Problems Of Linear Viscoelasticity Research Articles

A Viscoelastic Model for the Long‐Term Deflection of Segmental Prestressed Box Girders

AbstractMost of segmental prestressed concrete box girders exhibit excessive multidecade deflections unforeseeable by past and current design codes. To investigate such a behavior, mainly caused by creep and shrinkage phenomena, an effective finite element (FE) formulation is presented in this article. This formulation is developed by invoking the stationarity of an energetic principle for linear viscoelastic problems and relies on the Bazant creep constitutive law. A case study representative of segmental prestressed concrete box girders susceptible to creep is also analyzed in the article, that is, the Colle Isarco viaduct. Its FE model, based on the aforementioned energetic formulation, was successfully validated through the comparison with monitoring field data. As a result, the proposed 1D FE model can effectively reproduce the past behavior of the viaduct and predict its future behavior with a reasonable run time, which represents a decisive factor for the model implementation in a decision support system.

Solving the problem of linear viscoelasticity for multiply connected anisotropic plates

This paper describes the method for solving the problems of linear viscoelasticity for thin plates under the influence of bending moments and transverse forces. The small parameter method was used to reduce the original problem to a sequence of boundary-value problems solved via complex potentials of the bending theory of multiply connected anisotropic plates. The general representations of complex potentials and boundary conditions for their determination are obtained. The method for determining the stress state of the plate at any time with respect to complex approximation potentials is developed by replacing the powers of the small parameter by the Rabotnov operators. The problem of a plate with elliptical holes is solved. The numerical calculation results in the case of a plate with one or two holes are given. The variation of bending moments in time until stationary condition is reached is studied, and the influence of geometric characteristics of the plate on these variable is described.

Solution of the Problem of Linear Viscoelasticity for Multiply Connected Anisotropic Plates

Solution of the Problem of Linear Viscoelasticity for Multiply Connected Anisotropic Plates

On the Stabilization of a Non-Dissipative Cauchy Viscoelastic Problem

In this paper, we consider a linear Cauchy viscoelastic problem with an external source term. We find the critical weaker conditions on the source \({f(x,t)}\) needed to show that, for any compactly supported initial data and for an exponentially decaying relaxation function, the decay of the first energy of solution is polynomial.

Dynamic linear viscoelasticity problems for piecewise homogeneous bodies

We consider problems on transient wave processes in linearly viscoelastic piecewise homogeneous bodies in the case of small strains, a bounded perturbation propagation domain, and bounded creep of the materials forming the homogeneous components of the bodies. We study problems related to the construction of solutions of such problems by the method of Laplace integral transform with respect to time and the subsequent inversion. We state assertions about the properties of Laplace transforms of the solutions, which simplify the process of determining the original functions. We also consider relations of correspondence between relaxation kernels that belong to different function classes but still affect transient wave processes in a similar way.

Second-order two-scale computational method for ageing linear viscoelastic problem in composite materials with periodic structure

The correspondence principle is an important mathematical technique to compute the non-ageing linear viscoelastic problem as it allows to take advantage of the computational methods originally developed for the elastic case. However, the correspondence principle becomes invalid when the materials exhibit ageing. To deal with this problem, a second-order two-scale (SOTS) computational method in the time domain is presented to predict the ageing linear viscoelastic performance of composite materials with a periodic structure. First, in the time domain, the SOTS formulation for calculating the effective relaxation modulus and displacement approximate solutions of the ageing viscoelastic problem is formally derived. Error estimates of the displacement approximate solutions for SOTS method are then given. Numerical results obtained by the SOTS method are shown and compared with those by the finite element method in a very fine mesh. Both the analytical and numerical results show that the SOTS computational method is feasible and efficient to predict the ageing linear viscoelastic performance of composite materials with a periodic structure.

A Three-Dimensional Boundary Element Approach for Transient Anisotropic Viscoelastic Problems

This paper presents a three-dimensional direct boundary element approach for solving transient problems of linear anisotropic elasticity and viscoelasticity. In order to take advantage of the correspondence principle between viscoelasticity and elasticity the formulation is given in the Laplace domain. Anisotropic viscoelastic fundamental solutions are obtained using the correspondence principle and anisotropic elastic Green’s functions. The standard linear solid model is used to represent the mechanical behavior of viscoelastic material. Solution in time domain is calculated via numerical inversion by modified Durbin’s method. Numerical example is provided to validate the proposed boundary element formulation.

