www.elsevier.com/locate/jmbbm Available online at www.sciencedirect.com Research Paper A novel strategy to identify the critical conditions for growth-induced instabilities A. Javili a,n , P. Steinmann a , E. Kuhl b a University of Erlangen–Nuremberg, Egerlandstr. 5, 91058 Erlangen, Germany b Stanford University, 496 Lomita Mall, Stanford, CA 94305, USA article info Article history: Received 11 April 2013 Received in revised form 6 August 2013 Accepted 13 August 2013 Available online 29 August 2013 Keywords: Growth Morphogenesis Residual stress Prestress Stability analysis Instabilities abstract Geometric instabilities in living structures can be critical for healthy biological function, and abnormal buckling, folding, or wrinkling patterns are often important indicators of disease. Mathematical models typically attribute these instabilities to differential growth, and characterize them using the concept of fictitious configurations. This kinematic approach toward growth-induced instabilities is based on the multiplicative decomposi- tion of the total deformation gradient into a reversible elastic part and an irreversible growth part. While this generic concept is generally accepted and well established today, the critical conditions for the formation of growth-induced instabilities remain elusive and poorly understood. Here we propose a novel strategy for the stability analysis of growing structures motivated by the idea of replacing growth by prestress. Conceptually speaking, we kinematically map the stress-free grown configuration onto a prestressed initial configuration. This allows us to adopt a classical infinitesimal stability analysis to identify critical material parameter ranges beyond which growth-induced instabilities may occur. We illustrate the proposed concept by a series of numerical examples using the finite element method. Understanding the critical conditions for growth-induced instabilities may have immediate applications in plastic and reconstructive surgery, asthma, obstruc- tive sleep apnoea, and brain development. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction Structural instabilities in the form of creases, folds, or wrinkles are inherent to living matter. In many living sys- tems, the formation of structural instabilities is critical to biological function, e.g., to increase the surface-to-volume ratio of the system (Wyczalkowski et al., 2012). Typical examples are wrinkling of skin (Buganza Tepole et al., 2011), villi formation in the intestine (Balbi and Ciarletta, 2013), and folding of the developing brain (Xu et al., 2010). In other biological systems, however, the formation of structural instabilities can be a critical hallmark of disease, e.g., when associated with a narrowing lumen. The most prominent example of this latter category is the folding of the mucous membrane in asthmatic airways (Wiggs et al., 1997). It is thus not surprising that the mathematical modeling of folding in tubular organs (Ciarletta and Ben Amar, 2012), in particular the modeling of the folding mucous membrane (Moulton and Goriely, 2011; Li et al., 2011; Xie et al., 2013), has drawn increasing scientific attention within the past decade. Continuum approaches toward the formation of geometric instabilities in living systems typically adopt the concept of 1751-6161/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jmbbm.2013.08.017 n Corresponding author. Tel.: þ49 9131 85 28502; fax: þ49 9131 85 28503. E-mail address: ali.javili@ltm.uni-erlangen.de (A. Javili). journal of the mechanical behavior of biomedical materials29 (2014) 20–32