# On cavitation in transparent structural silicone adhesive: TSSA

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## Abstract

Cavitation in rubber-like materials describes sudden void growth of an initially voided material under hydrostatic tension until the material fails. To study the cavitation effect numerically, classical cavitation criteria are coupled with a continuum damage formulation of a Neo-Hookean material. A cavitation criterion defines a failure surface in three-dimensional stress or strain space, which represents the onset of excessive void growth and therefore the strong degradation of the bulk modulus. To account for this special case of material softening, a novel continuum damage formulation at finite strains is presented, where the initially constant bulk modulus of a hyperelastic material is reduced after satisfying a cavitation criterion. Since this formulation leads to an abrupt damage initiation, additionally a continuously volumetric damage formulation is proposed and compared with it. Therefore, novel void growth criteria are developed, which describe the cavitation effect even under smallest volumetric strains. For numerical validation, a single element test is simulated under hydrostatic tension. Furthermore, pancake tests are numerically analysed. The results with regard on the chosen cavitation criterion and the abrupt/continuously damage formulation are compared with each other analysing TSSA.

## Keywords

TSSA Compressible hyperelasticity Cavitation criteria Continuum damage formulation## List of symbols

- \({\left( \bullet \right) _{{\mathrm{iso}}}}\)
Isochoric/volume-preserving

- \({\left( \bullet \right) _{{\mathrm{vol}}}}\)
Volumetric

- tr\(\left( \bullet \right) \)
Trace of argument

- \({\nabla _0}\left( \bullet \right) \)
Gradient of argument

**F**Deformation gradient

*J*Relative volume

- \(J_{\mathrm{cr}}\)
Critical relative volume

**C**Right Cauchy–Green tensor

**b**Left Cauchy–Green tensor

- \({\mathbf{U}}\)
Right material stretch tensor

**R**Rotation tensor

- \({{{\bar{\mathbf{b}}}}}\)
Isochoric left Cauchy–Green tensor

- \(\lambda _i\)
Principal stretches

- \(\varepsilon _i^{{\mathrm{eng}}}\)
Principal engineering strain

- \(I_{1,(\bullet )}\)
First invariant of its argument

- \(I_{2,(\bullet )}\)
Second invariant of its argument

- \(I_{3,(\bullet )}\)
Third invariant of its argument

- \(\varvec{\sigma }\)
General Cauchy stress tensor

- \(\varvec{\sigma }_{\mathrm{prin}}\)
Principal Cauchy stress tensor

- \({\varepsilon _{\mathrm{vol}}}\)
Volume strain with engineering strains

- \({\varepsilon _{\mathrm{cr,vol}}}\)
Critical volume strain for damage initiation

- \({\varepsilon _{\mathrm{eqn,vol}}}\)
Equivalent volume strain with true strains

*p*Hydrostatic pressure

- \(\varPsi ( \bullet )\)
Helmholtz free energy

- \(\xi \)
Internal scalar damage variable

- \(\xi _{\mathrm{mod}}\)
Modified internal scalar damage variable

- \(\mu \)
Initial shear modulus

*K*Initial bulk modulus

- \(\varvec{\Upphi }\)
Orthogonal matrix

- \(I_{2,\sigma }'\)
Second invariant of stress deviator

- \(I_{3,\sigma }'\)
Third invariant of stress deviator

- \(\xi _1,\xi _2,\xi _3\)
Transformed coordinate system

- \(\theta \)
Stress angle

- \(\psi \)
Elevation

- tan\(\psi \)
Stress triaxiality

- \(\alpha ,\beta ,\gamma \)
Prefactors of void growth criteria

## 1 Introduction

As bonding materials, transparent structural silicones have prevailed because of their translucent appearance and mechanical properties (Sitte et al. 2011). Under quasi-static loading, these structural silicones belong to the group of hyperelastic materials, which are characterized by strain energy density functions respectively Helmholtz free energy functions. To characterize a Helmholtz free energy function, material parameters can be identified by regression analysis based on experimental investigations (Drass et al. 2017c, d; Kraus et al. 2017). A further characteristic of hyperelastic materials lies in the incompressible material behaviour. Hence, hyperelastic materials deform under a constant volume. Since structural silicones exhibit low stiffness under shear loading and high stiffness under volumetric deformations as shown and extensively studied in Drass et al. (2017a, b), Santarsiero et al. (2017), this high stiffness can lead to high hydrostatic stresses in the bulk material for special applications. Regarding classical connection systems, where a glass pane is bonded to a point fixing using structural silicones, the bonding material exhibits a disability for lateral contraction. Due to this, high hydrostatic stresses occur in the centre of bulk material, which lead to material failure in terms of the cavitation effect.

