A mathematical boundary integral equation analysis of standard viscoelastic solid polymers

Structural analysis of viscoelastic solid polymers is one of the most important subjects in engineering structures. Several attempts have been so far made for the integral equation approach to viscoelastic problems. From the basic assumptions of viscoelastic constitutive equations and weighted residual techniques, a simple but effective boundary element formulation (BEF) is implemented for the standard linear solid (SLS) viscoelastic models. The SLS model provides an approximate representation of the observed behavior of a real polymer in its viscoelastic range. This formulation needs only Kelvin’s fundamental solution of isotropic elastostatics with material constants prescribed as explicit functions of time. This approach avoids the use of relaxation functions and mathematical transformations, and it is able to solve the quasistatic viscoelastic problems with any load time-dependence and boundary conditions. As an application, a numerical example is provided to validate the proposed formulation. The problem of the pressurization of thick-walled cylindrical viscoelastic tanks made of PMMA polymer is completely analyzed by this approach.     

Alternative Kelvin viscoelastic procedure for finite elements

In this work, an alternative Kelvin viscoelastic formulation for the finite element method (FEM) is described. This formulation performs spatial approximations before considering time integration and makes use of differential viscoelastic relations. A matrix time differential equation arises from the proposed formulation. It is solved numerically by a time marching procedure. It is shown that, after a small simplification, this methodology can be employed together with existent dynamic FEM packages. The methodology is extended to dynamic analysis leading to a rheological explanation for the first order modal decomposition stiffness proportional damping matrix. Plates and shells applications are shown in order to demonstrate the proposed formulation accuracy and stability.     

粘弹性方程一种新的分裂正定混合元法

本文用分裂正定混合有限元方法研究二阶粘弹性方程. 首先构造一种新的分裂正定混合变分形式和基于这种分裂正定混合变分形式关于时间的半离散格式, 然后绕开关于空间变量的半离散化格式, 直接从时间半离散出发构造出全离散化的分裂正定混合有限元格式, 并给出这种分裂正定混合有限元解的误差估计. 这种研究思路使得理论论证变得更简单,这是处理二阶粘弹性方程的一种新的尝试.     

粘弹性方程全离散化有限体积元格式及数值模拟

本文利用有限体积元方法研究二维粘弹性方程, 给出一种时间二阶精度的全离散化有限体积元格式, 并给出这种全离散化有限体积元解的误差估计, 最后用数值例子验证数值结果与理论结果是相吻合的. 通过与有限元方法和有限差分方法相比较, 进一步说明了全离散化有限体积元格式是求解二维粘弹性方程数值解的最有效方法之一.     

Modelling of thin viscoelastic shell structures under Reissner–Mindlin kinematic assumption

In this paper, we study thin viscoelastic shell structures using a constitutive equation in hereditary integral form. An alternative mathematical formulations for several viscoelastic shell structures under the Reissner–Mindlin kinematical assumptions are obtained. The resulting equations are written as a Volterra equation of the second kind to allow further mathematical analysis. A locking-free finite element formulation, with selective reduced integration is used to approximate the equation. To perform numerical experiments we consider several situations suffering from locking in both cases dynamic and quasi-static. We show the good behavior of the model compared with other models from the literature.     

Degree of cure-dependent modelling for polymer curing processes at small strains

In this paper, a thermodynamically consistent small strain constitutive model is formulated that is directly based on the degree of cure, a key parameter in the curing (reaction) kinetics. The new formulation is also in line with the earlier proposed hypoelastic approach, cf. Hossain et al., 2010. The curing process of polymers is a complex phenomenon involving a series of chemical reactions which transform a viscoelastic fluid into a viscoelastic solid during which the temperature, the chemistry and the mechanics are coupled. Some representative numerical examples conclude the paper and show the capability of the newly proposed constitutive formulation to capture major phenomena observed during the curing processes of polymers. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A unified three-dimensional method for vibration analysis of the frequency-dependent sandwich shallow shells with general boundary conditions

In this paper, we present an accurate three-dimensional formulation for the vibrations of the laminated and sandwich shallow shells. The sandwich structure is characterized by a thick viscoelastic core and two thin composite faces. Frequency dependent viscoelastic models are introduced in the sandwiches. Without any change in solution procedure, the formulation makes it quite easy to change the boundary conditions. The solution can be obtained by means of Rayleigh–Ritz process combined with the three-dimensional modified Fourier series which are actually assumed displacement functions. These functions, without need to meet the boundary conditions in advance, take the form of the three-dimensional Fourier series with several closed-form auxiliary functions which are supplemented to deal with the discontinuities at the boundaries in terms of displacements and its derivatives. Besides, only three assumed displacement variables are employed in the formulation which effectively reduces the computation cost. The reliability and accuracy of the method are demonstrated by numerical comparisons and examples with the constant viscoelastic models as well as the frequency dependent ones. Modal analysis and parametric studies are conducted to examine the influences of the boundary condition, dimension, lamination scheme, temperature and frequency dependence of the materials.     

