Skip to main content
SpringerLink
Log in

A Single Crystal Beam Bent in Double Slip

  • Chapter
  • First Online:
Generalized Models and Non-classical Approaches in Complex Materials 1

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 89))

Abstract

A theory of plastic bending of single crystal beam with two active slip systems accounting for continuously distributed excess dislocations is proposed. If the resistance to dislocation motion is negligibly small, then excess dislocations pile up against the beam’s neutral line, leaving two small layers near the lateral faces dislocation-free. The threshold value at the onset of plastic yielding, the dislocation density, as well as the moment-curvature curve are found. If the energy dissipation is taken into account, excess dislocations at the beginning of plastic yielding occupy two thin layers, leaving the zone near the middle line as well as two layers near the beam’s lateral faces dislocation-free. The threshold bending moment at the dislocation nucleation and the hardening rate are higher than those in the case of zero dissipation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Ashby MF (1970) The deformation of plastically non-homogeneous materials. The Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics 21(170):399–424

    Google Scholar 

  • Berdichevsky VL (1983) Variational Principles of Continuum Mechanics (in Russ.). Nauka, Moscow

    Google Scholar 

  • Berdichevsky VL (2006a) Continuum theory of dislocations revisited. Continuum Mechanics and Thermodynamics 18(3):195–222

    Google Scholar 

  • Berdichevsky VL (2006b) On thermodynamics of crystal plasticity. Scripta Materialia 54(5):711–716

    Google Scholar 

  • Berdichevsky VL, Le KC (2007) Dislocation nucleation and work hardening in anti-plane constrained shear. Continuum Mechanics and Thermodynamics 18(7):455–467

    Google Scholar 

  • Bilby BA (1955) Types of dislocation source. In: Defects in crystalline solids. Report of 1954 Bristol conference, The Physical Society, London, pp 123–133

    Google Scholar 

  • Bilby BA, Gardner LRT, Smith E (1958) The relation between dislocation density and stress. Acta Metallurgica 6(1):29–33

    Google Scholar 

  • Cleveringa HHM, van der Giessen E, Needleman A (1999) A discrete dislocation analysis of bending. International Journal of Plasticity 15(8):837–868

    Google Scholar 

  • Evans JT (1995) Bending produced by dislocations in a beam. International Journal of Engineering Science 33:1321–1329

    Google Scholar 

  • Gurtin ME (2002) A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. Journal of the Mechanics and Physics of Solids 50(1):5–32

    Google Scholar 

  • Gurtin ME, Anand L, Lele SP (2007) Gradient single-crystal plasticity with free energy dependent on dislocation densities. Journal of the Mechanics and Physics of Solids 55(9):1853–1878

    Google Scholar 

  • Kaluza M, Le KC (2011) On torsion of a single crystal rod. International Journal of Plasticity 27(3):460–469

    Google Scholar 

  • Kochmann DM, Le KC (2008) Dislocation pile-ups in bicrystals within continuum dislocation theory. International Journal of Plasticity 24(11):2125–2147

    Google Scholar 

  • Kochmann DM, Le KC (2009a) A continuum model for initiation and evolution of deformation twinning. Journal of the Mechanics and Physics of Solids 57(6):987–1002

    Google Scholar 

  • Kochmann DM, Le KC (2009b) Plastic deformation of bicrystals within continuum dislocation theory. Mathematics and Mechanics of Solids 14(6):540–563

    Google Scholar 

  • Kröner E (1955) Der fundamentale Zusammenhang zwischen Versetzungsdichte und Spannungsfunktionen. Zeitschrift für Physik 142(4):463–475

    Google Scholar 

  • Kysar JW, Saito Y, Oztop MS, Lee D, Huh WT (2010) Experimental lower bounds on geometrically necessary dislocation density. International Journal of Plasticity 26(8):1097–1123

    Google Scholar 

  • Le KC (1999) Vibrations of Shells and Rods. Springer, Berlin

    Google Scholar 

  • Le KC, Nguyen BD (2012) Polygonization: Theory and comparison with experiments. International Journal of Engineering Science 59:211–218

    Google Scholar 

  • Le KC, Nguyen BD (2013) On bending of single crystal beam with continuously distributed dislocations. International Journal of Plasticity 48:152–167

