A 2D Daubechies finite wavelet domain method for transient wave response analysis in shear deformable laminated composite plates

Abstract

An efficient numerical method is developed for the simulation of dynamic response and the prediction of the wave propagation in composite plate structures. The method is termed finite wavelet domain method and takes advantage of the outstanding properties of compactly supported 2D Daubechies wavelet scaling functions for the spatial interpolation of displacements in a finite domain of a plate structure. The development of the 2D wavelet element, based on the first order shear deformation laminated plate theory is described and equivalent stiffness, mass matrices and force vectors are calculated and synthesized in the wavelet domain. The transient response is predicted using the explicit central difference time integration scheme. Numerical results for the simulation of wave propagation in isotropic, quasi-isotropic and cross-ply laminated plates are presented and demonstrate the high spatial convergence and problem size reduction obtained by the present method.

Appendix

The displacement field through the thickness according to the FSDT takes the form

$$\begin{aligned} \ u(x,y,z,t)= & {} u^0(x,y,t)+\beta _x(x,y,t)\cdot z \end{aligned}$$

(27)

$$\begin{aligned} \ v(x,y,z,t)= & {} v^0(x,y,t)+\beta _y(x,y,t)\cdot z \end{aligned}$$

(28)

$$\begin{aligned} \ w(x,y,z,t)= & {} w^0(x,y,t) \end{aligned}$$

(29)

where $u^0$, $v^0$, $w^0$ are the axial and transverse displacements at the mid-plane of the plate; $\beta _x$, $\beta _y$ are the rotations of the cross-section; and z is the local thickness coordinate. The in-plane strains $\varepsilon _x$, $\varepsilon _y$, $\varepsilon _{xy}$ and the out of plane shear strains $\varepsilon _{yz}$, $\varepsilon _{xz}$ are shown below

$$\begin{aligned} \ \varepsilon _x(x,z,t)= & {} \varepsilon _{x}^{0} + k_x \cdot z \end{aligned}$$

(30)

$$\begin{aligned} \ \varepsilon _y(y,z,t)= & {} \varepsilon _{y}^{0} + k_y \cdot z \end{aligned}$$

(31)

$$\begin{aligned} \ \varepsilon _{xy}(x,y,t)= & {} \varepsilon _{xy}^{0} + k_{xy} \end{aligned}$$

(32)

$$\begin{aligned} \ \varepsilon _{yz}(y,z,t)= & {} \varepsilon _{yz}^{0} \end{aligned}$$

(33)

$$\begin{aligned} \ \varepsilon _{xz}(x,z,t)= & {} \varepsilon _{xz}^{0} \end{aligned}$$

(34)

In the previous Eqs. (30–32), the generalized strains of the laminated plate cross section are defined as

(35)

where

$\varepsilon _{x}^{0}, \varepsilon _{y}^{0}, \varepsilon _{xy}^{0}$ are implying membrane strains, $k_x, k_y, k_{xy}$ are the curvatures and $\varepsilon _{yz}^{0}, \varepsilon _{xz}^{0}$ are the out of plain strains. The comma in the subscript indicates differentiation. $A's$, $B's$ and $D's$ are the extensional, bending and bending-extensional coupling stiffnesses. The principle of virtual work for a two-dimensional solid defined in terms of axial and transverse coordinates can be recast as

$$\begin{aligned} \delta V - \delta W + \delta T = 0 \end{aligned}$$

(36)

where $\delta V$, $\delta W$ and $\delta T$ are the virtual strain energy, the virtual work induced by external applied forces and the virtual kinetic energy respectively. Each of the term in Eq. (36) is given by

$$\begin{aligned} \delta V&= \int _{\varOmega _0} \Bigg \{ \int _{-h/2}^{h/2} \left[ \left( \delta \varepsilon _x+z\delta k_{x}\right) \sigma _{xx} + \left( \delta \varepsilon _{y}+z\delta k_{y}\right) \sigma _{yy} + \right. \nonumber \\&\quad \left. \left( \delta \varepsilon _{xy}+z\delta k_{xy}\right) \sigma _{xy}+ \delta \varepsilon _{xz}\sigma _{xz}+ \delta \varepsilon _{yz}\sigma _{yz} \right] dz \Bigg \} dxdy \nonumber \\&= \int _{\varOmega _0}\underset{\thicksim }{\delta \varepsilon _{L}^{T} }[\mathbf {K}_L]\underset{\thicksim }{ \varepsilon _{L} }dxdy \end{aligned}$$

(37)

$$\begin{aligned} \delta W=&\int _{\varOmega _0}\delta w_0 \left[ \left( q_b+q_t\right) \right] dxdy+\int _{\varGamma _{\sigma }}\int _{-h/2}^{h/2} \left[ \left( \delta u_n + \right. \right. \nonumber \\&\left. \left. z\delta \beta _n\right) \hat{\sigma }_{nn} +\left( \delta u_s +z\delta \beta _s \right) \hat{\sigma }_{ns}+\delta w_0\hat{\sigma }_{nz}\right] dzds \end{aligned}$$

(38)

$$\begin{aligned} \delta T&=\int _{\varOmega _0} \int _{-h/2}^{h/2}\rho \bigg [ \left( \delta {u}_0+z\delta {\beta }_x\right) \left( \ddot{u}_0 +z \ddot{\beta }_x\right) \nonumber \\&+\left( \delta {v}_0+z \delta {\beta }_y\right) \left( \ddot{v}_0 +z\ddot{\beta }_y\right) + \delta {w}_0\ddot{w}_0 \bigg ]dz\; dxdy\nonumber \\&=\int _{\varOmega _0}\underset{\thicksim }{\delta u_{L}^{T} } [\varvec{\rho }_L]\underset{\thicksim }{ \ddot{u}_{L} }dxdy \end{aligned}$$

(39)

where $\varGamma _{\sigma }$ denotes a portion of the boundary $\varGamma$ and

(40)

and

(41)

where $\rho _A$, $\rho _B$, $\rho _D$ are the areal mass, $1\mathrm{st}$ order inertia and the rotational inertia of the laminate respectively. For symmetric laminates $\rho _B =0$.