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<p><span class="math notranslate nohighlight">\(\renewcommand{\AA}{\text{Å}}\)</span></p>
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<section id="calculate-elastic-constants">
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<h1><span class="section-number">8.3.5. </span>Calculate elastic constants<a class="headerlink" href="#calculate-elastic-constants" title="Link to this heading"></a></h1>
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<p>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</p>
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<div class="math notranslate nohighlight">
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\[s_{ij} = C_{ijkl} e_{kl}\]</div>
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<p>where
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the repeated indices imply summation. <span class="math notranslate nohighlight">\(s_{ij}\)</span> are the elements of the
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symmetric stress tensor. <span class="math notranslate nohighlight">\(e_{kl}\)</span> are the elements of the symmetric strain
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tensor. <span class="math notranslate nohighlight">\(C_{ijkl}\)</span> are the elements of the fourth rank tensor of elastic
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constants. In three dimensions, this tensor has <span class="math notranslate nohighlight">\(3^4=81\)</span> elements. Using
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Voigt notation, the tensor can be written as a 6x6 matrix, where <span class="math notranslate nohighlight">\(C_{ij}\)</span>
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is now the derivative of <span class="math notranslate nohighlight">\(s_i\)</span> w.r.t. <span class="math notranslate nohighlight">\(e_j\)</span>. Because <span class="math notranslate nohighlight">\(s_i\)</span> is itself a
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derivative w.r.t. <span class="math notranslate nohighlight">\(e_i\)</span>, it follows that <span class="math notranslate nohighlight">\(C_{ij}\)</span> is also symmetric, with at
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most <span class="math notranslate nohighlight">\(\frac{7 \times 6}{2}\)</span> = 21 distinct elements.</p>
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<p>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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<a class="reference internal" href="change_box.html"><span class="doc">change_box</span></a> 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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<code class="docutils literal notranslate"><span class="pre">examples/ELASTIC</span></code> directory described on the <a class="reference internal" href="Examples.html"><span class="doc">Examples</span></a>
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doc page.</p>
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<p>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. There are at least
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3 ways to do this in LAMMPS. The most reliable way to do this is
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by exploiting the relationship between elastic constants, stress
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fluctuations, and the Born matrix, the second derivatives of energy
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w.r.t. strain <a class="reference internal" href="#ray"><span class="std std-ref">(Ray)</span></a>.
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The Born matrix calculation has been enabled by
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the <a class="reference internal" href="compute_born_matrix.html"><span class="doc">compute born/matrix</span></a> command,
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which works for any bonded or non-bonded potential in LAMMPS.
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The most expensive part of the calculation is the sampling of
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the stress fluctuations. Several examples of this method are
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provided in the <code class="docutils literal notranslate"><span class="pre">examples/ELASTIC_T/BORN_MATRIX</span></code> directory
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described on the <a class="reference internal" href="Examples.html"><span class="doc">Examples</span></a> doc page.</p>
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<p>A second way is to measure
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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. An example of this method is provided in the
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<code class="docutils literal notranslate"><span class="pre">examples/ELASTIC_T/DEFORMATION</span></code> directory
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described on the <a class="reference internal" href="Examples.html"><span class="doc">Examples</span></a> doc page.</p>
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<p>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 <a class="reference internal" href="#shinoda1"><span class="std std-ref">(Shinoda)</span></a>.
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We do not provide an example of this method.</p>
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<p>A nice review of the advantages and disadvantages of all of these methods
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is provided in the paper by Clavier et al. <a class="reference internal" href="#clavier"><span class="std std-ref">(Clavier)</span></a>.</p>
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<hr class="docutils" />
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<p id="ray"><strong>(Ray)</strong> J. R. Ray and A. Rahman, J Chem Phys, 80, 4423 (1984).</p>
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<p id="shinoda1"><strong>(Shinoda)</strong> Shinoda, Shiga, and Mikami, Phys Rev B, 69, 134103 (2004).</p>
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<p id="clavier"><strong>(Clavier)</strong> G. Clavier, N. Desbiens, E. Bourasseau, V. Lachet, N. Brusselle-Dupend and B. Rousseau, Mol Sim, 43, 1413 (2017).</p>
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