Masataka Fukunaga

A nonlinear fractional derivative model of impulse motion for viscoelastic materials

Generally, force can be described as a function of displacement in the mechanical model. A nonlinear fractional derivative model with respect to displacement is proposed to describe force for a viscoelastic material based on the measured data of impulsive motion. In the model, the nonlinearity is assumed to appear in the term of the fractional derivative. Three types of nonlinearity in the fractional derivative term are considered as candidates for a suitable model for reproducing the impulsive responses of the measured data. The first one is the case where the nonlinearity appears in the coefficient of the fractional derivative and the second in the fractionally differentiated term. The third one is the case where the nonlinearity appears as the combination of the above two types. The equation of motion and the initial conditions are derived by employing the above nonlinear models for head-on collisions of a rigid body onto the viscoelastic material. The property of the impulsive responses for the system that is derived above is characterized by the time when the acceleration shows its maximum. The symmetry property of increasing and decreasing acceleration response about the time of maximum acceleration is also considered. The second-type nonlinearity in the model seems to be adequate for reproducing the measured response.

Analysis of Impulse Response of a Gel by Nonlinear Fractional Derivative Models

In the separated paper in this conference, we proposed two fractional derivative models for stress-strain relation of viscoelastic materials. In this paper, these models are tried to analyze the experimental results on impulse response of a gel. It is shown that one model reproduces well the impulse response to relatively small impact velocity, while the other explains the response to larger impact velocity. It is also shown that the empirical nonlinear fractional derivative model proposed by authors at FDA08 has nearly identical impulse response to one of the models.Copyright

Fractional Derivative Constitutive Models for Finite Deformation of Viscoelastic Materials

A methodology to derive fractional derivative constitutive models for finite deformation of viscoelastic materials is proposed in a continuum mechanics treatment. Fractional derivative models are generalizations of the models given by the objective rates. The method of generalization is applied to the case in which the objective rate of the Cauchy stress is given by the Truesdell rate. Then, a fractional derivative model is obtained in terms of the second Piola–Kirchhoff stress tensor and the right Cauchy-Green strain tensor. Under the assumption that the dynamical behavior of the viscoelastic materials comes from a complex combination of elastic and viscous elements, it is shown that the strain energy of the elastic elements plays a fundamental role in determining the fractional derivative constitutive equation. As another example of the methodology, a fractional constitutive model is derived in terms of the Biot stress tensor. The constitutive models derived in this paper are compared and discussed with already existing models. From the above studies, it has been proved that the methodology proposed in this paper is fully applicable and effective.

Comparison of fractional derivative models for finite deformation with experiments of impulse response

In constructing a three-dimensional fractional derivative model for viscoelastic materials, difference of interpretation of strain in the fractional derivative results in many types of nonlinear models. In this paper, some nonlinear models are compared with experimental results of a polymer gel. In the experiment a weight was collided in the vertical direction on the upper free surface of a cylindrically shaped gel with the bottom side fixed. Then, the acceleration and the displacement of the contact surface between the weight and the gel are observed. In the models, deformation of the gel is approximated by the uniaxial compression of incompressible materials. Each model considered in this paper reproduces well the acceleration data in the early stage. However, differences between the calculated responses among the models begin to appear around the maximum point of the acceleration when the compression speed slows down. It is shown that incompressible models based on the weakly compressible approximations give good fits.

A high-speed algorithm for computation of fractional differentiation and fractional integration.

A high-speed algorithm for computing fractional differentiations and fractional integrations in fractional differential equations is proposed. In this algorithm, the stored data are not the function to be differentiated or integrated but the weighted integrals of the function. The intervals of integration for the memory can be increased without loss of accuracy as the computing time-step n increases. The computing cost varies as , as opposed to n2 of standard algorithms.

High Speed Algorithm for Computation of Fractional Differentiation and Integration

A high speed algorithm for computing fractional differentiations and fractional integrations in fractional differential equations is proposed. In this algorithm the stored data is not the history of the function to be differentiated or integrated but the history of the weighted integrals of the function. It is shown that, by the computational method based on the new algorithm, the integration time only increases in proportion to n log n, different from n2 by a standard method, for n steps of integrations of a differential integration.Copyright

Initial Condition Problems of Fractional Viscoelastic Equations

The 2nd order differential equation with fractional derivatives describing dynamic behavior of a single-degree-of-freedom viscoelastic oscillator, referred to as fractional viscoelastic equation (FVE), is considered. Some types of viscoelastic damped mechanical systems may be described by FVE. The differential equation with fractional derivatives is often called the fractional differential equation (FDE). FDE can be solved for zero initial values, but it can not generally be solved for non-zero initial values. How to solve the problem is one of the key issues in this field. This is called “Initial condition (value) problems” of FDE. In this paper, initial condition problems of FVE are solved by making use of the prehistory functions of unknowns which are specified before the initial instance (referred to as the initial functions) starts. Introduction of initial functions into FDE reflects the physical state in giving the initial values. In this paper, several types of initial function are used to solve unique solutions for a type of FVE (referred to as FVE-I). The solutions of FVE-I are obtained by means of both numerical and analytical methods. Implication of the solutions to viscoelastic material will also be discussed.Copyright

Three-Dimensional Fractional Derivative Models for Finite Deformation

Fractional derivative stress-strain relations are derived for compressible viscoelastic materials based on the continuum mechanics. Several types of stress tensor and strain tensors are specified to describe the dynamics of continuous media. Consequently there are many equivalent expressions of stress-strain relations. If memory effect is not taken into account, these relations are equivalently transformed from one to another by suitable tensor operations. However, if memory effect is included in the mechanics of the materials, different types of stress-strain relations can be derived depending on the choice of the type of stress tensor, or equivalently the choice of the strain energy function. In this paper, several types of fractional derivative stress-strain relations are proposed.Copyright

Nonlinear Fractional Derivative Stress-Strain Relations for Polymer Gels Based on the Generalized Maxwell Model

In this paper, we formulate two nonlinear stress-strain relations including memory effect in the dynamical behavior of gels that are the kind of viscoelastic materials. The basic assumption of the model is made that the gels consist of blobs of high polymers. Hereditary response of blobs to the stress determines the average stress-strain relation of the material. Two stress-strain relations are derived for different models of gels. These stress-strain relations are compared with the fractional derivative version of Lodge’s rubber-like liquids and the empirical nonlinear fractional derivative model proposed by Fukunaga et. al. at FDA08.Copyright

Fractional Derivative Consideration on Nonlinear Viscoelastic Dynamical Behavior Under Statical Pre-Displacement

Nonlinear fractional calculus model for the viscoelastic material is examined for oscillation around the off-equilibrium point. The model equation consists of two terms of different order fractional derivatives. The lower order derivative characterizes the slow process, and the higher order derivative characterizes the process of rapid oscillation. The measured difference in the order of the fractional derivative of the material, that the order is higher when the material is rapidly oscillated than when it is slowly compressed, is partly attributed to the difference in the frequency dependence between the two fractional derivatives. However, it is found that there could be possibility for the variable coefficients of the two terms with the rate of change of displacement.Copyright