Lattice structure: Scaling of strain related energy density

• A discrete form of SED criterion used for discrete structure is derived.
• The micro fracture mechanism is analyzed through the SED-based scaling law.
• The limit strain of micro bond is related to the lattice size and the SED factor.
• The lattice structure can be modeled by the energy density and scaling of strain.
• The discrete form of SED criterion can eliminate the lattice size-dependency problem of lattice model.

The strain energy density (SED) was used as the criterion to characterize stable and unstable crack growth in the fracture mechanics of linear and non-linear constitutive relations. It applies equally well for continuum mechanics problems in general. It is free from the restrictions of theories that implicate linear superposition. The energy density combines strain and stress and is not exclusive to strain alone nor stress alone. “Energy” and “density” for a point like element are the basic ingredients. The element can be microscopic and visualized as atoms or molecules. In this sense, the energy density criterion can be applied to the lattice structure and can also predict macro fracture. In this work, material is modeled with a lattice bond structure. Through the SED-based scaling law, the discrete form of SED criterion, which is specially used to lattice structure, is derived. The fracture mechanism of the lattice structure is analyzed. It is shown that the limit strain of the micro bond is related to the lattice size. The smaller the lattice size is, the larger the limit bond strain. It presents the same singularity order as the macro strain field near crack. The manifestation of scaling of strain is that micro bond strain is always much higher than the macro strain. Bond rupture is associated with the bond strain and the SED factor. The simulation results by the lattice structure with the SED criterion are almost free of lattice size dependency. It is suggested that the lattice structure can be modeled by the energy density and scaling of strain. The strain energy density approach can be applied to develop scale shifting laws in general.