## Abstract

We revisit the classic problem of elastic cavitation within the framework of stochastic elasticity. For the deterministic elastic problem, involving homogeneous isotropic incompressible hyperelastic spheres under radially symmetric tension, there is a critical dead-load traction at which cavitation can occur for some materials. In addition to the well-known case of stable cavitation post-bifurcation at the critical dead load, we show the existence of unstable snap cavitation for some isotropic materials satisfying Baker-Ericksen inequalities. For the stochastic problem, we derive the probability distribution of the deformations after bifurcation. In this case, we find that, due to the probabilistic nature of the material parameters, there is always a competition between the stable and unstable states. Therefore, at a critical load, stable or unstable cavitation occurs with a given probability, and there is also a probability that the cavity may form under smaller or greater loads than the expected critical value. We refer to these phenomena as ‘likely cavitation’. Moreover, we provide examples of homogeneous isotropic incompressible materials exhibiting stable or unstable cavitation together with their stochastic equivalent.

## Keywords

Stochastic hyperelastic models Stable or unstable cavitation Isotropic incompressible spheres Baker-Ericksen inequalities Dead-load traction Probability## Mathematics Subject Classification

74B20 33B15 94A15## 1 Introduction

Experiments carried out by Gent and Lindley in 1958 [13], on rubber cylinders, revealed that some materials can rupture under relatively small tensile dead loads by opening an internal cavity. Following this work, the theoretical analysis of Ball (1982) [6] provided an explanation for the formation of a spherical cavity at the centre of a sphere of isotropic hyperelastic incompressible material in radially symmetric tension under prescribed surface displacements or dead loads. There, the word ‘cavitation’ was used to describe such void-formation within a solid by analogy to the similar phenomenon observed in fluids. As cavitation in solids is an inherently nonlinear mechanical effect, not captured by the linear elasticity theory, many different studies have been devoted to the modelling of this effect within the finite elasticity framework. For instance: spheres of particular homogeneous isotropic incompressible materials were discussed in [9]; homogeneous anisotropic spheres with transverse isotropy about the radial direction were examined in [2, 31, 41]; concentric homogeneous spheres of different hyperelastic material were analysed in [18, 42, 46]; non-spherical cavities were investigated in [21]; cavities with non-zero pressure were presented in [47]; cavitation in an elastic membrane was studied in [55]; the homogenisation problem of nonlinear elastic materials was treated in [26, 27]; growth-induced cavitation in nonlinearly elastic solids was explored in [15, 30, 40]. Recent experimental results on the onset, healing and growth of cavities in elastomers were reported in [43, 44]. For many other results on cavitation in solids, we refer to the review articles [11, 12, 19], focusing on rubberlike materials, and the references therein.

The present work focuses on the phenomenon of “cavitation” contained within the theoretical context of finite elastostatics. Finite elasticity theory covers the simplest case where internal forces only depend on the current deformation of the material and not on its history, and is based on average data values. Within this framework, *hyperelastic materials* are the class of material models described by a strain-energy function characterised by a set of *deterministic* model parameters. In addition, for solid elastic materials, uncertainties in the observational data generally arise from the inherent variation in material properties and testing protocols [7, 10, 20, 38]. In view of these uncertainties, recently, stochastic representations of isotropic incompressible hyperelastic materials characterised by a stochastic strain-energy function, for which the model parameters are random variables following standard probability laws, were proposed in [51], while compressible versions of these models were constructed in [52]. Ogden-type stochastic strain-energy functions were calibrated to experimental data for rubber and soft tissue materials in [36, 53], and anisotropic stochastic models with the model parameters as spatially-dependent random field variables were calibrated to vascular tissue data in [54]. These models employ the maximum entropy principle for a discrete probability distribution introduced by Jaynes (1957) [22, 23, 24] and based on the notion of entropy (or uncertainty) defined by Shannon (1948) [45, 50]. Such models can be useful for stochastic finite element implementations [3, 4, 16, 17].

For stochastic hyperelastic models, the immediate question is: *what is the influence of the random model parameters on the predicted nonlinear elastic responses?* This question was previously considered by us in [37], for the stochastic Rivlin cube, and in [35], for the symmetric inflation of internally pressurised stochastic spherical shells and tubes. These idealised problems illustrate some important effects on the likely elastic responses of stochastic hyperelastic materials under large strains.

