Wave propagation in viscoelastic composite materials with long-memory effects

Abstract

The homogenization of viscoelastic composite materials gives rise to long-memory effects, reflected by the presence in the homogenized viscoelastic constitutive law of a delayed stress response expressed as a convolution integral involving the past material strain history. Computations reveal that the long-memory has the ability to delay or even absorb waves depending on the fraction of reinforcement and relative properties of the building phases with the composite. An approximation by a one-term Prony series of the computed spectrum of retardation times is shown to provide an accurate representation of the continuous spectrum of retardation times when simulating creep experiments. The long-memory effects entail shifts of the frequency band diagrams for both the natural and damped frequencies due to the presence of damping. Decreasing the effective retardation time of the composite leads to a faster attenuation of propagating longitudinal and shear waves.

Appendices

Appendix A: Longitudinal wave propagation analysis with long-memory effects

In order to analyze wave propagation in 1D composites to assess the impact of the long-memory, we consider the general constitutive law in the case of a tensile load given in component format by:

$$\begin{aligned} \sigma _{11}^{\hom } (y,t)=C_{11}^{\hom } \varepsilon (t)+\eta _{11}^{\hom } \frac{\partial }{\partial t}\varepsilon (t)+\int _0^t {C_{11} (t-s)} \frac{\partial \varepsilon }{\partial t}(s)\mathrm{d}s \end{aligned}$$

(A.1)

The dynamical equations of motion for the homogenized 1D viscoelastic continuum writes in terms of the Cauchy stress components, adopting the Cartesian in-plane coordinates x, y:

$$\begin{aligned} \frac{{\partial \sigma _{{xx}} }}{{\partial x}} = \rho ^{*}\ddot{u} \Rightarrow \frac{{\partial \left( {C_{{11}}^{{\hom }} \varepsilon (t) + \eta _{{11}}^{{\hom }} \frac{\partial }{{\partial t}}\varepsilon (t) + \int _{0}^{t} {C_{{11}} (t - s)} \frac{{\partial \varepsilon }}{{\partial t}}(s)\mathrm{d}s} \right) }}{{\partial x}} = \rho ^{*}\ddot{u} \end{aligned}$$

(A.2)

In order to obtain the displacement formulation of the equations of motion, the macrostrain is expressed in terms of the displacement, thus leading to

$$\begin{aligned} C_{11}^{\hom } \frac{\partial ^{2}u}{\partial x^{2}}+\eta _{11}^{\hom } \frac{\partial ^{3}u}{\partial t\partial x^{2}}+\int _0^t {C_{11} (t-s)} \frac{\partial ^{3}u}{\partial x\partial t}(s)\mathrm{d}s=\rho ^{*}\ddot{u} \end{aligned}$$

(A.3)

The solution is sought in the form of a 1D harmonic planar wave as a solution of the previous differential equation of motion:

$$\begin{aligned} u(x,t)=U\,\exp (\uplambda t-i(k_{1}\mathrm{x})) \end{aligned}$$

(A.4)

Inserting the harmonic planar wave (A.4) into the equilibrium equation (A.3) leads to

$$\begin{aligned} \begin{aligned}&C_{{11}}^{{\hom }} \frac{{\partial ^{2} u}}{{\partial x^{2} }} + \eta _{{11}}^{{\hom }} \frac{{\partial ^{3} u}}{{\partial t\partial x^{2} }} + \int _{0}^{t} {C_{{11}} (t - s)} \frac{{\partial ^{3} u}}{{\partial x^{2} \partial t}}(s)\mathrm{d}s = \rho ^{*} \ddot{u} \\&\quad = C_{{11}}^{{\hom }} \left( { - k_{1}^{2} \exp ^{{\left( {i\,\lambda t - k_{1}^{{}} x} \right) }} } \right) + \eta _{{11}}^{{\hom }} \left( { - \lambda k_{1}^{2} \exp ^{{\left( {i\,\lambda t - k_{1}^{{}} x} \right) }} } \right) \\&\qquad + \int _{0}^{t} {C_{{11}} (t - s)} \left( { - \lambda k_{1}^{2} \exp ^{{\left( {i\,\lambda s - k_{1}^{{}} x} \right) }} } \right) \mathrm{d}s = \rho ^{*} \left( {\lambda ^{2} \exp ^{{\left( {i\,\lambda t - k_{1}^{{}} x} \right) }} } \right) \end{aligned} \end{aligned}$$

(A.5)

