Problems Of Linear Viscoelasticity Research Articles

Prediction of Viscoelastic Behavior in Asphalt Concrete Using the Fast Multipole Boundary Element Method

This paper describes an application of the fast multipole boundary element method (FMBEM) in simulating the viscoelastic behavior of asphalt concrete (AC). FMBEM has some merits including easy meshing of complicated geometries, accuracy in solving singular fields, high calculation efficiency, and low data storage requirements. These merits lead to many applications in multiple time-step and multi-inclusion problems, such as prediction of AC creep behavior. In this paper, a fast multipole formulation and an algorithm for two-dimensional linear viscoelastic problems are introduced first. To consider the bonding between the asphalt matrix and aggregates, a viscoelastic interface model is used to simulate the interfacial imperfection. A numerical model of asphalt concrete is then constructed based on image processing techniques, and the model is capable of taking into account the effects of the size and content of the air voids. The creep compliance of AC is predicted by the developed FMBEM arithmetic and subsequently compared with the experimental results. It has been shown that the developed method can describe and simulate the correct creep evolutionary trend of AC. DOI: 10.1061/(ASCE)MT.1943- 5533.0000603. © 2013 American Society of Civil Engineers. CE Database subject headings: Asphalts; Concrete; Micromechanics; Viscoelasticity; Imaging techniques; Boundary element method. Author keywords: Asphalt concrete; Micromechanics; Fast multipole boundary element method; Viscoelastic properties; Image processing technique.

An approximate algorithm for solving the problems of linear viscoelasticity

An approximate algorithm for solving the problems of linear viscoelasticity

Solution of a viscoelastic boundary-value problem on the action of a concentrated force in an infinite plane

We formulate a theorem containing the solution of a boundary-value problem of isotropic linear viscoelasticity on the action of a concentrated force in an infinite plane. The two creep functions that are used in the constitutive relations and correspond to the shear and bulk expansion states and assumed to be independent; the general forms of these functions are not specified. Formulas are presented for the stress, strain, and displacement components.

Application of fractional derivative models in linear viscoelastic problems

Appropriate knowledge of viscoelastic properties of polymers and elastomers is of fundamental importance for a correct modelization and analysis of structures where such materials are present, especially when dealing with dynamic and vibration problems. In this paper experimental results of a series of compression and tension tests on specimens of styrene-butadiene rubber and polypropylene plastic are presented; tests consist of creep and relaxation tests, as well as cyclic loading at different frequencies. Experimental data are then used to calibrate some linear viscoelastic models; besides the classical approach based on a combination in series or parallel of standard mechanical elements as springs and dashpots, particular emphasis is given to the application of models whose constitutive equations are based on differential equations of fractional order (Fractional Derivative Model). The two approaches are compared analyzing their capability to reproduce all the experimental data for given materials; also, the main computational issues related with these models are addressed, and the advantage of using a limited number of parameters is demonstrated.

On the uniform decay in Cauchy viscoelastic problems

In this paper we consider a linear Cauchy viscoelastic problem with an external source term. We show that, for compactly supported initial data and for an exponentially decaying relaxation function, the decay of the first energy of solution is polynomial. The finite-speed propagation is used to compensate for the lack of Poincare’s inequality in \({\mathbb{R}^{n}}\).

Two-dimensional version of Sternberg and Al-Khozaie fundamental solution for viscoelastic analysis using the boundary element method

In this work a general and concise two-dimensional fundamental solution is obtained for quasi-static linear viscoelastic problems using the boundary element method. For this purpose, the three-dimensional fundamental displacement, derived by Sternberg and Al-Khozaie from the generalization of Navier equation, is integrated with respect to z-coordinate. A time formulation is constructed from the viscoelastic Reciprocity Principle, defined in terms of the Stieltjes integral and the material functions are acquired by means of Boltzmann's rheological model. The collocation method and a semi-analytical procedure for the singular boundary integral are employed to the numerical analysis of the boundary integral. The Gaussian quadrature, the analytical method and an incremental approach are used to deal with the convolution integral. As the latter has presented the best performance, it is employed in most analyses of the examples. Finally, numerical results of problems, found in the literature, are presented in order to validate the formulation and the two-dimensional fundamental solution.

