Eur. Phys. J. E (2019) 42: 47
https://doi.org/10.1140/epje/i2019-11807-x

Regular Article

Disentangling geometrical, viscoelastic and hyperelastic effects in force-displacement relationships of folded biological tissues

¹ Physics Institute, University of Zurich, Zurich, Switzerland
² Institute of Molecular Life Sciences, University of Zurich, Zurich, Switzerland
³ Life Science Zurich Graduate School, ETH Zurich and University of Zurich, Zurich, Switzerland

^* e-mail: aegerter@physik.uzh.ch

Received: 27 June 2018
Accepted: 7 March 2019
Published online: 18 April 2019

Abstract

Drosophila wing discs show a remarkable variability when subject to mechanical perturbation. We developed a stretching bench that allows accurate measurements of instantaneous and time-dependent material behaviour of the disc as a whole, while determining the exact three-dimensional structure of the disc during stretching. Our experiments reveal force relaxation dynamics on timescales that are significant for development, along with a surprisingly nonlinear force-displacement relationship. Concurrently our imaging indicates that the disc is a highly heterogeneous tissue with a complex geometry. Using image-based 3D finite element modelling we are able to identify the contributions of size, shape and materials parameters to the measured force-displacement relations. In particular, we find that simulating the stretching of a disc with stiffness patterns in the extra-cellular matrix (ECM) recapitulates the experimentally found stretched geometries. In our simulations, linear hyperelasticity explains the measured nonlinearity to a surprising extent. To fully match the experimental force-displacement curves, we use an exponentially elastic material, which, when coupled to material relaxation also explains time-dependent experiments. Our simulations predict that as the disc develops, two counteracting effects, namely the discs foldedness and the hardening of the ECM lead to force-relative displacement curves that are nearly conserved during development.