• • When distinguishing between elastic, viscous, and forms of viscoelastic behavior, it is helpful to reference the time scale of the measurement relative to the relaxation
    times of the material being observed, known as the Deborah number (De) where:[3] • where is the relaxation time of the material • is time Dynamic modulus Viscoelasticity is studied using dynamic mechanical analysis, applying a small oscillatory
    stress and measuring the resulting strain.Purely elastic materials have stress and strain in phase, so that the response of one caused by the other is immediate.

  • [3] Necessarily, the history experienced by the material is needed to account for time-dependent behavior, and is typically included in models as a history kernel K.[9] Second-order
    fluid[edit] Main article: Second-order fluid The second-order fluid is typically considered the simplest nonlinear viscoelastic model, and typically occurs in a narrow region of materials behavior occurring at high strain amplitudes and Deborah
    number between Newtonian fluids and other more complicated nonlinear viscoelastic fluids.

  • [1] Different types of responses () to a change in strain rate () Depending on the change of strain rate versus stress inside a material, the viscosity can be categorized
    as having a linear, non-linear, or plastic response.

  • A family of curves describing strain versus time response to various applied stress may be represented by a single viscoelastic creep modulus versus time curve if the applied
    stresses are below the material’s critical stress value.

  • Viscoelastic creep data can be presented by plotting the creep modulus (constant applied stress divided by total strain at a particular time) as a function of time.

  • Measurement Shear rheometry[edit] Shear rheometers are based on the idea of putting the material to be measured between two plates, one or both of which move in a shear direction
    to induce stresses and strains in the material.

  • The testing can be done at constant strain rate, stress, or in an oscillatory fashion (a form of dynamic mechanical analysis).

  • The elastic components, as previously mentioned, can be modeled as springs of elastic constant E, given the formula: where σ is the stress, E is the elastic modulus of the
    material, and ε is the strain that occurs under the given stress, similar to Hooke’s law.

  • For the isothermal conditions the model can be written as: where: • is the Cauchy stress tensor as function of time t, • p is the pressure • is the unity tensor • M is the
    memory function showing, usually expressed as a sum of exponential terms for each mode of relaxation: where for each mode of the relaxation, is the relaxation modulus and is the relaxation time; • is the strain damping function that depends
    upon the first and second invariants of Finger tensor .The strain damping function is usually written as: If the value of the strain hardening function is equal to one, then the deformation is small; if it approaches zero, then the deformations
    are large.

  • The model can be represented by the following equation: Under this model, if the material is put under a constant strain, the stresses gradually relax.

  • This is, however, specific to idealised flow; in the case of a cross-slot geometry the extensional flow is not ideal, so the stress, although singular, remains integrable,
    although the stress is infinite in a correspondingly infinitely small region.

  • Polymers remain a solid material even when these parts of their chains are rearranging in order to accommodate the stress, and as this occurs, it creates a back stress in
    the material.

  • When a material is put under a constant stress, the strain has two components.

  • The resulting stress vs. time data can be fitted with a number of equations, called models.

  • The second is a viscous component that grows with time as long as the stress is applied.

  • [5][6] Conversely, for low stress states/longer time periods, the time derivative components are negligible and the dashpot can be effectively removed from the system – an
    “open” circuit.

  • The viscous components can be modeled as dashpots such that the stress–strain rate relationship can be given as,where σ is the stress, η is the viscosity of the material,
    and dε/dt is the time derivative of strain.

  • Effect of temperature on viscoelastic behavior The secondary bonds of a polymer constantly break and reform due to thermal motion.

  • If, on the other hand, it is a viscoelastic solid, it may or may not fail depending on the applied stress versus the material’s ultimate resistance.

  • In purely viscous materials, strain lags stress by a 90 degree phase.Viscoelastic materials exhibit behavior somewhere in the middle of these two types of material, exhibiting
    some lag in strain.

  • For high stress or strain rates/short time periods, the time derivative components of the stress–strain relationship dominate.

  • For this model, the governing constitutive relations are: This model incorporates viscous flow into the standard linear solid model, giving a linearly increasing asymptote
    for strain under fixed loading conditions.

  • [citation needed] A viscoelastic material has the following properties: • hysteresis is seen in the stress–strain curve • stress relaxation occurs: step constant strain causes
    decreasing stress • creep occurs: step constant stress causes increasing strain • its stiffness depends on the strain rate or the stress rate Elastic versus viscoelastic behavior Unlike purely elastic substances, a viscoelastic substance has
    an elastic component and a viscous component.

  • [24] Shear rheometers are typically limited by edge effects where the material may leak out from between the two plates and slipping at the material/plate interface.

  • If the material exhibits a non-linear response to the strain rate, it is categorized as non-Newtonian fluid.

  • For instance, if the material is used to cope with short interaction time purpose, it could present as ‘hard’ material.

  • [15] An alternative form is obtained noting that the elastic modulus is related to the long term modulus by Therefore, This form is convenient when the elastic shear modulus
    is obtained from data independent from the relaxation data, and/or for computer implementation, when it is desired to specify the elastic properties separately from the viscous properties, as in Simulia (2010).

