method for structural analysis and in case of the analysis of smart structures,the same is also true.A brief review of the finite element analysis of the smart structures is now in order.

    Shi and Atluri(1990)developed finite element models of smart beams and frames,undergoing large deformations,using a complementary energy approach.Robbins and Reddy(1991)developed a finite element model of an aluminum beam actuated by a piezoelectric layer using a “l(fā)ayer wise displacement theory”.Ha et al.(1992)derived a finite element model of laminated composite plates containing distributed piezoelectric sensors and actuators,using an eight noded brick element augmented with incompatible modes.Hwang and Park(1993)presented a finite element formulation for control of vibration of laminated plates integrated with piezoelectric sensors and actuators.In 1994,Ray,Bhattacharyya and Samanta first derived a finite element model for three dimensional analysis of smart composite plates employing a“high order shear deformation theory”proposed by Lo,Christensen,Wu(1978).Saravanos and Heyliger(1995)derived a finite element model for static and free vibration analysis of composite beams with embedded piezoelectric sensors and actuators using “l(fā)ayer wise displacement theories”.Lin,Hsu and Huang(1996)derived a finite element model for analyzing the deflection control of plates with piezoelectric actuators.Saravanos,Heliger and Hopkins(1997)employed layer wise displacement and electric potential theories for the finite element analysis of laminated composite plates integrated with piezoelectric sensors and actuators.B-hattacharya,Suhail and Sinha(1998)developed a finite element model for the free vibration analysis of laminated composite plates coupled with piezoelectric sensors and actuators and the model is based on the “first order shear deformation theory”(FSDT).Chee,Tong and Steven(1999)derived a finite element model based on a“high order displacement field”and a “l(fā)ayer wise linear electric potential theory”for the static analysis of smart composite beams.Varadarajan,Chandrashekhara and Agarwal(2000)derived a finite element model of composite beam based on a “high order shear deformation theory”and implemented the LQG/LTR method for studying robust control of the beams using piezoelectric actuator layer.Valoor,Chandrashekhara and Agarwal(2001)derived a finite element model of composite beams integrated with piezoelectric sensors and actuators and employed neural network for robust control of the beams.Chee,Tong and Steven(2002)again derived a finite element model of smart composite plates based on a“high order shear deformation theory”and a “l(fā)ayer wise electric potential theory”and optimized the piezoelectric actuator orientations for static shape control of the plates.Kulkarni and Bajoria(2003)derived a finite element model using a“high order shear deformation theory”for analyzing active control of curved beams integrated with piezoelectric sensors and actuators.The finite element model derived by Luo and Tong(2004)is based on the Timoshenko beam theory and capable of detecting debonding of the piezoelectric sensors and actuators.Gupta,Seshu and Issac(2004)derived a finite element model of piezoelectrically actuated shells and experimentally verified the model.Ahmed,Upadhyay and Venkatesan(2005)developed a layer-by-layer finite element model of cantilever beam actuated by a piezoelectric layer capturing the continuity of shear stress across the interface between the piezoelectric layer and the host beam.Trindade and Benjeddou(2006)derived a finite element model of smart beams with embedded shear mode piezoceramic actuators and sensors using high order shear deformation theory.Using a layer wise displacement theory and employing an optimal control strategy,Zabihollah,Sedagahti and Ganesan(2007)derived a finite element model for analyzing active vibration control of smart laminated beams.Al-Ajmi and Benjeddu(2008)proposed a discrete layer finite element model for detecting the damage in smart beams.Neto,Yu and Roy(2009)proposed two finite elements for the static analysis of smart beams with piezoelectric actuators.Bendary,Elshafei and Riad(2010)proposed a finite element model of beams coupled with piezoelectric actuators which involved one dimensional isoperimetric hermite cubic shape functions and the lagrange interpolation function.In order to monitor the health of smart structures,Umesh and Ganguli(2011)developed a finite element model of smart composite plates using “first order shear deformation theory”and investigated the control gains as the damage indicators.Park and Lee(2012)derived spectral finite element model in frequency domain for the dynamic analysis of smart composite beams based on the Euler-Bernouli beam theory.Elshafei and Alraien(2013)presented a finite element formulation of smart composite beams based on a“high order shear deformation theory”.Zhang and Schmidt(2014)carried out geometrically nonlinear finite element analysis of smart composite structures using“first order shear deformation theory”.Song,Kim,Park and Lee(2015)derived a finite element model based on the“first order shear deformation theory”for investigating the guided waves in smart composite beams.

    The above review of literature indicates that all the finite element models of smart structures presented so far in the literature are based on some“displacement theories”which include “classical theories”,the “first order shear deformation theory”,“high order shear deformation theories”and “l(fā)ayer-wise theories”.In practice,the smart structures are thin.So,the use of high order shear deformation theories is not essential for finite element modeling of smart structures.Although the layerwise displacement theory provides accurate results for laminated structures when the material properties of the adjacent layers differ significantly,the finite element model based on the layer-wise theory involves excessively large number of nodal degrees of freedom increasing the computational cost of the model.On the other hand,if“first order shear deformation theory”is used,the finite element model needs to introduce the shear correction factor for alleviating the shear locking problem.

    Recently,Dong,EI-Gizawy,Juhany and Atluri(2014)developed an efficient locking-free 4-noded finite element for analyzing the laminated beams,based on simple and conventional 2D elasticity theories.This work motivated the authors to develop a new simple 4-noded finite element for analyzing the smart composite structures without using any higher order or layer-wise deformation theories.This paper is concerned with the derivation of such a new smart finite element.Laminated composite beams integrated with a piezoelectric layer at their top surfaces are considered for deriving this new smart finite element.Exact solutions of the overall smart beams are also derived here for validating the new finite element model.Two more finite element models of the overall beams based on an equivalent single layer high order shear deformation theory and a layer-wise high order shear deformation theory are also derived for the purpose of comparison,and for determining the number of the new smart elements required across the thickness of the overall beam.

    2 Basic Equations

    Figure 1:Schematic diagram of a simply supported laminated composite beam integrated with a piezoelectric actuator layer.

    Figure 1 illustrates a simply supported laminated composite beam integrated with a layer of piezoelectric material at its top surface.The length and the thickness of the beam are designated by L and h,respectively.The thickness of the piezoelectric layer is denoted by hp.The top surface of the beam is subjected to a distributed mechanical load q(x).The piezoelectric layer acts as a distributed actuator layer of the substrate beam.For actuating the substrate beam,the distributed electric potential(voltage)is applied on the top surface of the piezoelectric layer while the surface of the piezoelectric layer being in contact with the top surface of the substrate beam is grounded.The origin of the coordinate system(x-z)is located at one end of the beam such that the lines given by x=0 and x=L represent the ends of the beam and the plane given by z=0 denotes the mid-plane of the beam.The constitutive relations for the converse and the direct piezoelectric effects appropriate for the beam analysis are given by

    In Eqs.(1)and(2),the state of stresses{σp},the state of assumed strains{∈p?},the electric field vector{E},the electric displacement vector{D},the elastic coefficient matrix[Cp],the piezoelectric constant matrix[e]and the dielectric constant matrix[ε]are given by

    where upand wpare the displacements at any point in the piezoelectric layer along the x-and the z-directions,respectively while u and w are the same in the substrate beam.Based on the displacement fields,the states of strains{∈p}and{∈}at any point in the piezoelectric layer and in the substrate beam,respectively are given by

    The electric potential function φ(x,z)at any point in the piezoelectric layer is related to the electric fields as follows:

    The total potential energy(Π)of the overall smart composite beam of width b is given by

    In Eq.(8),Φ(x)andˉσ(x)are the applied distributed electric potential and charge on the top surface of the piezoelectric layer,respectively.Also,hkand hk+1represent the z coordinates of the bottom and the top surfaces of any orthotropic layer of the substrate beam,respectively.

