Stress of a Spatially Uniform Dislocation Density Field

  • Amit Acharya


It can be shown that the stress produced by a spatially uniform dislocation density field in a body comprising a linear elastic material under no loads vanishes. We prove that the same result does not hold in general in the geometrically nonlinear case. This problem of mechanics establishes the purely geometrical result that the \(\mathop{\mathrm{curl}}\nolimits \) of a sufficiently smooth two-dimensional rotation field cannot be a non-vanishing constant on a domain. It is classically known in continuum mechanics, stated first by the brothers Cosserat (SIAM J. Appl. Math. 25(3):483–491, 1973), that if a second order tensor field on a simply connected domain is at most a curl-free field of rotations, then the field is necessarily constant on the domain. It is shown here that, at least in dimension 2, this classical result is in fact a special case of a more general situation where the curl of the given rotation field is only known to be at most a constant.


Dislocations Stress-free Nonlinear elasticity 

Mathematics Subject Classification (2000)




It is a pleasure to acknowledge discussions with Reza Pakzad, who was able to construct a proof of the result presented here in a setting with weaker regularity. I acknowledge the support of grants ARO W911NF-15-1-0239 and NSF-CMMI-1435624.


  1. 1.
    Head, A.K., Howison, S.D., Ockendon, J.R., Tighe, S.P.: An equilibrium theory of dislocation continua. SIAM Rev. 35(4), 580–609 (1993) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Müller, S., Scardia, L., Zeppieri, C.I.: Geometric rigidity for incompatible fields, and an application to strain-gradient plasticity. Indiana Univ. Math. J. 63, 1365–1396 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Mura, T.: Impotent dislocation walls. Mater. Sci. Eng. A 113, 149–152 (1989) CrossRefGoogle Scholar
  4. 4.
    Shield, R.T.: The rotation associated with large strains. SIAM J. Appl. Math. 25(3), 483–491 (1973) CrossRefzbMATHGoogle Scholar
  5. 5.
    Willis, J.R.: Second-order effects of dislocations in anisotropic crystals. Int. J. Eng. Sci. 5(2), 171–190 (1967) CrossRefzbMATHGoogle Scholar
  6. 6.
    Yavari, A., Goriely, A.: Riemann–Cartan geometry of nonlinear dislocation mechanics. Arch. Ration. Mech. Anal. 205(1), 59–118 (2012) MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Dept. of Civil & Environmental EngineeringCarnegie Mellon UniversityPittsburghUSA

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