×

zbMATH — the first resource for mathematics

Transition waves in bistable structures. I: Delocalization of damage. (English) Zbl 1146.74335
Summary: We consider chains of dimensionless masses connected by breakable bistable links. A non-monotonic piecewise linear constitutive relation for each link consists of two stable branches separated by a gap of zero resistance. Mechanically, this model can be envisioned as a “twin-element” structure which consists of two links (rods or strands) of different lengths joined by the ends. The longer link does not resist to the loading until the shorter link breaks. We call this construction the waiting link structure. We show that the chain of such strongly nonlinear elements has an increased in-the-large stability under extension in comparison with a conventional chain, and can absorb a large amount of energy. This is achieved by two reasons. One is an increase of dissipation in the form of high-frequency waves transferring the mechanical energy to heat; this is a manifestation of the inner instabilities of the bonds. The other is delocalization of the damage of the chain. The increased stability is a consequence of the distribution of a partial damage over a large volume of the body instead of its localization, as in the case of a single neck formation in a conventional chain. We optimize parameters of the structure in order to improve its resistance to a slow loading and show that it can be increased significantly by delocalizing a damage process. In particular, we show that the dissipation is a function of the gap between the stable branches and find an optimal gap corresponding to maximum energy consumption under quasi-static extension. The results of numerical simulations of the dynamic behavior of bistable chains show that these chains can withstand without breaking the force which is several times larger than the force sustained by a conventional chain. The formulation and results are also related to the modelling of compressive destruction of a porous material or a frame construction which can be described by a two-branched diagram with a large gap between the branches. We also consider an extension of the model to multi-link chain that could imitate plastic behavior of material.
Part II see J. Mech. Phys. Solids 53, No. 2, 407–436 (2005; Zbl 1146.74336).

MSC:
74J99 Waves in solid mechanics
74R99 Fracture and damage
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Balk, A.M.; Cherkaev, A.V.; Slepyan, L.I., Dynamics of chains with non-monotone stress – strain relations. I. model and numerical experiments, J. mech. phys. solids, 49, 131-148, (2001) · Zbl 1005.74046
[2] Balk, A.M.; Cherkaev, A.V.; Slepyan, L.I., Dynamics of chains with non-monotone stress – strain relations. II. nonlinear waves and waves of phase transition, J. mech. phys. solids, 49, 149-171, (2001) · Zbl 1005.74047
[3] Charlotte, M.; Truskinovsky, L., Linear chains with a hyper-pre-stress, J. mech. phys. solids, 50, 217-251, (2002) · Zbl 1035.74005
[4] Cherkaev, A.; Slepyan, L., Waiting element structures and stability under extension, Int. J. damage mech., 4, 58-82, (1995)
[5] Cherkaev, A., Zhornitskaya, L., 2003. Dynamics of damage in two-dimensional structures with waiting links. In: Morchan, A.B. (editor), IUTAM Symposium on Asymptotics, Singularities and Homogenisation in Problems of Mechanics, Kluwer Academic Press, pp. 273-283.
[6] Friesecke, G.; Matthies, K., Atomic-scale localization of high-energy solitary waves on lattices, Physica D, 171, 4, 211-220, (2002) · Zbl 1064.82022
[7] Ngan, S.-C.; Truskinovsky, L., Thermal trapping and kinetics of martensitic phase boundaries, J. mech. phys. solids, 47, 141-172, (1999) · Zbl 0959.74052
[8] Puglisi, G.; Truskinovsky, L., Mechanics of a discrete chain with bi-stable elements, J. mech. phys. solids, 48, 1-27, (2000) · Zbl 0973.74060
[9] Slepyan, L.I., Dynamic factor in impact, phase transition and fracture, J. mech. phys. solids, 48, 931-964, (2000) · Zbl 0988.74050
[10] Slepyan, L.I., Feeding and dissipative waves in fracture and phase transition. II. phase-transition waves, J. mech. phys. solids, 49, 513-550, (2001) · Zbl 1003.74007
[11] Slepyan, L.I., Models and phenomena in fracture mechanics, (2002), Springer Berlin · Zbl 1047.74001
[12] Slepyan, L.I.; Troyankina, L.V., Fracture wave in a chain structure, J. appl. mech. techn. phys., 25, 6, 921-927, (1984)
[13] Slepyan, L.I., Troyankina, L.V., 1988. Impact waves in a nonlinear chain. In: Goldstein, R.V. (editor), Plasticity and fracture of solids, Academy of Science USSR, Navka, Moscow, pp. 175-186 (in Russian).
[14] Zel’dovich, Ya.B., Raizer, Yu.B., 1966, 1967. Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena 1, 2. Academic Press, New York, London.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.