Fractional Derivative Constitutive Model Research Articles

Fractional derivative Burgers models describing dynamic viscoelastic properties of asphalt binders

The Burgers model is ineffective in the representation of dynamic viscoelastic properties of asphalt binders; the existing fractional derivative Burgers (FDB I) model has limitations in its application and construction concept. Consequently, this study started by deriving the constitutive equations of the FDB I model and proposed a new application idea of Abel dashpot to construct the second fractional derivative Burgers (FDB II) model. Then, dynamic modulus formulas for the FDB II model were derived. On this basis, the frequency sweep tests of base and two polyphosphoric acid (PPA) modified asphalt binders were carried, and the dynamic viscoelastic properties of three different asphalt binders were described using three different models, and in-depth comparisons and analysis are carried out. The results show that both fractional derivative Burgers (FDB) models can simultaneously express the dynamic viscoelastic properties of asphalt binders with a set of parameters. Both FDB models perform better than traditional Burgers model in the representation of dynamic viscoelastic properties, and fitting results of master curves no longer exhibit oscillation properties like the traditional Burgers model. The R2 of FDB models could reach above 0.99, while R2 of traditional Burgers model was only 0.9349 at minimum. Among them, the FDB II model performs best. Five parameters of the FDB II model have relatively clear physical meanings. The new application concept of fractional dashpots was verified its theoretical feasibility and application advantages, which will provide a new idea for further application and development of fractional derivative constitutive models.

Study on the modified fractional derivative constitutive model of viscoelastic dampers

Viscoelastic (VE) dampers can effectively mitigate the vibration of structures under dynamic loading depending on better energy dissipating capability and are widely applied in practical engineering. On the basis of the studies on the mechanical properties of the VE material, the nonlinear behaviors of the VE damper are frequency dependent which can be elucidated by the changes in storage modulus and loss factor parameters. In this paper, experimental results of the VE damper reveal that the mechanical performance of the VE damper is correlative with the displacement amplitude. Therefore, a modified fractional derivative model is proposed to reflect the softening behavior of displacement amplitude on the VE damper. This result demonstrates that the modified model is reasonably practicable and can accurately evaluate the characteristic parameters of the VE damper. In order to apply the proposed constitutive model to finite element software, an equivalent method is presented to transform this model into the generalized Maxwell model. A three-dimensional finite element model of the VE damper is established with ABAQUS for numerical analysis. And compared with the experimental results, it is indicated that the equivalent generalized Maxwell model is available. This work verifies the conversion of the fractional derivative model to the linear VE model is feasible and provides a useful approach for the utilization of the fractional derivative model of the VE damper in finite element software.

Analytical analysis of sound transmission in passive damped multilayered shells

An analytical shell model is here proposed for the sound transmission analysis of composite multilayered structures embedding viscoelastic sheets. The present analytical solution, based on the Navier approach, is developed starting from the Principle of Virtual Displacements, through the use of advanced layer-wise higher-order models and by considering real viscoelastic frequency-dependent materials with fractional derivative constitutive models. Several simply-supported shell structures with isotropic and cross-ply composite layers embedding viscoelastic frequency-dependent sheets are considered and the results of the passive noise reduction analysis are given in terms of acoustic parameters based on Rayleigh integral method. Moreover a comprehensive study on different types of time loads is taken into account, considering impulse time load, time-window load histories, space–time dependent loads.

Investigation of a fractional derivative creep model of clay and its numerical implementation

The fractional calculus has been successfully applied to characterize the rheological property of elastic viscoplastic materials. However, soils were seldom involved in fractional derivative constitutive models, and the topic of triaxial creep tests is rarely discussed through fractional calculus. This study presents a fractional derivative model that describes the time-dependent behavior of Shanghai marine clay. The fractional derivative Merchant (FDM) model is established by introducing fractional element into the Merchant model. The comparison of the calculated values of the FDM model and the Merchant model shows that the FDM model can better describe the instant elastic deformation, attenuation creep and steady-state creep stages of the clay. The FDM model was implemented into a finite-element code in ABAQUS, which is used to simulate the creep tests. The simulations demonstrate that the development model has good ability to predict the time-dependent behavior of clay.

Fractional derivative-based creep constitutive model of deep artificial frozen soil

A fractional derivative constitutive model of deep artificial frozen soil was established based on the Nishihara model by analyzing the triaxial creep and shear test results of deep artificial frozen soil under different confining pressures and temperatures. Considering the physical and mechanical properties of frozen soil, linear Newton equations were substituted with nonlinear Newton equations, and a flexibility matrix was derived for numerical calculations. A relevant constitutive finite element program was developed and used to simulate a deep artificial frozen curtain of a coal mine through a user subroutine of the ADINA commercial finite element software. The theoretical calculations agreed well with the experimental results of the triaxial creep tests, and the numerical simulation results were consistent with the field measurement data. This fractional derivative constitutive model was verified to describe the non-decaying creep deformation characteristics of deep artificial frozen soil under high confining pressures. It can also be applied to the design and calculation of artificial freezing in deep strata.

