Multi-scale interaction potentials (F − r) for describing fracture of brittle disordered materials like cement and concrete

Fracture processes in brittle disordered materials like many geo-materials (rock, ice, concrete, cement, etc.) are a trade off between local stress concentrations caused by the heterogeneity of such materials, and local strength. At those locations where the ratio between stress and strength exceeds a critical threshold value, cracking may initiate. Depending on the size of the cracks they can be arrested by stronger and stiffer elements in the structure of the material, or they will propagate and become critical. Critical cracks lead to localisation of deformations and to softening. In currently popular cohesive crack models still some continuum ideas remain, namely the notion of stress, whereas the localisation of deformations is handeled correctly by means of displacements. During softening the macro-crack traverses the specimen's cross-section, thereby gradually decreasing the effective load-carrying area. This growth process is affected both by structure (specimen) size and boundary conditions, and a better description of softening may be achieved by using load and displacement as state variables. In this paper, a new method of modelling fracture is proposed by using fracture potentials (F - r relations) at various observation scales, from atomistic and molecular to macroscopic. The virtual material can be interpreted as being built up from spherical elements; the fracture potential describes the interactions between the spheres. Since the spherical elements interact at their contacts-points only, a force-separation law (F - r) suffices. Size/scale effects are dealt with directly in the F - r relation; size/scale effects on strength are merely a special point in the entire description and do not require a separate law. © Springer Science+Business Media, Inc. 2007.