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Journal Club May 2010: Cavitation in Elastomeric Solids

Oscar Lopez-Pamies's picture

Experimental evidence has shown that loading conditions with sufficiently large triaxilities can induce the sudden appearance of internal cavities within elastomeric (and other soft) solids. The occurrence of such instabilities, commonly referred to as cavitation, has been attributed to the growth of pre-existing defects. In a typical elastomer, defects are expected to appear randomly distributed and to have a wide range of sizes with average diameters fluctuating around 0.1 μm, but beyond these geometrical features not much is known about their nature — they may possibly correspond to actual holes, particles of dust, and/or even weak regions of the polymer network [1].

From a mechanics perspective, the occurrence of cavitation is an important phenomenon because it may signal the initiation of material failure, since upon continuing loading a number of “nucleated cavities” may grow, coalesce, and eventually form large enclosed cracks [I]. Alternatively, the post-cavitation growth of the cavities can be used to an advantage. A prominent example is that of rubber-toughened hard brittle polymers, where the cavitation and post-cavitation behavior of the rubber particles provides a critical toughening mechanism for these material systems (see, e.g., [2]). From a more fundamental perspective, the study of cavitation is of significant interest in order to gain further insight into the influence of defects in solids.

Here I will discuss the classical onset-of-cavitation result from the celebrated papers of Gent and Lindley (1959) and of Ball (1982) , and will outline some of the key remaining open problems in cavitation of elastomeric solids.

1.    Classical result for radially symmetric cavitation

We begin by recalling the well-known elastostatics problem of the radially symmetric deformation of a spherical shell (see, e.g., Section 5.3.2 in [3]). Specifically, let us consider a spherical shell that is made up of an incompressible isotropic material with stored-energy function Φ (λ_1, λ_2, λ_3), where λ_1, λ_2, and λ_3 are the principal stretches. In its undeformed stress-free configuration, the shell has outer radius Ro = 1 and inner radius Ri = f0^1/3, where we note that the prescribed quantity f0 corresponds to the initial porosity in the shell, i.e., the initial volume fraction of the cavity f0 =  Ri^3 /Ro^3 (see Fig. 1).

Upon applying a nominal hydrostatic pressure P on the outer boundary, the shell deforms with radial symmetry into another shell with outer radius ro = λ and inner radius ri = f0  + λ^3 – 1, so that the porosity in the deformed shell is given by

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Here, the size λ of the deformed outer radius is directly related to the applied hydrostatic pressure P via the relation

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Expressions (1) and (2) are valid for shells with any value of initial porosity in the physical range f0 in [0, 1]. We now focus on the special subclass of shells where the initial porosity is taken to be vanishingly small, that is,  f0 → 0+. In this limiting case, the cavity in the shell reduces to a zero-volume cavity or defect. Upon loading, the porosity (i.e., the size) of this defect can grow from its initially infinitesimal value of  f = f0 → 0+ in the undeformed configuration (P = 0) to finite values at some sufficiently large critical pressure Pcr (see Fig. 2). This event corresponds to the onset of cavitation. According to (1) and (2), the critical stress at which cavitation occurs is simply given by

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The result (3) as it stands was first derived by Ball (1982) using a rather different approach than the one illustrated here. Earlier, Gent and Lindley (1959) had obtained the specialized version of (3) for Neo-Hookean materials Φ = μ/2(λ_1^2 +  λ_2^2+ λ_3^2 – 3):

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2.    Open Problems

The result (3) was derived based on the restrictions that:

i) The material behavior is incompressible and isotropic.

ii) The applied loading is hydrostatic.

iii) The pre-existing defect is assumed to be a single spherical vacuous cavity.

In the last 30 years, numerous efforts have been devoted to extend the results of Gent and Lindley and of Ball to more general material behaviors, loading conditions, and types of defects. These generalizations have proved to be remarkably challenging and as such have prompted and continue to prompt the development of new mathematics. In the sequel, we recall the underlying physical motivation to carry out such generalizations, and outline the major findings of hitherto efforts together with key remaining open problems.

Material behavior
. Elastomers, as well as any other types of soft solids like biological tissues, are not incompressible, but in actuality they exhibit finite bulk moduli that range from 1 to up to 4 orders of magnitude larger than their shear moduli. In an attempt to make contact with this experimental evidence, extensions of the works of Gent and Lindley and of Ball have been put forward to account for material compressibility, but mostly in the special case of hydrostatic loading conditions (see, e.g., [4,5,6]). In the more general context of loadings with arbitrary 2D triaxiality, Lopez-Pamies [7] has recently derived a variational approximation for the onset of cavitation in compressible isotropic materials. Moreover, depending on their processes of synthesis/fabrication, elastomers can exhibit sizable degrees of anisotropy. And because of their intrinsic growth conditions, most soft biological tissues can also exhibit strong anisotropy. However, with the exception of a few highly idealized studies (see, e,g, [8]) in the context of radially symmetric cavitation, very little progress has been made thus far in the mathematical analysis of cavitation in anisotropic materials, especially those with physically relevant anisotropies (e.g., transverse isotropy, orthotropy).

Loading conditions. The occurrence of cavitation is expected to depend very intricately on the entire state of the applied loading conditions, not just on the hydrostatic component (see, e.g., [9]). Yet, the vast majority of cavitation studies to date have been almost exclusively limited to hydrostatic loading conditions, presumably because of the simpler tractability of this relevant but overly restricted case. Among the exceptions, Hou and Abeyaratne [10] have made use of variational arguments to derive an explicit upper bound for the onset of cavitation in incompressible isotropic solids subjected to generic loading conditions. More recently, Sivaloganathan, Spector and co-workers [11,12,13] (see also [14]) have established, via energy-minimization techniques, existence results for the onset of cavitation in isotropic polyconvex solids under fairly general loadings. In addition, as already mentioned above, there is the variational approximation put forward in [7] for the onset of cavitation in isotropic materials subjected to loadings with arbitrary 2D triaxiality. The fundamental problem of quantitatively predicting the onset of cavitation in a given nonlinear elastic solid under 3D arbitrary loading conditions remains open.

