Second-order scaling of cracked quasibrittle materials under dynamic loading

The new scaling theory finite similitude has made possible the design of multiple scaled experiments for the analysis of systems and structures. The theory has brought into existence an unlimited number of new similitude rules and alternative strategies for scaling analysis and experimental design. The recent application of the approach to fracture has revealed that two scaled experiments are sufficient for the capture of defect-size related size effects under quasistatic loading. However, for fracture under dynamic loading, it is shown necessary to invoke the second-order finite similitude rule, which can feature three scaled experiments or additional equations for the purposes of scaling analysis. This is demonstrated in the paper both analytically and numerically using models scaled down to three sizes or involving additional equations housing scale derivatives up to second order. It is confirmed that tests at three scales provide sufficient information for the reconstruction of dynamic fracture at full size. The practicality of the approach is demonstrated through replication of behaviour of a concrete gravity dam damaged during the Koyna earthquake. Showcased is the practical benefit of targeting invariances (e.g., length, gravity, material properties), enabling scaled experiments with unchanged materials under realistic loading. Length invariance under scaling is shown critically to underpin the application of power-law relationships to quasibrittle materials. The presented results confirm that dynamic fracture mechanics at full-scale can be replicated to low single-digit percentage errors at second order, providing improved accuracy on zeroth- and first-order rules.