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Damage gradient models approximate fracture mechanics using a modulation ofthe material stiffness. To this aim a single scalar field, the damage, is usedto degrade as a whole the elastic energy. If applied to the structural modelsof beams and shells, where the elastic energy is the sum of the stretching andbending contributions, a similar approach is not able to capture some importantfeatures. For instance, the coupling between axial and bending strains inducedby cracks non-symmetric with respect to the center line is completely missed.In this contribution, we deduce a constitutive law for a beam having a cracknon-symmetric with respect to the center line. To achieve this, we perform anasymptotic coarse-grained procedure from a 2D problem, using a sharp interfacemodel. We deduce a homogenized 1D elastic energy, coupling the axial andbending strains, the constitutive parameters of which depend in a different wayon the crack depth and we state how precisely they do. This should pave the wayto a rational development of a phase-field gradient model for thin structures.

Towards a phase-field model for thin structures: a coarse-grained constitutive law for brittle fracture of beams