Glass Struct. Eng. (2020) 5:41–65https://doi.org/10.1007/s40940-019-00115-4

SI: GLASS PERFORMANCE PAPER

Pseudo-elastic cavitation model: part I—finite elementanalyses on thin silicone adhesives in façades

M. Drass · P. A. Du Bois · J. Schneider ·S. Kolling

Received: 17 March 2019 / Accepted: 9 December 2019 / Published online: 14 January 2020© The Author(s) 2020

Abstract This study investigates the structural behav-ior of adhesive bonds of glass and metal using thin,structural silicones in heavily constrained applications.This special type of connection may lead to triaxialstress conditions under axial loading, which can leadto dilatation failure due to the abrupt growth of cavi-ties (cavitation effect). Cavitation failure leads to sig-nificant stress softening and loss of stiffness; how-ever, it increases connection’s ductility. These mate-rial deformations should be considered when design-ing glass-metal connections. Therefore, a constitu-tive model is developed to account for cavitation inhyperelastic materials. The volumetric component ofthe model is equipped with a non-linear Helmholtzfree energy function that accounts for isotropic voidgrowth under hydrostatic loading. An energy cou-pling term is then added that numerically explicatesstrain energy under isochoric deformation, while alsoguaranteeing physical material behavior. The energy

M. Drass (B) · J. SchneiderInstitute of Structural Mechanics and Design, TechnischeUniversität Darmstadt, Franziska-Braun-Str. 3, 64287Darmstadt, Germanye-mail: drass@ismd.tu-darmstadt.deschneider@ismd.tu-darmstadt.de

P. A. Du BoisAlzenau, Germany

S. KollingInstitute of Mechanics and Materials, TechnischeHochschule Mittelhessen, Wiesenstraße 14, 35390 Giessen,Germanye-mail: stefan.kolling@me.thm.de

contribution is calculated internally by analysing thegeometric evolution of inherent voids. The extendedvolumetric–isochoric split enables one to numericallycalculate heavily constrained silicone joints under arbi-trary deformationmodes. Three-dimensional finite ele-ment calculations on uniaxial tension, bulge, and pan-cake tests validate the constitutive model. All exper-iments could be validated with one set of materialparameters through numerical simulations. The numer-ical calculations were robust and efficient without anyunderlying mesh dependencies.

Keywords Cavitation · Transparent structural siliconeadhesive · Poro-hyperelastic materials · Finite porosity

Abbreviations

TSSA Transparent structural silicone adhesiveUT Uniaxial tensile testBT Biaxialx tension testSPC Shear pancake testPC Pancake test

List of symbols

tr(•) Trace of argumentΨ (•) Helmholtz free energyF Deformation gradientb Left Cauchy–Green tensorb̄ Isochoric left Cauchy–Green tensor

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λi Principal stretchesεengi Engineering strainJ Relative volumeIb First principal strain invariant of bIIb Second principal strain invariant of bIIIb Third principal strain invariant of bP First Piola–Kirchhoff stress tensorσ Cauchy stress tensorp Hydrostatic stressΩ Shape functionDcav Dissipated energy due to void growthΠ Isoperimetric inequalityΘ Equivalent void growth measureμ Initial shear modulusK Initial bulk modulusκ0, κ1, κ2, κ3 Volumetric material parameters

1 Introduction

1.1 Motivation and problem statement

When constructing conventional glass façades, a criti-cal design element is the connection between the glasspane and the supporting secondary load bearing mem-bers. To avoid stress peaks in the glass, the jointmust beductile and be able to withstand environmental influ-ences (wind, rain, sun, etc.), while also being archi-tecturally appealing. In modern construction, one typ-ical example of a ductile connection is achieved bydirectly bonding glass to metal with thin silicone adhe-sives (see Fig. 1). This is also known as a laminatedconnection due to a special production process (Bedonand Santarsiero 2018). When an adhesive joint is sub-jected to an axial load, the disability of lateral strainsmay cause abrupt cavity growth in the silicone. Thisbehavior is known as cavitation (Drass et al. 2017) andunder its effects the connection experiences significantincreases in ductility before final dilation failure occurs(see Fig. 2).

At present, such thin adhesive joints may not beemployed or studied because they do not meet thegeometric requirements of the Guideline for EuropeanTechnical Approval (ETAG 002 2012). The adhesivejoint height to width ratio must be in the range of0.33 < H/B < 1 to prevent cavitation failure. In con-trast to thick silicone adhesive joints—which are typi-cally used in façade constructions—thin adhesive jointsoffer clear advantages in terms of design, rigidity, struc-

Fig. 1 Point fixing connected to a glass pane with TSSA - Per-mission of Dow Corning Europe SA (2017)

tural diversity and architectural aesthetics. So far thereare no simple numericalmodels to describe thematerialbehavior of thin silicone adhesives in façade applica-tions, so analytical models were developed in order torealistically depict the structural behavior (Descampset al. 2017). Ongoing research in the field of struc-tural silicones and MS polymers lies in creating physi-cally accurate, constitutive models of structural behav-ior (Dispersyn et al. 2017; Drass et al. 2018d) and limitstate analyses of such connections (Staudt et al. 2018;Rosendahl et al. 2018; Santarsiero et al. 2018).

The attempt to approximate the special structuralbehaviorwith classical compressible hyperelasticmate-rial models using the volumetric–isochoric split inaccordance to Flory (1961) leads to non-physicaleffects (Ehlers and Eipper 1998; Li et al. 2007). Thisfact was proved by the work of Danielsson et al.(2004) and Drass et al. (2017), who analysed multi-voided representative volume elements (RVE) underhydrostatic tension loading. A key component is thatthe structural behavior exhibits a high initial stiffnessfollowed by strong stress softening caused by exces-sive void growth. Except the constitutive models pro-posed by Danielsson et al. (2004), Li et al. (2007) andDrass et al. (2017), none of the classical volumetricHelmholtz free energy functions (Hartmann and Neff2003) are able to represent this physical phenomenon(see “AppendixB”). Hence, conventional compressiblehyperelastic material models are not suitable to repre-sent volumetric stress softening effects caused by voidgrowth. Even the Gurson model, which accounts forvoid growth in metals, cannot be applied to polymericadhesive connections (Silvestru et al. 2018).

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Pseudo-elastic cavitation model: part I 43

Simulation of Point Fixing with Silicone Adhesive

Fig. 2 Experimental and numerical results of an axially loadedpoint fixing using a compressible Neo-Hookean material model(μ = 2.66 MPa and K = 945 MPa)

Focusing on point fixing connections in modernfaçades, classical finite element models accompaniedwith a compressible hyperelastic material formulationlead to an overestimation of stresses, thus, underesti-mating the ductility of this special type of connection.This fact is illustrated in Fig. 2, which presents theexperimental and numerical results of an axially loadedpoint fixing. From Fig. 2, it is obvious that the appar-ent stress softening effects due to cavitation cannot besimulated utilizing conventional hyperelastic materialmodels. Considering these drawbacks, the design pro-cess for thin adhesive glass to metal connections infaçades design issues a challenge for structural andfaçade engineers.

1.2 State of the art

Considerable effort has been put forth analysing exper-imentally thin, structural silicone adhesives as a bond-ing material for point fixed connections (Hagl et al.2012a, b; Drass et al. 2018b). Analytical predictionmodels based on extensive experimental investiga-tions were proposed by Santarsiero et al. (2016) todescribe the tensile resistance. Regarding the numeri-cal modelling of such connections, especially the stresssoftening under constrained axial loading, only Drasset al. (2017, 2018b) proposed a nonlinear volumetricHelmholtz free energy function and a continuum dam-age formulation accounting for the degradation of thebulk modulus due to void growth. A modified versionof the model of Ogden and Roxburgh (1999) was uti-lized to calculate isochoric softening due to theMullins

effect (Ioannidou-Kati et al. 2018). Yet still, the pro-posed models lack either a robust and efficient numer-ical calculation providing mesh-independent results orare not able to account for stress softening due to cav-itation.

