DEM Extensions: Acoustical Pre-Processing

Abstract

In numerous industries, particle-laden fluids are a key part of the fabrication of products such as (1) casted machine parts, (2) additively manufactured and 3D printed electronics and medical devices, and even (3) slurry processed food to name a few.

11.6   Chapter Appendix: Basics of Acoustics

In our approach, we model the individual particles as being rigid, and the material surrounding the particles as being isotropic and having a relatively low shear modulus, in the zero limit becoming an acoustical medium. Generally, for an isotropic material, one has the classical relationship between the components of infinitesimal strain (${\varvec{\epsilon }}$) to the Cauchy stress (${\varvec{\sigma }}$)

$$\begin{aligned} {\varvec{\sigma }}=\displaystyle {{{\varvec{I}}}\!{{\varvec{E}}}:{\varvec{\epsilon }}=3\kappa \frac{\mathrm{tr} {\varvec{\epsilon }}}{3}{} \mathbf{1}+2\mu {\varvec{\epsilon }}^{\prime }}, \end{aligned}$$

(11.14)

where ${{\varvec{I}}}\!{{\varvec{E}}}$ is the elasticity tensor and where ${\varvec{\epsilon }}^{\prime }$ is the strain deviator. The corresponding strain energy density is

$$\begin{aligned} \displaystyle {W=\frac{1}{2}{\varvec{\epsilon }}:{{{\varvec{I}}}\!{{\varvec{E}}}}:{\varvec{\epsilon }}= \frac{1}{2}\left( 9\kappa (\frac{\mathrm{tr} {\varvec{\epsilon }}}{3})^2+2\mu {\varvec{\epsilon }}^{\prime }:{\varvec{\epsilon }}^{\prime }\right) }. \end{aligned}$$

(11.15)

We focus on the dilatational deformation in the low shear modulus matrix surrounding the particles. This naturally leads to an idealized “acoustical” material approximation, $\mu \approx 0$. Hence, Eq. 11.14 collapses to ${\varvec{\sigma }}=-p \mathbf{1}$, where the pressure is $p=-3\, \kappa \,\frac{\mathrm{tr} {{\varvec{\epsilon }}}}{3} \mathbf{1}$ and with a corresponding strain energy of $\displaystyle {W=\frac{1}{2}\frac{p^2}{\kappa }}$. By inserting the simplified expression of the stress ${\varvec{\sigma }}=-p\mathbf{1}$ into the equation of equilibrium, we obtain

$$\begin{aligned} \nabla \cdot {\varvec{\sigma }}=-\nabla p=\rho \ddot{{\varvec{u}}}, \end{aligned}$$

(11.16)

where ${\varvec{u}}$ is the displacement. By taking the divergence of both sides, and recognizing that $\nabla \cdot {\varvec{u}}=-\frac{p}{\kappa }$, we obtain

$$\begin{aligned} \nabla ^2 p=\frac{\rho }{\kappa }\ddot{p}=\frac{1}{c^2}\ddot{p}. \end{aligned}$$

(11.17)

If we assume a harmonic solution, we obtain

$$\begin{aligned} p=Pe^{j({\varvec{k}}\cdot {\varvec{r}}-\omega t)} \Rightarrow \dot{p}=Pj\omega e^{j({\varvec{k}}\cdot {\varvec{r}}-\omega t)} \Rightarrow \ddot{p}=-P\omega ^2 e^{j({\varvec{k}}\cdot {\varvec{r}}-\omega t)}, \end{aligned}$$

(11.18)

and

$$\begin{aligned} \nabla p= & {} Pj(k_x{\varvec{e}}_x+k_y{\varvec{e}}_y+k_z{\varvec{e}}_z)e^{j({\varvec{k}}\cdot {\varvec{r}}-\omega t)} \Rightarrow \nabla \cdot \nabla p= \nabla ^2 p\nonumber \\ {}= & {} -P\underbrace{(k^2_x+k^2_y+k^2_z)}_{||{\varvec{k}}||^2} e^{j({\varvec{k}}\cdot {\varvec{r}}-\omega t)}. \end{aligned}$$

(11.19)

