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On nonlinear viscoelastic deformations: a reappraisal of Fung's quasi-linear viscoelastic model.

De Pascalis R, Abrahams ID, Parnell WJ - Proc. Math. Phys. Eng. Sci. (2014)

Bottom Line: The approach outlined herein is shown to yield improved behaviour and offers a straightforward scheme for solving a wide range of models.In the latter case, a numerical solution to a Volterra integral equation is required to obtain the results.This is achieved by a high-order discretization scheme.

View Article: PubMed Central - PubMed

Affiliation: School of Mathematics , University of Manchester , Oxford Road, Manchester M13 9PL, UK.

ABSTRACT
This paper offers a reappraisal of Fung's model for quasi-linear viscoelasticity. It is shown that a number of negative features exhibited in other works, commonly attributed to the Fung approach, are merely a consequence of the way it has been applied. The approach outlined herein is shown to yield improved behaviour and offers a straightforward scheme for solving a wide range of models. Results from the new model are contrasted with those in the literature for the case of uniaxial elongation of a bar: for an imposed stretch of an incompressible bar and for an imposed load. In the latter case, a numerical solution to a Volterra integral equation is required to obtain the results. This is achieved by a high-order discretization scheme. Finally, the stretch of a compressible viscoelastic bar is determined for two distinct materials: Horgan-Murphy and Gent.

No MeSH data available.


Related in: MedlinePlus

The imposed stretch history is shown in graph (a). The resultant dimensionless stress T/μ is plotted in graph (b), found for the Yeoh model (4.14) (dotted) and for the Mooney–Rivlin material (4.19) (dashed) where the solid curve is the neo-Hookean limit α=0 (or γ=1/2). (c) T/μ is plotted from the predictions of (4.24) (dotted), (4.23) (dashed) and (4.14) (solid), respectively. (d) The dimensionless stress, T/μ, is plotted against stretch, λ, from the predictions of (4.24) (dotted), (4.23) (dashed) and (4.14) (solid).
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RSPA20140058F1: The imposed stretch history is shown in graph (a). The resultant dimensionless stress T/μ is plotted in graph (b), found for the Yeoh model (4.14) (dotted) and for the Mooney–Rivlin material (4.19) (dashed) where the solid curve is the neo-Hookean limit α=0 (or γ=1/2). (c) T/μ is plotted from the predictions of (4.24) (dotted), (4.23) (dashed) and (4.14) (solid), respectively. (d) The dimensionless stress, T/μ, is plotted against stretch, λ, from the predictions of (4.24) (dotted), (4.23) (dashed) and (4.14) (solid).

Mentions: These two viscoelastic models can be compared by considering an ‘experiment’ where the stretch is imposed and the stress is measured. To specify matters, the stress relaxation function is chosen to be the classical one-term Prony series4.21D(t)=μ∞μ+(1−μ∞μ)e−t/τ,where μ, are the infinitesimal shear modulus and the long-time infinitesimal shear modulus, respectively, and τ is the relaxation time. These are set as4.22μ∞μ=0.5andτ=1.0s.A dynamic imposed stretch history (shown in figure 1a) is applied. The time variation is assumed slow so that inertial terms in the balance equations can be neglected. Figure 1b shows the resultant stress predictions T/μ for the Yeoh hyperelastic model (4.14) (dotted curves with α=1,2) and for the Mooney–Rivlin hyperelastic model (4.19) (dashed curves with γ=1/6,−1/3). It is interesting to observe that the two hyperelastic models depart from the solid curve, obtained when the strain energy function is of the neo-Hookean type, i.e. α=0,γ=1/2, in different ways. The Yeoh material is found to harden as the parameter α increases, whereas the effect of decreasing γ from 1/2 leads to a softening of the Mooney–Rivlin material's behaviour.Figure 1.


On nonlinear viscoelastic deformations: a reappraisal of Fung's quasi-linear viscoelastic model.

De Pascalis R, Abrahams ID, Parnell WJ - Proc. Math. Phys. Eng. Sci. (2014)

The imposed stretch history is shown in graph (a). The resultant dimensionless stress T/μ is plotted in graph (b), found for the Yeoh model (4.14) (dotted) and for the Mooney–Rivlin material (4.19) (dashed) where the solid curve is the neo-Hookean limit α=0 (or γ=1/2). (c) T/μ is plotted from the predictions of (4.24) (dotted), (4.23) (dashed) and (4.14) (solid), respectively. (d) The dimensionless stress, T/μ, is plotted against stretch, λ, from the predictions of (4.24) (dotted), (4.23) (dashed) and (4.14) (solid).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4042724&req=5

RSPA20140058F1: The imposed stretch history is shown in graph (a). The resultant dimensionless stress T/μ is plotted in graph (b), found for the Yeoh model (4.14) (dotted) and for the Mooney–Rivlin material (4.19) (dashed) where the solid curve is the neo-Hookean limit α=0 (or γ=1/2). (c) T/μ is plotted from the predictions of (4.24) (dotted), (4.23) (dashed) and (4.14) (solid), respectively. (d) The dimensionless stress, T/μ, is plotted against stretch, λ, from the predictions of (4.24) (dotted), (4.23) (dashed) and (4.14) (solid).
Mentions: These two viscoelastic models can be compared by considering an ‘experiment’ where the stretch is imposed and the stress is measured. To specify matters, the stress relaxation function is chosen to be the classical one-term Prony series4.21D(t)=μ∞μ+(1−μ∞μ)e−t/τ,where μ, are the infinitesimal shear modulus and the long-time infinitesimal shear modulus, respectively, and τ is the relaxation time. These are set as4.22μ∞μ=0.5andτ=1.0s.A dynamic imposed stretch history (shown in figure 1a) is applied. The time variation is assumed slow so that inertial terms in the balance equations can be neglected. Figure 1b shows the resultant stress predictions T/μ for the Yeoh hyperelastic model (4.14) (dotted curves with α=1,2) and for the Mooney–Rivlin hyperelastic model (4.19) (dashed curves with γ=1/6,−1/3). It is interesting to observe that the two hyperelastic models depart from the solid curve, obtained when the strain energy function is of the neo-Hookean type, i.e. α=0,γ=1/2, in different ways. The Yeoh material is found to harden as the parameter α increases, whereas the effect of decreasing γ from 1/2 leads to a softening of the Mooney–Rivlin material's behaviour.Figure 1.

Bottom Line: The approach outlined herein is shown to yield improved behaviour and offers a straightforward scheme for solving a wide range of models.In the latter case, a numerical solution to a Volterra integral equation is required to obtain the results.This is achieved by a high-order discretization scheme.

View Article: PubMed Central - PubMed

Affiliation: School of Mathematics , University of Manchester , Oxford Road, Manchester M13 9PL, UK.

ABSTRACT
This paper offers a reappraisal of Fung's model for quasi-linear viscoelasticity. It is shown that a number of negative features exhibited in other works, commonly attributed to the Fung approach, are merely a consequence of the way it has been applied. The approach outlined herein is shown to yield improved behaviour and offers a straightforward scheme for solving a wide range of models. Results from the new model are contrasted with those in the literature for the case of uniaxial elongation of a bar: for an imposed stretch of an incompressible bar and for an imposed load. In the latter case, a numerical solution to a Volterra integral equation is required to obtain the results. This is achieved by a high-order discretization scheme. Finally, the stretch of a compressible viscoelastic bar is determined for two distinct materials: Horgan-Murphy and Gent.

No MeSH data available.


Related in: MedlinePlus