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Viscoelastic Contacts

MichelleLOyen's picture

I was a little bit surprised in the introduction of this new forum to see mention of elastic and plastic contacts but no specific mention of viscoelastic contacts.

In the era of commercially-available instruments for indentation testing, the examination of viscoelastic contact mechanics, both in the context of polymers and biological tissues, seems to have taken on new life. To a first approximation, for indentation testing in the time domain, the fundamental mechanics has not much advanced beyond a few classic papers of the 1960s: Lee and Radok, J. Appl. Mech. 27 (1960) 438 and Ting TCT, J. Appl. Mech. 88 (1966) 845. However, the implementation of techniques for analysis of experimental data has progressed substantially. With spherical indenters the use of linearly viscoelastic models for characterization of a material creep or relaxation function is straightforward. Recent experimental studies have confirmed this, while more lingering questions remain for sharp contacts including Berkovich pyramidal indenters (most commonly shipped with commercial indenters). Sharp contacts seem to give rise to nonlinearly viscoelastic responses. Other topics of recent interest include frequency-domain measurements and examination of oscillating contacts and adhesion. (Although not mentioned in the listing of KLJ's most-loved topics in contact mechanics, viscoelastic contact has been the subject of several recent KLJ publications!)  Although research in viscoelastic contact mechanics has been strong in recent years, perhaps a challenge remains in the dissemination of information and the establishment of approachable experimental techniques for use by non-experts.

Henry Tan's picture

Seems that most papers are on the behavior of a viscoelastic material penetrated by a rigid indenter. Is there any work (theoretical or experimental) on the contact between two viscoelastic surfaces? Thanks.

MichelleLOyen's picture

Great point; my post emphasized one part of viscoelastic contact far more than other topics on this subject. For materials characterization purposes such as in measurements of viscoelasticity by indentation, the main subject of my posting, the contact is between a compliant and time-dependent layer and an indenter that is in comparison stiff (or rigid) and time-independent in mechanical response. Thus the viscoelasticity is entirely on one side of the problem.

Contact between two viscoelastic surfaces most commonly comes up in two places that I can think of: (1) biomechanical studies of contact in "diarthrodial" joints, such as the knee or hip, in which the contact is between two viscoelastic/poroelastic cartilage layers (a few mm thick on the surface of long bones), and (2) adhesion of two viscoelastic bodies--a nice paper on the subject is this one by Falsafi et al, J. Rheology 1997 as well as much recent work from Ken Johnson including this .

I just want to chime in here, that, for a number of reasons, this is a very challenging problem computationally.  My former postdoc Ilinca Stanciulescu (now at Illinois) and I have been working on this to study some interesting experimental results we've observed for contact between stimulus-responsive hydrogels. I believe we'll have something completed this summer that may be of great interest to the community of researchers working on this class of problems.  

Hi Prof. Dolbow. I am also interested on this topic. One of the challenging problems is to model the hydrogel in the impact process. Hyperelasticity model is an ideal candidate to model its nonlinear elastic behavior. In addition, I also want to know more knowledge about the the failure of hydrogel. Due to lacking experimental research, I find, to my knowledge, few work is done on the failure mechanism. I think it is a gap where some excellent work can be done.

MichelleLOyen's picture

This is a very interesting problem and one with lots of clinical relevance from the perspective of military and automotive injury biomechanics.

While hyperelastic models may be useful here, for a hydrogel the poroelastic response (considering the fluid-solid interactions for a saturated porous solid) might also be quite important. I find that much more work is published on the nonlinear elasticity aspects of soft materials and potentially important time effects get missed; especially in impact loading the time element is a critical consideration.

From a damage and failure perspective, I agree completely that failure mechanisms have also been largely under the radar and there are significant research opportunities here.

