Volume 3, Issue 4

We derive a cohesive fracture model by homogenizing a periodic composite material whose microstructure is characterized by the presence of brittle inclusions in a reticulated unbreakable elastic structure.

We improve an estimate given by Acerbi and Dal Maso in 1994, concerning the area of the graph of a singular map from the disk of into , taking only three values, and jumping on three half-lines meeting at the origin in a triple junction.

We study weak lower semicontinuity of integral functionals in W 1, p under standard p -growth conditions, with integrands whose negative part may have p -growth as well. A characterization is obtained which, besides quasiconvexity of the integrand, involves a second condition that in general is weaker than boundary quasiconvexity at zero as defined by Ball and Marsden, although for p -homogeneous integrands, it reduces to the latter.

We prove the uniqueness of viscosity solutions to a differential equation involving the infinity-Laplacian with a variable exponent. We also derive a version of Harnack's inequality for this minimax problem.

A homogenization result is given for a material with brittle periodic inclusions, under the requirement that the interpenetration of matter is forbidden. According to the ratio between the softness of the inclusions and the size of the microstucture, three different limit models are deduced via Γ-convergence. In particular it is shown that in the limit the non-interpenetration constraint breaks the symmetry between states where the material is in extension and in compression.