Characterizing three-dimensional creep of Beishan granite by the variable-coefficient Abel dashpot

Accurate description of creep behavior is of great significance to the safety evaluation of high-level radioactive waste disposal in granite. In this study, a fractional derivative constitutive model is proposed to depict the creep process of Beishan granite, based on the conformable derivative. A variable-coefficient Abel dashpot considering the effect of damage evolution on creep properties is introduced to describe the nonlinear dilatancy strain in the accelerating stage. A new model is proposed and further generalized to three-dimensional creep equations by adopting Drucker–Prager criterion, and the analytic solutions are given employing the Laplace transform. The parameters of new creep model are determined based on experimental data and fitting results are compared. The fitting curves present a good agreement with experimental data, indicating that the conformable derivative creep model achieves a good performance in describing all three creep stages. The proposed model is validated by the comparison of different experiments and degradation of fractional derivative model. In addition, a sensitivity analysis is carried out to show the effects of stress level, fractional derivative order, damage exponent, and other parameters.

Three-Dimensional Finite Element Simulations on Impact Responses of Gels With Fractional Derivative Models

Fractional derivative constitutive models, developed by the present authors (CND, vol.10, 061002, 2015), are implemented into a commercial finite element (FE) software, abaqus (referred to as a computational model) for solving dynamic problems of gel-like materials. This software is used to solve impact responses of gels, and the solutions are compared with the experimental results. The FE results reproduce well the experimental acceleration and displacement data from different types of gels whose properties are characterized by the fractional order and material parameters. Thus, the computational model presented here was validated. The fractional derivative model is compared with the Simo model (Computer Method in Applied Mechanics and Engineering, 60:153–173, 1987), which is an integer order derivative model. The response of the fractional derivative model can be approximated well when appropriate parameters of the Simo model are used. In the finite element method (FEM), compressibility is introduced artificially for simulations. Interpretations are given on the compressibility of materials in the FEM.

Anisotropy Characteristics of Stress Relaxation in Coal: An Improved Fractional Derivative Constitutive Model

To study the anisotropy characteristics of coal during stress relaxation, a series of uniaxial compressive stress relaxation experiments are conducted on samples cut perpendicular and parallel to bedding. Some conclusions are as follows: (1) the stress of the specimens undergoes rapid relaxation, transient relaxation and steady relaxation, approximately represented with a negative exponential law, before reaching the final strain. The stress decreases discontinuously at the final strain. (2) Anisotropy is observable from stress relaxation tests. The failure strengths and elastic moduli of the bedding-perpendicular samples are generally greater than those of the bedding-parallel samples due to the hardening effect. At the same strain and before the final strain, the relaxation stress of the bedding-perpendicular samples is also greater than that of the bedding-parallel samples. At the final strain, for the bedding-parallel samples, the relaxation stress increases more evidently with higher initial stress relaxation rate, and the residual stress ratio decrease more remarkably. (3) An improved nonlinear fractional derivative generalized Kelvin model is proposed by replacing the Newton dashpot by the Abel dashpot, and the parameters in the model reflect the sample anisotropy well. This new model is used to fit the stress relaxation data, and the results show an improved accuracy in depicting the nonlinear stress relaxation characteristics of raw lean coal in relation to anisotropy over that of other models.

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.

粘弾性体の分数階微分有限変形理論と非線形有限要素法 : 減衰マトリックスと運動方程式の定式化(機械力学,計測,自動制御)

Fractional derivative finite deformation theory and nonlinear finite element method were developed for evaluating dynamic characteristics of viscoelastic bodies used in shock and vibration reduction. Fractional differential orders from the static and dynamic testing of viscoelastic material with small strain were obtained. Then a fractional derivative constitutive model for finite deformation theory was constructed and nonlinear finite element equations of motion, involving a newly developed viscoelastic damping matrix as well as standard mass and stiffness matrices, were formulated. Based on this work, a nonlinear finite element computer program for the evaluation of vibration characteristics of viscoelastic products was developed. Simple numerical examples were computed and compared with the experimental results. Both results are in good agreement.

Optimal design and parameter estimation of frequency dependent viscoelastic laminated sandwich composite plates

Recent developments in optimization and parameter estimation of frequency dependent passive damping of sandwich structures with viscoelastic core are presented in this paper. A finite element model for anisotropic laminated plate structures with viscoelastic frequency dependent core and laminated anisotropic face layers has been formulated, using a mixed layerwise approach, by considering a higher order shear deformation theory (HSDT) to represent the displacement field of the viscoelastic core, and a first order shear deformation theory (FSDT) for the displacement fields of adjacent laminated face layers. The complex modulus approach is used for the viscoelastic material behaviour, and the dynamic problem is solved in the frequency domain, using viscoelastic material data for the core, assuming fractional derivative constitutive models. Constrained optimization of passive damping is conducted for the maximisation of modal loss factors, using the Feasible Arc Interior Point Algorithm (FAIPA). Identification of the frequency dependent material properties of the sandwich core is conducted by estimating the parameters that define the fractional derivative constitutive model. Optimal design and parameter estimation applications in sandwich structures are presented and discussed.

Viscoelastic material response with a fractional-derivative constitutive model

A rubber coupon is modeled as a viscoelastic material by a fractional-derivative constitutive law to capture the material behavior over a broad frequency range. The coupon, of finite length, is clamped at one end and subject to either a forced loading or a deflection at the other end. The physical response profile of the material for this distributed model is examined in terms of a relaxation response, associated with a branch-cut integration, and a modal response.