To implement the spark plasma sintering (SPS) model proposed by Eugene A. Olevsky in COMSOL, especially the stress-strain analysis part, follow these steps:

1. Set Up the Basic Modules

• Electrical Module: For electrical current.
• Heat Transfer Module: For heat transfer in solids.
• PDE Modules: For densification and grain growth using two coefficient form PDEs.

2. Nonlinear Elastic Material Node in Solid Mechanics Module

• Stress-Strain Relationship: Implement stress as a function of strain rate, which is key for the sintering process.

Steps for Stress-Strain Implementation

1. Activate Nonlinear Elastic Material Model

• Go to the "Solid Mechanics" module.
• Select the "Nonlinear Elastic Material" node.

2. Define the Stress-Strain Rate Relationship

• Create a custom constitutive model by defining the stress tensor (σ\sigmaσ) as a function of the strain rate tensor (ϵ˙\dot{\epsilon}ϵ˙).
• Use the creep subnode if direct implementation is not possible.

3. Implementing Sintering Stress

• Add an "Inelastic Strain Rate" subnode.
• Define the inelastic strain rate (ϵ˙inel\dot{\epsilon}_{\text{inel}}ϵ˙inel​) based on the sintering model equations from Olevsky’s paper.
• Use the variables and expressions to link the strain rate to the densification and grain growth mechanisms.

Example: Custom Constitutive Equation

codeσ_ij = f(ε_dot_ij)

Define this in the nonlinear elasticity settings by inputting the mathematical expressions derived from Olevsky’s model.

4. Parameter Definition and Coupling

• Define all necessary parameters (e.g., material properties, activation energy, etc.).
• Ensure proper coupling between the electrical, thermal, and mechanical models to reflect the SPS process.

Final Notes

• Validation: Validate your model by comparing with experimental data or results from literature.
• Adjustments: Be prepared to fine-tune the model parameters and settings for accurate simulation results.

By following these steps, you should be able to correctly implement the sintering stress-strain analysis part of the SPS model in COMSOL.