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# Journal Club May 2010: Cavitation in Elastomeric Solids

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 *R*o = 1 and inner radius *R*i = *f*0^1/3, where we note that the prescribed quantity *f*0 corresponds to the initial porosity in the shell, i.e., the initial volume fraction of the cavity *f*0 = *R*i^3 /*R*o^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 *r*o = λ and inner radius *r*i = *f*0 + λ^3 – 1, so that the porosity in the deformed shell is given by

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

Expressions (1) and (2) are valid for shells with any value of initial porosity in the physical range *f*0 in [0, 1]. We now focus on the special subclass of shells where the initial porosity is taken to be vanishingly small, that is, *f*0 → 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* = *f*0 → 0+ in the undeformed configuration (*P* = 0) to finite values at some sufficiently large critical pressure *P*cr (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

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):

**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.

## Two experimental papers on cavitation

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.

## Dear Zhigang, Thank you very

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.

## Cavitation in materials of various types

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).

## Cavitation-to-fracture transition

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."

## Cavitation in fluids

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?

## Dear LG, Actually, the

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.

## Advantageous Cavitation

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?

Thanks,

Matt

## Advantages of cavitation

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.

## On a related subject, a "cavity model" exists for indentation

Quasi-static normal indentation of an

elasto-plastic half-space by a rigid sphere—II. Results

International,Journal of Solids and Structures

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

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| Related

Articles

2.

Indentation of foamed plastics

International,Journal of Mechanical Sciences

Volume 17, Issue 7,July,1975

Pages 457-460, IN5-IN6M. Wilsea, K.L. Johnson, M.F.

Ashby

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3.

The correlation of indentation experiments

Journal,of the Mechanics and Physics of Solids

Volume 18, Issue 2,April,1970

Pages 115-126K.L. Johnson

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Articles

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.

http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

Associate Editor, Ferrari Millechili Journal, http://imechanica.org/node/7878

## Indentation and the "constitutive behavior" of defects

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.

## For indentation of elastomers, which are the key papers?

I am not an expert of indentation of elastomers, but I have some "limited" knowledge of indentation.

You refer to some limited work on indentation. Which are the key papers? A random search produces

Determination of the viscoelastic properties of

elastomeric

materials by the dynamicindentationmethod…

Force Microscopy. Importance of the

IndentationDepth andReduced Tip− Sample Energy Dissipation in Tapping Mode Atomic Force

Microscopy Study of

ElastomersDetermination of the viscoelastic properties of

elastomeric

materials by the dynamicindentationmethodIndentationof freestanding circularelastomerfilmsusing spherical indenters

Viscoelastic

characterization of polymers using instrumented

indentation. II.Dynamic testing

Characterizing viscoelastic properties of thin

elastomeric

membraneThe effect of indenter geometry on the elastic response to

indentationIn

situ mechanical characterization of square microfabricated

elastomeric

membranes using an improved microindentationA Numerical and Experimental Investigation of the

Machinability of

ElastomersI do see a number of interesting applications, ranging from Human skin, to eye contact lenses, etc. etc.

Please comment...

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

Associate Editor, Ferrari Millechili Journal, http://imechanica.org/node/7878

## The work that I was

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.

## Oscar the papers I referred to seemed more appropriate then!

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:-

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 curvefor 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.

http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

Associate Editor, Ferrari Millechili Journal, http://imechanica.org/node/7878

## Thanks for the paper you send me on email. Some comments

Oscar

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

http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

Associate Editor, Ferrari Millechili Journal, http://imechanica.org/node/7878

## Mike, I believe that the

Mike,

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…

## Oscar, this conversation seems like a Babel's tower!

Oscar

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.

http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

Associate Editor, Ferrari Millechili Journal, http://imechanica.org/node/7878

## Here are some (I hope) more specific answers

Mike,

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.

## Oscar we continue to diverge! Or maybe slowly converge??

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!

