CN103886165A - Analogue simulation method for electromagnetic elastic coupling of layering shell made of electromagnetic materials - Google Patents

Abstract

The invention provides an analogue simulation method for electromagnetic elastic coupling of a layering shell made of electromagnetic materials. A three-dimensional energy equation including the electricity-magnetism-elastic coupling effect is established based on the Hamilton extension principle; three-dimensional energy is asymptotically extended into serial two-dimensional recursion energy based on the variation asymptotic method, a leading variation item in the two-dimensional recursion energy is corrected asymptotically through inherent small parameters of the shell, and therefore a correction model close to the original three-dimensional energy as much as possible is obtained and is converted into the Reissner-Mindlin model form commonly used in the engineering; a three-dimensional variation reconstruction relation is inferred based on the obtained two-dimensional overall response and all levels of warping functions. The apriority hypothesis is not needed in the model, the structural electromagnetic elastic coupling performance under the function of various fields can be accurately predicted, the calculated amount is small, calculating efficiency is higher than that of the high-order set theory and the three-dimensional finite element solution, and the number of resources occupied in a computer is small.

Description

A kind of electromagnetic material stratiform housing electromagnetism bullet coupling Simulation analogy method

Technical field

The present invention relates to material mechanical performance analysis field, be specifically related to a kind of realistic model of the electromagnetic material stratiform housing electromagnetism bullet coupling performance based on variation method of approach, can effectively predict electricity-magnetic-bullet coupling performance of structure under many field actions.

Background technology

Intelligent composite by piezoelectric phase and the phase composition of pressure magnetic can produce single-phase piezoelectricity and the unexistent magnetoelectric effect of piezomagnetic material.Utilize this characteristic, can improve piezoelectricity and press the little frequency range of magnetic monophase materials, the shortcoming of little actuating, be therefore widely used in the fields such as aviation, automobile, electronics, building, medical science.

Van Suchtelen proposed piezoelectricity in 1972, press the combination meeting of magnetic to produce new material behavior is magnetoelectric effect.20th century the mid-90s, Nan, Huang and Kuo have proposed theoretically the mesomechanics model of piezoelectricity, piezomagnetic material and have estimated its coupling effect.Benveniste utilizes the concept of uniform field to obtain the exact relationship between the different piece in fibrous magneto-electro-elastic media coupling effect.Phase late 1990s, Li has further developed mesomechanics and has studied and between static(al) field, electric field, magnetic field, exist equally coupling for couplant heterogeneous, and the bicylindrical that has simultaneously obtained unlimited magneto-electro-elastic media is mingled with the solution with anisotropic body problem.Wu and Huang work out the sealing solution of piezoelectricity, piezomagnetic composite material magnetoelectric effect.At home, for starting late of piezoelectricity, piezomagnetic composite material research, calendar year 2001, Liu Jinxi etc. have studied the key property of piezoelectricity, piezomagnetic material two-dimensional problems Green function.2003, Wang Jianguo etc. studied the unlimited piezoelectricity of transverse isotropy, have pressed the state variable solution of magnetic.2006, Yao Weian etc. studied the Element BEM of MagnetoelectroelastiSolids Solids; In the same year, Zhou Zhengong etc. have studied the crackle of magnetoelectric composites to the scattering problems of elastic wave.

As the above analysis, during the intelligent structure forming for electromagnetic material is analyzed, need to there be the multi-disciplinary knowledge such as mathematics, mechanics, electricity and electromagnetics, and due to the complicacy being coupled between electricity, magnetic and elastic medium, problem solving has been brought to very large difficulty, although had some useful work, still had many aspects to need further perfect.In simplified model, suppose the displacement of linearity or high-order and electricity, magnetic potential as existing document great majority and distribute, ignore or partly ignored the heterogeneity of local deformation, thereby can not reflect the local stress and local electricity, magnetic field of electromagnetic device and structure joint portion.

Summary of the invention

For above shortcomings in prior art, the invention provides a kind of calculated amount little, take computer resource few, and efficiency high based on asymptotic variational method electromagnetic material stratiform housing electromagnetism bullet coupling performance simulation method.

For solving the problems of the technologies described above, realize goal of the invention, the technical solution used in the present invention is as follows:

A kind of electromagnetic material stratiform housing electromagnetism bullet coupling Simulation analogy method, comprises the following steps:

1) set up the three-dimensional energy equation containing electricity-magneto-elastic coupling effect in electromagnetic material stratiform housing based on Hamilton principle and rotation tensor decomposition of concept;

2) based on the asymptotic variational method, the analysis of three-dimensional energy equation split is obtained to two-dimentional energy functional, and utilize in the two-dimentional energy functional of the asymptotic correction of small parameter intrinsic in electromagnetic material stratiform housing the leading variation item containing warpage item, obtain correction model, correction model is converted to Reissner-Mindlin model form;

3) the three dimensional field Remodeling based on deriving, utilize the overall situation response of the two-dimentional shell face that Reissner-Mindlin model analysis obtains and the warping function reconstruct field variable through-thickness distribution that Dimension Reduction Analysis obtains, by constitutive relation reconstruct triaxiality, electric displacement, magnetic induction gesture; To electromagnetic material stratiform housing electromagnetism bullet, analogue simulation is carried out in coupling.

Further, described step 1 is specially: set up the three-dimensional energy equation containing electricity-magneto-elastic coupling effect in electromagnetic material stratiform housing based on Hamilton principle and rotation tensor decomposition of concept:

${\∫}_{t_1}^{t_2}[\mathrm{\δ}(K_{2D}+K^{*}-U)+\mathrm{\δ}{\stackrel{\‾}{W}}_{2D}+\mathrm{\δ}{\stackrel{\‾}{W}}^{*}]\mathrm{dt}=0;$

Wherein, t ₁, t ₂for any 2 fixed time points; K ^*for the broad sense complementary energy that dynamic load produces, K _2Dfor the two-dimensional structure kinetic energy of dynamic load generation;

the two-dimentional virtual work of doing for load, electricity/magnetic field;

for the residue virtual work that do in load, electricity/magnetic field, upper line is used for showing that this virtual work does not need the accurate variation of functional, and δ is variation symbol, and U is the interior energy that load, electricity/magnetic field produce in housing, and its concrete form is:

$U=\frac{1}{2}{\∫}_v[{\mathrm{\Γ}}^T{C}^{E,H}\mathrm{\Γ}-E^T{d}^{\mathrm{\Γ},H}E-H^T{\mathrm{\μ}}^{\mathrm{\Γ},E}H-2E{e}^H\mathrm{\Γ}-2H{q}^E\mathrm{\Γ}-2E{a}^{\mathrm{\ΓH}}]\mathrm{dv}$

In formula: Γ, E, H is respectively strain intensity, electric field intensity and the magnetic field intensity of load, the generation of electricity/magnetic field lower house; e ^h, q ^e, a ^Γbe respectively the pressure magnetic constant matrix of piezoelectric constant matrix, the electric field intensity of magnetic field intensity when constant when constant and the strain intensity magnetoelectricity constant matrices when constant; C ^e,H, d ^{Γ, H}, μ ^{Γ, E}be respectively the specific inductive capacity matrix of electric field intensity and magnetic field intensity elastic constant matrix, strain intensity and the magnetic field intensity when constant when constant and strain intensity and the electric field intensity magnetic permeability constant matrices when constant; V is shell space volume.