Method of fundamental solutions without fictitious boundary for plane time harmonic linear elastic and viscoelastic wave problems

This study makes the first attempt to apply a recently developed modified method of fundamental solutions (MFS) without fictitious boundary, which is named as the singular boundary method (SBM), to the solution of plane linear elastic and viscoelastic wave problems. Like the standard MFS, the SBM applies the fundamental solutions of the governing equations of interest as the basis functions. Unlike the standard MFS, the SBM, however, does not require the fictitious boundary outside physical domain to avoid the singularity of the fundamental solution and instead directly places the source points on the physical boundary coinciding with collocation points via the concept of origin intensity factors. To demonstrate the effectiveness of the SBM for plane elastic and viscoelastic wave problems, several numerical examples are given in comparison with analytical solutions, and numerical results of the MFS and the finite element method.

Introdução às técnicas do cálculo fracionário para estudar modelos da física matemática

Neste trabalho, recorremos à metodologia da transformada de Laplace a fim de mostrar a sua importância na abordagem de uma classe de equações diferenciais fracionárias. Em particular, apresentamos aplicações desta metodologia ao discutirmos possíveis generalizações de certos problemas físicos no campo da viscoelasticidade linear e osciladores harmônicos, comprovando que o uso do cálculo fracionário em modelagem e resolução de problemas usualmente abordados pelo cálculo de ordem inteira oferece vantagens promissoras para nos fornecer formulações mais consistentes com os dados experimentais.

Wave Propagation in Anisotropic Viscoelasticity

The theory of complete Bernstein functions is extended to matrix-valued functions and applied to analyze Green’s function of an anisotropic multi-dimensional linear viscoelastic problem. Green’s function is given by the superposition of plane waves. Each plane wave is expressed in terms of matrix-valued attenuation and dispersion functions given in terms of a matrix-valued positive semi-definite Radon measure. More explicit formulae are obtained for 3D isotropic viscoelastic Green’s functions. As an example of an anisotropic medium the transversely isotropic medium with a constant symmetry axis is considered.

Variational formulations for the linear viscoelastic problem in the time domain

Abstract Under the assumption of small displacements and strains, we formulate new variational principles for the linear viscoelastic hereditary problem, extending the well-known Hu-Washizu, Hellinger-Reissner, Total Potential Energy, and Complementary Energy principles related to the purely elastic problem. In addition, a new global minimum formulation is derived, giving an energetic interpretation. The new formulations are based on a convolutive bilinear form of the Stieltjes type and on the division of the time domain into two equal parts, with the resulting decomposition of the variables and of the equations governing the problem. In particular, the global minimum principle is achieved by virtue of the positive definiteness of a part of the split constitutive law operator and by means of a partial Legendre transform, and is then used to provide bounds of the overall mechanical properties of viscoelastic composite materials.

Integral Representation of Constitutive Equations in Linear Theory of Viscoelasticity

The presented paper delas with constitutive equations of the linear theory of viscoelasticity. The integral representations for homogeneous isotropic medium located in isothermal conditions are introduced. Relations are derived for the relaxation and creep processes, as well as relations between these functions in the Laplace images. In the solution we assume quasi-static external force loading under which the analyzed viscoelastic body in each time t shows infinitesimal deformations. General isotropic tensor of fourth degree Gijkl(t) we use to derive the relationship between deviator stress and deformation, as well as the relations between stresses and distortions that produce volumetric changes. It is shown that for the linear viscoelastic problem can be used Stieltjes convolution and its algebraic properties. Derived relations and procedures followed to the results of differential-operator's representation of constitutive equations of linear viscoelasticity theory that have been published in the previous work of Authors.

Comparative Analysis of 3D Viscoelastic and Poroelastic Dynamic Boundary-Element Modelling

3D dynamic boundary-value problems of linear viscoelasticity and poroelasticity are considered. Laplace integral transform and its numerical inversion are used. Classical viscoelastic models, such as Maxwell model, KelvinVoigt model, standard linear solid model, and model with weakly singular kernel (Abel type) are considered. Boundary integral equations (BIE) method is developed to solve three-dimensional boundary-value problems. A numerical modelling of wave propagation is done by means of boundary element approach.

