Identification of the Fractional Zener Model Parameters for a Viscoelastic Material over a Wide Range of Frequencies and Temperatures

The paper presents an analysis of the rheological properties of a selected viscoelastic material, which is dedicated to the reduction of vibrations in structures subjected to dynamic loads. A four-parameter, fractional Zener model was used to describe the dynamic behavior of the tested material. The model parameters were identified on the basis of laboratory tests performed at different temperatures and for different vibration frequencies. After proving that the material is thermoreologically simple, the so-called master curves were created using a horizontal shift factor. The Williams–Landel–Ferry formula was applied to create graphs of the master curves, the constants of which were determined for the selected temperature. The resulting storage and loss module functions spanned several decades in the frequency domain. The parameters of the fractional Zener model were identified by fitting the entire range of the master curves with the gradientless method (i.e., Particle Swarm Optimization), consisting in searching for the best-fitted solution in a set of feasible solutions. The parametric analysis of the obtained solutions allowed for the formulation of conclusions regarding the effectiveness of the applied rheological model.

Comprehensive Study on Shape Shifting Behaviors of Thermally Activated Hinges in FDM-based 4D Printing

4D printing of shape shifting structures, aka “hinges”, has raised a new standard in many fields. By using these hinges in certain parts of a 3D printed structures, a pre designed complex 3D shape with potential multifunctional application can be achieved from flat structure. This paper proposes a comprehensive semi-empirical model to predict the final shape shifting behavior and magnitude of the hinges printed by FDM process. First, all FDM main parameters are selected and reduced by design of experiment to printing speed, lamina thickness, nozzle temperature as well as printing pattern. In order to develop the model, a time-dependent constitutive model with these four process parameters were extracted for strain of an SMP homogeneous single layer structure using a fractional Zener model accompanied with Multiple Linear Regression (MLR) technique. Thereafter, the mathematical relations for shape shifting behavior of bilayer 4D printed structures were developed for beam bending and twisting by modifying Timoshenko’s constitutive equations. A comprehensive shape-shifting model was established including 3D printing parameters, angles, thickness ratios, activation time and temperature which was compared to the experimental data and results predicted both shape shifting behavior and magnitude of the hinges with good agreement. In addition, a novel flowchart was suggested to design and achieve the desired shape shifting behaviors. The proposed model and flowchart are novel tools to design 4D structures through desired shape-shifting of the hinges.

Transient vibrations of a fractional Zener viscoelastic cantilever beam with a tip mass

The presented work concerns the kinematically excited transient vibrations of a cantilever beam with a mass element fixed to its free end. The Euler–Bernoulli beam theory and the fractional Zener model of the beam material are assumed. A fractional Caputo derivative is used to formulate a viscoelastic material law. A characteristic equation, modal frequencies, eigenfunction and orthogonality conditions are achieved for the beam considered. The equations of motion of the system are solved numerically. A numerical solution of a multi-term fractional differential equation is obtained by means of a conversion to a mixed system of ordinary and fractional differential equations, each of the order of $$0 < \gamma \le 1$$ 0 < γ ≤ 1 . The transient time histories of the beam vibrations during the passage through resonance are calculated. A comparison between the beam responses obtained with a fractional and an integer viscoelastic material model is presented. The calculations performed reveal that use of the fractional damping affects on the time histories of the system. The calculated beam responses show that for some values of the order of the fractional derivative $$\gamma$$ γ , the amplitudes occurring in the area of the second resonance are greater than those obtained in the area of the first resonance, which does not occur in the case of the integer order of the fractional derivative. Moreover, an evaluation is made of the difference between the results obtained for the calculations using the fractional Zener model and the fractional Kelvin model. It is shown that for some physical beam parameters, the calculation results obtained using both models are virtually the same for both models, which means that the the simpler, fractional Kelvin–Voigt material can be used instead of the fractional Zener material model. This simplifies the solution and decreases the time needed to make the numerical calculations.

Simultaneous Characterization of Relaxation, Creep, Dissipation, and Hysteresis by Fractional-Order Constitutive Models

We considered relaxation, creep, dissipation, and hysteresis resulting from a six-parameter fractional constitutive model and its particular cases. The storage modulus, loss modulus, and loss factor, as well as their characteristics based on the thermodynamic requirements, were investigated. It was proved that for the fractional Maxwell model, the storage modulus increases monotonically, while the loss modulus has symmetrical peaks for its curve against the logarithmic scale log(ω), and for the fractional Zener model, the storage modulus monotonically increases while the loss modulus and the loss factor have symmetrical peaks for their curves against the logarithmic scale log(ω). The peak values and corresponding stationary points were analytically given. The relaxation modulus and the creep compliance for the six-parameter fractional constitutive model were given in terms of the Mittag–Leffler functions. Finally, the stress–strain hysteresis loops were simulated by making use of the derived creep compliance for the fractional Zener model. These results show that the fractional constitutive models could characterize the relaxation, creep, dissipation, and hysteresis phenomena of viscoelastic bodies, and fractional orders α and β could be used to model real-world physical properties well.

Glass Transition Behavior of Wet Polymers

We have performed a systematical investigation on the glass transition behavior of amorphous polymers with different solvent concentrations. Acrylate-based amorphous polymers are synthesized and treated by isopropyl alcohol to obtain specimens with a homogenous solvent distribution. The small strain dynamic mechanical tests are then performed to obtain the glass transition behaviors. The results show that the wet polymers even with a solvent concentration of more than 60 wt.% still exhibit a glass transition behavior, with the glass transition region shifting to lower temperatures with increasing solvent concentrations. A master curve of modulus as a function of frequency can be constructed for all the polymer–solvent systems via the time–temperature superposition principle. The relaxation time and the breadth of the relaxation spectrum are then obtained through fitting the master curve using a fractional Zener model. The results indicate that the breadth of the relaxation spectrum has been greatly expanded in the presence of solvents, which has been rarely reported in the literature. Thus, this work can potentially advance the fundamental understanding of the effects of solvent on the glass transition behaviors of amorphous polymers.

Mathematical modeling of wave propagation in viscoelastic media with the fractional Zener model

The question of interest for the presented study is the mathematical modeling of wave propagation in dissipative media. The generalized fractional Zener model in the case of dimension d (d=1,2,3) is considered. This work is devoted to the mathematical analysis of such model: existence and uniqueness of the strong and weak solution and energy decay result which guarantees the wave dissipation. The existence of the weak solution is shown using a priori estimates for solutions which are also presented.

Wave propagation dynamics in a fractional Zener model with stochastic excitation

Equations of motion for a Zener model describing a viscoelastic rod are investigated and conditions ensuring the existence, uniqueness and regularity properties of solutions are obtained. Restrictions on the coefficients in the constitutive equation are determined by a weak form of the dissipation inequality. Various stochastic processes related to the Karhunen-Loéve expansion theorem are presented as a model for random perturbances. Results show that displacement disturbances propagate with an infinite speed. Some corrections of already published results for a non-stochastic model are also provided.