A numerical analytic method for solving boundary-value problems of linear anisotropic viscoelasticity

The paper proposes an approach to solving boundary-value problems for linear viscoelastic orthotropic bodies based on the method of operator continued fractions and the boundary-element method. A problem-solving algorithm and a procedure to estimate the possible error are outlined. The solution for a viscoelastic orthotropic plate with a rigid circular inclusion under uniaxial tension is obtained as an example and compared with available ones

A fast multipole boundary element method for 2D viscoelastic problems

A fast multipole formulation for 2D linear viscoelastic problems is presented in this paper by incorporating the elastic–viscoelastic correspondence principle. Systems of multipole expansion equations are formed and solved analytically in Laplace transform domain. Three commonly used viscoelastic models are introduced to characterize the time-dependent behavior of the materials. Since the transformed multipole formulations are identical to those for the 2D elastic problems, it is quite easy to implement the 2D viscoelastic fast multipole boundary element method. Besides, all the integrals are evaluated analytically, leading to highly accurate results and fast convergence of the numerical scheme. Several numerical examples, including planar viscoelastic composites with single inclusion or randomly distributed multi-inclusions, as well as the problem of a crack in a pressured viscoelastic plane, are presented. The results are verified by comparison with the developed analytical solutions to illustrate the accuracy and efficiency of the approach.

A decay result to a viscoelastic problem in Rn with an oscillating kernel

In this paper we consider a linear Cauchy viscoelastic problem. We show that, for compactly supported initial data and for not necessarily decreasing (oscillating) relaxation function, the decay of the first energy of solutions is polynomial. The proof relies on some appropriately chosen functionals. The combination of which satisfies a certain integral inequality.

Inversion of the Laplace transform for some special functions

Methods for inversion of the Laplace transform oriented to solving problems of linear viscoelasticity, which describe long slow deformation processes, are suggested. A class of weakly singular kernels providing a more accurate description of experimental data than those commonly used is introduced. Computational schemes of the methods and ways to accelerate their convergence are described.

On the uniform decay in viscoelastic problems in [formula omitted

In this paper we consider a linear Cauchy viscoelastic problem. We show that, for compactly supported initial data and for an exponentially decaying relaxation function, the decay of the first energy of solution is polynomial. The finite-speed propagation is used to compensate for the lack of Poincaré’s inequality in R n .

A two dimensional mixed boundary-value problem in a viscoelastic medium

A fundamental solution for the transient, quasi-static, plane problems of linear viscoelasticity is introduced for a specific material. An integral equation has been found for any problem as a result of dynamic reciprocal identity which is written between this fundamental solution and the problem to be solved. The formulation is valid for the first, second and mixed boundary-value problems. This integral equation has been solved by BEM and algorithm of the BEM solution is explained on a sample, mixed boundary-value problem. The forms of time-displacement curves coincide with literature while time-surface traction curves being quite different in the results. The formulation does not have any singularity. Generalized functions and the integrals of them are used in a different form.

Completely Relaxed Solution of Linear Viscoelastic Problems

Completely Relaxed Solution of Linear Viscoelastic Problems

Existence and Stability Estimate for the Solution of the Ageing Hereditary Linear Viscoelasticity Problem

The paper is concerned with the existence and stability of weak (variational) solutions for the problem of the quasistatic evolution of a viscoelastic material under mixed inhomogenous Dirichlet-Neumann boundary conditions. The main novelty of the paper relies in dealing with continuous-in-time weak solutions and allowing nonconvolution and weak-singular Volterra's relaxation kernels.