InternationalJournal of Solids and Structures

Volume 45, Issue 24,

1 December 2008,

Pages 6034-6044

doi:10.1016/j.ijsolstr.2008.07.016 | How to Cite or Link Using DOI

Copyright © 2008 Elsevier Ltd All rights reserved.

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Cracks

in rubber

References and further reading

may be available for this article. To view references and further

reading you must purchase

this article.

P. Trappera and K.Y. Volokh, a,

aFaculty of Civil and

Environmental Engineering, Technion – Israel Institute of Technology,

Haifa 32000, Israel

Received 15 April 2008;

revised 6 July 2008.

Available online 31 July

2008.

Abstract

The onset of crack

propagation in rubber

is studied computationally by using the softening hyperelasticity

approach. The basic idea underlying the approach is to limit the

capability of a material model to accumulate energy without failure. The

latter is done by introducing a limiter for the strain energy density,

which results from atomic/molecular considerations and can be

interpreted as the average bond energy or the failure energy. Including

the energy limiter in a constitutive description of material it is

possible to enforce softening and, consequently, allow tracking the

onset of structural instability corresponding to the onset of material

failure. Specifically, initiation of crack

propagation is studied in the case of a thin sheet of a rubber-like

solid under the hydrostatic tension. The large deformation neo-Hookean

material model enhanced with the energy limiter is used for finding the

critical tension corresponding to the onset of static instability of the

sheet, i.e. the onset of fracture propagation. The influence of the crack

sharpness and length on the critical load is analyzed. It is found that

material is sensitive to the crack

sharpness when the shear modulus is significantly greater than the

average bond energy. The sensitivity declines when the value of the

shear modulus approaches the value of the failure energy. Roughly

speaking, softer materials are less sensitive to cracks

than more brittle materials where the brittleness is defined as a ratio

of the shear modulus to the failure energy. It is also found that the

critical tension is proportional to the inverse square root of the crack

length for more brittle materials. The latter means that the Griffith

theory based on the linearized elasticity is also applicable to softer

materials undergoing large deformations. Unfortunately, the

applicability of the Griffith theory is restricted to cracks

with equivalent sharpness only.

Keywords:Rubber; Crack;Fracture; Failure; Hyperelasticity; Softening

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## Even fatigue has the terminology you mention

InternationalJournal 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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Crack

initiation and propagation under multiaxial fatigue in a natural rubber

References and further reading

may be available for this article. To view references and further

reading you must purchase

this article.

N. Saintiera, , , 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

2005.

Abstract

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 crack

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 rubber.

Rigid

inclusions were systematically found at the crack

initiation. Depending on the type of inclusion (identified by EDS), cavitation

at the poles or decohesion are the very first damage processes observed.

Cracks

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, cracks

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; Cracknucleation; Crack

growth; Life prediction; Rubber

Article Outline

Material

and mechanical testing

procedure

and volume transport: large strain formulation

and stress tensors

energy density

Micromechanisms

of crack initiation

at crack initiation

damage initiation

Cavitation

Crack propagation mode

loading

loading

Material

rotations

torsion loading

reverse torsion loading

and static torsion loading

life prediction

of a pertinent mechanical parameter

results

results

Fig.

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

View Within Article

Fig.

2. Comparison between computed and experimental behavior.

View Within Article

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## And "crazing" is yet another terminology for formation of pores

Theoretical study of formation of pores in

elastic solids: particulate composites, rubber toughened polymers, crazing

International,Journal of Solids and Structures

Volume 39, Issue 11,June,2002

Pages 3079-3104Klaus P. Herrmann, Victor G.

Oshmyan

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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.

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## Indentation of an elastomer

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. SeeRecently, 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.

## Zhigang, you need no distinction in the case of selfsimilarity

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.

InternationalJournal 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.

Panagiotopoulos, a,

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.

Abstract

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

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## On the concept of selfsimilarity in indentation see e.g. Fleck

InternationalJournal 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

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Frictionless indentation of dissimilar elastic–plastic spheres

Sinisa Dj Mesarovica and Norman A Fleck, , 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.