You might be interested in this paper from APS Selvadurai, "Stationary damage modelling of poroelastic contact"

Zhigang Suo's picture

I have just started learning poroelasticity, hydrogel, elastic solvent, etc.  I have had some experience with the last topic before, but a case can be made that all three are applications of the same idea.  In a week or two, I'll post a section of notes on the subject in my Advanced Elasticity course.  This way I can learn the subject with my students, and with fellow iMechanicians.  Thank you, Michelle, for repeatedly coming back to the point of viscoelasticiy vs. poroelasticity.

On the schedule of the jClub, July is still open. 

  1. Will the Mechanics of Hydrogels make a suitable Theme of the Month?
  2. If yes to 1., any suggestion for a Discussion Leader for the theme?
Pradeep Sharma's picture

Zhigang: I know very little about "mechanics of hydrogels". It would be hard for me to say whether this topic would be of wide interest. I will read a bit more about this topic.....perhaps I will wait for your notes!

In any event, if there are interesting mechanics issues in this topic (like say, in flexible electronics) then it will certainly be a nice July theme.

Digressing slightly, regarding J-club, one of the reasons I have been keeping a slot unfilled is that I have been mulling around with a very specific thought in mind which I will share with all of you here and solicit feedback. As you will note on the J-club issue list, most topics (with few exceptions) are somewhat specialized e.g. bio, nano etc. I have been wondering whether a theme loosely centered around "foundations of rational continuum mechanics" would be of interest. In other words, are there any open or controversial or "basic" issues in mechanics that merit discussion?

MichelleLOyen's picture

I like your idea of the basic-continuum mechanics topic for an upcoming jClub slot but believe a hydrogels topic would be of interest, and thus would recommend it for later in the year. The hydrogels topic could be combined with a basic tutorial in linear poroelasticity, which also appears to be a topic of some interest in this community.

Kaifeng Liu's picture


I agree with Michelle. In soft hydrogels, water content varies from 50%-90%. Fluid flow by no means shall be ignored. Modeling of hydrogel mechanical response is not a easy thing. Current mechanical constitutive models can be classified in two big categories, to my understanding, single phase and multi phase. Among single phase models, various nonlinear elastic (e.g. hyperelastic) and viscoelastic models are out there. Hydrogels show definitely time-dependent behavior in experiments such as compressive creep test. Nonlinear elastic models certainly have limitations in this sense.

Biphasic model that accounts for the fluid flow in a porous media has been used to model cartilage for over 20 years (dated back to Mow V.C. 1980).

It is just recent that this idea is brought into hydrogel mechanics. Some studies used biphasic elastic (Broom and Oloyede 1998; Silva et al 2005;Lei and Szeri 2007) or biphasic viscoelastic model (Olberding and Suh 2006) in their study. More experimental observations are need to understand the fluid flow behavior in hydrogel, especially in surface layer. It is very important not only to the stress-strain response but also to the surface tribological behavior, since hdyrogels are experted to replace articulating load bearing tissues.

I am working on a biphasic viscoelastc model for hdyrogels right now. I would be happy to keep communicating with the community on this forum.

BTW, I love the idea of hydrogel mechanics for the jounal club! Dr. Dolbow, nice to see you here. I enjoy reading your recent publications on hydrogels.


Broom, N. D. and A. Oloyede (1998). "The importance of physicochemical swelling in cartilage illustrated with a model hydrogel system." Biomaterials 19(13): 1179-1188.

Lei, F. and A. Z. Szeri (2007). "Inverse analysis of constitutive models: Biological soft tissues." Journal of Biomechanics 40(4): 936-940.

Mow V.C., Kuci S.C.Lai W.M.and Armstrong C.G.,Biphasic Creep and Stress Relaxation of Artieular Cartilagc in Compression:Thcory and Experiments,J.Biomechanical Engincering,Vol.102,1980,pp73~84

Olberding, J. E. and J. K. F. Suh (2006). "A dual optimization method for the material parameter identification of a biphasic poroviscoelastic hydrogel: Potential application to hypercompliant soft tissues." Journal of Biomechanics 39(13): 2468-2475.