Further, described step 2 is specially: based on the asymptotic variational method, three-dimensional energy Dimension Reduction Analysis is obtained to two-dimentional energy functional: utilize the leading variation item containing warpage item in the two-dimentional energy functional of the asymptotic correction of small parameter intrinsic in electromagnetic material stratiform housing, obtain the asymptotic correction model of zeroth order and single order:

2Π ₀=∈ ^TA∈；

$2{\mathrm{\Π}}_1={\∈}^T{A}_R\∈+{\∈}_{;\mathrm{\α}}^T{B}_{\mathrm{\α\β}}{\∈}_{;\mathrm{\β}}+2{\∈}^{T}F;$

Wherein, Π ₀, Π ₁be respectively zeroth order, the energy functional that first approximation obtains; ∈ is broad sense two dimension dependent variable; A is two-dimentional stiffness matrix; A _r, B _{α β}for considering the revised stiffness matrix of housing initial curvature;

α=β=1,2, A _αfor the Lame parameter of two-dimentional shell face base vector in housing, x _αfor the plane coordinate system of setting up on electromagnetic material stratiform housing, x ₁, x ₂axle is respectively curved surface direction, the length direction along housing reference surface; F is load continuous item, and subscript T represents transposed matrix;

Asymptotic single order correction model is converted to Reissner-Mindlin model form:

Wherein,

for the dependent variable of Reissner-Mindlin model; $\mathrm{\γ}={\left[\begin{array}{cc}2{\mathrm{\γ}}_{13}& 2{\mathrm{\γ}}_{23}\end{array}\right]}^T,$ γ ₁₃and γ ₂₃for horizontal shear capacity; G is shearing rigidity matrix;

be respectively the stiffness matrix and the load continuous item that are converted to Reissner-Mindlin model.

Further, described step 3 is specially: based on the three dimensional field Remodeling of deriving, utilize the overall situation response of the two-dimentional shell face that Reissner-Mindlin model analysis obtains and the warping function reconstruct field variable through-thickness distribution that Dimension Reduction Analysis obtains:

$U_i=u_i+x_3{\left[\begin{array}{ccc}{C}_{31}& C_{32}& C_{33}\end{array}\right]}^T+S({\mathrm{<figure-callout\; id="0"\; label="V"\; filenames="HDA0000491546900000051.png,HDA0000491546900000061.png"\; state="\{\{state\}\}">V</figure-callout>}}_0+{V_1}_{});$

Wherein, U _i, u _ibe respectively three-dimensional housing distortion and two-dimentional housing distortion array; C _ijfor overall rotation tensor; x ₃for the coordinate on the thickness direction along housing reference surface, V ₀, V ₁for zeroth order and the asymptotic correction warpage of single order nodal value, S is shape function;

By constitutive relation reconstruct triaxiality, electric displacement, magnetic induction gesture:

$\left\{\begin{array}{c}\mathrm{\&sigma;}\\ D\\ B\end{array}\right\}=\begin{array}{c}\left[\begin{array}{ccc}C& -e& -q\\ {-e}^T& -d& -a\\ {-q}^T& {-a}^T& -\mathrm{\&mu;}\end{array}\right]\mathrm{\&Gamma;}\end{array};$

Wherein, σ, D, B is respectively load, the triaxiality of electricity/magnetic field lower house, electric displacement, magnetic induction gesture; C, e, q, d, a, μ is respectively the matrix containing elasticity, piezoelectricity, pressure magnetic, dielectric, magnetoelectricity and magnetoconductivity constant; Γ is generalized strain moment matrix.

Than prior art, tool of the present invention has the following advantages:

1, electromagnetic material stratiform housing electromagnetism bullet coupling Simulation analogy method provided by the invention, by showing with the comparative analysis of three-dimensional finite element exact solution: the three dimensional field distribution degree of accuracy that employing variation method of approach solves the shell structure being made up of heterogeneous body, anisotropy two-phase electromagnetic material is high.

2, the electromagnetic material stratiform housing electromagnetism bullet coupling model that the present invention sets up belongs to individual layer shell model, and calculated amount is little, and counting yield is higher than rationally opinion and three-dimensional finite element solution of high-level.

3, electromagnetic material stratiform housing electromagnetism bullet coupling Simulation analogy method provided by the invention, only need set up individual layer shell model, and division unit and node greatly reduce compared with three-dimensional finite element, take computer resource few.

4, electromagnetic material stratiform housing electromagnetism bullet coupling Simulation analogy method provided by the invention, does not need any apriority hypothesis, has mathematical tightness.

5, electromagnetic material stratiform housing electromagnetism bullet coupling Simulation analogy method provided by the invention, can simulate electricity-magnetic-bullet coupling performance of complete anisotropy layer of electro-magnetic material housing, broken through finite element and can only process the limitation of macroscopical orthotropic material.

Accompanying drawing explanation

Fig. 1 is the process flow diagram of electromagnetic material stratiform housing electromagnetism bullet coupling Simulation analogy method provided by the invention.

Fig. 2 is three layers of electromagnetic material housing geometric configuration in specific embodiment.

Fig. 3 is the coordinate system of setting up on three layers of electromagnetic material housing in specific embodiment.

Fig. 4 be in specific embodiment operating mode one middle shell along the potential distribution plan of thickness of shell direction.

Fig. 5 be in specific embodiment operating mode one middle shell along the magnetic potential gesture distribution plan of thickness of shell direction.

Fig. 6 be in specific embodiment operating mode one middle shell along the magnetic induction gesture distribution plan of thickness of shell direction.

Fig. 7 be in specific embodiment operating mode one middle shell along the electric displacement distribution plan of thickness of shell direction.

Fig. 8 be in specific embodiment operating mode one middle shell along the horizontal normal stress distribution figure of thickness of shell direction.

Fig. 9 be in specific embodiment operating mode one middle shell along the elastic displacement distribution plan of thickness of shell direction.

Figure 10 be in specific embodiment operating mode two middle shells along the potential distribution plan of thickness of shell direction.

Figure 11 be in specific embodiment operating mode two middle shells along the magnetic potential gesture distribution plan of thickness of shell direction.

Figure 12 be in specific embodiment operating mode two middle shells along the magnetic induction gesture distribution plan of thickness of shell direction.

Figure 13 be in specific embodiment operating mode two middle shells along the electric displacement distribution plan of thickness of shell direction.

Figure 14 be in specific embodiment operating mode two middle shells along the horizontal normal stress distribution figure of thickness of shell direction.

Figure 15 be in specific embodiment operating mode two middle shells along the elastic displacement distribution plan of thickness of shell direction.

Embodiment

Three layers of electromagnetic material housing (Fig. 2 intermediate cam shape mark place is provided with hold-down support, and circle marker place is provided with movable bearing support) as shown in Figure 2,3, the face in housing of getting is reference surface, x ₁, x ₂, x ₃axle is respectively curved surface direction, length direction and the thickness direction along housing reference surface, along x ₁the initial curvature of axle is k ₁₁, h, R, l is respectively thickness, radius-of-curvature and the length value of housing.The flow process of electromagnetic material stratiform housing electromagnetism bullet coupling Simulation analogy method as shown in Figure 1.

1, three-dimensional energy equation.

Comparatively complicated because relating to sign of operation, be first following statement: subscript i=1,2,3, j=1,2,3, α=1,2, β=1,2, a _αfor the Lame parameter relevant with two-dimentional shell face base vector,

<> represents along the definite integral of thickness of shell direction,

represent to act on the physical quantity of housing top, bottom surface.

