Uniaxial dynamic problem for a rod made with a material with yielding peak is formulated. The model used for formulating the problem takes into account an experimental observation that during the process of yielding a slow wave exists which divides a specimen into domains of elastic and plastic behavior. Process of yielding is described assuming the yielding peak and softening behavior is connected to the process of dislocation release. Discrepancies created during this process in the form of Lüders strips are described as shear lines between structural layers (e.g. ferrite-pearlite boundary, interatomic lattice) under an applied loading in a state of plastic yielding.

1. Corona E., Shaw J. A., Iadicola M. A. (2002) Buckling of steel bars with Lüders bands. International Journal of Solids and Structures, vol. 39, pp. 3313–3336.

2. Shaw J., Kyriakides S. (1997) Initiation and propagation of localized deformation in elasto-plastic strips under uniaxial tension. International Journal of Plasticity, vol. 13, pp. 837–871.

3. Liu Y., Kyriakides S. (2017) Effect of geometric and material discontinuities on the reeling of pipelines. Applied Ocean Research, 65, pp. 238–250.

4. Liu Y., Kyriakides S., Hallai J. (2015) Reeling of pipe with Lüders bands. International Journal of Solids and Structures, 72, pp. 11–25.

5. Broberg K. B. (1999). Cracks and fracture. Academic Press. 761 pp.

6. Hallai J., Kyriakides S. (2013) Underlying material response for Lüders-like instabilities. International Journal of Plasticity, vol. 47, pp. 1–12.

7. Barenblatt G.I. (1959) Concerning equilibrium cracks forming during brittle fracture. The stability of isolated cracks. Relationships with energetic theories. Journal of Applied Mathematics and Mechanics, vol. 23, no. 5, pp. 1273–1282.

8. Broberg K.B. (1978) On transient sliding motion. Geophysical Journal International, vol. 52, no. 3, pp. 397–432.

9. Yoffe E. H. (1951) The moving Griffith crack. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. 42, no. 330, pp. 739–750.

10. Labibov R. R. [et al]. (2018) Strips of localization of plastic deformation. Archive of Applied Mechanics, no. 12 (88), pp. 2221–2230.

11. Labibov R. R., Chernyakov Y. A. (2016). Modelling of slow plasticity waves Kharkov. Ukraine: KhPI.

12. Novozhilov V. V. (1969) On a necessary and sufficient criterion for brittle strength. Journal of Applied Mathematics and Mechanics, vol. 33, no. 2, pp. 201–210.

13. Barenblatt G. I. (1959) The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axially-symmetric cracks. Journal of Applied Mathematics and Mechanics, vol. 23, no. 3, pp. 622–636.
14. Barenblatt G. I. (1959) Equilibrium cracks formed during brittle fracture rectilinear cracks in plane plates. Journal of Applied Mathematics and Mechanics, vol. 23, no. 4, pp. 1009–1029.
15. Bažant Z. P., Belytschko T. B. (1985) Wave Propagation in a Strain-Softening Bar: Exact Solution, vol. 111, no. 3, pp. 381–389.
16. Yoshida F., Kaneda Y., Yamamoto S. (2008) A plasticity model describing yield-point phenomena of steels and its application to FE simulation of temper rolling. International Journal of Plasticity, no. 10 (24), pp. 1792–1818.

1. Corona E., Shaw J. A., Iadicola M. A. (2002) Buckling of steel bars with Lüders bands. International Journal of Solids and Structures, vol. 39, pp. 3313–3336.

2. Shaw J., Kyriakides S. (1997) Initiation and propagation of localized deformation in elasto-plastic strips under uniaxial tension. International Journal of Plasticity, vol. 13, pp. 837–871.

3. Liu Y., Kyriakides S. (2017) Effect of geometric and material discontinuities on the reeling of pipelines. Applied Ocean Research, 65, pp. 238–250.

4. Liu Y., Kyriakides S., Hallai J. (2015) Reeling of pipe with Lüders bands. International Journal of Solids and Structures, 72, pp. 11–25.

5. Broberg K. B. (1999). Cracks and fracture. Academic Press. 761 pp.

6. Hallai J., Kyriakides S. (2013) Underlying material response for Lüders-like instabilities. International Journal of Plasticity, vol. 47, pp. 1–12.

7. Barenblatt G.I. (1959) Concerning equilibrium cracks forming during brittle fracture. The stability of isolated cracks. Relationships with energetic theories. Journal of Applied Mathematics and Mechanics, vol. 23, no. 5, pp. 1273–1282.

8. Broberg K.B. (1978) On transient sliding motion. Geophysical Journal International, vol. 52, no. 3, pp. 397–432.

9. Yoffe E. H. (1951) The moving Griffith crack. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. 42, no. 330, pp. 739–750.

10. Labibov R. R. [et al]. (2018) Strips of localization of plastic deformation. Archive of Applied Mechanics, no. 12 (88), pp. 2221–2230.

11. Labibov R. R., Chernyakov Y. A. (2016). Modelling of slow plasticity waves Kharkov. Ukraine: KhPI.

12. Novozhilov V. V. (1969) On a necessary and sufficient criterion for brittle strength. Journal of Applied Mathematics and Mechanics, vol. 33, no. 2, pp. 201–210.

13. Barenblatt G. I. (1959) The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axially-symmetric cracks. Journal of Applied Mathematics and Mechanics, vol. 23, no. 3, pp. 622–636.
14. Barenblatt G. I. (1959) Equilibrium cracks formed during brittle fracture rectilinear cracks in plane plates. Journal of Applied Mathematics and Mechanics, vol. 23, no. 4, pp. 1009–1029.
15. Bažant Z. P., Belytschko T. B. (1985) Wave Propagation in a Strain-Softening Bar: Exact Solution, vol. 111, no. 3, pp. 381–389.
16. Yoshida F., Kaneda Y., Yamamoto S. (2008) A plasticity model describing yield-point phenomena of steels and its application to FE simulation of temper rolling. International Journal of Plasticity, no. 10 (24), pp. 1792–1818.