You need to account for the evolution of material properties over time, as well as the effects of thermal cycling or holding at elevated temperatures during the annealing process. In COMSOL, this would typically require incorporating both material property changes (e.g., thermal expansion, yield strength, etc.) as a function of time and temperature, and time-dependent processes that simulate the annealing behavior.

Here are the steps you can take and the COMSOL modules and techniques to use:

1. Use of Time-Dependent Physics

• You likely need to employ the Time-Dependent solver to simulate the evolution of material properties over time during the annealing process.
• Heat Transfer in Solids (HT) module or Thermal Stress module can be used to model the thermal history during the annealing. Set up your model so that the temperature distribution is time-dependent, which will affect the thermal expansion behavior.

2. Consider Material Property Evolution

• Copper's material properties, such as thermal expansion coefficient, yield strength, and elastic modulus, can change as a function of temperature and time during the annealing process.
• Create temperature- and time-dependent material properties:You can define a temperature-dependent coefficient of thermal expansion (CTE). However, for true annealing behavior, the CTE may also change over time as the material undergoes microstructural changes, such as grain growth or phase transformation. This can be implemented by defining CTE as a function of both temperature and time. Stress-Strain Relations: Incorporate time-dependent yield strength or plasticity models if plastic deformation is expected during annealing.
• For the time-dependent evolution of material properties, you may use the Material Property Functions in COMSOL to describe the change of properties over time. You can either input experimental data or use a function like an exponential or power law to represent the annealing process.

3. Incorporate Microstructure Evolution

• The Phase Field or Structural Mechanics module (depending on the detailed nature of the bonding and annealing process) can be used to simulate changes in the microstructure, such as grain growth, that would influence material properties like thermal expansion.
• To model the effects of grain coarsening or phase transformations (if applicable), use a time-dependent phase field model to simulate the microstructural evolution during the annealing process.

4. Thermal Expansion as a Function of Time

• If thermal expansion is not showing a significant change over time, ensure that the thermal expansion is not simply being treated as a constant for the temperature range you are studying. Instead, make it time-dependent:Define a time-dependent thermal expansion coefficient (CTE) that evolves over the duration of the simulation. You might need to consider the effect of recrystallization and how it alters the thermal expansion properties as the material anneals. Recrystallization models are typically time- and temperature-dependent, and they might require coupling with the Structural Mechanics module to capture strain hardening or softening effects.

5. Couple with Other Physical Effects

• If there is plastic deformation during the bonding or annealing process, use the Plasticity feature in the Structural Mechanics module to model time-dependent plastic behavior.
• Consider if creep or stress relaxation is significant in your process, and if so, use the Creep module to model time-dependent deformation.

6. Simulation Setup

• Time-Dependent Solvers: Set up the solver to run over a long enough period to capture the effects you are interested in. Typically, you'd want a transient simulation (using the Time-Dependent solver).
• Ensure that time steps are small enough to accurately capture the material changes at each step. If the time steps are too large, you might miss important dynamics of the annealing process.
• For better accuracy, refine the mesh around regions where you expect temperature gradients or significant material behavior changes (e.g., near bonding interfaces).

7. Post-Processing and Interpretation

• Analyze the evolution of thermal stresses and thermal strain over time.
• Track the changes in thermal expansion and other material properties as a function of both time and temperature.

8. Additional COMSOL Modules to Consider

• Structural Mechanics Module: For capturing the mechanical stresses due to thermal expansion and changes in material properties over time.
• Fatigue Module (if cyclic loading is involved): For capturing the effects of thermal cycles, which can contribute to material property changes and fatigue behavior during annealing.
• Optimization Module: If you are interested in optimizing the annealing process or bonding conditions to achieve the desired material properties.

By combining the time-dependent physics with evolving material properties and possibly microstructural effects, you should be able to better capture the effects of annealing on copper bonding. This approach should allow you to simulate the changes in thermal expansion and other material behaviors over time, leading to more accurate predictions of the bonding process.

Hi Reza Pirdadeh Beiranvand ,

Thank you so much for responding. I already followed the steps you mentioned in the above message. I use piecewise functions for time-dependent analysis and apply time-dependent physics. I also apply Heat Transfer in Solids, and Solid Mechanics is used to analyze the thermal expansion in copper. I recheck the material properties as well. I run the simulation for 2 hours with small time steps, but it doesn’t show much effect of annealing with time; the results appear constant.