Based on Hamilton principle (being Hamilton principle), the elastic dynamic performance of electromagnetic material housing can be expressed as:

${\&Integral;}_{t_1}^{t_2}[\mathrm{\&delta;}(K-U)+\mathrm{\&delta;}\stackrel{\&OverBar;}{W}]\mathrm{dt}=0---\left(1\right)$

In formula: t ₁, t ₂for any two fixed times; K, U is respectively the structure kinetic energy of dynamic load generation and the interior energy that load, electricity/magnetic field produce in housing,

by being done virtual work in load, electricity/magnetic field, upper line represents not need to this accurate variation, and δ is variation symbol.

To being in the shell structure in multiple physical field, in it, can functional be the energy functional of piezoelectricity, the broad sense potential energy of pressing magnetoelasticity dielectric material and piezoelectricity energy, pressure magnetic energy, electromagnetic energy composition:

$U=\frac{1}{2}{\&Integral;}_v[{\mathrm{\&Gamma;}}^T{C}^{E,H}\mathrm{\&Gamma;}-E^T{d}^{\mathrm{\&Gamma;},H}E-H^T{\mathrm{\&mu;}}^{\mathrm{\&Gamma;},E}H-2E{e}^H\mathrm{\&Gamma;}-2H{q}^E\mathrm{\&Gamma;}-2E{a}^{\mathrm{\&Gamma;H}}]\mathrm{dv}---\left(2\right)$

In formula: Γ, E, H is respectively strain, the Electric and magnetic fields intensity of load, the generation of electricity/magnetic field lower house; e ^h, q ^e, a ^Γbe respectively the pressure magnetic constant matrix of piezoelectric constant matrix, the electric field intensity of magnetic field intensity when constant when constant and the strain magnetoelectricity constant matrices when constant; C ^e,H, d ^{Γ, H}, μ ^{Γ, E}be respectively the specific inductive capacity matrix of elastic constant matrix, strain and the magnetic field intensity of Electric and magnetic fields intensity when constant when constant and strain and the electric field intensity magnetic permeability constant matrices when constant; V is shell space volume.

Based on rotation tensor decomposition of concept, stain vector Γ i in formula (2) _jmay be defined as:

Γ _ij=(F _ij+ F _ji)/2-△ _ij(3) in formula: △ i _jfor Kronecker symbol (being Kronecker symbol); F _ij, F _jibe the mixed base component of deformation gradient vector.

In formula (2), Electric and magnetic fields intensity can be by potential φ _ijwith magnetic potential gesture Ψ _ijdetermine.If disregard volume charge and body electric current, potential and magnetic potential gesture can represent with the change of variable of following form:

$\begin{array}{c}\mathrm{\&phi;}\left(x_i\right)=\mathrm{\&phi;}({\mathrm{<figure-callout\; id="1"\; label="x"\; filenames="HDA0000491546900000061.png"\; state="\{\{state\}\}">x</figure-callout>}}_1,x_2)+w_{\mathrm{\&phi;}}({\mathrm{<figure-callout\; id="1"\; label="x"\; filenames="HDA0000491546900000061.png"\; state="\{\{state\}\}">x</figure-callout>}}_1,x_2,x_3),\\ \mathrm{\&psi;}\left(x_i\right)=\mathrm{\&psi;}({\mathrm{<figure-callout\; id="1"\; label="x"\; filenames="HDA0000491546900000061.png"\; state="\{\{state\}\}">x</figure-callout>}}_1,x_2+)w_{\mathrm{\&psi;}}({\mathrm{<figure-callout\; id="1"\; label="x"\; filenames="HDA0000491546900000061.png"\; state="\{\{state\}\}">x</figure-callout>}}_1,x_2,x_3)\end{array}---\left(4\right)$ In formula: φ is two-dimentional electromotive force, value is the mean value of three-dimensional electromotive force φ through-thickness; Ψ is two-dimentional magnetic potential, and value is the mean value of three-dimensional electromotive force Ψ through-thickness.This means electric warping function w _φwith magnetic warping function w _Ψneed to meet:

<w _φ(x ₁,x ₂,x ₃)>=0,<w _ψ(x ₁,x ₂,x ₃)>=0 (5)

Three-dimensional electric field and magnetic field intensity E _i, H _imay be defined as:

$\begin{array}{c}{E}_{\mathrm{\&alpha;}}=E_{2D_{\mathrm{\&alpha;}}}-w_{\mathrm{\&phi;};\mathrm{\&alpha;}},{\mathrm{<figure-callout\; id="3"\; label="E"\; filenames="HDA0000491546900000061.png"\; state="\{\{state\}\}">E</figure-callout>}}_3=-w_{\mathrm{\&phi;},3}\\ H_{\mathrm{\&alpha;}}=H_{2D_{\mathrm{\&alpha;}}}-w_{\mathrm{\&Psi;};\mathrm{\&alpha;}},{\mathrm{<figure-callout\; id="3"\; label="H"\; filenames="HDA0000491546900000061.png"\; state="\{\{state\}\}">H</figure-callout>}}_3=-w_{\mathrm{\&Psi;},3}\end{array}---\left(6\right)$ In formula: be respectively two dimensional electric field and two-dimensional magnetic field,

Obtained containing h/R, (h/l) by formula (3) and (5) ²the Three Dimensional Generalized strain field of order item

for:

$\widehat{\mathrm{\&Gamma;}}={\mathrm{\&Gamma;}}_h\hat{w}+{\mathrm{\&Gamma;}}_{\&Element;}+{\mathrm{\&Gamma;}}_{\mathrm{Rh}}\hat{w}{\mathrm{\&Gamma;}}_{R\&Element;}+{\mathrm{\&Gamma;}}_{l_{\mathrm{\&alpha;}}}{\hat{w}}_{;\mathrm{\&alpha;}}---\left(7\right)$

In formula:

$\widehat{\mathrm{\&Gamma;}}={\left[\begin{array}{cccccccccccc}{\mathrm{\&Gamma;}}_{11}& 2{\mathrm{\&Gamma;}}_{12}& {\mathrm{\&Gamma;}}_{22}& 2{\mathrm{\&Gamma;}}_{13}& 2{\mathrm{\&Gamma;}}_{23}& {\mathrm{\&Gamma;}}_{33}& E_1& E_2& E_3& H_1& H_2& H_3\end{array}\right]}^T$

wherein, w ₁, w ₂, w ₃be respectively load along x ₁, x ₂, x ₃the buckling deformation value that axle produces; $\&Element;={\left[\begin{array}{cccccccccc}{\mathrm{\&epsiv;}}_{11}& 2{\mathrm{\&epsiv;}}_{12}& {\mathrm{\&epsiv;}}_{22}& K_{11}& K_{12}+K_{21}& K_{22}& E_{2D_1}& E_{2D_2}& H_{2D_1}& H_{2D_2}\end{array}\right]}^T$ Wherein

ε _{α β}, K _{α β}, (α, β=1,2) are respectively face internal strain and the face external strain of housing; Γ _h,

Γ _∈, Γ _rh, Γ _{r ∈}for corresponding integral operator matrix.

