Local Material Property Fields in Random Composites

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Local Material Property Fields in Random
Composites Research By Sarah C. Baxter,
M. Imran Hossain, M. Ohidul Islam, Frank D.
Ford
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Many critical damage phenomena in composite
materials, including cracking, are related to
local stresses which are linked to local
variations in material properties associated with
composite microstructure. Relatively little
research however, has been done on the effects of
randomness in microstructural configuration on
the material behavior of composites. For many
engineering applications, it is assumed that
small-scale fluctuations in material properties
are averaged when evaluating macroscopic
behavior. This approach does not provide any
information about local stresses.
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The analysis of local stresses requires a method
of characterizing material properties in terms of
material microstructure. This characterization
is made more difficult by the inherent randomness
in composite microstructure. In this work a
moving window technique, which includes
micromechanical analysis from the micromechanics
model known as the generalized method of cells,
(GMC), is used to produce material property
fields, for both elastic and inelastic material
properties, associated with the random
microstructure of a composite material.
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Using an image of the microstructure, assuming
that different phases are clearly delineated, the
effective properties of small overlapping areas
of the full sample, called windows, are
calculated using GMC. Effective properties of
each window are assigned the spatial coordinates
of the center-point of the window.
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The figure shows a cross-section of a composite
microstructure, where the black area corresponds
to matrix material and the white area corresponds
to fibers that run in the out-of-plane (z)
direction. The properties of the constituent
materials are assumed to be deterministic.
Both fiber and matrix are modeled as isotropic
the fiber elastic modulus (white areas) is taken
to be 413.7 GPa, and the matrix material elastic
modulus (black areas) is taken to be 91.04 Gpa.
The total fiber volume fraction for this sample
is 25.
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Micromechanics
GMC calculates the effective properties by
building global stress-strain relations from a
micro-level analysis.
GMC calculates effective material properties of
Windows 1,1 and 1,4 (e.g., Exx, Eyy, Ezz)
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Local fiber volume fraction ?(x,y), the axial
elastic modulus Ezz(x,y), and transverse elastic
moduli Exx(x,y) and Eyy(x,y), generated by the
moving window-GMC analysis of the sample
microstructure are plotted at right. The window
size is 5 x 5, (10 x 10 pixels), of the total
sample length (200 x 200 pixels).
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The axial modulus and the fiber volume fraction
appear well-correlated. The spatial variations
in the transverse elastic moduli are also
correlated to fiber volume fraction, but unlike
the axial elastic modulus do not seem to be a
direct function of the volume fraction.
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Probabilistic Analysis
From the material property fields, a
probabilistic material description, for each
property, can be developed in the form of
probability density functions, auto, and cross
correlation functions.
Probabilistic Density Function
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Auto Correlation Function
Cross Correlation Function of Axial Transverse
Moduli
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Inelastic Property Fields
Following the same approach, fields describing
the inelastic properties of the composite
material can be generated. The figures below
show the transverse yield stress, Eyy, and the
statistical characteristics of this field based
on a 5 x 5 moving window. The fibers were
assumed elastic and the yield stress and
hardening slope of the matrix were taken to be
300.5 MPa and 22.98E03 MPa, respectively. The
plastic strains were calculated at the
micromechanical level using a flow rule for the
matrix material based on the classical
incremental plasticity (Prandtl-Reuss) equations.

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0.2 Transverse Yield Stress - Y Direction
Auto Correlation Function
Cross-Correlation with Fiber Volume Fraction
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Finite Element Analysis
The basic assumption in Stochastic Finite Element
Analysis (SFEM) is that the material properties
are described by random fields.
Material property fields generated using the
moving window technique are used as input into
finite element codes to examine the influence of
random microstructure on the local elastic and
inelastic properties in composite materials.
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Deformed Shape
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Stress Distribution under uniform loading of
Random Composite
syy, max(red)0.39 MPa
txy, max(red)0.24 MPa
sxx, max(red)1.37 MPa
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CONCLUSIONS
Given the inherent randomness in composite
microstructure, composite constitutive behavior
is difficult to characterize using deterministic
techniques. Effective properties of the
composite provide insufficient detail when the
interest is in local behavior. A statistical
characterization of random material properties
associated with composite microstructure is
proposed.
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The moving window GMC technique generates
material property fields associated with complex
random material microstructures documented by
material micrographs. Random material property
fields are developed by smoothing the original
discrete material property field using
calculations of local effective properties. A
statistical description, specifically probability
density functions and auto- and cross-correlation
functions, for each property can be developed
from these fields. Simulations of additional
material property fields can be generated using
Monte Carlo techniques.
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The Monte Carlo simulations, in conjunction with
finite element models, will provide probabilistic
descriptions of local stresses that develop due
to random microstructure helping to establish
measures of material reliability (e.g., estimates
of the probability that stress will exceed
undesirable levels). GMC and finite element
methods can be applied to various fields where
system reliability is governed by small-scale
fluctuations in material/geometric properties.