A model of weak viscoelastic nematodynamics

The paper develops a continuum theory of weak viscoelastic nematodynamics of Maxwell type. It can describe the molecular elasticity effects in mono-domain flows of liquid crystalline polymers as well as the viscoelastic effects in suspensions of uniaxially symmetric particles in polymer fluids. Along with viscoelastic and nematic kinematics, the theory employs a general form of weakly elastic thermodynamic potential and the Leslie–Ericksen–Parodi type constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank (orientation) elasticity in liquid crystal polymers. In general case, even the simplest Maxwell model has many basic parameters. Nevertheless, recently discovered algebraic properties of nematic operations reveal a general structure of the theory and present it in a simple form. It is shown that the evolution equation for director is also viscoelastic. An example of magnetization exemplifies the action of non-symmetric stresses. When the magnetic field is absent, the theory is reduced to the symmetric, fluid mechanical case with relaxation properties for both the stress and director. Our recent analyses of elastic and viscous soft deformation modes are also extended to the viscoelastic case. The occurrence of possible soft modes minimizes both the free energy and dissipation, and also significantly decreases the number of material parameters. In symmetric linear case, the theory is explicitly presented in terms of anisotropic linear memory functionals. Several analytical results demonstrate a rich behavior predicted by the developed model for steady and unsteady flows in simple shearing and simple elongation.     

A novel nonlinear viscoelastic solid model

The standard linear solid (SLS) model has been widely used to describe linear viscoelastic materials. Although the SLS can be extended to describe more complex phenomena through the inclusion of additional components, a compact model of nonlinear viscoelasticity remains elusive. Here, using the framework of the three component SLS model, the stress-strain relationships of the individual components have been generalised. This work describes the derivation of the new generalised model, termed nonlinear viscoelastic solid model, or NVS, that can potentially describe nonlinear viscoelastic phenomena in a compact differential form and reduces to the familiar SLS model if linear components are selected. As a proof of concept, exponential functions of strain and strain rate were selected for the three components and major viscoelastic phenomena such as stress relaxation, creep and sawtooth strain loading were simulated. Finally, to demonstrate its efficacy in describing biological tissue, the NVS model was used to simulate the cyclic loading of mammalian stomach and cardiac muscle tissues.     

On weak solutions to evolution equations of viscoelastic fluid flows

We study a system of nonlinear equations describing unsteady flows of a viscoelastic fluid of the Oldroyd type in a bounded three-dimensional domain under the mixed boundary conditions: the Navier slip condition is set on one part of the boundary, while on the other part, the no-slip condition is set. We prove the theorem concerning the existence, uniqueness, and energy estimates for weak solutions.     

A nonlinear viscoelastic fractional derivative model of infant hydrocephalus

Infant communicating hydrocephalus is a clinical condition where the cerebral ventricles become enlarged causing the developing brain parenchyma of the newborn to be displaced outwards into the soft, unfused skull. In this paper, a hyperelastic, fractional derivative viscoelastic model is derived to describe infant brain tissue under conditions consistent with the development of hydrocephalus. An incremental numerical technique is developed to determine the relationship between tissue deformation and applied pressure gradients. Using parameter values appropriate for infant parenchyma, it is shown that pressure gradients of the order of 1 mm Hg are sufficient to cause hydrocephalus. Predicting brain tissue deformations resulting from pressure gradients is of interest and relevance to the treatment and management of hydrocephalus, and to the best of our knowledge, this is the first time that results of this nature have been established.     

Existence of periodic weak solutions to the three-dimensional nonlinear viscoelastic system

In this paper, we discuss the time-periodic solutions to a general three-dimensional nonlinear viscoelastic system with Dirichlet boundary condition. Employing the singularity of the integral kernel, we obtain the energy estimates in Sobolev spaces with fractional index and then show the existence of time-periodic solutions to the problem for viscoelastic solid and liquid models, respectively.     

A microstructurally motivated anisotropic viscoelastic model for soft tissues

In this paper, a microstructurally motivated approach to take into account the anisotropic viscoelastic behaviour of soft biological tissues is proposed. The constitutive model is based on the assumption that this behaviour results from an interaction between collagen fibres and surrounding matrix constituents. Accordingly, a non–linear viscoelastic one–dimensional model for fibres and the nearby ground substance is developed. This model is then generalised to the anisotropic three–dimensional case. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of a viscoelastic spring-mass model

In this paper we consider a linear wave equation with strong damping and dynamical boundary conditions as an alternative model for the classical spring-mass-damper ODE. Our purpose is to compare analytically these two approaches to the same physical system. We take a functional analysis point of view based on semigroup theory, spectral perturbation analysis and dominant eigenvalues.     

A finite element formulation of a viscoelastic model for dielectric elastomers

Smart materials by definition, are solids, fluids or gases which react independently on changing external conditions and modify one or more properties without external stimuli. Sensu lato an external energy can produce the reaction, such as stress, temperature, moisture, pH, magnetic or electric fields. The distinguishing characteristic for electroactive polymers (EAP) is, that they react with a deformation by the application of an electrical field. This contribution presents a nonlinear electro-viscoelastic model for dielectric elastomers and its finite element implementation. This type of smart materials belong to the group of EAP's and consists out of soft elastomer between compliant and conducting electrodes. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

General decay of solutions for a weak viscoelastic equation with acoustic boundary conditions

In this paper, we consider the weak viscoelastic equation $$u_{tt} - \Delta u + \alpha(t) \int\limits_{0}^{t} g(t-s)\Delta u(s)\, {\rm d}s=0$$ with a homogeneous Dirichlet condition on a portion of the boundary and acoustic boundary conditions on the rest of the boundary. We establish a general decay result, which depends on the behavior of both α and g, by using the perturbed energy functional technique. This is an extension and improvement of the previous result from Park and Park (Nonlinear Anal 74(3):993–998, 2011) (i.e., the similar problem with ${\alpha(t) \equiv 1}$ ) to the time-dependent viscoelastic case.     

A weak solvability of a system of thermoviscoelasticity for the Jeffreys model

We establish a weak solvability of the initial-boundary value problem for a dynamic model of thermoviscoelasticity. The problem under consideration is an extension of the Jeffreys model obtained with the help of a consequence of the energy balance equation. We study the corresponding initial-boundary value problem by splitting the problem and reducing it to an operator equation in a suitable Banach space.     

Global attractor for a nonlinear viscoelastic model

In this paper we establish the existence of global attractors of a semiflow and a semigroup for a nonlinear 1-d viscoelasticity in two frameworks. We exploit the properties of the analytic semigroup to show the compactness for the semiflow and semigroup generated by the weak global solutions.     

Combinatorial and accessible weak model categories

In a previous work, we have introduced a weakening of Quillen model categories called weak model categories. They still allow all the usual constructions of model category theory, but are easier to construct and are in some sense better behaved. In this paper we continue to develop their general theory by introducing combinatorial and accessible weak model categories. We give simple necessary and sufficient conditions under which such a weak model category can be extended into a left and/or right semi-model category. As an application, we recover Cisinski-Olschok theory and generalize it to weak and semi-model categories. We also provide general existence theorems for both left and right Bousfield localization of combinatorial and accessible weak model structures, which combined with the results above gives existence results for left and right Bousfield localization of combinatorial and accessible left and right semi-model categories, generalizing previous results of Barwick. Surprisingly, we show that any left or right Bousfield localization of an accessible or combinatorial Quillen model category always exists, without properness assumptions, and is simultaneously both a left and a right semi-model category, without necessarily being a Quillen model category itself.     

On the weak convergence of Wright-Fisher model

We prove that the sequence of stochastic processes obtained from Wright-Fisher models by transforming the time scales and state spaces in the usual way converges weakly to a diffusion process on the time interval [0,∞). Convergence of fixation probabilities and fixation time distributions are obtained as corollaries. These results extend a theorem of Watterson, who proved convergence in distribution to a diffusion at any given single time point for these processes.     

A contribution to the sintering model of viscoelastic liquid drops with solid kernels

A model for filled polymer sintering has been proposed. In this model the sintering process is reduced to coalescence of two viscoelastic drops with solid kernels. The mathematical calculations are based on the Ostwald empirical relation and the previous models for sintering of pure polymers. The results show that in the first approximation for calculations of filled polymer sintering we can use the same methods as for unfilled ones.Institute of Polymer Mechanics, Latvian Academy of Sciences, Riga, LV-1006 Latvia. Published in Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 564–569, July–August, 1997.     

A model-data weak formulation for simultaneous estimation of state and model bias

We introduce a Petrov–Galerkin regularized saddle approximation which incorporates a “model” (partial differential equation) and “data” (M experimental observations) to yield estimates for both state and model bias. We provide an a priori theory that identifies two distinct contributions to the reduction in the error in state as a function of the number of observations, M: the stability constant increases with M; the model-bias best-fit error decreases with M. We present results for a synthetic Helmholtz problem and an actual acoustics system.     

A general stability result for the semilinear viscoelastic wave model under localized effects

Our main goal in the present work is to address an integro-differential model under localized viscoelastic and frictional effects arising in the Boltzmann theory of viscoelasticity. More precisely, we consider a general version in the history context of the pioneer localized viscoelastic problem approached by Cavalcanti and Oquendo (2003) in the null history scenario, and more recently by Cavalcanti et al. (2018) in the history framework. By means of a new observability inequality, we prove a general stability result to the model under a weaker assumption on the localized frictional damping and a slower condition on the decreasing memory kernel (of polynomial type) than the previously mentioned works. To achieve such stability results, we still work in a general setting by removing the assumption on complementary damping mechanisms and show, in some reasonable situations concerning the density coefficient, that the localized viscoelastic effect is enough to ensure the general stability (of polynomial type) to the problem.