Creases in soft tissues generated by growth

Soft tissues growing under constraint often form creases. We adopt the model of growth that factors the deformation gradient into a growth tensor and an elastic deformation tensor, and show that the critical conditions for the onset of creases take a remarkably simple form. The critical conditions are illustrated with tubes of tissues growing either inside a rigid shell or outside a rigid core. By comparing the critical conditions for the onset of wrinkles, we show that the creases are the preferred type of instability. Furthermore, deep creases in a tube are simulated by using the finite-element method, and the number of creases in the tube is estimated by minimizing the free energy. Copyright c © EPLA, 2011 Soft tissues often grow under constraint. For example, many tubular organs —arteries, airways, esophagi and intestines— grow under the constraint of a stiffer tissue, the smooth muscle [1,2]. Some tumors grow under the constraint of necrotic cores [3]. Even in the absence of any stiff tissues, soft tissues develop internal constraint when the growth is inhomogeneous [4,5]. Constrained growth can cause the soft tissues to undulate and fold. These mechanical instabilities have physiological and pathological consequences [3,6–11]. When the smooth muscle of airways shortens, the mucosa folds and obstructs the airways; the amount of obstruction increases in asthma due to the thickening of the airway walls [6,7]. Buckling can enable the invagination of embryos [9], and the primordial development of sunflowers [10]. Fingerprint patterns can result from the buckling of the layer of basal cells of the fetal epidermis [11]. The shape of a tumor can indicate its type [3]. Figure 1 illustrates several ubiquitous patterns of instability. For a soft material covered with a stiff thin layer, compression beyond a critical level leads to wrinkles [12]. Further compression will turn some of the wrinkles into folds [13,14]. For a soft material not covered with a stiff layer, however, compression beyond a critical level leads to creases, without forming wrinkles first [15–21]. While the wrinkles keep the surface of the material locally smooth, the folds and creases cause the surface to self-contact. The (a)E-mail: [email protected] (a) wrinkles (b) folds (c) creases soft tissue soft tissue soft tissue stiff layer stiff layer Fig. 1: (Color online) Schematic illustrations of three ubiquitous patterns of mechanical instability. tip of a fold remains open due to the bending rigidity of the stiff layer, but the tip of a crease is sharp. The onset of wrinkles corresponds to superposing a field of strain infinitesimal in amplitude, but finite in space. By contrast, the onset of creases corresponds to superposing a field of strain finite in amplitude, but infinitesimal in space. Consequently, the critical condition for the onset of wrinkles can be determined by a classical linear perturbation analysis, while the critical condition for the onset of creases cannot. Several recent papers have studied creases in soft materials caused by external forces [15–20]. This paper studies creases in tissues undergoing constrained growth. We show that the critical conditions for the onset of the growth-induced creases take a remarkably simple form. The critical conditions are illustrated with tubes of tissues growing either inside a rigid shell or outside a rigid