Narrow Results

We prove a global existence theorem for the nonlinear viscoelastic equation for small data in H². Moreover we prove that the solution decays exponentially in bounded domains when t goes to infinity.

A network was characterized by its viscoelastic properties. The viscoelastic property indicates the deformations or changes in the shape and in the internal structure during the evolution of a network. The change in the direction of motion was taken as elastic deformation and the change in the vertical direction as viscous deformation. These deformations were related to the change of geometry of internal structure and of shape. Thus it was possible to characterize a network by its storage and loss moduli. The change of the structure of a network during its evolution changes also its entropy. However entropy depends on the number of microstates of an already existing framework. As examples, two different systems (i) New York Stock Exchange and (ii) a melody were studied for their viscoelastic properties. The change of viscous property was compared with the change of different types of entropies such as configurational entropy, crossing entropy, and topological entropy. This last entropy was introduced and explained in the text. It was found out that there is no direct correspondence between the increase of entropy and the increase of viscous property of a network although they sometimes correlate with each other.

The rise of empires can be elucidated by treating them as living organisms, and the celebrated Verhulst or Lotka–Volterra dynamics can be used to understand the growth mechanisms of empires. The fast growth can be expressed by an exponential function as in the case of Macedonian empire of the Alexander the Great whereas a sigmoidal growth can be expressed by power-law equation as in the case of Roman and Ottoman empires. The superpowers Russia and the USA follow somehow different mechanisms, Russia displays two different exponential growth behaviors whereas the USA follows two different power-law behaviors. They did not disturb and mobilize their social capacity much during the course of their rise. The decline and the collapse of an empire occur through a kind of fragmentation process, and the consequently formed small states become rather free in their behavior. The lands of the new states formed exhibit a hierarchical pattern, and the number of the states having an area smaller than the largest one can be given either by an exponential or power-law function. The exponential distribution pattern occurs when the states are quite free in their pursuits, but the power-law behavior occurs when they are under the pressure of an empire or a strong state in the region. The geological and geographical conditions also affect whether there occurs exponential or power-law behavior. The new unions formed such as the European Union and the Shanghai Cooperation increase the power-law exponent implying that they increase the stress in the international affairs. The viscoelastic behavior of the empires can be found from the scattering diagrams, and the storage $(G^{\prime })$and loss modulus $(G^{\prime \prime })$, and the associated work-like and heat-like terms can be determined in the sense of thermodynamics. The $G^{\prime }$ of Ottomans was larger than that of Romans implying that they confronted severe resistance during their expansion. The $G^{\prime }$ of Russia is also larger than that of the USA; in fact the USA did not face severe resistance as they had an overwhelming superiority over native Americans. The $G^{\prime }>G^{\prime \prime }$ indicates solidity in the social structure and Romans, Ottomans, and Russians all have $G^{\prime }$ larger than $G^{\prime \prime }$. The $G^{\prime }$ is slightly larger than $G^{\prime \prime }$ for the USA indicating that they have had a very flexible social structure. By the same token the ratio of the work-like term to the internal energy is larger for Ottomans than that of Romans, and larger for the USA than that of Russia. That means the fraction of the total energy allocated to improve the social capacity is larger for Romans than that of Ottomans, and is larger for Russians than that of the USA.

A new methodology was introduced to investigate the pattern formation in time series systems due to their viscoelastic behavior. Four stochastic processes, uniform distribution, normal distribution, Nasdaq-100 stock market index, and a melody were studied within this context. The time series data were converted into vectorial forms in a scattering diagram. The sequential vectors can be split into its in-line (or conservative) and out-of-line (or dissipative) components. Thus, one can define the storage and loss modulus for conservative, and dissipative components, respectively. Instead of using the geometric Brownian equation which involves Wiener noise term, the changes were taken into consideration at every step by introducing “lethargy” concept and the deviation from it. Thus, the mathematics is somehow simplified, and the dynamical behavior of time series systems can be elucidated at every step of change. The viscoelastic behavior of time series systems reveals patterns of the viscoelastic parameters such as storage and loss modulus, and also of thermodynamic work-like and heat-like properties. Besides, there occur some minima and maxima in the distribution of the angles between the sequential vectors in the scattering diagram. The same is true for the change of entropy of the system.

Dynamic fracture in viscoelastic solids has been studied computationally using a mesoscale Finite Element model. In order to study crack propagation in homogeneous or amorphous materials the locations of nodes are selected regularly or randomly, respectively. In both cases results show oscillations in crack velocity above a critical velocity of about one-half the Raleigh velocity. The complicated topology of cracks obeys the scaling law found in experimental works. Dissipative systems are found to bear a larger maximum strain than purely elastic systems before macroscopic fracture.

We propose a model for two identical granular fluids separated by a piston that can present clustering (volume tending to zero) for a range of parameters. This model is based on point granular particles. For a convenient range of parameters, a granular fluid-cluster collapse is then possible and permits us to get an insight on the physics of granular clusters based on the behavior of the fluid phase itself.

