Hill-Mandel
Use of Hill-Mandel feature
FEAP contains code to compute Hill-Mandel averages. To use the feature you need a periodic RVE mesh (matching nodes on opposing faces). The corner nodes should have boundary code 1 for all degrees of freedom.
Options for boundary conditions
Dirchlet
To set Dirchlet boundary conditions on the faces of the RVE, simply set the boundary codes to 1 for all face nodes.
Periodic
For periodic boundary conditions, use the ELINK command to force the solution to be the same on opposing faces of the mesh; see the example below.
Setting the average strain in the RVE
Single average strain
If you have just one average strain to compute or wish to compute a radial path in strain space then you can specify the strain in the mesh input using the PERIodic data command (note this does not mean that the boundary conditions are necessarily periodic). The input is a 3x3 matrix which is the displacement gradient or the small strain tensor, respectively for finite- and small-deformation problems; see the manual. If you define a proportional load you can explore a radial path in the direction of the given tensor.
Complex path
Alternately you can place the tensors in a file. One line per average strain you wish to impose. The formation is time-value followed by the tensor components of the displacement gradient or the small strain tensor, respectively for finite- and small-deformation problems. Note that for small strain tensors the shears should be given as engineering shear strains. For finite strains the ordering is , for small strains the ordering is . The command HILL,READ is used to have FEAP read the next line in the file.
Example
Here is a sample input file which has a 3 material microstructure that first pulled in the 1-direction, followed by an oscillating shear.
feap ** Example Hill-Mandel Homogenization **
0 0 0 2 2 9 ! Use nine node quadratic elements
parameter
f = 4 ! Mesh density
d = f*2 ! Nodes per block
L = 1 ! Edge dimension
material 1
solid
elastic isotropic 1000 0.25
plastic mises 1.5
mixed ! Use mixed for plasticity problems (Q2P1)
hist plot 1 1 ! Put ebarp in slot 1 for plotting
material 2
solid
elastic isotropic 5000 0.15
plastic mises 3.75
mixed
hist plot 1 1
material 3
solid
elastic isotropic 2500 0.3
plastic mises 2.0
mixed
hist plot 1 1
block
cart d d
quad 9
material 1
1 L L
2 2*L L
3 2*L 2*L
4 L 2*L
block
cart d 3*d
quad 9
material 2
1 0 0
2 L 0
3 L 3*L
4 0 3*L
block
cart d 3*d
quad 9
material 2
1 2*L 0
2 3*L 0
3 3*L 3*L
4 2*L 3*L
block
cart d d
quad 9
material 2
1 L 0
2 2*L 0
3 2*L L
4 L L
block
cart d d
quad 9
material 3
1 L 2*L
2 2*L 2*L
3 2*L 3*L
4 L 3*L
periodic cauchy file
loading.txt ! File with eps_avg(t)
! per line t e11 e22 e33 2e12 2e23 2e31
cboun ! Dirichlet BC for the corner nodes
node 0 0 1 1
node 3*L 0 1 1
node 3*L 3*L 1 1
node 0 3*L 1 1
end
tie ! Tie the blocks together
elink ! Link the motion on the edges
1 0 3*L 0 0
2 0 3*L 0 0
batch
loop,,40 ! There are 40 eps_avg in the file
time ! Advance the time (updates history variables)
hill read ! Read the next eps_avg
loop,,20
tang,,1 ! Form and solve
next
plot,wipe ! Wipe the plot
plot,defo,.5e3 ! Magnify the deformation
plot,hist,1,,1 ! Plot ebarp without the mesh
hill ! Compute Hill-Mandel stress and tangent,
next ! output goes to Ohill-ex
end
inte
stop
The contents of loading.txt are
0.1 0.00008 0 0 0 0 0 0.2 0.00016 0 0 0 0 0 0.3 0.00024 0 0 0 0 0 0.4 0.00032 0 0 0 0 0 0.5 0.0004 0 0 0 0 0 0.6 0.00048 0 0 0 0 0 0.7 0.00056 0 0 0 0 0 0.8 0.00064 0 0 0 0 0 0.9 0.00072 0 0 0 0 0 1 0.0008 0 0 0 0 0 1.1 0.0008 0 0 0.00007 0 0 1.2 0.0008 0 0 0.00014 0 0 1.3 0.0008 0 0 0.00021 0 0 1.4 0.0008 0 0 0.00028 0 0 1.5 0.0008 0 0 0.00035 0 0 1.6 0.0008 0 0 0.00042 0 0 1.7 0.0008 0 0 0.00049 0 0 1.8 0.0008 0 0 0.00056 0 0 1.9 0.0008 0 0 0.00063 0 0 2 0.0008 0 0 0.0007 0 0 2.1 0.0008 0 0 0.00063 0 0 2.2 0.0008 0 0 0.00056 0 0 2.3 0.0008 0 0 0.00049 0 0 2.4 0.0008 0 0 0.00042 0 0 2.5 0.0008 0 0 0.00035 0 0 2.6 0.0008 0 0 0.00028 0 0 2.7 0.0008 0 0 0.00021 0 0 2.8 0.0008 0 0 0.00014 0 0 2.9 0.0008 0 0 0.00007 0 0 3 0.0008 0 0 -9.32E-15 0 0 3.1 0.0008 0 0 -0.00007 0 0 3.2 0.0008 0 0 -0.00014 0 0 3.3 0.0008 0 0 -0.00021 0 0 3.4 0.0008 0 0 -0.00028 0 0 3.5 0.0008 0 0 -0.00035 0 0 3.6 0.0008 0 0 -0.00042 0 0 3.7 0.0008 0 0 -0.00049 0 0 3.8 0.0008 0 0 -0.00056 0 0 3.9 0.0008 0 0 -0.00063 0 0 4 0.0008 0 0 -0.0007 0 0