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jnuz73hbn
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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
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
- Is this a reasonable physical model for dentin as a heterogeneous composite?
- Is the assumption ##\varphi = (\omega_1 - \omega_2)t## acceptable, or should phase lag be derived differently
- Would a viscoelastic wave equation or phonon-based model be more appropriate?
- What would you change or improve in this approach?
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