On the advantages of mixed formulation and higher-order elements for computational morphoelasticity

In this paper, we present a mixed displacement–pressure finite element formulation that can successively model compressible as well as truly incompressible behaviour in growth-induced deformations significantly observed in soft materials. Inf–sup stable elements of various shapes based on quadratic Bézier elements are employed for spatial discretisation. At first, the capability of the proposed framework to accurately model finite-strain growth-induced deformations is illustrated using several examples of plate models in which numerical results are directly compared with analytical solutions. The framework is also compared with the classical Q1/P0 finite element that has been used extensively for simulating the deformation behaviour of soft materials using the quasi-incompressibility assumption. The comparisons clearly demonstrate the superior capabilities of the proposed framework. Later, the effect of hyperelastic constitute models and compressibility on the growth-induced deformation is also studied using the example of a bilayered strip in three dimensions. Finally, the potential of the proposed finite element framework to simulate growth-induced deformations in complex three-dimensional problems is illustrated using the models of flower petals, morphoelastic rods, and thin cylindrical tubes