壓電力學

10-06

壓電力學

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學習筆記，陸續更新……

NONLINEAR ELECTROELASTICITY FOR STRONG FIELDS

(1)the nonlinear theory of electroelasticty for large deformations

(2)strong electric fields

1.DEFORMATION AND MOTION OF A CONTINUUM

(1)reference or material coordinates:

$X=X_KI_K$

(2)present or spatial coordinates:

$y=y_ki_k$

(3)kronecker delta:

$I_kI_L=delta_{KL}$ $i_ki_l=delta_{kl}$

SPECIAL NOTE:

The two coordinate systems are chosen to be coincident, i.e.,

$o=O,i_1=I_1, i_2=I_2, i_3=I_3$

(4)A line element $dX$ at $t_0$ deforms into the following line element at $t$ :

$dy_i|_{t_{fixed}}=y_{i,K}dX_K$

deformation gradient tensor (two-point tensor): $y_{k,K}$

Jacobian determinant of the deformation: $det(y_{k,K})$

$det(y_{k,K})=det(y_{i，1}，y_{j，2}，y_{k，1})$

$=[y_{i，1}，y_{j，2}，y_{k，1}] =varepsilon_{ijk}y_{i，1}y_{j，2}y_{k，1}$

$det(y_{k,K})=det(y_{1，K}，y_{2，L}，y_{3，M})^T$

$=[y_{1，K}，y_{2，L}，y_{3，M}] =varepsilon_{KLM} y_{1，K}y_{2，L}y_{3，M}$

$det(y_{k,K})=frac{1}{6}varepsilon_{klm}varepsilon_{KLM}y_{k,K}y_{l,L}y_{m,M}$

(5)The length of a material line element before and after deformation is given by

$(dS)^2=dX_KdX_K =delta_{KL}dX_KdX_L$

$(ds)^2=dy_idy_i =y_{iK}dX_Ky_{iL}dX_L =C_{KL}dX_KdX_L$

deformation tensor: $C_{KL}=C_{LK}$

(6)At the same material point consider two non-collinear material line elements $dX^{(1)}$ and $dX^{(2)}$ which deform into $dy^{(1)}$ and $dy^{(2)}$ .

$N_LdA=dA_L=varepsilon_{LMN}dX^{(1)}_MdX^{(2)}_N$

$n_ida=da_i=varepsilon_{ijk}dy^{(1)}_jdy^{(2)}_k$

They are related by $da_i=JX_{L,i}dA_L$ .

(7)At the same material point consider three non-coplanar material line elements $dX^{(1)},dX^{(2)}$ and $dX^{(3)}$ which deform into $dy^{(1)},dy^{(2)}$ and $dy^{(3)}$ .

$dv=(dy^{(1)} imes dy^{(2)}) cdot dy^{(3)} =J (dX^{(1)} imes dX^{(2)})cdot dX^{(3)}$

(8)The deformation rate tensor $d_{ij}$ and the spin tensor $omega_{ij}$ are introduced by decomposing the velocity gradient into symmetric and anti-symmetric parts.

$partial_j v_i =v_{ij}=d_{ij}+omega_{ij}$

2.GLOBAL BALANCE LAWS

(1)When a dielectric is placed in an electric field, the electric charges in its molecules redistribute themselves microscopically, resulting in a macroscopic polarization.

$P=lim_{ riangle V ightarrow 0}{frac{sum_{ riangle V}P_{Micro}}{ riangle V}}$

(2)Piezoelectric Effects：Whether a material is piezoelectric depends on its microscopic charge distribution.

Experiments show that in certain materials polarization can also be induced by mechanical loads.This is called the direct piezoelectric effect.

When a voltage is applied to a material possessing the direct piezoelectric effect, the material deforms. This is called the converse piezoelectric effect.

(3) Electric Body Force, Couple and Power

When a mechanically deformable and electrically polarizable material is subjected to an electric field, a differential element of the material experiences body force and couple due to the electric field.

$F_j^E= ho_e E_j+E_{j,i}P_i$