From 75d7e2fa6757b14f56c20fdb7dead140acc24773 Mon Sep 17 00:00:00 2001 From: athomps Date: Wed, 12 May 2010 03:31:22 +0000 Subject: [PATCH] elastic constants example git-svn-id: svn://svn.icms.temple.edu/lammps-ro/trunk@4125 f3b2605a-c512-4ea7-a41b-209d697bcdaa --- doc/Section_example.html | 3 ++- doc/Section_example.txt | 1 + doc/Section_howto.html | 49 ++++++++++++++++++++++++++++++++++++---- doc/Section_howto.txt | 44 ++++++++++++++++++++++++++++++++++-- 4 files changed, 90 insertions(+), 7 deletions(-) diff --git a/doc/Section_example.html b/doc/Section_example.html index c52d848d18..b8be694bbe 100644 --- a/doc/Section_example.html +++ b/doc/Section_example.html @@ -30,10 +30,11 @@ Site.

These are the sample problems in the examples sub-directories:

-
+
+ diff --git a/doc/Section_example.txt b/doc/Section_example.txt index f1698e17c9..234198d380 100644 --- a/doc/Section_example.txt +++ b/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 diff --git a/doc/Section_howto.html b/doc/Section_howto.html index f5e26496cb..252cb6ba60 100644 --- a/doc/Section_howto.html +++ b/doc/Section_howto.html @@ -30,7 +30,8 @@ certain kinds of LAMMPS simulations. 4.14 Extended spherical and aspherical particles
4.15 Output from LAMMPS (thermo, dumps, computes, fixes, variables)
4.16 Thermostatting, barostatting and computing temperature
-4.17 Walls
+4.17 Walls
+4.18 Elastic constants

The example input scripts included in the LAMMPS distribution and highlighted in this section 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:

-
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
+
c_ID entire scalar, vector, or array
c_ID[I] one element of vector, one column of array
c_ID[I][J] 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.

-
+
@@ -1370,7 +1371,7 @@ thermodynamic output.

-

4.16 Walls +

4.17 Walls

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.

+

4.18 Elastic constants +

+

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 +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. +

+

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) +

+
+
@@ -1470,4 +1507,8 @@ Phys, 79, 926 (1983).

(Price) Price and Brooks, J Chem Phys, 121, 10096 (2004).

+ + +

(Shinoda) Shinoda, Shiga, and Mikami, Phys Rev B, 69, 134103 (2004). +

diff --git a/doc/Section_howto.txt b/doc/Section_howto.txt index e4ec66aae0..ec5fc1fa34 100644 --- a/doc/Section_howto.txt +++ b/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).
Command Input Output
thermo_style custom global scalars screen, log file
dump custom per-atom vectors dump file