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The nanoscale dimension known as D-spacing describes the staggering of collagen molecules, which are fundamental to the biphasic makeup of bone tissue. This dimension was long assumed to be constant, but recent studies have shown that the periodicity of collagen is variable. Given that the arrangement of collagen molecules is closely related to the degree of bone mineralization, recent studies have begun to look at D-spacing as a potential factor in the ongoing effort to battle postmenopausal osteoporosis. The theoretical models presented by previous studies have only opted to model a single collagen-hydroxyapatite period, so the creation of an intricate computational approach that more exhaustively models a network of collagen and mineral is well-warranted. Sheep present an excellent opportunity to examine metabolic disorders, as their bone structure similar to that of the human skeleton. Six Rambouillet-cross ewes were used for the purpose of gathering experimental data. Three ewes underwent a sham surgery (controls), while an ovariectomy (OVX) was performed on the remaining three sheep. Each sheep was sacrificed after 12 months and their radius and ulna were harvested for atomic force microscopy and mechanical testing. Each sheep bone produced up to 25 beam samples that were available for analysis, and two were randomly selected from each test sheep. The cranial anatomical sector was selected for testing as it replicates the tensile loading condition characteristically experienced by collagen molecules and its exclusive examination removes any unintended variation due to bone section. Experimental D-spacing measurements were used in a finite element software, Abaqus, to create the ``Complex Model'': a large-scale, 2-D staggered array representation of collagen and hydroxyapatite periodicity. D-spacings intrinsic variability was mimicked through a Gaussian distribution that randomly determined periodic lengths based on provided experimental data. The model was generated with these random conditions for 2 x 100 units. Safeguards were implemented to ensure appropriate ratios of collagen to hydroxyapatite throughout the randomization. Collagen was assigned viscoelastic material properties originally developed by Dr. Frank Richter and modified by Miguel Mendoza. Hydroxyapatite was modeled as an elastic isotropic material. Four models were created using randomized D-spacings from control sheep and four separate models were created based on OVX sheep. Tangent delta--a damping characteristic--was recorded to evaluate bone viscoelasticity across four test frequencies: 1, 3, 9, and 15 Hz. Results strongly suggest that the Complex Model matches experimental findings more accurately than previous computational approaches. These results indicate the complicated network of many collagen units is an essential parameter of adequate modeling. A repeated measures analysis of variance was performed to examine the differences between control and OVX sheep. After adjusting for all other predictors, at the 1% significance level, after adjusting for all other variables, there is not enough evidence to convince this study that the Surgical Treatment alone has a significant impact on output tangent delta. This finding leads this study to conclude that OVX is fully accounted for within the Complex Model through the inclusion of its D-spacing, and the answers to bone's complicated mechanical properties during estrogen loss may lie in how OVX changes collagen viscoelasticity. Significant interactions were found between the Model Type and the Test Frequency. A Tukey-Kramer pairwise comparison was performed between Complex and Experimental data, which determined the Complex Model did not behave statistically differently from experimental findings at 15 Hz. This result suggests the Complex Model may begin to be validated to experimental results in a statistically meaningfully way that is a first for this style of FEA approach. The flexibility implemented in the randomization of the Complex Model welcomes refinement primarily in modeling viscoelasticity and fine-tuning the representation of mineralization. Through adjusting these material characteristics, the Complex Model may become an even more powerful tool in examining bone viscoelasticity and metabolic disorders.

