Archive of Applied Mechanics

, Volume 88, Issue 1–2, pp 141–173 | Cite as

Geometrically linear continuum theory of dislocations revisited from a thermodynamical perspective

  • C. B. Silbermann
  • J. Ihlemann


Continuum dislocation theory (CDT) allows the consideration of dislocation ensembles by introducing the dislocation density tensor. Though the kinematics of geometrically linear CDT are well established, the closure of governing field equations is not finished yet. The present study now brings together different principles for such a closure: It is shown how the field equations for the CDT can be obtained from potential energy minimization and from the phase field approach. These two energetic methods are integrated into a generic thermodynamic framework with twofold benefit: First, the rigorous thermodynamic treatment allows clarifying physical consequences of the energetic methods, among them the proof of thermodynamic consistency. Second, the framework provides a basis for consistent extensions of CDT. In this way, a new dynamic formulation of CDT is presented, which enables the analysis of the evolution of dislocation structures during plastic deformation. Moreover, a variety of possible dissipative phenomena is considered and the mechanical balance laws are deduced. For two special cases, the field equations are derived in the strong form and the stability of the solution is analyzed. Next, a flexible numerical solution algorithm is presented using the finite difference method. Solutions of various initial boundary value problems are presented for the case of plane deformations. Therefore, some of the dissipative phenomena are further investigated and two distinct sources of the Bauschinger effect are identified. Special attention is also given to different boundary conditions and their effect on the solution. For the case of uniaxial compression, the numerical results are confronted with experimental data. Thus, the simulations are validated and a new consistent interpretation of the experimental results is achieved.


Dislocations Dislocated crystal Continuum theory Dissipation Phase field method Finite difference method 


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Supplementary material

419_2017_1296_MOESM1_ESM.gif (8.9 mb)
Supplementary material 1 (gif 9147 KB) Plastic slip field \(\beta_p/ \permille \) for \(\varphi=0^\circ\), final state plotted on the deformed crystal domain (magnified by a factor of 20): the fields are inhomogeneous.
419_2017_1296_MOESM2_ESM.gif (6.8 mb)
Supplementary material 2 (gif 6942 KB) Plastic slip field \(\beta_p/ \permille\) for \(\varphi=45^\circ\), final state plotted on the deformed crystal domain (magnified by a factor of 20): the fields are inhomogeneous.
419_2017_1296_MOESM3_ESM.gif (9.6 mb)
Supplementary material 3 (gif 9830 KB) Dislocation density field \(\rho_d/ {\upmu m}^{-2} {\rm for} \varphi=0^\circ\), final state plotted on the undeformed crystal domain: dislocation structures with a high density have emerged.
419_2017_1296_MOESM4_ESM.gif (6.6 mb)
Supplementary material 4 (gif 6787 KB) Dislocation density field \(\rho_d/ {\upmu m}^{-2} {\rm for} \varphi=45^\circ\), final state plotted on the undeformed crystal domain: dislocation structures with a high density have emerged.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Institute of Mechanics and Thermodynamics, Chair of Solid MechanicsChemnitz University of TechnologyChemnitzGermany

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