Overview
This research investigates the relationship between polyconvexity and true-stress-true-strain monotonicity in hyperelastic potentials for incompressible material behavior in three dimensions. The study specifically examined whether polyconvexity, as a constitutive condition, guarantees true-stress-true-strain monotonicity.
Approach
The investigation utilized a counterexample to explore the implications of polyconvexity on the mechanical response of incompressible hyperelastic materials. The counterexample was designed to test the sufficiency of polyconvexity in ensuring monotonic true-stress-true-strain behavior. The analysis was conducted within the framework of three-dimensional incompressible elasticity.
Findings
The study found that polyconvexity does not imply true-stress-true-strain monotonicity. This conclusion was drawn using a specific counterexample that demonstrates a scenario where a polyconvex potential does not exhibit monotonic true-stress-true-strain behavior. Consequently, the research indicates that polyconvexity, when considered in isolation, is not a strong enough condition to guarantee a physically reasonable response for idealized elasticity.