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Analysis of smart damping of laminated composite beams using mesh free method

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Abstract

This paper is concerned with the development of mesh free model for the performance analysis of active constrained layered damping (ACLD) treatments on smart laminated composite beams. The overall structure is composed of a substrate laminated composite beam integrated with a viscoelastic layer and a piezoelectric layer attached partially or fully at the top surface of the substrate beam. The piezoelectric layer acts as the active constraining layer of the smart beam and the viscoelastic layer acts as the constrained layer. A layer wise displacement theory has been used to derive the models. Both symmetric cross-ply and antisymmetric angle-ply laminated beams are considered for the numerical analysis. It is observed that ACLD treatment significantly improves the active damping properties of the substrate beam. The numerical results also reveal that the triangular ACLD treatment is more effective than the rectangular ACLD treatment of same thickness and volume for active damping of smart composite beams.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, 721302, India

    S. R. Sahoo & M. C. Ray

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  1. S. R. Sahoo
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  2. M. C. Ray
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Correspondence to M. C. Ray.

Appendices

Appendix 1

In Eq. (4), the matrices [Z 1 ], [Z 2 ] and [Z 3 ] are given by:

$$\left[ {\varvec{z}_{{\mathbf{1}}} } \right] = \left[ {\begin{array}{*{20}c} \varvec{z} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}} & {\mathbf{0}} & {\mathbf{0}} & \varvec{z} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right],$$
$$\left[ {\varvec{z}_{{\mathbf{2}}} } \right] = \left[ {\begin{array}{*{20}c} {\frac{\varvec{h}}{{\mathbf{2}}}} & {\varvec{z} - \frac{\varvec{h}}{{\mathbf{2}}}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}} & {\mathbf{0}} & {\frac{\varvec{h}}{{\mathbf{2}}}} & {\varvec{z} - \frac{\varvec{h}}{{\mathbf{2}}}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right],$$
$$\left[ {\varvec{z}_{{\mathbf{3}}} } \right] = \left[ {\begin{array}{*{20}c} {\frac{\varvec{h}}{{\mathbf{2}}}} & {\varvec{h}_{\varvec{v}} } & {\varvec{z} - \frac{\varvec{h}}{{\mathbf{2}}} - \varvec{h}_{\varvec{v}} } & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}} & {\frac{\varvec{h}}{{\mathbf{2}}}} & {\varvec{h}_{\varvec{v}} } & {\varvec{z} - \frac{\varvec{h}}{{\mathbf{2}}} - \varvec{h}_{\varvec{v}} } & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right]$$

Appendix 2

In Eq. (22), the matrices \([\varvec{L}_{{\varvec{tb}}} ]\) and \([\varvec{L}_{{\varvec{rb}}} ]\) are given by:

$$[\varvec{L}_{{\varvec{tb}}} ] = \left[ {\begin{array}{*{20}l} {[\varvec{L}_{{\varvec{tn}{\mathbf{1}}}} ]} \hfill & {[\varvec{L}_{{\varvec{tb}{\mathbf{2}}}} ]} \hfill & {[\varvec{L}_{{\varvec{tb}{\mathbf{3}}}} ]} \hfill & \ldots \hfill & {[\varvec{L}_{{\varvec{tbn}}} ]} \hfill \\ \end{array} } \right],$$
$$[\varvec{L}_{{\varvec{rb}}} ] = \left[ {\begin{array}{*{20}l} {[\varvec{L}_{{\varvec{rn}{\mathbf{1}}}} ]} \hfill & {[\varvec{L}_{{\varvec{rb}{\mathbf{2}}}} ]} \hfill & {[\varvec{L}_{{\varvec{rb}{\mathbf{3}}}} ]} \hfill & \ldots \hfill & {[\varvec{L}_{{\varvec{rbn}}} ]} \hfill \\ \end{array} } \right],\;[\varvec{L}_{{\varvec{tbi}}} ] = \left[ {\begin{array}{*{20}c} {\frac{{\mathbf{\partial }}}{{{\mathbf{\partial }}\varvec{x}}}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\frac{{\mathbf{\partial }}}{{{\mathbf{\partial }}\varvec{z}}}} \\ \end{array} } \right],$$
$$[\varvec{L}_{{\varvec{rbi}}} ] = \left[ {\begin{array}{*{20}c} {\frac{{\mathbf{\partial }}}{{{\mathbf{\partial }}\varvec{x}}}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\frac{{\mathbf{\partial }}}{{{\mathbf{\partial }}\varvec{x}}}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\frac{{\mathbf{\partial }}}{{{\mathbf{\partial }}\varvec{x}}}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{1}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{1}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\frac{{\mathbf{\partial }}}{{{\mathbf{\partial }}\varvec{z}}}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\frac{{\mathbf{\partial }}}{{{\mathbf{\partial }}\varvec{z}}}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\frac{{\mathbf{\partial }}}{{{\mathbf{\partial }}\varvec{z}}}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}} \\ \end{array} } \right],\quad \varvec{i} = {\mathbf{1}},{\mathbf{2}},{\mathbf{3}},{\mathbf{ \ldots }},\varvec{n},$$

