A higher-order morphoelastic beam model for tubes and filaments subjected to biological growth

Abstract

Slender structures, in which width and thickness are much smaller than the length, are common in the natural world, such as wheat straws, bean sprouts, and biofilaments. Due to biological growth, these structures exhibit diverse morphological profiles with curling hairs, curling leaves, and twining plants as some examples. Accurate prediction of growth induced instabilities and corresponding shear deformation require a higher-order morphoelastic beam theory for growing tubes and filaments, for which we present a theory here. In the theory, the deformation gradient is decomposed into elastic deformation and growth using multiplicative decomposition. Appropriate assumptions for slender structures with circular cross-section are introduced and the displacement field and constitutive relations are derived from three-dimensional morphoelasticity. Corresponding variational principle is established and equilibrium equations are obtained. Our higher-order beam with growth theory can model growth-induced instability and shear deformation of growing tubes and filaments with circular cross-sections. Importantly, it can predict shear stress distribution and locations of maximum transverse shear stress in the circular cross-section. The shear stress, critical growth and post-buckling predictions of our proposed model and analytical solutions were validated by comparing its predictions against results from three-dimensional finite element simulations. Growth-induced instability and corresponding shear stress distributions were analyzed and the results are discussed. Our model can provide improved prediction for critical growth induced instabilities which are overestimated if the shear deformation is not considered.