Can functional be expressed as interior:

$U=\frac{1}{2}{\&Integral;}_{\mathrm{\&Omega;}}<{\widehat{\mathrm{\&Gamma;}}}^T\left[\begin{array}{ccc}{C}^{E,H}& -e^H& -q^E\\ -{\left(e^H\right)}^T& -d^{\mathrm{\&Gamma;},H}& -a^{\mathrm{\&Gamma;}}\\ -{\left(q^E\right)}^T& -{\left(a^{\mathrm{\&Gamma;}}\right)}^T& -{\mathrm{\&mu;}}^{\mathrm{\&Gamma;},E}\end{array},{\widehat{\mathrm{\&Gamma;}}}_{\mathrm{\&mu;}}\right]{\widehat{\mathrm{\&Gamma;}}}^{\mathrm{\&mu;}}>\mathrm{d\&Omega;}=\frac{1}{2}{\&Integral;}_{\mathrm{\&Omega;}}<{\widehat{\mathrm{\&Gamma;}}}^{T}D\widehat{\mathrm{\&Gamma;}}{\mathrm{\&mu;}}^{}>\mathrm{d\&Omega;}---\left(8\right)$

In formula: Ω represents with reference to shell face face territory; μ is housing Geometric corrections coefficients, μ=1+x ₂(k ₁₁)+O (h ²/ R ²), O (h ²/ R ²) represent higher than h ²/ R ²rank item; D is the matrixes of 12 × 12 rank containing elasticity, piezoelectricity, pressure magnetic, dielectric, magnetoelectricity and magnetoconductivity constant, if the symmetrical material of monocline, and around self neutral axis rotation, some elements in matrix are always zero, obtain:

In formula: C ^e,Hrepresenting matrix $\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& 0& 0& 0\\ C_{12}& C_{22}& C_{23}& 0& 0& 0\\ C_{13}& C_{23}& C_{33}& 0& 0& 0\\ 0& 0& 0& C_{44}& 0& 0\\ 0& 0& 0& 0& C_{55}& 0\\ 0& 0& 0& 0& 0& C_{66}\end{array}\right],$ C _klthe elastic constant when constant for electric field strength E and magnetic field intensity H;-e ^hrepresenting matrix $\left[\begin{array}{ccc}0& 0& e_{31}\\ 0& 0& e_{32}\\ 0& 0& e_{33}\\ 0& e_{24}& 0\\ e_{15}& 0& 0\\ 0& 0& 0\end{array}\right],$ E _klfor the negative value of the piezoelectric constant of magnetic field intensity H when constant;-q ^erepresenting matrix $\left[\begin{array}{ccc}0& 0& q_{31}\\ 0& 0& q_{32}\\ 0& 0& q_{33}\\ 0& q_{24}& 0\\ q_{15}& 0& 0\\ 0& 0& 0\end{array}\right],$ Q _klfor the negative value of the pressure magnetic constant of electric field strength E when constant;-(e ^h) ^trepresenting matrix $\left[\begin{array}{cccccc}0& 0& 0& 0& e_{15}& 0\\ 0& 0& 0& e_{24}& 0& 0\\ e_{31}& e_{32}& e_{33}& 0& 0& 0\end{array}\right];$ -d ^{Γ, H}representing matrix $\left[\begin{array}{ccc}{d}_{11}& 0& 0\\ 0& d_{22}& 0\\ 0& 0& d_{33}\end{array}\right],$ D _klthe negative value of the specific inductive capacity when constant for strain Γ and magnetic field intensity H;-a ^Γrepresenting matrix $\left[\begin{array}{ccc}{a}_{11}& 0& 0\\ 0& a_{22}& 0\\ 0& 0& a_{33}\end{array}\right],$ A _klfor the negative value of the magnetoelectricity constant of strain Γ when constant;-(q ^e) ^trepresenting matrix $\left[\begin{array}{cccccc}0& 0& 0& 0& q_{15}& 0\\ 0& 0& 0& q_{24}& 0& 0\\ q_{31}& q_{32}& q_{33}& 0& 0& 0\end{array}\right];$ -(a ^Γ) ^trepresenting matrix $\left[\begin{array}{ccc}{a}_{11}& 0& 0\\ 0& a_{22}& 0\\ 0& 0& a_{33}\end{array}\right];$ -μ ^{Γ, E}representing matrix $\left[\begin{array}{ccc}{\mathrm{\&mu;}}_{11}& 0& 0\\ 0& {\mathrm{\&mu;}}_{22}& 0\\ 0& 0& {\mathrm{\&mu;}}_{33}\end{array}\right],$ μ _klthe negative value of magnetic permeability constant when strain Γ and electric field strength E are constant; Subscript k=l=1,2,3,4,5,6.

Formula (5) electromagnetism warping constraint adds 3 field of force warping constraints, finally obtains being constrained to of many coupling Layered housings:

$<{\mathrm{\&Gamma;}}_c\hat{w}>=0---\left(10\right)$

In formula: Γ _cbe 5 × 5 rank unit matrixs.

The virtual work that load, electricity/magnetic field are done can be expressed as:

$\mathrm{\&delta;}\stackrel{\&OverBar;}{W}={\&Integral;}_{\mathrm{\&Omega;}}(<P\&CenterDot;\mathrm{\&delta;}\hat{R}>+\mathrm{\&tau;}\&CenterDot;\mathrm{\&delta;}{\hat{R}}^{+}+\mathrm{\&beta;}\&CenterDot;\mathrm{\&delta;}{\hat{R}}^{-}-D^{\&PlusMinus;}\mathrm{\&delta;}{\mathrm{\&phi;}}^{\&PlusMinus;}-B^{\&PlusMinus;}\mathrm{\&delta;}{\mathrm{\&Psi;}}^{\&PlusMinus;})\mathrm{d\&Omega;}---\left(11\right)$

In formula: P, τ, β is respectively the muscle power of housing, the surface force of housing end face, the surface force of housing bottom surface.

P=P _ib _i, τ=τ _ib _i, β=β _ib _ip _ifor physical load intensity, τ _ifor the load intensity of housing end face surface force, β _irepresent the load intensity of housing bottom surface power, B _ifor the triad vector after distortion; D ^±for the electric displacement (electric flux density) of housing top, bottom surface, B ^±be respectively the magnetic induction gesture (magnetic flux density) of housing top, bottom surface; φ ^±, Ψ ^±be not potential and the magnetic potential gesture of housing top, bottom surface, for the Lagrangian variation of displacement field (containing virtual displacement and rotation), for the Lagrangian variation of the displacement field of housing end face,

for the Lagrangian variation of the displacement field of housing bottom surface.