Here, we address this question by employing a similar approach as in [35, 37] to revisit, in the context of stochastic elasticity, the cavitation problem of incompressible spheres of stochastic homogeneous isotropic hyperelastic materials under uniform radial tensile dead loads. Moreover, for all homogeneous isotropic hyperelastic models considered so far in the literature, cavitation appears as a supercritical bifurcation, where typically, after bifurcation, the cavity radius monotonically increases as the applied load increases (see, e.g., [9]). However, as we demonstrate here, the usual restriction that a material satisfies the Baker-Ericksen (BE) inequalities [5] is not sufficient to exclude the possibility of a subcritical bifurcation. In this case, one expects a *snap cavitation* for which there is a jump in the radius of the cavity immediately after bifurcation. Indeed, we obtain the general conditions under which a cavitation can appear through a supercritical or subcritical bifurcation and construct explicitly, for the first time, examples of isotropic incompressible hyperelastic models that exhibit snap cavitation. The stochastic version of these models are then explored. In this case, we find that, due to the probabilistic nature of the model parameters, supercritical or subcritical bifurcation occurs with a given probability, and there is also a probability that the cavity may form under smaller or greater loads that the expected critical value. We refer to these phenomena as ‘likely cavitation’.

We begin, in Sect. 2, with a detailed presentation of the stochastic elastic framework. Then, in Sect. 3, for the stochastic sphere, after we review the elastic solution to the cavitation problem under uniformly applied tensile dead load, we recast the problem in the stochastic setting, and find the probabilistic solution. Concluding remarks are provided in Sect. 4.

## 2 Stochastic Isotropic Hyperelastic Models

*deterministic*model parameters [14, 39, 56]. In contrast, a stochastic homogeneous hyperelastic model is defined by a stochastic strain-energy function, for which the model parameters are

*random variables*that satisfy standard probability laws [36, 51, 52, 53]. In this case, each model parameter is described in terms of its

*mean value*and its

*variance*, which contains information about the range of values about the mean value. While it is rarely possible if ever to obtain complete information about a random quantity in an elastic sample of material, the partial information provided by the mean value and the variance is the most commonly used in many practical applications [8, 20, 29]. Here, we combine finite elasticity and information theory, and rely on the following general hypotheses [35, 36, 37]:

- (A1)
Material objectivity: The principle of material objectivity (frame indifference) states that constitutive equations must be invariant under changes of frame of reference. It requires that the scalar strain-energy function, \(W=W(\textbf{F})\), depending only on the deformation gradient \(\textbf{F}\), with respect to the reference configuration, is unaffected by a superimposed rigid-body transformation (which involves a change of position) after deformation, i.e., \(W(\textbf{R}^{T}\textbf{F})=W(\textbf{F})\), where \(\textbf{R}\in SO(3)\) is a proper orthogonal tensor (rotation). Material objectivity is guaranteed by considering strain-energy functions defined in terms of invariants.

- (A2)
Material isotropy: The principle of isotropy requires that the strain-energy function is unaffected by a superimposed rigid-body transformation prior to deformation, i.e., \(W(\textbf{F}\textbf{Q})=W( \textbf{F})\), where \(\textbf{Q}\in SO(3)\). For isotropic materials, the strain-energy function is a symmetric function of the principal stretches \(\{\lambda _{i}\}_{i=1,2,3}\) of \(\textbf{F}\), i.e., \(W(\textbf{F})=\mathcal{W}(\lambda _{1},\lambda _{2},\lambda _{3})\).

