Using a path-following algorithm to analyze a quasi-static nonlinear structural problem involves selecting an appropriate constraint function. This function should improve the desired performance targets of the path-following algorithm such as robustness, speed, accuracy, and smoothness. In order to be able to draw a fair objective selection of a constraint function, it is necessary to collect adequate constraint equations as well as to define the performance of nonlinear methods. In this paper, three new path-following constraints applicable for damage analysis of quasi-brittle materials are proposed. Additionally, performance criteria and their numerical measures for a posteriori assessment of robustness, smoothness, accuracy, and speed of solving nonlinear problems by a path-following method are proposed. Based on the proposed criteria, the performance of the three new constraints and two existing ones is compared for two example problems. As a result, the performance measures are shown to possess an ability to clearly explore the strengths of each constraint. They establish a firm basis for the assessment of not only path-following methods but also other methods for solving nonlinear structural problems.

Two anisotropic stress-based gradient-enhanced damage models are proposed to address the issue of spurious damage growth typical of continuous standard gradient-enhanced damage models. Both models are based on a decreasing interaction length upon decreasing stresses and do not require additional model parameters or extra degrees of freedom when compared to standard gradient-enhanced models. It is observed that with the proposed models damage spreading is significantly reduced due to the occurrence of non-physical oscillations in the nonlocal strain field near the strain localization band. Model improvements to eliminate these strain oscillations upon vanishing length scale values are proposed. The capability of the models and their patched versions to correctly simulate damage initiation and propagation is investigated by means of mode-I failure, shear band and four-point bending tests.