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2021 | OriginalPaper | Chapter

2. Nonlinear Kinematics of a Continuously Dislocated Crystal

Authors : Christian B. Silbermann, Matthias Baitsch, Jörn Ihlemann

Published in: Introduction to Geometrically Nonlinear Continuum Dislocation Theory

Publisher: Springer International Publishing

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Abstract

This chapter presents the basics of geometrically nonlinear kinematics for continuously dislocated crystals. It involves the multiplicative split of the deformation and the slip-system-based decomposition of the velocity gradient. Eventually, appropriate measures for the crystal’s strain and geometrically necessary dislocation densities are derived.

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Footnotes
1
If decomposition (2.2) corresponded to a real (temporal) sequence, the incompatibility of https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-63696-8_2/501613_1_En_2_IEq22_HTML.gif could be seen as a reaction to that of https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-63696-8_2/501613_1_En_2_IEq23_HTML.gif . In reality, however, there is no such order.
 
2
Reference configuration and initial configuration are in general not identical [9, p. 20].
 
3
However, note that https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-63696-8_2/501613_1_En_2_IEq49_HTML.gif is transformed again multiplicatively by https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-63696-8_2/501613_1_En_2_IEq50_HTML.gif such that this is no pure additive decomposition.
 
4
The index variables https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-63696-8_2/501613_1_En_2_IEq56_HTML.gif mark the slip systems. A summation over slip systems is always indicated by a \(\sum \) sign. For i, j there is no summation convention.
 
5
This meets the expectation as long as only wandering of dislocations in the slip planes is taken into account, but not the possible climbing of edge dislocations under thermal activation.
 
Literature
1.
Kleber, W., Bautsch, H.J., Bohm, J.: Einführung in die Kristallographie. Oldenbourg Verlag, München (1998)CrossRef
2.
Silbermann, C.B., Ihlemann, J.: Geometrically linear continuum theory of dislocations revisited from a thermodynamical perspective. Arch. Appl. Mech. 88(1–2), 141–173 (2017)
3.
Le, K.C., Günther, C.: Nonlinear continuum dislocation theory revisited. Int. J. Plast. 53, 164–178 (2014)CrossRef
4.
Shutov, A.V., Kuprin, C., Ihlemann, J., Wagner, M.F.X., Silbermann, C.: Experimentelle Untersuchung und numerische Simulation des inkrementellen Umformverhaltens von Stahl 42CrMo4. Materialwiss. Werkstofftech. 41(9), 765–775 (2010)CrossRef
5.
Silbermann, C.B., Shutov, A.V., Ihlemann, J.: On operator split technique for the time integration within finite strain viscoplasticity in explicit FEM. PAMM 14(1), 355–356 (2014)CrossRef
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Gurtin, M.E., Reddy, B.D.: Some issues associated with the intermediate space in single-crystal plasticity. J. Mech. Phys. Solids 95, 230–238 (2016)CrossRef
7.
Krawietz, A.: Materialtheorie : Mathematische Beschreibung des phänomenologischen thermomechanischen Verhaltens. Springer, New York, Tokyo (1986)CrossRef
8.
Kröner, E.: Kontinuumstheorie der Versetzungen und Eigenspannungen. Springer, Berlin (1958)CrossRef
9.
Wriggers, P.: Nonlinear Finite Element Methods. Springer, Berlin (2008)
10.
Gurtin, M.E.: A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50(1), 5–32 (2002)CrossRef
11.
Silbermann, C.B., Ihlemann, J.: Analogies between continuum dislocation theory, continuum mechanics and fluid mechanics. IOP Conf. Ser. Mater. Sci. Eng. 181, 012,037+ (2017)
12.
Silbermann, C.B., Ihlemann, J.: Kinematic assumptions and their consequences on the structure of field equations in continuum dislocation theory. IOP Conf. Ser. Mater. Sci. Eng. 118, 012, 034+ (2016)
Metadata
Title
Nonlinear Kinematics of a Continuously Dislocated Crystal
Authors
Christian B. Silbermann
Matthias Baitsch
Jörn Ihlemann
Copyright Year
2021
Book
Introduction to Geometrically Nonlinear Continuum Dislocation Theory

Print ISBN: 978-3-030-63695-1

Electronic ISBN: 978-3-030-63696-8

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-3-030-63696-8_2

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