- (A3)Baker-Ericksen inequalities: In addition to the fundamental principles of objectivity and material symmetry, in order for the behaviour of a hyperelastic material to be physically realistic, there are some universally accepted constraints on the constitutive equations. Specifically, for a hyperelastic body, the Baker-Ericksen (BE) inequalities, which state that
*the greater principal (Cauchy) stress occurs in the direction of the greater principal stretch*, are [5]:where \(\{\lambda _{i}\}_{i=1,2,3}\) and \(\{T_{i}\}_{i=1,2,3}\) denote the principal stretches and the principal Cauchy stresses, respectively, and “≥” replaces the strict inequality “>” if any two principal stretches are equal [5, 28]. The BE inequalities (1) take the equivalent form$$ (T_{i}-T_{j} ) (\lambda _{i}- \lambda _{j} )>0 \quad \mbox{if } \lambda _{i}\neq \lambda _{j},\quad i,j=1,2,3, $$(1)where the strict inequality “>” is replaced by “≥” if any two principal stretches are equal.$$ \biggl(\lambda _{1}\frac{\partial \mathcal{W}}{\partial \lambda _{1}}- \lambda _{2}\frac{\partial \mathcal{W}}{\partial \lambda _{2}} \biggr) (\lambda _{1}-\lambda _{2} )>0\quad \mbox{if } \lambda _{i}\neq \lambda _{j},\quad i,j=1,2,3, $$(2) - (A4)
Finite mean and variance for the random shear modulus: We assume that, for any given deformation, the random shear modulus, \(\mu \), and its inverse, \(1/\mu \), are second-order random variables, i.e., they have finite mean value and finite variance [51, 52, 53].

As it is well known, the deformation of an homogeneous isotropic hyperelastic material under uniaxial tension is a simple extension in the direction of the tensile force if and only if the BE inequalities hold [28]. Under these conditions, the shear modulus is positive, but the individual coefficients may be either positive or negative, allowing for some interesting nonlinear elastic effects to be captured (see [32, 33, 34, 37] and the references therein). In particular, in the present paper, the initiation of either stable or unstable snap cavitation in a homogeneous isotropic sphere will be presented.

Our aim here is to analyse the radially symmetric finite deformations of a sphere of stochastic hyperelastic material defined by (3), under tension, when subject to prescribed surface dead loads applied uniformly in the radial direction. One can view the stochastic sphere as an ensemble (or population) of spheres, where each sphere has the same initial radius and is made from a homogeneous isotropic incompressible hyperelastic material, with the elastic parameters not known with certainty, but drawn from known probability distributions. Then, for every hyperelastic sphere, the finite elasticity theory applies. For the stochastic hyperelastic body, the question is: *what is the probability distribution of stable radially symmetric deformation under a given surface dead load?*

## 3 Incompressible Spheres

In this section, we consider a sphere of stochastic incompressible hyperelastic material described by (3), subject to a radially symmetric deformation, caused by the sole action of a given radial tensile dead load. As for the deterministic elastic sphere [6], we obtain conditions on the constitutive law, such that, setting the internal pressure equal to zero, where the radius tends to zero, the required external dead load is finite, and therefore cavitation occurs. We further analyse the stability of the cavitated solution, and distinguish between supercritical cavitation, where the cavity radius monotonically increases as the dead load increases, and subcritical (snap) cavitation, with a sudden jump to a finite internal radius immediately after initiation. To the best of our knowledge, in the deterministic elastic case, the onset of snap cavitation in a homogeneous isotropic sphere has not been discussed before. Therefore, we start our analysis in the deterministic elastic context before extending it to the stochastic case.

Before considering the stochastic setting, we briefly revisit the deterministic elastic case.

### 3.1 Deterministic Elastic Spheres

Another theoretical possibility is that the bifurcation could be subcritical (i.e., the cavitated solution exists locally for values less than \(P_{0}\) and is unstable). One would then expect an unstable *snap cavitation* with a sudden jump to a cavitated solution with a finite internal radius. This subcritical behaviour of the homogeneous isotropic elastic sphere has not been explicitly demonstrated in the literature before. Here, we show that, depending on the model parameters, the family of materials (24) can exhibit both behaviours. General conditions for a given material to exhibit either a subcritical or supercritical bifurcation are provided in the Appendix.

### 3.2 Stochastic Elastic Spheres

- (a)
In Fig. 6(a), \(b=0\) in (8), and the random variable \(R_{1}=\mu _{1}/\mu \) is drawn from a Beta distribution with \(\xi _{1}=287\) and \(\xi _{2}=36\). In this case, \(\underline{\mu }_{1}=3.6<5.4=4\underline{\mu }/3\), and stable cavitation, with supercritical bifurcation after the spherical cavity opens, is expected.

