General nonlocal Kelvin–Voigt viscoelasticity: application to wave propagation in viscoelastic media

Abstract

In this paper, we develop a new relaxation model that accounts for the nonlocal fields of Kelvin–Voigt viscoelastic materials and assumes different nonlocal-attenuations of the material’s elastic and viscous properties. The new relaxation model is then implemented to study the influence of the nonlocal, non-neighbor interactions on the wave propagation in viscoelastic media, while elucidating how the contrast between longitudinal and transverse nonlocal fields contributes to the dispersion of the propagating waves. The numerical results reveal two mechanisms of viscoelastic wave damping, as in addition to the viscosity—which is an explicit wave damping—viscoelastic waves are also damped implicitly by the nonlocal effect. We therefore foresee that the new relaxation model will be used for the design of non-reciprocal and non-Hermitian material systems.

Appendix

Equation (10) can be rewritten as

$$t_{11} (x,t) = T_{ - \lambda } + T_{ + \lambda } + T_{ - \mu } + T_{ + \mu } + T_{ - \lambda v} + T_{ + \lambda v} + T_{ - \mu v} + T_{ + \mu v},$$

(A1)

where

$$T_{ - \lambda } = \frac{{\overline{\lambda }}}{{2e_{\lambda } a}}\int\limits_{ - \infty }^{x} {\exp \left( {\frac{\xi - x}{{e_{\lambda } a}}} \right)\varepsilon_{11} (\xi ,t)\,{\text{d}}\xi },$$

(A2)

$$\begin{gathered} T_{ + \lambda } = \frac{{\overline{\lambda }}}{{2e_{\lambda } a}}\int\limits_{x}^{\infty } {\exp \left( {\frac{x - \xi }{{e_{\lambda } a}}} \right)\varepsilon_{11} (\xi ,t)\,{\text{d}}\xi }, \hfill \\ \, \hfill \\ \end{gathered}$$

(A3)

$$T_{ - \mu } = \frac{{\overline{\mu }}}{{e_{\mu } a}}\int\limits_{ - \infty }^{x} {\exp \left( {\frac{\xi - x}{{e_{\mu } a}}} \right)\varepsilon_{11} (\xi ,t)\,{\text{d}}\xi },$$

(A4)

$$T_{ + \mu } = \frac{{\overline{\mu }}}{{e_{\mu } a}}\int\limits_{x}^{\infty } {\exp \left( {\frac{x - \xi }{{e_{\mu } a}}} \right)\varepsilon_{11} (\xi ,t)\,{\text{d}}\xi },$$

(A5)

$$T_{ - \lambda v} = \frac{{\lambda^{\prime}}}{{2e_{\lambda } a}}\int\limits_{ - \infty }^{x} {\exp \left( {\frac{\xi - x}{{e_{\lambda } a}}} \right)\dot{\varepsilon }_{11} (\xi ,t)\,{\text{d}}\xi },$$

(A6)

$$T_{ + \lambda v} = \frac{{\lambda^{\prime}}}{{2e_{\lambda } a}}\int\limits_{x}^{\infty } {\exp \left( {\frac{x - \xi }{{e_{\lambda } a}}} \right)\dot{\varepsilon }_{11} (\xi ,t)\,{\text{d}}\xi },$$

(A7)

$$T_{ - \mu v} = \frac{{\mu^{\prime}}}{{e_{\mu } a}}\int\limits_{ - \infty }^{x} {\exp \left( {\frac{\xi - x}{{e_{\mu } a}}} \right)\dot{\varepsilon }_{11} (\xi ,t)\,{\text{d}}\xi },$$

(A8)

$$T_{ + \mu v} = \frac{{\mu^{\prime}}}{{e_{\mu } a}}\int\limits_{x}^{\infty } {\exp \left( {\frac{x - \xi }{{e_{\mu } a}}} \right)\dot{\varepsilon }_{11} (\xi ,t)\,{\text{d}}\xi }.$$

(A9)

Taking the first, second, third and fourth derivatives of Eq. (A1) with respect to x, we have

$$\frac{{\partial t_{11} }}{\partial x} = \frac{1}{{e_{\lambda } a}}(T_{ + \lambda } - T_{ - \lambda } ) + \frac{1}{{e_{\mu } a}}(T_{ + \mu } - T_{ - \mu } ) + \frac{1}{{e_{\lambda } a}}(T_{ + \lambda v} - T_{ - \lambda v} ) + \frac{1}{{e_{\mu } a}}(T_{ + \mu v} - T_{ - \mu v} ),$$

