Influence of First to Second Gradient Coupling Tensors Terms with Surface Effects on the Wave Propagation of 2D Network Materials

Abstract

The influence of surface energy terms on the wave propagation characteristics of network materials is analyzed in this contribution. The asymptotic homogenization technique is extended to account for additional surface properties of network materials made of the periodic repetition of a unit cell consisting of beam type elements. The presence of a thin coating with specific properties give rise to surface effects that are accounted for by a strain gradient behavior at the mesoscopic level of an equivalent continuum. These effects emerge in the asymptotic expansion of the effective stress and hyperstress tensors versus the small scale parameters and the additional small parameters related to surface effects. The role of the coupling tensors between first and second order gradient kinematic terms, obtained by the homogenization, materials is pursued in this contribution, accounting for surface effects arising from the presence of a thin coating on the surface of the structural beam elements of the network. The lattice beams have a viscoelastic behavior described by a Kelvin voigt model and the homogenized second gradient viscoelasticity model reflects both the lattice topology, anisotropy and microstructural features in terms of its geometrical and micromechanical parameters. We formulate the dynamic equilibrium equations and compute the network materials’ wave propagation attributes. We compute the influence of the coupling tensor terms on the wave propagation characteristics as a function of the propagation direction. Considerable differences between second gradient media description with and without the consideration of the coupling energy term contributions are observed for propagating modes along the non-centrosymmetric inner material direction. We assess the effect of coupling energy over a wide range of propagating directions, deriving useful overall conclusions on its role in the wave propagation features of 3D architectured media.

