Harmonic plane waves in isotropic micropolar medium based on two-parameter nonlocal theory

Abstract

In this paper, the system of equations for nonlocal micropolar elastic materials is developed taking into account the assumption that the attenuation functions for the elastic and micropolar material coefficients are different, and applied for harmonic body waves. The dispersion equations of harmonic body waves propagating in a micropolar medium and their cutoff frequencies are obtained in simple form based on the new assumption. The obtained dispersion relations are potentially useful in an inverse problem by fitting the data of elastic and micropolar harmonic waves speed to estimate the elastic and micropolar nonlocal parameters of the medium. Some concerning remarks about the difference between the two-parameter nonlocal theory and the one-parameter nonlocal theory of Eringen are numerically discussed to show the necessary of the developed theory in the problem of wave propagation.

Appendices

Appendices

Apendix A: Dispersion equations in one-parameter nonlocal theory

In special case, when $\epsilon _1=\epsilon _2=\epsilon$

The longitudinal acoustic branch (LA)

LA wave is dispersive wave propagating with $\omega _1=\omega _1(k_0)$ determined by

$$\begin{aligned} \dfrac{\omega _1^2}{k_0^2}=\dfrac{c_1^2}{1+{\bar{\epsilon }}} \end{aligned}$$

(A.1)

where ${\bar{\epsilon }}=(k_0\epsilon )^2$.

The phase velocity $x_1$ depends on angular frequency ${\bar{\omega }}$ as

$$\begin{aligned} x_1=\dfrac{1}{g}-2e{\bar{\omega }}^2{\bar{e}}^2 \end{aligned}$$

(A.2)

where ${\bar{e}}=\epsilon /\sqrt{j}$.

The cutoff frequency of LA waves is:

$$\begin{aligned} \omega _1^{\textrm{cutoff}}=\sqrt{\dfrac{1}{2eg{\bar{e}}^2}}\;\omega _0. \end{aligned}$$

(A.3)

The longitudinal optic branch (LO)

LO wave is dispersive wave propagating with $\omega _3$ determined by

$$\begin{aligned} \dfrac{\omega _3^2}{k_0^2}=\dfrac{c_3^2}{1 + {\bar{\epsilon }}}+\dfrac{\omega _0^2}{k_0^2(1 + {\bar{\epsilon }})}. \end{aligned}$$

(A.4)

The phase velocity $x_3$ depends on angular frequency ${\bar{\omega }}$ as

$$\begin{aligned} x_3=\dfrac{{\bar{\omega }}^2({\bar{\alpha }}+{\bar{\beta }} +{\bar{\gamma }}-2e{\bar{\omega }}^2{\bar{e}}^2)}{{\bar{\omega }}^2-1}. \end{aligned}$$

(A.5)

The cutoff frequency of LO waves is:

$$\begin{aligned} \omega _3^{\textrm{cutoff}}=\sqrt{\dfrac{{\bar{\alpha }}+{\bar{\beta }} +{\bar{\gamma }}}{2e{\bar{e}}^2}}\;\omega _0. \end{aligned}$$

(A.6)

The transverse acoustic branch (TA) propagating with $\omega _2$ and the transverse optic branch (TO) propagating with $\omega _4$ determined by

$$\begin{aligned} \omega ^4-2pk_0^2\omega ^2+qk_0^4=0 \end{aligned}$$

(A.7)

where

$$\begin{aligned} \begin{aligned} 2p&=\dfrac{1}{1 + {\bar{\epsilon }}}\left( \dfrac{\omega _0^2}{k_0^2}+c_2^2+c_4^2\right) ,\\ q&=\dfrac{1}{(1 + {\bar{\epsilon }})^2}\left[ \dfrac{\omega _0^2}{k_0^2} \left( c_2^2-\dfrac{j\omega _0^2}{4}\right) +c_2^2c_4^2\right] . \end{aligned} \end{aligned}$$

(A.8)

