Characterization of material response in multiple fiber
composite systems using AFM and FEM
Harsha Yejju, Xin Xu, Arvind Raman, Marisol Koslowski
Center for Predictive Materials, Modeling and Simulation, Purdue University, West Lafayette, IN 47906, USA.
Abstract
The difference in bulk matrix properties of polymer and the fiber-matrix interphase region is
pronounced at the nanoscale level. Here we present finite elements simulations that show the effect
of the variation of the materials properties of this fiber-matrix interphase region. We carry
simulations involving high density of fibers aligned in close proximity to each other. We will show
the material response to indentation in the fiber-matrix interphase region with multiple fibers in
close proximity to each other and the dependence of the material response on the volume of fibers
and on the vicinity of the indentation probe.
1. Introduction
2
2. AFM (Xin)
3
3. Processed AFM Results Topography, Amplitude, Effective
Stiffness (Harsha)
3.1. Topography
Fig. 3. Topography Image (Dataset8) for cantilever
stiffness kc = 18.4 N/m, preload force Fpl = 230 nN,
and average amplitude on fiber Zfiber = 0.5 nm.
Fig. 1. Topography Image (Dataset5) for cantilever
stiffness kc = 19.8 N/m, preload force Fpl = 221 nN,
and average amplitude on fiber Zfiber = 0.5 nm.
Fig. 4. Topography Image (Dataset9) for cantilever
stiffness kc = 18.4 N/m, preload force Fpl = 230 nN,
and average amplitude on fiber Zfiber = 0.5 nm.
Fig. 2. Topography Image (Dataset7) for cantilever
stiffness kc = 18.4 N/m, preload force Fpl = 230 nN,
and average amplitude on fiber Zfiber = 0.5 nm.
4
3.2. Amplitude
Fig. 5. AFM amplitude data for matrix regions
(Dataset5) for cantilever stiffness kc = 19.8 N/m,
preload force Fpl = 221 nN, average amplitude on
fiber Zfiber = 0.3972 nm.
Fig. 6. AFM amplitude data for matrix regions
(Dataset7) for cantilever stiffness kc = 18.4 N/m,
preload force Fpl = 230 nN, average amplitude on
fiber Zfiber = 0.5 nm.
Fig. 7. AFM amplitude data for matrix regions
(Dataset8) for cantilever stiffness kc = 18.4 N/m,
preload force Fpl = 230 nN, average amplitude on
fiber Zfiber = 0.5 nm.
Fig. 8. AFM amplitude data for matrix regions
(Dataset9) for cantilever stiffness kc = 18.4 N/m,
preload force Fpl = 230 nN, average amplitude on
fiber Zfiber = 0.5 nm.
3.3. Effective Stiffness
5
N/m, preload force Fpl = 230 nN, average amplitude
on fiber Zfiber = 0.5 nm.
Fig. 9. Normalized 2-dimensional effective stiffness
plot (Dataset 5) for cantilever stiffness kc = 19.8
N/m, preload force Fpl = 221 nN, average amplitude
on fiber Zfiber = 0.3972 nm.
Fig. 10. Normalized 2-dimensional effective stiffness
plot (Dataset 7) for cantilever stiffness kc = 18.4
N/m, preload force Fpl = 230 nN, average amplitude
on fiber Zfiber = 0.5 nm.
Fig. 11. Normalized 2-dimensional effective stiffness
plot (Dataset 8) for cantilever stiffness kc = 18.4
Fig. 12. Normalized 2-dimensional effective stiffness
plot (Dataset 9) for cantilever stiffness kc = 18.4
N/m, preload force Fpl = 230 nN, average amplitude
on fiber Zfiber = 0.5 nm.
6
4. Finite Element Procedure and
Results (Harsha)
4.a. Finite Element Method Data Analysis
Procedure
Normalized Stiffness:
Normalized stiffness, a measure of the
localized stiffness of the composite RVE
relative to a bulk matrix sample (i.e.
entire RVE with identical mesh is filled
with matrix material) was previously
coined by Bogetti et al. to characterize the
indentation response in the interphase
region in a single fiber composite system.
(Need to clean up following section)
Displacement at point of application of
load for a half-space is given by
d
P 1  2 
 Er
(1)
In equation (1) it is clear to see that as ‘r’
approaches zero the displacement at the
point approaches infinity. Furthermore,
7
oscillations due to mesh sensitivity were
observed in the preliminary results for
displacement. To address this mesh
sensitivity it was necessary to relate the
element size and nodal displacement at
the point of application of the load. The
relationship between element size, Le, and
node displacement d for a unit point load
were found by Bogetti et. Al. [1] to be:
d
AP
ELe
(2)
To solve the problem of the singularity,
taking the ratio of displacements of two
different materials with same
concentrated load and same Poisson’s
ratio shows that the ratio of the finite
element displacements will converge to
the same limit as that based on the theory
of elasticity.
P (1   2 )
d1
 E1r
E

 2
2
d 2 P (1   ) E1
 E2 r
(3)
AP
d1 E1Le E2


AP
E1
d2
E2 Le
(4)
Hence, the following formulation was
used to normalize stiffness [1]:
E1 
P
d1
(1)
The stiffness of the composite system, E1 ,
is calculated using equation (1) where P is
the applied load and d1 is the deflection at
point where load is applied on the
composite system.
E2 
P
d2
(2)
The stiffness of the bulk matrix system,
E2 , is calculated using equation (2) where
P is the applied load and d2 is the
deflection at point where load is applied
on the bulk matrix system. Normalized
stiffness formula is displayed as equation
(3).
E
d
 1 2
E2 d1
(3)
The bulk sample is the exact same FEM
(with identical meshing) as shown in
geometry except with uniform bulk
matrix properties assigned to the whole
model, i.e. volume of mesh previously
containing fiber is also assigned bulk
matrix properties.
Effective stiffness:
The effective stiffness is an extension of
the normalized stiffness to accommodate
the cantilever stiffness of the AFM. The
normalized stiffness characterization
remains valid at this adjusted level due to
the small magnitude of the forces
(~0.05μN-0.10 μN), which are well within
the elastic regime of the constitutive
model.
(Note: Show some proof of the test
conducted to prove response of analytic
model is in the linear elastic regime. By
performing FEM runs it was found that
material response was linear for forces in
the range of 0.01μN-1μN.)
4.b. Finite Element Mesh and Boundary
Conditions
8
The meshes generated in PATRAN/ANSYS
were subsequently solved using the open
source Finite Element Analysis Program
(FEAP) software. 8-noded threedimensional linear brick elements were
used for the analysis. Since the loading
force is in the axial direction, the relevant
periodicity boundary conditionsi for
normal loading case were imposed on the
RVE to simulate the presence of other
fibers (i.e. displacements normal to the
face of the RVE on the 4 transverse faces
and back face were constrained to be
zero).
The dimension of the fiber in both RVE
configurations is the same 3 m and is
well within the specified range for the
experimental sample. The hexagonal
array configurations considered in this
analysis matched the volume fraction of
0.58 provided by the manufacturer.
However, the square array RVE had a
volume fraction significantly smaller
(~0.29). Despite this significant
difference in volume fraction the
predicted material response is nearly
identical for both square and hexagonal
RVE’s. For a perfectly bonded system,
this result proves the increased stiffness
gradient in the matrix region, arising due
to fiber constraint, is highly localized to a
narrow thickness around the fiber.
Figure: Normalized stiffness vs. radial
distance from fiber for different volume
fractions.
Furthermore, for same RVE dimensions
and mesh the material response due to
the indentation response for a
transversely isotropic fiber and isotropic
fiber are nearly identical. Table ___ shows
the material properties and dimensions of
geometry used for each model and figure
___ shows the corresponding material
response respectively.
Figure: Normalized stiffness vs. radial
distance from fiber for different fiber 3D
constitutive equations
Modeling of imperfect bond
Subsequently, the mesh was changed by
inserting regions of constant radial
thickness around the fiber to represent
the imperfectly bonded interphase. The
following characteristics of (a) interphase
thickness (varied between 200-1000 nm
for each interphase) (b) Young’s modulus
of interphase EI (Em premultiplied by
some factor) and (c) number of
interphases were parameterized to
simulate the different gradients observed
in the AFM experiments. The parameters
used to simulate the imperfectly bonded
zones have been listed in the tables
below.
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4.c. FEM Results
Fig. 15. Effective stiffness against distance to closest
fiber in matrix regions (Dataset8) for cantilever
stiffness kc = 18.4 N/m, preload force Fpl = 230 nN,
average amplitude on fiber Zfiber = 0.5 nm.
Fig. 13. Effective stiffness against distance to closest
fiber in matrix regions (Dataset5) for cantilever
stiffness kc = 19.8 N/m, preload force Fpl = 221 nN,
average amplitude on fiber Zfiber = 0.5 nm.
Fig. 14. Effective stiffness against distance to closest
fiber in matrix regions (Dataset7) for cantilever
stiffness kc = 18.4 N/m, preload force Fpl = 230 nN,
average amplitude on fiber Zfiber = 0.5 nm.
Fig. 16. Effective stiffness against distance to closest
fiber in matrix regions (Dataset9) for cantilever
stiffness kc = 18.4 N/m, preload force Fpl = 230 nN,
average amplitude on fiber Zfiber = 0.5 nm.
10
4.d. Averaged Effective Stiffness
Fig. 20. Dataset 7
Fig. 17. Dataset 5
Fig. 21. Dataset 8
Fig. 18. Dataset 5
Fig. 22. Dataset 8
Fig. 19. Dataset 7
11
Fig. 23. Dataset 9
Fig. 26. Dataset 5
Fig. 24. Dataset 9
Fig. 27. Dataset 5
4.e. Imperfectly bonded zone
Fig. 25. Dataset 5
Fig. 28. Dataset 5
12
Fig. 29. Dataset 5
5. Conclusion
Note: It was necessary to use this
normalization method because the
application of a point load that was used
to simulate the indentation probe is in
reality a distributed load. The next stage
is to simulate the indentation with a
distributed load using contact analysis.
Note: The present model successfully
characterizes the increased stiffness in the
matrix region observed experimentally from
AFM experiments. It still remains of academic
interest to characterize the response of the
AFM in the fiber region of the composite. The
limitation of the present model stems from a
lack of an analytical model to predict the
displacement due to a point force on a
transversely isotropic material medium.
Volume fraction does not affect the
effective stiffness values. This can be
explained by the very narrow width that
the interphase region has in the unsized
carbon fiber reinforced composite system
being studied.
13
Acknowledgements
References
14
Sun C.T. , Vaidya R.S. Prediction of
composite properties from a
representative volume element
i