    2.1 Derivation of new 4-noded smart finite elements

    Figure 2:Typical 4-noded finite elements:(a)one piezoelectric smart element and one laminated beam element across the thickness of the overall smart beam(b)one laminated smart element across the thickness of the overall smart beam

    The overall smart composite beam is discretized by four noded isoparametric elements.The finite element mesh can be generated by using two elements across the thickness of the overall beam as shown in Fig.2(a).In this case,the bottom element is composed of the orthotropic layers of the substrate beam and is called the beam element,while the top element is composed of the piezoelectric material only and is called the piezoelectric element.The other option for generating the mesh is to use one element across the thickness of the overall beam as shown in Fig.2(b).This element is a laminated smart finite element in which the top layer is the piezoelectric layer while the other layers are the orthotropic layers of the substrate beam.Thus three types of elements namely the piezoelectric smart element,laminated smart finite element and the beam element are to be formulated for the mesh described by Fig.2.The piezoelectric smart element is characterized by its height hpand length L/n with n being the number of elements along the length of the beam.The length and height of the beam element are L/n and h,respectively.The assumed strain fields in the piezoelectric and the beam elements are given by[Dong,EI-Gizawy,Juhany and Atluri(2014)]:

    in which{γp}and{γ}are the matrices of unknown constants for the assumed strain distributions and[A]is a matrix describing the distribution of the assumed strains in the elements.These are given by

    It may be noted from Eqs.(9)and(10)that the assumed normal strains vary linearly with respect to the element-local Cartesian coordinates and the assumed transverse shear strain is constant.It is also to be noted that the assumed bending strain is not coupled with the shear deformation.On the other hand,the displacements at any point in the respective element are given by nodal interpolation:

    in which[N]is the shape function matrix,{dpe}and{de}are the nodal displacement degrees of freedom for the piezoelectric element and the beam element,respectively and their explicit forms are as follows:

    Figure 3:4-noded element with five points of collocation.

    Thus the compatibility of the assumed strains with the mesh based strains based on the displacement fields at the above mentioned five points given by Eq.(13)results in the following conditions:

    Piezoelectric element:

    Beam element:

    It may be noted that the coordinates,,andare the local Cartesian coordinates of the points of collocation in the piezoelectric element while xC,xD,zAand zBare the local Cartesian coordinates in the beam element.Using the straindisplacement relations given by Eq.(6)in Eqs.(14)and(15),the unknown constants{γp}and{γ}can be determined as follows:

    where,

    It may again be noted that each row of the matrix[B]is to be computed at each of the five collocated points given by Eq.(13).On substitution of Eq.(16)into Eq.(9),the assumed strains can be expressed in terms of the nodal displacement degrees of freedom as follows:

    where

    Using Eq.(5)in Eq.(3),it can be written that

    where

    The materials being studied here are linear.Thus substituting Eqs.(1),(2),(4),(11),(18),(5)and(6)in Eq.(4)and subsequently applying the principle of minimum potential energy i.e.δΠ =0,the following elemental governing equilibrium equations are obtained:

    where[Kpe]and[Ke]are the elemental stiffness matrices for the piezoelectric element and the substrate beam element,respectively;andare the elemental electro-elastic coupling matrix and the elemental dielectric stiffness matrix of the piezoelectric layer,respectively.Also,and{Fe}are the elemental electrical load vector due to the applied distributed charge and the elemental mechanical load vector,respectively.The forms of these matrices are given by

    As discussed in[Dong,EI-Gizawy,Juhany and Atluri(2014)],the technique of“over integration”is needed to accurately evaluate the stiffness matrices of laminated elements.In order to take care of the different material properties of each lamina in the substrate,a layer-wise two-point Gauss quadrature in the thickness direction is adopted in this study.In this way,we consider another variable(-1≤ηk≤1)as the natural coordinate in the thickness direction of any(k-th)individual layer,which can be related to the natural coordinate(-1≤η≤1)of the whole beam element as follows:

    Thus the elemental stiffness matrix for the substrate beam is to be evaluated as:

    The elemental equations as derived above are assembled in a straight forward manner to obtain the global equations of equilibrium as follows:

    It may be noted that although two sets of equations given by Eqs.(27)and(28)are derived,Eq.(27)is required to compute the nodal displacements if the electric potential is prescribed and Eq.(28)estimates the corresponding nodal charges.

    2.2 Exact Solutions of the Smart Composite Beam

    Replacing the assumed strains{∈p?}and{∈p?}by the displacement field based strains{∈p}and{∈},respectively in Eq.(8)and applying the principle of minimum potential energy i.e.δΠ=0,the following governing equilibrium equations for the piezoelectric layer and the orthotropic layers of the substrate beam are obtained:

    The simply supported boundary conditions obtained from the variational principle are

    Also,the variational principle yields the following interface continuity conditions:

    2.2.1 Exact solutions for the piezoelectric layer

    For a particular mode of deformation,the displacement functions and the electric potential function for the piezoelectric solid which satisfy the boundary conditions at the edges of the beam given by Eq.(31)are assumed as

    where Up(z),Wp(z)and Φ(z)are unknown functions of z and p=mπ/L with m being the mode number.It may further be assumed that

    where U0p,W0pand Φ0are unknown constants to be determined and s is a characteristic parameter.Considering{∈p}and{∈}in place of{∈p?}and{∈p?},respectively and subsequently using Eqs.(1),(2),(6),(33)and(34)into the governing equations given by Eq.(29),the following set of homogeneous algebraic equations are obtained:

    in which

    For non-trivial solutions of U0p,W0pand Φ0,the determinant of the co-efficient matrix of Eq.(36)must vanish.This leads to the following sixth degree polynomial equation:

    where,

    For the geometrical parameters and material properties of the piezoelectric material that is considered for evaluating the numerical results,two real roots and two pairs of complex conjugate roots of Eq.(37)are obtained and these are denoted as

    Now using the roots of Eq.(37)and the relations given by Eq.(40)in Eqs.(33)and(34),the exact solutions for the displacement fields(up,wp)and the electric potential function(φ)in the piezoelectric layer are derived as follows:

    in which Ui(i=1,2,3,..,6)are the unknown constants.The various constants appearing in Eqs.(43)and(44)are given by

    Next,using the constitutive relations given by Eq.(1),the exact solutions for the stresses in the piezoelectric layer are obtained as follows:

    The various coefficients Ti(i=1,2,3,...,12)appearing in Eqs.(46)to(48)are given by

    2.2.2 Exact Solutions for the substrate Beam

    For a particular mode of deformation,the displacement field for any(k-th)layer of the substrate beam satisfying the boundary conditions given by Eq.(31)can be assumed as

    in whichU0k,W0kare the unknown constants for the k-th layer and ris a characteristic parameter.Substitution of Eq.(50)into the governing equations for the k-th

    layer of the substrate beam given by Eq.(30)and the use of strain displacement relations result into the following characteristics equation for the layer:

    The roots of Eq.(51)are given by

    Using Eqs.(4),(50)and(52)and carrying out some algebraic manipulations,the exact solutions for the displacement fileds and the state of stresses at any point in the k-th layer of the substrate beam can be derived as follows:

