Why Introducing Compressibility is Necessary for Incompressible Elements like Rubber in Abaqus/Explicit Analysis

In the context of Abaqus/Explicit analysis, providing a small amount of compressibility for materials like rubber, which are theoretically incompressible, is a critical step in ensuring accurate and stable simulations. This practice is rooted in several fundamental principles and benefits.

Numerical Stability

The primary reason for introducing compressibility is to improve numerical stability in the simulation. Incompressible materials, by definition, maintain a constant volume under any deformation. However, this ideal condition can lead to numerical instabilities, especially in dynamic explicit analyses. When the material is treated as perfectly incompressible, the computational solver may struggle to converge, potentially resulting in unrealistic or erratic simulation outcomes.

Pressure Calculation

In explicit finite element analysis, the pressure field is often derived from the volume change of the material. For perfectly incompressible materials, there is no volume change, making it challenging to calculate pressure accurately. Introducing a small amount of compressibility allows for a more realistic pressure calculation, ensuring that the simulation results are reliable and physically meaningful.

Material Behavior

Real-world materials, including rubber, exhibit a certain degree of compressibility under specific conditions, especially under high pressure. By incorporating a small amount of compressibility, the model can more accurately simulate the behavior of rubber under various loading conditions. This approach ensures that the simulation results more closely match real-world behavior, enhancing the predictive capabilities of the analysis.

Avoiding Zero Eigenvalues

In specific formulations, such as those used in finite element analysis, treating certain materials as perfectly incompressible can lead to the emergence of zero eigenvalues in the stiffness matrix. These zero eigenvalues can complicate the solution process, leading to convergence issues and inaccurate results. A small amount of compressibility helps to avoid these zero eigenvalues, ensuring a more robust and accurate solution.

Implementation of Incompressibility

Most modern computational software, including Abaqus, use mixed formulations to handle incompressibility. Techniques such as Lagrange multipliers or penalty methods are employed to introduce and manage the constraints associated with incompressibility. A small compressibility parameter helps these formulations to function more effectively, ensuring that the solver can manage these constraints efficiently and produce reliable results.

Conclusion

While rubber and other materials are often treated as incompressible in engineering applications, the introduction of a small amount of compressibility is essential for achieving numerical stability, accurate pressure calculations, and realistic material behavior in simulations. This approach ensures that the simulation results are both reliable and meaningful, enhancing the predictive capabilities of the analysis and improving overall confidence in the design process.