Geometry and mechanical properties of defects
. In addition to the type of material behavior and loading conditions, it is reasonable to expect that the initial shape and spatial distribution of pre-existing defects are geometrical features that might significantly impact when cavitation occurs. However, the greater part of existing cavitation studies have been overwhelmingly focused on material systems containing a single defect of spherical (or circular) shape [2, 11]. It is only recent that studies on single cavities of non-spherical shape [12] and on large number of point defects at which cavitation can initiate [11,12,13,14] have begun to be pursued. In addition to their geometrical attributes, defects posses mechanical properties as well, although very little is known about them experimentally as already pointed out above. The simplest hypothesis is to consider that they are vacuous (i.e., traction-free cavities), as in fact was assumed in the original works of Gent and Lindley [I] and of Ball[II], as well as in most of subsequent efforts [2, 15]. Under suitable circumstances, however, it is known that defects may contain a non-zero pressure [17]. How pressurized defects, and, more generally, defects with more complex mechanical properties, impact cavitation is yet to be thoroughly examined.

We conclude by remarking that in addition to the above considerations on material behavior, loading conditions, and the geometric and constitutive nature of defects, surface-energy, fracture, and dynamic effects could also play a role on the cavitation in elastomeric solids.

Main References

[I]     Gent, A.N., Lindley, P.B., 1959. Internal rupture of bonded rubber cylinders in tension. Proc. R. Soc. Lond. A. 249, 195–205. doi:10.1098/rspa.1959.0016

[II]     Ball, J.M. 1982. Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. R. Soc. A 306, 557–611. doi:10.1098/rsta.1982.0095

Other References

[1]     Gent, A.N., 1991. Cavitation in rubber: a cautionary tale. Rubber Chem. Technol. 63, G49–G53.

[2]     Fond, C., 2001. Cavitation criterion for rubber materials: A review of void-growth models. Journal of Polymer Science: Part B 39, 2081–2096.

[3]     Ogden, R.W., 1997. Non-linear elastic deformations. Dover Publications Inc. Mineola, N.Y.

[4]     Stuart, C.A., 1985. Radially symmetric cavitation for hyperelastic materials. Ann. Inst. Henri Poincare, Analyse non lineaire 2, 33–66.

[5]     Sivaloganathan, J., 1986. Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity. Archive for Rational Mechanics and Analysis 96, 97–136.

[6]     Horgan, C.O., Abeyaratne, R., 1986. A bifurcation problem for a compressible nonlinearly elastic medium: growth of a micro-void. Journal of Elasticity 16, 189–200.

[7]     Lopez-Pamies, O., 2009. Onset of cavitation in compressible, isotropic, hyperelastic solids. Journal of Elasticity 94, 115–145.

[8]     Antman, S.S., Negron-Marrero, P.V., 1987. The remarkable nature of radially symmetric equilibrium states of aleotropic nonlinearly elastic bodies. Journal of Elasticity 18, 131–164.

[9]     Chang, Y.-W., Gent, A.N., Padovan, J., 1993. Expansion of a cavity in a rubber block under unequal stresses. International Journal of Fracture 60, 283–291.

[10]     Hou, H.-S., Abeyaratne, R., 1992. Cavitation in elastic and elastic-plastic solids. Journal of the Mechanics and Physics of Solids 40, 571–592.

[11]     Sivaloganathan, J., Spector, S.J., 2000. On the existence of mimimizers with prescribed singular points in nonlinear elasticity. Journal of Elasticity 59, 83–113.

[12]     Sivaloganathan, J., Spector, S.J., 2002. A construction of infinitely many singular weak solutions to the equations of nonlinear elasticity. Proc. R. Soc. Ed.  132A, 985–992.

[13]     Sivaloganathan, J., Spector, S.J., Tilakraj, V., 2006. The convergence of regularized minimizers for cavitation problems in nonlinear elasticity. SIAM J. Appl. Math. 66, 736–757.

[14]     Henao, D., 2009. Cavitation, invertibility, and convergence of regularized minimizers in nonlinear elasticity. Journal of Elasticity 94, 55–68.

[15]     Horgan, C.O., Polignone, D.A., 1995. Cavitation in nonlinearly elastic solids: a review. Applied Mechanics Reviews 48, 471–485.

[16]     James, R.D., Spector, S.J., 1991. The formation of filamentary voids in solids. Journal of the Mechanics and Physics of Solids 39, 783–813.

[17]     Gent, A.N., Tompkins, D.A., 1969. Surface energy effects for small holes or particles in elastomers. J. Polym. Sci. Part A2 7, 1483–1487.

Zhigang Suo's picture

Dear Oscar:  Thank you very much for this timely post on a timeless phenomenon.  Here are two experimental papers on cavitation:

M.F. Ashby, F.J. Blunt and M. Bannister, Flow characteristics of highly constrained metal wires.  Acta Metallurgica 37, 1847-1857 (1989).  The paper shows the expansion of a single cavity in a metal under a triaxial stress.

Santanu Kundu, Alfred J. Crosby, Cavitation and Fracture Behavior of Polyacrylamide Hydrogels. Soft Matter 5, 425-431 (2009).  This paper demonstrates the "cavitation rheology"--the use of cavitation as a technique to determine elastic modulus of a soft material.  The technique enables the determination of modulus of a small part of a tissue, beneath the surface.

Oscar Lopez-Pamies's picture

Dear Zhigang,

Thank you very much for pointing out this two experimental works — I did not know the article of Ashby et al. and I am very glad that I do know. Indeed, cavitation can also occur in metals as well as in gels.

In metals, it appears that the first reported work on cavitation is that of Bishop, Hill, and Mott (1945)   who while studying indentation and hardness tests ended up examining the problem of radially symmetric cavitation!  Since then, many researchers have studied this type of instability in metals. Hou and Abeyaratne (1992) provide a very nice summary of these works in their article.