Bone is a largely bipartite viscoelastic composite. Its mechanical behavior is determined by strain rate and the relative proportions of its principal constituent elements, hydroxyapatite and collagen, but is also largely dictated by their geometry and topology. Collagen fibrils include many segments of tropocollagen in staggered, parallel sequences. The physical staggering of this tropocollagen allows for gaps known as hole-zones, which serve as nucleation points for apatite mineral. The distance between adjacent repeat units of tropocollagen is known as D-Spacing and can be measured by Atomic Force Microscopy (AFM). This D-Spacing can vary in length slightly within a bundle, but by an additional order of magnitude within the same specimen, and can significantly alter the proportion of hydroxyapatite. Previous researchers have built and refined a Finite Element Analysis “Complex Model” to capture the consequences of adjusting D-Spacing and the viscoelastic parameters. This will ultimately serve to elucidate and perhaps predict the mechanical consequences of biological events that alter these parameters. This study aims to further refine the model’s precision by accounting for crosslinking between fibrils, the presence of which serves to add mechanical strength. This study also looks to refine the currently used rheological models by way of frequency dependent parameters in the hopes of improving model accuracy over a wider frequency range. Hormonal factors such as estrogen can significantly determine the composition of bone. Menopause marks a significant reduction in circulating estrogen and has been shown to factor heavily in the development of conditions like osteoporosis. Because sheep feature a hormonal cycle and skeletal structure similar to humans, three of six mature Columbia-Rambouillet ewes were randomly selected to undergo an ovariectomy, the remainder serving as sham-operated controls. Twelve months later twenty-five beam samples were harvested from their radius bones for mechanical analysis and other testing, including atomic force microscopy (AFM) and dynamic mechanical analysis (DMA). The data gleaned from these tests provide an experimental basis of comparison with The Complex Model. A 2-D Finite Element Analysis model in Abaqus was first created by Miguel Mendoza, which enforced viscoelasticity and a realistic proportion and placement of hydroxyapatite and collagen. The viscoelasticity was modeled using a Standard Linear Solid involving springs and a dashpot element. Crosslinks of varying number and location were arranged within the former model configuration as node to surface tie-constraints to explore the treatment of the FEA Model as a more realistic assembly of parts. Frequencies utilized for this model included 1, 3, 9 and 12 Hz. This approach is referred to in this research as the Intermolecular Forces (IMF) Scheme. The model was subsequently refined by Christopher Ha and Austin Cummings. The model was characterized by 2x100 unit half-cells, the lengths of which were randomly generated by a Python script. This script ingested the mean and standard deviation D-Spacing length to generate a model geometrically similar to a real specimen bearing those dimensions. A frequency dependent value for the dashpot element in the rheological model used for tropocollagen was developed using this latter FEA model, named the Complex Model. Dashpot values explored for this variable dashpot included 0.0125, 0.125, 0.3125, 0.45, 0.5875, 0.725, 0.8625 and 1.25 GPa-s, some values chosen for their high performance in past studies and others to further narrow the search for the best performing dashpot. All dashpot values were investigated over the previously stated frequencies in addition to 2, 5, 7 and 12 Hz. The best fit dashpot values were plotted against the frequencies in which they best performed and a polynomial trend line was fitted to establish an equation, and that equation was used to modify an existing user material subroutine for tropocollagen to provide an automatic frequency dependent dashpot value to Abaqus. This approach is referred to in this research as the Variable Dashpot (VD) Scheme. Results for the IMF scheme generally performed poorly, with the fully tie-constrained model performing best with 0.77 and 0.024 for R2 and RMSE respectively. Of the randomized crosslink models, that with the lowest number (N=20) of randomly placed non-enzymatic crosslinks performed best with 0.81 and 0.051 for R2 and RMSE respectively. Increasing the number of randomized crosslinks reduced model fit, and the remaining three variants exhibited mean R2 and RMSE values of 0.66-0.67 and 0.052 respectively. For the VD scheme, models running custom modified variable dashpot UMATs yielded R2 and RMSE values of 0.87 and 0.012 for C2207, and 0.89 and 0.008 for C1809. This is a notable fit considering all other material property parameters are held constant throughout each frequency. In the rheological model, this research also found a striking difference between the frequency dependent viscous element values that made each model perform best. This indicates that differences in D-Spacing standard deviations between OVX and control may be associated with distinct strain-rate dependent mechanical responses.

