An Ogden-like formulation incorporating phase-field fracture in elastomers: from brittle to pseudo-ductile failures.

Over the past 50 years the Ogden model has been widely used in material modelling owing to its ability to match accurately the experimental data on elastomers at large strain, as well as its mathematical properties, such as polyconvexity. In this paper, these characteristics are exploited to formulate a finite-strain model that incorporates, through the phase-field approach recently proposed by Wu (Wu 2017 J. Mech. Phys. Solids 103, 72-99) for small strains, a cohesive damage mechanism which leads to the progressive degradation of the material stiffness and to failure under tension. By properly tailoring the constitutive parameters, the model is capable of encompassing a wide range of effects, from brittle to pseudo-ductile failure modes. A plane stress problem is formulated to test the model against experiments on double-network elastomers, which display a pseudo-ductile damage behaviour at large strain, and on conventional rubber compounds with brittle failure. The results show that the proposed model is applicable to fracture coalescence and propagation in a wide range of materials. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

Torque-induced reorientation in active fibre-reinforced materials.

We introduce a continuum model for a fibre reinforced material in which the reference orientation of the fibre may evolve with time, under the influence of external stimuli. The model is formulated in the framework of large strain hyperelasticity and the kinematics of the continuum is described by both a position vector and by a remodelling tensor which, in the present context, is an orthogonal tensor representing the fibre reorientation process. By imposing suitable thermodynamical restrictions on the constitutive equation, we obtain an evolution equation of the remodelling tensor governed by the Eshelby torque, whose stationary solutions are studied in absence of any external source terms. It is shown that the fibres reorient themselves in a configuration that minimises the elastic energy and get aligned along a direction that may or may not be of principal strain. The explicit analysis of the Hessian of the strain energy density allows us to discriminate among the stationary solutions, which ones are stable. Examples are given for passive reorientation processes driven by applied strains or external boundary tractions. Applications of the proposed theory to biological tissues, nematic or magneto-electro active elastomers are foreseen.