Generalisation of non-iterative methods for the modelling of structures under non-proportional loading

Localisation of initially distributed cracking is a numerical challenging task, which is difficult to accomplish with conventional iterative methods, e.g. the Newton-Raphson method (Crisfield in Comput Aided Anal Des Concr Struct (1):331-358, 1984). A total approach, such as the sequentially linear approach (SLA), has been used to overcome convergence problems. However, the use of a total approach in combination with non-proportional loading raises important difficulties due to: (1) an incomplete description of the material loading history; and (2) incremental nonlinear behaviour due to the rotation of the principal stress directions. In this manuscript, two existing methods adopting combined total and incremental approaches are extended to non-proportional loading conditions. In these methods, preferential use of the incremental approach is made and the total approach is adopted only when critical bifurcations points are found. Comparison with a purely total non-iterative method (SLA) is performed, and the numerical results are validated using experimental tests available in literature.