Pioneering experimental work focusing on the cav-itation effect in rubber-like materials was presentedby Busse (1938), Yerzley (1939) and Gent and Lindley(1959). To obtain an overview of ongoing studies onthe cavitation phenomenon in an analytical and numer-ical framework, the authors refer to Fond (2001). Dalet al. (2018) categorized the constitutive descriptionof hyperelastic materials accounting for cavitation intothree essential thematic categories. The first subsectionsplits the Helmholtz free energy function Ψ into anisochoric part Ψiso and a volumetric part Ψvol. Regard-ing volumetric Helmholtz free energy functions, thereexist only a few proposals in literature to describevolumetric deformations under a physical viewpoint(Bischoff et al. 2001; Danielsson et al. 2004; Drasset al. 2018b). The main focus in the development ofthe volumetric Helmholtz free energy functions wason an incompressible and nearly incompressible mate-rial behaviour, which ensures the element’s stabilityeven under small isochoric stiffness (Bonet and Wood2008, p. 171). The penalty function method describesa procedure where the strain energy potential is cal-culated as a sum of an incompressible material plus apenalty term enforcing the incompressibility constraintreading J = 1 (Simo and Taylor 1982). Following(Belytschko et al. 2013, p. 253), the penalty parameterK should be chosen large enough to enforce the incom-pressibility constraint, but not so large that numericalill-conditioning occurs. An overview on the volumetricHelmholtz free energy function is given in Hartmannand Neff (2003).

Since the present paper utilizes the volumetric–isochoric split to analyse cavitation, the second andthird categories in accordance to Dal et al. (2018) areonly briefly summarized. The second category ana-lytically analyses the evolution of a single void asit reaches critical stresses in an undamaged medium.Additionally, the kinematics of a macroscopic con-tinuum with an inherent void used as a representa-tive volume element were examined by Ball (1982)and Hou and Abeyaratne (1992). The third categorymakes use of homogenization schemes accounting foran incompressible medium with a given void frac-tion. Danielsson et al. (2004) captured the idea of Hou

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and Abeyaratne (1992) to determine a homogenizedHelmholtz free energy function for poro-hyperelasticmaterials. Li et al. (2007) developed a compressiblehyperelastic material law accounting for cavitation andcavity growth using a Voigt-type homogenization.

1.3 Scope and outline

The present study utilizes the Flory-type volumetric–isochoric split (Flory 1961) to develop a constitutivemodelling approach accounting for stress softening instructural silicones under triaxial loading. Stress soft-ening due to the Mullin’s effect under isochoric defor-mation modes is excluded in this paper, but part ofthe investigations of Drass et al. (2019a). Section 2provides a short overview on the pseudo-elasticityapproach, which is used to account for modelling cav-itation. In Sect. 3, the novel pseudo-elastic cavita-tion model is presented with a detailed derivation ofthe approach, including single-element tests as bench-mark tests and the algorithmic setting. The new modelprovides an extension of the classical volumetric–isochoric split by adding a coupling term Ψvol,couple.With this approach, cavitation and the correspond-ing stress softening effect due to void growth in sili-cone adhesives can be analysed numerically for heav-ily constrained axially loaded connections. To validatethe present constitutive model, Sect. 4 discusses threeexperiments from literature that are simulated withthe novel constitutive model. Here, three-dimensionalfinite element calculations are performed simulating asimple uniaxial tension test (Santarsiero et al. 2016),a biaxial tension test in terms of a bulge test (Drasset al. 2018c), and a nearly triaxial tension test (whichcan be accomplished by testing the so-called pancaketest) (Hagl et al. 2012a, b). In addition, pancake testswith two different diameters are simulated to prove theaccuracy of the model.

2 General concept of pseudo-elasticity

In the investigations of Fung (1980) on living tis-sues, it was found that the loading and unloadingpaths under cyclic loading differ significantly. Eachcan be described separately by a unique relationshipbetween stresses and strains. Since both load pathswere represented independently by two single classical

hyperelastic material models or with different mate-rial parameters for one hyperelastic material model,but the unloading path is strictly speaking inelastic,this phenomenological concept was termed pseudo-elasticity. Lazopoulos and Ogden (1998) proposed thegeneral concept of pseudo-elasticity by introducing aHelmholtz free energy function Ψ (F, η) that is depen-dent on the deformation gradient F and additionallyon an internal scalar variable η. The extension of Ψ

by a continuous or discontinuous internal variable η

additionally motivates the designation of this theory aspseudo-elasticity.

For the extended functionalΨ (F, η), the first Piola–Kirchhoff stress tensor P can be calculated through

P = ∂Ψ

∂F(F, η) + ∂Ψ

∂η(F, η)

∂η

∂F(F) (1)

assuming an elastic body, which is mapped from thereference to the current configuration by the deforma-tion gradient F. Since the internal variable dependsadditionally on F, the functional must be differenti-ated according to η, whereby η must be subsequentlydifferentiated with respect to F. In the absence of bodyforces the first and at the same time also usual equilib-rium equation in the Lagrangian description reads

∇−→X

· P = 0 (2)

where∇−→X

·(•) represents the divergence operator withrespect to the reference configuration. Lazopoulos andOgden (1998) formulated an additional equilibriumcondition derived from a variational principle (station-ary energy principle), which reads

∂Ψ

∂η(F, η) = 0. (3)

This equilibrium equation implicitly relates the internalvariableη toF and thus continuouslymodifies the strainenergy density function with progressing deformation.Based on these basic equations, Ogden and Roxburgh(1999) defined a pseudo-elastic energy function of type

Ψ (F, η) = η Ψ̂ (F) + φ (η) , (4)

where φ (η) describes a damage function to describethe Mullin’s effect. The inclusion of η and φ (η) offersthe possibility to change the formof the energy functionduring deformation.

According toDorfmann andOgden (2003), the dam-age parameter was assumed to be inactive during load-ing conditions and active for unloading with η ∈ [0, 1].

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Pseudo-elastic cavitation model: part I 45

The internal variablewas explicitly derived by insertingEq. (4) in Eq. (3), which reads

− φ′ (η) = Ψ̂ (F) . (5)

Since the damage function φ (η) is intended to deter-mine η depending on the state of deformation, φ (η)

can be chosen arbitrarily. However, Eq. (3) must befulfilled.

3 Pseudo-elastic cavitation model

A pseudo-elastic Helmholtz free energy function isderived in the following, which takes into accountthe effective stress softening caused by cavitation inrubber-like materials (Gent and Lindley 1959; Drasset al. 2018a). In contrast to the approach of Ogdenand Roxburgh (1999), in which the theory of pseudo-elasticity was used to calculate Mullin’s damage infilled rubbers, here the contribution of the strain energydue to the anisotropic geometrical evolution of inher-ent cavities in the bulk material for any deforma-tion is determined. The pseudo-elasticity approach inaccordance to Ogden and Roxburgh (1999) was delib-erately chosen to avoid multi-scale modelling (Dalet al. 2018), homogenization methods (Lopez-Pamiesand Castañeda 2007) or even the FE2 method (Feyel2003) for reasons of simplicity and time saving due tothe above-mentioned numerically complex calculationmethods.

The advantages are that the hyperelastic Helmholtzfree energy function can be explicitly described at themacro-scale, the Flory-type volumetric–isochoric splitcan be used, which is ideal for the separate determina-tion ofmaterial parameters, damage and healing effectscan be modelled without great effort and the renounce-ment of time-consuming multi-scale modelling meth-ods. Especially the description of damage and heal-ing effects under cyclic loading can be described veryclearly by this type of formulation. A detailed descrip-tion on the extension of this approach towards cyclicloading is presented in part II of this paper.

3.1 General concept

To define a macroscopic Helmholtz free energy func-tion that calculates the volumetric Helmholtz freeenergy of a porous rubber-like material under any

deformationphenomenologically, the pseudo-elasticityapproach according to Lazopoulos andOgden (1998) isused. In principle, we assume a Flory-type hyperelasticmaterial lawwith the special feature that the volumetricpart of the functional is able to describe isotropic voidgrowth. Drass et al. (2018b) presented a nonlinear vol-umetric Helmholtz free energy function (see Eq. (43)),which is capable of representing the cavitation effectand thus effective stress softening due to growing voidscaused by triaxial loading (see “Appendix A”). Hence,the strain energy density function reads

Ψ (b) = Ψiso(b̄) + Ψvol,ND (J ) . (6)

The advantages of this formulation lie in the applica-tion of the Flory-type volumetric–isochoric split, thepossibility of representing stress softening due to cav-itation and the adaptability of the volumetric formula-tion. The disadvantage are described in “Appendix B”for reasons of clarity. However, in summary the maindisadvantage lies in the fact that under isochoric loadsanomalies can be observed in the structural responsethat only arise from the application of the Flory-typesplit of the Helmholtz free energy function. Therefore,the pseudo-elastic formulation is chosen in order toavoid the disadvantage mentioned above.