Cavitation in gels appears to be a much more recent area of research. In fact, to the best of my knowledge, the paper by Kundu and Crosby is one of the first pieces of experimental evidence of cavitation in gels.

The main difference (which is actually an interesting challenge) between cavitation in elastomeric solids and cavitation in metals and gels is that of accounting for plasticity and nonlinear viscoelasticity effects.

Zhigang Suo's picture

A prerequisite for substantiual growth of a cavity in a material is that the material is capable of large deformation before fracture.  The ratio of the radius of the grown cavity over that of the initial cavity, by definition, is the stretch of the material on the surface of the cavity.

Cavitation in a creeping material.  In a creeping material (a liquid, or a metal at elevated temperature), the material is capable of unlimited strain by rearragements of atoms or molecules, without buiding up elastic energy.  Growth of cavity is often observed.  Because the liquid may flow to unlimited strain at any small stress, the cavity can grow at any small stress.  No critical stress exists, unless the cavity is so small that the surface energy plays a role.  It's just a matter of time. 

Growth of a cavity in a creeping material is analyzed in the following paper:

Budiansky, B., Hutchinson, J.W., Slutsky, S., "Void Growth and Collapse in Viscous Solids." in Mechanics of Solids edited by H. G. Hopkins and M. J. Sewell, Pergamon Press, 13-45 (1982).

The role of surface energy in caviation is discussed in the following paper:

T-j. Chuang, K. I. Kagawa, J. R. Rice and L. B. Sills, "Non-equilibrium Models for Diffusive Cavitation of Grain Interfaces", Acta Metallurgica, Overview Paper No. 2, 27, 1979, pp. 265-284.

Cavitation in a material with a yield strength.  By contrast, for a material with a yield strength, a cavity grows substantially only when the applied stress exceeds several times the yield strength, as first shown by Bishop, Hill, and Mott (1945).

Cavitation-to-fracture transition in elastomers.  For a cavity in an elastomer, the growth will be limitd by the contour length of the polymer chains.  When stressed further, fracture is expected.  The transition from cavitation to fracture seems to be less studied.  In addition to the experimental paper by Kundu and Crosby, the following theoretical paper is interesting:

Y.Y. Lin and C.Y. Hui, Cavity growth from crack-like defects in soft materials.  International Journal of Fracture 126, 205-221 (2004).

Oscar Lopez-Pamies's picture

Dear Zhigang,

You raised a very important point: the cavitation-to-fracture transition is crucial to understand how defects actually grow in elastomers.

However, very little is known about this transition. Here is one of the few papers (which is discussed by Lin and Hui, 2004) that addresses this issue:  “Williams and Schapery, 1965. Spherical flaw instability in hydrostatic tension. International Journal of Fracture Mechanics 1, 64-72."


LG's picture

Dear Prof. Oscar,

Glad to see the discussion of phenomenon about cavitation that i am interested in. Could you please show us some clue about the cavitation in fluids, for instance "gas bubble expansion". Alternatively, if there are some literatures about cavitation in fluids from you that can show  us?


Oscar Lopez-Pamies's picture

Dear LG,

Actually, the phenomenon of “cavitation” in solids owes its name to the phenomenon of cavitation in fluids, which had been observed and investigated much earlier.

Cavitation in fluids has been and continues to be a very active area of research. As a result, many articles and books have been written about it. Here I mention a couple of classical works of Lord Rayleigh and Batchelor:

Rayleigh, Lord 1917. On the pressure developed in a liquid during the collapse of a spherical cavity. Philosophical Magazine Series 6 V34, 94-98.

Sections 6.11 and 6.12 in  Batchelor, G.K., 1967. "An Introduction to Fluid Dynamics". Cambridge University Press.

A more recent contribution is the monograph “Cavitation and Bubble Dynamics” by Christopher Brennen.

Matt Pharr's picture

Hi Oscar,

 I am curious about advantageous cavitation.  You say that in rubber cavitation can be beneficial.  Do people know the mechanism for this?   Does this occur in any systems other than rubber?



Oscar Lopez-Pamies's picture

Hi Matt,

Thanks for your comment.

The most standard case in which cavitation is used to an advantage is that of rubber-toughened hard polymers. The idea is to embed small (in the order of microns) rubber particles in hard brittle polymers. Upon loading, the rubber particles eventually cavitate allowing the material to accommodate larger macroscopic deformations, as opposed to fracturing. Here are a couple of articles (with some nice pictures) describing the process in more detail.

Fond, C., 2001. Cavitation criterion for rubber materials: A review of void-growth models. Journal of Polymer Science: Part B 39, 2081–2096.

Cheng, C., Hiltner, A., Baer, E., 1995. Cooperative cavitation in rubber-toughened polycarbonate. Journal of Materials Science 30, 587–595.

In general, cavitation in elastomeric solids need not be detrimental. In addition to the application of rubber-toughened polymers, cavitation can be used — as pointed out above by Zhigang — to indirectly measure the Young’s modulus of soft materials. It seems reasonable to expect that there are other applications where the phenomenon of cavitation can be used to an advantage (in general, applications that involve/require a sudden change from a stiff to a soft response would be candidates).

Also, as discussed above with Zhigang, cavitation has been observed in many different types of materials, not just rubber. This makes perfect sense given that cavitation corresponds to the growth of pre-existing little tiny defects into large defects, and most real materials contain defects.


Mike Ciavarella's picture


Quasi-static normal indentation of an
elasto-plastic half-space by a rigid sphere—II. Results

Journal of Solids and Structures
, Volume 21, Issue 8, 1985,
Pages 865-888
G. B. Sinclair, P. S. Follansbee, K. L. Johnson

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Indentation of foamed plastics
Journal of Mechanical Sciences
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, Pages 457-460, IN5-IN6
M. Wilsea, K.L. Johnson, M.F.

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The correlation of indentation experiments
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K.L. Johnson

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 These above, and the 1970 JMPS paper in particular, describe one classical model on how an indentation test deforms a halfspace by producing an expanding "cavity".  A simple equation is derived under the approximation that the cavity is like in a full space, rather than halfspace.

I wonder if  this connection makes you think of possible extensions of your interests.  For example, indentation of elastomers is usually done?   Also, what is the connection between "real" cavities, and "cavities" as models?

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
Editor, Italian Science Debate,
Associate Editor, Ferrari Millechili Journal,

Oscar Lopez-Pamies's picture

Dear Mike,

Thanks for your comments and the references.

As suggested by the work of Crosby and co-workers in gels, a potential use of cavitation is as a means of indirectly measuring the mechanical response of elastomeric solids. In this connection, the technique of indentation could prove ideal — an interesting advantage of soft solids is that their indentation can be done pretty much with indenters of any length scale.

I have seen some works on indentation in elastomers, but not many and not related yet to cavitation.  

Your second question is still a key open problem. Indeed, what the “constitutive behavior” of the types of defects that constitute the precursors of cavitation really is remains unclear. Certainly, idealizing defects as tiny spherical holes that are vacuous may seem at first too crude. Yet this idealization has proved in some cases to be a fair representation of actual defects. At some level, this makes sense since it is reasonable to expect that the two main ingredients that characterize these defects are that (i) they are very small and (ii) they are extremely soft under a change in volume. At any rate, certainly more work needs to be done in the modeling of defects.

Mike Ciavarella's picture

Oscar Lopez-Pamies's picture

The work that I was referring to was actually not a paper but a presentation by a group from Southern Methodist University (Wei Tong et al.) with title “Mechanical Characterization of Soft Materials by a Novel Indentation Technique” in the 2008 IMECE ASME conference in Boston. I include the abstract below:

“An experimental technique is presented to measure the mechanical properties of soft materials. It consists of an indenter made of glass or other hard but transparent materials and a digital microscope for direct viewing of the contact surface of the indented material through a transparent indenter. As the digital microscope looks through the transparent indenter directly during the test so the actual contact area can be measured at each indent load step based on the digital images acquired. By using a robust digital image correlation deformation mapping analysis tool, the optical distortion of the transparent indenter is quantified first and subsequently corrected so the surface deformation of the soft materials under the indenter can be accurately measured. A finite element analysis on the surface deformation characteristics of soft materials under indentation along with some preliminary experimental results will be given. Extension of such a technique to the study of adhesion and friction of soft materials will also be discussed.”

I am not sure if these authors have already published this work in journals or not.

I am yet to think thoroughly about the applications, but I suspect that there are a number of  “cool” ones as the ones that you have indicated. Right now I am finishing the write up of a new theoretical strategy that permits to determine the onset of cavitation in nonlinear elastic solids containing random distributions of fairly general types of defects under general 3D loading conditions. In case it is of your interest, I have attached (if I manage) a two-page abstract where the new theory is used (as a first application) to determine a criterion for the onset of cavitation in Neo-Hookean solids containing an isotropic distribution of vacuous defects.

Mike Ciavarella's picture

I would not put all my money in this paper you refer to.  There seemed to be many more, which I included in my "random" list.

Please have a look at them, to see if you find an interest.

I am not understanding now what is the focus of your research or of your article here:  are we obsessed by cavitation, and someone is forcing us to look at this problem?  Or we have a reason for?


In any case, Suo referred to a paper from Ashby which you were enthusiast Suo pointed to, but in fact I found it has little to do with elastomers and even cavities, so I wonder if there is some confusion here.  In fact, see Asbhy's abstract:-



Brittle solids can be toughened by
incorporating ductile inclusions into them. The inclusions bridge the
crack and are stretched as the crack opens, absorbing energy which
contributes to the toughness. To calculate the contribution to the
toughness it is necessary to know the force-displacement curve
for an inclusion, constrained (as it is) by the stiff, brittle matrix.
Measured force-displacement curves for highly constrained metal wires
are described and related to the unconstrained properties of the wire.
The constraint was achieved by bonding the wire into a thick-walled
glass capillary, which was then cracked in a plane normal to the axis of
the wire and tested in tension. Constraint factors as high as 6 were
found, but a lesser constraint gives a larger contribution to the
toughness. The diameter of the wires (or of the inclusions) plays an
important role. Simple, approximate, models for the failure of the wires
are developed. The results allow the contribution of ductile particles
to the toughness of a brittle matrix composite to be calculated.



It seems to me completely unrelated!!

So please repeat what is that you are looking for please??


Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
Editor, Italian Science Debate,
Associate Editor, Ferrari Millechili Journal,

Mike Ciavarella's picture



  thanks for sending me the paper on email..  Maybe you could post something like an image here easily.


Now, you do seem "obsessed by cavitation". but that could be good. 

indentation is almost hydrostatic, but not quite.  so you could
concentrate there, where it is easy to make experiments also.   you
could make it interesting if the indentation is self-similar, like with a
cone, but self-similar is also a crack.  In that case, you should have a
steady-state regime when crack is advancing or indentation, which makes
many models simpler and powerful.

I confess I know nothing about cavitation ahead of a crack.  there must
be some literature, please check.

Best, Mike

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
Editor, Italian Science Debate,
Associate Editor, Ferrari Millechili Journal,

Oscar Lopez-Pamies's picture


I believe that the paper referred to by Zhigang, but more importantly the paper by Bishop Hill, and Mott (1945), makes a connection with the growth of a cavity in an infinite medium that is subjected to hydrostatic loading on its boundary, i.e., at infinity. It is in that way that it is connected to cavitation.

My main motivation to study cavitation is to understand precisely (quantitatively) the effect that pre-existing defects have on elastomeric solids. The hope is that whatever we learn in this study will be of use to understand the effect of other types of defects in more general types of solids. Also, the hope is that the mathematical tools that will be developed will also be useful to study problems of more general types of heterogeneous solids, where the heterogeneities are not defects, but rather particles, fibers, grains, etc…

Mike Ciavarella's picture


 probably the fault is mine here, but we seem to talk two different languages.  I don't see you read my suggestions nor I understand where Zhigang's suggested paper has nothing to do with expanding cavities in elastomers.  If you like, please reply to my simple questions/suggestions:  I simply asked


1) is there a solution for a crack propagating with an elastomer, and hence provoquing cavity expansions ahead of it?

