A nonlocal anisotropic damage model is proposed for quasi-brittle materials, such as concrete and rock. The local anisotropic damage model is formulated by combining a free energy derived from micromechanics with phenomenological yield criteria and damage potentials. The trace of the total strain is used to distinguish tensile and compressive loading paths, and to account for the influence of the confining pressure on the propagation of compression cracks. Yield criteria in tension and compression are expressed in terms of equivalent strains, which depend on the difference between principal strain components. A non-local measure of strain is used to avoid localization. Constitutive parameters are calibrated against published experimental data for concrete and shale. Simulations of three-point bending tests show that non-local enhancement is necessary and efficient to avoid mesh dependency upon strain softening. Simulations of borehole excavation damage zone show that the damage model is not mesh dependent upon stress hardening. Numerical predictions are in agreement with experimental observations and the model can capture unilateral effects, tensile softening, compressive hardening and confinement dependent compressive behavior.It is still a challenge to formulate a constitutive model that captures damage induced anisotropy and stiffness reduction, unilateral effects due to crack closure and a transition from brittle to ductile behavior at increasing confining stress (

Krajcinovic et al., 1991; Chiarelli et al., 2003

). What is the most appropriate way to express the free energy so that the direction dependence of crack inception and propagation can be captured? How to define the anisotropic damage variable to ensure straightforward physical interpretation and efficient computation: a second order or fourth order damage tensor (

Leckie et al., 1981

), or discrete crack densities (

Jin & Arson, 2017

)? How to address the salient material behavior differences between compressive and tensile loading (

Lubarda et al., 1994; Comi & Perego, 2001)?

In the case of stain softening behavior, what are the best techniques to regularize localization problems: the crack band theory (

Bazant & Oh, 1983

), the strain gradient (

Geers et al., 1998

) or an integration based nonlocal approach (

Pijaudier-Cabot & Bazant, 1987


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