    2.2.3 Solutions of unknown constants

    In order to solve the unknown constants(Ui,i=1,2,3,..6;,i=1,2,3,4 and k=1,2,3,...,N),the prescribed boundary conditions are considered as follows:

    in which q0is the amplitude of the prescribed mechanical load and V is the amplitude of the prescribed electric potential at the top surface of the piezoelectric layer.Satisfaction of the prescribed boundary conditions given by Eqs.(31)and(58)and the continuity conditions given by Eq.(32)leads to the following system of algebraic equations:

    in which Kijis the coefficient of Γjwhile the vectors{Γ}and{Q}are given by

    3 Finite element model using LHSDT

    For the beam analysis,an equivalent single layer high order shear deformation theory(HSDT)proposed by Lo,Christensen and Wu(1978)is given by

    in which u0and w0are the translational displacement of any point on the midplane(z=0)of the substrate beam along x and z directions,respectively;θxand θzare the first order rotational variables while φx,γxand φzare high order rotational variables.In this section,a finite element model of the overall smart beam being studied here is derived using a“l(fā)ayer-wise high-order shear deformation theory(LHSDT)”.According to thisLHSDT,the displacement field at any point in the substrate beam is given by Eq.(61)while the displacement field at any point in the piezoelectric layer is considered as follows:

    in which ψx,lx,mx,ψzand lzare the generalized rotational coordinates for the piezoelectric layer.The generalized displacement coordinates at any point in the overall beam are expressed in a vector form as follows:

    A three-noded bar element is used for implementing thisLHSDTto discretize the overall beam.Thus the generalized displacement coordinate vector for the i-th node of the element is given by

    and the generalized displacement vectorj5i0abt0bat any point in the element can be expressed in terms of the nodal generalized displacement vector as follows:

    in which the shape function matrix[N]and the nodal generalized displacement vector{de}for the element are given by

    while[Ni]=niI with niand I being the shape function associate with the i-th node of the element and a(12x12)identity matrix,respectively.The state of strains at any point in the substrate beam and that in the piezoelectric layer of the element can be expressed in terms of the nodal generalized degrees of freedom as follows:

    in which the matrices[Z1]and[Z2]and the nodal strain-displacement matrices[B1]and[B2]are presented in theAppendix.The electric potential function which is zero at the interface between the piezoelectric layer and the substrate beam may be assumed as

    wherein φ0is the electric potential distribution at the top surface of the piezoelectric layer and can be expressed in terms of the nodal electric potential degrees of freedom{φe}as follows:

    Using Eqs.(7),(68)and(69),the electric filed vector at any point in the piezoelectric layer of the element can expressed as

    in which the matrices[Zp]and[B3]are presented in the Appendix.Using Eqs.(8),(65),(67),(69)and(70)and carrying out the explicit integration with respect to z,the total potential energy of a typical element can be expressed as follows:

    in which,

    Now applying the principle of minimumpotential energy i.e.δΠ=0,thefollowing elemental governing equilibrium equations of the overall smart beam based on theLHSDTare obtained:

    In Eq.(72),the various elemental matrices are given by

    The global equations of equilibrium of the overall smart beam are obtained by assembling the elemental equations in a straight forward manner.However,they can be representd by Eqs.(27)and(28).Using the “equivalent single layerHSDT”given by Eq.(61),the above finite element model(FEM)has been suitably augmented to derive anotherFEMof the overall smart beam.However,for the sake of brevity the derivation of thisFEMis not presented here.

    4 Computation of transverse shear stress

    The bending stress(σx)computed by the finite element models as derived above can be utilized to compute the transverse shear stress in the overall smart beams by numerically integrating the governing equilibrium equation as follows:

    5 Results and discussions

    In this section,numerical results are presented to investigate the performance of the new 4-noded smart finite elements derived here.The thickness of the substrate beam and that of the piezoelectric layer are considered as 5mm and 250μm,respectively,while the aspect ratio(L/h)of the substrate beam is considered as 50.The elastic and piezoelectric material properties of the piezoelectric layer(PZT5H)are used as follows[Smith and Auld(1991)]:

    The material properties of the orthotropic layers of the substrate beam are used as follows[Pagano(1970)]:

    in which the symbols have their usual meaning.The top surface of the overall smart beam is subjected to the sinusoidally distributed upward mechanical load given by Eq.(58)while the value of the amplitude(q0)of the applied load is 50N/m2.Unless otherwise mentioned,the overall smart beam is discretized considering one element for the substrate beam and one piezoelectric finite element across the thickness of the overall smart beam as shown in Fig.2(a).Also 10 elements are considered

    along the length of the beam.First a smart beam is considered in which the substrate beam is a single layered beam.Figure 4 illustrates the static responses of this beam when the piezoelectric layer is activated with the prescribed electric potential(voltage)on its top surface.Displayed in the figure are the deflections w(x,0)of the mid-plane of the beam computed by the present new smart finite element model(FEM)and exact solutions.It may be observed from this figure that the piezoelectric layer activated by a negative voltage counteracts the upward deflection of the beam due to the mechanical load only.The responses obtained by the present new smartFEMalmost identically match with the exact solutions when the piezoelectric layer is passive(V=0)and active(V 6=0).The distributions of the axial normal stress σx(a/2,z)and the transverse shear stress σxz(L/30,z)across the thickness of the substrate of this smart beam have been illustrated in Figs.5 and 6,respectively.It may be observed from these figures that the flexural stress and transverse shear stress computed by the present newFEMmatch excellently with the exact solutions for the flexural stress and the transverse shear stress when the piezoelectric layer is active(V 6=0)and passive(V=0).Figures 7 to 9 illustrate that the deflections w(x,0)of the mid-plane of the cross-ply substrate beams when the lamination sequence in the beams are(0°/90°),(0°/90°/0°)and(0°/90°/0°/90p),respectively.It may be observed from these figures that the present new smartFEMaccurately computes the deflections of the mid-plane of these substrate beams.Figures 10 and 11 illustrate the comparison of the distributions of the axial normal stress σx(a/2,z)across the thickness of the symmetric(0°/90°/0°)and antisymmetric(0°/90°/0°/90p)cross-ply substrate beams computed by the present new smartFEMwith that obtained by the exact solutions.It may be observed that the present new smartFEMaccurately computes the bending stress in the multi-layered composite beams when the piezoelectric layer is active(V 6=0)and passive(V=0).When compared with the exact solutions,the transverse shear stress σxz(L/30,z)across the thickness of the multilayered symmetric(0°/90°/0°)and antisymmetric(0°/90°/0°/90p)crossply substrate beams computed by the present new smartFEMare indistinguishable from those obtained by the exact solutions as shown in Figs.12 and 13,respectively.

    Figure 4:Active static control of shape of a single layered composite beam(L/h=50,hp=250μm,h=0.005m,

    Figure5:Distribution of axial stress across the thickness of the beam with and without actuated by a piezoelectric layer(L/h=50,hp=250μm,h=0.005m,q0=50N/m2).

    Figure 6:Distribution of the transverse shear stress across the thickness of the single layered composite beam(L/h=50,hp=250μm,h=0.005m,q0=50N/m2).

    Figure 7:Active static control of shape of a two layered(0°/90°)composite beam(L/h=50,hp=250μm,h=0.005m,

    Figure 8:Active static control of shape of a three layered(0°/90°/0°)composite beam(L/h=50,hp=250μm,h=0.005m,

    Figure 9:Active static control of shape of a four layered(0°/90°/0°/90°)composite beam(L/h=50,hp=250μm,h=0.005m,

    Figure 10:Distribution of the axial stress across the thickness of the three layered(0°/90°/0°)composite beam with and without actuated by a piezoelectric layer(L/h=50,hp=250μm,h=0.005m,q0=50N/m2).