Silva, P., S. Crozier, et al. (2005). "An experimental and finite element poroelastic creep response analysis of an intervertebral hydrogel disc model in axial compression." Journal of Materials Science-Materials in Medicine 16(7): 663-669.





Thanks for your comments.  In fact, we've been studying hydrogel failure (in stimulus-responsive gels) for about a year and a half now, and have seen some very interesting things. Some of our early results are discussed here.  

I've also a number of papers discussing the chemo-mechanical and thermo-mechanical modeling of stimulus-responsive gels.  A discussion and link to papers is provided here.  

Prof. Dolbow and Michelle

Thank you so much. I think another problem is the thermal effect during viscoelastic contact ( e.g. impact with hydrogel). The heat has to be produced from the energy dissipation during the impact. So it is close related to the thermal mechanical properties of hydrogels(heat conductivity, time scale etc.) Could you talk something about this in the transient impact process from your understanding ? Also I welcome any comments from others.



MichelleLOyen's picture

I had not thought of this interesting question before. One of the things that makes hydrogels and biological materials critically different from classic engineering materials is water. Wet materials are affected by the weird and wonderful properties of water, which is quite an anomolous liquid. Most importantly in the context of this question, the high thermal conductivity of water (highest of any liquid) helps prevent local thermal fluctuations and temperature changes in "wet" organisms like ourselves and thus I would not necessarily expect a large thermal effect in hydrogel impact.

Is there any method to find work of adhesion for a non-control film in which JKR equation is not applicable(linear fit)?Thank you

I am doing experiments of adhesion by using both film and indenter(PDMS hemi sphere) made by PDMS. I want to know about the validity of application of JKR theory for this and please suggest me some papers regarding this.

thank you

Aaron Goh's picture


Good to see that you have brought up the subject on viscoelastic materials.  I am also hoping that more is done on non-linear viscoelastic materials, which I think is important for soft matter including biological tissues and gel-based systems such as food products.  Also, for such materials, the geometrical effects are rather important but most theories relate to ideal conditions e.g. infinite/semi-infinite sample size.  For elastic-plastic materials, the deformation is concentrated locally so a decent sized sample is sufficient, but for an elastic body undergoing large deformation, this is less clear.  I believe there are several publications on this subject (e.g. Anand and Scanlon 2002, Karduna et al. 1997, Costa and Yin 1999) but I don't think a simple scheme to delineate between geometry and material properties exist as yet.  Having said that, I definitely have lost touch on the publications in this area since switching to industry in 2005-I notice your flurry of publications since 2005.  Before, I used to work on indentation of non-linear viscoelasticity of food products at Imperial and had drafted a manuscript using FE to look into the geometrical effects. However, the manuscript was a casualty of the switching of jobs.  There are, of course, other areas in which the indentation understanding will be helpful such as the injection work at Cambridge and the cutting work at Imperial.



Thanks for bringing up the topic of indenting a nonlinear viscoelastic solid. I work in rubber industry and really interested in this topic. I do understand that there is very little literature available on this topic. I would really appreciate if you can direct me to some of the articles, which will help me in understanding some of the challenges of this problem, and some possible solutions. Both FEA and experimental work will help me.




MichelleLOyen's picture

Indentation of linearly viscoelastic materials has been reasonably well-explored, both in mathematics and in experiments. However, as several comments have noted here, there are challenges for analysis of indentation of nonlinearly viscoelastic materials, including rubbers, soft tissues and hydrogels. The latter of these materials are further complicated by the viscoelastic-vs-poroelastic issue as well.

Commonly in the biomechanics community, this type of analysis relies on FE implementation and parameter estimation techniques based on reverse modeling. There are, however, analytical approaches as well. In either FE implementation or in analytical approaches, either in the case of linear or nonlinear viscoelasticity, some a priori form of the constitutive model must be assumed since step-loading (the condition that would allow for unambiguous viscoelastic measurements) is experimentally impossible.