Simulation of Deep Drawing of Carbon Fibre Prepregs

The simulation of prepregs must regard highly anisotropic, viscoelastic and thermal-chemical properties. To this end a constitutive model is split into an anisotropic elastic part, which represents the fibre fraction and an isotropic, viscoelastic part, representing the matrix. The second part also contains curing, causing a dependency on time and temperature. During real deep-drawing processes large deformations up to 50 % occur, which is considered in a formulation at large strains. This model contains an anisotropic elastic part based on a Neo-Hooke law enhanced by an anisotropic part. A viscoelastic part is added using Hencky-strains and the work-conjugate Hill-stress to transfer a model for small strains into large strains. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational treatment of free convection effects on perfectly conducting viscoelastic fluid

We introduced a magnetohydrodynamic model of boundary-layer equations for a perfectly conducting viscoelastic fluid. This model is applied to study the effects of free convection currents with one relaxation time on the flow of a perfectly conducting viscoelastic fluid through a porous medium, which is bounded by a vertical plane surface. The state space approach is adopted for the solution of one-dimensional problems. The resulting formulation together with the Laplace transform technique is applied to a thermal shock problem and a problem for the flow between two parallel fixed plates, both without heat sources. Also a problem for the semi-infinite space in the presence of heat sources is considered. A discussion of the effects of cooling and heating on a perfectly conducting viscoelastic fluid is given. Numerical results are illustrated graphically for each problem considered.     

A new mixed finite element method for the Stokes problem

We propose a new mixed formulation of the Stokes problem where the extra stress tensor is considered. Based on such a formulation, a mixed finite element is constructed and analyzed. This new finite element has properties analogous to the finite volume methods, namely, the local conservation of the momentum and the mass. Optimal error estimates are derived. For the numerical implementation of this finite element, a hybrid form is presented. This work is a first step towards the treatment of viscoelastic fluid flows by mixed finite element methods.     

A reduced FVE formulation based on POD method and error analysis for two-dimensional viscoelastic problem

Proper orthogonal decomposition (POD) method has been successfully used in the reduced-order modeling of complex systems. In this paper, we extend the applications of POD method, i.e., combine a classical finite volume element (FVE) method with POD method to establish a reduced FVE formulation with lower dimensions and sufficiently high accuracy for two-dimensional viscoelastic problem with real practical applied background, and analyze the errors between the reduced POD FVE solution and the classical FVE solution so as to provide scientific theoretic basis for service applications. Some numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is also shown that the reduced FVE formulation based on POD method is feasible and efficient for solving two-dimensional viscoelastic problem.     

Parametric oscillations of a viscoelastic cylindrical shell

The formulation of the problem of parametric oscillations of viscoelastic shells is given. It was shown that this problem is reduced to the study of the equations, investigated in [4, 5].Moscow Institute of Geodesy, Aerophotography, and Cartography Engineers. Translated from Mekhanika Polimerov, No. 3, pp. 479–483, May–June, 1974.     

Galerkin-based multi-scale time integration for nonlinear structural dynamics

This paper deals with a GALERKIN-based multi-scale time integration of a viscoelastic rope model. Using HAMILTON's dynamical formulation, NEWTON's equation of motion as a second-order partial differential equation is transformed into two coupled first order partial differential equations in time. The considered finite viscoelastic deformations are described by means of a deformation-like internal variable determined by a first order ordinary differential equation in time. The corresponding multi-scale time-integration is based on a PETROV-GALERKIN approximation of all time evolution equations, leading to a new family of time stepping schemes with different accuracy orders in the state variables. The resulting nonlinear algebraic time evolution equations are solved by a multi-level NEWTON-RAPHSON method. Realizing this transient numerical simulation, we also demonstrates a parallelized solution of the viscous evolution equation in CUDA©. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A dynamic unilateral contact problem with adhesion and friction in viscoelasticity

The aim of this paper is to study an interaction law coupling recoverable adhesion, friction and unilateral contact between two viscoelastic bodies of Kelvin–Voigt type. A dynamic contact problem with adhesion and nonlocal friction is considered and its variational formulation is written as the coupling between an implicit variational inequality and a parabolic variational inequality describing the evolution of the intensity of adhesion. The existence and approximation of variational solutions are analysed, based on a penalty method, some abstract results and compactness properties. Finally, some numerical examples are presented.     