Cavitation in rubber-like materials is commonly known as sudden void growth under hydrostatic tension (Drass et al. 2017a, b). The damaging effect of cavitation was analysed by the pioneering experimental work of Busse (1938), Yerzley (1939) and Gent and Lindley (1959), where flat rubber cylinders were vulcanized to plane metal end pieces and afterwards tensioned. In the experimental results a yield point in the load displacement behaviour was observed due to internal rupture, e.g. void nucleation (Busse 1938; Yerzley 1939). Gent and Lindley (1959) interpreted the discontinuity in the load displacement response as sudden appearance of internal flaws, which was determined by (i) audible popping and (ii) cut open the pancake test specimen before and after reaching the breaking point. In the experimental test set-up of Lindsey (1967), a single void at the centre of the material was observed, which rapidly grows spherically and finally interacts with the boundary. The critical load under hydrostatic tension (HT) was determined to \({p_{\mathrm{cr,HT}}} = {5 /2}\,\mu \), which corresponds to the results of Gent and Lindley (1959). In this context, \(\mu \) represents the shear modulus. A brief summary for experimental investigations on cavitation in elastomers is given in Hamdi et al. (2014), where the critical hydrostatic load of \({p_{\mathrm{cr,HT}}} = {5/2}\,\mu \) was confirmed by several authors.

Regarding analytical work on cavitation, Ball (1982) studied a class of non-smooth bifurcation problems, which accounts for cavity nucleation at the centre of a unit sphere consisting of an incompressible matrix in n-dimensional space. He found that after reaching a critical stress level, the solution bifurcates. For a Neo-Hookean material with \(n=2\), the solution for critical homogeneous hydrostatic loading corresponds to the results of Gent and Lindley (1959). In contrast to Ball’s idea, where cavitation was understood as internal rupture of an undamaged incompressible material, Hou and Abeyaratne (1992) proposed a kinematically admissible deformation field in a general three-dimensional framework, where solutions for arbitrary triaxial loading conditions were calculated to determine the onset of cavitation in incompressible isotropic solids. In their work, the critical hydrostatic load of \({p_{\mathrm{cr,HT}}} = {5/2}\,\mu \) for homogeneous loading was also confirmed. Based on the obtained results, Hou and Abeyaratne (1992) proposed a cavitation criterion, which determines the critical load for void growth under arbitrary triaxial loading conditions. A further closed-form cavitation criterion was developed by Lopez-Pamies et al. (2011b). The proposed analytical criterion of Lopez-Pamies et al. (2011b) represents an approximated solution of a more complex criterion proposed in (Lopez-Pamies et al. 2011a), however, for all practical purposes the closed-form criterion can be utilized. Regarding the classical cavitation criteria proposed by Gent and Lindley (1959), Hou and Abeyaratne (1992) and Lopez-Pamies et al. (2011b), all proposed criteria assume an initial void fraction of approximately zero. In this context, the initial void fraction describes the ratio between the void volume and the rubber volume in the undeformed space. Hence, the proposed classical criteria lead to the same result of \({p_{\mathrm{cr,HT}}} = {5/2}\,\mu \) under homogeneous hydrostatic loading.

As a numerical example, a single element test is simulated under hydrostatic tension. The results of the single element test clearly show the differences between the discontinuous and continuous damage formulation. Furthermore, to validate the constitutive damage approaches, the pancake test under axial tension is numerically analysed, which represents a typical loading condition for laminated point fixings as shown by Drass et al. (2017a, b), Santarsiero et al. (2017). As bonding material the transparent structural silicone adhesive TSSA is investigated. All numerical examples are analysed as three-dimensional numerical models.

The present paper is structured by giving a short overview on continuum mechanics (Sect. 2) and compressible hyperelasticity accounting for damage processes (Sect. 3) to provide a brief summary of the governing equations. Additionally, an extension of a damage formulation is proposed, where damage can be analysed as a discontinuous or continuous reduction of the volumetric stiffness, i.e. the bulk modulus. Followed by giving an introduction concerning classical cavitation criteria (Sect. 4) in hyperelasticity, where damage is represented as a discontinuous approach, in Sect. 5, new developed cavitation criteria are presented, which give the opportunity to model damage as a continuous phenomenon. A numerical validation of the discontinuous and continuous damage formulations are presented and compared with each other in Sect. 6. In Sect. 7, the main findings of the present study are summarized and a short outlook is given.

