The continuous Airy-based for stress-singularities (CASS) method: an energy-based numerical formulation for unilateral materials

Unreinforced masonry buildings represent a large part of the world's architectural heritage. Despite their significant diffusion, their mechanical behaviour is still subjected to increasing scientific investigations as modelling difficulties arise from the negligible tensile strength in tension. Because of this unilateral response, masonry constructions can flexibly accommodate different loading and displacement-like boundary conditions in a new deformed configuration through the formation of crack patterns. The internal stress state changes accordingly and can be represented through stress singularities. Defining the internal stress states compatible with general boundary conditions is primarily essential from an engineering perspective. This paper presents the Continuous Airy-based for Stress-Singularities (CASS) method, a new numerical strategy to model the internal stress state in unilateral materials, whose main peculiarity is the ability in describing singular stress fields independently of the discretisation. The internal equilibrium is guaranteed using the Airy stress potential discretising the domain with plate-type finite elements. The compatibility with boundary displacements is satisfied through a variational formulation based on the minimum of the Total Complementary Energy. The boundary value problem is then reduced to a second-order cone programming where the linear objective function represents the work of the emerging stress for the boundary displacements, and the constraints enforce the boundary equilibrium and the material compatibility. The CASS method is benchmarked against analytical solutions and other numerical strategies on several 2D problems, also used to illustrate and discuss its main features. Lastly, a real masonry facade is considered to show the method's effectiveness clearly.