Computational Viscoelasticity

Viscoelastic relations may be expressed in both integral and differential forms. Integral forms are very general and appropriate for theoretical work. Differential forms are related to rheological models that provide a more direct physical interpretation of viscoelastic behavior. In this chapter we describe the most usual rheological models, deduce their differential equations and, by solving them, we find the corresponding integral representations. These relations will be set in a more computational friendly form in Chap. 3 and extended to three-dimensional situations in Chap. 4 and then used in analytical and computational solutions.

The State Variables approach is an alternative to the history dependent integral representation given in Chap.​ 2. It has a physical basis because of its origin in thermodynamic formulations and shows computational advantages. The creep and relaxation functions are approximated by exponential series. The introduction of state variables leads to n differential equations of first order in place of the differential equation of order n linked to a generalized model. This formulation leads to exponential expressions that make incremental integration easier, allowing the determination of the viscoelastic strains at time t + Δt as a function of the viscoelastic strains and stresses at time t. Then, there is no need to store the whole history of stress or strain. In this chapter, we introduce the basic formulation that is later extended to 3D, aging and nonlinear situations.

In this chapter we extend the viscoelastic constitutive relations to a three dimensional setting. An important subject is the determination of parameters to be used in the solution of real problems. Viscoelastic behavior is determined experimentally, mainly through uniaxial creep tests. To obtain 3D relations, some simplifying assumptions are made. In the case of concrete, it is assumed that the Poisson ratio does not change with time. In the case of polymers, as the creep in shear is more important than volumetric creep, this one is disregarded and the material is considered as elastic in bulk. The viscoelastic equations are here presented in a general form and then simplified for isotropic materials. Finally, a procedure based on the state variables approach is presented for a general anisotropic linear viscoelastic material.

Laplace Transform is a useful tool in solving important problems in different areas of science and engineering. Usually, it is employed to convert differential or integral equations into algebraic equations, simplifying the problem solutions. Particularly, in linear nonageing viscoelasticity, interesting applications have been found for Laplace transform techniques. Many computational solutions are also based on the use of Laplace transforms. As already mentioned, an important task in viscoelasticity consists of determining relations between the different constitutive viscoelasticity functions of a material. In this chapter, we show procedures based on Laplace transforms that allow us to obtain relaxation function given the corresponding creep function, or vice versa. Also, we show equivalence conditions between the integral and differential representations of the constitutive viscoelastic relations. In many practical situations, we know the creep function, which is evaluated in uniaxial tension or compression tests, and we need to determine the viscoelastic constitutive functions for multiaxial states of stress or strain. This problem is also focused in the present chapter. Finally, using the similarity between the mathematical formulations of the linear elastic and linear viscoelastic mechanical problems in the Laplace domain, the Correspondence Principle is stated and applied.

The viscoelastic constitutive relations presented so far were developed under the hypothesis of isothermal conditions. However, most viscoelastic materials, particularly polymers, have temperature dependent constitutive relations. The mechanisms responsible for these thermal effects have micro-structural origin and are, consequently, complex. In this chapter we present a brief description on temperature effects on the linear viscoelasticity behavior of polymers and concrete and a simplified formulation that is adequate for the so called thermo-rheologically simple materials.

Viscoelastic behavior may show physical and/or geometrical nonlinearity. Physical nonlinearity corresponds to situations in which the linear behavior described in Chap.​ 1 (Sect.​ 1.​3.​2) is not observed, even in small strain situations. Geometrical nonlinearity corresponds to situations of large deformations (large displacements and/or large strain). Both effects can appear combined in some problems (e.g. polymers, biomechanics). Alternative nonlinear or quasi-linear single integral representations have been proposed, some of which are described in Sect. 8.2. In Sect.8.3, a nonlinear state variables formulation proposed by Simo is described. The situation involving large displacements associated with small strains that is particularly important in the analyses of materials and structures is addressed in detail in Chap.​ 9.

The finite element method is the most popular numerical procedure for the analysis of solids and structures, including those with time dependent properties. In this chapter, we present an incremental viscoelastic finite element formulation for problems with geometrical nonlinearity characterized by large displacements and rotations with small strains. The formulation is based on a total Lagrangian kinematic description. We begin with a brief presentation on the principle of virtual displacements for geometrically nonlinear problems. Procedures used for the computational implementation of the nonlinear viscoelastic model are also presented. We assume that the reader has a basic knowledge of the finite element method and of nonlinear continuum mechanics.

The Boundary Element Method (BEM) is derived through the discretization of an integral equation (the classical Somigliana identity, first published in 1886). An interesting account of BEM early development may be found in (Cheng and Cheng 2005). This formulation can only be derived for certain classes of problems and hence, is not as widely applicable as the finite element method. However, when applicable, it often results in numerical methods that are easier to use and computationally more efficient. The advantages of the BEM arise from the fact that only the boundary of the domain requires sub-division. In cases where the domain is exterior to the boundary (e.g. the atmosphere surrounding an airplane, the soil surrounding a tunnel, the material surrounding a crack tip) the advantages of the BEM are even greater as the equation governing the infinite domain is reduced to an equation over the (finite) boundary. In this chapter we shortly review two alternative procedures for the solution of problems in linear viscoelasticity: the solution in the Laplace transformed domain and the use of a general inelastic formulation. For the latter, we make reference to the use of the Dual Reciprocity Method (DRM) that allows a pure boundary formulation.

The finite-volume theory is a well-known technique frequently used for analysis of boundary-value problems in fluid mechanics [7]. Its excellent performance has motivated attempts to extend it to solid mechanics problems. Thus, in the past two decades, several authors presented formulations based on this technique. Here, we present one of these finite-volume formulations, known as the Parametric Finite-Volume Formulation. It uses the Finite Volume Direct Averaged Method—FVDAM [1] as a basis and is summarized for the case of linear elastic problems in Cavalcante et al. [2, 3]. An extension of the Parametric Finite-Volume Formulation in order to include linear viscoelastic effect, here presented, can be found in a more detailed form in Escarpini Filho [5].

To help the reader to practice with a professional computer code, we use Abaqus to solve a few problems in viscoelasticity (small and large strains). First we relate Abaqus procedure to the general formulation given in this book and then we provide detailed instructions to run the code.