Skip to main content
Log in

A Remark on Stress of a Spatially Uniform Dislocation Density Field

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Acharya, A.: Stress of a spatially uniform dislocation density field. J. Elast. 137(2), 151–155 (2019)

    Article  MATH  Google Scholar 

  2. Acharya, A., Knops, R.J., Sivaloganathan, J.: On the structure of linear dislocation field theory. J. Mech. Phys. Solids 130, 216–244 (2019)

    Article  MATH  Google Scholar 

  3. Bryant, R.L., Clelland, J.N.: Flat metrics with a prescribed derived coframing. SIGMA 16, 004 (2020)

    MATH  Google Scholar 

  4. Constantin, P., Foias, C.: Navier–Stokes Equations. University of Chicago Press, Chicago (1988)

    Book  MATH  Google Scholar 

  5. Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun. Pure Appl. Math. 55, 1461–1506 (2002)

    Article  MATH  Google Scholar 

  6. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  7. Head, A.K., Howison, S.D., Ockendon, J.R., Tighe, S.P.: An equilibrium theory of dislocation continua. SIAM Rev. 35, 580–609 (1993)

    Article  MATH  Google Scholar 

  8. Kröner, E., Seeger, A.: Nicht-lineare Elastizitätstheorie der Versetzungen und Eigenspannungen. Arch. Ration. Mech. Anal. 3, 97–119 (1959)

    Article  MATH  Google Scholar 

  9. Liouville, J.: Théorème sur l’équation \(dx^{2} + dy^{2} + dz^{2} = \lambda d\alpha ^{2} + d\beta ^{2}+d \gamma ^{2}\). J. Math. Pures Appl. 15, 103 (1850)

    Google Scholar 

  10. Mura, T.: Impotent dislocation walls. Mater. Sci. Eng. A 113, 149–152 (1989)

    Article  Google Scholar 

  11. Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. AMS Chelsea Publishing, New York (2001)

    MATH  Google Scholar 

  12. Willis, J.R.: Second-order effects of dislocations in anisotropic crystals. Int. J. Eng. Sci. 5, 171–190 (1967)

    Article  MATH  Google Scholar 

  13. Yavari, A., Goriely, A.: Riemann–Cartan geometry of nonlinear dislocation mechanics. Arch. Ration. Mech. Anal. 205, 59–118 (2012)

    Article  MATH  Google Scholar 

Download references

Acknowledgement

The author acknowledges MOE-LSC Project AF0710029/011, NSFC (National Science Foundation of China) Youth Project BC0711045, and Shanghai Frontier Science Center of Modern Analysis for their support. We are very grateful to the anonymous referees for their careful reading and constructive remarks. We are also deeply indebted to Amit Acharya for kind communications and insightful discussions, and to Janusz Ginster for pointing out a fallible argument in an earlier version of the draft.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Siran Li.

Ethics declarations

Competing Interests

The author declares no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, S. A Remark on Stress of a Spatially Uniform Dislocation Density Field. J Elast 153, 155–160 (2023). https://doi.org/10.1007/s10659-022-09974-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10659-022-09974-5

Keywords

Mathematics Subject Classification (2010)

Navigation