If we divide (A.7) by $\omega _0^4$, we can express $\omega _2$ and $\omega _4$ in the form

$$\begin{aligned} \left( \dfrac{w_2}{\omega _0}\right) ^2=\dfrac{{\bar{k}}}{2e(1+{\bar{k}} {\bar{\epsilon }}^2)}\left( {\bar{p}}-\sqrt{{\bar{\Delta }}}\right) ,\quad \left( \dfrac{w_4}{\omega _0}\right) ^2=\dfrac{{\bar{k}}}{2e(1+{\bar{k}} {\bar{\epsilon }}^2)}\left( {\bar{p}}+\sqrt{{\bar{\Delta }}}\right) \end{aligned}$$

(A.9)

where

$$\begin{aligned} \begin{aligned} {\bar{p}}&=1+\dfrac{c_4^2}{c_2^2}+\dfrac{\omega _0^2}{k_0^2c_2^2}=1+{\bar{\gamma }}+\dfrac{2e}{{\bar{k}}},\\ {\bar{\Delta }}&=\sqrt{{\bar{p}}^2-{\bar{q}}}=\left( 1-\dfrac{c_4^2}{c_2^2} -\dfrac{\omega _0^2}{k_0^2c_2^2}\right) ^2+\dfrac{j\omega _0^4}{k_0^2c_2^4}= \left( 1-{\bar{\gamma }}-\dfrac{2e}{{\bar{k}}}\right) ^2+\dfrac{4e^2}{{\bar{k}}}. \end{aligned} \end{aligned}$$

(A.10)

This equation is the same with Eringen (1984; Eq. (3.10) [46]). Note that there is a typo in Eringen with notation $k_1=(\rho j c_2^2/2\kappa )^{1/2}k$ instead of $(\rho j c_2^2/2\kappa )k$.

The equation defining phase velocity $x_2$ and $x_4$ of TA and TO waves depending on angular frequency ${\bar{\omega }}$ is

$$\begin{aligned} x^2+bx+c=0 \end{aligned}$$

(A.11)

where

$$\begin{aligned} \begin{aligned} b&=-\dfrac{-2+e+2(1+{\bar{\gamma }}){\bar{\omega }}^2+4e{\bar{\omega }} ^2(1-2{\bar{\omega }}^2){\bar{e}}_2^2}{2({\bar{\omega }}^2-1)},\\ c&=\dfrac{{\bar{\omega }}^2(-1+2e{\bar{\omega }}^2{\bar{e}}^2) (-{\bar{\gamma }}+2e{\bar{\omega }}^2{\bar{e}}^2)}{{\bar{\omega }}^2-1}. \end{aligned} \end{aligned}$$

(A.12)

The cutoff frequency of TA and TO waves is:

$$\begin{aligned} \omega _2^{\textrm{cutoff}}=\sqrt{\dfrac{{\bar{\gamma }}}{2e{\bar{e}}^2}}\;\omega _0\quad \text {and}\quad \omega _4^{\textrm{cutoff}}=\sqrt{\dfrac{1}{2e{\bar{e}}^2}}\;\omega _0. \end{aligned}$$

(A.13)

Appendix B: dispersion equations in local theory

In special case, when $\epsilon _1=\epsilon _2=0$

The longitudinal acoustic branch (LA) propagating with $\omega _1$ determined by

$$\begin{aligned} \dfrac{\omega _1^2}{k_0^2}=c_1^2\quad \text {or}\quad V_1=c_1. \end{aligned}$$

(B.1)

In local theory, LA wave is not dispersive.

The longitudinal optic branch (LO) propagating with $\omega _3$ determined by

$$\begin{aligned} \dfrac{\omega _3^2}{k_0^2}=c_3^2+\dfrac{\omega _0^2}{k_0^2} \end{aligned}$$

(B.2)

or with phase velocity depending on angular frequency

$$\begin{aligned} x_3=\dfrac{{\bar{\omega }}^2({\bar{\alpha }}+{\bar{\beta }}+{\bar{\gamma }})}{{\bar{\omega }}^2-1}. \end{aligned}$$

(B.3)

When $\omega \rightarrow \infty$ or $k_0\rightarrow \infty$, we have $x_3\rightarrow {\bar{\alpha }}+{\bar{\beta }}+{\bar{\gamma }}$ or $V_3\rightarrow c_3$.

