Proposal of an adaptive stiffness-scaled analysis procedure for dynamic analysis of rate-dependent fracture in quasi-brittle materials

The analysis of fracture in brittle and quasi-brittle materials can pose significant challenges to solution-finding algorithms. With this in mind, we present a novel adaptive path-scaled analysis method that enables a purely incremental, forward-only solution strategy. The tangent path is continuously followed without losses of energy and convergence issues. The proposed method is particularly suitable for dynamic problems involving rate-dependent fracture in quasi-brittle materials. The key novelties include the introduction of an energy-controlled solution strategy using a binary pathway vector that systematically captures all admissible combinations of loading and unloading at integration points, and the incorporation of the dynamic increase factor directly into the traction-crack opening constitutive relationship to model rate-dependent behaviour. The method formulates dynamic equilibrium equations within the discrete strong discontinuity approach and employs an adaptive stiffness technique that combines tangent and secant stiffness matrices. The proposed method is validated using experimental data of three-point bending tests on notched beams, an L-specimen, and thick cylinders subject to varying loading rates, including impact. Results show an overall good agreement with experimental data adequately reproducing the crack mouth opening displacements, as well as the rate-dependent increase in peak loads and sharper post-peak responses. Key fracture parameters, such as crack mouth opening velocity, crack-tip velocity, crack pattern, fragmentation, traction responses, and quantitative energy-balance measures across loading rates (including inferred DIF amplification), are also adequately predicted.