ER fluids under shear are often approximated by the Bingham model with the electric field induced yield stress. This approach, however, neglects the properties of the solid phase created. The oscillation measurements taken under field below the yield point and leading to calculation of storage and loss moduli of the electrified solid do not allow for determination of the full stress–strain relationship of the solidified ER suspension. In order to overcome this problem and collect information about the mechanical properties of this solid a dedicated rheometer was constructed capable of measuring small strains under moderate stresses and under electric field. Full shapes of the stress–strain curves were recorded leading to determination of storage modulus, maximum deformation and yield point. Looking for relations between the viscoelastic behavior of ER suspensions and properties of their material components a series of fluids comprising silicone oil dispersions of conductive polymers (polyaniline, poly(p-phenylene), pyrolyzed polyacrylonitrile) and polymer electrolytes (polyacrylonitrile complexes with inorganic salts) was studied. It was found that the viscoelastic responses of ER fluids were affected by material characteristics of the dispersed phase.

The rheological properties of novel MR fluids are characterized using a parallel plate MR shear rheometer. In these MR fluids the surface of iron particles is coated with a polymer. The rheological properties are measured and compared at various magnetic field strengths, shear rates and strain amplitudes. It has been shown that these MR fluids exhibit stable and desirable rheological properties such as, low viscosity and high yield stress.

A mini-Split Hopkinson Tensile bar (mSHTB) system is developed. The system employs small diameter polymeric bars, which can achieve a closer impedance match with the specimens, thus it provides a lower noise-to-signal ratio and a longer duration of tensile pulse, which results in a higher maximum strain. With the three element viscoelastic model, a characteristic method for reconstruction of the profiles of strain, particle velocity, and stress in an arbitrary cross section of a viscoelastic bar on the basis of the strain signal measured in one section is developed. Experiments using the mSHTB enable us to appropriately characterize the dynamic behavior of small-sized, low-impedance material under a particular range of high strain rate.

In this study we analyzed the deformation of the polymeric rod impacting on the rigid wall which is called "Taylor impact test."" We simulated three-dimensional Taylor impact test depending on the various polymeric materials using the explicit finite element method by employing DYNA3D code. In simulation, polymeric materials were modeled using viscoelastic constitutive relations with the relaxation time and shear modulus. We have carried out the numerical simulation for the transient deformation characteristics and discussed effects of the viscoelastic constants on the deformation of the polymeric rod under impact.

Plastics is commonly used in consumer electronics because of it is high strength per unit mass and good productivity, but plastic components may often become distorted after injection molding due to residual stress after the filling, packing, and cooling processes. In addition, plastic deteriorates depending on various temperature conditions and the operating time, which can be characterized by stress relaxation and creep. The viscoelastic behavior of plastic materials in the time domain can be expressed by the Prony series using the ABAQUS commercial software package. This paper suggests a process for predicting post-production deformation under cyclic thermal loading. The process was applied to real plastic panels, and the deformation predicted by the analysis was compared to that measured in actual testing, showing the possibility of using this process for predicting the post-production deformation of plastic products under thermal loading.

A new hindered phenol with branched structure (chemical formula: 2-(((3,5-ditert-butyl-4-hydroxybenzoyl)oxy)methyl)-2-ethylpropane-1,3-diyl bis(3,5-ditert-butyl-4-hydroxybenzoate)) was synthesized and characterized, and its thermo-oxidative stability as antioxidant into nitrile rubber (NBR) was assessed by the measurements of accelerated aging, with two antioxidants (AO-80 and AO-2246) as a comparison. By measuring and analyzing the oxidative induction temperature, oxidative induction time and dynamic mechanical properties of NBR with addition of 0.5 wt.% antioxidants, it is confirmed that new hindered phenol with branched structure could effectively inhibit the oxidation degradation of NBR.

In this paper, the thermal effect on wave dispersion characteristic induced by the spinning and longitudinal motions in the viscoelastic carbon nanotubes (CNTs) conveying fluid is presented. Hamilton’s principle is utilized to derive the governing equation of this nanotube based on the non-local strain gradient and Euler–Bernoulli beam theories. Then, the dispersion solution is found by using the Naiver method. Based on this, the influences of the spinning and longitudinal motion velocities, structural damping, temperature and flow velocity on dispersion relation of the nanotubes are discussed according to numerical simulation. In view of the results of numerical examples, some interesting conclusions can be drawn. The existence of spinning motion leads to the coupling between the vibration in the $y$ and $z$ directions, which induces that the first-order transverse wave frequency increases/decreases for small/large wave number and the second-order one increases. The important solutions presented in the work will provide the useful information for the designation of the nanotubes conveying fluid with the spinning and longitudinal motion.

We study a system of nonlinear partial differential equations governing the motion of an incompressible viscoelastic fluid of Oldroyd type in a bounded domain, with a non-Newtonian viscosity depending on the second invariant of the rate of deformation tensor. Considering the equations in a suitably decomposed form, we establish, for small and suitably regular data, existence of a unique solution using a fixed point argument in an appropriate functional setting. This model includes the classical Oldroyd-B fluid as a particular case.

We consider a class of abstract evolutionary variational inequalities arising in the study of frictionless contact problems for linear viscoelastic materials with long-term memory. We prove an existence and uniqueness result, by using arguments of time-dependent elliptic variational inequalities and Banach's fixed point theorem. We then consider numerical approximation of the problem by introducing spatially semi-discrete, time semi-discrete and fully discrete schemes. For both schemes, we show the existence of a unique solution and derive error estimates. Finally, we apply the abstract results to the analysis and numerical approximation of the Signorini frictionless contact problem between two viscoelastic bodies with long-term memory.

We consider a class of abstract second-order evolutionary inclusions involving a Volterra-type integral term, for which we prove an existence and uniqueness result. The proof is based on arguments of evolutionary inclusions with monotone operators and the Banach fixed point theorem. Next, we apply this result to prove the solvability of a class of second-order integrodifferential hemivariational inequalities and, under an additional assumption, its unique solvability. Then we consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is dynamic, the material behavior is described with a viscoelastic constitutive law involving a long memory term and the contact is modelled with subdifferential boundary conditions. We derive the variational formulation of the problem which is of the form of an integrodifferential hemivariational inequality for the displacement field. Then we use our abstract results to prove the existence of a unique weak solution to the frictional contact model.

Small deformations of a viscoelastic body are considered through the linear Maxwell and Kelvin–Voigt models in the quasi-static equilibrium. A robust mixed finite element method, enforcing the symmetry of the stress tensor weakly, is proposed for these equations on simplicial tessellations in two and three dimensions. A priori error estimates are derived and numerical experiments presented. The approach can be applied to general models for linear viscoelasticity and thus offers a unified framework.

In this work we investigate a phase field model for damage processes in two-dimensional viscoelastic media with non-homogeneous Neumann data describing external boundary forces. In the first part we establish global-in-time existence, uniqueness, a priori estimates and continuous dependence of strong solutions on the data. The main difficulty is caused by the irreversibility of the phase field variable which results in a constrained PDE system. In the last part we consider an optimal control problem where a cost functional penalizes maximal deviations from prescribed damage profiles. The goal is to minimize the cost functional with respect to exterior forces acting on the boundary which play the role of the control variable in the considered model. To this end, we prove existence of minimizers and study a family of "local" approximations via adapted cost functionals.

We introduce a new phase field model for tumor growth where viscoelastic effects are taken into account. The model is derived from basic thermodynamical principles and consists of a convected Cahn–Hilliard equation with source terms for the tumor cells and a convected reaction–diffusion equation with boundary supply for the nutrient. Chemotactic terms, which are essential for the invasive behavior of tumors, are taken into account. The model is completed by a viscoelastic system consisting of the Navier–Stokes equation for the hydrodynamic quantities, and a general constitutive equation with stress relaxation for the left Cauchy–Green tensor associated with the elastic part of the total mechanical response of the viscoelastic material. For a specific choice of the elastic energy density and with an additional dissipative term accounting for stress diffusion, we prove existence of global-in-time weak solutions of the viscoelastic model for tumor growth in two space dimensions $d=2$ by the passage to the limit in a fully-discrete finite element scheme where a CFL condition, i.e. $\mathrm{\Delta }t\le C{h}^2$, is required.

Moreover, in arbitrary dimensions $d\in \{2,3\}$, we show stability and existence of solutions for the fully-discrete finite element scheme, where positive definiteness of the discrete Cauchy–Green tensor is proved with a regularization technique that was first introduced by Barrett and Boyaval [Existence and approximation of a (regularized) Oldroyd-B model, Math. Models Methods Appl. Sci. 21 (2011) 1783–1837]. After that, we improve the regularity results in arbitrary dimensions $d\in \{2,3\}$ and in two dimensions $d=2$, where a CFL condition is required. Then, in two dimensions $d=2$, we pass to the limit in the discretization parameters and show that subsequences of discrete solutions converge to a global-in-time weak solution. Finally, we present numerical results in two dimensions $d=2$.

Despite the existence of respiratory mechanics models in the literature, rarely one finds analytical expressions that predict the work of breathing (WOB) associated with natural breathing maneuvers in non-ventilated subjects. In the present study, we develop relations that explicitly identify WOB, based on a proposed nonlinear model of respiratory mechanics. The model partitions airways resistance into three components (upper, middle and small), includes a collapsible airways segment, a viscoelastic element describing lung tissue dynamics and a static chest wall compliance. The individual contribution of these respiratory components on WOB is identified and analyzed. For instance, according to model predictions, during the forced vital capacity (FVC) maneuver, most of the work is expended against dissipative forces, mainly during expiration. In addition, expiratory dissipative work during FVC is almost equally partitioned among the upper airways and the collapsible airways resistances. The former expends work at the beginning of expiration, the latter at the end of expiration. The contribution of the peripheral airways is small. Our predictions are validated against laboratory data collected from volunteer subjects and using the esophageal catheter balloon technique.