# zbMATH — the first resource for mathematics

Reformulation of elasticity theory for discontinuities and long-range forces. (English) Zbl 0970.74030
Summary: Some materials may naturally form discontinuities such as cracks as a result of deformation. As an aid to the modeling of such materials, a new framework for the basic equations of continuum mechanics, called the ‘peridynamic’ formulation, is proposed. The propagation of linear stress waves in the new theory is discussed, and wave dispersion relations are derived. Material stability and its connection with wave propagation is investigated. It is demonstrated by an example that the reformulated approach permits the solution of fracture problems using the same equations either on or off the crack surface or crack tip. This is an advantage for modeling problems in which the location of a crack is not known in advance.

##### MSC:
 74J05 Linear waves in solid mechanics 74B99 Elastic materials
Full Text:
##### References:
 [1] Abraham, F.F.; Brodbeck, D.; Rudge, W.E.; Xu, X., A molecular dynamics investigation of rapid fracture mechanics, J. mech. phys. solids, 45, 1595-1619, (1997) · Zbl 0974.74558 [2] Daw, M.S.; Foiles, S.M.; Baskes, M.I., The embedded-atom method: a review of theory and applications, Mat. sci. rep., 9, 251-310, (1993) [3] Eringen, A.C.; Edelen, D.G.B., On nonlocal elasticity, Int. J. engng. sci., 10, 233-248, (1972) · Zbl 0247.73005 [4] Eringen, A.C.; Kim, B.S., Relation between non-local elasticity and lattice dynamics, Crystal lattice defects, 7, 51-57, (1977) [5] Eringen, A.C.; Speziale, C.G.; Kim, B.S., Crack-tip problem in non-local elasticity, J. mech. phys. solids, 25, 339-355, (1977) · Zbl 0375.73083 [6] Gao, H.; Klein, P., Numerical simulation of crack growth in an isotropic solid with randomized internal cohesive bonds, J. mech. phys. solids, 46, 187-218, (1998) · Zbl 0974.74008 [7] Hadamard, J., Leçons sur la propagation des ondes et LES equations de l’hydrodynamique, (1903), A. Hermann Paris, pp. 241-262, Reprinted by Chelsea, New York (1949) [8] Hellan, K., (), 7-47 [9] Knowles, J.K.; Sternberg, E., On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics, J. elast., 8, 329-379, (1978) · Zbl 0422.73038 [10] Krumhansl, J.A., Some considerations of the relation between solid state physics and generalized continuum mechanics, (), 298-311 · Zbl 0188.58902 [11] Love, A.E.H., 1944. Mathematical Theory of Elasticity. Dover, New York, pp. 616-627 [12] Miller, R.; Tadmor, E.B.; Phillips, R.; Ortiz, M., Quasicontinuum simulation of fracture at the atomic scale, Modelling simul. mater. sci. eng., 6, 607-638, (1998) [13] Timoshenko, S.P., 1953. History of the Strength of Materials. McGraw-Hill, New York, pp. 104-107 [14] Todhunter, I. 1886. A History of the Theory of Elasticity and of the Strength of Materials. Vol. 1, pp. 138-139, 223-224, 283-284. Cambridge University Press, Cambridge. Reprinted by Dover, New York (1960) [15] Willis, J.R., A comparison of the fracture criteria of griffith and Barenblatt, J. mech. phys. solids, 15, 151-162, (1967)
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.