2) is there a simple theory of intendation of elastomers,and hence provoquing cavity expansions behind the indenter?


These seem to me the most basic problems open if this cavity problem has any sense.

Otherwise, sorry I don't follow, and I will not try to distract you more with my Off Track contributions.


Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
Editor, Italian Science Debate,
Associate Editor, Ferrari Millechili Journal,

Oscar Lopez-Pamies's picture


Your comments are not distractions at all, I do appreciate your different point of view on the subject, which certainly enriches my point of view.

Here are my answers to the questions:

1) is there a solution for a crack propagating with an elastomer, and hence provoquing cavity expansions ahead of it?

Unfortunately, I do not know of a solution or experiment dealing with cavitation ahead of a propagating crack in an elastomer. This is opposite to metals, where there has been a lot of work on this problem, mostly from a numerical perspective.

On the other hand, there has been quite a bit of work on the study of cavitation in particle and fiber-reinforced elastomers. A series of papers by Gent and co-workers in “Journal of Materials Science” show very impressive pictures of this phenomenon. Because of the much stiffer behavior of the reinforcing particles and fibers, the hydrostatic component of the local stress within the elastomeric matrix phase can reach high values which in turn lead to cavitation.

2) is there a simple theory of intendation of elastomers,and hence provoquing cavity expansions behind the indenter?

To my knowledge, there is not much done on that front. However, it seems that this is an interesting problem worth studying. Incidentally, this problem is related to the studies of Gent on particle-reinforced elastomers.

Since we have now generated a criterion (equation (2) in the paper that I e-mailed you) that allows determining cavitation under general loading conditions (not under just purely hydrostatic loading), it should be relatively simple to use that criterion to study whether there is cavitation during an indentation test. Of course, it would be great to have the experiments as well.

Mike Ciavarella's picture


Oscar, I understand you know equation 2 of your paper, and this is a starting point!

Now, I am not sure there is nothing on cracks in elastomers or indentation on them.  For one thing, I have posted you at least 10 papers, which I invite you, once again, to have a look at.

On cracks, there are also a number of papers, see e.g. only one random example below.

I start to think that you know the literature on "cavitation" , whee by "cavitation" it is simply meant the detachment of hard particles from a elastomeric material.  But that I find an unfortunate terminology.   I suspect there is no "cavitation" as such, and as intended from the fluid terminology, in a rubber or elastomer.

So we are at a dead point, unless you change direction somewhere!

Journal of Solids and Structures

Volume 45, Issue 24
1 December 2008,
Pages 6034-6044

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materials undergoing large deformations. Unfortunately, the
applicability of the Griffith theory is restricted to previous termcracksnext term
with equivalent sharpness only.

Keywords: previous termRubber; Cracknext term;
Fracture; Failure; Hyperelasticity; Softening



Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
Editor, Italian Science Debate,
Associate Editor, Ferrari Millechili Journal,

Mike Ciavarella's picture

Journal of Fatigue

Volume 28, Issue 1
January 2006,
Pages 61-72

doi:10.1016/j.ijfatigue.2005.03.006 | How to Cite or Link Using DOI

Copyright © 2005 Elsevier Ltd All rights reserved.

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Cracknext term
initiation and propagation under multiaxial fatigue in a natural previous termrubbernext term

References and further reading
may be available for this article. To view references and further
reading you must purchase
this article.

N. Saintiera, Corresponding Author Contact Information, E-mail The Corresponding Author, G. Cailletaudb and R. Piquesb

aLAMEFIP, Ecole Nationale Supérieure
des Arts et Métiers, EA CNRS 2727, Esplanade des Arts et Métiers, 33405
Talence Cedex, France

bCentre des
matériaux P.M. FOURT, Ecole Nationale Supérieure des Mines de Paris, UMR
CNRS 7633, Evry cedex 91003, France

Received 25 February 2004; 

revised 9 February 2005; 

accepted 23 March 2005. 

Available online 20 June


The ever growing use of elastomers and
polymers in structures leads to the need of pertinent multiaxial fatigue
life criteria for such materials. Thus, the understanding of the
fatigue previous termcracknext term
initiation micro-mechanisms and their link to the local stress and/or
strain history is essential. Scanning electron microscopy and Energy
Dispersive Spectroscopy (EDS) have been used to investigate those
micromechanisms on a natural previous term

inclusions were systematically found at the previous termcracknext term
initiation. Depending on the type of inclusion (identified by EDS), previous termcavitationnext term
at the poles or decohesion are the very first damage processes observed.
previous termCracksnext term
orientations are compared to local principal stress orientation history,
the later being obtained from finite element calculations (FE). It is
shown that if large strain conditions are correctly taken into account, previous termcracksnext term
are found to propagate systematically in the direction given by the
maximal first principal stress reached during a cycle, even under
non-proportional loading. A fatigue life criterion is proposed.

Keywords: Multiaxial fatigue; previous termCracknext term
nucleation; previous termCracknext term
growth; Life prediction; previous termRubbernext term

Article Outline

1. Introduction
and mechanical testing
3. Numerical
3.1. Surface
and volume transport: large strain formulation
3.2. Strain
and stress tensors
3.3. Strain
energy density
3.4. Identification
of crack initiation
4.1. Flaws
at crack initiation
4.2. Fatigue
damage initiation
4.2.1. Decohesion
4.2.3. Micro-propagation
Crack propagation mode
5.1. One-dimensional
5.2. Multiaxial
5.2.1. PSDR
5.2.3. Crack orientations and PSD
5.2.4. Alternate
torsion loading
5.2.5. Fully
reverse torsion loading
5.2.6. Push–pull
and static torsion loading
6. Fatigue
life prediction
6.1. Choice
of a pertinent mechanical parameter
6.2. One-dimensional
6.3. Multiaxial
7. Conclusions

Thumbnail image

1. Specimens geometries used for fatigue tests (dimension in mm).

View Within Article

Thumbnail image

2. Comparison between computed and experimental behavior.

View Within Article



Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
Editor, Italian Science Debate,
Associate Editor, Ferrari Millechili Journal,

Mike Ciavarella's picture

Theoretical study of formation of pores in
elastic solids: particulate composites, rubber toughened polymers, crazing

Journal of Solids and Structures
, Volume 39, Issue 11, June
, Pages 3079-3104
Klaus P. Herrmann, Victor G.