Nonetheless, to take the benefits of the Flory-typehyperelastic material and the nonlinear volumetricHelmholtz free energy function Ψvol,ND, which is alsocapable of representing stress softening due to cavita-tion, an additional strain energy couple term Ψcouple isadded. This couple term expresses the amount of thestrain energy due to anisotropic geometric evolutionof a void in relation to the dissipated energy due toisotropic void growth. Keeping this in mind, the mod-ified Helmholtz free energy function reads

Ψ (b) = Ψiso(b̄) + Ψvol,ND (J ) + Ψcouple (J,Ω) .

(7)

To be more specific, the strain energy coupling term isdefined by a shape function Ω ∈ [0, 1] based on thegeometrical evolution of a deforming void. The vari-able Ω has therefore two main objectives. On the onehand, it governs the amount of dissipated strain energyDcav due to void growth, and on the other hand, it basi-cally decides whether there is a volumetric or isochoricdeformation. In the case of a purely volumetric defor-mation, Ψvol,ND alone is able to calculate the strainenergy as a result of void growth, so that the additionaldissipated strain energy Dcav due to cavitation must

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Fig. 3 Illustration of the total energy dissipation Dcav due tovoid growth

become zero. Consequently, Ω must be zero for a vol-umetric load. Looking at isochoric deformations, Ω

must assume values of > 0 to compensate for the lossstrain energy caused by pore growth, as no significantpore growth is expected under isochoric deformations.The dissipated strain energy Dcav is defined as the dif-ference between an unvoided and voided hyperelasticmaterial under hydrostatic loading. For reasons of sim-plicity, it is assumed that the dissipated strain energyreads

Dcav = Ψvol,classic − Ψvol,ND, (8)

which serves as a working hypothesis. In this context,Ψvol,classic represents the classical formulation of thevolumetric part of a Helmholtz free energy functionwhich is explicitly defined in “AppendixA”byEq. (37).

An exemplary illustration of the dissipated strainenergy due to void growth is shown in Fig. 3. Sincethe coupling term depends on the global deformationgradient F, i.e. on isochoric and volumetric deforma-tions, it can be understood as an intermediate config-uration between the volumetric and isochoric config-urations. To illustrate this, Fig. 4 shows the volumet-ric, isochoric and intermediate configuration and theassociated Helmholtz free energy functions. Since thevolumetric deformation gradient is a second order unittensor weighted with a scalar value, the order of thedecomposition of the total deformation gradient is arbi-trary. To clarify that the computation of the Helmholtzfree energies for the respective configuration is con-cerned, the undeformed and deformed material bodies(B0, BJ , B̄ and B) including a deforming cavity forthe respective configuration are shown.

To be in line with the theory of pseudo-elasticity,a function φ depending on Ω must also be added,which serves to implicitly calculate the shape func-tion Ω in terms of the state of deformation. In con-trast to the approach of Ogden and Roxburgh (1999),in which φ (Ω) was understood as a kind of damagefunction to describe the Mullin’s effect under cyclicloading of filled rubbers, φ (Ω) represents the counter-part of a damage function in this case and thus servesonly to determine the shape functionΩ . This approachis consistent with the proposed ansatz of Lazopou-los and Ogden (1998), Ogden and Roxburgh (1999)and Dorfmann and Ogden (2003). Accordingly, theentire pseudo-elastic Helmholtz free energy functioncan be written as

Ψ = Ψiso(b̄) +Ψvol,ND (J ) + ΩDcav (J ) +φ (Ω)

︸ ︷︷ ︸Ψcouple(J,Ω)

,

(9)

where now the strain energy couple termΨcouple(J,Ω)

is defined as the sum of Ω Dcav and φ (Ω).The calculation of the Cauchy stress tensor for a

Flory-type hyperelastic material is generally done by

σ general = 2

J

[∂Ψiso

∂b

(b̄) + ∂Ψvol

∂b(J )

]· b. (10)

For the stress calculation of the couple term, Ω mustalso be considered, since it depends on themacroscopicdeformation gradient F or on the left Cauchy Greentensor b respectively. Considering only Ψcouple fromEq. (9), the Cauchy stress tensor is calculated as fol-lows:

σ couple= 2

J

[∂Ψcouple

∂b(J,Ω) + ∂Ψcouple

∂Ω(J,Ω)

∂Ω

∂b(b)

]· b.

(11)

Keeping in mind that Lazopoulos and Ogden (1998)determined an additional equilibrium equation basedon equilibrium thermodynamics, which only resultsfrom the inclusion of an internal variable in the con-stitutive model, here Ω , the calculation of the Cauchystress tensor for the couple term simplifies consider-ably (Dorfmann and Ogden 2003). Since the additionalequilibrium equation

∂Ψcouple

∂Ω(J,Ω)

!= 0 (12)

must be fulfilled according to Lazopoulos and Ogden(1998), the Cauchy stress tensor is calculated after

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Pseudo-elastic cavitation model: part I 47

Fig. 4 Behavior of initiallyspherical void undervolumetric, isochoric andintermediate configurations

some mathematical manipulations with respect tothe pseudo-elastic Helmholtz free energy function ofEq. (9) to

σ =[2

J

∂Ψiso(b̄)

∂b̄· b̄

]dev

+[∂Ψvol

∂ J(J,Ω) + ∂Ψcouple

∂ J(J,Ω)

]

︸ ︷︷ ︸=p

I(13)

where p represents in general the hydrostatic pressure.Up to this point the shape function is still unknown,

but the constraint equation Eq. (12) implicitly definesthe variable Ω . Hence, the pseudo-elastic Helmholtzfree energy function is now differentiated with respectto Ω resulting in

∂Ψ

∂Ω= Dcav + φ′ (Ω) = 0. (14)

This equation is used in the following to derive theshape functionΩ . However, additional constraint equa-tions for φ must be defined beforehand. An additionalconstraint equation according to Ogden and Roxburgh(1999), which must be fulfilled to determine the shapefunction Ω , reads

φ (Ω)∣∣Ω=0 = 0. (15)

If Ω is inactive, i.e. Ω = 0, then the Helmholtz freeenergy describing isotropic void growth according toEq. (6) must come out. A third condition is describedby

φ′′ (Ω) ≤ 0, (16)

in particular, intended to provide a softer structuralresponse on the unloading path due to the Mullin’seffect. Since thepresentmaterialmodel initially ignoresmaterial softening due to cyclic loading, the condition

of Eq. (16) is not further considered. A self-evidentrequirement, which must also be met, is that the shapefunction is within the limits of Ω ∈ [0, 1].

Fulfilling the constraint equations, the choice ofφ (Ω) is arbitrary. In this case, φ (Ω) was chosen insuch a way that φ′ (Ω) reads

φ′ (Ω) = − tanh−1 (Ω − Θ (1 − Π))

ΘΠ, (17)

where Θ ∈ [0, 1] and Π ∈ [0, 1] are additional vari-ables that distinguish between volumetric and isochoricdeformations or describe the aspect ratio between adeformed to an undeformed cavity. The exact deriva-tion of both variables is described in the following sec-tions.

Returning to the shape function, it can be explicitlydescribed by inserting Eq. (17) into Eq. (14), whichleads to

Ω = Θ (1 − Π) + tanh (Dcav Θ Π) . (18)

SinceΘ andΠ are limited in their functional values, butthe dissipated energyDcav can take any positive valuesdepending on the applied deformation, the value rangeof the trigonometric function is additionally given withW = {tanh (•) ∈ [0, 1] | • ≥ 0 }. Looking at the rightside of Eq. (18), the first term aims for the value oneand is monotonically increasing under the condition ofincreasing isochoric deformation, whereas the secondterm becomes zero and monotonically decreases. Thisshows that Ω lies in the range of Ω ∈ [0, 1].

As the shape function is now fully described, theenergy functionφ (Ω) canbedeterminedby integratingEq. (17) with respect to Ω , which leads to

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φ (Ω) =1

2

− ln(1 − (Ω − Θ (1 − Π))2

)

ΘΠ

+ (Θ (1 − Π) − Ω) tanh−1 (Ω − Θ (1 − Π))

ΘΠ.

