:: Department of Mechanical and Aerospace Engineering

William W. Roberts - Director

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.

Figure 1: 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).

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.

Figure 2: 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).

Figure 3: Frictional energy dissipated during the first compression - release cycle, plotted against time, for a representative fibrous assembly case computed, where the compression and decompression are assumed to be performed at a constant rate.

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.

Figure 4: Pressure Poisson's Ratio - the ratio of the average pressure acting on the four side walls of the fibrous assembly to the applied pressure â plotted as a function of the volume, as the assembly is compressed to 70% of its initial volume, for five representative cases computed.

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.

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.

Figure 5: 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).