Because buckling deformation is less, can ignore safely

the product term of middle warpage and virtual rotation, is reduced to formula (11):

$\mathrm{\&delta;}\stackrel{\&OverBar;}{W}=\mathrm{\&delta;}{\stackrel{\&OverBar;}{W}}_{2D}+\mathrm{\&delta;}{\stackrel{\&OverBar;}{W}}^{*}---\left(12\right)$

In formula,

$\mathrm{\&delta;}{\stackrel{\&OverBar;}{W}}^{*}={\&Integral;}_{\mathrm{\&Omega;}}[<P_i{\mathrm{\&delta;w}}_i>+{\mathrm{\&tau;}}_i\mathrm{\&delta;}{w}_i^{+}+{\mathrm{\&beta;}}_i\mathrm{\&delta;}{w}_i^{-}{D}^{\&PlusMinus;}\mathrm{\&delta;}{w}_{\mathrm{\&phi;}}^{\&PlusMinus;}-B^{\&PlusMinus;}\mathrm{\&delta;}{w}_{\mathrm{\&psi;}}^{\&PlusMinus;}]\mathrm{d\&Omega;}---\left(14\right)$

In formula, be respectively virtual displacement and rotation that housing produces; D ⁺for the electric displacement of housing end face, D ^-for the electric displacement of housing bottom surface, B ⁺for the magnetic induction gesture of housing end face, B ^-for the magnetic induction gesture of housing bottom surface,

for load is along x _ithe buckling deformation value that axle produces at housing end face, for load is along x _ithe buckling deformation value that axle produces in housing bottom surface, for the electric buckling deformation value of housing top, bottom surface,

for the magnetic buckling deformation value of housing top, bottom surface, generalized force f _iwith moment of flexure m _αmay be defined as:

f _i=<P _i>+τ _i+β _i；m _α=<x ₃P _α>+h(τ _α-β _α)/2 (15)

For the kinetic energy of housing, can be expressed as:

$K=\frac{1}{2}{\&Integral;}_v\mathrm{\&rho;}{\hat{V}}^T\mathrm{dv}=K_{2D}+K^{*}---\left(16\right)$

In formula,

for the absolute velocity of particle on shell face;

$\begin{array}{c}{K}_{2D}=\frac{1}{2}{\&Integral;}_{\mathrm{\&Omega;}}(\stackrel{\&OverBar;}{\mathrm{\&mu;}}{\hat{V}}^T+\hat{V}+2{\mathrm{\&omega;}}^T\mathrm{\&mu;}\stackrel{\&OverBar;}{\mathrm{\&xi;}}\hat{V}+{\mathrm{\&omega;}}^T\mathrm{j\&omega;})\mathrm{d\&Omega;}\\ K^{*}=\frac{1}{2}{\&Integral;}_v\mathrm{\&rho;}[{(\widetilde{\mathrm{\&omega;}}\tilde{w}+\&PartialD;\tilde{w}/\&PartialD;t)}^T+2{(\hat{V}+\widetilde{\mathrm{\&omega;}}\mathrm{\&xi;})}^T(\widetilde{\mathrm{\&omega;}}\hat{w}+\&PartialD;\hat{w}/\&PartialD;t\left)\mathrm{dv}\right]\end{array}---\left(17\right)$

In formula, ρ is mass density, for the antisymmetric matrix of inertia angular velocity omega; j is the conventional inertia constant of performance analysis: $\stackrel{\&OverBar;}{\mathrm{\&mu;}}=<\mathrm{\&rho;}>,\mathrm{\&mu;}\stackrel{\&OverBar;}{\mathrm{\&xi;}}={\left[\begin{array}{ccc}0& 0& <x_3\mathrm{\&rho;}>\end{array}\right]}^T,j\left[\begin{array}{ccc}<x_3^2\mathrm{\&rho;}>& 0& 0\\ 0& <x_3^2\mathrm{\&rho;}>& 0\\ 0& 0& 0\end{array}\right]$

Utilize formula (12) and (16), formula (1) can be written as:

${\&Integral;}_{t_1}^{t_2}[\mathrm{\&delta;}(K_{2D}+K^{*}-U)+\mathrm{\&delta;}{\stackrel{\&OverBar;}{W}}_{2D}+\mathrm{\&delta;}{\hat{W}}^{*}]\mathrm{dt}=0---\left(1\right)$

In formula:

the two-dimensional structure kinetic energy and the two-dimentional virtual work done of load, electricity/magnetic field that produce for dynamic load;

be respectively the broad sense complementary energy of dynamic load generation and the residue virtual work that do in load, electricity/magnetic field.

So far, set up the energy functional equation based on reference surface displacement, but this is only another expression-form of former dimensional multi-field coupled problem, direct solution difficulty is larger.Main difficulty is unknown warping function

existing document is supposed in advance according to two-dimentional variable conventionally

version, be directly two-dimentional shell surface model by former three-dimensional continuous model dimensionality reduction.But for the shell structure being formed by heterogeneous body, anisotropy two-phase electromagnetic material, introduce this hypothesis and can produce obvious error.The present invention is by variation method of approach, utilize the asymptotic calculating warping function of the intrinsic small parameter of housing, suppose without apriority, and can make to calculate simplification, can simulate complete anisotropy layer of electro-magnetic material and close electricity-magnetic-bullet coupling performance of shell, break through finite element and can only process the limitation of macroscopical orthotropic material.

2 Dimension Reduction Analysis.

Many coupling lower houses vibrate take low-frequency vibration as main, and the buckling deformation that load produces is relatively little, in Dimension Reduction Analysis, can ignore safely the K in formula (19) ^*with

only to carrying out asymptotic analysis containing the energy functional of interior energy and virtual work.According to variational principle, unknown warping function can obtain by energy functional Π being got in value:

δ Π=0 (20) is in order effectively to analyze layer structure, and is connected mutually with the two dimensional panel shell solver in finite element, application finite element method by discrete three dimensional warped function be One Dimensional Finite n-ary form n:

W (x _i)=S (x ₃) V (x ₁, x ₂) in (21) formula: S is shape function; V is the warpage field nodal value matrix along horizontal normal direction.

The discrete form that can obtain energy functional is:

$\begin{array}{c}2\mathrm{\&Pi;}=V^T\mathrm{EV}+{2V}^T(D_{h\&Element;}+D_{\mathrm{Rh}\&Element;}\&Element;+D_{\mathrm{hRh}}V+D_{\mathrm{hR}\&Element;}\&Element;+D_{{\mathrm{hl}}_{\mathrm{\&alpha;}}}{V}_{;\mathrm{\&alpha;}})\\ +{\&Element;}^T(D_{\&Element;\&Element;}+2D_{\&Element;R\&Element;})\&Element;+V_{;\mathrm{\&alpha;}}^T{D}_{l_{\mathrm{\&alpha;}}{l}_{\mathrm{\&beta;}}}{V}_{;\mathrm{\&beta;}}+V_{;\mathrm{\&alpha;}}^T{D}_{l_{\mathrm{\&alpha;}}\&Element;}+{2V}^{T}L\end{array}---\left(22\right)$

In formula:

l is load continuous item, L=-S ^{+ T}τ-S ^-Tβ-<S ^tp>-S ^{± T}φ-S ^{± T}Ψ, S ⁺for the shape function of housing end face, S ^-for the shape function of housing bottom surface; The new matrix relevant with geometric configuration and material properties of introducing is:

