Multiscale design of a rectangular sandwich plate with viscoelastic core and supported at extents by viscoelastic materials

Abstract

This work presents a multiscale model of viscoelastic constrained layer damping treatments for vibrating plates/beams. The approach integrates a finite element (FE) model of macroscale vibrations and a micromechanical model to include effects of microscale structure and properties. The FE model captures the shear deformation of the viscoelastic core, rotary inertial effects of all layers, and viscoelastic boundaries of the plate. Comparison with analytical and FE results validates the proposed FE model. A self-consistent (SC) model makes the micro to macro scale transition to approximate the effective behavior a heterogeneous core. Modal damping resulting from the presence of voids and negative stiffness regions in the core material is modeled. Results show that negative stiffness regions in the viscoelastic core material, even at low volume fractions, yield superior macroscopic damping behavior. The coupled SC and FE models provide a powerful multiscale predictive design tool for sandwich beams and plates.

Appendix: Elements mass and stiffness matrices and other defined matrices

The finite element matrices defined in Eqs. 18, and 20 are defined as follows:

$$ \begin{aligned} \left[{\mathbf{K}}_{e1}\right]=&\iint\limits_{(x,y)_e} \left(\sum^{3}_{i=1} \left([{\mathbf{N}}_{im}]^T[{\mathbf{C}}_{im}] [{\mathbf{N}}_{im}]+[{\mathbf{N}}_{ib}]^T[{\mathbf{C}}_{ib}] [{\mathbf{N}}_{ib}]\right)\right.\\ &\left.+[{\mathbf{N}}_{2s}]^T[{\mathbf{C}}_{2s}][{\mathbf{N}}_{2s}]\right) \hbox{d} x \hbox{d} y,\\ \left[{\bf M}_{e}\right]= & \iint\limits_{(x,y)_{e}} \left(\sum^{3}_{i=1}\rho_{i}h_{i}\left[\left({\mathbf{N}}^{T}_{ui}{\mathbf{N}}_{ui}+{\mathbf{N}}^{T}_{vi}{\mathbf{N}}_{vi}+{\mathbf{N}}^{T}_{w}{\mathbf{N}}_{w}\right)\right.\right.\\&\left.\left.+\frac{h^{2}_{i}}{12}\left({\mathbf{N}}^{T}_{\alpha i} {\mathbf{N}}_{\alpha i}+{\mathbf{N}}^{T}_{\beta i}{\mathbf{N}}_{\beta i}\right)\right]\right)\hbox{d} x\hbox{d} y. \end{aligned} $$

(28)

Writing Eqs. 3, 6, and 7, terms in Eq. 28 yields the following expressions:

$$ \begin{aligned} {[}{\mathbf{N}}_{1m}]=&\left[\begin{array}{c} {\mathbf{N}}_{u1,x}\\ {\mathbf{N}}_{v1,y}\\ {\mathbf{N}}_{u1,y}+{\mathbf{N}}_{v1,x}\end{array}\right],\\ [{\mathbf{N}}_{3m}]=&\left[\begin{array}{c} {\mathbf{N}}_{u3,x}\\ {\mathbf{N}}_{v3,y}\\ {\mathbf{N}}_{u3,y}+{\mathbf{N}}_{v3,x}\end{array}\right],\\ [{\mathbf{N}}_{1b}]=&\left[\begin{array}{c} -{\mathbf{N}}_{w,xx}\\ -{\mathbf{N}}_{w,yy}\\ -2{\mathbf{N}}_{w,xy}\end{array}\right], \end{aligned} $$

(29)

$$ \begin{aligned} {[}{\mathbf{N}}_{2m}]&=\frac{[{\mathbf{N}}_{1m}]+[{\mathbf{N}}_{3m}]}{2}+\frac{h_{3}-h_{1}}{4}[{\mathbf{N}}_{1b}],\\ [{\mathbf{N}}_{2b}]&=\frac{[{\mathbf{N}}_{1m}]-[{\mathbf{N}}_{3m}]}{h_{2}}-\frac{h_{3}+h_{1}}{2h_{2}}[{\mathbf{N}}_{1b}],\\ [{\mathbf{N}}_{2s}]&=\frac{1}{h_{2}}\left[\begin{array}{c} {\mathbf{N}}_{u1}-{\mathbf{N}}_{u3}+h_{0}{\mathbf{N}}_{w,x}\\ {\mathbf{N}}_{v1}-{\mathbf{N}}_{v3}+h_{0}{\mathbf{N}}_{w,y}\end{array}\right],\\ [{\mathbf{N}}_{3b}]&=[{\mathbf{N}}_{1b}],\quad{\mathbf{N}}_{\alpha 1}={\mathbf{N}}_{\alpha 3}={\mathbf{N}}_{w,x},\\ {\mathbf{N}}_{\alpha 2}&=\frac{{\mathbf{N}}_{u1}-{\mathbf{N}}_{u3}}{h_{2}}+\frac{h_{3}+h_{1}}{2h_{2}}{\mathbf{N}}_{w,x},\\ {\mathbf{N}}_{\beta 2}&=\frac{{\mathbf{N}}_{v1}-{\mathbf{N}}_{v3}}{h_{2}}+\frac{h_{3}+h_{1}}{2h_{2}}{\mathbf{N}}_{w,y},\\ {\mathbf{N}}_{u2}&=\frac{{\mathbf{N}}_{u1}+{\mathbf{N}}_{u3}}{2}+\frac{h_{1}-h_{3}}{4}{\mathbf{N}}_{w,x},\\ {\mathbf{N}}_{v2}&=\frac{{\mathbf{N}}_{v1}+{\mathbf{N}}_{v3}}{2}+\frac{h_{1}-h_{3}}{4}{\mathbf{N}}_{w,y},\\ h_{0}&=h_{2}+\frac{h_{1}+h_{3}}{2},\\ {\mathbf{N}}_{\beta 1}&={\mathbf{N}}_{\beta 3}={\mathbf{N}}_{w,y}. \end{aligned} $$

(30)

The other terms in Eqs. 29 and 30 are the classical shape functions employed in Sect. 2.5 (Dhatt et al. (2005)).