In search of constitutive conditions in isotropic hyperelasticity: polyconvexity versus true-stress-true-strain monotonicity

The polyconvexity of a strain-energy function is nowadays increasingly presented as the ultimate material stability condition for an idealized elastic response. While the mathematical merits of polyconvexity are clearly understood, its mechanical consequences have received less attention. In this contribution we contrast polyconvexity with the recently rediscovered true-stress-true-strain monotonicity (TSTS-M${}^{++}!$) condition. By way of explicit examples, we show that neither condition by itself is strong enough to guarantee physically reasonable behavior for ideal isotropic elasticity. In particular, polyconvexity does not imply a monotone trajectory of the Cauchy stress in unconstrained uniaxial extension which TSTS-M${}^{++}!$ ensures. On the other hand, TSTS-M${}^{++}!$ does not impose a monotone Cauchy shear stress response in simple shear which is enforced by Legendre-Hadamard ellipticity and in turn polyconvexity. Both scenarios are proven through the construction of appropriate strain-energy functions. Consequently, a combination of polyconvexity, ensuring Legendre-Hadamard ellipticity, and TSTS-M${}^{++}!$ seems to be a viable solution to Truesdell’s Hauptproblem. However, so far no isotropic strain-energy function has been identified that satisfies both constraints globally at the same time. Although we are unable to deliver a valid solution here, we provide several results that could prove helpful in the construction of such an exceptional strain-energy function.

💡 Research Summary

The paper investigates two fundamental constitutive constraints that have been proposed to address Truesdell’s “Hauptproblem” in finite elasticity: polyconvexity and true‑stress‑true‑strain monotonicity (TSTS‑M⁺⁺). Polyconvexity, introduced as a mathematically convenient surrogate for quasiconvexity, guarantees existence of minimizers and implies rank‑one convexity and Legendre‑Hadamard ellipticity. However, its mechanical interpretation is vague, and it does not directly enforce monotonic stress‑strain behaviour in specific deformation modes.

TSTS‑M⁺⁺, recently revived, is an objective‑rate based monotonicity condition: the inner product of the Cauchy stress rate (taken with the Zaremba‑Jaumann rate) and the rate of deformation D must be positive for all admissible deformations. This condition is equivalent to the pointwise monotonicity ⟨σ−σ′, log V−log V′⟩ > 0 for any two states, i.e. the Cauchy stress is a strictly monotone function of the logarithmic strain. Consequently, TSTS‑M⁺⁺ guarantees a strictly increasing true‑stress response in unconstrained uniaxial extension‑compression.

The authors formulate four challenge problems that probe the sufficiency of each condition. They provide complete solutions to challenges (ii) and (iv): (ii) a polyconvex (indeed rank‑one convex) compressible energy that nevertheless exhibits a non‑monotonic true‑stress curve in uniaxial extension, and (iv) a compressible energy satisfying TSTS‑M⁺⁺ but showing a non‑monotonic true‑shear‑stress response in simple shear. Explicit families of strain‑energy functions are constructed, and rigorous proofs are given that the respective conditions hold while the opposite monotonicity fails.

These results demonstrate that polyconvexity alone does not ensure physically reasonable behaviour in uniaxial stretch, and TSTS‑M⁺⁺ alone does not guarantee shear‑stability. Polyconvexity, through Legendre‑Hadamard ellipticity, enforces monotone shear stress, whereas TSTS‑M⁺⁺ enforces monotone normal stress. Hence, a combination of both constraints appears necessary to satisfy Truesdell’s desiderata: existence of solutions, incremental stability, and physically plausible stress‑strain responses in all basic deformation modes.

The paper further discusses the difficulty of finding a single isotropic, compressible strain‑energy function that is globally polyconvex and globally satisfies TSTS‑M⁺⁺. No such function is known, and the authors provide several analytical observations that may guide future construction, such as the role of chain‑limited energies and convexity in the logarithmic strain. They also comment on the implications for data‑driven and neural‑network‑based material models, warning that imposing only polyconvexity may be insufficient to prevent unphysical responses.

In conclusion, the work clarifies the distinct mechanical consequences of polyconvexity and TSTS‑M⁺⁺, supplies concrete counter‑examples to their individual sufficiency, and argues that a synergistic enforcement of both conditions is a promising route toward a comprehensive solution of the Hauptproblem in isotropic hyperelasticity. Future research is encouraged to seek explicit energy functions that satisfy both constraints globally, possibly by exploiting the analytical tools introduced in this study.