We insert these relations into Eq. 11.17 and obtain an expression for the magnitude of the wave number vector

$$\begin{aligned} -P||{\varvec{k}}||^2e^{j({\varvec{k}}\cdot {\varvec{r}}-\omega t)}=-\frac{\rho }{\kappa } P\omega ^2e^{j({\varvec{k}}\cdot {\varvec{r}}-\omega t)} \Rightarrow ||{\varvec{k}}||=\frac{\omega }{c}. \end{aligned}$$

(11.20)

Equation 11.16 (balance of linear momentum) implies

$$\begin{aligned} \rho \ddot{{\varvec{u}}}=-\nabla p=-Pj(k_x{\varvec{e}}_x+k_y{\varvec{e}}_y+k_z{\varvec{e}}_z)e^{j({\varvec{k}}\cdot {\varvec{r}}-\omega t)}. \end{aligned}$$

(11.21)

Now we integrate once, which is equivalent to dividing by $-j\omega$, and obtain the velocity

$$\begin{aligned} \dot{{\varvec{u}}}=\frac{Pj}{\rho \omega }(k_x{\varvec{e}}_x+k_y{\varvec{e}}_y+k_z{\varvec{e}}_z)e^{j({\varvec{k}}\cdot {\varvec{r}}-\omega t)}, \end{aligned}$$

(11.22)

and do so again for the displacement

$$\begin{aligned} {\varvec{u}}=\frac{Pj}{\rho \omega ^2}(k_x{\varvec{e}}_x+k_y{\varvec{e}}_y+k_z{\varvec{e}}_z)e^{j({\varvec{k}}\cdot {\varvec{r}}-\omega t)}. \end{aligned}$$

(11.23)

Thus, we have

$$\begin{aligned} ||\dot{{\varvec{u}}}||=\frac{P}{c\rho }. \end{aligned}$$

(11.24)

The reflection of a plane harmonic pressure wave at an interface is given by enforcing continuity of the (acoustical) pressure and disturbance velocity at that location; this yields the ratio between the incident and reflected pressures. We use a local coordinate system (Fig. 11.3) and require that the number of waves per unit length in the $x-\mathrm{direction}$ must be the same for the incident, reflected, and refracted (transmitted) waves,

$$\begin{aligned} {\varvec{k}}_i\cdot {\varvec{e}}_x={\varvec{k}}_r\cdot {\varvec{e}}_x= {\varvec{k}}_t\cdot {\varvec{e}}_x. \end{aligned}$$

(11.25)

From the pressure balance at the interface, we have

$$\begin{aligned} P_ie^{j({\varvec{k}}_i\cdot {\varvec{r}}-\omega t)}+ P_re^{j({\varvec{k}}_r\cdot {\varvec{r}}-\omega t)}= P_te^{j({\varvec{k}}_t\cdot {\varvec{r}}-\omega t)}, \end{aligned}$$

(11.26)

where $P_i$ is the incident pressure ray, $P_r$ is the reflected pressure ray, and $P_t$ is the transmitted pressure ray. This forces a time-invariant relation to hold at all parts on the boundary, because the arguments of the exponential must be the same. This leads to ($k_i=k_r$)

$$\begin{aligned} k_i sin \theta _i=k_r sin \theta _r\Rightarrow \theta _i=\theta _r, \end{aligned}$$

(11.27)

and

$$\begin{aligned} k_i sin \theta _i=k_t sin \theta _t\Rightarrow \frac{k_i}{k_t} =\frac{sin\theta _t}{sin \theta _i}=\frac{\omega /c_t}{\omega /c_i} =\frac{c_i}{c_t} =\frac{v_i}{v_t} =\frac{n_t}{n_i}. \end{aligned}$$

(11.28)

Equations 11.25 and 11.26 imply

$$\begin{aligned} P_ie^{j({\varvec{k}}_i \cdot {\varvec{r}})}+ P_re^{j({\varvec{k}}_r \cdot {\varvec{r}})}= P_te^{j({\varvec{k}}_t \cdot {\varvec{r}})}. \end{aligned}$$

(11.29)