  • The Prony series for the shear relaxation is where is the long term modulus once the material is totally relaxed, are the relaxation times (not to be confused with in the
    diagram); the higher their values, the longer it takes for the stress to relax.

  • Whereas elasticity is usually the result of bond stretching along crystallographic planes in an ordered solid, viscosity is the result of the diffusion of atoms or molecules
    inside an amorphous material.

  • Viscous materials, like water, resist shear flow and strain linearly with time when a stress is applied.

  • [26] This method uses a constant sample length throughout the experiment, and supports the sample in between the rollers via an air cushion to eliminate sample sagging effects.

  • Viscoelastic creep When subjected to a step constant stress, viscoelastic materials experience a time-dependent increase in strain.

  • In other words, it takes less work to stretch a viscoelastic material an equal distance at a higher temperature than it does at a lower temperature.

  • Once the parameters of the creep model are known, produce relaxation pseudo-data with the conjugate relaxation model for the same times of the original data.

  • The Maxwell model for creep or constant-stress conditions postulates that strain will increase linearly with time.

  • In addition, when the stress is independent of this strain rate, the material exhibits plastic deformation.

  • However, polymers for the most part show the strain rate to be decreasing with time.

  • [13][14] Prony series In a one-dimensional relaxation test, the material is subjected to a sudden strain that is kept constant over the duration of the test, and the stress
    is measured over time.

  • First, fit the creep data with a model that has closed form solutions in both compliance and relaxation; for example the Maxwell-Kelvin model (eq.

  • When a stress is applied to a viscoelastic material such as a polymer, parts of the long polymer chain change positions.

  • [25] Because this typically makes use of capillary forces and confines the fluid to a narrow geometry, the technique is often limited to fluids with relatively low viscosity
    like dilute polymer solutions or some molten polymers.

  • Although the Standard Linear Solid Model is more accurate than the Maxwell and Kelvin–Voigt models in predicting material responses, mathematically it returns inaccurate results
    for strain under specific loading conditions.

  • Hysteresis is observed in the stress–strain curve, with the area of the loop being equal to the energy lost during the loading cycle.

  • Plastic deformation results in lost energy, which is uncharacteristic of a purely elastic material’s reaction to a loading cycle.

  • At constant stress (creep), the model is quite realistic as it predicts strain to tend to σ/E as time continues to infinity.

  • At time , a viscoelastic material is loaded with a constant stress that is maintained for a sufficiently long time period.

  • First, an elastic component occurs instantaneously, corresponding to the spring, and relaxes immediately upon release of the stress.

  • Generally speaking, an increase in temperature correlates to a logarithmic decrease in the time required to impart equal strain under a constant stress.

  • When the original stress is taken away, the accumulated back stresses will cause the polymer to return to its original form.

  • [28] It then calculates the sample viscosity using the well known equation where is the stress, is the viscosity and is the strain rate.

  • When the stress is maintained for a shorter time period, the material undergoes an initial strain until a time , after which the strain immediately decreases (discontinuity)
    then gradually decreases at times to a residual strain.

  • Materials lie in this region would exist long-range elasticity driven by entropy.

  • The initial stress is due to the elastic response of the material.

  • While using for long interaction time purposes, it would act as ‘soft’ material.

  • The model is extremely good with modelling creep in materials, but with regards to relaxation the model is much less accurate.

  • The temperature in this region for a given polymer is too low to endow molecular motion.

  • When the back stress is the same magnitude as the applied stress, the material no longer creeps.

  • So, if one has creep data, it is not easy to get the coefficients of the (relaxation) Prony series, which are needed for example in.