(A10)

$$\begin{aligned} \frac{{\partial^{2} t_{11} }}{{\partial x^{2} }} & = - \left( {\frac{{\overline{\lambda }}}{{e_{\lambda }^{2} a^{2} }} + \frac{{2\overline{\mu }}}{{e_{\mu }^{2} a^{2} }}} \right)\varepsilon_{11} (x,t) + \frac{1}{{e_{\lambda }^{2} a^{2} }}(T_{ + \lambda } + T_{ - \lambda } ) + \frac{1}{{e_{\mu }^{2} a^{2} }}(T_{ + \mu } + T_{ - \mu } ) \\ & \quad - \left( {\frac{{\lambda^{\prime}}}{{e_{\lambda }^{2} a^{2} }} + \frac{{2\mu^{\prime}}}{{e_{\mu }^{2} a^{2} }}} \right)\dot{\varepsilon }_{11} (x,t) + \frac{1}{{e_{\lambda }^{2} a^{2} }}(T_{ + \lambda v} + T_{ - \lambda v} ) + \frac{1}{{e_{\mu }^{2} a^{2} }}(T_{ + \mu v} + T_{ - \mu v} ), \\ \end{aligned}$$

(A11)

$$\begin{aligned} \frac{{\partial^{3} t_{11} }}{{\partial x^{3} }} & = - \left( {\frac{{\overline{\lambda }}}{{e_{\lambda }^{2} a^{2} }} + \frac{{2\overline{\mu }}}{{e_{\mu }^{2} a^{2} }}} \right)\frac{{\partial \varepsilon_{11} }}{\partial x} + \frac{1}{{e_{\lambda }^{3} a^{3} }}(T_{ + \lambda } - T_{ - \lambda } ) + \frac{1}{{e_{\mu }^{3} a^{3} }}(T_{ + \mu } - T_{ - \mu } ) \\ & \quad - \left( {\frac{{\lambda^{\prime}}}{{e_{\lambda }^{2} a^{2} }} + \frac{{2\mu^{\prime}}}{{e_{\mu }^{2} a^{2} }}} \right)\frac{{\partial \dot{\varepsilon }_{11} }}{\partial x} + \frac{1}{{e_{\lambda }^{3} a^{3} }}(T_{ + \lambda v} - T_{ - \lambda v} ) + \frac{1}{{e_{\mu }^{3} a^{3} }}(T_{ + \mu v} - T_{ - \mu v} ), \\ \end{aligned}$$

(A12)

$$\begin{aligned} \frac{{\partial^{4} t_{11} }}{{\partial x^{4} }} & = - \left( {\frac{{\overline{\lambda }}}{{e_{\lambda }^{2} a^{2} }} + \frac{{2\overline{\mu }}}{{e_{\mu }^{2} a^{2} }}} \right)\frac{{\partial^{2} \varepsilon_{11} }}{{\partial x^{2} }} - \left( {\frac{{\overline{\lambda }}}{{e_{\lambda }^{4} a^{4} }} + \frac{{2\overline{\mu }}}{{e_{\mu }^{4} a^{4} }}} \right)\varepsilon_{11} + \frac{1}{{e_{\lambda }^{4} a^{4} }}(T_{ + \lambda } + T_{ - \lambda } ) + \frac{1}{{e_{\mu }^{4} a^{4} }}(T_{ + \mu } + T_{ - \mu } ) \\ & \quad - \left( {\frac{{\lambda^{\prime}}}{{e_{\lambda }^{2} a^{2} }} + \frac{{2\mu^{\prime}}}{{e_{\mu }^{2} a^{2} }}} \right)\frac{{\partial^{2} \dot{\varepsilon }_{11} }}{{\partial x^{2} }} - \left( {\frac{{\lambda^{\prime}}}{{e_{\lambda }^{4} a^{4} }} + \frac{{2\mu^{\prime}}}{{e_{\mu }^{4} a^{4} }}} \right)\dot{\varepsilon }_{11} + \frac{1}{{e_{\lambda }^{4} a^{4} }}(T_{ + \lambda v} + T_{ - \lambda v} ) + \frac{1}{{e_{\mu }^{4} a^{4} }}(T_{ + \mu v} + T_{ - \mu v} ). \\ \end{aligned}$$

(A13)

Using Eqs. (A1), (A11) and (A13) and after making some simplifications, Eq. (19) is obtained. In a similar way, Eqs. (20) and (21) are derived.