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AbstractIt is difficult to overestimate the
multi-functional role and practical meaning of the processes of the
formation of pores in solids, especially in polymers and polymer based
materials, which are capable to a noticeable plastic deformation.
Various mechanisms are responsible for this phenomenon in different
systems. Particularly, it is debonding in particulate filled composites,
elastomeric inclusions failure in rubber toughened polymers, nucleation of
microvoids at defects in glassy polymers. Two main effects of the
formation of pores should be underlined. The first is a decrease in the
material's stiffness, which is mostly emphasized for composites filled
by rigid inclusions. The second is an improvement in the fracture
toughness which is widely explored in practice. The nucleation of pores
affects the fracture toughness, firstly, absorbing the energy for the
new surface formation and, secondly, facilitating of a plastic flow of
the basic material. The paper proposed is partly a review of previously
obtained results and represents also the novel data and laws. It
concerns two aspects of the problem. An analysis of the conditions
advantageous for the appearance of a single pore and of the completeness
of this event is the first. This part of the paper is mostly a review,
but a novel comparable analysis of the regularities of a pore formation
by the way of a debonding along the surfaces of rigid particles in
particulate filled composites and caused by a failure of rubbery
inclusions will be presented. The second aspect of the problem is a
spatial cooperation in the nucleation of pores. Some results in this
field also have been obtained previously. However, the corresponding
part of the paper mostly represents new data as well as a new analysis.
Three types of systems will be analyzed from the cooperation point of
view: particulate filled composites, rubber toughened plastics and homogeneous
polymers for which a formation of micropores in a diffuse or a
cooperative manner is a well known phenomenon named as crazing. Certain
corrections of the previous conclusions concerning the cooperation
arising during the failure of rubbery particles have been performed.
Furthermore, the angles of the disposition of porous zones will be
estimated. In addition, it will be shown that the conditions
advantageous for an individual cavitation as well as the laws of a diffuse
or cooperative proceeding of the multiple crazing are qualitatively the
same. However, the different features will also be stated.


Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
Editor, Italian Science Debate,
Associate Editor, Ferrari Millechili Journal,

Zhigang Suo's picture

For indentation of an elastomer, it seems signifigant to separately consider shallow indentation and deep indentation.

Shallow indentation.  When the depth of indentation is small compared to the radius of contact, linear elastic solution seems to be an excellent approximation.  The indentation hardness of an elastomer is related to elastic modulus.  Hardness of an elastomer has been routinely measured for a long time.  See

Recently, we and others have begun to use shallow indentation to measure poroelastic constants of gels.  The experimental data are very encouraging.  See

Yuhang Hu, Xuanhe Zhao, Joost Vlassak, Zhigang Suo, Using indentation to characterize the poroelasticity of gels. Applied Physics Letters 96, 121904 (2010).

As mentioned above, shallow indentation is analyzed by using theories of infinitesimal strains, and is perhaps unrelated to cavitation.

Deep indentation.  When the depth of indentation is large compared to the radius of contact, metals can behave very differently from elastomers.  Deep indentation happens, for example, when a rod is pushed into a material. 

As a rod goes deeper into a metal, atoms of the metal flow, and a steady state is possibly.  As pointed out by Bishop, Hill and Mott, the situation is closely approximated by cavitation.

As a rod goes deeper into an elastomer, however, the polymer chains of the elastomer are stretched toward their extension limit.  No steady state is possibly.  Cracks form in the elastomer along the path of the rod.  Here is a recent experiment:

Wei-Chun Lin, Kathryn J. Otim, Joseph L. Lenhart, Phillip J. Cole, Kenneth R. Shull.  Indentation fracture of silicone gels.  J. Mater. Res. 24, 957 (2009).

For elastomers, crack after large deformation is a significant phenomenon.  The phenomenon may happen both during cavitation and deep indentation.


Mike Ciavarella's picture

dear Zhigang

  of course for indentation with a sphere, the deformation is proportional to a/R and hence as you say you need the distinction.  However, for the case I suggested, self-similar indentation (which is vaguely more similar to a self-similar situation in a crack) you get obviously only geometrical definition of strain, which is or a cone connected to the angle of the cone.  Hence, the only distinction there is shallow cone or sharp cone.  Indeed,after writing these 2 lines I searched "cone indentation rubber", and I found this paper which confirms our obvious remarks.


Journal of Solids and Structures

Volume 46, Issue 6
15 March 2009,
Pages 1436-1447

doi:10.1016/j.ijsolstr.2008.11.008 | How to Cite or Link Using DOI

Copyright © 2008 Elsevier Ltd All rights reserved.

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Conical indentation of incompressible rubber-like materials

A.E. Giannakopoulosa and D.I.
PanagiotopoulosCorresponding Author Contact Information, a, E-mail The Corresponding Author

aLaboratory for Strength of
Materials and Micromechanics, Department of Civil Engineering,
University of Thessaly, Volos 38336, Greece

Received 5 July 2008; 

revised 3 November 2008. 

Available online 18
November 2008.


