This paper contributes to the formulation of continuum damage models for orthotropic materials under plane stress conditions. Two stress transformation tensors, related to tensile and compressive stress states, respectively, are used to establish a one-to-one mapping relationship between the orthotropic behaviour and an auxiliary model. This allows the consideration of two individual damage criteria, according to different failure mechanisms, i.e. cracking and crushing. The constitutive model adopted in the mapped space makes use of two scalar variables which monitor the local damage under tension and compression, respectively. The model affords the simulation of orthotropic induced damage, while also accounting for unilateral effects, thanks to a stress tensor split into tensile and compressive contributions. The fundamentals of the method are presented together with the procedure utilized to adjust the model in order to study the mechanical behaviour of masonry material. The validation of the model is carried out by means of comparisons with experimental results on different types of orthotropic masonry at the material level.