Studies of Fiber Processing and Machine
Design: Mathematical Modeling, Computer Simulation, and Virtual
Prototyping (W. W. Roberts)
Mathematical modeling and computer simulation are underway
on fiber processing technologies with focus on the motion,
dynamics, transport, and collection of fibers within specific
representative classes of gas flow environments and machine
design configurations of importance in the industrial setting.
Desired capabilities are evident in capturing dominant physical
processes and fundamental dynamical mechanisms underlying
gas-machine, fiber-machine, fiber-fiber, and gas-fiber interactions.
Simulations on high-speed computers demonstrate powerful predictive
capabilities of optimizing fiber movement, manageability,
and control; optimizing input-fiber characteristics to produce
output products of higher quality; and virtual prototyping
and optimal design and redesign of fiber processing machinery.
Photographic time-snapshot of a three-dimensional model-predicted
transport of fibers in a computer-simulated two-phase turbulent
gas-fiber flow (from left to right), with electrostatic charging
of the fibers that effectively pins the fibers to a grounded
conveyor belt (lower right).
Studies of Fibrous Assemblies: Modeling / Computer Simulation
of Compressional and Recovery Behavior (Collaborators: W.
W. Roberts, N. B. Beil)
An important problem that has been researched by fiber and
textile scientists and engineers for over 50 years is modeling
the compression and recovery behavior of fibrous assemblies.
We are developing a three-dimensional model to relate the
mechanical properties of individual fibers and how they are
arranged in a fibrous assembly to the bulk properties of the
fibrous assembly. At the present stage of development, the
model allows the prediction of the bulk properties of the
fibrous assembly during compression from the physical properties
of its component individual fibers, taking into account both
static and kinetic friction at contacts between fibers. The
figure below depicts a representative model fibrous assembly
undergoing compression, as the top wall is slowly depressed.
Computer simulations are run for a number of cases with specific
friction conditions applied in order to compare predictions
of this model with experimental results and with van Wyk's
theory of the uniaxial compression of an initially random
fibrous assembly. These computer simulations demonstrate a
reasonable ability to predict the undetermined constant K
in van Wyk's theory. The computer simulations also show a
significantly greater number of fiber-fiber contacts being
formed than theories based only on the diameter and arrangement
of fibers have predicted. The predicted contacts have a wide
range of contact forces, while only a small percentage of
them do not slip. The model may be used to investigate phenomena
associated with the compression of fibrous assemblies, such
as fiber crimp and hysteresis. We track computationally the
potential energy in the assembly and the work done on the
assembly, and we are able to produce realistic looking hysteresis
plots and can predict the amount of frictional energy dissipated
as a function of time. We find that fiber crimp has a large
effect on the compressional properties of a fibrous assembly
in that more highly crimped fibers absorb more energy as they
are compressed. They also absorb a higher proportion of their
energy in the twisting mode, which has been neglected by previous
investigators.
 
Side views of a three-dimensional unit cube cell, depicting
a representative model fibrous assembly containing 50 fibers
that constitute a fiber volume fraction of 0.8%, before compression
(left panel) and during compression, as the top wall is depressed,
to 70% of its initial volume (right panel).
The model not only allows exploration of the characteristics
of a fibrous assembly under compression at a level of detail
impossible to achieve through experiment but also allows inclusion
of effects that are very difficult to account for quantitatively
through theory alone. Factors that can be accounted for, thus
far, include initial arrangement and configuration of the assembly,
fiber crimp, various types of friction, distribution of contact
forces, and steric exclusion of fibers. Applications of this
work include predicting the properties of wool or fiber fill
based on the fibers and on the processing used, designing insulation
that retains its insulating properties after being compressed,
developing materials for acoustic noise and vibration control,
understanding fibrous cytostructural invadopodia in malignant
tumor cancers, and simulating other medical fibrous malfunctions.
Studies involving the Multi-Scale Mathematical Modeling
/ Computer Simulation of Cancerous Tumor Invadopodia - Bridging
Nano-, Micro-, and Milli- Scales (Collaborators: W. W. Roberts,
G. T. Gillies, H. L. Fillmore, I. Chasiotis)
Certain types of cancer cells produce a variant of filopodia
that has been termed the "invadopodium." The cancerous cell
projects the invadopodium into the extracellular matrix, the
structure of which is subsequently degraded to enable cell
motility through it. The interaction between the cell and
its surroundings during the invasion process is a very complex
one, with the invadopodium playing a role that is not yet
fully understood. The mechanics of invadopodium self-assembly
occur on the nanoscale, while the extension and invasion of
the invadopodium into the extracellular matrix occur on the
micron to tenths-of-a-millimeter scale. This current work
is an attempt to move toward much-needed deeper fundamental
understanding through critical multi-scale mathematical modeling
and computer simulation that hopefully will bridge the required
nano-, micro-, and milli- scales spanned. A primary focus
in this research is the formulation of a three-dimensional
mathematical model, based on the momentum balance and moment
balance partial differential equations and constitutive equations
of 3-dimensional fibrous elastic microtubule structures of
appropriate bending stiffnesses within such an invadopodium,
to relate the mechanical properties of the nanometers-diameter
microtubules to the bulk properties of the tenths-of-a-millimeter-in-length
invadopodium.
Oblique end-view perspective of a 3-dimensional model invadopodium,
with a number of microtubule fibers readily apparent and distinguishable
at its near end (i.e., the right end).
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