The kernel $\hbox {C}_{11}(\mathrm{t}-\mathrm{s})$ in Eq. (A.5) is interpolated as $C_{11} (t-s)=A\exp \left( {b\left( {t-s} \right) } \right)$, then the characteristic equation is written as follows:

$$\begin{aligned}&C_{11}^{\hom } \left( {-k_{1}^{2} \exp ^{\left( {i\,\lambda t-k_{1} x} \right) }} \right) +\eta _{11}^{\hom } \left( {-\lambda k_{1}^{2} \exp ^{\left( {i\,\lambda t-k_{1} x} \right) }} \right) \nonumber \\&\qquad +\int _0^t {A\left( {\exp ^{\left( {b\left( {t-s} \right) } \right) }} \right) } \left( {-\lambda k_{1}^{2} \exp ^{\left( {i\,\lambda s-k_{1} x} \right) }} \right) \mathrm{d}s=\rho ^{*}\left( {\lambda ^{2}\exp ^{\left( {i\,\lambda t-k_{1} x} \right) }} \right) \nonumber \\&\quad =C_{11}^{\hom } \left( {-k_{1}^{2} \exp ^{\left( {i\,\lambda t} \right) }} \right) +\eta _{11}^{\hom } \left( {-\lambda k_{1}^{2} \exp ^{\left( {i\,\lambda t} \right) }} \right) -\lambda k_{1}^{2} A\exp ^{\left( {bt} \right) }\int _0^t {\left( {\exp ^{\left( {-b\left( s \right) +i\,\lambda s} \right) }} \right) } \mathrm{d}s=\rho ^{*}\left( {\lambda ^{2}\exp ^{\left( {i\,\lambda t} \right) }} \right) \nonumber \\&\quad =C_{11}^{\hom } \left( {-k_{1}^{2} } \right) +\eta _{11}^{\hom } \left( {-\lambda k_{1}^{2} } \right) -\frac{\lambda k_{1}^{2} A}{-b+i\,\lambda }\left( {\exp ^{\left( {-b\left( t \right) } \right) }} \right) \mathrm{d}s=\rho ^{*}\left( {\lambda ^{2}} \right) \end{aligned}$$

(A.6)

A Taylor expansion of $\exp ^{\left( {-b\left( t \right) } \right) }$ is done up to the second order:

$$\begin{aligned} = C_{{11}}^{{\hom }} \left( { - k_{1}^{2} } \right) + \eta _{{11}}^{{\hom }} \left( { - \lambda k_{1}^{2} } \right) - \frac{{\lambda k_{1}^{2} A}}{{ - b + i\,\lambda }}\left( {1 + \left( { - bt} \right) + \frac{{\left( { - bt} \right) ^{2} }}{2}} \right) \mathrm{d}s = \rho ^{*} \left( {\lambda ^{2} } \right) \end{aligned}$$

(A.7)

Equation (A.7) represents the characteristic equation describing the dispersion of longitudinal waves taking into account the long-memory effect.

Appendix B: Determination of the creep function of the homogenized medium and comparison with the response of the heterogeneous medium

We expose a mathematical formulation of the effective creep function for the homogenized Kelvin–Voigt medium under shear stress. In this model, we take a general 2D viscoelastic medium of Kelvin–Voigt type subjected to a shear constant stress. The general constitutive law writes in 2D

$$\begin{aligned} \left[ {{\begin{array}{c} {\sigma _{xx} } \\ {\sigma _{yy} } \\ {\sigma _{xy} } \\ \end{array} }} \right] =\left( {{\begin{array}{ccc} {\lambda +2\mu } &{} \mu &{} 0 \\ \mu &{} {\lambda +2\mu } &{} 0 \\ 0 &{} 0 &{} {2\mu } \\ \end{array} }} \right) \left( {{\begin{array}{c} {\varepsilon _{\mathrm{xx}} } \\ {\varepsilon _{\mathrm{yy}} } \\ {\varepsilon _{\mathrm{xy}} } \\ \end{array} }} \right) +\left( {{\begin{array}{ccc} {\eta _{v} +2\eta _{s} } &{} {\eta _{s} } &{} 0 \\ {\eta _{s} } &{} {\eta _{v} +2\eta _{s} } &{} 0 \\ 0 &{} 0 &{} {2\eta _{s} } \\ \end{array} }} \right) \left( {{\begin{array}{c} {\dot{{\varepsilon }}_{\mathrm{xx}} } \\ {\dot{{\varepsilon }}_{\mathrm{yy}} } \\ {\dot{{\varepsilon }}_{\mathrm{xy}} } \\ \end{array} }} \right) \end{aligned}$$

(B.1)

where $\lambda$, $\mu$ are the two lame coefficients, and $\eta _{v}$, $\eta _{s}$ the volume and shear viscosity, respectively. Under a constant applied stress, we obtain the corresponding creep function and then the corresponding viscosity.

When a constant shear stress is applied, the deformation – the creep function - expresses as:

$$\begin{aligned} \sigma _{\mathrm{xx}}^{0} ={{\textit{\textbf{E}}}}\,\varepsilon _{xx} +\eta _{s} \,\dot{{\varepsilon }}_{xx} \,\,\Rightarrow \varepsilon _{xx} =\frac{\sigma _{\mathrm{xx}}^{0} }{{{\textit{\textbf{E}}}}}\left( {1-\text{ e}^{-\frac{{{\textit{\textbf{E}}}}}{\eta _{s} }\text{ t }}} \right) \end{aligned}$$

(B.2)

where ${{\textit{\textbf{E}}}}$ is the elastic Young modulus, and $\eta _{s}$ is the shear viscosity. The structure under investigation is the tendon microstructure presented into details in Sect. 4.

Fig. 14

Comparison of the creep function between the homogenized model (blue line) and the response of the heterogeneous medium (red line) (color figure online)

Figure 14 shows that the creep response of the homogenized medium is close to that of the initially heterogeneous medium; the difference increases when time elapses.