In linear viscoelasticity the elastic response is assumed to be linear and the creep or relaxation function form must be chosen; in nonlinear viscoelasticity based on strain-time or stress-time separability (commonly termed "quasi-linear viscoelasticity" or QLV models after YC Fung) a separate functional form for the elastic and (normalized) time-dependent responses must be selected. This simplification of separability works well for some materials and poorly for others; a discussion is in this paper.

To ascertain if a material is behaving linearly or nonlinearly viscoelastic is a straightforward, albeit time-consuming, process in that indentation tests must be conducted over a range of load- or displacement-levels. If you assume a linearly viscoelastic analysis and then find that the obtained behavior is indeed independent of level, then the material behavior is linearly viscoelastic. This can be done two ways, (1) by using a fit-and-predict scheme with several sets of data, or (2) by independently calculating parameters for a series of tests at different levels.

If the parameters are not level-independent, such that either of the two tests above fails, then some nonlinearity is creeping in. The bigger problem here is then usually in that the nonlinearity can be from one of two sources: (1) actual material nonlinearity, or (2) nonlinearity arising due to the fact that the sample is a finite thickness resting on a substantially (several orders of magnitude) stiffer substrate. Deconvoluting these two nonlinearities, material versus geometrical, is the tricky part. Tips from the thin film mechanics field might prove helpful but work needs to be done.

There is indeed very little published on this subject; I have several things in final preparation that I will flag as soon as they are ready.


Aaron Goh's picture

Actually many gels used in foods e.g. agar, gelatin, are hardly viscoelastic (G' much larger than G'' or flat stress relaxation behaviour).  Does not mean that the 'porosity' is not important though as reflected in the syneresis of water amongst other issues.  I had looked at biological tissues before but their J-shaped stress-strain curves make them pretty atrocious to get a nicely shaped specimen no? 

MichelleLOyen's picture

Very interesting comments.

In the context of hydrogels and time-dependent responses, over what range of times/frequencies are you considering? I hesitate to ever say that something is not, or is hardly, viscoelastic since that sort of statement only applied over the range of times/frequencies of the experiment! An apparently elastic response is sometimes just the result of being at one (adiabatic or isothermal) modulus limit, whereas the bulk of time-dependent behavior may be dominant over a different range of times/frequencies.

With regards to the preparation of biological samples, great point. A very old joke is that actual dog bones are not of "engineering dog bone" shape for good tensile test specimens! It is one of the reasons why indentation tests are so popular for biological samples: it's far easier to perform the tests. Soft tissues are of course even more difficult than bone; for bones you can at least try machine a "dog bone" specimen although I have seen some discussion over the potential for microcracking damage as a result of the specimen machining.

I am not a fan of the term "J-shaped" responses for soft tissues; many of them have an overall quadratic response at low to moderate strains and potentially (but not always, depending on the collagen orientation distribution) a "linear" region when the "recruitment" phase is complete, especially in aligned tissues such as tendon. But in membranes and tissues with a planar collagen orientation, failure can occur prior to any distinguishable "linear region".

Aaron Goh's picture


You are right, I was tempted to add 'in the timeframe of a typical Instron test' but decided not to and I realise that is a mistake. It is pretty much like the concept of 'yield stress' of soft solids and the duration to measure it which my more enlightened colleagues like to argue about (think shelf-life of products and doing a test to match that time frame).

I think another reason why indentation is popular is that in-vivo it is probably the only test can that can be done!! Imagine probing the swelling of a brain by taking a piece of it to do a compression test....

