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Henry Tan's picture


Henry Tan's picture

Close-form analytical solution for linear elastic-viscoeloastic problems can be easily derived using the correspondence principle.

Nonlinear elastic-viscoelastic problems can be solved similarly?

MichelleLOyen's picture


no, the correspondence principle is not applicable to stress analysis for nonlinearly viscoelastic materials. The most typical solution for nonlinear viscoelastic stress analysis is a multiple-integral representation; Findley, Lai and Onaran go through a few examples in Chapter 12 of "Creep and Relaxation of Nonlinear Viscoelastic Materials" while noting that historically far more effort has gone into material characterization of nonlinear materials than stress analysis.


Henry Tan's picture

Thank you, Michelle,

Your information on multiple-integral viscoelastic constitutive equations are of great help to me.

What's your experience in stress analysis and material characterization of viscoelastic materials?

MichelleLOyen's picture


my research for the last decade has been primarily concerned with time-dependent mechanical behavior, including viscoelasticity and more recently poroelasticity, mainly for mechanical property measurements and constitutive behavior characterization of both linearly and nonlinearly viscoelastic materials including bulk polymers, polymer thin films, and all different sorts of hydrated biological tissues.

My interests in stress analysis and more advanced topics in viscoelasticity are more recent, and started when I became quite interested in indentation techniques and the associated non-homogeneous stress fields compared with simple uniaxial or biaxial tension.

I primarily use analytical viscoelasticity formulations, and mainly emphasize integral equations for their real-world flexibility and do not tend to use finite element approaches as much. I cannot recommend a better resource than

Findley, W. N., Onaran, K. and Lai, W. J. Creep and Relaxation of Nonlinear Viscoelastic Materials: With an Introduction to Linear Viscoelasticity

although I also really do like the Lakes book "Viscoelastic Solids" (CRC 1998) as another good resource.


Henry Tan's picture

I am also searching analytical stress analysis for viscoelastic materials.

Rohit Khanna's picture

hi! Michelle,

Do you think that we can correlate the viscoelasticity  with toughness. ?


Henry Tan's picture

Toughness is the resistance to fracture of a material when stressed. It is defined as the amount of energy per volume that a material can absorb before rupturing.
A viscoelastic material causes energy dissipation during fracture, and therefore, increases the amount of energy before rupture.
So, what you said is right. A viscoelastic material is tougher than the a material with the same properties  but without viscoelasticity.

vicky.nguyen's picture

Assuming that the failure process is rate-independent (that is the intrinsic fracture energy is constant) you can calculate the increase in the energy release rate from bulk viscous dissipation.  For a 3 parameter (standard solid) model, the energy release rate for a fast steady-state growing crack (no inertia) scales with E0/Einfity where E0 and Einfity are the instantaneous and equilibrium Young's moludus.

My paper below includes a bunch of references on this topic. 

T.D. Nguyen and S. Govindjee (2006) Numerical study of geometric constrain and cohesive parameters ini steady-state viscoelastic crack growth, Int. J. Fracture, 141, 255-269. 


Thao (Vicky) Nguyen
Assistant Professor
Mechanical Engineering Department
Johns Hopkins University

MichelleLOyen's picture

I think this is still quite an open question.  Historically it is typical to lump plastic deformation in with toughness such that plasticity is a toughening mechanism.  However, since viscoelastic processes are recoverable--just perhaps on a time-scale long relative to the experiment itself--we chose to separately account for viscoelastic dissipation explicitly when looking at fracture of soft tissues.  In this case, we subtracted off the viscoelastic part of a hysteresis loop in which energy was dissipated by both viscoelastic and fracture modes.  There are relatively few papers on fracture in soft tissues and I think it will take some time before we understand fully whether this is a more useful approach than lumping the two parameters together.  But in some unpublished preliminary results we found that the viscous deformation and fracture resistance did not directly correlate, supporting the idea that these are two very different things.  For details on our method see the paper linked here .


Dr V. K. Gupta
Associate Professor (Mechanical Engg.)
University College of Engg.
Punjabi University, Patiala (Punjab)

Zhigang Suo's picture

Hi everyone, if you are looking for more discussions on viscoelasticity, you may wish to look at this thread on viscoelastic contact initiated by Michelle Oyen.  The discussion went beyond contact, and touched on several fundamental issues.

Henry Tan's picture

Already linked in March last year. I would say that this is one of the best blogs in iMechanica.

More and more people are paying attention to viscoelasticity since many interested materials have this property: solid rocket propellants, explosives, hydrogel polymers, some MEMS chips, biomaterials, polymeric composites, etc.

