Computational aspects of growth-induced instabilities through eigenvalue analysis

Abstract

The objective of this contribution is to establish a computational framework to study growth-induced instabilities. The common approach towards growth-induced instabilities is to decompose the deformation multiplicatively into its growth and elastic part. Recently, this concept has been employed in computations of growing continua and has proven to be extremely useful to better understand the material behavior under growth. While finite element simulations seem to be capable of predicting the behavior of growing continua, they often cannot naturally capture the instabilities caused by growth. The accepted strategy to provoke growth-induced instabilities is therefore to perturb the solution of the problem, which indeed results in geometric instabilities in the form of wrinkles and folds. However, this strategy is intrinsically subjective as the user is prescribing the perturbations and the simulations are often highly perturbation-dependent. We propose a different strategy that is inherently suitable for this problem, namely eigenvalue analysis. The main advantages of eigenvalue analysis are that first, no arbitrary, artificial perturbations are needed and second, it is, in general, independent of the time step size. Therefore, the solution obtained by this methodology is not B C. Linder subjective and thus, is generic and reproducible. Equipped with eigenvalue analysis, we are able to compute precisely the critical growth to initiate instabilities. Furthermore, this strategy allows us to compare different finite elements for this family of problems. Our results demonstrate that linear elements perform strikingly poorly, as compared to quadratic elements.

Computational homogenization of nano‐materials accounting for size effects via surface elasticity

“…[18][19][20][21]. Computational aspects of surface elasticity theory using the finite element method are detailed in [22][23][24].…”

Mixed isogeometric analysis of strongly coupled diffusion in porous materials

“…In addition, we provided a mathematical proof to show that the inf‐sup condition in mixed elasticity and Stokes flow is sufficient to prove stability in poromechanics problems. This work is expected to provide a framework for more application‐based problems such as geometrical instabilities() observed in swelling hydrogels due to coupled diffusion() or a diffusion‐induced fracture. ()…”

Nonlinear dynamics of uniformly loaded Elastica: Experimental and numerical evidence of motion around curled stable equilibrium configurations

“…Furthermore, the substrate provides a versatile medium to assign distributed loads on the beam similar to thin films on elastic foundations. [74,76,89] A further remark is needed here: the introduced discrete model is intrinsically nonlinear. It naturally incorporates geometrical and material non-linearities: this is the reason for which, also when one linearizes it, he can avoid all the issues implied by the loss of objectivity of the energy when, in the neighborhood of a certain equilibrium configuration, the linearization is performed.…”

“…The advantage of this approach is that the fictitious bulk material can regularize the behavior of the beam and this allows to analyze buckling in a computational framework. Furthermore, the substrate provides a versatile medium to assign distributed loads on the beam similar to thin films on elastic foundations …”