Cracking of quasi-brittle structures under monotonic and cyclic loadings: A d+/d− damage model with stiffness recovery in shear

Published in International Journal of Solids and Structures Vol. 135, pp. 148–171, 2017
DOI: 10.1016/j.ijsolstr.2017.11.017

Abstract

In the present paper, a new $d^{+}/d^{-}$ damage model apt for quasi-brittle materials is described and its validation is carried out considering unreinforced concrete, reinforced concrete and masonry elements.

Two independent scalar damage variables, $d^{+}$ and $d^{-}$, in combination with the split of the reversible strain tensor into its positive and negative counterparts, are adopted in order to simulate the pronounced dissimilar response under tension and compression, typical of these materials. An energy-equivalent framework 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. In addition to the degradation parameters, the permanent strain tensor is also contemplated by the model and a modification of the exponential softening modulus is proposed in order to treat the evolution of the two causes of dissipation, damage and irreversible deformations, in a coupled way.

The formulation is integrated with a multidirectional damage procedure, addressed to extend the microcrack closure-reopening (MCR) capabilities of the model to shear cyclic conditions, characterized by orthogonal (or however intersecting) sets of cracks. Maintaining unaltered the dependence of the constitutive law from two scalar indeces, $d^{+}$ 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. The adequacy of the proposed constitutive model in reproducing experimental results has been proven for both monotonic and cyclic loading conditions. The two examples of application involving cyclic loads, dominated by 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, evolution of stiffness degradation and dissipation capacity.