The transverse acoustic branch (TA) propagating with $\omega _2$ and the transverse optic branch (TO) propagating with $\omega _4$ determined by

$$\begin{aligned} \omega ^4-2pk_0^2\omega ^2+qk_0^4=0 \end{aligned}$$

(B.4)

where

$$\begin{aligned} \begin{aligned}&2p=\dfrac{\omega _0^2}{k_0^2}+c_2^2+c_4^2,\\&q=\dfrac{\omega _0^2}{k_0^2}\left( c_2^2-\dfrac{j\omega _0^2}{4}\right) +c_2^2c_4^2. \end{aligned} \end{aligned}$$

(B.5)

If we divide (B.4) by $k_0^4$, we can express $\omega _2$ and $\omega _4$ in the form [48]

$$\begin{aligned} \left( \dfrac{w_2}{k_0}\right) ^2=p-\sqrt{\Delta },\quad \left( \dfrac{w_4}{k_0}\right) ^2=p+\sqrt{\Delta } \end{aligned}$$

(B.6)

where $\Delta =p^2-q=\dfrac{1}{4}\left( \dfrac{\omega _0^2}{k_0^2}+c_4^2+c_2^2\right) ^2+j\dfrac{\omega _0^4}{4k_0^2}$.

When $k\rightarrow \infty$, we have $V_2\rightarrow \min \{c_2,c_4\}$ and $V_4\rightarrow \max \{c_2,c_4\}$.

Appendix C: eigen-vector of body waves

We express the displacements and stresses of plane waves in the form

$$\begin{aligned} \begin{aligned} u_i&=U_ie^{\text {i} k_0(p_1x_1+p_2x_2-Vt)},\; \phi _i= -\text {i} k_0\Sigma _ie^{\text {i} k_0(p_1x_1+p_2x_2-Vt)},\\ t_{ij}&=\text {i} k_0(\mu +\kappa )T_{ij}e^{\text {i} k_0(p_1x_1+p_2x_2-Vt)},\;m_{ij}=(\mu +\kappa )M_{ij}e^{\text {i} k_0(p_1x_1+p_2x_2-Vt)}. \end{aligned} \end{aligned}$$

(C.1)

Then, the eigen-vectors are computed from the constitutive equations (12) and (13) as:

LA wave:

$$\begin{aligned} \begin{aligned} U_1&=1, U_2=\frac{p_2}{p_1},\\ L_1L_2[T_{11}]&=\frac{e({\bar{\varepsilon }}_1-{\bar{\varepsilon }}_2)g\,p_1^2 +(1+{\bar{\varepsilon }}_2)\left[ 1 + (-2 + e) g p_2^2\right] }{g\,p_1},\\ L_1L_2[T_{12}]&=L_1L_2[T_{21}]=\left[ e({\bar{\varepsilon }}_1-{\bar{\varepsilon }}_2) +(2 - e)(1+{\bar{\varepsilon }}_2)\right] p_2,\\ L_1L_2[T_{22}]&=\frac{e({\bar{\varepsilon }}_1 -{\bar{\varepsilon }}_2)g\,p_2^2+(1+{\bar{\varepsilon }}_2)\left[ 1 + (-2 + e) g p_1^2\right] }{g\,p_1}. \end{aligned} \end{aligned}$$

(C.2)

The other amplitudes of displacements and stresses are zero. We can see that the displacement unit vector ${\textbf {d}}$ and propagation unit vector ${\textbf {p}}$ of LA wave are the same.