Strict and practical bounds through a non-intrusive and goal-oriented error estimation method for linear viscoelasticity problems

In this work, we set up a non-intrusive procedure that yields for strict and high-quality error bounds of quantities of interest in linear viscoelasticity problems solved by means of the finite element method (FEM). The goal-oriented error estimation approach uses the concept of dissipation error and classical duality techniques involving the solution of an adjoint problem. The non-intrusive feature of this approach is achieved by introducing enrichment functions, via a partition of unity, when solving the adjoint problem numerically (handbook techniques), so that the discretization parameters defined for the primal problem can be reused. The resulting local error estimation method is thus highly effective, easy to implement in a finite element code, and it enables to consider discretization error on truly pointwise quantities of interest.

A non-intrusive approach of goal-oriented error estimation for evolution problems solved by the finite element method

In this article, we set up a non-intrusive procedure that yields for strict and highquality error bounds of quantities of interest in linear viscoelasticity problems solved by means of the Finite Element Method. The non-intrusive feature is achieved by introducing, via a partition of unity, enrichment functions in the solution of the adjoint problem (handbook techniques). The resulting goal-oriented error estimation method is thus easy to implement in a FE code and enables to consider trully pointwise quantities of interest.

Saint-Venant problem of three-dimensional linear viscoelasticity in the Hamiltonian system

The traditional Saint-Venant problem of three-dimensional viscoelasticity is discussed under the Hamiltonia system with the use of the Laplace integral transformation, and the original problem is transformed into finding eigenvalues and eigenvectors of the Hamiltonia operator matrix. Since local effect near the boundary is usually neglected, all solutions of Saint-Venant problems can be obtained directly by the combinations of zero eigenvectors. Moreover, the adjoint relationships of the symplectic orthogonality of zero eigenvectors in the Laplace domain are generalized to the time domain. Therefore the problem can be discussed directly in the eigenvector space of the time domain, and the iterative application of Laplace transformation is not needed. Simply by applying the adjoint relationships of the symplectic orthogonality, an effective method for boundary condition is given. Based on this method, some typical examples are discussed, in which the whole character of total creep and relaxation of viscoelasticity is clearly revealed.

Polynomial stability without polynomial decay of the relaxation function

AbstractWe consider a linear viscoelastic problem and prove polynomial asymptotic stability of the steady state. This work improves previous works where it is proved that polynomial decay of solutions to the equilibrium state occurs provided that the relaxation function itself is polynomially decaying to zero. In this paper we will not assume any decay rate of the relaxation function. In case the kernel has some flat zones then we prove polynomial decay of solutions provided that these flat zones are not too big. If the kernel is strictly decreasing then there is no need for this assumption. Copyright © 2008 John Wiley & Sons, Ltd.

Space-time finite element approximation andnumerical solution of hereditary linearviscoelasticity problems

In this paper we suggest a fast numerical approach to treat problems of the hereditary linear viscoelasticity, which results in the system of elliptic partial differential equations in space variables, who's coefficients are Volterra integral operators of the second kind in time. We propose to approximate the relaxation kernels by the product of purely time- and space-dependent terms, which is achieved by their piecewise-polynomial space-interpolation. A priori error estimate was obtained and it was shown, that such approximation does not decrease the convergence order, when an interpolation polynomial is chosen of the same order as the shape functions for the spatial finite element approximation, while the computational effort is significantly reduced.

A non-intrusive approach of goal-oriented error estimation for evolution problems solved by the finite element method

In this article, we set up a non-intrusive procedure that yields for strict and highquality error bounds of quantities of interest in linear viscoelasticity problems solved by means of the Finite Element Method. The non-intrusive feature is achieved by introducing, via a partition of unity, enrichment functions in the solution of the adjoint problem (handbook techniques). The resulting goal-oriented error estimation method is thus easy to implement in a FE code and enables to consider trully pointwise quantities of interest.