$\begin{array}{c}E=<{\left[{\mathrm{\&Gamma;}}_{h}S\right]}^{T}D\left[{\mathrm{\&Gamma;}}_{h}S\right]\mathrm{\&mu;}>,D_{h\&Element;}=<{\left[{\mathrm{\&Gamma;}}_{h}S\right]}^T\mathrm{D\&Gamma;}{\&Element;}_{}\mathrm{\&mu;}>\\ D_{{\mathrm{hl}}_{\mathrm{\&alpha;}}}=<{\left[{\mathrm{\&Gamma;}}_{h}S\right]}^{T}D\left[{\mathrm{\&Gamma;}}_{l_{\mathrm{\&alpha;}}}\right]>,D_{l_{\mathrm{\&alpha;}}{l}_{\mathrm{\&beta;}}}=<{\left[{\mathrm{\&Gamma;}}_{l_{\mathrm{\&alpha;}}}S\right]}^{T}D\left[{\mathrm{\&Gamma;}}_{l_{\mathrm{\&beta;}}}S\right]>\\ D_{\&Element;\&Element;}=<{{\mathrm{\&Gamma;}}_{\&Element;}}^T{\mathrm{D\&Gamma;}}_{\&Element;}\mathrm{\&mu;}>,D_{l_{\mathrm{\&alpha;}}\&Element;}=<{\left[{\mathrm{\&Gamma;}}_{l_{\mathrm{\&alpha;}}}S\right]}^T{\mathrm{D\&Gamma;}}_{\&Element;}>\\ D_{\mathrm{hRh}}=<{\left[{\mathrm{\&Gamma;}}_{h}S\right]}^{T}D\left[{\mathrm{\&Gamma;}}_{\mathrm{Rh}}S\right]>,D_{\mathrm{hR}\&Element;}=<{\left[{\mathrm{\&Gamma;}}_{h}S\right]}^T{\mathrm{D\&Gamma;}}_{R\&Element;}>\\ D_{\&Element;R\&Element;}=<{{\mathrm{\&Gamma;}}_{\&Element;}}^T{\mathrm{D\&Gamma;}}_{\&Element;}>,D_{\mathrm{Rh}\&Element;}=<{\left[{\mathrm{\&Gamma;}}_{\mathrm{Rh}}S\right]}^T{\mathrm{D\&Gamma;}}_{\&Element;}>\end{array}---\left(23\right)$

Wherein,

for integral operator matrix.

2.1 zero-order approximation.

For verifying that variation method of approach is applied to the validity of intellectual material housing modeling, first build the classic laminated model of electromagnetic material stratiform housing based on variation method of approach.

The discrete form of formula (10) warping function is:

V ^tin the formula of H Ψ=0 (24): H=S ^ts; Ψ is E ₀the orthogonalization kernel matrix of (zero initial curvature), Ψ ^th Ψ=I, I is unit matrix.Problem is converted into formula (24) constraint following formula (22) minimization problem like this.

Application variation method of approach, first needs to find leading term according to the different rank of functional.Owing to only having warpage to change, only need find the leading term containing warpage.Functional after formula (22) zero-order approximation

leading term be

${2\mathrm{\&Pi;}}_0^{*}=V^T{E}_{0}V+{2V}^T{D}_{h\&Element;0}+{\&Element;}^T{D}_{\&Element;\&Element;0}\&Element;---\left(25\right)$

In formula: E ₀for the homography of μ=1 (without Geometric corrections) in formula E in formula (23), Dh ò 0 is formula D in formula (23) _{h ò}the homography of middle μ=1 (without Geometric corrections), D _{ò ò 0}for formula D in formula (23) _{ò ò}the homography of middle μ=1 (without Geometric corrections).

Minimize formula (25), can obtain zeroth order warping function and be

$V={\hat{V}}_0\&Element;{=V}_0---\left(26\right)$

In formula

zeroth order broad sense warpage nodal value, V ₀for the asymptotic correction warpage of the zeroth order along horizontal normal direction nodal value matrix.

In formula (26) generation, is returned to formula (25), can obtain the asymptotic two-dimentional energy functional Π that is adapted to zeroth order ₀for

2 Π ₀=∈ ^tin A ∈ (27) formula:

for two-dimentional stiffness matrix.

Formula (27) is identical with classic shell theory form, but do not introduce Kirchhoff hypothesis (in before and after distortion, the normal of face is straight line, and with middle perpendicular).

2.2 first approximation.

The classic axisymmetrical laminated shell body Model that zero-order approximation obtains can predict shell structure the overall situation and face internal variable distribute, but centering thick shell also needs to utilize the intrinsic small parameter h/R of housing, h/l carries out more high-order approximation, distributes with Accurate Prediction face exogenousd variables.

For considering the impact of housing initial curvature, first to energy functional Π _rleading term carries out the correction of h/R rank:

2 Π _r=∈ ^ta _rin ∈ (28) formula:

$A_R=A+{\hat{V}}_0^T{E}^{*}{\hat{V}}_0^T+D_{\&Element;\&Element;}^{*}+{2\hat{V}}_0^T(D_{h\&Element;}^{*}+F_{\mathrm{hR}\&Element;}+D_{\mathrm{Rh}\&Element;})+2D_{\&Element;R\&Element;}+2{\hat{V}}_0^T{D}_{\mathrm{hRh}}{\hat{V}}_0---\left(29\right)$

In formula: E ^*for the homography of μ=μ-1 in formula E in formula (23),

for formula D in formula (23) _{h ∈}the homography of middle μ=μ-1,

for formula D in formula (23) _{∈ ∈}the homography of middle μ=μ-1.

Secondly, be the transverse shear deformation of accurate description housing, need derive (h/l) ²rank modified energy functional.For this reason, by simple zeroth order warpage perturbation be:

V=V ₀+ V ₁(30) wherein, V ₁for the asymptotic correction warpage of the single order along horizontal normal direction nodal value matrix.

Can obtain first approximation energy functional

leading term is

$2{\mathrm{\&Pi;}}_1^{*}={V_1}^{\mathrm{<figure-callout\; id="1"\; label="T\; EV"\; filenames="HDA0000491546900000061.png"\; state="\{\{state\}\}">T</figure-callout>}}{\mathrm{EV}}_{}{}_1+2{V_1}^T{D}_{\mathrm{\&alpha;}}{\&Element;}_{;\mathrm{\&alpha;}}+{{2V}_1}^{T}L---\left(31\right)$

In formula:

Similar with zero-order approximation, single order warpage can be solved by following formula

V ₁=V _{1 α}∈ _{; α}+ V _1L(32) in formula: V _{1 α}, V _1Lbe respectively single order correction warpage nodal value and single order load nodal value.

Formula (32) substitution formula (31), obtains asymptotic being adapted to (h/l) ², (h/R) rank gross energy functional Π ₁for

${2\mathrm{\&Pi;}}_1={\&Element;}^T{A}_R\&Element;+{\&Element;}_{;\mathrm{\&alpha;}}^T{B}_{\mathrm{\&alpha;\&beta;}}{\&Element;}_{;\mathrm{\&beta;}}+2{\&Element;}^{T}F---\left(33\right)$

In formula:

$\begin{array}{c}{B}_{\mathrm{\&alpha;\&beta;}}={\hat{V}}_0^T{D}_{l_{\mathrm{\&alpha;}}{l}_{\mathrm{\&beta;}}}{\hat{V}}_0+{\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="HDA0000491546900000061.png"\; state="\{\{state\}\}">B</figure-callout>}}_{1\mathrm{\&alpha;}}^T{D}_{\mathrm{\&beta;}},\\ F={\stackrel{Y}{V}}_0^{T}L-(D_{\mathrm{\&alpha;}}^T{V}_{1L;\mathrm{\&alpha;}}+{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA0000491546900000061.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1\mathrm{\&alpha;}}^T{L}_{;\mathrm{\&alpha;}})/2\end{array}---\left(34\right)$

The conversion of 2.3Reissner model form.

