Active elasticity drives the formation of periodic beading in damaged axons

In several pathological conditions, such as coronavirus infections, multiple sclerosis, Alzheimer's and Parkinson's diseases, the physiological shape of axons is altered and a periodic sequence of bulges appears. Experimental evidences suggest that such morphological changes are caused by the disruption of the microtubules composing the cytoskeleton of the axon. In this paper, we develop a mathematical model of damaged axons based on the theory of continuum mechanics and nonlinear elasticity. The axon is described as a cylinder composed of an inner passive part, called axoplasm, and an outer active cortex, composed mainly of F-actin and able to contract thanks to myosin-II motors. Through a linear stability analysis we show that, as the shear modulus of the axoplasm diminishes due to the disruption of the cytoskeleton, the active contraction of the cortex makes the cylindrical configuration unstable to axisymmetric perturbations, leading to a beading pattern. Finally, the non-linear evolution of the bifurcated branches is investigated through finite element simulations

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

The mathematical modeling of the contraction of a muscle is a crucial problem in biomechanics. Several different models of muscle activation exist in literature. A possible approach to contractility is the so-called active strain: it is based on a multiplicative decomposition of the deformation gradient into an active contribution, accounting for the muscle activation, and an elastic one, due to the passive deformation of the body. We show that the active strain approach does not allow to recover the experimental stress-stretch curve corresponding to a uniaxial deformation of a skeletal muscle, whatever the functional form of the strain energy. To overcome such difficulty, we introduce an alternative model, that we call mixture active strain approach, where the muscle is composed of two different solid phases and only one of them actively contributes to the active behavior of the muscle

Shape transitions in a soft incompressible sphere with residual stresses

Residual stresses may appear in elastic bodies, owing to the formation of misfits in the microstructure, driven by plastic deformations or thermal or growth processes. They are especially widespread in living matter, resulting from dynamic remodelling processes aimed at optimizing the overall structural response to environmental physical forces. From a mechanical viewpoint, residual stresses are classically modelled through the introduction of a virtual incompatible state that collects the local relaxed states around each material point. In this work, we employ an alternative approach based on a strain energy function that constitutively depends only on the deformation gradient and the residual stress tensor. In particular, our objective is to study the morphological stability of an incompressible sphere, made of a neo-Hookean material, and subjected to given distributions of residual stresses. The boundary value elastic problem is studied with analytic and numerical tools. Firstly, we perform a linear stability analysis on the prestressed solid sphere using the method of incremental deformations. The marginal stability conditions are given as a function of a control parameter, which is the dimensionless variable that represents the characteristic intensity of the residual stresses. Secondly, we perform finite-element simulations using a mixed formulation in order to investigate the postbuckling morphology in the fully nonlinear regime. Considering different initial distributions of the residual stresses, we find that different morphological transitions occur around the material domain, where the hoop residual stress reaches its maximum compressive value. The loss of spherical symmetry is found to be controlled by the mechanical and geometrical properties of the sphere, as well as the spatial distribution of the residual stress. The results provide useful guidelines for designing morphable soft spheres, for example by controlling residual stresses through active deformations. They finally suggest a viable solution for the nondestructive characterization of residual stresses in soft tissues, such as solid tumours

Tunable morphing of electroactive dielectric-elastomer balloons

Designing smart devices with tunable shapes has important applications in industrial manufacture. In this paper, we investigate the nonlinear deformation and the morphological transitions between buckling, necking, and snap-through instabilities of layered DE balloons in response to an applied radial voltage and an inner pressure. We propose a general mathematical theory of nonlinear electro-elasticity able to account for finite inhomogeneous strains provoked by the electro-mechanical coupling. We investigate the onsets of morphological transitions of the spherically symmetric balloons using the surface impedance matrix method. Moreover, we study the nonlinear evolution of the bifurcated branches through finite element numerical simulations. Our analysis demonstrates the possibility to design tunable DE spheres, where the onset of buckling and necking can be controlled by geometrical and mechanical properties of the passive elastic layers. Relevant applications include soft robotics and mechanical actuators

Flattened and wrinkled encapsulated droplets: Shape-morphing induced by gravity and evaporation

We report surprising morphological changes of suspension droplets (containing class II hydrophobin protein HFBI from Trichoderma reesei and water) as they evaporate with a contact line pinned on a rigid solid substrate. Both pendant and sessile droplets display the formation of an encapsulating elastic film as the bulk concentration of solute reaches a critical value during evaporation, but the morphology of the droplet varies significantly: for sessile droplets, the elastic film ultimately crumples in a nearly flattened area close to the apex while in pendant droplets, circumferential wrinkling occurs close to the contact line. These different morphologies are understood through a gravito-elasto-capillary model that predicts the droplet morphology and the onset of shape changes, as well as showing that the influence of the direction of gravity remains crucial even for very small droplets (where the effect of gravity can normally be neglected). The results pave the way to control droplet shape in several engineering and biomedical applications.Comment: 5 pages, 4 figure

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

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