On a path-following method for non-linear solid mechanics with applications to structural and cardiac mechanics subject to arbitrary loading scenarios

• Displacement-control method for arbitrary loading scenarios.
• Cavity-volume-control method for multiple independently acting cavity-pressure loadings.
• Adaptation of the displacement- and the cavity-volume control method to numerical modelling frameworks which do not possess the Kronecker delta property.
• Computational modelling of the plastic post-buckling behaviour of shells.
• Computational modelling of a bi-ventricular heart model.

In computational solid mechanics path-following methods have been proven useful when dealing with non-monotonously evolving loading magnitudes as encountered e.g. in instability or softening behaviour. The present work derives from a displacement-based method a cavity-volume-based path-following method and considers its application specifically to cardiac mechanics problems. Both methods are able to account for an arbitrary number of simultaneously acting loading conditions. When applied to the Newton-Raphson method, the corresponding loading increments are computed by means of volume increments or point-wise prescribed displacement increments, respectively. No additional variables are required and the physics of the problem at hands is not altered. Both methods are implemented in an in-house meshfree modelling software and successfully applied to non-linear elastic and inelastic problems in structural mechanics. The cavity-volume control method, in particular, is demonstrated to accurately predict the highly non-linear elastic and anisotropic material behaviour encountered when modelling the heart. Albeit, the proposed methods can be equally used e.g. in finite element methods, they are very well suited for meshfree methods where the Kronecker delta property does not apply.