Although the asymptotic second order that is adapted to of formula (33), because of the derivative ∈ containing Two-Dimensional Generalized strain _{; α}, be difficult to effectively application in Practical Project.For obtaining practical energy functional, can be Reissner-Mindlin model form conventional in Practical Project by approximate energy conversion.For this reason, need to increase by 2 horizontal shear capacities $\mathrm{\&gamma;}={\left[\begin{array}{cc}2{\mathrm{\&gamma;}}_{13}& 2{\mathrm{\&gamma;}}_{23}\end{array}\right]}^T$ As independence and freedom degree.Reissner-Mindlin dependent variable

with the pass of ∈ be:

in formula:

for the matrix containing classic generalized strain amount, wherein, ${\mathrm{\&epsiv;}}_{11}^{*},{\mathrm{\&epsiv;}}_{12}^{*},{\mathrm{\&epsiv;}}_{22}^{*},K_{11}^{*},K_{12}^{*},K_{22}^{*},E_{2{\mathrm{<figure-callout\; id="1"\; label="D"\; filenames="HDA0000491546900000061.png"\; state="\{\{state\}\}">D</figure-callout>}}_1}^{*},E_{2{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="HDA0000491546900000051.png,HDA0000491546900000061.png"\; state="\{\{state\}\}">D</figure-callout>}}_2}^{*},H_{2{\mathrm{<figure-callout\; id="1"\; label="D"\; filenames="HDA0000491546900000061.png"\; state="\{\{state\}\}">D</figure-callout>}}_1}^{*},H_{2{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="HDA0000491546900000051.png,HDA0000491546900000061.png"\; state="\{\{state\}\}">D</figure-callout>}}_2}^{*}$ Be classic generalized strain amount.

By formula (35) substitution formula (33), can obtain the gross energy functional that is adapted to second order that Reissner strain represents and be:

For the partial derivative of variable in cancelling (37)

can utilize two moment of flexure balance equations of formula (15) derive γ with

relational expression be:

In formula: G is shearing rigidity matrix, F _γfor shearing force, m ₁, m ₂be respectively along x ₁, x ₂the moment of flexure of direction.

Utilize formula (38), formula (37) can be rewritten as

In formula:

The form of broad sense Reissner-Mindlin model is

If in formula (39)

Formula (39) and formula (41) equivalence.Order

u* non-vanishing in Practical Project, but can make U by optimisation techniques such as least square methods ^*level off to as far as possible zero, obtain accurate result of calculation, do not repeat them here.

2.4 Remodeling.

The electromagnetism stratiform housing broad sense Reissner-Mindlin model building can Accurate Prediction shell face overall situation response, for the distribution situation of the field variable through-thickness of more paying close attention to, need Remodeling to predict.For this reason, utilize the warping function reconstruct field variable through-thickness that the overall situation responds and Dimension Reduction Analysis obtains that two-dimentional shell surface analysis obtains to distribute:

$U_i=u_i+x_3{\left[\begin{array}{ccc}{C}_{13}& C_{32}& C_{33}-1\end{array}\right]}^T+S({\mathrm{<figure-callout\; id="0"\; label="V"\; filenames="HDA0000491546900000051.png,HDA0000491546900000061.png"\; state="\{\{state\}\}">V</figure-callout>}}_0+V_1)---\left(43\right)$ In formula: U _i, u _ibe respectively three-dimensional housing distortion and two-dimentional housing distortion array; C _ijfor overall rotation tensor.

By constitutive relation restructural triaxiality, electric displacement, magnetic induction gesture be

$\left\{\begin{array}{c}\mathrm{\&sigma;}\\ D\\ B\end{array}\right\}=\begin{array}{c}\left[\begin{array}{ccc}C& -e& -q\\ {-e}^T& -d& -a\\ {-q}^T& {-a}^T& -\mathrm{\&mu;}\end{array}\right]\mathrm{\&Gamma;}\end{array}---\left(44\right)$

In formula: σ, D, B is respectively load, the triaxiality of electricity/magnetic field lower house, electric displacement, magnetic induction gesture; Γ is generalized strain moment matrix, and C, e, q, d, a, μ are respectively the matrix containing elasticity, piezoelectricity, pressure magnetic, dielectric, magnetoelectricity and magnetoconductivity constant.

In order to verify validity and the accuracy of the method that the present invention sets up, utilize cylindrical bending problem to analyze.The shell structure that electromagnetic material stratiform housing adopts heterogeneous body, anisotropy two-phase electromagnetic material to form, upper and lower two-layer employing piezoelectric BaTiO3 (brief note is B); Internal layer is magnetic material CoFe2O4 (brief note is F).Each layer thickness is 0.1m, housing radius R=10mm, angle

the coordinate adopting is

x ₂∈ [0, l], x ₃∈ [h/2, h/2].Divide two kinds of operating modes:

Operating mode one: housing end face effect sinusoidal pattern load: without electric field, magnetic fields.

Operating mode two: piezoelectric layer and pressure magnetosphere surface of contact ground connection, piezoelectric layer outside surface effect electromotive force

machinery-free field, magnetic fields.Electromagnetic material constant is listed in table 1.

Operating mode one middle shell is along the field variable distribution plan of thickness of shell direction, as shown in Fig. 4～Fig. 9, operating mode two middle shells along the field variable distribution plan of thickness of shell direction as shown in Figure 10～Figure 15.Can find out, when the electromagnetism bullet coupling Simulation analogy method that adopts the application to provide is carried out analogue simulation to the housing in above-mentioned two kinds of different operating modes, the field variable of the housing of reconstruct distributes fine with three-dimensional finite element exact solution anastomose property.Wherein: the 1. elastic displacement u of operating mode one ₃be less than operating mode two, but both elastic displacement trend is completely contrary; 2. the magnetic potential gesture Ψ curve under two kinds of operating modes is continuous at intersection, but curve inclination angle slope discontinuous; 3. due to piezoelectricity, press magneto-coupling effect in zones of different generation effect, the horizontal normal stress σ under two kinds of operating modes ₃₃character, variation tendency are completely different; 4. the magnetic induction gesture B under two kinds of operating modes ₃there is identical variation tendency, be that piezoelectric layer magnetic induction is close to zero (piezomagnetic coefficient of piezoelectric layer is zero), the magnetic induction of magnetic material layer is nonlinearities change (magnetoelectric effect), if now still adopt linear hypothesis can cause larger error; 5. two kinds of operating modes press down the electric displacement D of magnetosphere ₃be zero (piezoelectric modulus is zero), but the piezoelectric layer electric displacement of operating mode one is nonlinearities change (piezoelectricity coupling effect); Operating mode two is irregular to be followed; 6. operating mode one piezoelectric layer potential is nonlinear Distribution (piezoelectricity coupling effect), and operating mode two piezoelectric layer potentials are linear distribution, and pressing magnetosphere potential is zero.

Table 2 provides calculating scale and the computing time of three-dimensional finite element solution and solution of the present invention.As can be seen from Table 2: while adopting the response of computation structure of the present invention, the unknown quantity number that needs the unknown quantity number of numerical solution need to solve much smaller than three-dimensional finite element method.The calculated amount of solution of the present invention is little, and calculating scale greatly reduces, and has improved counting yield.

Table 1 electromagnetic material constant

As can be seen here, electromagnetic material stratiform housing electromagnetism bullet coupling Simulation analogy method provided by the invention, by showing with the comparative analysis of three-dimensional finite element exact solution: the three dimensional field distribution degree of accuracy that employing variation method of approach solves the shell structure being made up of heterogeneous body, anisotropy two-phase electromagnetic material is high.Calculated amount is little, and counting yield is rationally discussed and three-dimensional finite element solution higher than high-level, and, only need set up individual layer shell model, division unit and node greatly reduce compared with three-dimensional finite element, take computer resource few.

