Continuum damage model for nonlinear analysis of masonry structures
Authors:   , Pelà Luca 
Published: October 2023

The present work focuses on the formulation of a Continuum Damage Mechanics model for nonlinear analysis of masonry structural elements. The material is studied at the macro-level, i.e. it is modelled as a homogeneous orthotropic continuum.
The orthotropic behaviour is simulated by means of an original methodology, which is based on nonlinear damage constitutive laws and on the concept of mapped tensors from the anisotropic real space to the isotropic fictitious one. It is based on establishing a one-to-one mapping relationship between the behaviour of an anisotropic real material and that of an isotropic fictitious one. Therefore, the problem is solved in the isotropic fictitious space and the results are transported to the real field. The application of this idea to strain-based Continuum Damage Models is rather innovative.
The proposed theory is a generalization of classical theories and allows us to use the models and algorithms developed for isotropic materials. A first version of the model makes use of an isotropic scalar damage model. The adoption of such a simple constitutive model in the fictitious space, together with an appropriate definition of the mathematical transformation between the two spaces, provides a damage model for orthotropic materials able to reproduce the overall nonlinear behaviour, including stiffness degradation and strain-hardening/softening response.
The relationship between the two spaces is expressed in terms of a transformation tensor which contains all the information concerning the real orthotropy of the material. A major advantage of this working strategy lies in the possibility of adjusting an arbitrary isotropic criterion to the particular behaviour of the orthotropic material. Moreover, orthotropic elastic and inelastic behaviours can be modelled in such a way that totally different mechanical responses can be predicted along the material axes.
The aforementioned approach is then refined in order to account for different behaviours of masonry in tension and compression. The aim of studying a real material via an equivalent fictitious solid is achieved by means of the appropriate definitions of two transformation tensors related to tensile or compressive states, respectively. These important assumptions permit to consider two individual damage criteria, according to different failure mechanisms, i.e. cracking and crushing. The constitutive model adopted in the fictitious space makes use of two scalar variables, which monitor the local damage under tension and compression, respectively. Such a model, which is based on a stress tensor split into tensile and compressive contributions that allows the model to contemplate orthotropic induced damage, permits also to account for masonry unilateral effects. The orthotropic nature of the Tension-Compression Damage Model adopted in the fictitious space is demonstrated. This feature, both with the assumption of two distinct damage criteria for tension and compression, does not permit to term the fictitious space as “isotropic”. Therefore, the proposed formulation turns the original concept of “mapping the real space into an isotropic fictitious one” into the innovative and more general one of “mapping the real space into a favourable (or convenient) fictitious one”. Validation of the model is carried out by means of comparisons with experimental results on different types of orthotropic masonry.
The model is fully formulated for the 2-dimensional case. However, it can be easily extended to the 3-dimensional case. It provides high algorithmic efficiency, a feature of primary importance when analyses of even large scale masonry structures are carried out. To account for this requisite it adopts a strain-driven formalism consistent with standard displacement-based finite element codes. The implementation in finite element programs is straightforward.
Finally, a localized damage model for orthotropic materials is formulated. This is achieved by means of the implementation of a crack tracking algorithm, which forces the crack to develop along a single row of finite elements. Compared with the smeared cracking approach, such an approach shows a better capacity to predict realistic collapsing mechanisms. The resulting damage in the ultimate condition appears localized in individual cracks. Moreover, the results do not suffer from spurious mesh-size or mesh-bias dependence. The numerical tool is finally validated via a finite element analysis of an in-plane loaded masonry shear wall.