    Figure 11:Distribution of the axial stress across the thickness of the four layered(0°/90°/0°/90°)composite beam with and without actuated by a piezoelectric layer(L/h=50,hp=250μm,h=0.005m,q0=50N/m2).

    Figure 12:Distribution of the transverse shear stress across the thickness of the three layered(0°/90°/0°)composite beam(L/h=50,hp=250μm,h=0.005m,q0=50N/m2).

    Figure 13:Distribution of the transverse shear stress across the thickness of the four layered(0°/90°/0°/90°)composite beam(L/h=50,hp=250μm,h=0.005m,q0=50N/m2).

    Figure 14:Comparisons of the responses due to in-plane actuation(e316=0,e33=0)and combined in-plane and transverse actuations(e316=0,e336=0)with those obtained by the exact solutions considering one element across the thickness of the overall single layered substrate composite beam(L/h=50,hp=250μm,h=0.005m,

    Figure 15:Comparison of the FEM based on the HSDT with the present smart FEM for active shape control of a single layered substrate composite beam(L/h=50,hp=250μm,h=0.005m,

    Figure 16:Comparison of the FEM based on the LHSDT with the present smart FEM for the active shape control of a single layered substrate composite beam(L/h=50,hp=250μm,h=0.005m,q0=50N/m2,

    Figure 17:Active shape control of single layered cantilever substrate composite beam(L/h=50,hp=250μm,h=0.005m,q0=50N/m2,

    At this juncture it may be recalled that the forgoing results are presented considering one element across the thickness of the substrate beam and one piezoelectric finite element across the thickness of the piezoelectric layer as shown in Fig.2(a).Thus two elements are used across the thickness of the overall smart beam.Since the beam element contains multiple orthotropic layers,a question naturally arises that if one element can be used across the thickness of the overall smart beam in which the top layer of the element is the piezoelectric layer as shown in Fig.2(b).Figure 14 illustrates such results for the deflection of the mid-plane of a single-layered substrate beam when one element is used across the thickness of the overall beam.It may also be noted that when e316=0 and e33=0,the piezoelectric layer causes in-plane actuation.On the other hand when e316=0 and e336=0,the piezoelectric layer causes both in-plane and transverse actuations simultaneously.Also,if the magnitude of e33is much larger than that of e31,the transverse actuation by the piezoelectric layer will be predominant must be modeled.It may be observed from Fig.14 that if the piezoelectric layer acts as the in-plane actuator,the responses obtained by the present smartFEMwith one element across the thickness of the overall beam excellently match with the exact solutions.If both in-plane and transverse actuations by the piezoelectric layer are modeled by using one element across the thickness of the overall beam,the responses of the actuated substrate beam do not match with the exact solutions.This may be attributed to the fact that the transverse displacement continuity at the interface between the piezoelectric layer and the substrate beam cannot be explicitly satisfied using one element across the thickness.But it may again be observed from Fig.4 that if one element for the substrate beam and one piezoelectric finite element for the piezoelectric layer are used across the thickness of the overall beam,the responses due to transverse actuation computed by the present newFEMexcellently match with the exact solutions.In order to be confirmed that a separate piezoelectric finite element is necessary across the thickness of the overall beam for the piezoelectric material characterized with large value of e33,the responses of the overall beam obtained by the different finite element models based on an equivalent single layer high order shear deformation theory(HSDT)and a layer-wise high order shear deformation theory(LHSDT)as derived in Section 3.0 are compared with the exact solutions as shown in Figs.15 and 16.For the results obtained by the present new smartFEMas displayed in Figs.15 and 16,two elements are considered across the thickness of the overall beam.It may be observed from Fig.15 that the responses obtained by theFEMbased on an equivalent single layerHSDTdiffer unacceptably from that obtained by the present new smartFEMand exact solutions.But the responses obtained by theFEMbased on theLHSDTexcellently match with that obtained by the present new smartFEMand exact solutions as shown in Fig.16.This ensures that at least two elements comprising one element for the substrate beam and one element for the piezoelectric layer across the thickness of the overall smart beam must be considered for deriving accurateFEMbased on the proposed new method.Also,for the results presented in Fig.16,the new smartFEMrequires 66 degrees of freedom for the beam,whereas theFEMbased of theLHSDTrequires 492 degrees of freedom.Thus the present new smartFEMis computationally much less costly than theFEMbased on theLHSDT.Similar results are also obtained for the substrate cross-ply beams with more number of laminae.However,for the sake of brevity they are not presented here.Finally,the deflections of a cantilever smart beam subjected to uniformly distributed load with intensity q0=50N/m2and actuated by the piezoelectric layer are presented in Fig.17.In this case,the piezoelectric layer is subjected to a uniformly distributed applied electric potential at its top surface.It may be observed from this figure that the responses obtained by the present newFEMmatch excellently with that obtained by theFEMbased on theLHSDT.Thus the present new 4-noded smart finite element can be efficiently used for accurate modeling of the smart composite beams without using the existing higher-order deformation theories or layer-wise deformation theories.

    6 Conclusions

    In this paper,new smart finite elements have been developed for the static analysis of smart laminated composite beams.The smart beam is composed of a laminated substrate composite beam integrated with a piezoelectric layer at its top surface which acts as the distributed actuator of the substrate beam.In case of simply supported beams,the top surface of the piezoelectric layer is subjected to the sinusoidally distributed mechanical load and a sinusoidally distributed applied electric potential.In case of the cantilever beam,the top surface of the piezoelectric layer is subjected to the uniformly distributed mechanical load and electrical potential.For the simply supported smart composite beams,exact solutions are derived to validate the proposed new smart finite elements.These smart beams are also modeled for comparison purpose by the conventional finite element method using an“equivalent high order shear deformation theory(HSDT)”and a “l(fā)ayer-wise high order shear deformation theory(LHSDT)”.Several examples are considered for presenting the numerical results.Two types of new smart finite elements are developed.One is purely piezoelectric called the “piezoelectric element”and is used to discretize the piezoelectric layer.The other element is a “l(fā)aminated smart element”in which the top layer is piezoelectric,while the other layers are the orthotropic layers of the substrate beam.The overall smart composite beam has been discretized by using either two elements or one element across the thickness of the beam.In case of two elements across the thickness,the top element is the“piezoelectric finite element”and the bottom element is the “l(fā)aminated beam element”containing only the orthotropic layers of the substrate beam.When the piezoelectric actuator layer is active and passive,and causes both transverse and in-plane actuations,the deflections,bending stresses and the transverse shear stresses of the smart composite beams computed by the present new smartFEMexcellently match with those obtained by the exact solutions if two elements are used across the thickness of the overall beams.If one laminated smart finite element is used across the thickness of the overall smart composite beams,the transverse actuation by the piezoelectric layer cannot be accurately modeled.The transverse actuation by the piezoelectric layer cannot be accurately modeled even if the overall smart beam is modeled by using the equivalent single layerHSDT.When the overall smart beam is modeled by using theLHSDT,the responses of the smart composite beams due to both transverse and in-plane actuations by the piezoelectric layer excellently match with those obtained by the exact solutions and the new smartFEMconsidering two elements across the thickness of the beams.This corroborates the fact that in case of the piezoelectric actuator that is characterized by the large value of the transverse piezoelectric coefficientat least one piezoelectric finite element and one beam element must be used across the thickness of the overall smart beams for accurate finite element modeling of the smart composite beams.The new smartFEMderived here also accurately computes the active and passive responses of the cantilever smart composite beams.As compared to the otherFEMs based on theHSDTand theLHSDT,the effort needed to derive the present new 4-noded smartFEMis negligible.Also,the present new smartFEMis computationally significantly less costly than theFEMs derived by using theHSDTand theLHSDT.The investigations carried out here suggests that the present new 4-noded smart finite element can be efficiently used for accurate modeling of smart composite beams without using the explicit forms of the displacement fields such as the classical beam theory,the first order and the high order shear deformation theories,the layer-wise theory and the like.