- (b)
In Fig. 6(b), \(b=-3\) in (8), and the random variable \(R_{1}=(\mu _{1}+3)/(\mu +6)\) draws its values from a Beta distribution with \(\xi _{1}=325\) and \(\xi _{2}=10\). Thus, \(4\underline{\mu}/3=5.4<\underline{\mu}_{1}=6.75<8.1=2\underline{\mu}\), and unstable cavitation, with subcritical bifurcation after the spherical cavity forms, is expected.

For the numerical examples shown in Fig. 6 also, the critical dead load is \(P_{0}=4\mu -3\mu _{1}/2\), as given by (39), with \(\mu \) and \(\mu _{1}\) following probability distributions. In each case, the expectation is that the onset of cavitation occurs at the mean value \(\underline{P_{0}}=4\underline{\mu }-3\underline{\mu }_{1}/2\), found at the intersection of the dashed black line with the horizontal axis. However, there is a chance that cavity can form under smaller or greater critical loads that the expected load value, as shown by the coloured interval about the mean value along the horizontal axis.

To summarise, for a stochastic elastic sphere under uniform tensile dead load, we obtain the probabilities of stable or unstable cavitation, given that the material parameters are generated from known probability density functions. In the deterministic elastic case, there is a single critical parameter value that strictly separates the cases where the initiation of either stable or unstable cavitation occurs. By contrast, in the stochastic case, there is a probabilistic interval, containing the deterministic critical value, where there is always a competition between the stable and unstable states in the sense that both have a quantifiable chance to be found. For the onset of cavitation, there is also a probabilistic interval where a cavity may form, with a given probability, under smaller or greater loads that the expected critical value.

## 4 Conclusion

This work is motivated by the fact that a crucial part in assessing the physical properties of many solid materials is to quantify the uncertainties in their mechanical responses, which cannot be ignored. In particular, the idealised problem of the formation of a spherical cavity at the centre of a solid sphere illustrates some important effects on the likely elastic responses of stochastic hyperelastic materials under large strains.

For homogeneous isotropic incompressible spheres of stochastic hyperelastic material, subject to radial tensile dead loads applied uniformly on the sphere surface, we examined the possible radially symmetric deformations and determined which of these deformations are stable. Homogeneous stochastic hyperelastic material models satisfying certain theoretical assumptions were recently introduced to capture the dispersion in experimental data in addition to the traditional mean-data values [36, 53].

For the deterministic elastic problem, where the model parameters are single-valued constants, non-trivial deformations, whereby a spherical cavity forms at the centre of the sphere, are possible for some classes of materials when the applied tensile dead loads are sufficiently large [6]. In some materials, cavitation is stable, in the sense that the cavity radius monotonically increases as the applied dead load increases [9]. Here, we showed that a sudden jump in the cavity opening, causing unstable snap cavitation, at the critical dead load can also occur in a homogeneous isotropic incompressible sphere, provided that the material satisfies Baker-Ericksen inequalities. If such a material could be found, a sphere made of this material would suddenly increase its volume at a critical load and show some form of hysteresis as the load is removed.

In the stochastic case, the probabilistic nature of the solution reflects the probability in the constitutive law, and bifurcation and stability can be quantified in terms of probabilities. By contrast to the deterministic elastic problem, where deterministic critical parameter values strictly separate the cases where either the stable or unstable cavitation occurs, for the stochastic problem, we obtained probabilistic intervals where both states have a quantifiable chance to exist. For the onset of cavitation, there is a probabilistic interval where the cavity may form, with a given probability, under smaller or greater loads that the expected critical value.

As a direct application of our approach, one could consider the cavitation of an inhomogeneous sphere made of concentric homogeneous spheres of different stochastic material, similar to the concentric homogeneous spheres of deterministic elastic material treated in [18] and [46]. Such composite spheres would require comparing both ensemble and spatial averages.

## Notes

### Acknowledgements

We thank John Ball for a discussion on the corrected version of Proposition 5.2 of [6], as presented here in the Appendix. The support for Alain Goriely by the Engineering and Physical Sciences Research Council of Great Britain under research grant EP/R020205/1 is gratefully acknowledged.

