Markus Lazar (Institute of Solid State Physics, Darmstadt University of Technology, Germany). Towards Nano-Elasticity based on Gradient Elasticity.

In this talk the theory of gradient elasticity is presented. The fundamental problem of non-singular dislocations in the framework of gradient elasticity will be investigated. A general theory of non-singular dislocations is developed for linearly elastic materials. Using the calculus of variations and the framework of incompatible elasticity, we derive the field equations of gradient elasticity, which are inhomogeneous partial differential equations of fourth order.

In order to solve "eigendistortion problems" in such a framework, we derive the Green tensor of anisotropic gradient elasticity with up to six independent length scale parameters as a special version of Mindlin's form II anisotropic gradient elasticity theory [1] and as the anisotropic generalization of gradient elasticity of Helmholtz type [2,3,4]. The framework models materials where anisotropy is twofold, namely the bulk material anisotropy and a weak non-local anisotropy relevant at the nano-scale [5,6]. The continuum theory of anisotropic gradient elasticity is an excellent candidate for eigenstrain-problems up to the nano-scale [7].

Using the Green tensor of the theory of gradient elasticity, the non-singular fields which are produced by dislocations are given. All obtained dislocation fields are non-singular due to the regularization of the classical singular fields. The results have a direct application to numerical implementation and computer simulation of non-singular dislocations within the so-called (discrete) dislocation dynamics [8]. Therefore, these non-singular formulas of the dislocation fields are suitable for the numerical implementation in 3D dislocation dynamics without singularities. Such a dislocation dynamics without singularities offers the promise of predicting the dislocation microstructure evolution from fundamental principles and on sound physical grounds. Thus, a dislocation-based plasticity theory can be based on the gradient theory of non-singular dislocations.

References

[1] R.D. Mindlin, Micro-structure in linear elasticity, Arch. Rational. Mech. Anal. 16 (1964) 51-78.

[2] M. Lazar, G.A. Maugin, Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity, International Journal of Engineering Science 43 (2005) 1157-1184.

[3] M. Lazar, The fundamentals of non-singular dislocations in the theory of gradient elasticity: Dislocation loops and straight dislocations, Int. J. Solids Struct. 50 (2013) 352-362.

[4] M. Lazar, On gradient field theories: gradient magnetostatics and gradient elasticity, Phil. Mag. 94 (2014) 2840-2874.

[5] M. Lazar, G. Po, The non-singular Green tensor of gradient anisotropic elasticity of Helmholtz type, Eur. J. Mech. A Solids 50 (2015) 152-162.

[6] M. Lazar, G. Po, The non-singular Green tensor of Mindlin's anisotropic gradient elasticity with separable weak non-locality,  Physics Letters A 379 (2015) 1538-1543.

[7] D. Seif, G. Po, M. Mrovec, M. Lazar, C. Elsässer, P. Gumbsch, Atomistically enabled nonsingular anisotropic elastic representation of near-core dislocation stress fields in alpha-iron, Physical Review B 91 (2015) 184102.

[8] G. Po, M. Lazar, D. Seif, N. Ghoniem, Singularity-free dislocation dynamics with strain gradient elasticity, J. Mech. Phys. Solids 68 (2014) 161-178.