    Ahmad,S.N.;Upadhyay,C.S.;Venkatesan,C.(2005):Electroelastic analysis and layer-by-layer modeling of a smart beam.AIAA Journal,vol.43,no.12,pp.2606-2612.

    Al-Ajmi,M.A.;Benjeddou,A.(2008):Damage indication in smart structures using modal effective electromechanical coupling coefficients.Smart Materials and Structures,vol.17,art no.035023.

    Bailey,T.;Hubbard,J.E.(1985):Distributed piezoelectric polymer active vibration control of a cantilever beam.AIAA Journal of Guidance and Control,vol.8,no.5,pp.605-611.

    Bruke,S.E.;Hubbard,J.E.(1987):Active vibration control of a simply supported beam using a spatially distributed actuator.IEEE Control System Magazine,vol.8,pp.25-30.

    Bhattacharyya,P.;Suhail,H.;Sinha,P.K.(1998):Finite element free vibration analysis of smart laminated composite beams and plates.Journal of Intelligent Material Systems and Structures,vol.9,pp.20-32.

    Bendary,I.M.;Elshafei,M.A.;Riad,A.M.(2010):Finite element model of s-mart beams with distributed piezoelectric actuators.Journal of Intelligent Material Systems and Structures,vol.21,pp.747-754.

    Crawley,E.F.;Luis,J.D.(1987):Use of piezoelectric actuators as elements of intelligent structures.AIAA Journal,vol.27,pp.1801-1807.

    Chee,C.Y.K.;Tong,L.;Steven,G.(1999):A mixed model for composite beams with piezoelectric actuators and sensors.Smart Materials and Structures,vol.8,pp.417-432.

    Chee,C.;Tong,L.;Steven,G.P.(2002):Piezoelectric actuator orientation optimization for static shape control of composite plates.Composite Structures,vol.55,pp.169-184.

    Dong,L.;Atluri,S.N.(2011):A Simple Procedure to Develop Efficient&Stable Hybrid/Mixed Elements,and Voronoi Cell Finite Elements for Macro-&Micromechanics.Computers,Materials&Continuum,vol.24,no.1,pp.61-104.

    Dong,L.;EI-Gizawy,A.S.;Juhany,K.A.;Atluri,S.N.(2014):A simple locking-alleviated 4-node mixed-collocation finite element with over-integration,for homogenous or functionally-graded or thick-section laminated composite beams.Computers,Materials&Continuum,vol.40,no.1,pp.49-77.

    Elshafei,M.A.;Alraiess,F.(2013):Modeling and analysis of smart piezoelectric beams using simple higher order shear deformation theory.Smart Materials and Structures,vol.22,art no.035006.

    Forward,R.L.(1981):Electronic damping of orthogonal bending modes in a cylindrical mast-experimen.Journal of Spacecraft and Rocket,vol.18,no.1,pp.11-17.

    Gupta,V.K.;Seshu,P.;Issac,K.K.(2004):Finite element and experimental investigation of piezoelectric actuated smart shells.AIAA Journal,vol.42,pp.2112-2118.

    Ha,S.K.;Keilers,C.;Chang,F.K.(1992):Finite element analysis of composite structures containing distributed piezoceramic sensors and actuators.AIAA journal,vol.30,no.3,pp.772-780.

    Hwang,W.S.;Park,C.H.(1993):Finite element modeling of piezoelectric sensors and actuators.AIAA journal,vol.31,no.5,pp.930-936.

    Im,S.;Atluri,S.N.(1989):Effects of piezoactuator on a finitely deformed beam subjected to general loading.AIAA Journal,vol.25,pp.1373-1385.

    Kulkarni,S.A.;Bajoria,K.M.(2003):Finite element modeling of smart plates/shells using higher order shear deformation theory.Composite Structures,vol.62,pp.41-50.

    Lo,K.H.;Christensen,R.M.;Wu,E.M.(1978):Stress solution determination for high order plate theory.International Journal of Solids and Structures,vol.14,pp.655-662.

    Lin,C.;Hsu,C.;Huang,H.N.(1996):Finite element analysis on deflection control of plates with piezoelectric actuators.Composite Structures,vol.35,pp.423-433.

    Luo,Q.;Tong,L.(2004):An accurate laminated element for piezoelectric smart beams including peel stress.Computational Mechanics,vol.33,pp.108-120.

    Miller,S.E.;Hubbard,J.E.(1987):Obsevability of a Bernouli-Euler beam using PVF2as a distributed sensor.MIT Draper Laboratory Report.

    Neto,M.A.;Yu,W.;Roy,S.(2009):Two finite elements for general composite beams with piezoelectric actuators and sensors.Finite Elements in Analysis and Design,vol.45,pp.295-304.

    Pagano,N.J.(1970):Exact solutions for rectangular bidirectional composites and sandwich plates.Journal of Composite Materials,vol.4,pp.20-34.

    Park,H.;Lee,U.(2012):Dynamic analysis of smart composite beams by using the frequency-domain spectral elemental method.Journal of Mechanical Science and Technology,vol.26,pp.2511-2521.

    Robbins,D.H.;Reddy,J.N.(1991):Analysis of piezoelectrically actuated beams using a layer-wise displacement theory.Computers&Structures,vol.41,no.2,pp.265-279.

    Ray,M.C.;Bhattacharyya,R.;Samanta,B.(1994):Static analysis of an intelligent structure by the finite element method.Computers and Structures,vol.52,no.4,pp.617-631.

    Shi,G.;Atluri,S.N.(1990):Active control of nonlinear dynamic response of space-frames using piezo-electric actuators.Computers and Structures,vol.34,no.4,pp.549-564.

    Smith,W.A.;Auld,B.A.(1991):Modeling 1-3 composite piezoelectrics:Thickness mode oscillations.IEEE Transactions on Ultrasonics,Ferroelectrics and Frequency Control,vol.31,pp.40–47.

    Saravanos,D.A.;Heyliger,P.R.(1995):Coupled layerwise analysis of composite beams with embedded piezoelectric sensors and actuators.Journal of Intelligent Material Systems and Structures,vol.6,pp.350-362.

    Saravanos,D.A.;Hetliger,P.R.;Hopkins,D.A.(1997):Layerwise mechanics and finite element for the dynamic analysis of piezoelectric composite plates.International Journal of Solids and Structures,vol.34,no.3,pp.359-378.

    Song,Y.;Kim,S.;Park,I.;Lee,U.(2015):Dynamics of two-layer smart composite Timoshenko beams:frequency domain spectral element analysis.Thin-Walled Structures,vol.89,pp.84-92.

    Trindade,M.A.;Benjeddou,A.(2006):On higher-order modeling of smart beams with embedded shear-mode piezoceramic actuators and sensors.Mechanics of Advanced Materials and Structures,vol.13,pp.357-369.

    Umesh,K.;Ganguli,R.(2011):Composite material and piezoelectric coefficient uncertainty effects on structural health monitoring using feedback control gains as damage indicators.Structural Health Monitoring,vol.10,no.2,pp.115-129.

    Varadarajan,S.;Chandrashekhara,K.;Agarwal,S.(2000):LQG/LTR-based robust control of composite beams with piezoelectric devices.Journal of Vibration and Control,vol.6,pp.607-630.