## References

- 1.Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Applied Mathematics Series, vol. 55. US Government Printing Office, Washington (1964) zbMATHGoogle Scholar
- 2.Antman, S.S., Negrón-Marrero, P.V.: The remarkable nature of radially symmetric equilibrium states of aleotropic nonlinearly elastic bodies. J. Elast.
**18**, 131–164 (1987) CrossRefGoogle Scholar - 3.Arregui-Mena, J.D., Margetts, L., Mummery, P.M.: Practical application of the stochastic finite element method. Arch. Comput. Methods Eng.
**23**, 171–190 (2016) MathSciNetCrossRefGoogle Scholar - 4.Babuška, I., Tempone, R., Zouraris, G.E.: Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation. Comput. Methods Appl. Mech. Eng.
**194**, 1251–1294 (2005) ADSMathSciNetCrossRefGoogle Scholar - 5.Baker, M., Ericksen, J.L.: Inequalities restricting the form of stress-deformation relations for isotropic elastic solids and Reiner-Rivlin fluids. J. Wash. Acad. Sci.
**44**, 24–27 (1954) MathSciNetGoogle Scholar - 6.Ball, J.M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. R. Soc. A
**306**, 557–611 (1982) ADSMathSciNetCrossRefGoogle Scholar - 7.Bayes, T.: An essay towards solving a problem in the doctrine of chances. Philos. Trans.
**53**, 370–418 (1763) MathSciNetCrossRefGoogle Scholar - 8.Caylak, I., Penner, E., Dridger, A., Mahnken, R.: Stochastic hyperelastic modeling considering dependency of material parameters. Comput. Mech. (2018). https://doi.org/10.1007/s00466-018-1563-z CrossRefGoogle Scholar
- 9.Chou-Wang, M-S., Horgan, C.O.: Void nucleation and growth for a class of incompressible nonfinearly elastic materials. Int. J. Solids Struct.
**25**, 1239–1254 (1989) CrossRefGoogle Scholar - 10.Farmer, C.L.: Uncertainty quantification and optimal decisions. Proc. R. Soc. A
**473**, 20170115 (2017) ADSMathSciNetCrossRefGoogle Scholar - 11.Fond, C.: Cavitation criterion for rubber materials: a review of void-growth models. J. Polym. Sci., Part B
**39**, 2081–2096 (2001) CrossRefGoogle Scholar - 12.Gent, A.N.: Cavitation in rubber: a cautionary tale. Rubber Chem. Technol.
**63**, G49–G53 (1991) CrossRefGoogle Scholar - 13.Gent, A.N., Lindley, P.B.: Internal rupture of bonded rubber cylinders in tension. Proc. R. Soc. Lond. A
**249**, 195–205 (1959) ADSCrossRefGoogle Scholar - 14.Goriely, A.: The Mathematics and Mechanics of Biological Growth. Springer, New York (2017) CrossRefGoogle Scholar
- 15.Goriely, A., Moulton, D.E., Vandiver, R.: Elastic cavitation, tube hollowing, and differential growth in plants and biological tissues. Europhys. Lett.
**91**, 18001 (2010) ADSCrossRefGoogle Scholar - 16.Hauseux, P., Hale, J.S., Bordas, S.PS.: Accelerating Monte Carlo estimation with derivatives of high-level finite element models. Comput. Methods Appl. Mech. Eng.
**318**, 917–936 (2017) ADSMathSciNetCrossRefGoogle Scholar - 17.Hauseux, P., Hale, J.S., Cotin, S., Bordas, S.P.S.: Quantifying the uncertainty in a hyperelastic soft tissue model with stochastic parameters. Appl. Math. Model. (2018). https://doi.org/10.1016/j.apm.2018.04.021 MathSciNetCrossRefGoogle Scholar
- 18.Horgan, C.O., Pence, T.J.: Cavity formation at the center of a composite incompressible nonlinearly elastic sphere. J. Appl. Mech.