    Google Scholar 

  • Le KC, Nguyen QS (2010) Polygonization as low energy dislocation structure. Continuum Mechanics and Thermodynamics 22(4):291–298

    Google Scholar 

  • Le KC, Sembiring P (2008a) Analytical solution of plane constrained shear problem for single crystals within continuum dislocation theory. Archive of Applied Mechanics 78(8):587–597

    Google Scholar 

  • Le KC, Sembiring P (2008b) Plane constrained shear of single crystal strip with two active slip systems. Journal of the Mechanics and Physics of Solids 56(8):2541–2554

    Google Scholar 

  • Le KC, Sembiring P (2009) Plane constrained uniaxial extension of a single crystal strip. International Journal of Plasticity 25(10):1950–1969

    Google Scholar 

  • Maugin GA (1990) Infernal variables and dissipative structures. Journal of Non-Equilibrium Thermodynamics 15(2):173–192

    Google Scholar 

  • Maugin GA (1992) The Thermomechanics of Plasticity and Fracture, Cambridge Texts in Applied Mathematics, vol 7. Cambridge University Press, Cambridge

    Google Scholar 

  • Motz C, Weygand D, Senger J, Gumbsch P (2008) Micro-bending tests: A comparison between three-dimensional discrete dislocation dynamics simulations and experiments. Acta Materialia 56(9):1942–1955

    Google Scholar 

  • Mura T (1987) Micromechanics of Defects in Solids. Kluwer Academic Publishers, Oxford

    Google Scholar 

  • Nye JF (1953) Some geometrical relations in dislocated crystals. Acta Metallurgica 1(2):153–162

    Google Scholar 

  • Ortiz M, Repetto EA (1999) Nonconvex energy minimization and dislocation structures in ductile single crystals. Journal of the Mechanics and Physics of Solids 47(2):397–462

    Google Scholar 

  • Read WT (1957) Dislocation theory of plastic bending. Acta Metallurgica 5(2):83–88

    Google Scholar 

  • Sandfeld S, Hochrainer T, Gumbsch P, Zaiser M (2010) Numerical implementation of a 3D continuum theory of dislocation dynamics and application to micro-bending. Philosophical Magazine 90(27–28):3697–3728

    Google Scholar 

  • Sedov LI (1965) Mathematical methods of constructing models of continuum media (in Russ.). Usp Matem Nauk 20:123–182

    Google Scholar 

  • Wang W, Huang Y, Hsia KJ, Hu KX, Chandra A (2003) A study of microbend test by strain gradient plasticity. International Journal of Plasticity 19(3):365–382

    Google Scholar 

Download references

Acknowledgements

The financial support by the German Science Foundation (DFG) through the research project LE 1216/4-2 is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Computational Engineering, Ruhr University Bochum, D-44780, Bochum, Germany

    Xiangyu Cui

  2. Lehrstuhl für Mechanik - Materialtheorie, Ruhr-Universität Bochum, D-44780, Bochum, Germany

    Khanh Chau Le

Authors
  1. Xiangyu Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Khanh Chau Le
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiangyu Cui .

Editor information

Editors and Affiliations

  1. Institut für Mechanik, Otto-von-Guericke-Universität, Magdeburg, Germany

    Holm Altenbach

  2. Université Pierre et Marie Curie, Institut Jean Le Rond d’Alembert, Paris, France

    Joël Pouget

  3. Université Pierre et Marie Curie, Institut Jean Le Rond d’Alembert, Paris, France

    Martine Rousseau

  4. Université Pierre et Marie Curie, Institut Jean Le Rond d'Alembert, Paris Cedex 05, France

    Bernard Collet

  5. Institut Jean le Rond d'Alembert, Université Pierre et Marie Curie, Paris, France

    Thomas Michelitsch

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Cui, X., Le, K.C. (2018). A Single Crystal Beam Bent in Double Slip. In: Altenbach, H., Pouget, J., Rousseau, M., Collet, B., Michelitsch, T. (eds) Generalized Models and Non-classical Approaches in Complex Materials 1. Advanced Structured Materials, vol 89. Springer, Cham. https://doi.org/10.1007/978-3-319-72440-9_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-72440-9_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-72439-3

  • Online ISBN: 978-3-319-72440-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation