In this work we develop a variable order (VO) differential equation of motion for a spherical particle sedimenting in a quiescent viscous liquid. In particular, we examine the various force terms in the equation of motion and propose a new form for the history drag acting on the particle. We show that the variable order formulation allows for an effective way to express the dynamic transition of the dominant forces over the entire time of the motion of the particle from rest to terminal velocity. The use of VO operators also allows us to examine the evolving dynamics of the wake during sedimentation. Using numerical data from a finite element simulation of a sedimenting particle, we first solve for the order of the derivative that returns the correct decay of the history force. We then propose a relatively simple expression for the history force that is a function of the Reynolds number and particle-to-fluid density ratio. The new history drag expression correlates very well (R2>0.99) with the numerical data for terminal Reynolds numbers ranging from 2.5 to 20, and for particle-to-fluid density ratios of interest in practice (1<β<10).

We introduce the use of fractional differentiation for simulating cloth deformations underwater. The proposed approach is able to achieve realistic underwater deformations without simulating the Eulerian body of water in which the cloth is immersed. Instead, we propose a particle-based cloth model where half-derivative viscoelastic elements are included for describing both the internal and external dynamics of the cloth. These elements model the cloth responses to fluid stresses and are also able to emulate the memory-laden behavior of particles in a viscous fluid. As a result, we obtain fractional clothes , which are able to correctly depict the dynamics of the immersed cloth interacting with the fluid even though the fluid is not simulated. The proposed approach produces realistic underwater cloth deformations and has obvious advantages in simplicity and speed of computation in comparison to volumetric fluid simulation approaches.

We review the application of differential operators of noninteger order to the modeling of dynamic systems. We compare all the definitions of Variable Order (VO) operators recently proposed in literature and select the VO operator that has the desirable property of continuous transition between integer and non-integer order derivatives. We use the selected VO operator to connect the meaning of functional order to the dynamic properties of a viscoelastic oscillator. We conclude that the order of differentiation of a single VO operator that represents the dynamics of a viscoelastic oscillator in stationary motion is a normalized phase shift. The normalization constant is found by taking the difference between the order of the inertial term (2) and the order of the spring term (0) and dividing this difference by the angular phase shift between acceleration and position in radians (𝜋), so that the normalization constant is simply 2/𝜋.

Author(s): Ramirez, Lynnette E. S. | Advisor(s): Coimbra, Carlos | Abstract: This work demonstrates the practicality of using variable order (VO) derivative operators for modeling the dynamics of complex systems. First we review the various candidate VO integral and derivative operator definitions proposed in the literature. We select a definition that is appropriate for physical modeling based on the following criteria: the VO operator must be able to return all intermediate values between 0 and 1 that correspond to the argument of the order of differentiation in addition to the integer order derivatives, and the derivative of a true constant function should be 0. Then we apply the chosen operator to 3 different problems: a stationary analysis of viscoelastic oscillators, the formulation of a Lagrangian equation of motion for a sedimenting particle in a viscous fluid, and the development of a constitutive equation for viscoelastic materials. In the first problem we obtain an analytical solution for the order of the operator and connect the meaning of functional order to the dynamic properties of a viscoelastic oscillator. We replace the multi-term differential equation for the viscoelastic oscillator with a single-term VO equation. We determine that the order of differentiation for a single operator describing all dynamic elements in the stationary equation of motion (mass, damping and spring) is equal to the normalized phase shift. The normalization constant is found by taking the difference between the order of the inertial term (2) and the order of the spring term (0) and dividing this difference by the angular phase shift between acceleration and position in radians (π), so that the normalization constant is simply 2/π. For the second problem we focus on the transient equation of motion for a spherical particle sedimenting in a quiescent viscous liquid. In particular, we examine the various force terms in the equation of motion and propose a new form for the history drag acting on the particle at finite Reynolds numbers. This new form equates the history drag to the VO derivative of the velocity of the particle. Using numerical results from a finite element simulation of the particle we solve for order of the derivative q and evaluate how the order changes over time. Based on these results we propose a simple form for q and obtain a correlation for the history drag acting on the particle that is in good agreement with the numerical data for terminal Reynolds numbers ranging from 2.5 to 20. In the final problem we present a simple constitutive equation for linear viscoelastic materials strained at constant strain rates. We propose a relationship in which the stress is related to the q(t) derivative of strain, where q(t) in this case is a function of normalized time. This order function is postulated to be proportional to the rate of change of disorder within the material. From a statistical mechanics based theory, we find that q(t) is proportional to tInt. Using experimental data for an epoxy resin and carbon/epoxy composite undergoing compression, we determine the final form for the constitutive equation that models the linear viscoelastic deformation in time. The resulting dimensionless constitutive equation agrees well all the normalized data.

A constitutive equation for linear viscoelasticity is presented. The equation is formulated using a Variable-Order (VO) integro-differential operator in which the order of the derivative q(t*) is allowed to be a function of the dependent or independent variables. We propose a relationship in which the stress is related to the q-th derivative of strain, where q(t*) and the normalized time t* characterize the viscoelastic response of the material. We assume that the function q(t*) is a measure of the rate of change of disorder within the material and develop a statistical mechanical model. The resulting model correlates well with experimental results for strain rate values varying over eight orders of magnitude. Using experimental data for a carbon/epoxy composite and an epoxy resin undergoing constant rate compression in the linear range, we derive a semi-empirical functional relationship with the normalized time that is used in a VO constitutive equation to model the viscoelastic deformation in time. The resulting dimensionless constitutive equation requires a much smaller number of empirically determined coefficients to adequately represent the data.

A constitutive relation for linear viscoelasticity of composite materials is formulated using the novel concept of Variable Order (VO) differintegrals. In the model proposed in this work, the order of the derivative is allowed to be a function of the independent variable (time), rather than a constant of arbitrary order. We generalize previous works that used fractional derivatives for the stress and strain relationship by allowing a continuous spectrum of non-integer dynamics to describe the physical problem. Starting with the assumption that the order of the derivative is a measure of the rate of change of disorder within the material, we develop a statistical mechanical model that is in agreement with experimental results for strain rates varying more than eight orders of magnitude in value. We use experimental data for an epoxy resin and a carbon/epoxy composite undergoing constant compression rates in order to derive a VO constitutive equation that accurately models the linear viscoelastic deformation in time. The resulting dimensionless constitutive equation agrees well with all the normalized data while using a much smaller number of empirical coefficients when compared to available models in the literature.