Hi,
I've seen a lot of papers considering the application of the geometric phase analysis (GPA) method to experimental HRTEM images in order to extract the strain field of an edge dislocation in a material or between two materials (see for example, Acta Materialia 56 (2008) 2570–2575 by C.W. Zhao et al.). In these papers the Peierls-Nabarro (PN) and Foreman models of an edge dislocation are widely used. However, the expression for the displacement field of the PN model used, which has the form of
Ux(x,y)= - (b/2Pi) arctan(2(1-v)x/y)
is considerably different from the original expression of the model (see for example, Zeitschrift fur Physik, 126, 450-464, 1949 by Lebfried and Lucke)
Ux(x,y)= b/(2Pi) ( arctan( x / ( y +/- dzeta )) - x y / ( 2(1-v) ( x^2 + ( y +/- dzeta)^2))),
where dzeta=b/(2(1-v)), v being Poisson ratio and b interplane distance. The "+" sign is for y>0, "-" for y
Hi, Nabarro and Zhao do not contradict each other, the question would be more likely, how is Leibfried there come, see:
http://ac.els-cdn.com/S0921509397001846/1-s2.0-S0921509397001846-main.pdf?_tid=cf8c7d0a-2637-11e5-8182-00000aacb35d&acdnat=1436445865_e6348cd8d9d7703b57f14a45fb8d64cd
http://iopscience.iop.org/0959-5309/59/2/309/pdf/0959-5309_59_2_309.pdf
http://download.springer.com/static/pdf/785/art%253A10.1007%252FBF01669489.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2FBF01669489&token2=exp=1436447666~acl=%2Fstatic%2Fpdf%2F785%2Fart%25253A10.1007%25252FBF01669489.pdf%3ForiginUrl%3Dhttp%253A%252F%252Flink.springer.com%252Farticle%252F10.1007%252FBF01669489*~hmac=298d750d87638e27ffaaf6e4ccb54cdf44213cdb8e2e4959d326b54021c706ab
http://ac.els-cdn.com/S0375960107010274/1-s2.0-S0375960107010274-main.pdf?_tid=13d64212-263c-11e5-bebe-00000aacb360&acdnat=1436447697_73b3c4d5965baa670ddd3b3ec3646b39
Thank you, for your reply, Volker!
Let me explain in detail.
First. In original Nabarro paper (http://iopscience.iop.org/0959-5309/59/2/309/pdf/0959-5309_59_2_309.pdf) there's an expression for displacement along the dislocation glide plane (p.260)
Ux(x,0)= - (b/2Pi) arctan(2(1-v)x/b).
Nothing is said about 2D displacement field Ux(x,y). Seems like Zhao et al (http://www.sciencedirect.com/science/article/pii/S0375960107010274), as well as Chung-Wang et al (http://iopscience.iop.org/1674-1056/18/6/057) and Hojo et al (http://scitation.aip.org/content/aip/journal/apl/98/15/10.1063/1.3575566) and many others, have just changed "b" in the arctan term into "y", and thus derived 2D displacement field
Ux(x,y)= - (b/2Pi) arctan(2(1-v)x/y).
Is it acceptable?
Second. The expression for 2D field Ux(x,y) is given by Leibfried (http://link.springer.com/article/10.1007/BF01669489) on p. 461. It has been probably derived from the stress function, which could be found in the textbook "Theory of dislocations" by Hirth and Lothe (1982) on p. 222
Ux(x,y)= b/(2Pi) ( arctan( x / ( y +/- dzeta )) - x y / ( 2(1-v) ( x^2 + ( y +/- dzeta)^2)))
Far from the dislocation core it is in agreement with elastic theory, but the expression given by Zhao and others is not in agreement with elastic theory. However, the expression given by Zhao is in more or less agreement with experiment (GPA).
Seems like we should omit the second summand to come closer to the Zhao's expression, but still the result would not be the same. So something has been done more.
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