Henry Tan's picture

links to related blogs:

Viscoelastic Contacts,

Resistance to time-dependent deformation of polymer-based anocomposites,

Constitutive Modelling of Elastomers,

Henry Tan's picture

links to outside world:

Roderic Lakes,

Theory of polymer dynamics,

MichelleLOyen's picture

Please note that the shape of the hysteresis loop shown on the Wikipedia viscoelasticity page is unphysical; if there is dissipation taking place the loop will not return to the point (zero, zero) in that manner! If the loop is for displacement/strain control, for example, the unloading curve will hit the x-axis (y=0) at some non-zero (positive) value of x. This is one of the greatest and most common errors in schematic illustrations of hysteresis effects in viscoelasticity and biomechanics texts.

(The Wikipedia viscoelasticity page needs cleaning up in general; there are better references for basic concepts on this subject than a broken link to the MIT website. However, whether it is worth the effort to try to clean this up, as opposed to posting a non-wiki website with a better introduction on the subject, is perhaps an open question.)

Henry Tan's picture

Thanks for pointing out.

It can only be said to be correct if the origin of the coordinate system in that figure is not (0,0).

Hello Michelle and Henry,

My name is Milliyon. I am new to this website. I have got a good impression on what you guys are discussing. I am also starting a research on the same field and thought would benefit alot from your discussion. I am more on the finite element modeling but also more interested on the non linear stress dependent behavior of visco-elastic materials. Would you guys please let me know if anyone of you use ABAQUS aswell?



When I submitted a review article "Time Dependent Materials" to a Monograph Shock and Vibration, Ed. W and B Pilkey, University of Virginia, SAVIAC, 1995, pp.253-284,  I found that ABACUS included a well organized capability for viscoelasticity with also a well prepared manual and believe that ABACUS being remained as the most  proficient software for the subject by the finite element method. 

  As an self-introduction, please visit: Journal of Material Processing Technology, Articles in vol.140(1903), 1-5 Y.Yamada, Mechanics of Materials, I saw and participated: a reminiscence.

yoshiaki yamada

Henry Tan's picture

vol. 140 (2003), 1-5

Henry Tan's picture


I am reading your paper with great interest (it relates to one of the problem I am solving)

Analytical techniques for indentation of viscoelastic materials. Phil. Mag., 86: 5625-5641, 2006

I have a question: For a linear viscoelastic material, the bulk modulus K is often assumed to be a time-independent constant, but the shear modulus G is time-dependent.

The Poisson's ratio can be derived from (3K-2G)/(6K+2G).

Can we then conclude that the Poisson's ratio is also time-dependent?


Let me share my view on your question.

Bulk modulus,K, is often assumed time-indepenent. The reason behind this lies on the assumption that the material in subject is highly incompressible and has a high bulk modulus value.If the material is incompressible, then it implies that the possions ration is nearly 0.5 always and thus can be taken time-indepenent. The main point here is, the material has to be highly incompressible for the assumption to remain valid.


MichelleLOyen's picture


In the set-up I use in the paper you cite, the two elastic constants are assumed to be the shear modulus and Poisson's ratio; the Poisson's ratio is chosen to be time-independent so there is only one time-dependent function G(t).  Many time-dependent materials are effectively incompressible or only slightly compressible, and I've looked at both the incompressible case and the Poisson's ratio fixed case in that paper.  You can assume that there are two independent functions that are both time-dependent in the set-up of the problem, and some recent work has been done on this type of problem (see for example Cheng et al. 2000 Flat punch indentation of a viscoelastic material, J. Polymer Science part B).  However, for many experimental conditions where only scalar measurements are considered (here the axial force and axial displacement for indentation) there are good reasons related to uniqueness and sensitivity why only one time-dependent function is considered.

This results in there being time-dependent functions for E(t) and K(t), both directly related to G(t) through the constant value of Poisson's ratio (0.5 or not 0.5 but still constant).  If you begin the problem with assumptions of a time-independent K and a time-dependent G(t) you would get time-dependent functions for E(t) and nu(t) that would be slightly more complicated than the E(t) and K(t) functions you get from doing the problem as I have done it but still computed through the standard relationships assuming linear viscoelasticity is really holding.  For more complicated circumstances, the following paper is recommended as quite pedagogical:

Lu H, Zhang X and Knauss WG, Uniaxial, Shear and Poisson Relaxation and their Conversion to Bulk Relaxation: Studies on PMMA.  Polymer Engineering and Science, 37 (1997) 1053. 