For a rectangular patch of ACLD treatment, the various sub-matrices appearing in Eqs. (24) and (25) are given by:

$$[\varvec{K}_{{\varvec{tt}}} ] = \int\limits_{{\mathbf{0}}}^{\varvec{L}} {[\varvec{B}_{\varvec{t}} ]^{\varvec{T}} [\varvec{D}_{\varvec{t}}^{\varvec{b}} ][\varvec{B}_{\varvec{t}} ]} \varvec{dx} + \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {[\varvec{B}_{\varvec{t}} ]^{\varvec{T}} [\varvec{D}_{\varvec{t}}^{\varvec{v}} ][\varvec{B}_{\varvec{t}} ]} \varvec{dx} + \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {[\varvec{B}_{\varvec{t}} ]^{\varvec{T}} [\varvec{D}_{\varvec{t}}^{\varvec{p}} ][\varvec{B}_{\varvec{t}} ]} \varvec{dx} + \int\limits_{{\mathbf{0}}}^{\varvec{L}} {[\varvec{N}_{\varvec{t}} ]^{\varvec{T}} [\varvec{\alpha}_{\varvec{t}} ][\varvec{N}_{\varvec{t}} ]} \varvec{dx},$$
$$[\varvec{K}_{{\varvec{tr}}} ] = \int\limits_{{\mathbf{0}}}^{\varvec{L}} {[\varvec{B}_{\varvec{t}} ]^{\varvec{T}} [\varvec{D}_{{\varvec{tr}}}^{\varvec{b}} ][\varvec{B}_{\varvec{r}} ]} \varvec{dx} + \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {[\varvec{B}_{\varvec{t}} ]^{\varvec{T}} [\varvec{D}_{{\varvec{tr}}}^{\varvec{v}} ][\varvec{B}_{\varvec{r}} ]} \varvec{dx} + \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {[\varvec{B}_{\varvec{t}} ]^{\varvec{T}} [\varvec{D}_{{\varvec{tr}}}^{\varvec{p}} ][\varvec{B}_{\varvec{r}} ]} \varvec{dx} + \int\limits_{{\mathbf{0}}}^{\varvec{L}} {[\varvec{N}_{\varvec{t}} ]^{\varvec{T}} [\varvec{\alpha}_{{\varvec{tr}}} ][\varvec{N}_{\varvec{r}} ]} \varvec{dx},$$
$$[\varvec{K}_{{\varvec{rr}}} ] = \int\limits_{{\mathbf{0}}}^{\varvec{L}} {[\varvec{B}_{\varvec{r}} ]^{\varvec{T}} [\varvec{D}_{{\varvec{rr}}}^{\varvec{b}} ][\varvec{B}_{\varvec{r}} ]} \varvec{dx} + \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {[\varvec{B}_{\varvec{r}} ]^{\varvec{T}} [\varvec{D}_{{\varvec{rr}}}^{\varvec{v}} ][\varvec{B}_{\varvec{r}} ]} \varvec{dx} + \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {[\varvec{B}_{\varvec{r}} ]^{\varvec{T}} [\varvec{D}_{{\varvec{rr}}}^{\varvec{p}} ][\varvec{B}_{\varvec{r}} ]} \varvec{dx} + \int\limits_{{\mathbf{0}}}^{\varvec{L}} {[\varvec{N}_{\varvec{r}} ]^{\varvec{T}} [\varvec{\alpha}_{\varvec{r}} ][\varvec{N}_{\varvec{r}} ]} \varvec{dx},$$
$$\{ \varvec{F}\} = \int\limits_{{\mathbf{0}}}^{\varvec{L}} {\overline{\varvec{p}} \varvec{N}_{\varvec{t}}^{\varvec{T}} [\begin{array}{*{20}c} {\mathbf{0}} & {\mathbf{1}} \\ \end{array} ]^{\varvec{T}} } \varvec{dz},\quad \{ \varvec{F}_{{\varvec{tp}}} \} = \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {\varvec{B}_{\varvec{t}} \varvec{D}_{{_{{\varvec{tp}}} }}^{\varvec{p}} } \varvec{dz},\quad \{ \varvec{F}_{{\varvec{rp}}} \} = \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {\varvec{B}_{\varvec{r}} \varvec{D}_{{_{{\varvec{rp}}} }}^{\varvec{p}} } \varvec{dz},$$
$$[\varvec{M}] = \sum\limits_{{\varvec{k} = {\mathbf{1}}}}^{{\varvec{k} = \varvec{N}}} {\int\limits_{{\mathbf{0}}}^{\varvec{L}} {\int\limits_{{\varvec{h}_{\varvec{k}} }}^{{\varvec{h}_{{\varvec{k} + {\mathbf{1}}}} }} {[\varvec{N}_{\varvec{t}} ]^{\varvec{T}} [\varvec{\rho}^{\varvec{k}} ]} [\varvec{N}_{\varvec{t}} ]} \varvec{dzdx}} + \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {[\varvec{N}_{\varvec{t}} ]^{\varvec{T}} [\varvec{\rho}^{{\varvec{N} + {\mathbf{1}}}} \varvec{h}_{\varvec{v}} ]\varvec{b}(\varvec{x})[\varvec{N}_{\varvec{t}} ]} \varvec{dx} + \int\limits_{0}^{{\varvec{L}_{\varvec{a}} }} {[\varvec{N}_{\varvec{t}} ]^{\varvec{T}} [\varvec{\rho}^{{\varvec{N} + {\mathbf{1}}}} \varvec{h}_{\varvec{p}} ]\varvec{b}(\varvec{x})[\varvec{N}_{\varvec{t}} ]} \varvec{dx}$$

