elastic constants example

git-svn-id: svn://svn.icms.temple.edu/lammps-ro/trunk@4125 f3b2605a-c512-4ea7-a41b-209d697bcdaa

doc/Section_example.html

@@ -30,10 +30,11 @@ Site</A>.
 </P>
 <P>These are the sample problems in the examples sub-directories:
 </P>
-<DIV ALIGN=center><TABLE  BORDER=1 >
+<DIV ALIGN=center><TABLE  WIDTH="0%"  BORDER=1 >
 <TR><TD >colloid</TD><TD >  big colloid particles in a small particle solvent, 2d system</TD></TR>
 <TR><TD >crack</TD><TD >	  crack propagation in a 2d solid</TD></TR>
 <TR><TD >dipole</TD><TD >   point dipolar particles, 2d system</TD></TR>
+<TR><TD >elastic</TD><TD >  zero temperature elastic constant tensor of silicon</TD></TR>
 <TR><TD >ellipse</TD><TD >  ellipsoidal particles in spherical solvent, 2d system</TD></TR>
 <TR><TD >flow</TD><TD >	  Couette and Poiseuille flow in a 2d channel</TD></TR>
 <TR><TD >friction</TD><TD > frictional contact of spherical asperities between 2d surfaces</TD></TR>

doc/Section_example.txt

@@ -30,6 +30,7 @@ These are the sample problems in the examples sub-directories:
 colloid:  big colloid particles in a small particle solvent, 2d system
 crack:	  crack propagation in a 2d solid
 dipole:   point dipolar particles, 2d system
+elastic:  zero temperature elastic constant tensor of silicon
 ellipse:  ellipsoidal particles in spherical solvent, 2d system
 flow:	  Couette and Poiseuille flow in a 2d channel
 friction: frictional contact of spherical asperities between 2d surfaces

doc/Section_howto.html

@@ -30,7 +30,8 @@ certain kinds of LAMMPS simulations.
 4.14 <A HREF = "#4_14">Extended spherical and aspherical particles</A><BR>
 4.15 <A HREF = "#4_15">Output from LAMMPS (thermo, dumps, computes, fixes, variables)</A><BR>
 4.16 <A HREF = "#4_16">Thermostatting, barostatting and computing temperature</A><BR>
-4.17 <A HREF = "#4_17">Walls</A> <BR>
+4.17 <A HREF = "#4_17">Walls</A><BR>
+4.18 <A HREF = "#4_18">Elastic constants</A> <BR>
 
 <P>The example input scripts included in the LAMMPS distribution and
 highlighted in <A HREF = "Section_example.html">this section</A> also show how to
@@ -1024,7 +1025,7 @@ discussed below, it can be referenced via the following bracket
 notation, where ID in this case is the ID of a compute.  The leading
 "c_" would be replaced by "f_" for a fix, or "v_" for a variable:
 </P>
-<DIV ALIGN=center><TABLE  BORDER=1 >
+<DIV ALIGN=center><TABLE  WIDTH="0%"  BORDER=1 >
 <TR><TD >c_ID </TD><TD > entire scalar, vector, or array</TD></TR>
 <TR><TD >c_ID[I] </TD><TD > one element of vector, one column of array</TD></TR>
 <TR><TD >c_ID[I][J] </TD><TD > one element of array 
@@ -1197,7 +1198,7 @@ data and scalar/vector/array data.
 input, that could be an element of a vector or array.  Likewise a
 vector input could be a column of an array.
 </P>
-<DIV ALIGN=center><TABLE  BORDER=1 >
+<DIV ALIGN=center><TABLE  WIDTH="0%"  BORDER=1 >
 <TR><TD >Command</TD><TD > Input</TD><TD > Output</TD><TD ></TD></TR>
 <TR><TD ><A HREF = "thermo_style.html">thermo_style custom</A></TD><TD > global scalars</TD><TD > screen, log file</TD><TD ></TD></TR>
 <TR><TD ><A HREF = "dump.html">dump custom</A></TD><TD > per-atom vectors</TD><TD > dump file</TD><TD ></TD></TR>
@@ -1370,7 +1371,7 @@ thermodynamic output.
 </P>
 <HR>
 
-<A NAME = "4_17"></A><H4>4.16 Walls 
+<A NAME = "4_17"></A><H4>4.17 Walls 
 </H4>
 <P>Walls in an MD simulation are typically used to bound particle motion,
 i.e. to serve as a boundary condition.
@@ -1444,6 +1445,42 @@ frictional walls, as well as triangulated surfaces.
 </P>
 <HR>
 
