Nonlinear analysis of masonry and concrete structures under monotonic and cyclic loading: a regularized multidirectional d+/d− damage model

A rigorous structural analysis is fundamental in the safety assessment of the built heritage and in its efficient conservation and rehabilitation. In line with the necessity of refined techniques, the objective of the present thesis is to develop and validate, in a displacement-based finite element framework, a nonlinear model apt for the study of masonry and concrete structures under monotonic and cyclic loading. The proposed constitutive law adopts two independent scalar damage variables, d+ and d−, in combination with the spectral decomposition of the elastic strain tensor, to simulate the pronounced dissimilar response under tension and compression, typical of these materials. The assumption of energy-equivalence between the damaged solid and the effective (undamaged) one is considered for representing the orthotropy induced in the material by the degradation process, with the consequence that a thermodynamically consistent constitutive operator, positive definite, symmetric and strain-driven, is derived. The formulation is integrated with a multidirectional damage procedure, addressed to extend the microcrack closure-reopening (MCR) capabilities to generic cyclic conditions, especially shear cyclic conditions, making the model suitable for dealing with seismic actions. Maintaining unaltered the dependence of the constitutive law from d+ and d−, this approach activates or deactivates a tensile (compressive) damage value on the base of the current maximum (minimum) principal strain direction. In correspondence with damage activation (crack opening) or deactivation (crack closure), a smooth transition is introduced, in order to avoid abrupt changes in stiffness and enhance the numerical performance and robustness of the multidirectional procedure. Moreover, the mesh-objectivity of the numerical solutions is ensured by resorting to a nonlocal regularization technique, based on the adoption of damage variables driven by an averaged elastic strain tensor. To perform the averaging of the strain tensor, an internal length lRG is considered in the continuum. The strategy chosen to define the parameters affecting the softening behaviour consists in the modification of the local softening law on the base of the internal length, with the intent of ensuring the proper evaluation of the correct fracture energy Gf. The adequacy of the proposed constitutive model in reproducing experimental results is proven for both monotonic and cyclic loading conditions. Under monotonic loads, unreinforced concrete notched elements subjected to pure tension, pure bending and mixed-mode bending are studied. The two examples of application involving cyclic loads, a masonry and a reinforced concrete wall under in-plane cyclic shear, constitute a validation of the multidirectional damage approach, showing how the suitable representation of unilateral effects and permanent deformations is essential to model the observed structural response in terms of maximum resistance and dissipation capacity. The effectiveness of the regularized damage formulation is proven by successfully studying a masonry arch and reinforced and unreinforced concrete elements. Besides the validation of the numerical results with experimental or analytical data, each application is exploited to highlight one or more features of the formulation: the mesh-size and mesh-bias independence of the results, the effect of the choice of the variable to be averaged, the possibility to reproduce structural size effects, the influence of the internal length lRG. On this latter aspect, the almost null dependence of the regularized solutions on the internal length in terms of force-displacement curves, achieved thanks to the calibration strategy adopted to define the energy dissipation, suggests the interpretation of the internal length as a regularization parameter. On the one hand, this implies an analogy between the role played by the nonlocal internal length in a nonlocal model and the one’s of the mesh size in the crack band approach (Bažant and Oh, 1983). On the other hand, this translates in the versatility of the regularized damage model, which requires only the identification of the standard material properties (elastic constants, fracture energies and strengths). Finally, the d+/d− damage model is successfully applied to the study of a three-span masonry arch bridge subjected to a concentrated vertical load, in order to evaluate its carrying capacity and its failure mechanism. Numerical issues, usually neglected in large-scale applications, are also addressed proving the reliability of the regularized approach to provide mesh-independent results and its applicability.