    Valoor,M.T.;Chandrashekhara,K.;Agrawal,S.(2001):Self-adaptive vibration control of smart composite beams using recurrent neural architecture.International Journal of Solids and Structures,vol.38,pp.7857-7874.

    Zabihollah,A.;Sedagahti,R.;Ganesan,R.(2007):Active vibration suppression of smart laminated beams using layerwise theory and an optimal control strategy.Smart Materials and Structures,vol.16,pp.2190-2201.

    Zhang,S.Q.;Schmidt,R.(2014):Static and dynamic FE analysis of piezoelectric integrated thin-walled composite structures with large rotations.Composite Structures,vol.112,pp.345-357.

    Appendix

    The matrices[Z1],[Z2]and[Zp]appearing in Eqs.(67)and(70)are as follows:

    and

    The nodal strain-displacement matrices[B1]and[B2]are given by

    The nonzero elements of the submatrices[B1i],i=1,2,3 are as follows:

    B1i(5,9)=B1i(6,10)=B1i(7,3)=ni, B1i(8,4)=2niand B1i(9,5)=3ni

    The nonzero elements of the submatrices[B2i],i=1,2,3 are as follows:

    The nodal electric field-potential matrix[B3]appearing in Eq.(70)is given by

    1Department of Mechanical Engineering,Indian Institute of Technology,Kharagpur,India.

    2Corresponding Author,School of Aeronautic Science and Engineering,Beihang University,China.Email:dong.leiting@gmail.com.