**56**, 302–308 (1989) ADSMathSciNetCrossRefGoogle Scholar - 19.Horgan, C.O., Polignone, D.A.: Cavitation in nonlinearly elastic solids: a review. Appl. Mech. Rev.
**48**, 471–485 (1995) ADSCrossRefGoogle Scholar - 20.Hughes, I., Hase, T.PA.: Measurements and Their Uncertainties: A Practical Guide to Modern Error Analysis. Oxford University Press, Oxford (2010) zbMATHGoogle Scholar
- 21.James, R.D., Spector, S.J.: The formation of filamentary voids in solids. J. Mech. Phys. Solids
**39**, 783–813 (1991) ADSMathSciNetCrossRefGoogle Scholar - 22.Jaynes, E.T.: Information theory and statistical mechanics I. Phys. Rev.
**108**, 171–190 (1957) ADSMathSciNetCrossRefGoogle Scholar - 23.Jaynes, E.T.: Information theory and statistical mechanics II. Phys. Rev.
**106**, 620–630 (1957) ADSMathSciNetCrossRefGoogle Scholar - 24.Jaynes, E.T.: Probability Theory: The Logic of Science. Cambridge University Press, Cambridge (2003) CrossRefGoogle Scholar
- 25.Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions, vol. 1, 2nd edn. John Wiley & Sons, New York (1994) zbMATHGoogle Scholar
- 26.Lopez-Pamies, O.: Onset of cavitation in compressible, isotropic, hyperelastic solids. J. Elast.
**94**, 115–145 (2009) MathSciNetCrossRefGoogle Scholar - 27.Lopez-Pamies, O., Idiart, M.I., Nakamura, T.: Cavitation in elastomeric solids: I—a defect-growth theory. J. Mech. Phys. Solids
**59**, 1464–1487 (2011) ADSMathSciNetCrossRefGoogle Scholar - 28.Marzano, M.: An interpretation of Baker-Ericksen inequalities in uniaxial deformation and stress. Meccanica
**18**, 233–235 (1983) CrossRefGoogle Scholar - 29.McCoy, J.J.: A statistical theory for predicting response of materials that possess a disordered structure. Technical report ARPA 2181, AMCMS Code 5911.21.66022, Army Materials and Mechanics Research Center, Watertown, Massachusetts (1973) Google Scholar
- 30.McMahon, J., Goriely, A.: Spontaneous cavitation in growing elastic membranes. Math. Mech. Solids
**15**, 57–77 (2010) MathSciNetCrossRefGoogle Scholar - 31.Merodio, J., Saccomandi, G.: Remarks on cavity formation in fiber-reinforced incompressible non-linearly elastic solids. Eur. J. Mech. A, Solids
**25**, 778–792 (2006) ADSMathSciNetCrossRefGoogle Scholar - 32.Mihai, L.A., Goriely, A.: Positive or negative Poynting effect? The role of adscititious inequalities in hyperelastic materials. Proc. R. Soc. A
**467**, 3633–3646 (2011) ADSMathSciNetCrossRefGoogle Scholar - 33.Mihai, L.A., Goriely, A.: Numerical simulation of shear and the Poynting effects by the finite element method: an application of the generalised empirical inequalities in non-linear elasticity. Int. J. Non-Linear Mech.
**49**, 1–14 (2013) CrossRefGoogle Scholar - 34.Mihai, L.A., Goriely, A.: How to characterize a nonlinear elastic material? A review on nonlinear constitutive parameters in isotropic finite elasticity. Proc. R. Soc. A
**473**, 20170607 (2017). https://doi.org/10.1098/rspa.2017.0607 ADSMathSciNetCrossRefGoogle Scholar - 35.Mihai, L.A., Fitt, D., Woolley, T.E., Goriely, A.: Likely equilibria of stochastic hyperelastic spherical shells and tubes (2018). arXiv:1808.02110
- 36.Mihai, L.A., Woolley, T.E., Goriely, A.: Stochastic isotropic hyperelastic materials: constitutive calibration and model selection. Proc. R. Soc. A
**474**, 20170858 (2018) ADSMathSciNetCrossRefGoogle Scholar - 37.Mihai, L.A., Woolley, T.E., Goriely, A.: Likely equilibria of the stochastic Rivlin cube. Philos. Trans. R. Soc. A (2018). https://doi.org/10.1098/rsta.2018.0068 CrossRefGoogle Scholar
- 38.Oden, J.T.: Adaptive multiscale predictive modelling. Acta Numer.