The continuity of the displacement, and hence the velocity

$$\begin{aligned} {\varvec{v}}_i+{\varvec{v}}_r={\varvec{v}}_t, \end{aligned}$$

(11.30)

leads to, after use of Eq. 11.24,

$$\begin{aligned} -\frac{P_i}{\rho _ic_i}cos\theta _i+ \frac{P_r}{\rho _rc_r}cos\theta _r= -\frac{P_t}{\rho _tc_t}cos\theta _t. \end{aligned}$$

(11.31)

We solve for the ratio of the reflected and incident pressures to obtain

$$\begin{aligned} r=\frac{P_r}{P_i}= \frac{\hat{A}cos\theta _i-cos\theta _t}{\hat{A}cos\theta _i+cos\theta _t}, \end{aligned}$$

(11.32)

where $\hat{A}{\mathop {=}\limits ^\mathrm{def}}\frac{A_t}{A_i}=\frac{\rho _tc_t}{\rho _ic_i}$, where $\rho _t$ is the medium which the ray encounters (transmitted), $c_t$ is corresponding sound speed in that medium, $A_t$ is the corresponding acoustical impedance, $\rho _i$ is the medium in which the ray was traveling (incident), $c_i$ is corresponding sound speed in that medium $A_i$ is the corresponding acoustical impedance. The relationship (the law of refraction) between the incident and transmitted angles is $c_tsin\theta _t=c_isin\theta _i$. Thus, we may write the Fresnel relation

$$\begin{aligned} r=\frac{\tilde{c}\hat{A}cos\theta _i-(\tilde{c}^2-sin^2\theta _i)^{\frac{1}{2}}}{\tilde{c}\hat{A}cos\theta _i+(\tilde{c}^2-sin^2\theta _i)^{\frac{1}{2}}}, \end{aligned}$$

(11.33)

where $\tilde{c}{\mathop {=}\limits ^\mathrm{def}}\frac{c_i}{c_t}$. The reflectance for the (acoustical) energy $\mathcal{R}=r^2$ is

$$\begin{aligned} \mathcal{R}=\left( \frac{P_r}{P_i}\right) ^2= \left( \frac{\hat{A}cos\theta _i-cos\theta _t}{\hat{A}cos\theta _i+cos\theta _t}\right) ^2=\left( \frac{I_r}{I_i}\right) ^2. \end{aligned}$$

(11.34)

For the cases where $sin \theta _t=\frac{sin \theta _i}{\tilde{c}}>1$, one may rewrite the reflection relation as

$$\begin{aligned} r= \frac{\tilde{c}\hat{A}cos\theta _i-j(sin^2\theta _i-\tilde{c}^2)^{\frac{1}{2}}}{\tilde{c}\hat{A}cos\theta _i+j(sin^2\theta _i-\tilde{c}^2)^{\frac{1}{2}}}. \end{aligned}$$

(11.35)

where $j=\sqrt{-1}$. The reflectance is $\mathcal{R} {\mathop {=}\limits ^\mathrm{def}}r \bar{r}=1$, where $\bar{r}$ is the complex conjugate. Thus, for angles above the critical angle $\theta _i \ge \theta ^*_i$, all of the energy is reflected. We note that when $A_t=A_i$ and $c_i=c_t$, then there is no reflection. Also, when $A_t>>A_i$ or when $A_t<<A_i$, then $r\rightarrow 1$.

Remark

If one considers for a moment an incoming pressure wave (ray), which is incident on an interface between two general elastic media ($\mu \ne 0$), reflected shear waves must be generated in order to satisfy continuity of the traction, $[\!|{\varvec{\sigma }}\cdot {\varvec{n}}|\!]=\mathbf{0}$. This is due to the fact that

$$\begin{aligned}{}[\!|\left( 3\kappa \mathrm{tr} \frac{{\varvec{\epsilon }}}{3} \mathbf{1}+2\mu {\varvec{\epsilon }}^{\prime }\right) \cdot {\varvec{n}}|\!]=\mathbf{0}. \end{aligned}$$

(11.36)

For an idealized acoustical medium, $\mu =0$, no shear waves need to be generated to satisfy Eq. 11.36.