On the network orientational affinity assumption in polymers and the micro–macro connection through the chain stretch

Variations on Ogden’s model: close and distant relatives

“…The average over M = 3 chains aligned with the three principal directions of strain N k may be seen as one particular realization of this average and has implicitly been used in the early developments of rubber-elasticity theory [45]. It has been noted [16,35] that this choice is in agreement with the additive decomposition suggested by Valanis & Landel [7],…”

“…Amores et al [35] recently proposed to define the chain stretch by the projection of the right stretch tensor as λ r = M • UM, and the ad hoc generalization, which contains the original relation [35] as The angle variable θ is an extension of θ to the interval 0 < φ ≤ 2π , in order to uniquely represent all unit vectors in the φ = π/2 plane with a single variable. Note that the scale in (b) is three times larger than that in (a).…”

“…In fact, relation(3.5) includes the models (3.2)-(3.4) and the one in[35] for r = 2, r = 2/β, r = 1 and r = β = 1, respectively, which differ in the general case of arbitrary M. However, for chains oriented along the Lagrangian principal directions, i.e. for M = N k with k = 1, 2, 3 referring to the eigenvectors of U, the relations (3.2)-(3.5) coincide independently of the values of β and r, and give λ r → λ β k = |FN k | β (3.6)…”

Crossing Scales: Data-Driven Determination of the Micro-scale Behavior of Polymers From Non-homogeneous Tests at the Continuum-Scale

“…All the required dimensions and the mesh employed in the simulation is shown in Figure 2. The reader can find further details about the non-affine deformation chain material model in Supplementary Appendix SC , a discussion about the suitability of the non-affine stretch employed in the model (free-fluctuating network assumption) is presented in Amores et al (2021).…”

“…The MMM approach predicts well any general deformation mode requiring just a single experimental curve to characterize the chain behavior, see (Amores et al, 2020, Figure 3). Since a similar Data-Driven MMM framework using the affine deformation measure (λ C ch ) 2 = r ⋅ C ⋅ r was not able to offer the same results, in Amores et al (2021) we questioned the affine micro stretch assumption from theoretical grounds, which seems to be the most popular in micro-structural models; Treloar (1975), Arruda and Boyce (1993), Alastrué et al (2009), Sáez et al (2011), Khiêm and Itskov (2016). In that same work the non-affine measure of deformation λ ch = r ⋅ U ⋅ r did show to be in accordance with the "controversial" C 2 slope observed by Mooney that up to the date could not be successfully explained from the classical statistical theory.…”