    3Department of Mechanical Engineering,Texas Tech University,USA

    亚洲18禁久久av| 日韩免费av在线播放| 丰满人妻熟妇乱又伦精品不卡| 午夜福利视频1000在线观看| 好男人在线观看高清免费视频| 久久婷婷人人爽人人干人人爱| 少妇裸体淫交视频免费看高清| 伊人久久精品亚洲午夜| 脱女人内裤的视频| av黄色大香蕉| 久久久色成人| 久9热在线精品视频| 欧美成狂野欧美在线观看| 日韩欧美免费精品| 国产一区二区亚洲精品在线观看| 亚洲片人在线观看| 热99在线观看视频| av国产免费在线观看| 久久精品夜夜夜夜夜久久蜜豆| 精品国产三级普通话版| 国产精品精品国产色婷婷| 噜噜噜噜噜久久久久久91| 尤物成人国产欧美一区二区三区| 丰满乱子伦码专区| 老鸭窝网址在线观看| 欧美日韩福利视频一区二区| 又爽又黄无遮挡网站| 观看免费一级毛片| 成人欧美大片| 成人国产综合亚洲| 日本与韩国留学比较| 精品国产三级普通话版| a级毛片免费高清观看在线播放| 嫩草影院新地址| 国产欧美日韩一区二区三| 变态另类丝袜制服| 欧美黑人欧美精品刺激| 亚洲av美国av| 宅男免费午夜| www.999成人在线观看| 欧美三级亚洲精品| 好男人电影高清在线观看| 不卡一级毛片| 欧美一区二区精品小视频在线| 99久久久亚洲精品蜜臀av| 久久久久久九九精品二区国产| 两个人的视频大全免费| 午夜福利高清视频| 午夜福利高清视频| 国产成人福利小说| 亚洲成a人片在线一区二区| 人人妻人人看人人澡| 91字幕亚洲| 欧美黑人欧美精品刺激| 波多野结衣高清作品| 麻豆一二三区av精品| 蜜桃亚洲精品一区二区三区| 国产单亲对白刺激| 99热6这里只有精品| 男人狂女人下面高潮的视频| 一卡2卡三卡四卡精品乱码亚洲| 成年女人毛片免费观看观看9| 欧美黄色淫秽网站| 中出人妻视频一区二区| 中文字幕精品亚洲无线码一区| 日本撒尿小便嘘嘘汇集6| 成熟少妇高潮喷水视频| 一级av片app| 男女那种视频在线观看| 精品一区二区三区视频在线观看免费| 国产一区二区在线av高清观看| 别揉我奶头~嗯~啊~动态视频| 成人一区二区视频在线观看| 国内少妇人妻偷人精品xxx网站| 国产国拍精品亚洲av在线观看| 在现免费观看毛片| 午夜日韩欧美国产| 婷婷色综合大香蕉| 久久午夜福利片| 亚洲美女搞黄在线观看 | 啪啪无遮挡十八禁网站| 国产免费av片在线观看野外av| 日本与韩国留学比较| 久久久久久久久久黄片| 欧美bdsm另类| 欧美精品国产亚洲| 在线观看美女被高潮喷水网站 | 女人被狂操c到高潮| 婷婷丁香在线五月| 中文在线观看免费www的网站| 国产精品一区二区免费欧美| a级毛片a级免费在线| 最近视频中文字幕2019在线8| 欧美一区二区国产精品久久精品| 色综合站精品国产| 免费看美女性在线毛片视频| 18禁在线播放成人免费| 欧美日韩中文字幕国产精品一区二区三区| 床上黄色一级片| 午夜亚洲福利在线播放| 午夜免费激情av| 熟女人妻精品中文字幕| 99热6这里只有精品| 亚洲人成伊人成综合网2020| 免费看日本二区| 直男gayav资源| 麻豆一二三区av精品| 日韩国内少妇激情av| 成人特级黄色片久久久久久久| 日本成人三级电影网站| 又粗又爽又猛毛片免费看| www.色视频.com| 制服丝袜大香蕉在线| 三级国产精品欧美在线观看| 免费看a级黄色片| 国产精品一区二区三区四区久久| 亚洲av第一区精品v没综合| 99riav亚洲国产免费| 简卡轻食公司| 三级毛片av免费| 91麻豆av在线| 特大巨黑吊av在线直播| 丰满人妻熟妇乱又伦精品不卡| 真人做人爱边吃奶动态| 成人国产综合亚洲| 亚洲一区二区三区不卡视频| 成年女人看的毛片在线观看| 91久久精品电影网| 欧美+亚洲+日韩+国产| 色综合亚洲欧美另类图片| www.999成人在线观看| 一区二区三区免费毛片| 两人在一起打扑克的视频| 免费黄网站久久成人精品 | 亚洲在线观看片| 精品福利观看| 欧美又色又爽又黄视频| 国产高清激情床上av| 久久久久久久午夜电影| 亚洲av一区综合| 两个人视频免费观看高清| 欧美xxxx黑人xx丫x性爽| 51国产日韩欧美| 色综合欧美亚洲国产小说| 国产又黄又爽又无遮挡在线| 麻豆一二三区av精品| 欧美激情久久久久久爽电影| 欧美日韩综合久久久久久 | 老司机深夜福利视频在线观看| 欧美在线黄色| 别揉我奶头~嗯~啊~动态视频| 在线观看舔阴道视频| 国产一区二区在线观看日韩| 好男人在线观看高清免费视频| netflix在线观看网站| 久久精品国产自在天天线| 久久久久九九精品影院| 亚洲午夜理论影院| 在线免费观看的www视频| 熟女人妻精品中文字幕| 日本熟妇午夜| 少妇的逼好多水| aaaaa片日本免费| 国内精品一区二区在线观看| 精品午夜福利在线看| 免费在线观看亚洲国产| 精品一区二区三区视频在线观看免费| 99国产综合亚洲精品| 精品久久久久久成人av| 三级国产精品欧美在线观看| 亚洲av电影在线进入| 国产亚洲精品久久久com| 国产精品亚洲av一区麻豆| 欧美最黄视频在线播放免费| 9191精品国产免费久久| 欧美三级亚洲精品| 天美传媒精品一区二区| 99在线视频只有这里精品首页| 久久精品国产99精品国产亚洲性色| 亚洲人成网站在线播| 9191精品国产免费久久| av福利片在线观看| 亚洲美女视频黄频| 搡老妇女老女人老熟妇| 国产成人aa在线观看| 国产 一区 欧美 日韩| 免费无遮挡裸体视频| 亚洲av成人不卡在线观看播放网| 国产私拍福利视频在线观看| 性插视频无遮挡在线免费观看| 蜜桃久久精品国产亚洲av| 久久久久国产精品人妻aⅴ院| 自拍偷自拍亚洲精品老妇| 亚洲狠狠婷婷综合久久图片| 亚洲成av人片免费观看| 欧美中文日本在线观看视频| 淫妇啪啪啪对白视频| 熟妇人妻久久中文字幕3abv| 久久香蕉精品热| 神马国产精品三级电影在线观看| 亚洲美女视频黄频| 国产精品爽爽va在线观看网站| 欧美成人a在线观看| 老司机午夜十八禁免费视频| 级片在线观看| 国产午夜精品论理片| 精品人妻偷拍中文字幕| 日韩中字成人| 搡老妇女老女人老熟妇| 99久国产av精品| 欧美乱色亚洲激情| 精品人妻偷拍中文字幕| 香蕉av资源在线| 毛片一级片免费看久久久久 | 波多野结衣高清作品| 两个人视频免费观看高清| x7x7x7水蜜桃| 亚洲人成网站高清观看| 中文字幕高清在线视频| 动漫黄色视频在线观看| 最好的美女福利视频网| 麻豆国产97在线/欧美| 国产在线精品亚洲第一网站| 欧美+日韩+精品| av在线蜜桃| 国产高清激情床上av| 超碰av人人做人人爽久久| 亚洲精品456在线播放app | 国产淫片久久久久久久久 | 国产精品电影一区二区三区| 国产探花极品一区二区| 男女下面进入的视频免费午夜| 九九热线精品视视频播放| 国产精品爽爽va在线观看网站| 国产精品影院久久| 99久久精品热视频| 亚洲五月天丁香| 国产成人a区在线观看| 悠悠久久av| a级毛片a级免费在线| 免费看日本二区| 国内精品一区二区在线观看| 丰满人妻一区二区三区视频av| 麻豆成人午夜福利视频| 免费一级毛片在线播放高清视频| 午夜影院日韩av| 51午夜福利影视在线观看| 国产亚洲精品久久久com| 男女那种视频在线观看| 1000部很黄的大片| 最近视频中文字幕2019在线8| 色av中文字幕| 夜夜看夜夜爽夜夜摸| 国产精品自产拍在线观看55亚洲| 女人被狂操c到高潮| 白带黄色成豆腐渣| 欧美黄色片欧美黄色片| 美女 人体艺术 gogo| 看十八女毛片水多多多| 久久久久久久久久黄片| 欧美xxxx性猛交bbbb| 久久久久久久久久成人| 国产蜜桃级精品一区二区三区| a在线观看视频网站| 亚洲午夜理论影院| www.