**27**, 353–450 (2018) MathSciNetCrossRefGoogle Scholar - 39.Ogden, R.W.: Non-linear Elastic Deformations, 2nd edn. Dover, New York (1997) Google Scholar
- 40.Pence, T.J., Tsai, S.J.: Bulk cavitation and the possibility of localized deformation due to surface layer swelling. J. Elast.
**87**, 161–185 (2007) MathSciNetCrossRefGoogle Scholar - 41.Polignone, D.A., Horgan, C.O.: Cavitation for incompressible anisotropic nonlinearly elastic spheres. J. Elast.
**33**, 27–65 (1993) CrossRefGoogle Scholar - 42.Polignone, D.A., Horgan, C.O.: Effects of material unisotropy and inhomogeneity on cavitation for composite incompressible anisotropic nonlinearly elastic spheres. Int. J. Solids Struct.
**30**, 3381–3416 (1993) CrossRefGoogle Scholar - 43.Poulain, X., Lefèvre, V., Lopez-Pamies, O., Ravi-Chandar, K.: Damage in elastomers: nucleation and growth of cavities, micro-cracks, and macro-cracks. Int. J. Fract.
**205**, 1–21 (2017) CrossRefGoogle Scholar - 44.Poulain, X., Lopez-Pamies, O., Ravi-Chandar, K.: Damage in elastomers: healing of internally nucleated cavities and micro-cracks. Soft Matter
**14**, 4633–4640 (2018) ADSCrossRefGoogle Scholar - 45.Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J.
**27**, 379–423, 623–659 (1948) MathSciNetCrossRefGoogle Scholar - 46.Sivaloganathan, I.: Cavitation, the incompressible limit, and material inhomogeneity. Q. Appl. Math.
**49**, 521–541 (1992) MathSciNetCrossRefGoogle Scholar - 47.Sivaloganathan, J.: On cavitation and degenerate cavitation under internal hydrostatic pressure. Proc. R. Soc. A
**455**, 3645–3664 (1999) ADSMathSciNetCrossRefGoogle Scholar - 48.Soize, C.: A nonparametric model of random uncertainties for reduced matrix models in structural dynamics. Probab. Eng. Mech.
**15**, 277–294 (2000) CrossRefGoogle Scholar - 49.Soize, C.: Maximum entropy approach for modeling random uncertainties in transient elastodynamics. J. Acoust. Soc. Am.
**109**, 1979–1996 (2001) ADSCrossRefGoogle Scholar - 50.Soni, J., Goodman, R.: A Mind at Play: How Claude Shannon Invented the Information Age. Simon & Schuster, New York (2017) zbMATHGoogle Scholar
- 51.Staber, B., Guilleminot, J.: Stochastic modeling of a class of stored energy functions for incompressible hyperelastic materials with uncertainties. C. R., Méc.
**343**, 503–514 (2015) CrossRefGoogle Scholar - 52.Staber, B., Guilleminot, J.: Stochastic modeling of the Ogden class of stored energy functions for hyperelastic materials: the compressible case. J. Appl. Math. Mech.
**97**, 273–295 (2016) MathSciNetGoogle Scholar - 53.Staber, B., Guilleminot, J.: Stochastic hyperelastic constitutive laws and identification procedure for soft biological tissues with intrinsic variability. J. Mech. Behav. Biomed. Mater.
**65**, 743–752 (2017) CrossRefGoogle Scholar - 54.Staber, B., Guilleminot, J.: A random field model for anisotropic strain energy functions and its application for uncertainty quantification in vascular mechanics. Comput. Methods Appl. Mech. Eng.
**333**, 94–113 (2018) ADSMathSciNetCrossRefGoogle Scholar - 55.Steigmann, D.J.: Cavitation in elastic membranes—an example. J. Elast.
**28**, 277–287 (1992) MathSciNetCrossRefGoogle Scholar - 56.Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics, 3rd edn. Springer, New York (2004) CrossRefGoogle Scholar

## Copyright information

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.