Michelle, Thank you so much for shedding some light on this topic for me, Is there any particular paper I should start to learn about indentation of nonlinear visco-elastic materials. Thanks


MichelleLOyen's picture

I don't know offhand of a good single journal reference for the basics of indentation in nonlinearly viscoelastic materials (you may have identified a gap in the current literature!) However, it can't hurt to read up on indentation of linearly viscoelastic materials (section 6.5 in Johnson's Contact Mechanics) and indentation of nonlinear materials (section 6.6 in the same) although the treatment of nonlinearity here is focused on plasticity, not materials with strain-stiffening responses.

I also recommend the book "Creep and Relaxation of Nonlinear Viscoelastic Materials with an Introduction to Linear Viscoelasticity" by Findley, Lai and Onaran as a general text for nonlinear viscoelasticity. However, their treatment of (even linearly viscoelastic) indentation appears to suffer from a lack of awareness of the work of Lee, Radok, Ting (and others) who addressed the moving boundary condition issues that arise with spherical punch geometries.


Thank you so much for your promt response, your suggested reference will be of a great help. I would really appreciate if you let me know if you come acorss any more good references in this area.




Gang Huang's picture

It is really good to see some enlightening discussions on viscoelastic contacts in this forum.  Not like elastic indentation for which methods have been well established to measure mechanical properties, measuring viscoelastic properties using nanoindentation still remains as a topic not fully paid attention to. In recent years, efforts have been made in the study of viscoelastic indentation, to name a few, such as creep/relaxation behavior during viscoelastic indentation using flat punch indenter, measurements of creep compliance and relaxation modulus using nanoindentation, sharp viscoelastic indentation. For non-experts in this area, a free software from Dr. Hongbing Lu’s group is available for use to solve for creep compliance using constant rate/step loading conditions. When viscoelastic material has varying Poison’s ratio, nanoindentation can be used to measure two independent material functions. Nonetheless, work on viscoelastic indentation, especially on nonlinearly viscoelastic contacts is far from complete. Even in the regime of linear viscoelasticity, a lot of problems still remain unresolved to well understand viscoelastic indentation, such as entire unloading history, indentation on viscoelastic materials with anisotropy, and so on.

Seungtae Choi's picture

Viscoelastic contact is one of very interesting topics for me. We may encounter many situations involving contacts of soft materials in flexible electronics and bio-related applications. Recently, I have conducted indentation experiments of elastomer films with a flat-ended cylindrical tip. Among various shapes of indenter tips, a flat tip only provides constant area during indentations, which makes it easier than others to analyze the indentation experiments. The attached manuscript is the summary of our experiments, which was recently submitted to a Journal. The theory in the manuscript is based on the evelopment of our solution for the flat indentation on an elastic film bonded to a rigid substrate, which will also be submitted to a Journal shortly. Since PDMS film is so compliant that the head dynamics of Nano Indenter XP system is taken into account. Even though an elastomer film, PDMS (polydimethylsiloxane), was tested, and the indentation depth was approximately 10 % of the film thickness, we uses the linear theory of viscoelasticity to quantify the relaxation modulus in time domain. I hope this indentation method and the results will help our understanding of viscoelastic materials.

Seungtae Choi 
R&D Staff Member
Micro Systems Lab, SAIT
Republic of Korea


MichelleLOyen's picture

How much viscoelastic deformation were you seeing in PDMS?  Its response is the closest thing to a time-independent elastic material as I have ever seen!

A few other recent papers have considered indentation viscoelasticity in thin films:

Oyen ML, Cook RF, Moody NR, and Emerson JA:  Indentation Responses of Time-Dependent Films on Stiff Substrates,  Journal of Materials Research, 19 (2004) 2487-97.   Erratum in Journal of Materials Research, 19 (2004) 3120-1.

Zhang CY, Zhang YW, Zeng KY, Extracting the mechanical properties of a viscoelastic polymeric film on a hard elastic substrate.  J. Mater. Res. 2004, 19(10):3053-61.

And of course the classic paper for this problem (flat punch indentation of compliant film on soft substrate) is actually in the biomechanics literature:

Hayes WC, Keer LM, Herrmann G, Mockros LF, A mathematical analysis for indentation tests of articular cartilage. J Biomech 1972, 5(5):541-51. 