Finally explanation is, above embodiment is only unrestricted in order to technical scheme of the present invention to be described, although the present invention is had been described in detail with reference to preferred embodiment, those of ordinary skill in the art is to be understood that, can modify or be equal to replacement technical scheme of the present invention, and not departing from aim and the scope of technical solution of the present invention, it all should be encompassed in the middle of claim scope of the present invention.

Claims (4)

1. an electromagnetic material stratiform housing electromagnetism bullet coupling Simulation analogy method, is characterized in that, comprises the following steps:

1) set up the three-dimensional energy equation containing electricity-magneto-elastic coupling effect in electromagnetic material stratiform housing based on Hamilton principle and rotation tensor decomposition of concept;

2) based on the asymptotic variational method, the analysis of three-dimensional energy equation split is obtained to two-dimentional energy functional, and utilize in the two-dimentional energy functional of the asymptotic correction of small parameter intrinsic in electromagnetic material stratiform housing the leading variation item containing warpage item, obtain correction model, and be converted into Reissner-Mindlin model form;

3) the three dimensional field Remodeling based on deriving, utilize the overall situation response of the two-dimentional shell face that Reissner-Mindlin model analysis obtains and the warping function reconstruct field variable through-thickness distribution that Dimension Reduction Analysis obtains, by constitutive relation reconstruct triaxiality, electric displacement, magnetic induction gesture, to electromagnetic material stratiform housing electromagnetism bullet, analogue simulation is carried out in coupling.

2. electromagnetic material stratiform housing electromagnetism bullet coupling Simulation analogy method as claimed in claim 1, it is characterized in that, described step 1 is specially: set up the three-dimensional energy equation containing electricity-magneto-elastic coupling effect in electromagnetic material stratiform housing based on Hamilton principle and rotation tensor decomposition of concept:

${\&Integral;}_{t_1}^{t_2}[\mathrm{\&delta;}(K_{2D}+K^{*}-U)+\mathrm{\&delta;}{\stackrel{\&OverBar;}{W}}_{2D}+\mathrm{\&delta;}{\stackrel{\&OverBar;}{W}}^{*}]\mathrm{dt}=0;$

Wherein, t ₁, t ₂for any 2 fixed time points; K ^*for the broad sense complementary energy that dynamic load produces, K _2Dfor the two-dimensional structure kinetic energy of dynamic load generation; the two-dimentional virtual work of doing for load, electricity/magnetic field;

for the residue virtual work that do in load, electricity/magnetic field, upper line is used for showing that this virtual work does not need the accurate variation of functional, and δ is variation symbol, and U is the interior energy that load, electricity/magnetic field produce in housing, and its concrete form is:

$U=\frac{1}{2}{\&Integral;}_v[{\mathrm{\&Gamma;}}^T{C}^{E,H}\mathrm{\&Gamma;}-E^T{d}^{\mathrm{\&Gamma;},H}E-H^T{\mathrm{\&mu;}}^{\mathrm{\&Gamma;},E}H-2E{e}^H\mathrm{\&Gamma;}-2H{q}^E\mathrm{\&Gamma;}-2E{a}^{\mathrm{\&Gamma;H}}]\mathrm{dv}$

In formula: Γ, E, H is respectively strain intensity, electric field intensity and the magnetic field intensity of load, the generation of electricity/magnetic field lower house; e ^h, q ^e, a ^Γbe respectively the pressure magnetic constant matrix of piezoelectric constant matrix, the electric field intensity of magnetic field intensity when constant when constant and the strain intensity magnetoelectricity constant matrices when constant; C ^e,H, d ^{Γ, H}, μ ^{Γ, E}be respectively the specific inductive capacity matrix of electric field intensity and magnetic field intensity elastic constant matrix, strain intensity and the magnetic field intensity when constant when constant and strain intensity and the electric field intensity magnetic permeability constant matrices when constant; V is shell space volume.

3. electromagnetic material stratiform housing electromagnetism bullet coupling Simulation analogy method as claimed in claim 1, it is characterized in that, described step 2 is specially: based on the asymptotic variational method, three-dimensional energy Dimension Reduction Analysis is obtained to two-dimentional energy functional: utilize the leading variation item containing warpage item in the two-dimentional energy functional of the asymptotic correction of small parameter intrinsic in electromagnetic material stratiform housing, obtain the asymptotic correction model of zeroth order and single order:

2Π ₀=∈ ^TA∈；

$2{\mathrm{\&Pi;}}_1={\&Element;}^T{A}_R\&Element;+{\&Element;}_{;\mathrm{\&alpha;}}^T{B}_{\mathrm{\&alpha;\&beta;}}{\&Element;}_{;\mathrm{\&beta;}}+2{\&Element;}^{T}F;$

Wherein, Π ₀, Π ₁be respectively zeroth order, the energy functional that first approximation obtains; ∈ is broad sense two dimension dependent variable; A is two-dimentional stiffness matrix; A _r, B _{α β}for considering the revised stiffness matrix of housing initial curvature; α=β=1,2, A _αfor the Lame parameter of two-dimentional shell face base vector in housing, x _αfor the plane coordinate system of setting up on electromagnetic material stratiform housing, x ₁, x ₂axle is respectively curved surface direction, the length direction along housing reference surface; F is load continuous item, and subscript T represents transposed matrix;

Asymptotic single order correction model is converted to Reissner-Mindlin model form:

Wherein,

for the dependent variable of Reissner-Mindlin model; $\mathrm{\&gamma;}={\left[\begin{array}{cc}2{\mathrm{\&gamma;}}_{13}& 2{\mathrm{\&gamma;}}_{23}\end{array}\right]}^T,$ γ ₁₃and γ ₂₃for horizontal shear capacity; G is shearing rigidity matrix;

be respectively the stiffness matrix and the load continuous item that are converted to Reissner-Mindlin model.

4. electromagnetic material stratiform housing electromagnetism bullet coupling Simulation analogy method as claimed in claim 1, it is characterized in that, described step 3 is specially: based on the three dimensional field Remodeling of deriving, utilize the overall situation response of the two-dimentional shell face that Reissner-Mindlin model analysis obtains and the warping function reconstruct field variable through-thickness distribution that Dimension Reduction Analysis obtains:

$U_i=u_i+x_3{\left[\begin{array}{ccc}{C}_{31}& C_{32}& C_{33}\end{array}\right]}^T+S(V_0+{V_1}_{});$

Wherein, U _i, u _ibe respectively three-dimensional housing distortion and two-dimentional housing distortion array; C _ijfor overall rotation tensor; x ₃for the coordinate on the thickness direction along housing reference surface, V ₀, V ₁for zeroth order and the asymptotic correction warpage of single order nodal value, S is shape function;

By constitutive relation reconstruct triaxiality, electric displacement, magnetic induction gesture:

$\left\{\begin{array}{c}\mathrm{\&sigma;}\\ D\\ B\end{array}\right\}=\begin{array}{c}\left[\begin{array}{ccc}C& -e& -q\\ {-e}^T& -d& -a\\ {-q}^T& {-a}^T& -\mathrm{\&mu;}\end{array}\right]\mathrm{\&Gamma;}\end{array};$

Wherein, σ, D, B is respectively load, the triaxiality of electricity/magnetic field lower house, electric displacement, magnetic induction gesture; C, e, q, d, a, μ is respectively the matrix containing elasticity, piezoelectricity, pressure magnetic, dielectric, magnetoelectricity and magnetoconductivity constant; Γ is generalized strain moment matrix.

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