A Comparison Between Active Strain and Active Stress in Transversely Isotropic Hyperelastic Materials

Abstract

Active materials are media for which deformations can occur in absence of loads, given an external stimulus. Two approaches to the modeling of such materials are mainly used in literature, both based on the introduction of a new tensor: an additive stress Pact in the active stress case and a multiplicative strain Fa in the active strain one. Aim of this paper is the comparison between the two approaches on simple shears.Considering an incompressible and transversely isotropic material, we design constitutive relations for Pact and Fa so that they produce the same results for a uniaxial deformation along the symmetry axis. We then study the two approaches in the case of a simple shear deformation. In a hyperelastic setting, we show that the two approaches produce different stress components along a simple shear, unless some necessary conditions on the strain energy density are fulfilled. However, such conditions are very restrictive and rule out the usual elastic strain energy functionals. Active stress and active strain therefore produce different results in shear, even if they both fit uniaxial data.Our results show that experimental data on the stress-stretch response on uniaxial deformations are not enough to establish which activation approach can capture better the mechanics of active materials. We conclude that other types of deformations, beyond the uniaxial one, should be taken into consideration in the modeling of such materials. Date

An orthotropic electro-viscoelastic model for the heart with stress-assisted diffusion

“…We note that the curves produced without SAD are substantially lagged (as expected from the choice of diffusion parameters) with respect to the two other cases, that display no major discrepancies. The remaining panels in the figure show point-wise transients of the main mechanical and electrical fields measured on the point (x 0 , y 0 , z 0 ) = (25,25,10). The evolution of the electric and activation fields remains very similar in all three cases; for instance the shape of the action potential is almost not modified after adding SAD or viscous contribution and for the other fields also very subtle differences are observed (the calcium concentration was slightly shifted to the left in the hyperelastic and viscoelastic cases).…”

“…Another difference in the present contribution is that we employ a more accurate cellular model, tailored for recovering human action potential dynamics, restitution features under constant pacing as well as sustained fibrillation behaviours and spiral waves breakup [5]. While the active strain approach is adopted in many instances in the literature and is often favoured due to the practicality of measuring strains directly using imaging techniques [58], the active stress approach is somewhat simpler and more naturally incorporated in already existing models for passive deformation [25]. In this work, we will adopt both formulations, although we find that the active strain formulation better reproduces physiologically accurate deformation regimes in ventricular geometries.…”

Activation of a muscle as a mapping of stress–strain curves

“…Thus, we describe the muscle as a mixture of passive (like elastin, randomly distributed collagen) and active materials (like the sarcomeres). We call this approach mixture active strain [14,11,22,10].…”

“…where C = F T F is the right Cauchy-Green tensor. We choose to model the isotropic part of the muscle as a Gent material [9], so that the strain energy density is given by (10) ψ iso (F) = − µI max 2 log 1 − I 1 − 3 I max where µ is the shear modulus and I max is a parameter that sets the maximum value reachable by I 1 .…”

“…Nevertheless, the preservation of the well posedness of the mathematical problem is not always guaranteed by the latter approach [24,1] and could manifest itself in unexpected numerical issues. The active stress formulation is so general that includes the active strain as a particular case [10].…”