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Numerical investigation of the influence of fiber geometry on filtration performance with a coupled LB-DE method J. Mech December Finally, the stretch of a compressible viscoelastic bar is determined for two distinct materials: Horgan—Murphy and Gent. The study of nonlinear viscoelastic deformations of solid materials has a very long history, with a consequent proliferation of a diverse and extensive array of constitutive models.

## Exponential stability in linear viscoelasticity with almost flat memory kernels

For a comprehensive overview of the topic, the reader is directed to the recent paper by Wineman [ 1 ] and related references therein. The subject is still active, with models continuing to be developed across the field, from highly mathematical approaches where implementation is not a concern to very applied studies where ease of application is essential. Each approach has its advantages: the models that more accurately describe the microphysics tend to prove difficult to employ in practical engineering or biomedical situations, whereas simpler approaches often miss crucial details.

If the viscous model assumes a fading memory effect of the strain history, then this usually gives rise mathematically to Volterra-type integral equations, which in the linear case can generally be solved either analytically or numerically. However, in the nonlinear case of interest to describe finite deformations , these problems are difficult to solve by any methods, even finite-element approaches. It is therefore crucial to develop constitutive models that are simple enough to be amenable to straightforward and rapid solution methods, yet include enough detail to capture the important physics underlying these relevant materials.

His constitutive assumption, often called quasi-linear viscoelasticity QLV or Fung's model of viscoelasticity, assumes that the viscous relaxation rate is independent of the instantaneous local strain. Of course, as with any constitutive model for a complex nonlinear material, QLV has limitations. However, it can be expected to be appropriate for materials whose relaxation-rate coefficients are weakly dependent on deformation, or where the deformation comprises small perturbations about a large initial deformation.

Various experimental results have appeared in the literature that appear to confirm that QLV is, in practice, a reasonable model for a range of materials; see for example [ 4 — 6 ].

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The apparent deficiency of QLV will be discussed below, but the purpose of this paper is to reappraise Fung's approach, suggesting how it can be reformulated against existing interpretations in the literature. It will be seen that the form offered herein has similarities to other viscoelastic formulations, including that by Simo [ 7 ], and most importantly exhibits behaviour that makes it both relatively easy to use and physically useful.

Analysis will commence with the general tensorial form of QLV proposed by Fung [ 2 ]. This tensorial integral identity is the natural generalization of the simple one-dimensional relationship proposed by Fung [ 2 ], which preserves objectivity.

As mentioned, thanks to the relative simplicity of Fung's approach over more general nonlinear viscoelastic models, it has proved extremely popular. This is especially true in the biomechanics field where it has been employed to predict the large deformations of soft tissues. However, despite the widespread use of QLV, published studies especially in the incompressible limit appear to interpret 1.

## Analysis of models for viscoelastic wave propagation

In the paragraphs below, a number of common approaches are discussed, where the interpretations of the model may be questioned. The main aim of this paper is then to re-derive QLV, starting from basic principles and also to ensure consistency in the limit of infinitesimal deformations, thus recovering the Boltzmann theory of linear viscoelasticity. The simplest interpretation of Fung's relation is to assume a purely one-dimensional homogeneous deformation. This is attractive for its ease of use and has been employed extensively in biomedical applications [ 8 — 10 ], for example to model the shearing deformation of brain tissue [ 6 ].

However, in practical applications, especially for incompressible or near incompressible materials purely one-dimensional deformations, such as pure uniaxial extension, will not be realizable. Thus, a tensorial form of QLV must be employed. As mentioned, this relation must, at the minimum, satisfy objectivity, and hence the form chosen in 1.

Nevertheless, a number of authors have erroneously expressed Fung's relation not for the second Piola—Kirchhoff stress, which guarantees objectivity, but in terms of other stress measures. For example, see Fung's original discussion on the subject in [ 2 , page ], in which the QLV relation is expressed in terms of the Kirchhoff stress! In the latest version [6. Further technical aspects of objectivity are discussed in the paper by Liu [ 11 ].

Another approach to QLV sometimes employed in the literature e. Clearly, in this case the strain energy function must be dissipative and depend on the history of the strain, and so these authors express it as a fading-memory integral with integrand given as a hyperelastic strain energy function.