In case of triangular patch, the above mentioned matrices are to be augmented as follows:

$$[\varvec{K}_{{\varvec{tt}}} ] = \int\limits_{{\mathbf{0}}}^{\varvec{L}} {[\varvec{B}_{\varvec{t}} ]^{\varvec{T}} [\varvec{D}_{\varvec{t}}^{\varvec{b}} ][\varvec{B}_{\varvec{t}} ]} \varvec{dx} + \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {[\varvec{B}_{\varvec{t}} ]^{\varvec{T}} [\varvec{D}_{\varvec{t}}^{\varvec{v}} ][\varvec{B}_{\varvec{t}} ]\varvec{b}(\varvec{x})} \varvec{dx} + \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {[\varvec{B}_{\varvec{t}} ]^{\varvec{T}} [\varvec{D}_{\varvec{t}}^{\varvec{p}} ][\varvec{B}_{\varvec{t}} ]} \varvec{b}(\varvec{x})\varvec{dx} + \int\limits_{{\mathbf{0}}}^{\varvec{L}} {[\varvec{N}_{\varvec{t}} ]^{\varvec{T}} [\varvec{\alpha}_{\varvec{t}} ][\varvec{N}_{\varvec{t}} ]} \varvec{dx},$$
$$[\varvec{K}_{{\varvec{tr}}} ] = \int\limits_{{\mathbf{0}}}^{\varvec{L}} {[\varvec{B}_{\varvec{t}} ]^{\varvec{T}} [\varvec{D}_{{\varvec{tr}}}^{\varvec{b}} ][\varvec{B}_{\varvec{r}} ]} \varvec{dx} + \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {[\varvec{B}_{\varvec{t}} ]^{\varvec{T}} [\varvec{D}_{{\varvec{tr}}}^{\varvec{v}} ][\varvec{B}_{\varvec{r}} ]} \varvec{b}(\varvec{x})\varvec{dx} + \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {[\varvec{B}_{\varvec{t}} ]^{\varvec{T}} [\varvec{D}_{{\varvec{tr}}}^{\varvec{p}} ][\varvec{B}_{\varvec{r}} ]} \varvec{b}(\varvec{x})\varvec{dx} + \int\limits_{{\mathbf{0}}}^{\varvec{L}} {[\varvec{N}_{\varvec{t}} ]^{\varvec{T}} [\varvec{\alpha}_{{\varvec{tr}}} ][\varvec{N}_{\varvec{r}} ]} \varvec{dx},$$
$$[\varvec{K}_{{\varvec{rr}}} ] = \int\limits_{{\mathbf{0}}}^{\varvec{L}} {[\varvec{B}_{\varvec{r}} ]^{\varvec{T}} [\varvec{D}_{{\varvec{rr}}}^{\varvec{b}} ][\varvec{B}_{\varvec{r}} ]} \varvec{dx} + \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {[\varvec{B}_{\varvec{r}} ]^{\varvec{T}} [\varvec{D}_{{\varvec{rr}}}^{\varvec{v}} ][\varvec{B}_{\varvec{r}} ]} \varvec{b}(\varvec{x})\varvec{dx} + \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {[\varvec{B}_{\varvec{r}} ]^{\varvec{T}} [\varvec{D}_{{\varvec{rr}}}^{\varvec{p}} ][\varvec{B}_{\varvec{r}} ]} \varvec{b}(\varvec{x})\varvec{dx} + \int\limits_{{\mathbf{0}}}^{\varvec{L}} {[\varvec{N}_{\varvec{r}} ]^{\varvec{T}} [\varvec{\alpha}_{\varvec{r}} ][\varvec{N}_{\varvec{r}} ]} \varvec{dx},$$
$$\{ \varvec{F}\} = \int\limits_{{\mathbf{0}}}^{\varvec{L}} {\overline{\varvec{p}} \varvec{N}_{\varvec{t}}^{\varvec{T}} [\begin{array}{*{20}c} {\mathbf{0}} & {\mathbf{1}} \\ \end{array} ]^{\varvec{T}} } \varvec{dz},\quad \{ \varvec{F}_{{\varvec{tp}}} \} = \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {\varvec{B}_{\varvec{t}} \varvec{D}_{{_{{\varvec{tp}}} }}^{\varvec{p}} \varvec{b}(\varvec{x})} \varvec{dz},\quad \{ \varvec{F}_{{\varvec{rp}}} \} = \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {\varvec{B}_{\varvec{r}} \varvec{D}_{{_{{\varvec{rp}}} }}^{\varvec{p}} } \varvec{b}(\varvec{x})\varvec{dz},$$
$$[\varvec{M}] = \sum\limits_{{\varvec{k} = {\mathbf{1}}}}^{{\varvec{k} = \varvec{N}}} {\int\limits_{{\mathbf{0}}}^{\varvec{L}} {\int\limits_{{\varvec{h}_{\varvec{k}} }}^{{\varvec{h}_{{\varvec{k} + {\mathbf{1}}}} }} {[\varvec{N}_{\varvec{t}} ]^{\varvec{T}} [\varvec{\rho}^{\varvec{k}} ]} [\varvec{N}_{\varvec{t}} ]} \varvec{dzdx}} + \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {[\varvec{N}_{\varvec{t}} ]^{\varvec{T}} [\varvec{\rho}^{{\varvec{N} + {\mathbf{1}}}} \varvec{h}_{\varvec{v}} ]\varvec{b}(\varvec{x})[\varvec{N}_{\varvec{t}} ]} \varvec{dx} + \int\limits_{{\mathbf{0}}}^{{\varvec{L}_{\varvec{a}} }} {[\varvec{N}_{\varvec{t}} ]^{\varvec{T}} [\varvec{\rho}^{{\varvec{N} + {\mathbf{1}}}} \varvec{h}_{\varvec{p}} ]\varvec{b}(\varvec{x})[\varvec{N}_{\varvec{t}} ]} \varvec{dx}$$

where \(\varvec{b}(\varvec{x}) = \left( {{\mathbf{1}} - \frac{\varvec{x}}{{\varvec{L}_{\varvec{a}} }}} \right)\), \(\varvec{L}_{\varvec{a}}\) is the length of the patch. Various rigidity matrices and the rigidity vectors for electro-elastic coupling appearing in the above equations are given by:

$$[\varvec{D}_{\varvec{t}}^{\varvec{b}} ] = \sum\limits_{{\varvec{k} = {\mathbf{1}}}}^{\varvec{N}} {\int\limits_{{\varvec{h}_{\varvec{k}} }}^{{\varvec{h}_{{\varvec{k} + {\mathbf{1}}}} }} {[\varvec{C}^{\varvec{k}} ]} } \varvec{dz},\quad [\varvec{D}_{\varvec{t}}^{\varvec{v}} ] = \int\limits_{{\varvec{h}_{{_{{\varvec{N} + {\mathbf{1}}}} }} }}^{{\varvec{h}_{{\varvec{N} + {\mathbf{2}}}} }} {[\varvec{C}^{{\varvec{N} + {\mathbf{1}}}} ]} \varvec{dz},\quad [\varvec{D}_{\varvec{t}}^{\varvec{p}} ] = \int\limits_{{\varvec{h}_{{_{{\varvec{N} + {\mathbf{2}}}} }} }}^{{\varvec{h}_{{\varvec{N} + {\mathbf{3}}}} }} {[\varvec{C}^{{\varvec{N} + {\mathbf{2}}}} ]} \varvec{dz},\quad [\varvec{D}_{{\varvec{tr}}}^{\varvec{b}} ] = \sum\limits_{{\varvec{k} = {\mathbf{1}}}}^{\varvec{N}} {\int\limits_{{\varvec{h}_{\varvec{k}} }}^{{\varvec{h}_{{\varvec{k} + {\mathbf{1}}}} }} {[\bar{\varvec{C}}^{\varvec{k}} ]} } [\varvec{Z}_{{\mathbf{1}}} ]\varvec{dz},\quad [\varvec{D}_{{\varvec{tr}}}^{\varvec{v}} ] = \int\limits_{{\varvec{h}_{{\varvec{N} + {\mathbf{1}}}} }}^{{\varvec{h}_{{\varvec{N} + {\mathbf{2}}}} }} {[\bar{\varvec{C}}^{{\varvec{N} + {\mathbf{1}}}} ]} [\varvec{Z}_{{\mathbf{2}}} ]\varvec{dz},\quad [\varvec{D}_{{\varvec{tr}}}^{\varvec{p}} ] = \int\limits_{{\varvec{h}_{{\varvec{N} + {\mathbf{2}}}} }}^{{\varvec{h}_{{\varvec{N} + {\mathbf{3}}}} }} {[\bar{\varvec{C}}^{{\varvec{N} + {\mathbf{2}}}} ]} [\varvec{Z}_{{\mathbf{3}}} ]\varvec{dz},\quad [\varvec{D}_{{\varvec{rr}}}^{\varvec{b}} ] = \sum\limits_{{\varvec{k} = {\mathbf{1}}}}^{\varvec{N}} {\int\limits_{{\varvec{h}_{\varvec{k}} }}^{{\varvec{h}_{{\varvec{k} + {\mathbf{1}}}} }} {[\varvec{Z}_{{\mathbf{1}}} ]^{\varvec{T}} [\varvec{C}^{\varvec{k}} ]} } [\varvec{Z}_{{\mathbf{1}}} ]\varvec{dz},\quad [\varvec{D}_{{\varvec{rr}}}^{\varvec{v}} ] = \int\limits_{{\varvec{h}_{{\varvec{N} + {\mathbf{1}}}} }}^{{\varvec{h}_{{\varvec{N} + {\mathbf{2}}}} }} {[\varvec{Z}_{{\mathbf{2}}} ]^{\varvec{T}} [\varvec{C}^{{\varvec{N} + {\mathbf{1}}}} ]} [\varvec{Z}_{{\mathbf{2}}} ]\varvec{dz},\quad [\varvec{D}_{{\varvec{rr}}}^{\varvec{p}} ] = \int\limits_{{\varvec{h}_{{\varvec{N} + {\mathbf{2}}}} }}^{{\varvec{h}_{{\varvec{N} + {\mathbf{3}}}} }} {[\varvec{Z}_{{\mathbf{3}}} ]^{\varvec{T}} [\varvec{C}^{{\varvec{N} + {\mathbf{2}}}} ]} [\varvec{Z}_{{\mathbf{3}}} ]\varvec{dz},\quad [\varvec{D}_{{\varvec{tp}}}^{\varvec{p}} ] = \int\limits_{{\varvec{h}_{{\varvec{N} + {\mathbf{2}}}} }}^{{\varvec{h}_{{\varvec{N} + {\mathbf{3}}}} }} { - \{ \varvec{e}\} /\varvec{h}_{\varvec{p}} } \varvec{dz},\quad [\varvec{D}_{{\varvec{rp}}}^{\varvec{p}} ] = \int\limits_{{\varvec{h}_{{\varvec{N} + {\mathbf{2}}}} }}^{{\varvec{h}_{{\varvec{N} + {\mathbf{3}}}} }} { - [\varvec{Z}_{{\mathbf{3}}} ]^{{\mathbf{T}}} \{ \varvec{e}\} /\varvec{h}_{\varvec{p}} } \varvec{dz}$$

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Sahoo, S.R., Ray, M.C. Analysis of smart damping of laminated composite beams using mesh free method. Int J Mech Mater Des 14, 359–374 (2018). https://doi.org/10.1007/s10999-017-9379-0

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