The Statistical Two-Scale Method for Predicting Viscoelastic Properties of Composites with Consistent Random Distribution of Particles

This study presents a statistical two-scale method to predict the viscoelastic properties of composite materials with consistent random distribution of particles. The explicit formulation for predicting the effective viscoelastic relaxation modulus is given. At first, the Laplace transformation is used to the linear viscoelastic problem, the effective generalized relaxation modulus in Laplace domain for composites is derived. Then, the effective relaxation modulus in time domain is obtained by the least-square and inverse Laplace transformation. At the end of this paper, some numerical examples are given to validate that the presented method is feasible and effective.

A Numerical–Analytical Method for Solving Problems of Linear Viscoelasticity

A numerical method for solving problems of the linear viascoelasticity of an anisotropic body is proposed. The method does not require deriving explicit analytical expressions for the creep and relaxation kernels. The approximate solution of the integral equations is based on using experimental data preliminarily smoothed with filling in data gaps. The results of numerical analysis of the viscoelastic stress state of an orthotropic plate with an elliptic hole are presented

Contact Effect Evaluation using Stress Distribution in Viscoelastic Material Under Generalized Loading

Contact of a hard tool with soft tissues is very common in medical researches and clinical applications. To visualize the stress, in this work, we develop a set of analytical equations to compute the 3D stress distribution of a viscoelastic material under generalized force. Based on the facts that many researchers have demonstrated that soft tissues, such as liver, kidney, and so on, can be approximated as linear viscoelastic materials, and that in many cases only small deformation exists, we derive our solution from the classic elastic contact problem and apply the correspondence principle to address the linear viscoelastic problem. Such a method has the advantages of analytic form of solutions and thus avoids using the commonly used finite element method (FEM). In our model, the loading owns an arbitrary profile in space and time and comprises of both tangential and vertical components. To verify its efficiency as a replacement of FEM, we compare the results from our method and from FEM commercial software, with positive results obtained. A number of clinical cases are simulated, such as evaluating the effects of blade sharpness and slicing angle on the initial contact of needle insertion or punching. The results provide quantifiable insights to differentiate the difficulties of fracture initiation under various slicing angles or loading magnitudes.

A linear viscoelasticity problem with a singular memory kernel: an existence and uniqueness result

The existence and uniqueness of solution to a one-dimensional hyperbolic problem arising in viscoelasticity is here considered. Specifically, the case of a singular kernel is analyzed. This choice is suggested by applications according to the literature to model a wider variety of materials.

Model of growth and coalescence of two collinear cracks in a viscoelastic body

For the plane problem of linear viscoelasticity, a model of propagation and coalescence of two collinear cracks is proposed. We have constructed the model within the framework of the theory of propagation of cracks in a viscoelastic environment. As an illustration of using the model, we have obtained time dependences of the coordinates of the ends of the cracks for a wide range of parameters of the model, among which are geometric parameters of the problem, intensity of internal load, and crack-resistance parameters.

Influence of auxeticity of reinforcements on the overall properties of viscoelastic composite materials

This work aims to analyze the damping response of viscoelastic composite reinforced by elastic auxetic heterogeneities by means of micromechanical modeling. The linear viscoelastic problem can be transformed into the associated elastic one via the Carson-Laplace transform (C-LT). Loss factors are taken into account by the introduction of the frequency-dependent complex stiffness tensors of the viscoelastic phases. The micromechanical formalism, based on the kinematic integral equation, leads to the computation of effective storage modulus and its associated loss factor in the quasi-static domain. The possibility to enhance viscoelastic (VE) properties of a polymeric material such as PVB is examined through several mixing configurations. Thus, the use of elastic auxetic heterogeneities is analyzed in comparison with classical elastic and viscoelastic reinforcements. The model predictions for VE phases, confirm the possibility to improve the global material stiffness. Also, it is shown in the particular case of elastic and spherical heterogeneities, by a proper choice of phases’ stiffness ratio Q, that auxetic reinforcements represent a good compromise to have simultaneously enhanced stiffness and loss factor response in composite materials.

Special quadrature rules for Laplace transform inversion

Quadrature rules for Laplace transform inversion are studied that are adapted to the inversion of transforms corresponding to slowly varying long processes characteristic of linear viscoelasticity problems. The convergence of special quadrature rules for Laplace transform inversion is proved.