2024/09/30 by Neff, Patrizio, Holthausen, Sebastian, d'Agostino, Marco Valerio +4 · 1 citation
#15A24 #73G05 #73G99 #74B20 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2409.20051
We combine the rate-formulation for the objective, corotational Zaremba-Jaumann rate \frac\rm D\rm ZJ\rm D t [σ] = ℍ\rm ZJ(σ).D, D = \rm sym \rm D v , operating on the Cauchy stress σ, the Eulerian strain rate D and the spatial velocity v with the novel \enquotecorotational stability postulate (CSP)⟨ \frac\rm D\rm ZJ\rm D t[σ], D ⟩ gt; 0 ∀ D∈\rm Sym(3)∖\0\ to show that for a given isotropic Cauchy-elastic constitutive law B ↦ σ(B) in terms of the left Cauchy-Green tensor B = F FT, the induced fourth-order tangent stiffness tensor ℍ\rm ZJ(σ) is positive definite if and only if for \widehatσ(log B):=σ(B), the strong monotonicity condition (TSTS-M++) in the logarithmic strain is satisfied. Thus (CSP) implies (TSTS-M++) and vice-versa, and both imply the invertibility of the hypo-elastic material law between the stress and strain rates given by the tensor ℍ\rm ZJ(σ). The same characterization remains true for the corotational Green-Naghdi rate as well as the corotational logarithmic rate, conferring the corotational stability postulate (CSP) together with the monotonicity in the logarithmic strain tensor (TSTS-M++) a far reaching generality. It is conjectured that this characterization of (CSP) holds for a large class of reasonable corotational rates. The result for the logarithmic rate is based on a novel chain rule for corotational derivatives of isotropic tensor functions.