Inelastic material formulations based on a co-rotated intermediate configuration—Application to bioengineered tissues

Abstract

In the field of material modeling, there is a general trend to include more and more complex phenomena in the modeling, making the models’ theoretical derivation and numerical implementation extremely difficult. In particular, modeling inelastic material behavior under finite deformations in a continuum mechanical manner, e.g. to simulate soft biological tissues, remains one of the most challenging tasks. Unfortunately, the multiplicative decomposition of the deformation gradient usually utilized in this context suffers from an inherent rotational non-uniqueness, making it not straightforward to combine this approach with algorithmic differentiation (AD)—a very helpful tool in modern computational mechanics. To address this issue in the growth and remodeling model proposed herein, a novel co-rotated intermediate configuration is introduced. This configuration shares essential characteristics with the intermediate one, but is uniquely defined and applicable to a wide range of inelastic materials. In this regard, the concept of structural tensors, hardening effects, and a thermodynamically consistent derivation are discussed as well. Since the stress-driven growth model presented is based on the approach of homeostatic surfaces by Lamm et al. (2022), a large number of derivatives of potentials and energies are required, which can be elegantly implemented using AD due to the co-rotated formulation. Moreover, fiber remodeling of collagen fibers is taken into account in a stress-driven manner using AD. Finally, qualitative comparisons are made with recently published experiments by Eichinger et al. (2020) in uniaxial and multiaxial settings, revealing the efficient combination of the proposed framework and the material model.