Nonlinear multiscale simulation of instabilities due to growth of an elastic film on a microstructured substrate
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O R I G I NA L
Iman Valizadeh
· Oliver Weeger
Nonlinear multiscale simulation of instabilities due to growth of an elastic film on a microstructured substrate
Received: 30 December 2019 / Accepted: 27 June 2020 © The Author(s) 2020
Abstract The objective of this contribution is the numerical investigation of growth-induced instabilities of an elastic film on a microstructured soft substrate. A nonlinear multiscale simulation framework is developed based on the FE2 method, and numerical results are compared against simplified analytical approaches, which are also derived. Living tissues like brain, skin, and airways are often bilayered structures, consisting of a growing film on a substrate. Their modeling is of particular interest in understanding biological phenomena such as brain development and dysfunction. While in similar studies the substrate is assumed as a homogeneous material, this contribution considers the heterogeneity of the substrate and studies the effect of microstructure on the instabilities of a growing film. The computational approach is based on the mechanical modeling of finite deformation growth using a multiplicative decomposition of the deformation gradient into elastic and growth parts. Within the nonlinear, concurrent multiscale finite element framework, on the macroscale a nonlinear eigenvalue analysis is utilized to capture the occurrence of instabilities and corresponding folding patterns. The microstructure of the substrate is considered within the large deformation regime, and various unit cell topologies and parameters are studied to investigate the influence of the microstructure of the substrate on the macroscopic instabilities. Furthermore, an analytical approach is developed based on Airy’s stress function and Hashin–Shtrikman bounds. The wavelengths and critical growth factors from the analytical solution are compared with numerical results. In addition, the folding patterns are examined for two-phase microstructures and the influence of the parameters of the unit cell on the folding pattern is studied. Keywords Growth-induced instabilities · Microstructures · Concurrent multiscale simulation · Finite element squared method
1 Introduction Biological tissues are an important class of soft materials. Due to their low stiffness, they are subjected to large deformations, and in particular to morphological instabilities, which can be induced due to mechanical forces, changes in temperature, tissue growth, etc. As a result of having a low elastic modulus, soft materials are especially prone to surface instabilities, for example, induced by buckling. These instabilities mostly occur in the form of wrinkling and folding. The evolution of structural instabilities plays a vital role, which appropriates the biological function of a living system [1]. For instance, in brain development of mammals, I. Valizadeh (B) · O. Weeger Cyber-Physical Simulation Group, Department of Mechanical Engineering, Technical University of Darmstadt, Dolivostraße 15, 64293 Darmstadt, Germany E-mail: valiz
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