## 2 Basics on continuum mechanics

To understand the concept of damage formulations in a constitutive sense, a brief introduction on continuum mechanics with the most important formulas is given in this section. Hence, this section represents an overview with the governing equations, however, the main part and novelty of the present paper is placed in Sect. 4.

### 2.1 Incompressible hyperelasticity

*P*. Following Holzapfel (2000), the position vector respectively the material (or undeformed) curve of \({{P}_0}\) is denoted as \({{\varvec{X}}}\), whereas the spatial (or deformed) curve of

*P*is described by \({{\varvec{x}}}\). The relation between the spatial tangent vector \(d{{\varvec{x}}}\) and the material tangent vector \(d{{\varvec{X}}}\) is described by a second order tensor, which is called deformation gradient \(\mathbf F \) (see Fig. 2). The classical nomenclature for the deformation gradient \(\mathbf F \) and the Jacobian

*J*read

*J*. The scalar

*p*is defined as an indeterminate Lagrange multiplier, which must be calculated based on equilibrium equations and boundary conditions (Holzapfel 2000).

### 2.2 Compressible hyperelasticity

**b**, which lead to the strain invariants

## 3 Constitutive modelling of hyperelastic materials

### 3.1 Compressible Neo-Hookean material model at small volume strains

*n*represents the number of network chains per unit volume,

*k*is the Boltzmann’s constant and

*T*is the absolute temperature. Since no free energy should be available in the undeformed state, which is conterminous when the first isochoric invariant is \({I_{1,{{\bar{\mathbf{b}}}}}} = 3\), the originally proposed form of the micro-mechanically motivated Neo-Hookean (NH) constitutive law (Treloar 1975) is given by

*K*respectively \(\kappa _2\), which represents the initial bulk modulus respectively the inverse bulk modulus. The relative volume is described by

*J*. With differentiation of Eq. (18) with respect to the relative volume

*J*, one obtains the expression for the hydrostatic pressure

*p*, which reads

*J*, one obtains a relationship for the bulk modulus over deformation history, which is constant for the classical volumetric formulation. As an example, the Helmholtz free energy function \({\varPsi _{{\mathrm{vol,classic}}}}\left( J \right) \), the hydrostatic pressure \({p _{{\mathrm{vol,classic}}}}\left( J \right) \) and the bulk modulus \({K _{{\mathrm{vol,classic}}}}\left( J \right) \) is plotted with respect to the relative volume in Fig. 3a. For this, typical material parameters for rubber-like materials, i.e. structural silicones, were applied in the calculation, which are \(\mu = {\text {1}}\;{\text {MPa}}\) and \(K = {\text {1000}}\;{\text {MPa}}\).

### 3.2 Compressible Neo-Hookean material model at large volume strains

### 3.3 Compressible Neo-Hookean damage material model at large volume strains

*p*is plotted against the relative volume

*J*in Fig. 4. Additionally, the evolution of the internal scalar damage variable \({\xi }_{\mathrm{mod} }\) dependent on the chosen critical damage threshold value \({J_{\mathrm{cr}} }\) respectively \(\varepsilon _{\mathrm{cr,vol}}\) is also represented. As can be seen in Fig. 4 the damage threshold governs the damage initiation. Reaching the assumed critical damage threshold, a strong degradation of the bulk modulus takes place, which is represented as a strong stress softening.

## 4 Classical cavitation criteria for void growth

Relevant cavitation criteria from literature are discussed in this section, which represent the abrupt onset of cavitation in terms of an excessively void growth due to high hydrostatic stresses. The classical approaches were proposed by different researchers based on different theoretical backgrounds. Additionally, novel void growth criteria are introduced, which describe the initiation of void growth even for smallest volumetric deformations.

### 4.1 Cavitation criterion: Gent and Lindley

**U**, which reads

### 4.2 Cavitation criterion: Hou and Abeyaratne

### 4.3 Cavitation criterion: Lopez-Pamies

## 5 Novel cavitation criteria for void growth

In the context of a continuum damage formulation, a case differentiation is necessary to decide whether damage initiates or not. Therefore, novel void growth criteria defined in Cauchy stress space respectively Hencky strain space are introduced, which describe the onset of void growth even under smallest volume changes within the material.

### 5.1 Void growth criterion in stress space

Formulating a stress based void growth criterion, the information at which critical hydrostatic stress inherent voids start to grow is needed. Since even for smallest triaxial stresses inherent voids start to grow to an infinite extend, the critical hydrostatic stress is set to approximately zero. This stays in contrast to the solutions of the classical cavitation criteria, which is given by Eq. (1). However, under a physical viewpoint the solution of Eq. (1) is only valid for an infinite amount of void fraction, which stays in contrast regarding the microstructure of filled rubber-like materials.

