How a dissimilar-chain system is splitting: Quasi-static, subsonic and supersonic regimes

Abstract

We consider parallel splitting of a strip composed of two different chains. As a waveguide, the dissimilar-chain structure radically differs from the well-studied identicalchain system. It is characterized by three speeds of the long waves, c 1 and c 2 for the separate chains, and c + = (c 2 1 + c 2 2 )/2 for the intact strip where the chains are connected. Accordingly, there exist three ranges, the subsonic for both chains (0, c 2 ) (we assume that c 2 < c 1 ), the intersonic (c 2 , c + ) and the supersonic, (c + , c 1 ). The speed in the latter range is supersonic for the intact strip and at the same time, it is subsonic for the separate higherspeed chain. This fact allows the splitting wave to propagate in the strip supersonically.We derive steady-state analytical solutions and find that the splitting can propagate steadily only in two of these speed ranges, the subsonic and the supersonic, whereas the intersonic regime is forbidden. In the case of considerable difference in the chain stiffness, the lowest dynamic threshold corresponds to the supersonic regime. The peculiarity of the supersonic mode is that the supersonic energy delivery channel, being initially absent, is opening with the moving splitting point.Based on the discrete and related continuous models we find which regime can be implemented depending on the structure parameters and loading conditions. The analysis allows us to represent the characteristics of such processes and to demonstrate strengths and weaknesses of different formulations, quasi-static, dynamic, discrete or continuous.Analytical solutions for steady-state regimes are obtained and analyzed in detail. We find the force − speed relations and show the difference between the static and dynamic thresholds. The parameters and energy of waves radiated by the propagating splitting are determined. We calculate the strain distribution ahead of the transition point and check whether the steady-state solutions are admissible.

Dynamic fracture of a dissimilar chain

“…Let us recall that similar problems were considered in [19,20] where the dissimilar chains were also compared with the corresponding long wavelength approximation. In the latter, the authors unveiled the possibility of an intersonic fracture propagation and evaluated a simple formula to check admissibility regime border, which we have already mentioned in §4a.…”

“…In [20], simplified admissibility conditions were proposed, which reduce to the verification of the following: u ′(0) < 0. This condition works well for intermediate and high crack speeds but fails to identify the admissible steady states at low speeds, which has been discussed in [35].…”

“…The analogous problem for two chains of identical material is thoroughly studied in [35]. Two distinct problems appear to be formally mathematically equivalent to that discussed in this paper: mode II loading of a bi-material structural interface with buckling rods and the fracture of dissimilar chains in mode II configuration in [19,20]. Earlier works on fracture and phase transitions in one-dimensional structures [23,36] demonstrate that not every steady-state solution is admissible.…”

“…The theoretical study [18] examines bi-material solids in the framework of lattice structures but it only considers a quasi-static formulation. Steady-state fault propagation in a dissimilar structure is analysed in [19,20], where the technique developed by Slepyan for crack propagation in lattice structures has been used. In his paramount works [21,22], Slepyan developed a method for linking the microscopic discrete models of dynamic fracture and phase transitions to their macroscopic limits.…”

Dynamic phenomena and crack propagation in dissimilar elastic lattices

“…Analytical modelling of failure processes in lattice systems utilises the Wiener-Hopf technique, whereby the governing equations are reduced to a scalar Wiener-Hopf problem and solved as shown by Slepyan (2002), who developed this approach for homogeneous lattices. Only few attempts have been made to fully analyse the failure of certain dissimilar structured systems using this approach, with the most relevant works being carried out for dissimilar chains by Gorbushin and Mishuris (2017) and Berinskii and Slepyan (2017). In general, when the structure undergoing failure is heterogeneous, matrix Wiener-Hopf problems can arise, for which the general procedure to their solution is not known.…”