(19)

On the basis of this procedure, the first condition that∂Ψcouple/∂Ω must be zero is directly fulfilled, sincethis condition has just determined the shape function.Verifying the second constraint equation according toEq. (15) by inserting Ω = 0 gives

φ (Ω)∣∣Ω=0 =1

2

− ln(1 − (Θ (1 − Π))2

)

ΘΠ

− tanh−1 (Θ (1 − Π))Θ (1 − Π)

ΘΠ

(20)

To meet this condition, i.e. to obtain a constitutivemodel that represents isotropic void growth when Ω

is inactive, a new condition results for Θ , which gov-erns the shape function Ω . The new condition to fulfilEq. (15) is that the parameter Θ must be zero at purelyhydrostatic load for which φ achieves the limit valuezero. In contrast, it should take the value one for iso-choric loads, so that this parameter can be describedusing a Heaviside step function, for example. Since thegeneral concept of the Flory-type pseudo-elastic mate-rial model was presented, the parameters Ω and Θ ofthe shape function are defined more precisely in thefollowing.

3.2 Isoperimetric inequality

As described in the general concept of the pseudo-elastic cavitation model, there is a shape function thatgoverns the strain energy due to the anisotropic geomet-ric development of a deforming void in relation to thedissipated strain energy due to isotropic void growth.For this purpose, a scalar quantity must be found whichdefines the geometric evolution of a deforming cavityvia a shape parameter within the limits zero and one. Todo so, one of the fundamental problems in the classi-cal calculus of variations is employed, which is to findthe geometric figure with maximum area at a givenperimeter (Zwierzynski 2016). The proposed problemcan be described by formulating a so-called isoperi-metric inequality

ϑ = 4π As

L2s

≤ 1 (21)

defined in the Euclidean space R2. The isoperimetric

inequality holds if a simple closed curve s of lengthLs , enclosing a region of the area As , is a circle. Thus,ϑ describes the deviation between a deformed closedcurve s and a circle. The classical formulation of theinequality ϑ is defined inR2 with ϑ ∈ [0, 1]. To obtaina stricter criterion describing the deviation between adeformed closed curve s and a circle, the inequality canbe provided with an exponent n, which reads then

ϑ =(4π As

L2s

)n

≤ 1 with n ∈ N. (22)

N represents the set of natural numbers. Since thepresent phenomenological pseudo-elastic approach isintended to model cavitation in rubbers and rubber-likematerials, it is decisive to describe the geometrical evo-lution of an initially spherical cavity with one criterion.For this purpose, the isoperimetric inequality is con-verted into three-dimensional space inwhichϑ is evalu-ated in three unique geometrical planes. Since the shapeof the spherical cavity changes only due to an externaldeformation, it is obvious to describe the isoperimetricinequality as a function of a macroscopic deformationmeasure. Therefore the eigenvaluesλi with i ∈ [1, 2, 3]of the left stretch tensor

v = √b =

3∑

i=1

λi−→n i ⊗ −→n i , (23)

are used to evaluate the isoperimetric inequality. Thegeometrical planes, in which ϑ is evaluated, areuniquely defined by the cross product of the eigen-vectors −→n of the left stretch tensor v, which reads−→P i− j = −→n i × −→n j with i = j ∈ [1, 2, 3]. Duplicateplanes are excluded in the following considerations.For reasons of simplicity, a geometric mean value iscalculated to evaluate the isoperimetric inequality forthe three geometric planes (ϑ1−2, ϑ2−3 and ϑ3−1), i.e.

Π = 3√

ϑ1−2 ϑ2−3 ϑ3−1. (24)

This simplification is permissible, since for the presentpseudo-elastic Helmholtz free energy function, onthe one hand, the strain energy contribution throughanisotropic pore growth is considered purely phe-nomenologically and, on the other hand, a smearedisotropic material behavior is assumed. Both assump-tions are merely working hypotheses that can beimproved by integrating structural tensors to describeanisotropic material behavior.

As an example, the isoperimetric inequality is deter-mined for the

−→P 1−2 geometrical plane. Within this

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Pseudo-elastic cavitation model: part I 49

plane, the surface area of a cut void simply readsAs,1−2 = πλ1λ2, whereas the perimeter can onlybe calculated using an approximation. Utilizing theapproximate solution in accordance toRamanujan (Vil-larino 2005), the perimeter can be calculated with

Ls,1−2 ≈π (λ1 + λ2)

+ 3π(λ1 − λ2)2

10 (λ1 + λ2) +√

λ21 + 14λ1λ2 + λ22

(25)

To obtain the values for As and Ls for all other princi-pal stretch planes (1–2, 2–3 and 3–1), the indices of theprincipal axis have to be interchanged cyclically with-out considering duplicates. Consequently, the isoperi-metric inequality can easily be calculated with respectto Eq. (22) depending on the desired principal stretchplane (1–2, 2–3 and 3–1).

To better understand the behavior of the isoperimet-ric inequalityΠ , a three-dimensional representation ofit is shown in Fig. 5 depending on the exponent n. Inaddition, deformationpaths according to uniaxial, biax-ial and hydrostatic loading conditions are added in thisplot to clearly show the dependency between Π andthe applied deformation. It is obvious that the inequal-ity assumes the value one for the reference configura-tion regardless of the deformation applied. Followingthe hydrostatic loading path, the inequality is alwaysfulfilled and remains at the value one, even if the poregrows spherically. If the isochoric loading paths arefollowed, the value moves from one to zero for theinequality with increasing deformation. This behaviorcan be enhanced rapidly by increasing the exponent n.

3.3 Equivalent void growth measure

In this section, an equivalent void growth measure Θ

is introduced that distinguishes between isochoric andvolumetric deformations in terms of a scalar value. Todifferentiate between these deformations, a so-calledvoid growth criterion εeqn formulated in Hencky strainspace according to Drass et al. (2018b) is used andmathematicallymodified, so thatΘ takes the value zerofor volumetric deformations, whereas it becomes oneunder isochoric loading. The requirement that the cri-terion equals zero for volumetric deformations resultsfrom Eq. (15), which means that the pseudo-elasticHelmholtz free energy function degenerates to Eq. (6).That is, the Helmholtz free energy function is able to

represent stress softening due to cavitation under purehydrostatic loading.

Returning to the void growth criterion, it is definedby

εeqn = I1,ε + I2,ε + I3,ε, (26)

where I1,ε, I2,ε and I3,ε describe the first, second andthird invariants of the Hencky strain tensor ε with

I1,ε = ε1 + ε2 + ε3, (27)

I2,ε = ε1 ε2 + ε2 ε3 + ε1 ε3, (28)

I3,ε = ε1 ε2 ε3. (29)

As alreadymentioned, the equivalent void growthmea-sure is a kind of Heaviside step function, which is gen-erally defined by

Θc : R → K with εeqn �→⎧⎨

⎩

0 : εeqn > 0c : εeqn = 01 : εeqn < 0

, (30)

where [0, 1, c = 0] ∈ K. Keeping this in mind, theequivalent void growth measure can be fully describedby

Θ = 1

2

∣∣εeqn

∣∣ − εeqn∣∣εeqn

∣∣ . (31)

This special formulation shows that Θ = 1 applies toisochoric deformations since εeqn < 0 or takes a valueofΘ = 0 for the reference configuration and for purelyvolumetric deformations since εeqn > 0 .

3.4 Constitutive modelling and algorithmic settings

The calculation of the Cauchy stress tensor for thepseudo-elastic material model has already been pre-sented in Sect. 3.1 and is summarized in Eq. (13). Dueto the additional equilibrium equation resulting fromthe inclusion of an internal variable in the constitutivemodel according to Lazopoulos and Ogden (1998), thecalculation of the stress tensor could be significantlysimplified.

For reasons of completeness the spatial tangent stiff-ness will be additionally derived in the following. Asalready shown in the derivation of the Cauchy stresstensor, the components of the additional strain energycoupling term depend on J and b, but only the compo-nents of the derivative with respect to J are included inthe final equation of the Cauchy stress tensor due to theadditional equilibrium equation of ∂Ψcouple/∂Ω = 0.

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50 M. Drassa et al.

Fig. 5 Three dimensionalillustration of isoperimetricinequality ϑ1−2 in λ1–λ2space, with UT;BT and HT

Therefore, only the derivation terms according to Jare described below. According to the proposal ofHolzapfel (2000, p. 265), the volumetric spatial tan-gent of the strain energy couple term ccouple can bewritten directly in the Eulerian configuration by

ccouple = (pcouple + Jscouple

)I ⊗ I − 2 pcouple I,

(32)

where I ⊗ I represents the dyadic product of two sec-ond order identity tensors and I describes a fourth-order identity tensor. Based on the simplifications dueto the additional equilibrium equation of Eq. (12), the

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Pseudo-elastic cavitation model: part I 51

hydrostatic pressure of the strain energy couple termreads

pcouple = Ω∂Dcav

∂ J(J ) . (33)

Since pcouple is now defined by Eq. (33), the parameterscouple must be also computed. It describes the secondorder derivative of Ψcouple with respect to J , whichreads

scouple=∂ pcouple∂ J

(J,Ω) +∂ pcouple∂Ω

(J,Ω)∂Ω

∂ J(b) .

