Least-Square Atomic Strain

Ju Li, Futoshi Shimizu

June 12, 2005

Since strain is by definition a relative measure[1], usually one needs two atomistic configurations, the reference and the current, in order to compute the local strain. A specialized trick could work if the reference configuration is known to be of high symmetry, and that only provides the shear invariant[2, 3]. To compute the full transformation matrix J and strain tensor η, one needs two configurations.

Define integer N_{i} to be the number of neighbors of atom i in the present configuration. For each neighbor j of atom i, their present separation is

d_{ji} ≡ x_{j} − x_{i},  (1)

and their old separation was

d_{ji}^{0} ≡ x_{j}^{0} − x_{i}^{0}.  (2)

Note that in our present programs[4, 5, 6], d_{ji} and d_{ji}^{0} etc. are all considered to be row vectors.

We seek a locally affine transformation matrix J_{i}, that best maps

{d_{ji}^{0}} → {d_{ji}}, ∀j ∈ N_{i}  (3)

in other words, we seek J_{i} that minimizes:

∑_{j∈N_{i}} |d_{ji}^{0}J_{i} − d_{ji}|^{2} = ∑_{j∈N_{i}}(d_{ji}^{0}J_{i} − d_{ji})(d_{ji}^{0}J_{i} − d_{ji})^{T}
= ∑_{j∈N_{i}} Tr((d_{ji}^{0}J_{i} − d_{ji})^{T} (d_{ji}^{0}J_{i} − d_{ji}))
= Tr ∑_{j∈N_{i}}(J_{i}^{T} d_{ji}^{0T} − d_{ji}^{T})(d_{ji}^{0}J_{i} − d_{ji}).  (4)

Performing arbitrary matrix variation δJ_{i}^{T} in above, we get

0 = Tr ∑_{j∈N_{i}} δJ_{i}^{T} d_{ji}^{0T} (d_{ji}^{0}J_{i} − d_{ji})
= TrδJ_{i}^{T} ∑_{j∈N_{i}} d_{ji}^{0T} (d_{ji}^{0}J_{i} − d_{ji}).  (5)

For the above to be true for any δJ_{i}^{T}, the matrix

∑_{j∈N_{i}} d_{ji}^{0T} (d_{ji}^{0}J_{i} − d_{ji})  (6)

has to be zero. In other words, there must be:

(∑_{j∈N_{i}} d_{ji}^{0T} d_{ji}^{0}) J_{i} = ∑_{j∈N_{i}} d_{ji}^{0T} d_{ji}.  (7)

If we define

V_{i} ≡ ∑_{j∈N_{i}} d_{ji}^{0T} d_{ji}^{0}, W_{i} ≡ ∑_{j∈N_{i}} d_{ji}^{0T} d_{ji},  (8)

then there is simply

J_{i} = V_{i}^{−1}W_{i}  (9)

The Lagrangian strain matrix is then calculated as

η_{i} = \frac{1}{2} (J_{i}J_{i}^{T} − I) .  (10)

And we can then compute its hydrostatic invariant

η_{i}^{hydro} = \frac{1}{3}Trη_{i},  (11)

calibrated to η_{xx} = η_{yy} = η_{zz} = a, and the local shear invariant,

η_{i}^{Mises} = \sqrt{\frac{1}{2}Tr(η − η_{m}I)^{2}}
= \sqrt{η_{yz}^{2} + η_{xz}^{2} + η_{xy}^{2} + \frac{(η_{yy} − η_{zz})^{2} + (η_{xx} − η_{zz})^{2} + (η_{xx} − η_{yy})^{2}}{6}} . (12)

calibrated to η_{xy} = η_{yx} = b, and all others zero.

Note that

1. The order of J_{i} and J_{i}^{T} in (10) matrix product is important. The wrong product order would also give you a symmetric matrix, but that symmetric matrix has nothing to do with the strain.

2. The product order of (10) appears to be the reverse of that of [1], because we adopt row-based vector scheme now [4, 5, 6], while column-based scheme was adopted in [1].

References

[1] J. H. Wang, J. Li, S. Yip, S. Phillpot, and D. Wolf. Mechanical instabilities of homogeneous crystals. Phys. Rev. B, 52:12627–12635, 1995.

[2] J. Li. http://alum.mit.edu/www/liju99/Graphics/A/Doc/vonMisesInvariant.pdf.

[3] J. Li. Atomeye: an efficient atomistic configuration viewer. Model. Simul. Mater. Sci. Eng., 11:173–177, 2003.

[4] J. Li. Basic molecular dynamics. In S. Yip, editor, Handbook of Materials Modeling, chapter 2.8. Kluwer Academic Publishers, 2005.

[5] J. Li. Atomistic calculation of mechanical behavior. In S. Yip, editor, Handbook of Materials Modeling, chapter 2.19. Kluwer Academic Publishers, 2005.

[6] J. Li. Atomistic visualization. In S. Yip, editor, Handbook of Materials Modeling, chapter 2.31. Kluwer Academic Publishers, 2005.