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Linearized theory of peridynamic states. (English) Zbl 1188.74008
Summary: A state-based peridynamic material model describes internal forces acting on a point in terms of the collective deformation of all the material within a neighborhood of the point. In this paper, the response of a state-based peridynamic material is investigated for a small deformation superposed on a large deformation. The appropriate notion of a small deformation restricts the relative displacement between points, but it does not involve the deformation gradient (which would be undefined on a crack). The material properties that govern the linearized material response are expressed in terms of a new quantity called the modulus state. This determines the force in each bond resulting from an incremental deformation of itself or of other bonds. Conditions are derived for a linearized material model to be elastic, objective, and to satisfy balance of angular momentum. If the material is elastic, then the modulus state is obtainable from the second Fréchet derivative of the strain energy density function. The equation of equilibrium with a linearized material model is a linear Fredholm integral equation of the second kind. An analogue of Poincaré’s theorem is proved that applies to the infinite dimensional space of all peridynamic vector states, providing a condition similar to irrotationality in vector calculus.

74B15 Equations linearized about a deformed state (small deformations superposed on large)
Full Text: DOI
[1] Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E.: Peridynamic states and constitutive modeling. J. Elast. 88, 151–184 (2007) · Zbl 1120.74003 · doi:10.1007/s10659-007-9125-1
[2] Silling, S.A., Weckner, O., Askari, E., Bobaru, F.: Crack nucleation in a peridynamic solid. Int. J. Frac. (to appear) · Zbl 1425.74045
[3] Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000) · Zbl 0970.74030 · doi:10.1016/S0022-5096(99)00029-0
[4] Silling, S.A., Zimmermann, M., Abeyaratne, R.: Deformation of a peridynamic bar. J. Elast. 73, 173–190 (2003) · Zbl 1061.74031 · doi:10.1023/B:ELAS.0000029931.03844.4f
[5] Weckner, O., Abeyaratne, R.: The effect of long-range forces on the dynamics of a bar. J. Mech. Phys. Solids 53, 705–728 (2005) · Zbl 1122.74431 · doi:10.1016/j.jmps.2004.08.006
[6] Emmrich, E., Weckner, O.: Analysis and numerical approximation of an integro-differential equation modeling non-local effects in linear elasticity. Math. Mech. Solids 12, 363–384 (2007) · Zbl 1175.74013 · doi:10.1177/1081286505059748
[7] Emmrich, E., Weckner, O.: On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity. Commun. Math. Sci. 5, 851–864 (2007) · Zbl 1133.35098
[8] Alali, B.: Multiscale analysis of heterogeneous media for local and nonlocal continuum theories. Ph.D. thesis, Louisiana State University (2008)
[9] Bobaru, F., Yang, M., Alves, L.F., Silling, S.A., Askari, E., Xu, J.: Convergence, adaptive refinement, and scaling in 1D peridynamics. Int. J. Numer. Methods Eng. 77, 852–877 (2009) · Zbl 1156.74399 · doi:10.1002/nme.2439
[10] Silling, S.A., Lehoucq, R.B.: Convergence of peridynamics to classical elasticity theory. J. Elast. 93, 13–37 (2008) · Zbl 1159.74316 · doi:10.1007/s10659-008-9163-3
[11] Kunin, I.A.: Elastic Media with Microstructure I: One-Dimensional Models. Springer, Berlin (1982) · Zbl 0527.73002
[12] Kunin, I.A.: Elastic Media with Microstructure II: Three-Dimensional Models. Springer, Berlin (1983) · Zbl 0536.73003
[13] Rogula, D.: Nonlocal Theory of Material Media. Springer, Berlin (1982). pp. 137–149 and 243–278 · Zbl 0503.73001
[14] Rosen, B.W.: Mechanics of composite strengthening. In: Fiber Composite Materials, pp. 37–75. American Society of Metals, Metals Park (1964)
[15] Drapier, S., Grandidier, J.-C., Jochum, C., Potier-Ferry, M.: Structural plastic microbuckling and compressive strength of long-fibre composite materials. In: Maugin, G.A., et al. (eds.) Continuum Thermomechanics, pp. 125–136. Kluwer, New York (2000)
[16] Lehoucq, R.B., Silling, S.A.: Statistical coarse-graining of molecular dynamics into peridynamics. Sandia National Laboratories report, SAND2007-6410 (2007)
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