As noted above, in an actual experiment in can be difficult to measure two independent time-dependent functions if you are measuring scalar quantities in one dimension only, and so the simplification of "all time-dependence lumped into a single function" is often not a bad one.  It pays also at this stage also to note that all viscoelasticity is empirical! 

Henry Tan's picture

To your last sentence:

Is there a mechanism-based viscoelasticity law that is not empirical?

Temesgen Markos's picture

In viscoelastic materials, the poisson ratio is not constant through time, it can increase or decrease depending on how the bulk and shear relaxation functions. In polymeric solids, for example, it increases.

I am not sure if v = (3K-2G)/(6K + 2G) can be used for viscoelastic materials. I guess for the linear case we can use the correspondence princple and get some integral expression for v interms of K(t) and G(t).

Waiting for Michelle

Henry Tan's picture

Can inverse Laplace transform be applied directly to (3K-2G)/(6K+2G), where K and G are the expression for K ang G in the Laplace transform space, respectively?

kilishzin's picture


Can anyone tell me if there is a 3D source code for Eyring model in ABAQUS to model the non linear viscoelasicity in polymer?

If not how I can use ABAQUS to model the non linear viscoelaicity? is there another model which uses spring and dashpots?



Henry Tan's picture

I am studying the movement of a fast particle, with inertia effect, in a viscoelastic media. Are there comments in this field?

Hi Honglai,

I think the problem of viscoelastic flow around a rigid sphere was simulated by Sugeng and Tanner, 1986,J. Non-Newtonian Fluid Mech., 20, 281-292.  There are probably more recent results.

The best text on that sort of thing is, in my opinion, "Dynamic of Polymeric Fluids: Vol.1: Fluid Mechanics", by Bird, Armstrong, and Hassager published by Wiley-Interscience in 1987.

-- Biswajit 

vicky.nguyen's picture

Hi Henry,

I just stumbled on your blog regarding viscoelasticity. I work on  modeling viscoelasticity, most recently anisotoropic nonlinear viscoelasticity of fibrous tissues.   I'd like to address a few of your points:

1) I like Ferry's  book Viscoelasticity in Polymers.  It's a wonderful mechanics and materials treatment that combines experiments and modeling.  

2) Bulk modulus does indeed change with time for a lot of polymers.  It's often neglected because for most materials, it changes by a factor of 2 or so while the shear modulus can change by orders or magnitude.  Plus it's more difficult to characterize experimentally.  

3) If you're dealing with large deformation or with a material where the creep/relaxation rate is dependent on stress/strain you should consider a fully nonlinear treatment.  These treatments do not assume a separable time-dependence and strain-dependence of the stress response.  These can't be solved analytically, but let me know if you're interested and I can point you to some references.


Thao (Vicky) Nguyen
Assistant Professor
Mechanical Engineering Department
Johns Hopkins University

Hi I am looking for the development of the law tensoriel viscoelastic
Thank you

Can i know more about tensorial law for viscoelastic.., as i have developed material modeling and shift factor estimation.., for viscoelastic materials ..,



Hi all,

I have this confusing thought for a quite long time. 

Is there any relation between the time-independent Young's modulus with
the time-dependent Young's modulus (instantaneous and equilibrium
moduli or loss and storage moduli)?

If yes, what is the equation to relate them?

So many terminologies which make this confusion!

Please help.


For a viscoelastic mterial, the fracture energy is rate-dependent. Is there an analytical approach to predict fracture energy at one rate (or temperature) from that at another rate?



Are we considering glass (silicate glasses) as a viscoelastic material? If yes! If yes then how does it deforms? (Maybe obvious but i am a bit confused)



Hello everyone,

I would like to ask you if there is a way to adjust the Rayleigh damping in ABAQUS Standard to the frequency range under consideration. Is it possible to use multiple steps in a single input file, in which the materials' properties change from step to step?

Tahnk you very much in advance

Lixiang Yang's picture

Hi, I am new in the viscoelastic field. I had a question about nonlinear constitutive relationship.

Can this relationship be included into finite element analysis? Which solver(scheme) in commerical software is used to solve the corresponding time dependent problems?



Lixiang Yang



I am a PhD student. I am trying to figure out what sort of progress has been made in the field of nonlinear viscoelasticity till date experimentally, or by using FEM and inverse analysis. I could not find any state-of-the-art review paper on this subject published recently. If any one could help me with the information, I would greatly appreciate it.

Thank you,


Lixiang Yang's picture

trying to add some new idea to viscoelasticity modeling. 

A viscoelasticity model for polymers: Time, temperature, and hydrostatic pressure dependent Young's modulus and Poisson's ratio across transition temperatures and pressures


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