No / A dynamic system with memory is a system for which knowledge of the equations of motion, together with the state at a given time instant t0 is insufficient to predict the evolution of the state at time instants t>t0. To calculate the response of systems with memory starting from an initial time instant t0, complete knowledge of the history of the system for t<t0 is needed. This is because the state vector does not contain all the information necessary to fully characterize the state of the system, i.e., the state vector of the system is not complete. In this paper, a state space formulation of viscoelastic systems with memory is proposed, which overcomes the concept of memory by enlarging the state vector with a number of internal variables that bear the information about the previous history of the system. The number of these additional internal variables is in some cases finite, in other cases, it would need to be infinite, and an approximated model has to be used with a finite number of internal variables. First a state space representation of the generalized Maxwell model is shown, then a new state space model is presented in which the relaxation function is approximated with Laguerre polynomials. The accuracy of the two models is shown through numerical examples.

Flexible high-deflection strain gauges have been demonstrated as cost-effective and accessible sensors for capturing human biomechanical deformations. However, the interpretation of these sensors is notably more complex compared to conventional strain gauges, partially owing to the viscoelastic nature of the strain gauges. On top of the non-linear viscoelastic behavior, dynamic resistance response is even more difficult to capture due to spikes in resistance during strain changes. This research examines the relationships between stress, strain, and resistance in nanocomposite sensors during dynamic strain situations. Under the assumption that both macroscopic stress and resistance are governed by microscopic stress concentrations at the junctions between nanoparticles and silicone matrix, the stress-resistance relationship is analyzed. Both stress and resistance are found to exhibit aspects of viscoelastic behavior, including creep decay and relaxation during constant strains. However, the resistance spikes are found to be more complex than a simple stress-resistance model can capture. This research then develops a model that captures the strain-resistance relationship of the sensors, including resistance spikes, during cyclical movements. The forward model, which converts strain to resistance, is comprised of four parts to accurately capture the different aspects of the sensor response: a quasi-static linear model, a spike magnitude model, a long-term creep decay model, and a short-term decay model. An inverse problem approach is used to create an inverse model, which predicts the strain vs time data that would result in the observed resistance data. The model is calibrated for a particular sensor from a small amount of cyclic data from a single test. The resulting sensor-specific model is able to accurately predict the resistance output with an R-squared value of 0.90. The inverse model is able to accurately predict key strain characteristics with a percent error of 0.5. The model can be used in a wide range of applications, including biomechanical modeling and analysis. It is found that the resistance spikes are directly correlated to the strain acceleration in terms of timing and in terms of magnitude. Poisson contraction rates and voids in the material are possible causes for resistance spikes during dynamic strain movements.

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Thesis: M.S., Massachusetts Institute of Technology, Department of Civil Engineering, 1976 / Bibliography: leaf 139. / M.S. / M.S. Massachusetts Institute of Technology, Department of Civil Engineering

Yes / By means of die drawing technique at rubber-state, effect of the orientation of microstructure on dielectric properties of polypropylene/multi-walled carbon nanotubes nanocomposites (PP/MWCNTs) was emphasized in this work. Viscoelasticity behavior of PP/MWCNTs with MWCNTs weight loadings from 0.25 to 5 wt% and dielectric performance of the stretched PP/MWCNTs under different drawing speeds and drawing ratios were studied for seeking an insight of the influences of dispersion and orientation state of MWCNTs and matrix molecular chains. A viscosity decrease (ca. 30%) of the PP/MWCNTs-0.25wt% melt was obviously observed owing to the free volume effect. Differential scanning calorimetry (DSC) and wide angle X-ray diffraction (WAXD) were adopted to detect the orientation structure and the variation of crystal morphology of PP/MWCNTs. Melting plateau regions, which indicated the mixed crystallization morphology for the stretched samples, were found in the DSC patterns instead of a single-peak for the unstretched samples. It was found that the uniaxial stretching process broke the conductive MWCNTs networks and consequently increased the orientation of MWCNTs as well as molecular chains along the tensile force direction, leading to an improvement of the dielectric performance.