(34)

Keeping inmind thatΩ was already defined inEq. (18),the calculation of scouple can be simplified to

scouple=Ω∂2 Dcav

∂ J 2(J ) + Θ Π

cosh (Θ ΠDcav)2

[∂Dcav

∂ J(J )

]2.

(35)

An algorithmic box for the numerical treatment is givenin Table 1.

3.5 Parameter studies of isoperimetric extension ofvolumetric Helmholtz free energy function

To get more insight into the function ofΩ orΘ andΠ ,and to understand how the algorithm behaves with anydeformations, reference Fig. 6. Here Ω , Θ and Π areshown under different deformation modes with respectto the principal stretch λ1. In this context, λ1 is dis-played on a logarithmic scale to investigate the behav-ior of Ω or Θ and Π at small deformations. For thispurpose, a single element test under uniaxial, biaxialand hydrostatic load is numerically investigated. SinceΩ , Π and Θ depend only on the principal stretches λiand an acceleration parameter n, the results displayeddepend on the deformation λ1 and n. As mentioned atthe beginning, the parameter n controls how fast onemoves from the initial value ofϑ = 1 of the isoperimet-ric inequality to the value zero due to an isochoric load.However, to obtain the classical isoperimetric inequal-ity, n must be set to one.

To evaluate the described parameters Ω , Π and Θ

numerically, a normalized Neo-Hooke material modelwith μ = 1 MPa is used. The parameters of Ψvol,ND

are selected as examples to perform the numerical cal-culations of the single-element benchmark tests (κ0 =0.02181, κ1 = 0.67900 and κ2 = 0.09791).

As shown in Fig. 6, Θ starts at Θ = 0 and veryquickly reachesΘ = 1 for isochoric deformations (UT,BT). In contrast, for triaxial loading, Θ = 0 with theresult that only Ψvol,ND is active during numerical cal-culation. It should also be noted that the equivalentvoid growth measure Θ starts at a value of Θ = 0for zero deformations. The structural reaction of theisoperimetric inequality Π starts at zero deformations(λ1 = λ2 = λ3 = 1) with a value of Π = 1. Forthe homogeneous hydrostatic load state, Π = 1 for theentire deformation regime, while for isochoric defor-mations the isoperimetric inequality decreases fromone to zero depending on the parameter n. By increas-ing the parameter n, the proposed constitutive modelconverges to the classical formulation of Ψvol,classic

much faster, as the influence of Ψvol,ND is reduced sig-nificantly.

In summary, in this section thevolumetricHelmholtzfree energy function proposed by Drass et al. (2018b)was successfully transferred into the context of pseudo-elasticity in order to prevent anomalies in structuralbehaviour, especially in isochoric deformation (see“Appendix B”). Therefore, the volumetric compo-nent of the model was equipped with the nonlin-ear Helmholtz free energy function that accounts forisotropic void growth under hydrostatic loading. Anenergy coupling term was then added that numericallyexplicates strain energy under isochoric deformation,while also guaranteeing physical material behaviour.The energy contribution is calculated internally byanalysing the geometric evolution of inherent voids.On the basis of one-element tests at any deformation,parameter studieswere carried out to confirm the physi-cal behaviour and to analyse the influence of individualparameters.

4 Numerical validation of constitutive model

This chapter aims at the validation of the proposedmaterial model. First of all, the quasi-static uniaxialtensile test and the bulge test are investigated with thepseudo-elastic cavitationmodel presented in Sect. 3. Toshow the general validity of this model, pancake testswith three different diameters are numerically calcu-lated and compared with the experimental ones. Since,according to literature (Dal et al. 2018), it has not yetbeen possible to validate cavitation material models

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52 M. Drassa et al.

Table 1 An algorithmic box of the FE procedure during numerical simulations for the pseudo-elastic cavitation model

1. given: deformation gradient F at tn = t∗ and tn+1 = t∗ + dt

2. compute dissipated strain energy Dcav due to isotropic void growth

Dcav = Ψvol,classic (J ) − Ψvol,ND (J )

3. compute principal stretches λi from left stretch tensor v = √b =

3∑

i=1λi ni ⊗ ni at tn+1 and evaluate isoperimetric

inequality Π and equivalent void growth measure Θ

Π = 3√

ϑ1−2 ϑ2−3 ϑ3−1 with ϑ =(4π AsLs

)n ≤ 1 with n ∈ N

Θ = 12

|εeqn|−εeqn

|εeqn| with εeqn = I1,ε + I2,ε + I3,ε

4. compute shape function Ω based on (2) and (3)

Ω = Θ (1 − Π) + tanh (Dcav Θ Π)

5. compute strain energy couple term Ψcouple (J,Ω) and the resulting pseudo-elastic Helmholtz free energy function

Ψ = Ψiso(b̄) + Ψvol,ND (J ) + ΩDcav (J ) + φ (Ω)

︸ ︷︷ ︸Ψcouple(J,Ω)

6. compute Cauchy stress tensor σ at tn+1

σ =[2J

∂Ψiso(b̄)

∂b̄· b̄

]dev+

[∂Ψvol

∂ J(J,Ω) + ∂Ψcouple

∂ J(J,Ω)

]

︸ ︷︷ ︸=p

I

7. compute tangent moduli ciso and cvol at tn+1

c = ciso + cvol with

ciso = 23

μJ

[−b̄ ⊗ I − I ⊗ b̄ + I1,Nb I + I1,b̄

3 I ⊗ I]

cvol = (p + Js) I ⊗ I − 2p I .

8. compute tangent moduli ccouple = (pcouple + Jscouple

)I ⊗ I − 2 pcouple I, with

pcouple = Ω ∂Dcav∂ J (J )

scouple = Ω ∂2 Dcav∂ J 2

(J ) + Θ Π

cosh (Θ ΠDcav)2

[∂Dcav

∂ J (J )]2

over full-scale small component simulations, this ispresented here for the first time.

4.1 Transparent structural silicone adhesive

The product DOWSILTM TSSA, a transparent struc-tural silicone adhesive, promises however due to itsmaterial properties, its high resistance to external envi-ronmental influences and its transparent appearance ahigher application of silicones in this industry (Drasset al. 2019b). An interesting behavior can be observedwhen this material is exposed to isochoric or hydro-static loading conditions. Under such loading con-ditions, the initially transparent silicone shows anincreasingly opaque or white appearance, which usu-ally is taken as an indicator of material softening.

The micro-structure of TSSA consists of poly-dimethylsiloxanes (PDMS) and a high amount of nano-silica particles (ca. 20–30%). The nano-silica particlesform aggregates, where one nano-silica particle has adiameter of approximately 1nm (Drass et al. 2019b).At the boundaries of these aggregates a multitude ofnanocavities can develop due to themanufacturing pro-cess, which may lead to an increased porosity or alsocalled finite porosity. In addition, the initially constantporosity can increase during deformation, when vac-uoles arise at the boundaries of the nano-silica surfacesduring stretching due to weak or non-existing filler-matrix interaction. Especially in flat-bonded transpar-ent silicone adhesive joints under axial load, the so-called cavitation effect is present, in which a very dom-inant effective stress softening can be detected due tostrong void growth under hydrostatic tensile load. In

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Pseudo-elastic cavitation model: part I 53

Fig. 6 Evaluation of Ω , Π and Θ as function of n dependent on the applied deformation

experimental investigations on the cavitation effect onTSSA, which are usually carried out in so-called pan-cake tests, the effect of the white coloring was detectedand analyzed by Drass et al. (2018b). In the pancaketest, flat-bonded steel cylinders are pulled axially. Asa result of the hindered lateral contraction, a triaxialstress state is generated in the material, which leads toexcessive void growth.