Abstract

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

R1and sphere 2 of radius

R2 are pressed together in africtionless normal indentation, so that the contact radius is

aat a total overlap of

h.Full-size image (2K)

Fig.

1. Geometry of contact between two spherical particles with radii

R1and

R2. A contact radiusais generated for atotal overlap

h.View Within Article

The following simplifying assumptions

are introduced to obtain a self-similar

solution:

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 according to

(1.1)

σ=σ

i1/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 (1

m∞).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.

3.

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

u1at a radius

r, and the corresponding normal displacement ofsphere 2 is

u2, conformity of the two surfaces withinthe contact dictates that

(1.2)

with

hR1and

hR2.With

these restrictions, the indentation solution has the property of self-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, andan equivalent strength σe describes the combined strengths of

the two spheres,

(1.3)

1/

R0≡1/R1+1/R2,σe−

m≡σ1−m+σ2−m.The

average pressure is related to the contact radius

aby thepower-law relation

(1.4)

and the

contact area is proportional to the indentation depth,

(1.5)

where the

constants

c2(m) andk(m) dependon

m, but are independent of the indentation depth, and of thediameters and strengths of the bodies in contact. Biwa and Storakers (1995)

tabulated

c2(m) andk(m). Theirfinite element formulation is based on the assumption of self-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

haccording to(1.6a)

where

(1.6b)

In

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 inthe indentation regimes which we observe occur at smaller contacts

(0.1<

a/R<0.3), so that our results are not affectedby the difference between the spherical and the parabolic shape.

Stress

and strain distributions in Storakers,

B., Biwa, S. and Larsson, P.-L., 1997. Similarity analysis of inelastic

contact.

Int. J. Solids Struct.3424, pp. 3061–3083begin_of_the_skype_highlighting 3061–3083 end_of_the_skype_highlighting.

Article| PDF (1315 K)

| MathSciNet | View Record in Scopus | Cited By in Scopus (61)Storakers et al. (1997) self-similar

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.

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## On selfsimilarity and cracks there is obviously more, rubber too

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

1.

Observing ideal “self-similar” crack growth

in experiments

Engineering Fracture Mechanics,Volume,73, Issue 18

December 2006,Pages 2748-2755Kaiwen

Xia, Vijaya B. Chalivendra, Ares J. Rosakis

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2.

Self-similar crack expansion method for

analysis of

cracks in an infinite medium or semi-infinite medium

Computers,& Structures

Volume 74, Issue 3,January 2000,Pages309-317

Yonglin Xu, Brian Moran, Ted Belytschko

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3.

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,Pages401-429

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

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In particular for rubber, I find these

1.

Mesoscopic simulation of dynamic crack

propagation in

rubber materials

Polymer,Volume 43,,Issue 2

January 2002,Pages 395-401G. Heinrich,

J. Struve, G. Gerber

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Articles

2.

A literature survey on fatigue analysis

approaches for

rubber

International Journal of Fatigue,Volume,24, Issue 9

September 2002,Pages 949-961W. V.

Mars, A. Fatemi

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Articles

3.

Physics of fracture and mechanics of self-affine cracks

Engineering,Fracture Mechanics

Volume 57, Issues 2-3,May-June 1997,Pages 135-203Alexander S. Balankin

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Articles

Any useful?

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## Zhigang, to be more precise about selfsimilarity

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 notaffected

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?

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## Oscar so what is the status of this debate? We have lost spin??

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## I am back

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.

## OK, the only work I have on inclusions or cavities is this!

On the stress concentration around a hole in

a half-plane subject to moving contact loads

International,Journal of Solids and Structures

Volume 43, Issue 13,June,2006

Pages 3895-3904L. Afferrante, M. Ciavarella, G.

Demelio

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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 917-925M. Ciavarella, A. Baldini, J.R. Barber, A.

Strozzi

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Conditions

of yield and cyclic plasticity around inclusions

soton.ac.ukM[PDF]

Ciavarella- The Journal of Strain Analysis forEngineering …, 2000 - Prof Eng Publishing

see if they help at all....

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## Thanks

Mike,

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

Oscar