+<A NAME = "4_18"></A><H4>4.18 Elastic constants 
+</H4>
+<P>Elastic constants characterize the stiffness of a material. The formal
+definition is provided by the linear relation that holds between
+the stress and strain tensors in the limit of infinitesimal deformation.
+In tensor notation, this is expressed as s_ij = C_ijkl * e_kl, where 
+the repeated indices imply summation. s_ij are the elements of the
+symmetric stress tensor. e_kl are the elements of the symmetric
+strain tensor. C_ijkl are the elements of the fourth rank tensor
+of elastic constants. In three dimensions, this tensor has 3^4=81
+elements. Using Voigt notation, the tensor can be written
+as a 6x6 matrix, where C_ij is now the derivative of s_i
+w.r.t. e_j. Because s_i is itself a derivative w.r.t. e_i, it follows
+that C_ij is also symmetric, with at most 6*5/2 = 21 distinct elements.
+</P>
+<P>At zero temperature, it is easy to estimate these derivatives by
+deforming the cell in one of the six directions using 
+the command <A HREF = "displace_box.html">displace_box</A> 
+and measuring the change in the stress tensor. A general-purpose
+script that does this is given in the examples/elastic directory
+described in <A HREF = "Section_example.html">this section</A>.
+</P>
+<P>Calculating elastic constants at finite temperature is more challenging,
+because it is necessary to run a simulation that perfoms time averages
+of differential properties. One way to do this is to measure the change in 
+average stress tensor in an NVT simulations when the cell volume undergoes a
+finite deformation. In order to balance
+the systematic and statistical errors in this method, the magnitude of the
+deformation must be chosen judiciously, and care must be taken to fully
+equilibrate the deformed cell before sampling the stress tensor. Another
+approach is to sample the triclinic cell fluctuations that occur in an 
+NPT simulation. This method can also be slow to converge and requires
+careful post-processing <A HREF = "#Shinoda">(Shinoda)</A>
+</P>
+<HR>
+
 <HR>
 
 <A NAME = "Cornell"></A>
@@ -1470,4 +1507,8 @@ Phys, 79, 926 (1983).
 
 <P><B>(Price)</B> Price and Brooks, J Chem Phys, 121, 10096 (2004).
 </P>
+<A NAME = "Shinoda"></A>
+
+<P><B>(Shinoda)</B> Shinoda, Shiga, and Mikami, Phys Rev B, 69, 134103 (2004).
+</P>
 </HTML>

doc/Section_howto.txt

@@ -27,7 +27,8 @@ certain kinds of LAMMPS simulations.
 4.14 "Extended spherical and aspherical particles"_#4_14
 4.15 "Output from LAMMPS (thermo, dumps, computes, fixes, variables)"_#4_15
 4.16 "Thermostatting, barostatting and computing temperature"_#4_16
-4.17 "Walls"_#4_17 :all(b)
+4.17 "Walls"_#4_17
+4.18 "Elastic constants"_#4_18 :all(b)
 
 The example input scripts included in the LAMMPS distribution and
 highlighted in "this section"_Section_example.html also show how to
@@ -1359,7 +1360,7 @@ thermodynamic output.
 
 :line
 
-4.16 Walls :link(4_17),h4
+4.17 Walls :link(4_17),h4
 
 Walls in an MD simulation are typically used to bound particle motion,
 i.e. to serve as a boundary condition.
@@ -1431,6 +1432,42 @@ curved surfaces specified by the "fix wall/gran"_fix_wall_gran.html
 command.  At some point we plan to allow regoin surfaces to be used as
 frictional walls, as well as triangulated surfaces.
 
+:line
+
+4.18 Elastic constants :link(4_18),h4
+
+Elastic constants characterize the stiffness of a material. The formal
+definition is provided by the linear relation that holds between
+the stress and strain tensors in the limit of infinitesimal deformation.
+In tensor notation, this is expressed as s_ij = C_ijkl * e_kl, where 
+the repeated indices imply summation. s_ij are the elements of the
+symmetric stress tensor. e_kl are the elements of the symmetric
+strain tensor. C_ijkl are the elements of the fourth rank tensor
+of elastic constants. In three dimensions, this tensor has 3^4=81
+elements. Using Voigt notation, the tensor can be written
+as a 6x6 matrix, where C_ij is now the derivative of s_i
+w.r.t. e_j. Because s_i is itself a derivative w.r.t. e_i, it follows
+that C_ij is also symmetric, with at most 6*5/2 = 21 distinct elements.
+
+At zero temperature, it is easy to estimate these derivatives by
+deforming the cell in one of the six directions using 
+the command "displace_box"_displace_box.html 
+and measuring the change in the stress tensor. A general-purpose
+script that does this is given in the examples/elastic directory
+described in "this section"_Section_example.html.
+
+Calculating elastic constants at finite temperature is more challenging,
+because it is necessary to run a simulation that perfoms time averages
+of differential properties. One way to do this is to measure the change in 
+average stress tensor in an NVT simulations when the cell volume undergoes a
+finite deformation. In order to balance
+the systematic and statistical errors in this method, the magnitude of the
+deformation must be chosen judiciously, and care must be taken to fully
+equilibrate the deformed cell before sampling the stress tensor. Another
+approach is to sample the triclinic cell fluctuations that occur in an 
+NPT simulation. This method can also be slow to converge and requires
+careful post-processing "(Shinoda)"_#Shinoda
+
 :line
 :line
 
@@ -1452,3 +1489,6 @@ Phys, 79, 926 (1983).
 
 :link(Price)
 [(Price)] Price and Brooks, J Chem Phys, 121, 10096 (2004).
+
+:link(Shinoda)
+[(Shinoda)] Shinoda, Shiga, and Mikami, Phys Rev B, 69, 134103 (2004).