Heat generation under shock waves

[jnuz73hbn]

I want to describe physically the thermal response of dentin under laser-induced shock waves.
Dentin is not homogeneous. It consists of a stiff hydroxylapatite lattice and a softer, viscoelastic collagen phase.

My idea is:

mechanical wave enters dentin ⇒ different components react differently ⇒ phase shift + damping ⇒ energy loss ⇒ heat
The shock wave acts like an external driving force on the structure.



⇒ external force $F_0$ drives the atoms
⇒ dentin responds like a driven damped oscillator

Amplitude:

$$
A(\omega) = \frac{F_0}{\sqrt{(k - m\omega^2)^2 + (b\omega)^2}}
$$

Phase shift:

$$
\tan(\varphi) = \frac{b\omega}{k - m\omega^2}
$$

⇒ damping $b$ increases ⇒ phase shift $\varphi$ increases
Different parts of dentin have different stiffness:

$$
\omega = \sqrt{\frac{k}{m}}, \quad
k_{\text{lattice}} \gg k_{\text{collagen}} \Rightarrow \omega_1 \neq \omega_2
$$

⇒ lattice reacts fast (high $\omega$)
⇒ collagen reacts slower (low $\omega$)
$$
\varphi = (\omega_1 - \omega_2)t \neq 0
$$

⇒ same excitation, but different response speed
⇒ components are no longer synchronized

Relative motion :
$$
\Delta x(t) = A_1 \sin(\omega t) - A_2 \sin(\omega t + \varphi) \neq 0
$$

⇒ particles move against each other
⇒ internal friction appears

Energy dissipation -> Heat


Velocity:

$$
v(t) = \omega A \cos(\omega t)
$$

Power loss:

$$
P(t) = b v^2 = b \omega^2 A^2 \cos^2(\omega t)
$$

Average:

$$
\langle P \rangle = \frac{1}{2} b \omega^2 A^2
$$

⇒ damping converts mechanical energy into heat

$$
Q \propto \int \langle P \rangle dt \Rightarrow Q \propto b \omega^2 A^2
$$

At resonance:

$$
A \propto \frac{1}{b}
$$

⇒

$$
Q \propto \frac{\omega^2}{b}
$$

$$
b \uparrow \Rightarrow \varphi \uparrow
$$

$$
\varphi \neq 0 \Rightarrow \Delta x \neq 0
$$

$$
\Delta x \neq 0 \Rightarrow \text{friction} \Rightarrow \text{heat}
$$

My questions

1. Is this a reasonable physical model for dentin as a heterogeneous composite?
2. Is the assumption $\varphi = (\omega_1 - \omega_2)t$ acceptable, or should phase lag be derived differently
3. Would a viscoelastic wave equation or phonon-based model be more appropriate?
4. What would you change or improve in this approach?

Any feedback would help a lot.