In the last decade, the indentation test
has become a useful tool for probing mechanical properties of small
material volumes. In this context, little has been done for rubber-like
materials (elastomers), although there is pressing need to use
instrumented indentation in biomechanics and tissue examination. The
present work investigates the quasi-static normal instrumented
indentation of incompressible rubber-like substrates by sharp rigid
cones. A second-order elasticity analysis was performed in addition to
finite element analysis and showed that the elastic modulus at
infinitesimal strains correlates well with the indentation response that
is the relation between the applied force and the resulting vertical
displacement of the indentor’s tip. Three hyperelastic models were
analyzed: the classic Mooney–Rivlin model, the simple Gent model and the
one-term Ogden model. The effect of the angle of the cone was
investigated, as well as the influence of surface friction. For blunt
cones, the indentation response agrees remarkably well with the
prediction of linear elasticity and confirms available experimental
results of instrumented Vickers indentation.

Keywords: Rubber materials; Incompressibility;
Conical indentation; Hyperelasticity; Finite elements





Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
Editor, Italian Science Debate,
Associate Editor, Ferrari Millechili Journal,

Mike Ciavarella's picture

Journal of Solids and Structures

Volume 37, Issues 46-47
20 November 2000,
Pages 7071-7091

doi:10.1016/S0020-7683(99)00328-5 | How to Cite or Link Using DOI
Copyright © 2000 Elsevier Science Ltd. All rights reserved.

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Frictionless indentation of dissimilar elastic–plastic spheres

Sinisa Dj Mesarovica and Norman A Flecknext termCorresponding Author Contact Information, E-mail The Corresponding Author, b

a Department of Material Science and
Engineering, University of Virginia, Charlottesville VA 22903, USA

b Department of Engineering, Cambridge
University, Trumpington Street, Cambridge CB2 1PZ, UK

Received 30 May 1999.

Available online 6 September 2000.


A finite element study is performed on
the frictionless normal contact of elastic–plastic spheres and rigid
spheres. The effects of elasticity, strain hardening rate, relative size
of the spheres and their relative yield strength are explored.
Indentation maps are constructed, taking as axes the contact size and
yield strain, for a wide range of geometries. These show the competing
regimes of deformation mechanism: elastic, elastic–plastic, fully
plastic similarity and finite deformation regime. The boundaries of the
regimes depend upon the degree of strain hardening, relative size of the
bodies in contact and upon their relative yield strengths. The regime
of practical importance is the finite deformation regime for practical
applications such as powder compaction. The contact force–displacement
law, to be used as a part of the micromechanical constitutive model for
powder compaction, is constructed semi-empirically by scaling the
similarity contact law by a factor which depends on the relative size,
relative yield strength and the strain hardening exponent of the bodies
in contact. The accuracy of the assumption of independent contacts is
addressed for the isostatic compaction of an assembly of rigid and
deformable spheres, arranged in a B2 unit cell, based on two overlapping
simple cubic lattices. Provided that the relative density of the
compact is lower than about 0.82, the contacts deform independently.




1.1. The similarity solution

Storakers et al. (1997) and Storakers (1997) have developed a
similarity solution for the normal indentation of two viscoplastic
spheres by extending the Hill et al. (1989) solution for
the indentation of a half-space by a rigid sphere. Here, we summarise
the rate-independent version of their similarity solution. The
configuration studied by Storakers et al. (1997) is shown
in Fig. 1. Sphere 1 of radius R1
and sphere 2 of radius R2 are pressed together in a
frictionless normal indentation, so that the contact radius is a
at a total overlap of h.

Full-size image<br />
			(2K) - Opens new window
Full-size image (2K)

1. Geometry of contact between two spherical particles with radii R1
and R2. A contact radius a is generated for a
total overlap h.

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The following simplifying assumptions
are introduced to obtain a previous termself-similarnext term

1. The two spheres are composed of
rigid-plastic, power-law-solids in accordance with J2 flow
theory. In uniaxial tension, the stress σ is related to the strain var<br />
epsilon according to


σ=σivar<br />
epsilon1/m, i=1,2,

where sphere 1 has a reference strength σ1 and sphere 2 has a
reference strength σ2. Both solids have the same strain
hardening exponent (1less-than-or-equals, slantmless-than-or-equals, slant∞).

2. The contact radius is assumed to be
sufficiently small compared to the radius of each sphere that each
sphere can be treated as a semi-infinite half-space.

Strains and deformations are small, and the spherical profile of the
bodies in contact is approximated by a paraboloid of revolution. Then,
if the normal displacement of sphere 1 within the contact patch is u1
at a radius r, and the corresponding normal displacement of
sphere 2 is u2, conformity of the two surfaces within
the contact dictates that



with hmuch less-thanR1
and hmuch<br />

these restrictions, the indentation solution has the property of previous termselfnext term-similarity,
i.e., the geometry, stress and strain fields at any stage of
indentation can be expressed in terms of an invariant solution.
Moreover, the solution to the problem of contact between spherical
bodies is a generalisation of the solution for the contact between a
rigid sphere and a semi-infinite solid, and is obtained from the latter
by appropriate scaling. The method can be generalised to include
rate-dependent solids with a response describable by a power law–creep
law. The solution for the indentation of a semi-infinite solid by a
rigid sphere is provided by Hill and Bower and Storakers et al. (1997) for a
power-law creeping solid, and by Biwa and Storakers (1995) for a J2
flow theory solid.

The scaling law, relating the indentation of
the spheres to the indentation of a half-space by a rigid ball,
generalises the one used in elastic Hertzian contact. An equivalent
radius R0 suffices to describe a given geometry, and
an equivalent strength σe describes the combined strengths of
the two spheres,



average pressure is related to the contact radius a by the
power-law relation



and the
contact area is proportional to the indentation depth,



where the
constants c2(m) and k(m) depend
on m, but are independent of the indentation depth, and of the
diameters and strengths of the bodies in contact. Biwa and Storakers (1995)
tabulated c2(m) and k(m). Their
finite element formulation is based on the assumption of previous termselfnext term-similarity
and are essentially single-step solutions, where the history dependence
is replaced by a spatial (radial) dependence.

