Sintering (time- and temperature- dependent)

Sintering model in MercuryDPM referred to as SinterNormal (SinterNormalSpecies, SinterInteraction).

To model sintering, the elastoplastic and dissipative forces are linear functions of the effective, elastic overlap \( (\delta > \delta_p)\), and normal velocity \( \dot{\delta} \). This model assumes that the stiffness increases with the amount of plastic deformation, due to the increased contact surface, with \( k = k_1 \) for zero plastic overlap up to maximum of \( k = k_{2}^{max}\) at maximum plastic overlap \( \delta_2^{max}\), which is set to avoid solid volume fractions above one.

The unloading stiffness is updated according to

\( \hat{k}_2 = k_1 + (k^{max}_2 - k_1)\frac{\delta_p}{\delta^{*}} \)

Header:

#include <Species/SinterSpecies.h>

Interaction function:

void SinterInteraction::computeNormalForce() 

The algorithm to compute the normal force as a function of the plastic overlap is:

computeNormalForce
1. Set the relative velocity	relativeVelocity_
2. Set the normal relative velocity	setNormalRelativeVelocity_
3. If overlap	\( \delta^n > 0 \)
4. Calculate the effective diameter		\( D_{ij} = 2R_{ij} \)
5. Maximum plastic flow limit overlap		\( \delta^{*} = \phi_{penet} D_{ij} \)
6. Increase the maximum overlap if necessary		\( \delta_p^{min} = min(\delta^{*},\delta) \)
		\( \delta_p= max(\delta_p^{min},\delta_p) \)
7. Compute the unloading stiffness		\( \hat{k}_2 = k_1 + (k_2^{max} - k_1)\frac{\delta_p}{\delta^{*}} \)
8. Compute the normal force		\( f^n = \hat{k}_2(\delta - \delta_p) \)
9. Compute rate of plastic overlap			\( \dot{\delta_p} \)
10. Compute plastic overlap			\( \delta_p = \dot{\delta_p} t \)
11. Set normal force	setAbsoluteNormalForce(std::abs(normalForce))
12. Set force	setForce(getNormal() * normalForce)
13. Set torque	setTorque(Vec3D(0.0, 0.0, 0.0))

Plastic overlap:

sintering of two particles.

In MercuryDPM, the solid state of sintering is modelled updating the plastic overlap.

1. Frenkel SINTERTYPE::CONSTANT_RATE

   The viscous flow mechanism, which was proposed by Frenkel, can be operative in the sintering of viscoelastic materials like glass and polymers. If the material follows the behaviour of a Newtonian fluid, the neck growth and shrinkage kinetics are expressed as

   \( a^2 = \frac{\gamma_s R}{\eta} t \)

   where \( a \) is the contact radius, \( \gamma_s \) is the interfacial surface tension, \( \eta \) zero-shear viscosity, \( t \) is time.

   The characteristic sintering time is defined as

   \( \tau = \frac{R \eta}{\gamma_s} \)

   \( \tau \) highlights the importance influence of the particle radius, the smaller the particle, the faster the sintering process.

2. Temperature dependent Frenkel SINTERTYPE::TEMPERATURE_DEPENDENT_FRENKEL

Unit test: SinterForceUnitTest.cpp

Reference:
R. Fuchs et. al., Initial stage sintering of polymer particles – Experiments and modelling of size-, temperature- and time-dependent contacts