4.2 Simulation of dumbbell-shaped tensile test

Dumbbell-shaped tensile tests, also known as uniaxialtensile tests, are themost common testswhen analyzingthe structural behavior of any materials, since they arerelatively simple to perform. To proof that the pseudo-elastic cavitation model of Sect. 3 is able to approxi-mate this behavior correctly—not leading into a non-physical behavior due to disproportional void growth—

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54 M. Drassa et al.

a full-scale simulation of the dumbbell-shaped testspresented by Drass et al. (2018c) is performed.

Since the present constitutive model only consid-ers isothermal processes at a constant strain rate of ,this study investigates the stress-strain behavior undera constant temperature of T = 23 ◦C and a quasi-static displacement rate vUT = 5mm/s. Based on theresults of Drass et al. (2018c), TSSA exhibits a non-linear elastic behavior until fracture occurs—a char-acteristic property for rubber-like materials—however,TSSA does not exhibit strain stiffening due to lockingstretches of themacro-molecules in the polymermatrix.Consequently, when considering isochoric deforma-tions, simple isochoric constitutive models, like theNeo-Hookean model, can be used for TSSA (Drasset al. 2018d). The parameters of the pseudo-elastic cav-itation model were obtained using inverse numericaloptimization methods. Since the focus of this work isnot on determining the parameters, reference is madeto the work of Drass et al. (2017) to determine them.The material parameters used to characterize TSSAare given separately in Table 2 for the isochoric andvolumetric Helmholtz free energies. The parametershave been determined simultaneously based on theuniaxial, biaxial and pancake tests. Since the mean-ing of the shear modulus is simple, the fully metricmaterial parameters of Ψvol,ND are described vividly in“Appendix C” in order to better understand and illus-trate their meaning.

Based on the determinedmaterial parameters for thepseudo-elastic cavitationmodel, the critical hydrostaticpressure pcr at which cavitation ensues is reached for acritical relative volume of Jcr = 1.041. During the uni-axial tension test the deformations are mainly volume-constant because the voids drastically change shape butnot in volume. This means the numerical calculation ofthe test should not give way to substantial void growth.

This can be confirmed regarding Fig. 7, where theresults of the numerical calculation are visually com-pared with the experimental results. Figure 7a com-pares the structural response of both the experiment andthe finite element calculations. The min. / max. valuesof the experiments are gray curves, the mean valuesare red circles and the standard deviation (1 x SD) is anerror bar. All numerical calculations were performedwith an increasing amount of finite elements (328–23,142 Finite Elements) to analyze a potential meshdependency. The results show nomesh-dependency forthe simulations of the uniaxial tensile tests.

The numerical model in Fig. 7b displays the bound-ary conditions at a constant mesh density. Since three-dimensional numerical calculations were performed,three-dimensional, higher-order volume elements withpurely quadratic displacement behaviors were used. Asolid element consists of 20 nodes with three degreesof translational freedom. In order to shorten the cal-culation time, three symmetry axes were used to ana-lyze one eighth of a sample. A displacement of u =75.00 mm was applied incrementally to the top of thenumerical model. Thus, the simulation was carried outdisplacement-controlled. The numerical results of thepseudo-elastic cavitation model and the experimentalstructural response calculations correspond well. Sincethe constitutive approach takes void growth and thegeometric evolution of cavities into account, the cav-ity evolution at various positions and load-steps canbe seen in Fig. 7c. The geometrical evolution of thecavities is calculated by evaluating the isoperimetric,volumetric shape functionΩ (Π,Θ). Based on numer-ically calculated values of the equivalent void growthmeasureΘ and the isoperimetric inequalities t1−2, t2−3

and t3−1, the geometric evolution of an initially spher-ical cavity is calculated and visualized. Figure7c illus-trates how the initially spherical cavities deform withminimal change in volume: they actually morph into a“cigar-like” shape. This is confirmed by evaluating theactual void fraction f in the current configuration. Thevoid fraction f can be calculated by

f = f0J

(1 + J − 1

f0

). (36)

The relative volume J is calculated at each Gaussianpoint and at each load step. TSSA has an approxi-mate initial pore content of approximately f0 ≈ 3.0 %(Drass et al. 2019b).

Figure 7 demonstrates that when a compressiblehyperelastic material is loaded isochoric, there is aslight increase of void fraction, especially at largestrains. This is natural, since the bulk modulus is notinfinite for the analyzed material. Stress softening dueto excessive void growth is not observed in the experi-ment or in the simulation. Consequently, the presentedconstitutive modeling approach, where an isochoricNeo-Hookean material model was coupled with thepseudo-elastic cavitation model from Sect. 3, is ableto validate a simple tensile test without leading to non-physical results.

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Pseudo-elastic cavitation model: part I 55

Table 2 Optimisedmaterial parameters forTSSA for pseudo-elasticcavitation model

Material model Parameters

Isochoric Neo-Hooke model μ = 2.6652

Pseudo-elastic κ0 = 0.0004

Cavitation model κ1 = 0.2699

κ2 = 0.2501

κ3 = − 0.195

Fig. 7 Validation ofuniaxial tensile tests andschematic visualization ofcavity evolution

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56 M. Drassa et al.

4.3 Simulation of bulge-test

So-called bulge tests on TSSAwere presented byDrasset al. (2018c), where the foil-like TSSA was inflated ina balloon shape using a special test device. In order tomeasure local strains, a stochastic speckle pattern wasapplied to thematerial to be tested.By this experimentalset-up, a strain state near equi-biaxial tension can beexamined.

The numerical model of the bulge tests was build upaccording to the test set-up. Rotational symmetries inthe test set-up were employed to reduce the calculationtime (see Fig. 8b). In accordance with Sect. 4.2, higherorder, three-dimensional solid elements were used tocalculate the three-dimensional numerical model of thebulge test. The anchor plate and the edge of the TSSAwere fixed supported. A pressure p was applied to thebottom of the TSSA film to inflate it like a balloon.The finite element calculationwas accomplished force-controlled. As a result of the force-controlled simula-tion of the bulge test, an instability problem occurs atvery large deformations, so that the arc length methodmust be applied in the simulation. With increasingdeformation, the TSSA film leans against the anchorplate so that a contact formulation must be used inthe numerical model. The normal Lagrangian detectionmethod was utilized for contact modeling The contactformulation defines a surface contact that can only beopen or closed, and the normal Lagrangian formula-tion closes all gaps and eliminates penetration betweenthe contact and target surfaces. The investigation of theeffects of friction determined that the coefficient of fric-tion has no influence on the force-displacement behav-ior. Therefore, the contact algorithm is formulated witha frictionless approach.

Figure 8a compares the structural response betweenthe bulge test and the finite element analyses. Hereit becomes clear that the structural responses of thenumerical models lie within the standard deviation ofthe experiment. This proves that the novel constitu-tive model approach is ideally suited to take biaxialdeformations into account. The increasing mesh den-sity shows no significant deviation between the struc-tural responses of the finite element calculations.

Figure 8c references the cavity evolution duringbiaxial deformation.According to the results of the uni-axial tensile tests (see Fig. 7c), the initially sphericalcavities deform under almost volume constant condi-tions. This is obvious, since the bulge test is supposed

to represent a nearly isochoric deformation. Softeningin the experimental and numerical structural responsesis a typical phenomenon for the bulge test. However,the observed softening has nothing to do with stress-softening due to cavitation. This can be explained bythe void volume’s lack of change, even at large defor-mations. Due to the isochoric, biaxial deformation dur-ing the bulge test, the spherical cavities deform into apenny shape. As already mentioned, stress softeningdue to cavitation can be excluded when analysing thebulge test. ConsideringFig. 8c the actual void volume isshown at different positions and at different load steps.Here, a slight increase in void volume can be seen atvery large deformations, but this is primarily due to thecompressible material formulation and not to cavita-tion.

4.4 Simulation of pancake tests

In this section numerical simulations of pancake testsfrom literature (Hagl et al. 2012a, b; Drass et al. 2018b)are carried out. Starting with the simulation of the pan-cake tests with a diameter of 20 mm based on theinvestigations byHagl et al. (2012a, b), two flat-bondedsteel cylinders were loaded in axial tension. Due to thelack of lateral contraction in the experimental set-upand the almost incompressible material behavior, largevolumetric deformations occur in the adhesive layer.These conditions lead to the cavitation effect, and con-sequently to severe stress softening in the structuralbehavior.

