On the Load–Unload (L–U) and Force–Release (F–R) Algorithms for Simulating Brittle Fracture Processes via Lattice Models

Abstract

General summaries on the load–unload and force–release methods indicate that the two methods are efficient for different-charactered quasi-static failures; therefore, it is important to choose the right one for different applications. Then we take, as an example, the case where the release of the ruptured element's internal force is infinitely slower than the relaxation of the lattice system and analyze why the force–release method works better than the load–unload method in this particular case. Different trial deformation fields are used by them to track the next equilibrium state. Force–release method ensures that the deformation throughout the whole failure process coincides exactly with the controlled-displacement boundary conditions and we utilize the ‘left modulus’ concept to prove that this method satisfies the energetic evolution in the force–displacement diagram; both of which are not satisfied by the load–unload method. To illustrate that the force–release method is not just another form of the load–unload method, a tensile test on a specifically designed system is analyzed to further compare the above two methods, showing that their predicted sequences of elemental failures can be different. In closing, we simulate the uniaxial tensile test on a beam lattice system by the load–unload and force–release methods and exploit the details of the resulting fracture processes.