Works Cited

[‘1. Meyers and Chawla (1999): “Mechanical Behavior of Materials”, 98-103.
2. ^ Jump up to:a b c McCrum, Buckley, and Bucknell (2003): “Principles of Polymer Engineering,” 117-176.
3. ^ Jump up to:a b c d e f g Macosko, Christopher W. (1994). Rheology
: principles, measurements, and applications. New York: VCH. ISBN 978-1-60119-575-3. OCLC 232602530.
4. ^ Biswas, Abhijit; Manivannan, M.; Srinivasan, Mandyam A. (2015). “Multiscale Layered Biomechanical Model of the Pacinian Corpuscle”. IEEE Transactions
on Haptics. 8 (1): 31–42. doi:10.1109/TOH.2014.2369416. PMID 25398182. S2CID 24658742.
5. ^ Jump up to:a b Van Vliet, Krystyn J. (2006). “3.032 Mechanical Behavior of Materials”
6. ^ Jump up to:a b Cacopardo, Ludovica (Jan 2019). “Engineering
hydrogel viscoelasticity”. Journal of the Mechanical Behavior of Biomedical Materials. 89: 162–167. doi:10.1016/j.jmbbm.2018.09.031. hdl:11568/930491. PMID 30286375. S2CID 52918639 – via Elsevier.
7. ^ Tanner, Roger I. (1988). Engineering Rheologu.
Oxford University Press. p. 27. ISBN 0-19-856197-0.
8. ^ Roylance, David (2001); “Engineering Viscoelasticity”, 14–15
9. ^ Drapaca, C.S.; Sivaloganathan, S.; Tenti, G. (2007-10-01). “Nonlinear Constitutive Laws in Viscoelasticity”. Mathematics
and Mechanics of Solids. 12 (5): 475–501. doi:10.1177/1081286506062450. ISSN 1081-2865. S2CID 121260529.
10. ^ Oldroyd, James (February 1950). “On the Formulation of Rheological Equations of State”. Proceedings of the Royal Society of London. Series
A, Mathematical and Physical Sciences. 200 (1063): 523–541. Bibcode:1950RSPSA.200..523O. doi:10.1098/rspa.1950.0035. S2CID 123239889.
11. ^ Owens, R. G.; Phillips, T. N. (2002). Computational Rheology. Imperial College Press. ISBN 978-1-86094-186-3.
12. ^
Jump up to:a b Poole, Rob (October 2007). “Purely elastic flow asymmetries”. Physical Review Letters. 99 (16): 164503. Bibcode:2007PhRvL..99p4503P. doi:10.1103/PhysRevLett.99.164503. hdl:10400.6/634. PMID 17995258.
13. ^ Wagner, Manfred (1976).
“Analysis of time-dependent non-linear stress-growth data for shear and elongational flow of a low-density branched polyethylene melt”. Rheologica Acta. 15 (2): 136–142. doi:10.1007/BF01517505. S2CID 96165087.
14. ^ Wagner, Manfred (1977). “Prediction
of primary normal stress difference from shear viscosity data usinga single integral constitutive equation”. Rheologica Acta. 16 (1977): 43–50. doi:10.1007/BF01516928. S2CID 98599256.
15. ^ E. J. Barbero. “Time-temperature-age Superposition Principle
for Predicting Long-term Response of Linear Viscoelastic Materials”, chapter 2 in Creep and fatigue in polymer matrix composites. Woodhead, 2011.
16. ^ Jump up to:a b Simulia. Abaqus Analysis User’s Manual, 19.7.1 “Time domain vicoelasticity”,
6.10 edition, 2010
17. ^ Computer Aided Material Preselection by Uniform Standards
18. ^ Jump up to:a b E. J. Barbero. Finite Element Analysis of Composite Materials. CRC Press, Boca Raton, Florida, 2007.
19. ^ S.A. Baeurle, A. Hotta, A.A. Gusev,
Polymer 47, 6243-6253 (2006).
20. ^ Aklonis., J.J. (1981). “Mechanical properties of polymer”. J Chem Educ. 58 (11): 892. Bibcode:1981JChEd..58..892A. doi:10.1021/ed058p892.
21. ^ I. M., Kalogeras (2012). “The nature of the glassy state: structure
and glass transitions”. Journal of Materials Education. 34 (3): 69.
22. ^ I, Emri (2010). Time-dependent behavior of solid polymers.
23. ^ Rosato, et al. (2001): “Plastics Design Handbook”, 63-64.
24. ^ Magnin, A.; Piau, J.M. (1987-01-01). “Shear
rheometry of fluids with a yield stress”. Journal of Non-Newtonian Fluid Mechanics. 23: 91–106. doi:10.1016/0377-0257(87)80012-5. ISSN 0377-0257.
25. ^ Jump up to:a b Dealy, J.M. (1978-01-01). “Extensional Rheometers for molten polymers; a review”.
Journal of Non-Newtonian Fluid Mechanics. 4 (1–2): 9–21. doi:10.1016/0377-0257(78)85003-4. ISSN 0377-0257.
26. ^ Meissner, J.; Hostettler, J. (1994-01-01). “A new elongational rheometer for polymer melts and other highly viscoelastic liquids”. Rheologica
Acta. 33 (1): 1–21. doi:10.1007/BF00453459. ISSN 1435-1528. S2CID 93395453.
27. ^ Bach, Anders; Rasmussen, Henrik Koblitz; Hassager, Ole (March 2003). “Extensional viscosity for polymer melts measured in the filament stretching rheometer”. Journal
of Rheology. 47 (2): 429–441. Bibcode:2003JRheo..47..429B. doi:10.1122/1.1545072. ISSN 0148-6055. S2CID 44889615.
28. ^ Sentmanat, Martin L. (2004-12-01). “Miniature universal testing platform: from extensional melt rheology to solid-state deformation
behavior”. Rheologica Acta. 43 (6): 657–669. doi:10.1007/s00397-004-0405-4. ISSN 1435-1528. S2CID 73671672.
29. ^ Rod Lakes (1998). Viscoelastic solids. CRC Press. ISBN 0-8493-9658-1.
• Silbey and Alberty (2001): Physical Chemistry, 857. John
Wiley & Sons, Inc.
• Alan S. Wineman and K. R. Rajagopal (2000): Mechanical Response of Polymers: An Introduction
• Allen and Thomas (1999): The Structure of Materials, 51.
• Crandal et al. (1999): An Introduction to the Mechanics of Solids
• J. Lemaitre and J. L. Chaboche (1994) Mechanics of solid materials
• Yu. Dimitrienko (2011) Nonlinear continuum mechanics and Large Inelastic Deformations, Springer, 772p
Photo credit:’]