The peridynamic theory has been used to model and simulate numerically various kinds of mechanical behavior of solids. This work is devoted to analytical solutions of the elastodynamic behavior of linearized isotropic state-based peridynamic materials. First, we present the solutions of the dispersion relations, group velocities, and phase velocities of longitudinal and transverse waves, and examine in detail the effects of the Poisson’s ratio on these properties. It is shown that the elastodynamic behavior of the state-based peridynamic material with a negative Poisson’s ratio is remarkably different from that of the material with a positive Poisson’s ratio. We then derive the general solutions of initial-value problems, and obtain the Green’s function in a closed form. Finally, we study the evolution of a displacement discontinuity in the state-based peridynamic medium, and find that each component of the discontinuity in the three-dimensional theory varies independently according to the same vibrational mode. The results may have implications in investigations of wave propagations, including discontinuities such as phase transitions and kink propagations.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Le and Bobaru  classify the linearized state-based peridynamic theories into two types: One is objective, and the other is not. We use the objective type.
Silling, S.A.: Reformation of elasticity theory for discontinuities and longrange force. J. Mech. Phys. Solids 48, 175–209 (2000)
Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E.: Peridynamic states and constitutive modeling. J. Elast. 88, 151–184 (2007)
Silling, S.A.: Linearized theory of peridynamic states. J. Elast. 99, 85–111 (2010)
Mindlin, R.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)
Kröner, E.: Elasticity theory of materials with long range cohesive forces. Int. J. Solids Struct. 3, 731–742 (1967)
Willis, J.: Variational and related methods for the overall properties of composites. Adv. Appl. Mech. 21, 1–78 (1981)
Askes, H., Aifantis, E.: Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int. J. Solids Struct. 48, 1962–1990 (2011)
Maugin, G.: Continuum Mechanics Through the Twentieth Century—A Concise Historical Perspective. Springer, Dordrecht (2013)
S̆ilhavý, M.: Higher gradient expansion for linear isotropic peridynamic materials. Math. Mech. Solids 22, 1483–1493 (2017)
Silling, S.A., Lehoucq, R.B.: Peridynamic theory of solid mechanics. Adv. Appl. Mech. 44, 73–168 (2010)
Oterkus, S., Madenci, E., Agwai, A.: Peridynamic thermal diffusion. J. Comput. Phys. 265, 71–96 (2014)
Wang, L.J., Xu, J., Wang, J.: A peridynamic framework and simulation of non-Fourier and nonlocal heat conduction. Int. J. Heat Mass Transf. 118, 1284–1292 (2018)
Gerstle, W., Silling, S.A., Read, D., Tewary, V., Lehoucq, R.B.: Peridynamic simulation of electromigration. Comput. Mater. Continua 8, 75–92 (2008)
Dayal, K., Bhattacharya, K.: Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. J. Mech. Phys. Solids 54, 1811–1842 (2006)
Aguiar, A.R., Royer-Carfagni, G.F., Seitenfuss, A.B.: Wiggly strain localizations in peridynamic bars with non-convex potential. Int. J. Solids Struct. 138, 1–12 (2018)
Weckner, O., Abeyaratne, R.: The effect of long-range forces on the dynamics of a bar. J. Mech. Phys. Solids 53, 705–728 (2005)
Zimmermann, M.: A continuum theory with long-range forces for solids. Ph.D. Thesis, Massachusetts Institute of Technology (2005)
Weckner, O., Brunk, G., Epton, M.A., Silling, S.A., Askari, E.: Green’s functions in non-local three-dimensional linear elasticity. Proc. R. Soc. A 465, 3463–3487 (2009)
Wang, L.J., Xu, J., Wang, J.: Static and dynamic Green’s functions in peridynamics. J. Elast. 126, 95–125 (2017)
Mikata, Y.: Analytical solutions of peristatic and peridynamic problems for a 1D infinite rod. Int. J. Solids Struct. 49, 2887–2897 (2012)
Seleson, P., Parks, M.L., Gunzburger, M., Lehoucq, R.B.: Peridynamics as an upscaling of molecular dynamics. Multiscale Model. Simul. 8, 204–227 (2009)
Seleson, P., Parks, M.L.: On the role of the influence function in the peridynamic theory. Int. J. Multiscale Comput. Eng. 9, 689–706 (2011)
Wildman, R.A., Gazonas, G.A.: A finite difference-augmented peridynamics method for reducing wave dispersion. Int. J. Fract. 190, 39–52 (2014)
Bažant, Z.P., Luo, W., Chau, V.T., Bessa, M.A.: Wave dispersion and basic concepts of peridynamics compared to classical nonlocal damage models. J. Appl. Mech. 83, 111004 (2016)
Butt, S.N., Timothy, J.J., Meschke, G.: Wave dispersion and propagation in state-based peridynamics. Comput. Mech. 60, 1–14 (2017)
Le, Q.V., Bobaru, F.: Objectivity of state-based peridynamic models for elasticity. J. Elast. 131, 1–17 (2017)
Sarego, G., Le, Q.V., Bobaru, F., Zaccariotto, M., Galvanetto, U.: Linearized state-based peridynamics for 2-D problems. Int. J. Numer. Methods Eng. 108, 1174–1197 (2016)
Madenci, E.: Peridynamic integrals for strain invariants of homogeneous deformation. Z. Angew. Math. Mech. 97, 1236–1251 (2017)
Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (2011)
Eringen, A.C.: Continuum Physics II: Continuum Mechanics of Single-Substance Bodies. Academic Press, New York (1975)
Huang, Z.P.: Fundamentals of Continuum Mechanics, 2nd edn. Higher Education Press, Beijing (2012), in Chinese
The work is supported by the National Natural Science Foundation of China under Grant 11521202. Part of the work was completed when Linjuan Wang was visiting the Department of Mechanical Engineering at Massachusetts Institute of Technology under support of the Chinese Scholarship Council, and Professor Rohan Abeyaratne.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Wang, L., Xu, J. & Wang, J. Elastodynamics of Linearized Isotropic State-Based Peridynamic Media. J Elast 137, 157–176 (2019). https://doi.org/10.1007/s10659-018-09723-7
- Nonlocal continuum theory
- Dispersion relation
- Wave propagation
- Green’s function
Mathematics Subject Classification