Peridynamics is a
recently developed formulation for continuum mechanics which describes material
deformation using a nonlocal approach. Unlike Classical Continuum Mechanics
(CCM) where the conservation equations are cast into partial differential
equations, Peridynamics describes the deformation in terms of
integro-differential equations. Additionally, peridynamics permits a natural
length scale that is absent in CCM. This facilitates the modeling of complex
material behavior and fracture which is not dependent on the numerical
discretization length scale. In this paper, we develop a Peridynamic
formulation for a Timoshenko beam. Full details and numerical examples are
presented for both bending and axial behavior. While the development in this
paper is limited to elastic, infinitesimal deformations, the approach can be
extended to finite inelastic deformations.
References
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[2]
Emmrich, E. and Weckner, O. (2006) The Peridynamic Equation of Motion in Non-Local Elasticity Theory. III European Conference on Computational Mechanics: Solids, Structures and Coupled Problems in Engineering, Springer, Lisbon.
[3]
Silling, S.A. and Lehoucq, R.B. (2010) Peridynamic Theory of Solid Mechanics. Sandia National Laboratory Technical Report SAMD 2010-1233J.
[4]
Emmrich, E., Lehoucq, R.B. and Puhst, D. (2013) Peridynamics: A Nonlocal Continuum Theory. Meshfree Methods for Partial Differential Equations VI: Lecture Notes in Computational Science & Engineering, 89, 45-65. http://dx.doi.org/10.1007/978-3-642-32979-1_3
[5]
Kilic, B. (2008) Peridynamic Theory for Progressive Failure Prediction in Homogeneous and Heterogeneous Materials, Ph.D. Thesis, University of Arizona, Tucson.
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Agwai, A. (2011) A Peridynamic Approach for Coupled Fields. Ph.D. Thesis, University of Arizona, Tucson.
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Tupek, M.R., Rimoli, J.J. and Radovitzky, R. (2013) An Approach for Incorporating Classical Continuum Damage Models in State-Based Peridynamics. Computational Methods in Applied Mechanical Engineering, 263, 20-26. http://dx.doi.org/10.1016/j.cma.2013.04.012
[8]
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[9]
Duangpanya, M. (2011) A Peridynamic Formulation for Transient Heat Conduction in Bodies with Evolving Discontinuities. Ph.D. Thesis, University of Nebraska-Lincoln, Lincoln.
[10]
Bobaru, F., Yang, M., Alves, L.F., Silling, S., Askari, E. and Xu, J. (2009) Convergence, Adaptive Refinement, and Scaling in 1D Peridynamics. International Journal for Numerical Methods in Engineering, 77, 852-877. http://dx.doi.org/10.1002/nme.2439
[11]
Feynman, R.P., Leighton, R.B. and Samds, M. (1963) Chapter 9, Newton’s Laws of Dynamics. The Feynman Lectures on Physics, 1, Addison-Wesley.
[12]
Demmie, P.N. and Silling, S.A. (2007) An Approach to Modeling Extreme Loading of Structures Using Peridynamics. Journal of Mechanics of Materials and Structures, 2, 1921-1945. http://dx.doi.org/10.2140/jomms.2007.2.1921