The problem of a viscoelastic cylinder rolling on a rigid half-space

The problem of a viscoelastic cylinder rolling on a rigid base, propelled by a line force acting at its centre, is solved in the noninertial approximation. The method used is based on a decomposition of hereditary integrals developed by the authors in previous work, and on the viscoelastic Kolosov-Muskhelishvili equations which are used to generate a Hilbert problem. In this formulation, the problem reduces to a nonsingular integral equation in space and time, which simplifies under steady-state conditions and for exponential decay materials, to algebraic form. There are also two subsidiary conditions.In the case of a standard linear model, explicit analytic results and numerical examples are given for the pressure function, for surface displacements, and also for hysteretic friction.     

Study on vibration and stability of an axially translating viscoelastic Timoshenko beam: Non-transforming spectral element analysis

Engineering systems, such as rolled steel beams, chain and belt drives and high-speed paper, can be modeled as axially translating beams. This article scrutinizes vibration and stability of an axially translating viscoelastic Timoshenko beam constrained by simple supports and subjected to axial pretension. The viscoelastic form of general rheological model is adopted to constitute the material of the beam. The partial differential equations governing transverse motion of the beam are derived from the extended form of Hamilton's principle. The non-transforming spectral element method (NTSEM) is applied to transform the governing equations into a set of ordinary differential equations. The formulation is similar to conventional FFT-based spectral element model except that Daubechies wavelet basis functions are used for temporal discretization. Influences of translating velocities, axial tensile force, viscoelastic parameter, shear deformation, beam model and boundary condition types are investigated on the underlying dynamic response and stability via the NTSEM and demonstrated via numerical simulations.     

Analysis and experiments on the spreading dynamics of a viscoelastic drop

We study the spreading dynamics of a sessile viscoelastic drop on a horizontal surface, where a simplified Phan–Thien–Tanner (sPTT) model is considered to represent the rheology of viscoelastic drop. We have adopted a macroscopic approach to obtain the temporal evolution of the spreading drop, while to establish the efficacy of the theoretical model, we have validated the results obtained from the mathematical formulation with the experimental results for both the Newtonian (Si-oil) and viscoelastic (PDMS and aqueous solution of CMC and glycerin) drops. Following the framework of Seaver–Berg approximation, the spherical shape of the drop is assumed as a cylindrical disk here. We observe from this study that an increment in the elasticity of the fluid enhances the velocity gradient and increases the viscous dissipation in the drop volume, leading to a reduction in the spreading rate.     

A geometrically exact viscoplastic membrane-shell with viscoelastic transverse shear resistance avoiding degeneracy in the thin-shell limit. Part I: The viscoelastic membrane-plate

We reduce a viscoelastic finite-strain continuum model to a two-dimensional membrane-plate. The reduction is based on assumed kinematics, analytical integration through the thickness and physically motivated simplifications. The resulting formulation is observer-invariant and accounts for thickness stretch and finite rotations.The membrane energy is a quadratic, uniformly Legendre-Hadamard elliptic, first order energy in contrast to classical membrane models and the corresponding system of balance equations remains of second order. An evolution equation for some independent rotation is appended (already in the bulk-model) introducing viscoelastic transverse shear resistance. It can be shown that this reduced membrane formulation is locally well-posed. Use is made of a dimensionally reduced version of an extended Korns first inequality.     

On one mathematical model of creep in superalloys

In [Sv1] a new micromechanical approach to the prediction of creep flow in composites with perfect matrix/particle interfaces, based on the nonlinear Maxwell viscoelastic model, taking into account a finite number of discrete slip systems in the matrix, has been suggested; high-temperature creep in such composites is conditioned by the dynamic recovery of the dislocation structure due to slip/climb motion of dislocations along the matrix/particle interfaces. In this article the proper formulation of the system of PDE's generated by this model is presented, some existence results are obtained and the convergence of Rothe sequences, applied in the specialized software CDS, is studied.     

Fundamental solution of generalized thermo-viscoelasticity using the finite element method

The present paper is aimed at an investigation of the temperature, displacement, and stress in a viscoelastic half space of Kelven–Voigt type. The formulation is applied according to three theories of generalized thermoelasticity: Lord–Shulman with one relaxation time, Green–Lindsay with two relaxation times, as well as the coupled theory. The nondimensional governing equations are solved by the finite element method. Numerical results for the temperature distribution, displacement, and thermal stress are represented graphically. Comparisons are made with the results predicted by the CD, L-S, and G-L theories in the presence and absence of the viscoelastic relaxation time.