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 loadunload and forcerelease 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 forcerelease method works better than the loadunload method in this particular case. Different trial deformation fields are used by them to track the next equilibrium state. Forcerelease 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 forcedisplacement diagram; both of which are not satisfied by the loadunload method. To illustrate that the forcerelease method is not just another form of the loadunload 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 loadunload and forcerelease methods and exploit the details of the resulting fracture processes.

Influences of Domain Symmetry on Supersonic Combustion Modeling

“…In Case 2 where there is no viscosity, the post-peak ductility is very weak. Actually, this has been a fundamental issue in lattice modeling for a few decades [11,20,21,24] . If the time-dependence in stress redistribution is not considered, the post-peak ductility remains unrealistically low whether the force-release or load-unload algorithm is adopted [11,13,20,21] .…”

“…. So far, the operation of stress redistribution is the same as that discussed in [20] . The above releasing of E2's internal forces is assumed to happen during an infinitesimal time, i.e.…”

“…11 also indicates a temporal energy dissipation associated with local rupture. In previous non-iterative sequentially linear algorithms like the F-R method (e.g., [17,20] ), the material is assumed to be purely immediately elastic (without any viscosity), so the local rupture-induced energy dissipation rate is infinite. Nevertheless, in Fig.…”

“…For this end, in the development of lattice-type models, there have been two main kinds of non-iterative algorithms, i.e. the load-unload method and the force-release method [17,20] , which are based on two totally different assumptions in terms of characteristic time scales during progressive failures.…”

“…Even though massive successful applications of lattice models in analyzing failure phenomena have been achieved, the macroscopic softening-related issue, i.e. that the load-displacement curve obtained by the lattice model shows a much lower ductility than experimental observations [20] particularly during the post-peak stage, calls for further investigations. Lilliu and van Mier [21] built a three-dimensional lattice model which was still not able to fully recur the realistic ductility.…”