Likely equilibria of the stochastic Rivlin cube

Area of Science:

• Continuum Mechanics
• Solid Mechanics
• Applied Mathematics

Background:

• The Rivlin cube problem addresses the stability of homogeneous equilibria in hyperelastic bodies under specific loads.
• Deterministic analysis provides a baseline for understanding material behavior but doesn't account for real-world variations.

Purpose of the Study:

• To investigate the stochastic version of the Rivlin cube problem, incorporating random elastic parameters.
• To determine the influence of parameter uncertainty on the stability of equilibria in hyperelastic materials.
• To derive the probability distribution of deformations based on parameter variability.

Main Methods:

• Modeling elastic parameters as random variables following standard probability laws.
• Analyzing the stability of homogeneous equilibria under equitriaxial dead loads in a stochastic framework.
• Investigating the relationship between material constitutive laws and equilibrium stability.

Main Results:

• Identified likely equilibria for the stochastic Rivlin cube problem.
• Quantified the dependence of equilibrium stability on material constitutive laws with random parameters.
• Derived probability distributions for deformations under parameter uncertainty.

Conclusions:

• Stochastic analysis is crucial for understanding the stability of hyperelastic bodies with variable material properties.
• Parameter uncertainty significantly impacts the prediction of equilibria and their stability.
• The study contributes to the understanding of material behavior in the presence of inherent variations, relevant to Rivlin's legacy.