The aim of this study is to numerically reproducethe structural response of the pancake test includingthe pronounced cavity growth that leads to a strongsoftening of the material. Symmetry boundary condi-tions were used to numerically model an angle ratioϕ between a full cylindrical model and a specific sec-tion (see Fig. 9b). The lower surface of the numericalmodel is only supported in the z-direction. Because ofthis, r and ϕ are free in their deformation. The radialand tangential degrees of freedom (r and ϕ) were fixedfor the upper side of the numerical model, while anaxial displacement corresponding to the deformation ofthe pancake test was applied in the vertical z-direction.As a result, the finite element calculation was con-trolled by a continuous increase of the displacement.In accordance with the three-dimensional models forthe uniaxial tensile test and the bulge test, the pan-cake test was also modeled three-dimensionally with

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Pseudo-elastic cavitation model: part I 57

Fig. 8 Validation of biaxialtension tests and schematicvisualization of cavityevolution

three-dimensional volume elements. Since the struc-tural response of the numerical model and the exper-imental results agree, the novel constitutive modelingapproach can adequately represent the strongly nonlin-ear structural behavior of the pancake test (see Fig. 9a).Furthermore, no mesh-dependent results are observedin the numerical analyses. Due to the initially almostincompressible material behavior, void growth must betaken into account for the numerical analysis of the pan-

cake test. Cavity evolution is easy to prove: The initiallyspherical cavities begin to grow in both axial and lateraldirections, and a closer look at Fig. 9c reveals that thevoid volume inside the adhesive increases significantlybecause of the constrained axial loading. At the edgeof the adhesive, where the disability of lateral contrac-tion is not very pronounced, the void volume increasesonly slightly. This demonstrates that there is a clearsoftening inside of the material due to excessive void

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58 M. Drassa et al.

Fig. 9 Validation ofpancake tests performedby Hagl et al. (2012a, b) andschematic visualization ofcavity evolution

growth, while the marginal areas are not influenced bythe cavitation effect. Therefore, the deformation of thepancake test is not comparable with the almost volume-preserving deformation of uniaxial tensile tests and thebiaxial tensile test.

Since it has already been shown that the pseudo-elastic cavitation model is well suited to approximatethe pancake tests performed by Hagl et al. (2012a, b),additional pancake tests with other diameters are sim-

ulated in the following. All simulation models are pro-vided with the same set of material parameters, whichare summarized in Table 2. The simulation results forthree different diameters are shown in Fig. 10a. It isobvious that the pseudo-elastic cavitationmodel is verywell suited to approximate the behavior of TSSA in thepancake test for different diameters. The model is ableto represent the increase in initial stiffness for largerdiameters of the pancake test specimen. Furthermore,

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Pseudo-elastic cavitation model: part I 59

Fig. 10 Validation of pancake tests performed by Hagl et al. (2012a, b) and Drass et al. (2018b) with (a) pseudo-elastic cavitation modeland (b) classical volumetric Helmholtz free energy function Ψvol,classic

the occurrence of the cavitation effect and the asso-ciated strong material softening is well approximated.With increasing deformation, the load can be increased,which can also be modeled by the new model.

In order to further show that the classical formula-tion of the volumetric Helmholtz free energy functionΨvol,classic, which is generally implemented by defaultin commercial FE codes, is not sufficient to repre-sent the material softening in the pancake test, thesewere simulated again with the classical formulation(Fig. 10b).Here, only the initial stiffness can be approx-imated, which is not sufficient to represent the stresssoftening and accordingly the ductility of the bond.

5 Conclusions

This paper analyses cavitation in rubber-like materi-als, commonly understood as sudden cavity growthunder hydrostatic stress. The experimental, analyticaland numerical treatment of this phenomenon is an inte-gral part of materials and mechanics research. How-ever, many numerical approaches lack a user-friendlyenvironment in terms of finite element calculations andnumerical robustness. This paper presented a novelconstitutive modeling approach for cavitation in rub-bers and rubber-like materials, in which the volumetricpart is equipped with a strongly non-linear Helmholtz-free energy function (Drass et al. 2018b), which isresponsible for isotropic void growth under hydrostaticloading conditions.

The classic formulation ofΨvol,ND was extended to ageneralized form (see Eq. 43) so that the set of volumet-

ric material parameters can be controlled by the run-ning index n. Furthermore, an extension of the classicvolumetric–isochoric split was carried out, where theaddition of a coupling termΨvol,couple makes it possibleto consider the cavitation effect in numerical calcula-tions using a simple approach. The coupling term cre-ates the consideration of cavitation in the form of stresssofteningunder volumetric loads.However, if isochoricdeformations occur, such as uniaxial or biaxial tension,the coupling term prevents extreme pore growth andconsequently material softening due to pore growth.The key element of this constitutivemodeling approachis the geometric analysis of the development of initiallyspherical cavities under any deformations. For this pur-pose, Ω was defined as an isoperimetric volumetricform function that takes into account whether a voidgrows and how the void grows geometrically. SinceΩ must explain if and how a void grows, it was con-structed as a function of equivalent volumetric strainand isoperimetric inequality.

To validate the novel constitutive model approach,three experiments from the literature were simulatedwith the novelmodel. Three-dimensional finite elementcalculations were then performed on a simple uniaxialtensile test and a biaxial tensile test (in the form of abulge test) Drass et al. (2018c) to validate the materiallaw for isochoric deformations. The numerical anal-yses presented that the material behavior, as well asthe evolution of inherent voids, could be well approxi-mated. Furthermore, it was shown that the newmaterialmodel does not soften the material due to void growthin isochoric deformations, in accordance with physi-cal behavior. In contrast to the isochoric tests, the so-

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60 M. Drassa et al.

called pancake test was also modeled numerically, inwhich two flat-bonded steel cylinders are pulled axi-ally (Hagl et al. 2012a, b). Due to the almost incom-pressible material behavior in the heavily constrainedtensile test, large volumetric deformation occurs dueto void growth, which leads to an enormous materialsoftening. The experimental structural response as wellas the geometry evolution of inherent pores could besimulated verywell with the novelmaterial model. Fur-thermore, the new material model made it possible tocalculate material softening, which was previously notpossible with commercial FE codes.

Future work will deal with an extension of thedeveloped model with respect to the viscoelastic andtemperature-dependent behavior in that direction thatthe models can be applied for structure calculation ofthin silicone adhesive joints in high temperature envi-ronments such as the Middle East.

Acknowledgements Open Access funding provided by Pro-jektDEAL.Wewould like to gratefully thankTheDowChemicalCompany (“DOW”) for their support during our studies. Addi-tionally, we would like to thank Julia Schultz and Joseph Yostfor the help with the linguistic fine-tuning of the paper.

Open Access This article is licensed under a Creative Com-mons Attribution 4.0 International License, which permits use,sharing, adaptation, distribution and reproduction in anymediumor format, as long as you give appropriate credit to the originalauthor(s) and the source, provide a link to the Creative Com-mons licence, and indicate if changes were made. The images orother third partymaterial in this article are included in the article’sCreative Commons licence, unless indicated otherwise in a creditline to thematerial. If material is not included in the article’s Cre-ative Commons licence and your intended use is not permitted bystatutory regulation or exceeds the permitted use, you will needto obtain permission directly from the copyright holder. To viewa copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

A Volumetric hyperelastic constitutive models

In contrast to the brief description of isochoric hypere-lastic models and the restriction of the detailed descrip-tion to only three essential models due to the enor-mous number of isochoric models, almost all volu-metric energy functions occurring in the literature arepresented in detail in the following, since only fewapproaches exist here. An overview of the various volu-metricHelmholtz free energy functions for hyperelasticmaterials is given by Hartmann and Neff (2003). Thesimplest form of a volumetric Helmholtz free energy

is dependent on the bulk modulus K and the relativevolume J only. Due to its simple form, the functionalis denoted as the classical volumetric Helmholtz freeenergy function, which reads

Ψvol,classic = K

2(J − 1)2. (37)

Since the classical volumetric Helmholtz free energyfunction violates the postulate of convexity for J → 0,which reads

limJ→+∞ Ψvol = lim

J→0Ψvol = ∞, (38)

Pence and Gou (2015) suggested a volumetric strainenergy function that does not exhibit this disadvantageresulting in

Ψvol,ext = K

8

(J 2 + 1

J 2− 2

). (39)

Simo and Taylor (1982) also proposed an improve-ment of the classical volumetric Helmholtz free energyfunction by adding the natural logarithm to the powerof two. The enhanced volumetric strain energy functionreads

Ψvol,ST = K

4

[(J − 1)2 + ln (J )2

]. (40)

Regardingvariationalmethods for the volume-constraintin finite deformation elasto-plasticity, Simo et al.(1985) proposed a variation of Eq. (40) by consider-ing the natural logarithm only, i.e.