Appendix: Transition from Curvilinear to Cartesian Coordinates

After development of Eq. (19.14) in Cartesian coordinates, one obtains

$$ \mathop {\text{lim}}\limits_{\varepsilon \to 0} P = \int\limits_{\Omega } {\left[ {\frac{1}{g}\sum\limits_{b} {\left[ {\begin{array}{*{20}l} {} \hfill \\ {\left( {\begin{array}{*{20}l} {T_{E}^{1} \left( {L_{1} \delta_{1b} \left( {c_{{\theta_{1} }} \frac{{\partial \dot{V}_{o} }}{\partial x} + s_{{\theta_{1} }} \frac{{\partial \dot{V}_{o} }}{\partial y}} \right) + L_{2} \delta_{2b} \left( {c_{{\theta_{2} }} \frac{{\partial \dot{V}_{o} }}{\partial x} + s_{{\theta_{2} }} \frac{{\partial \dot{V}_{o} }}{\partial y}} \right)} \right)} \hfill \\ { + N_{E}^{1} \left( {L_{1} \delta_{1b} \left( {c_{{\theta_{1} }} \frac{{\partial \dot{U}_{o} }}{\partial x} + s_{{\theta_{1} }} \frac{{\partial \dot{U}_{o} }}{\partial y}} \right) + L_{2} \delta_{2b} \left( {c_{{\theta_{2} }} \frac{{\partial \dot{U}_{o} }}{\partial x} + s_{{\theta_{2} }} \frac{{\partial \dot{U}_{o} }}{\partial y}} \right)} \right)} \hfill \\ \end{array} } \right)} \hfill \\ { + \varepsilon \left( {\begin{array}{*{20}l} {T_{E}^{1} \left( \begin{aligned} \frac{{L_{1}^{2} \delta_{1b}^{2} }}{2}\left( {c_{{\theta_{1} }}^{2} \frac{{\partial^{2} \dot{V}_{o} }}{{\partial x^{2} }} + s_{{\uptheta_{1} }}^{2} \frac{{\partial^{2} \dot{V}_{o} }}{{\partial y^{2} }} + 2s_{{\theta_{1} }} c_{{\theta_{1} }} \frac{{\partial^{2} \dot{V}_{o} }}{\partial x\partial y}} \right) \hfill \\ + \frac{{L_{2}^{2} \delta_{2b}^{2} }}{2}\left( {c_{{\theta_{2} }}^{2} \frac{{\partial^{2} \dot{V}_{o} }}{{\partial x^{2} }} + s_{{\uptheta_{2} }}^{2} \frac{{\partial^{2} \dot{V}_{o} }}{{\partial y^{2} }} + 2s_{{\theta_{2} }} c_{{\theta_{2} }} \frac{{\partial^{2} \dot{V}_{o} }}{\partial x\partial y}} \right) \hfill \\ \end{aligned} \right)} \hfill \\ { + T_{E}^{2} \left( {L_{1} \delta_{1b} \left( {c_{{\theta_{1} }} \frac{{\partial \dot{V}_{o} }}{\partial x} + s_{{\theta_{1} }} \frac{{\partial \dot{V}_{o} }}{\partial y}} \right) + L_{2} \delta_{2b} \left( {c_{{\theta_{2} }} \frac{{\partial \dot{V}_{o} }}{\partial x} + s_{{\theta_{2} }} \frac{{\partial \dot{V}_{o} }}{\partial y}} \right)} \right)} \hfill \\ { + N_{E}^{1} \left( \begin{aligned} \frac{{L_{1}^{2} \delta_{1b}^{2} }}{2}\left( {c_{{\theta_{1} }}^{2} \frac{{\partial^{2} \dot{U}_{o} }}{{\partial x^{2} }} + s_{{\uptheta_{1} }}^{2} \frac{{\partial^{2} \dot{U}_{o} }}{{\partial y^{2} }} + 2s_{{\theta_{1} }} c_{{\theta_{1} }} \frac{{\partial^{2} \dot{U}_{o} }}{\partial x\partial y}} \right) \hfill \\ + \frac{{L_{2}^{2} \delta_{2b}^{2} }}{2}\left( {c_{{\theta_{2} }}^{2} \frac{{\partial^{2} \dot{U}_{o} }}{{\partial x^{2} }} + s_{{\uptheta_{2} }}^{2} \frac{{\partial^{2} \dot{U}_{o} }}{{\partial y^{2} }} + 2s_{{\theta_{2} }} c_{{\theta_{2} }} \frac{{\partial^{2} \dot{U}_{o} }}{\partial x\partial y}} \right) \hfill \\ \end{aligned} \right)} \hfill \\ { + N_{E}^{2} \left( {L_{1} \delta_{1b} \left( {c_{{\theta_{1} }} \frac{{\partial \dot{U}_{o} }}{\partial x} + s_{{\theta_{1} }} \frac{{\partial \dot{U}_{o} }}{\partial y}} \right) + L_{2} \delta_{2b} \left( {c_{{\theta_{2} }} \frac{{\partial \dot{U}_{o} }}{\partial x} + s_{{\theta_{2} }} \frac{{\partial \dot{U}_{o} }}{\partial y}} \right)} \right)} \hfill \\ \end{array} } \right)} \hfill \\ { + \varepsilon^{2} \left( {\begin{array}{*{20}l} {T_{E}^{2} \left( \begin{aligned} \frac{{L_{1}^{2} \delta_{1b}^{2} }}{2}\left( {c_{{\theta_{1} }}^{2} \frac{{\partial^{2} \dot{V}_{o} }}{{\partial x^{2} }} + s_{{\uptheta_{1} }}^{2} \frac{{\partial^{2} \dot{V}_{o} }}{{\partial y^{2} }} + 2s_{{\theta_{1} }} c_{{\theta_{1} }} \frac{{\partial^{2} \dot{V}_{o} }}{\partial x\partial y}} \right) \hfill \\ + \frac{{L_{2}^{2} \delta_{2b}^{2} }}{2}\left( {c_{{\theta_{2} }}^{2} \frac{{\partial^{2} \dot{V}_{o} }}{{\partial x^{2} }} + s_{{\uptheta_{2} }}^{2} \frac{{\partial^{2} \dot{V}_{o} }}{{\partial y^{2} }} + 2s_{{\theta_{2} }} c_{{\theta_{2} }} \frac{{\partial^{2} \dot{V}_{o} }}{\partial x\partial y}} \right) \hfill \\ \end{aligned} \right)} \hfill \\ { + N_{E}^{2} \left( \begin{aligned} \frac{{L_{1}^{2} \delta_{1b}^{2} }}{2}\left( {c_{{\theta_{1} }}^{2} \frac{{\partial^{2} \dot{U}_{o} }}{{\partial x^{2} }} + s_{{\uptheta_{1} }}^{2} \frac{{\partial^{2} \dot{U}_{o} }}{{\partial y^{2} }} + 2s_{{\theta_{1} }} c_{{\theta_{1} }} \frac{{\partial^{2} \dot{U}_{o} }}{\partial x\partial y}} \right) \hfill \\ + \frac{{L_{2}^{2} \delta_{2b}^{2} }}{2}\left( {c_{{\theta_{2} }}^{2} \frac{{\partial^{2} \dot{U}_{o} }}{{\partial x^{2} }} + s_{{\uptheta_{2} }}^{2} \frac{{\partial^{2} \dot{U}_{o} }}{{\partial y^{2} }} + 2s_{{\theta_{2} }} c_{{\theta_{2} }} \frac{{\partial^{2} \dot{U}_{o} }}{\partial x\partial y}} \right) \hfill \\ \end{aligned} \right)} \hfill \\ \end{array} } \right)} \hfill \\ \end{array} } \right]} } \right]} dV $$

(A.1)

$c_{\theta } ,s_{\theta } ,c_{\theta }^{2}$ and $s_{\theta }^{2}$ stand for ${ \cos }\theta ,{ \sin }\theta , {\text{cos}}^{2} \theta$ and ${ \sin }^{2} \theta$, they are the components of the periodicity vectors [43].