Seungtae Choi's picture


The strain was approximately 10 % in average sense. Your point is quite right since PDMS is elastic up to quite large strain level, but it can be viscoelastic. We did find time-dependent behavior of PDMS. Even we observed residual deformation (we can say it as viscoplastic behavior). You can see the results from the link. 


Seungtae Choi

R&D Staff Member

Micro Systems Lab, SAIT

Republic of Korea

MichelleLOyen's picture

I agree that viscoelastic indentation has received little historical interest but this does seem to be changing quickly!  The recent burst in activity in the literature seems to be driven separately by interests in polymer (and polymer-matrix composite) characterisation and by biological material studies. My own motivations were originally in characterising biological materials, and we are finally starting to make some progress.

A study on variability in sharp (Berkovich) viscoelastic nanoindentation on bone was just published:

Oyen ML and Ko C-C:  Examination of Local Variations in Viscous, Elastic, and Plastic Indentation Responses in Healing Bone, Journal of Materials Science: Materials in Medicine, 18 (2007) 623-8.

(Inexplicably it has not yet appeared on the journal's website but I posted a preprint previously here .) 

This is in addition to two recent works on spherical indentation viscoelasticity in bone:

Bembey AK, Oyen ML, Bushby AJ, Boyde A: Viscoelastic properties of bone as a function of hydration state determined by nanoindentation.  Philosophical Magazine, 86(33-35) (2006) 5691 - 5703.

Bembey AK, Bushby AJ, Boyde A, Ferguson VL, Oyen ML: Hydration Effects on the Micro-Mechanical Properties of Bone.  Journal of Materials Research, 21 (2006) 1962-8.

And an invited review paper:

Oyen ML and Bushby AJ, Viscoelastic Effects in Small-Scale Indentation of Biological Materials.

published in International Journal of Surface Science and Engineering 2007 - Vol. 1, No.2/3 pp. 180 - 197

I am doing experiments of adhesion by using both film and indenter(PDMS hemi sphere) made by PDMS. I want to know about the validity of application of JKR theory for this and please suggest me some papers regarding this.

thank you

MichelleLOyen's picture

A good set of papers come recently out of the lab of Kathy Wahl, including these:

Ebenstein DM, Wahl KJ. "A comparison of JKR-based methods to analyze quasi-static and dynamic indentation force curves", Journal of Colloid and Interface Science, 298(2):652-662 (2006).

Wahl KJ, Asif SAS, Greenwood JA, Johnson KL, Oscillating adhesive contacts between micron-scale tips and compliant polymers. Journal of Colloid and Interface Science, Volume 296, Issue 1, 1 April 2006, Pages 178-188.

I also enjoyed:

Basire C and Fretigny C, Kinetics of adhesion on a viscoelastic sample by force microscopy.  Tribology Lett. 10 (2001) 189-93. 


Hope that helps! 


I went through some of your papers cited above and other related articles. I found them very interesting. when we assume elastic K and viscoelastic G, we will get time dependent E and nu. K and E are related with nu. My question is:

Material behavior in tension and compression could vary. so when we compute E from K, which E( tension or compression) are we referring to? or are we assuming that the tension and compression E values for the material are the same? In practice, which experimental E modulus corrospond to the E value we obtain from K ?

I am looking forward to read your comments


Hi! I am working on the modeling of PDMS. What I focused on previously was based on the viscoelasticity. However, on today's group meeting, somebody mentioned other models based on the hyperelasticity. Now i am not sure which one is better for modeling of PDMS this kind of material? Thanks a lot!

Hello all, i am a master student in Structural engineering in Jordan. My thesis is about FEA of Thermoplastic Bridges, all the models part are made of thermoplastic which is a viscoelastic material, i calculated the material properties from relaxtion curve, I wanna ask if there is a method to check that i computed them Right?

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