Ψvol,STP = K

2

[ln (J )2

]. (41)

Miehe (1994) proposed also an alteration of the above-mentioned volumetric Helmholtz free energies, whichreads

Ψvol,M = K [J − ln (J ) − 1] . (42)

Following Hartmann and Neff (2003) and Li et al.(2007), most of the models were developed withouta physical background. The main focus was set onenforcing the incompressibility constraint under ensur-ing the element stability (Bonet and Wood 2008, p.171). In contrast, Drass et al. (2018b) proposed amicro-mechanically motivated volumetric Helmholtzfree energy function accounting for cavitation in struc-tural silicones. The novel approach reads in a general-ized form

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Pseudo-elastic cavitation model: part I 61

Fig. 11 Approximation of conventional volumetric Helmholtzfree energy functions via Ψvol,ND with K = 100 MPa and μ =1 MPa

Ψvol,ND =∫

J

(J − 1)∑m

i=0 κi (J − 1)id J + A with i ∈ N.

(43)

A represents an integration constant ensuring a stress-free reference configuration.

In Fig. 11, the hydrostatic pressure p dependingon the relative volume J is illustrated with respect toall proposed volumetric Helmholtz free energy func-tions. In addition, Ψvol,ND is presented as a fittingresult of all conventional volumetric strain energy func-tions to demonstrate its flexibility. From Fig. 11 itis obvious that Ψvol,ND includes all classical energyfunctions, since Ψvol,ND adequately approximates thestructural responses of all classical models. RegardingFig. 11, all proposed models show a similar structuralresponse. The classic Helmholtz free energy functionshows a linear dependence between p and J , whileall other strain energy functions show a moderate to astrong nonlinearity. Regarding hydrostatic tensile load-ing (J > 1), moderate softening of structural behaviorforΨvol,ext,Ψvol,ST andΨvol,M can be observed. In con-trast, Ψvol,STP shows a distinct softening of the struc-tural behavior, which is not due to the cavity growthunder hydrostatic loading and the associated macro-scopic material softening, but only due to the chosenmathematical approach. In summary, the classical volu-metric approaches are good in the sense of a stable finiteelement calculation, but processes such as stress soften-ing due to the cavitation effect in rubbers under hydro-static tensile loading cannot be reproduced. However,with the micro-mechanical approach of Drass et al.

(2017, 2018b) void growth due to cavitation can bemodeled.

B Remarks on volumetric Helmholtz free energyfunctions

In order to study the behavior of the proposed volu-metric strain energy density function not only in purehydrostatic load cases, simple tension problems using aFlory-type hyperelastic material model are analyzed inthe following. The compressible Neo-Hookean mate-rial model combined with the novel nonlinear volumet-ric Helmholtz free energy function reads

Ψ = μ

2

(Ib̄ − 3

) +∫

J

(J − 1)∑m

i=0 κi (J − 1)idJ . (44)

Dealing with a compressible Flory-type hyperelasticmaterial formulation, anomalies within the structuralbehavior may occur due to the volumetric–isochoricsplit of an arbitrary Helmholtz free energy function(Ehlers and Eipper 1998). Since the proposed nonlinearvolumetric Helmholtz free energy function accountsfor stress softening due to the cavitation effect, it islogical that it may lead to an abnormal structural per-formance under isochoric deformation modes.

To do so, simple tension problems for a nearlyincompressible behavior (κ0 → 0) and highly com-pressible material behavior have been analyzed, whereclassical formulations of the volumetricHelmholtz freeenergy function were applied. In contrast, in this studywe analyze the novel, strongly nonlinear volumetricstrain energy density function, where the initial bulkmodulus is varied between 10 ≤ K ≤ 1000 MPa.Since the present study does not investigate the influ-ence of the bifurcation point for the onset of cavita-tion and softening behavior, these parameters are setconstant. Thus, the bifurcation point is governed byκ1 = 0.7 and the stress softening parameters readκ2 = 0.3 and κ3 = 0.

In Fig. 12 the results for the simple tension problemare illustrated under different viewpoints. Figure 12ashows the first and second principal stretch of a unit cellunder uniaxial tension. Keeping in mind that the sec-ond principal stretch λ2 must be calculated as implicitfunction of the first principal stretch λ1, an anomaly inthe structural behavior is obvious. With an increase ofthe first principal stretch λ1, a decrease of the lateralstretch can be regarded. However, reaching a critical

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62 M. Drassa et al.

deformation level, the lateral stretch increases with anincreasing longitudinal deformation. This means thatthe specimens becomes thicker under uniaxial tensiondeformation, which is not physical. This can be alsoobserved regarding Fig. 12b, where a strong increaseof the relative volume can be regarded for large longi-tudinal stretches. The strong increase of relative vol-ume leads to stress softening effects under a uniax-

ial deformation. Considering Fig. 12c, where the axialCauchy stress is plotted against the Hencky strain, thestructural responses show a physical behavior until acritical strain is reached. After reaching the criticalpoint, a decrease of stress can be observed for all ana-lyzed case studies. This can be also confirmed withregard to Fig. 12d, where a distinct stress softening canbe observed for the Cauchy stress plotted against the

Fig. 12 Parameter studies on novel volumetric Helmholtz free energy function in accordance to Eq. (44) analyzing a unit cell underuniaxial tension loading: a λ1 − λ2, b λ1 − J , c σ − ε and d σ − (J − 1)

Fig. 13 Presentation of theexperimental results of thepancake tests in theengineering stress-straindiagram, annotation of themeaning of the materialparameters in the diagramand approximate equationsfor the volumetricparameters of the newvolumetric Helmholtz freeenergy function

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Pseudo-elastic cavitation model: part I 63

dilatation J − 1. Since this behavior shows evidentlyan anomaly in the structural performance in accordanceto Ehlers and Eipper (1998), the proposed Helmholtzfree energy function must be enhanced to exclude suchsoftening effect under isochoric deformations. For purevolumetric deformations, however, the new volumetricHelmholtz free energy function is well suited to rep-resent a stiff structural response followed by excessivestress softening. Even stress softening with subsequentstress stiffening can be easily adjusted with the newlyproposed volumetric Helmholtz-free energy function.

To avoid this non-physical mechanical behavior,Drass et al. (2018b) presented a so-called void growthcriterion. The void growth criterion differentiatesbetween isochoric and volumetric deformations basedon case differentiation an “if-request” during finite ele-ment calculations. Another possibility is to formulatea condition for the start of volumetric softening. Thisapproach was followed by Drass et al. (2017, 2018a)in terms of a continuum damage formulation. How-ever, it is advantageous to avoid case differentiationin the algorithm in order to obtain an efficient androbust numerical calculation without obtaining mesh-dependent results (Waffenschmidt et al. 2014). Thisprocedure is applied and presented in detail in Sect. 3,where a continuous pseudo-elastic formulation of thenon-linear volumetric Helmholtz free energy functionis formulated without an if-request in the algorithmicsetting.

C Definition of volumetric material parameters forΨvol,ND

In this section, the volumetric material parameters ofΨvol,ND are vividly explained based on the structuralresponse of a pancake test under axial loading. Con-verting the force-displacement curves of the pancaketests into engineering stress strain curves results in thefollowing diagram.

As can be seen from Fig. 13, the initial stiffnessor the volumetric material parameter κ0 respectivelycan be approximately calculated by the reciprocal ofthe initial engineering stress σ0 divided by the initialstrain ε0. This is indicated in the diagram by the gra-dient at the origin. The second parameter describesthe bifurcation point or bifurcation stress pcr in thestress-strain diagram. From this the second volumet-ric material parameter κ1 can be read approximately

by forming the reciprocal of the bifurcation stress pcr.Since the last two volumetric material parameters arerelated to each other, they cannot be read directly fromthe engineering stress–strain diagram of the pancaketests. However, for such pancake diagrams the mate-rial parameter κ2 must be within the limits κ2 ∈ [0, 1]and the parameter κ3 within the limits κ3 ∈ [−1, 0]. Atthis point it should be noted that the material parame-ter κ2 describing the inclination after the occurrence ofcavitation failure reflects the effective stress softeningdue to excessive void growth, from which it is positiveaccording to the definitionofNelder (1966). In contrast,the material parameter κ3 describes the effective stressstiffening behaviour after cavitation has occurred, sothat this is given a negative sign.

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