Relations (1.4)
and (1.5) imply that the indentation force depends upon the indentation
depth h according to






the present article, we use the spherical, rather than the parabolic
shape of the bodies in contact. The differences in the profiles of a
sphere and a paraboloid with the same curvature at the apex become
significant only for large contacts a/R>0.4. Changes in
the indentation regimes which we observe occur at smaller contacts
(0.1<a/R<0.3), so that our results are not affected
by the difference between the spherical and the parabolic shape.

and strain distributions in Storakers,
B., Biwa, S. and Larsson, P.-L., 1997. Similarity analysis of inelastic
contact. Int. J. Solids Struct. 34 24, pp. 3061–3083
              3061–3083      end_of_the_skype_highlighting
. Article
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Storakers et al. (1997) previous termself-similarnext term
solution for the contact between the power-law, incompressible,
rigid-plastic spheres are, apart from scaling (1.3), identical to the
solution for the indentation of a half-space. This can be contrasted
with the Sternberg and Rosenthal (1952)
singular solution for anti-diametric concentrated load on a linear
elastic compressible sphere. Whereas in the limit of infinite radius,
the Sternberg and Rosenthal solution does reduce to the Boussinesq
solution for half-space, the stress distributions for finite radii have a
different singularity from the Boussinesq distribution.


Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
Editor, Italian Science Debate,
Associate Editor, Ferrari Millechili Journal,

Mike Ciavarella's picture

In general, it is difficult to mention the best papers, but I find these for example


You are entitled to access the full text of this document

Observing ideal “self-similar crack growth
in experiments

Engineering Fracture Mechanics, Volume
73, Issue 18
, December 2006, Pages 2748-2755
Xia, Vijaya B. Chalivendra, Ares J. Rosakis

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Self-similar crack expansion method for
analysis of
in an infinite medium or semi-infinite medium

& Structures
, Volume 74, Issue 3, January 2000, Pages

Yonglin Xu, Brian Moran, Ted Belytschko

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Self-similar analysis of plasticity-induced
closure of small fatigue cracks

Journal of the Mechanics and
Physics of Solids
, Volume 49, Issue 2, February 2001, Pages

L. R. F. Rose, C. H. Wang

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In particular for rubber, I find these

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Mesoscopic simulation of dynamic crack
propagation in

Polymer, Volume 43,
Issue 2
, January 2002, Pages 395-401
G. Heinrich,
J. Struve, G. Gerber

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A literature survey on fatigue analysis
approaches for

International Journal of Fatigue, Volume
24, Issue 9
, September 2002, Pages 949-961
W. V.
Mars, A. Fatemi

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Physics of fracture and mechanics of self-affine cracks
Fracture Mechanics
, Volume 57, Issues 2-3, May-June 1997,
Pages 135-203
Alexander S. Balankin

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Any useful?

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
Editor, Italian Science Debate,
Associate Editor, Ferrari Millechili Journal,

Mike Ciavarella's picture

A small correction:  when I said you need a selfsimilar profile to have a selfsimilar solution, I meant not necessarily a cone, but also a sphere.  More precisely, a parabolae.  The parabolae is self.similar as Fleck paper and the Storakers previous ones show. This is why Fleck says

The differences in the profiles of a
sphere and a paraboloid with the same curvature at the apex become
significant only for large contacts a/R>0.4.
Changes in
the indentation regimes which we observe occur at smaller contacts
(0.1<a/R<0.3), so that our results are not
by the difference between the spherical and the parabolic shape. 

You also need a power law material, as otherwise, you loose self-similarity of the solution.

But with power laws you can do a lot, including creep etc.

For example, the approximate solution for indentation of a J2 power law Hertz indenter is very simple, and reported in Johnsons' book.  I am just using it rigth now for rough contact, when you have a statistical distribution of them, to show that many "surprising" results of complicated atomistic simulations done with brute force in USA (Mark Robbins's group) can be explained quite simply with 2-3 simple equations...


Probably the same is true for complicated brute force atomistic simulations of rubber-like materials  :)

But to converge into something precise, what exactly are you and Oscar trying to do ?  Two separate issues?


Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
Editor, Italian Science Debate,
Associate Editor, Ferrari Millechili Journal,

Mike Ciavarella's picture

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
Editor, Italian Science Debate,
Associate Editor, Ferrari Millechili Journal,

Oscar Lopez-Pamies's picture

Hi Mike,

Sorry for being disconnected for a few days (I was out of town).

Here is a clarification: what I mean by cavitation is the sudden growth into finite sizes of very tiny defects that are naturally present in solids. This sort of “instability” happens when the solid is subjected to loading conditions with sufficiently large triaxialities.

Now, when an elastomer is reinforced with stiff particles (which is usually the case for most applications), the stresses near (not at the particle/matrix interface) the reinforcing particles often reach high values of hydrostatic stress. That is why cavitation occurs near stiff particles. Another different “failure” mechanism in these material systems is particle/matrix detachment. This is due to the fact that the bonding between the elastomer and the particles is not sufficiently strong. These 2 different failure mechanisms in elastomeric solids have been investigated by Gent and co-workers in the series of papers that I sent you.

I am yet to become more familiar with how cavitation phenomena enter in indentation problems and materials with cracks.  I have not had the chance to carefully read the many papers that you have suggested, but I will do so shortly.

Mike Ciavarella's picture

On the stress concentration around a hole in
a half-plane subject to moving contact loads

Journal of Solids and Structures
, Volume 43, Issue 13, June
, Pages 3895-3904
L. Afferrante, M. Ciavarella, G.
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 Reduced dependence on loading parameters in
almost conforming contacts

International Journal of
Mechanical Sciences
, Volume 48, Issue 9, September 2006,
Pages 917-925
M. Ciavarella, A. Baldini, J.R. Barber, A.
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of yield and cyclic plasticity around inclusions
M Ciavarella - The Journal of Strain Analysis for
Engineering …, 2000 - Prof Eng Publishing


see if they help at all....



Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
Editor, Italian Science Debate,
Associate Editor, Ferrari Millechili Journal,

Oscar Lopez-Pamies's picture


Thanks for the papers. They do help since we are right now carrying out a systematic study of fields around stiff inclusions.


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