Advertisement

Dynamic Damage Propagation with Memory: A State-Based Model

  • Robert LiptonEmail author
  • Eyad Said
  • Prashant K. Jha
Reference work entry

Abstract

A model for dynamic damage propagation is developed using nonlocal potentials. The model is posed using a state-based peridynamic formulation. The resulting evolution is seen to be well posed. At each instant of the evolution, we identify a damage set. On this set, the local strain has exceeded critical values either for tensile or hydrostatic strain, and damage has occurred. The damage set is nondecreasing with time and is associated with damage state variables defined at each point in the body. We show that a rate form of energy balance holds at each time during the evolution. Away from the damage set, we show that the nonlocal model converges to the linear elastic model in the limit of vanishing nonlocal interaction.

Keywords

Damage model Nonlocal interactions Energy dissipation State-based peridynamics 

Notes

Acknowledgements

This material is based upon the work supported by the U S Army Research Laboratory and the U S Army Research Office under contract/grant number W911NF1610456.

References

  1. A. Agwai, I. Guven, E. Madenci, Predicting crack propagation with peridynamics: a comparative study. Int. J. Fract. 171, 65–78 (2011)CrossRefGoogle Scholar
  2. F. Bobaru, W. Hu, The meaning, selection, and use of the peridynamic horizon and its relation to crack branching in brittle materials. Int. J. Fract. 176, 215–222 (2012)CrossRefGoogle Scholar
  3. F. Bobaru, J.T. Foster, P.H. Geubelle, S.A. Silling, Handbook of Peridynamic Modeling (CRC Press, BOCA Ratone, 2016)zbMATHGoogle Scholar
  4. K. Dayal, K. Bhattacharya, Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. J. Mech. Phys. Solids 54, 1811–1842 (2006)MathSciNetCrossRefGoogle Scholar
  5. Q. Du, Y. Tao, X. Tian, A peridynamic model of fracture mechanics with bond-breaking. J Elast. (2017). https://doi.org/10.1007/s10659-017-9661-2 MathSciNetCrossRefGoogle Scholar
  6. E. Emmrich, D. Phust, A short note on modeling damage in peridynamics. J. Elast. 123, 245–252 (2016)MathSciNetCrossRefGoogle Scholar
  7. E. Emmrich, O. Weckner, On the well-posedness of the linear peridynamic model and its convergencee towards the Navier equation of linear elasticity. Commun. Math. Sci. 5, 851–864 (2007)MathSciNetCrossRefGoogle Scholar
  8. J.T. Foster, S.A. Silling, W. Chen, An energy based failure criterion for use with peridynamic states. Int. J. Multiscale Comput. Eng. 9, 675–688 (2011)CrossRefGoogle Scholar
  9. W. Gerstle, N. Sau, S. Silling, Peridynamic modeling of concrete structures. Nuclear Eng. Des. 237, 1250–1258 (2007)CrossRefGoogle Scholar
  10. Y.D. Ha, F. Bobaru, Studies of dynamic crack propagation and crack branching with peridynamics. Int. J. Fract. 162, 229–244 (2010)CrossRefGoogle Scholar
  11. P. K. Jha, R. Lipton, Numerical analysis of peridynamic models in Hölder space, arXiv preprint arXiv:1701.02818 (2017)Google Scholar
  12. R. Lipton, Dynamic brittle fracture as a small horizon limit of peridynamics. J. Elast. 117, 21–50 (2014)MathSciNetCrossRefGoogle Scholar
  13. R. Lipton, Cohesive dynamics and brittle fracture. J. Elast. 124(2), 143–191 (2016)MathSciNetCrossRefGoogle Scholar
  14. R. Lipton, S. Silling, R. Lehoucq, Complex fracture nucleation and evolution with nonlocal elastodynamics. arXiv preprint arXiv:1602.00247 (2016)Google Scholar
  15. R. Lipton, E. Said, P.K. Jha, Free damage propagation with memory. J. Elast. To appear in J. Elasticity (2018)CrossRefGoogle Scholar
  16. T. Mengesha, Q. Du, Nonlocal constrained value problems for a linear peridynamic Navier equation. J. Elast. 116, 27–51 (2014)MathSciNetCrossRefGoogle Scholar
  17. E. Oterkus, I. Guven, E. Madenci, Fatigue failure model with peridynamic theory,in IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems (ITherm), Las Vegas, June 2010, pp. 1–6Google Scholar
  18. K. Pham, J.J. Marigo, From the onset of damage to rupture: construction of responses with damage localization for a general class of gradient damage models. Contin. Mech. Thermodyn. 25, 147–171 (2013)MathSciNetCrossRefGoogle Scholar
  19. S.A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)MathSciNetCrossRefGoogle Scholar
  20. S.A. Silling, E. Askari, A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83, 1526–1535 (2005)CrossRefGoogle Scholar
  21. S.A. Silling, E. Askari, Peridynamic model for fatigue cracking. Sandia Report, SAND2014-18590, 2014Google Scholar
  22. S.A. Silling, F. Bobaru, Peridynamic modeling of membranes and fibers. Int. J. Nonlinear Mech. 40, 395–409 (2005)CrossRefGoogle Scholar
  23. S.A. Silling, M. Epton, O. Weckner, J. Xu, E. Askari, Peridynamic states and constitutive modeling. J. Elast. 88, 151–184 (2007)MathSciNetCrossRefGoogle Scholar
  24. S.A. Silling, R.B. Lehoucq, Convergence of peridynamics to classical elasticity theory. J. Elast. 93, 13–37 (2008)MathSciNetCrossRefGoogle Scholar
  25. S. Silling, O. Weckner, E. Askari, F. Bobaru, Crack nucleation in a peridynamic solid. Int. J. Fract. 162, 219–227 (2010)CrossRefGoogle Scholar
  26. O. Weckner, R. Abeyaratne, The effect of long-range forces on the dynamics of a bar. J. Mech. Phys. Solids 53, 705–728 (2005)MathSciNetCrossRefGoogle Scholar
  27. O. Weckner, E. Emmrich, Numerical simulation of the dynamics of a nonlocal, inhomogeneous, infinite bar. J. Comput. Appl. Mech, 6, 311–319 (2005)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Center for Computation and TechnologyLouisiana State UniversityBaton RougeUSA
  2. 2.Department of MathematicsLouisiana State UniversityBaton RougeUSA

Personalised recommendations