色视频.com| 99精品久久久久人妻精品| 国产精品不卡视频一区二区 | 午夜福利在线观看吧| 国产综合懂色| 在线观看av片永久免费下载| 国产精品女同一区二区软件 | 国产 一区 欧美 日韩| 欧美一区二区精品小视频在线| 观看美女的网站| 高清在线国产一区| 成人av一区二区三区在线看| 亚洲av电影在线进入| 欧美日韩亚洲国产一区二区在线观看| av福利片在线观看| 97超视频在线观看视频| 成人性生交大片免费视频hd| 国产精品人妻久久久久久| eeuss影院久久| 欧美成人a在线观看| 国产精品一区二区免费欧美| 欧美区成人在线视频| 国产av不卡久久| 亚洲人成伊人成综合网2020| 欧美不卡视频在线免费观看| 能在线免费观看的黄片| 日本三级黄在线观看| 一区二区三区激情视频| 日本 av在线| 国产一级毛片七仙女欲春2| 99久久精品热视频| a级毛片a级免费在线| 国产av一区在线观看免费| 久久久久亚洲av毛片大全| 夜夜爽天天搞| 在线观看舔阴道视频| 午夜福利欧美成人| 听说在线观看完整版免费高清| 国产亚洲欧美98| 国产精品不卡视频一区二区 | 国产一区二区三区在线臀色熟女| 能在线免费观看的黄片| 18+在线观看网站| 精品久久久久久,| 日日干狠狠操夜夜爽| 中文字幕精品亚洲无线码一区| 一区二区三区四区激情视频 | 性色av乱码一区二区三区2| 久久午夜亚洲精品久久| 乱码一卡2卡4卡精品| av在线观看视频网站免费| 岛国在线免费视频观看| 女人被狂操c到高潮| 亚洲欧美日韩高清在线视频| 久久久久久九九精品二区国产| avwww免费| 免费人成视频x8x8入口观看| 亚洲av不卡在线观看| 国产激情偷乱视频一区二区| 亚洲国产高清在线一区二区三| 88av欧美| 99在线视频只有这里精品首页| 国产大屁股一区二区在线视频| 97碰自拍视频| 亚洲性夜色夜夜综合| 亚洲自偷自拍三级| 一级黄片播放器| 波多野结衣巨乳人妻| 大型黄色视频在线免费观看| www.999成人在线观看| 久久久久九九精品影院| 老师上课跳d突然被开到最大视频 久久午夜综合久久蜜桃 | 一本综合久久免费| 国内毛片毛片毛片毛片毛片| 91九色精品人成在线观看| 90打野战视频偷拍视频| 国产精品久久电影中文字幕| 不卡一级毛片| a级毛片a级免费在线| 国产精品久久久久久久久免 | 国内精品美女久久久久久| 国产欧美日韩精品亚洲av| 男人狂女人下面高潮的视频| 91av网一区二区| 色视频www国产| av天堂在线播放| 国内精品久久久久久久电影| 99国产极品粉嫩在线观看| 国产中年淑女户外野战色| 嫩草影视91久久| bbb黄色大片| 99久久久亚洲精品蜜臀av| 99在线视频只有这里精品首页| 999久久久精品免费观看国产| 国产亚洲av嫩草精品影院| 亚洲电影在线观看av| 国产成人影院久久av| 国产老妇女一区| 日韩高清综合在线| 一区二区三区激情视频| 久9热在线精品视频| 国产免费一级a男人的天堂| 午夜精品在线福利| 午夜福利在线观看免费完整高清在 | 亚洲人成伊人成综合网2020| 日韩精品青青久久久久久| 午夜激情欧美在线| 亚洲美女黄片视频| 精品日产1卡2卡| 国产一区二区在线观看日韩| 色尼玛亚洲综合影院| 国产蜜桃级精品一区二区三区| 欧美国产日韩亚洲一区| 高清日韩中文字幕在线| 最近最新中文字幕大全电影3| 丰满人妻一区二区三区视频av| 九色国产91popny在线| 亚州av有码| 国内精品久久久久精免费| 少妇的逼好多水| 国产伦精品一区二区三区四那| 国产三级黄色录像| 国产精品不卡视频一区二区 | 变态另类成人亚洲欧美熟女| 国产精华一区二区三区| 国产爱豆传媒在线观看| 成年免费大片在线观看| www日本黄色视频网| 99热这里只有精品一区| 色综合亚洲欧美另类图片| 国产又黄又爽又无遮挡在线| 哪里可以看免费的av片| 国产亚洲精品av在线| 免费看美女性在线毛片视频| 少妇高潮的动态图| 俺也久久电影网| 亚洲精品在线美女| 亚洲人成电影免费在线| 2021天堂中文幕一二区在线观| 麻豆av噜噜一区二区三区| 搡老岳熟女国产| 国产精品久久电影中文字幕| 在线免费观看的www视频| 人妻丰满熟妇av一区二区三区| 国产一区二区激情短视频| 人人妻人人看人人澡| 偷拍熟女少妇极品色| 亚洲欧美日韩东京热| 久久亚洲真实| 动漫黄色视频在线观看| 国产精品嫩草影院av在线观看 | 自拍偷自拍亚洲精品老妇| 久久精品国产99精品国产亚洲性色| 五月伊人婷婷丁香| 尤物成人国产欧美一区二区三区| 一二三四社区在线视频社区8| 99久久精品一区二区三区| 久久久久久大精品| 99国产精品一区二区三区| 男女视频在线观看网站免费| 精品国产亚洲在线| 18禁黄网站禁片午夜丰满| 久久欧美精品欧美久久欧美| 国产单亲对白刺激| www.色视频.com| 久久久精品大字幕| 亚洲狠狠婷婷综合久久图片| 国产精品一区二区三区四区免费观看 | 97人妻精品一区二区三区麻豆| 亚洲人成网站在线播| avwww免费| 一本精品99久久精品77| 女生性感内裤真人,穿戴方法视频| 在线播放无遮挡| 国产高清有码在线观看视频| 女同久久另类99精品国产91| 久久亚洲真实| 亚洲av五月六月丁香网| 高清毛片免费观看视频网站| 午夜福利在线观看免费完整高清在 | 国产精品99久久久久久久久| 亚洲五月天丁香| 亚洲精品在线美女| 久久久久性生活片| 精品国产三级普通话版| 天堂网av新在线| 亚洲最大成人av| 757午夜福利合集在线观看| 日韩欧美精品免费久久 | 老司机午夜福利在线观看视频| 一a级毛片在线观看| 在线播放国产精品三级| 国产一区二区在线av高清观看| 亚洲无线观看免费| 久久天躁狠狠躁夜夜2o2o| 国产极品精品免费视频能看的| 无人区码免费观看不卡| 亚洲精品日韩av片在线观看| 亚洲av不卡在线观看| 免费电影在线观看免费观看| 精品久久久久久久人妻蜜臀av| 天美传媒精品一区二区| 国产视频一区二区在线看| 欧美色视频一区免费| 国产精品影院久久| 亚洲精品乱码久久久v下载方式| 国产成年人精品一区二区| 亚洲av中文字字幕乱码综合| 亚洲欧美日韩卡通动漫| 国产一区二区三区视频了| 亚洲欧美激情综合另类| 最新中文字幕久久久久| 亚洲三级黄色毛片| 日本免费a在线| 在线观看66精品国产| 国产亚洲精品久久久com| 99视频精品全部免费 在线| 18禁裸乳无遮挡免费网站照片| 亚洲av二区三区四区| 欧美成人a在线观看| 99久久九九国产精品国产免费| 久久久久久久精品吃奶| 国产精品亚洲美女久久久| 免费看美女性在线毛片视频| 日日摸夜夜添夜夜添av毛片 | 欧美日韩乱码在线| 搡老熟女国产l中国老女人| 乱人视频在线观看| 91在线观看av| 嫩草影院精品99| 在线观看舔阴道视频| 亚洲性夜色夜夜综合| 91在线观看av| 欧美性感艳星| 免费av观看视频| 舔av片在线| 久久久久免费精品人妻一区二区| 成人特级av手机在线观看| 午夜视频国产福利| 噜噜噜噜噜久久久久久91| 亚洲午夜理论影院| 黄色视频,在线免费观看| 国产熟女xx| 一个人免费在线观看电影| 亚洲va日本ⅴa欧美va伊人久久| 精品免费久久久久久久清纯| 欧美激情久久久久久爽电影| 91久久精品国产一区二区成人| 日韩精品中文字幕看吧| 老鸭窝网址在线观看| 变态另类成人亚洲欧美熟女| 久久午夜福利片| 国产精品亚洲一级av第二区| 国产午夜福利久久久久久| 久久午夜福利片| 麻豆成人av在线观看| 十八禁人妻一区二区| 日韩大尺度精品在线看网址| 88av欧美| 男女那种视频在线观看| 美女高潮喷水抽搐中文字幕| 久久久久久久亚洲中文字幕 | 久久久久精品国产欧美久久久| 高清日韩中文字幕在线| 亚洲av成人精品一区久久| 国产精品一区二区三区四区久久| 日本一本二区三区精品| 色吧在线观看| 真实男女啪啪啪动态图| 国产精品一区二区三区四区久久| 中文字幕久久专区| 亚洲国产欧美人成| 欧美高清成人免费视频www| 极品教师在线免费播放| 欧美+日韩+精品| 一区二区三区四区激情视频 | 午夜a级毛片| 丰满人妻熟妇乱又伦精品不卡| 国产欧美日韩一区二区三| 国产在线精品亚洲第一网站| 一二三四社区在线视频社区8| 天堂av国产一区二区熟女人妻| 国产精品精品国产色婷婷| 欧美激情国产日韩精品一区| 69av精品久久久久久| 精品不卡国产一区二区三区| 国产亚洲精品久久久com| 国产精品综合久久久久久久免费| 三级毛片av免费| 国产男靠女视频免费网站| www.熟女人妻精品国产| 90打野战视频偷拍视频| 99视频精品全部免费 在线| 夜夜夜夜夜久久久久| 黄片小视频在线播放| 内射极品少妇av片p| 99在线视频只有这里精品首页| 一a级毛片在线观看| 看片在线看免费视频| 久久午夜福利片| 2021天堂中文幕一二区在线观| 亚洲avbb在线观看| 丰满的人妻完整版| 老女人水多毛片| 十八禁网站免费在线| 国产探花在线观看一区二区| 国产精品一区二区三区四区久久| 国内精品久久久久精免费| 国产一区二区亚洲精品在线观看| 国产熟女xx| 亚洲国产精品成人综合色| 午夜两性在线视频| 成人av在线播放网站| 国产av在哪里看| 午夜福利成人在线免费观看| 久久精品国产清高在天天线| 高潮久久久久久久久久久不卡| 亚洲无线在线观看| 熟妇人妻久久中文字幕3abv| 毛片一级片免费看久久久久 | 色5月婷婷丁香| 一个人观看的视频www高清免费观看| 亚洲中文字幕日韩| 国产美女午夜福利| av国产免费在线观看| 中文字幕人妻熟人妻熟丝袜美| 久久精品影院6| 国产成+人综合+亚洲专区| 99久久无色码亚洲精品果冻| 999久久久精品免费观看国产| 色av中文字幕| 久久久精品欧美日韩精品| 永久网站在线| 国产成人啪精品午夜网站| 99国产综合亚洲精品| 每晚都被弄得嗷嗷叫到高潮| 国产精品永久免费网站| 亚洲av一区综合| 精品人妻偷拍中文字幕| 观看免费一级毛片| 久久久久久久精品吃奶| 人人妻,人人澡人人爽秒播| 高潮久久久久久久久久久不卡| 天天躁日日操中文字幕| 无遮挡黄片免费观看| 国产精品野战在线观看| 嫁个100分男人电影在线观看| 97人妻精品一区二区三区麻豆| 赤兔流量卡办理| 午夜精品在线福利|