You're encountering a common and challenging problem in metal forming simulations: accurately modeling the interfacial behavior between the workpiece (copper) and the tooling (steel), specifically the sticking phenomenon. This is further complicated by the hydrostatic nature of the extrusion process. The discrepancy between your simulated and experimental forces suggests that the interface conditions are not being adequately represented in your Abaqus model.

Here's a breakdown of potential issues and strategies to address them:

1. Understanding the Sticking Phenomenon:

Sticking occurs when the interfacial friction is so high that the material at the interface effectively adheres to the tooling, preventing relative slip. This is often observed in metal forming processes involving high temperatures and pressures, where metallurgical bonding or severe plastic deformation can occur at the interface.

2. Addressing Sticking in Abaqus:

Abaqus offers several approaches to model interfacial behavior, but accurately capturing sticking requires careful consideration:

• Penalty Friction Formulation (with High Friction Coefficient): This is the most common approach. You can define a friction coefficient (μ) between the copper and steel surfaces. For sticking, you would use a very high value of μ (e.g., μ > 1.0, or even a very large number like 10 or 100). However, simply increasing μ may not be sufficient to fully capture sticking. At very high friction values, numerical instabilities can arise, leading to convergence problems.
• "Rough" Contact: In Abaqus/Standard, using the "rough" friction formulation can be a good approximation of sticking. This formulation enforces no relative motion at the interface until the shear stress exceeds a critical value. This critical value is related to the contact pressure.
• Kinematic Contact with Tie Constraint: You can use a "tie" constraint to completely bond the copper and steel surfaces together in the regions where sticking is expected. This is the most direct way to model sticking, but it's important to apply it only to the areas where sticking is physically plausible. Overusing tie constraints can lead to overly constrained models.
• Cohesive Elements: Cohesive elements can be inserted between the copper and steel to explicitly model the interfacial bonding and potential failure. This approach is more computationally expensive but can provide a more accurate representation of the interface behavior, including the transition from sticking to sliding. You'd need to define the cohesive properties (e.g., stiffness, strength, fracture energy) based on experimental data or material models.
• User-Defined Subroutines (UMAT or FRIC): For the most complex scenarios, you can use user-defined subroutines to implement custom friction models that better capture the sticking phenomenon. This requires advanced knowledge of Abaqus and material modeling.

3. Addressing the Hydrostatic Extrusion Discrepancy:

The fact that your simulated force is significantly lower than the experimental value suggests that the interface conditions are not adequately capturing the resistance to material flow caused by sticking. Here's how to address this within the context of hydrostatic extrusion:

• Accurate Pressure Application: Ensure that the hydrostatic pressure is being applied correctly to the copper billet. The pressure should be applied uniformly to all surfaces of the billet except for the extrusion exit.
• Die Geometry and Mesh Refinement: The die geometry and mesh refinement in the deformation zone are crucial. A finer mesh in this region will improve the accuracy of the stress and strain calculations, especially near the interface.
• Interface Definition at Die Entry: The interface conditions at the die entry are particularly important. This is where the material first comes into contact with the die and where sticking is most likely to initiate. Consider using a tie constraint or a very high friction coefficient in this region.
• Effect of Temperature: If the extrusion process involves significant temperature rise due to plastic deformation, consider performing a coupled temperature-displacement analysis. Temperature can significantly affect the material properties and the interfacial behavior.
• Mesh Distortion: In severe deformation problems, mesh distortion can lead to inaccurate results or convergence problems. Consider using adaptive meshing or remeshing techniques to maintain mesh quality throughout the simulation.

Specific Recommendations:

By systematically addressing these points, you should be able to improve the accuracy of your Abaqus simulations and obtain results that are closer to your experimental values. Remember that accurately modeling sticking is a complex problem, and it may require a combination of different techniques and careful calibration of model parameters.