LO Wave:

$$\begin{aligned} \begin{aligned} \Sigma _1&=1, \Sigma _2=\frac{p_2}{p_1},\\ L_1L_2[T_{13}]&=-(1+{\bar{\varepsilon }}_1)e\frac{p_2}{p_1},L_1L_2[T_{31}] =(1+{\bar{\varepsilon }}_1)e\frac{p_2}{p_1},\\ L_1L_2[T_{23}]&=(1+{\bar{\varepsilon }}_1)e,\;L_1L_2[T_{32}] =-(1+{\bar{\varepsilon }}_1)e,\\ L_1L_2[M_{11}]&=(1+{\bar{\varepsilon }}_1){\bar{k}}\left[ \dfrac{{\bar{\alpha }}}{p_1} +({\bar{\beta }} + {\bar{\gamma }}) p_1\right] ,\\ L_1L_2[M_{21}]&=L_1L_2[M_{12}]=(1+{\bar{\varepsilon }}_1){\bar{k}} ({\bar{\beta }} + {\bar{\gamma }}) p_2,\\ L_1L_2[M_{22}]&=(1+{\bar{\varepsilon }}_1)\dfrac{{\bar{k}}}{p_1}\left[ {\bar{\alpha }}+({\bar{\beta }} + {\bar{\gamma }}) p_2^2\right] . \end{aligned} \end{aligned}$$

(C.3)

The rotational displacement vector ${\textbf {d}}$ and propagation vector ${\textbf {p}}$ of LO wave are the same.

TA and TO waves:

$$\begin{aligned} \begin{aligned} U_1&=\frac{p_2}{p_1}, U_2=-1,\Sigma _3=\frac{e({\bar{\varepsilon }}_1 -{\bar{\varepsilon }}_2)+(1+{\bar{\varepsilon }}_2)\left[ 1 - (1 + {\bar{\varepsilon }}_1) x\right] }{e(1+{\bar{\varepsilon }}_1)\, p_1},\\ L_1L_2[T_{11}]&=\left[ e({\bar{\varepsilon }}_1-{\bar{\varepsilon }}_2) +(1+{\bar{\varepsilon }}_2)(2 - e)\right] p_2,\\ L_1L_2[T_{12}]&=\dfrac{\left[ e({\bar{\varepsilon }}_1-{\bar{\varepsilon }}_2) +(1+{\bar{\varepsilon }}_2)(2 - e)\right] p_2^2-(1 + {\bar{\varepsilon }}_1) (1 + {\bar{\varepsilon }}_2) x}{p_1},\\ L_1L_2[T_{21}]&=\left[ e({\bar{\varepsilon }}_1-{\bar{\varepsilon }}_2) +(1+{\bar{\varepsilon }}_2)(2 - e)\right] p_1 +\dfrac{(1 + {\bar{\varepsilon }}_1)(1 + {\bar{\varepsilon }}_2) x}{p_1},\\ L_1L_2[T_{22}]&=-\left[ e({\bar{\varepsilon }}_1-{\bar{\varepsilon }}_2) +(1+{\bar{\varepsilon }}_2)(2 - e)\right] p_2,\\ L_1L_2[M_{13}]&=\dfrac{{\bar{k}} {\bar{\gamma }} \left[ e({\bar{\varepsilon }}_1 -{\bar{\varepsilon }}_2)-(1+{\bar{\varepsilon }}_2)(-1 + x + {\bar{\varepsilon }}_1 x)\right] }{e},\\ L_1L_2[M_{23}]&=\dfrac{{\bar{k}} {\bar{\gamma }} p_2 \left[ e({\bar{\varepsilon }}_1-{\bar{\varepsilon }}_2)-(1+{\bar{\varepsilon }}_2)(-1 + x + {\bar{\varepsilon }}_1 x)\right] }{e\,p_1}. \end{aligned} \end{aligned}$$

(C.4)

The displacement vector ${\textbf {d}}$ and propagation vector ${\textbf {p}}$ of TA and TO wave are perpendicular.