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50 lines
2.1 KiB
ReStructuredText
50 lines
2.1 KiB
ReStructuredText
Calculate elastic constants
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===========================
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Elastic constants characterize the stiffness of a material. The formal
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definition is provided by the linear relation that holds between the
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stress and strain tensors in the limit of infinitesimal deformation.
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In tensor notation, this is expressed as s\_ij = C\_ijkl \* e\_kl, where
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the repeated indices imply summation. s\_ij are the elements of the
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symmetric stress tensor. e\_kl are the elements of the symmetric strain
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tensor. C\_ijkl are the elements of the fourth rank tensor of elastic
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constants. In three dimensions, this tensor has 3\^4=81 elements. Using
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Voigt notation, the tensor can be written as a 6x6 matrix, where C\_ij
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is now the derivative of s\_i w.r.t. e\_j. Because s\_i is itself a
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derivative w.r.t. e\_i, it follows that C\_ij is also symmetric, with at
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most 7\*6/2 = 21 distinct elements.
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At zero temperature, it is easy to estimate these derivatives by
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deforming the simulation box in one of the six directions using the
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:doc:`change\_box <change_box>` command and measuring the change in the
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stress tensor. A general-purpose script that does this is given in the
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examples/elastic directory described on the :doc:`Examples <Examples>`
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doc page.
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Calculating elastic constants at finite temperature is more
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challenging, because it is necessary to run a simulation that performs
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time averages of differential properties. One way to do this is to
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measure the change in average stress tensor in an NVT simulations when
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the cell volume undergoes a finite deformation. In order to balance
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the systematic and statistical errors in this method, the magnitude of
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the deformation must be chosen judiciously, and care must be taken to
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fully equilibrate the deformed cell before sampling the stress
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tensor. Another approach is to sample the triclinic cell fluctuations
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that occur in an NPT simulation. This method can also be slow to
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converge and requires careful post-processing :ref:`(Shinoda) <Shinoda1>`
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----------
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.. _Shinoda1:
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**(Shinoda)** Shinoda, Shiga, and Mikami, Phys Rev B, 69, 134103 (2004).
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.. _lws: http://lammps.sandia.gov
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.. _ld: Manual.html
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.. _lc: Commands_all.html
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