This approach may be somewhat hard to justify and does not allow the user to consider the problem in terms of an auxiliary instantaneous measure of strain, i.

## Boundary Value Problems in Linear Viscoelasticity

However, this point is often not recognized, especially for incompressible materials, as can be seen, for example, by inspection of the integrands setting the stretch equal to unity in the articles by [ 15 ] see eqns 8 and 12 therein , [ 16 ] eqns 9 and 13 and [ 17 ] eqn 9. Note that this requirement is satisfied in the one-dimensional models in [ 8 , 9 ] but not in [ 10 ]. However, for incompressible materials, an arbitrary pressure term Lagrange multiplier can be adjusted so that the stress field is indeed causal, but then it must take a specific form at later times.

There are other variants of QLV employed in the literature that offer slightly modified forms of the governing equation to that suggested by Fung. The reader is referred, for example, to articles [ 18 — 20 ]. In this paper, the method employed follows closely that proposed by Fung but does not exhibit any of the limitations just discussed. The paper is organized as follows. It is helpful to commence analysis by recapping the theory of linear viscoelasticity. Under the assumption of isotropy, infinitesimal elastic deformations can be described by the constitutive law. To incorporate viscoelastic behaviour, the most natural extension of 2.

Integrating 2. Note that this expression incorporates any jump discontinuity when the motion starts. Moreover, the first term in 2. Now, in the elastic case 2. Thus, in the limit of incompressibility, equations 2. When the strain is not infinitesimal, linear theory becomes inappropriate to describe deformations; hence, a nonlinear constitutive law has to be considered. As discussed in the Introduction, Fung's hypothesis 1.

In his renowned work [ 2 ], Fung introduced this quasi-linear constitutive model in order to capture the nonlinear stress—strain relationship of living tissues; however, it also has applicability to elastomeric materials. Before deriving the quasi-linear theory, it is useful to introduce some standard definitions and notations.

The deformation gradient tensor F is defined by. Now, Fung [ 2 ] makes the assumption that the QLV stress depends linearly on the superposed time history of a related nonlinear elastic response a nonlinear instantaneous measure of strain. This allows, for example, for incorporation of a finite hyperelastic theory in the analysis. In index notation, Fung's theory 1. Fung refers to G ijkl as a reduced relaxation function tensor. Moreover, if the material is isotropic then G , a tensor of rank four, has just two independent components, being therefore consistent with linear theory.

Following the analysis of the previous section, for isotropic materials it is convenient to split the equivalent instantaneous Cauchy stress into two parts, one which accounts for microscopic isochoric deformations of the material and the other that measures purely compressive deformations. It is assumed that this decomposition can be achieved by taking the deviatoric and hydrostatic components of the equivalent elastic Cauchy stress:.

The split of equation 3.

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However, it cannot be generalized to viscoelasticity, as in 2. Instead, the second Piola—Kirchhoff stress tensor associated with 3. It must be emphasized that the subscripts D and H do not refer to the deviatoric and hydrostatic parts of the second Piola—Kirchhoff stress, but correspond to the second Piola—Kirchhoff stress of the deviatoric and hydrostatic Cauchy stress components, respectively. Assuming a superposition principle as for the linear case, it is now possible to introduce an objective viscoelastic law, relating the second Piola—Kirchhoff stress to the past history of the nonlinear rate of strain measure.

This is taken as. The latter relaxation functions relate to the inherent viscous processes involved with instantaneous shear and compressional volumetric deformations, respectively. Clearly, if the material was anisotropic, then a more complex tensorial relaxation function would be required.

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Let us then specialize equations 3. The general form of elastic Cauchy stress may be written e. In terms of the strain energy function, they are given by. From this, the second Piola—Kirchhoff counterparts, 3. The viscoelastic stress is obtained from 3. Note that as required objectivity is now preserved for the viscoelastic model 3. In fact, in general both the compressive and shear components of the stress history contribute to the deviatoric and hydrostatic parts of T. However, the main point to note is that when there is no deformation, i. Similarly, T e H vanishes as the strain energy function has always to satisfy the additional relation e.

The final consideration of this section is the constraint of incompressibility for all possible deformations, i.