Reinforced rubber under a simple tensile test

[ZAARAOUI999]

Hello, Mr. Andreas, I wish you were well, so as is mentioned by the title of this discussion, I'm trying to simulate a reinforced rubber under a simple tensile test. So, the flow of my work is:

• Creating a rectangular geometry (20x20) containing randomly distributed inclusions (fibers)
• Extracting of the regions (so we will have a region for the matrix and a region for each inclusion)
• Creating Fields (mixed field with n=1 (no volume change, no pressure on boundaries will be taken into consideration))
• Linking all the inclusions's fields to the matrix field
• Creating materials and solidbodies (MatrixSolid (NeoHook: C10=0.7 and k=2.0 (with plane strain flag from Matadi)) and five InclusionSolid (Linear Elastic Plane Strain: E=2000.0 and nu=0.3))
• Applying uniaxial test boundaries on the matrix
• Setting the load (displacement of 10 mm)
• Creating a step and fire up the job
  Here are the regions and the mesh
  
  

Then, I got this!!!

It looks like I have misunderstood things or this simulation needs the definition of a relation between fibers and matrix I mean at the level of the interface. And it's clear that the behavior of the fibers isn't normal. Any idea about what's happening here?
Thanks in advance

[adtzlr]

Hi, thats indeed a bit tricky. Here is an example. Hope that helps.

import felupe as fem
import numpy as np

# create a rectangle
m = fem.Rectangle(n=21)

# take some points from the inside for the fiber-reinforced area
eps = 1e-3
mask = np.arange(m.npoints)[np.logical_and.reduce([
    m.points[:, 0] >= 0.3,
    m.points[:, 0] <= 0.7 + eps,
    m.points[:, 1] >= 0.3,
    m.points[:, 1] <= 0.7 + eps,
])]

# copies of the mesh
mesh = [m.copy(), m.copy()]

# create sub-meshes (fiber, matrix)
mesh[0].update(m.cells[ np.all(np.isin(m.cells, mask), axis=1)])
mesh[1].update(m.cells[~np.all(np.isin(m.cells, mask), axis=1)])

# a global and two sub-regions
region = fem.RegionQuad(m)
regions = [fem.RegionQuad(me) for me in mesh]

# a global and two sub-fields
field = fem.FieldsMixed(region, n=1, planestrain=True)
fields = [
    fem.FieldsMixed(regions[0], n=1, planestrain=True),
    fem.FieldsMixed(regions[1], n=1, planestrain=True),
]

# two material model formulations
neo_hooke = fem.NeoHooke(mu=1, bulk=1)
linear_elastic = fem.LinearElastic(E=100, nu=0.3)

# the solid bodies
fiber = fem.SolidBody(umat=linear_elastic, field=fields[0])
matrix = fem.SolidBody(umat=neo_hooke, field=fields[1])

# boundary conditions; applied on the global field
# which must be the x0-argument in job.evaluate()
boundaries = dict(
    fixed=fem.Boundary(field[0], fx=0),
    move=fem.Boundary(field[0], fx=1),
)

# prepare a step with substeps
move = fem.math.linsteps([0, 0.5], num=10)
step = fem.Step(
    items=[matrix, fiber],
    ramp={boundaries["move"]: move}, 
    boundaries=boundaries
)

# take care of the x0-argument
job = fem.Job(steps=[step])
job.evaluate(x0=field, filename="result.xdmf")

Have a nice evening,
Andreas

[adtzlr]

This also works with plane-strain materials from matadi:

import felupe as fem
import numpy as np

# create a rectangle
m = fem.Rectangle(n=21)

# take some points from the inside for the fiber-reinforced area
eps = 1e-3
mask = np.arange(m.npoints)[np.logical_and.reduce([
    m.points[:, 0] >= 0.3,
    m.points[:, 0] <= 0.7 + eps,
    m.points[:, 1] >= 0.3,
    m.points[:, 1] <= 0.7 + eps,
])]

# copies of the mesh
mesh = [m.copy(), m.copy()]

# create sub-meshes (fiber, matrix)
mesh[0].update(m.cells[ np.all(np.isin(m.cells, mask), axis=1)])
mesh[1].update(m.cells[~np.all(np.isin(m.cells, mask), axis=1)])

# a global and two sub-regions
region = fem.RegionQuad(m)
regions = [fem.RegionQuad(me) for me in mesh]

# a global and two sub-fields
field = fem.FieldsMixed(region, n=1)
fields = [
    fem.FieldsMixed(regions[0], n=1),
    fem.FieldsMixed(regions[1], n=1),
]

# two material model formulations
import matadi as mat

neo_hooke = mat.MaterialHyperelasticPlaneStrain(mat.models.neo_hooke, C10=0.5, bulk=1)
linear_elastic = fem.LinearElasticPlaneStrain(E=100, nu=0.3)

# the solid bodies
fiber = fem.SolidBody(umat=linear_elastic, field=fields[0])
matrix = fem.SolidBody(umat=neo_hooke, field=fields[1])

# boundary conditions; applied on the global field
# which must be the x0-argument in job.evaluate()
boundaries = dict(
    fixed=fem.Boundary(field[0], fx=0),
    move=fem.Boundary(field[0], fx=1),
)

# prepare a step with substeps
move = fem.math.linsteps([0, 0.5], num=10)
step = fem.Step(
    items=[matrix, fiber],
    ramp={boundaries["move"]: move}, 
    boundaries=boundaries
)

# take care of the x0-argument
job = fem.Job(steps=[step])
job.evaluate(x0=field, filename="result.xdmf")

[ZAARAOUI999]

That's worked, I'm very thankful for you, Thanks.

[adtzlr]

I'll update the docs when I have some free time. These parts are missing (define a global field for sub-fields) because I'm still trying to enhance the API - but this is a long-term project.

[ZAARAOUI999]

My results after your help, that's great!

[adtzlr]

Congratulations 👏🏻 , finally it looks like a realistic strain distribution!