### 5.2 Void growth criterion in strain space

A three-dimensional illustration of the proposed strain based void growth criterion is presented under different viewpoints in Fig. 9.

### 5.3 Comparison of cavitation-criteria

Regarding the Burzyński-planes for the stress and strain based void growth criteria no intersection points for isochoric deformations can be observed. However, considering hydrostatic loading conditions, a point of intersection can be observed for both criteria at the origin of the Burzyński-plane (see Fig. 11e, g). This is essential to describe a continuous damage formulation. In contrast, it is obvious that a discontinuous damage arises if the point of intersection within the Burzyński-planes is not located at the origin (see Fig. 11a, c).

With regard to the \(\pi \)-planes illustrated in Fig. 11, where three different cutting planes for different \({I_1=\mathrm{const.} }\) are represented, all failure surfaces are similar. The failure surface of the stress based void growth criterion is more “spiky”, whereas the cavitation criterion of Hou and Abeyaratne (1992) and Lopez-Pamies et al. (2011b) are more rounded down. Comparing the cavitation criteria of Hou and Abeyaratne (1992) and Lopez-Pamies et al. (2011b), it is to note that the cavitation criterion of Hou and Abeyaratne (1992) is smaller at same values for \({I_1 }\). Hence, the criterion of Lopez-Pamies et al. (2011b) exhibits much more hydrostatic load combinations at which cavitation initiates. In this context, a hydrostatic load combination is defined as a superposition of a pure hydrostatic deformation with a isochoric deformation, where the critical relative volume is assumed to be constant (Lopez-Pamies et al. 2011b).

In conclusion, the presented transformation of a failure surfaces defined in the three-dimensional Haigh and Westergaard-space into the context of a two-dimensional illustration is pragmatic without any loss of information. This fact was proved by the failure surfaces, which represents the onset of cavitation for rubber-like materials, where the proposed transformation by Altenbach and Kolupaev (2014) was accomplished.

## 6 Numerical investigations

Numerical investigations are presented analysing the proposed volumetric damage model coupled with classical cavitation criteria as well as the proposed void growth criteria. To show the behaviour of the constitutive formulations simple single element tests under hydrostatic loading conditions were analysed. Furthermore, numerical studies for all presented constitutive descriptions were performed considering the pancake test. To understand the applied boundary conditions, both numerical models are illustrated schematically in Fig. 12.

### 6.1 Numerical comparison: single element test

To get more insight into the damage formulations dependent on the presented cavitation criteria, Fig. 13 can be adduced. Here, the structural response of a unit cell containing one finite element is illustrated. Since the presented constitutive models describe volumetric damage only, the unit cell was loaded hydrostatically, where the hydrostatic pressure *p*, which is dependent on the relative volume *J*, was numerically studied. A schematic illustration of the numerical model is given in Fig. 12a, where a three-dimensional, higher order solid element was utilized. It exhibits a pure quadratic displacement behaviour incorporating 20 nodes with three translational degrees of freedom.

Regarding the constitutive description, a normalized Neo-Hookean material model was applied, where the initial shear modulus was set to \(\mu =1 \; \mathrm{MPa}\). Regarding the damage parameters of the internal scalar damage variable \(\xi \), the optimized damage parameters of Drass et al. (2017a) were applied exemplarily in the numerical calculations with \(\kappa _0 = 0.002953\), \(\kappa _1= 0.819\) and \(\kappa _2 = 0.337875\) .

Finally, it turns out that the critical hydrostatic loads, which describe the onset of cavitation, are totally different between the classical approaches considering cavitation in rubber-like materials and the novel conception for cavitation, where cavitation is understood as a continuous progress of void growth. Regarding large volume changes all approaches converge to each other.

### 6.2 Numerical comparison: pancake test

A typical loading condition for laminated point fixings can be analysed experimentally by the pancake test. The pancake test specimen consists of a thin bonding material, here TSSA, which is bonded between a point fixing and a glass pane and afterwards tensioned. Different kinds of pancake tests were conducted and documented in (Hagl 2016; Drass et al. 2017b; Santarsiero et al. 2017; Dispersyn et al. 2017).

Drass et al. (2017b) tested TSSA experimentally by pancake tests, where during the experimental testing the cavitation effect was analysed by visible eye. Therefore, a special testing device was utilized, where a glass pane was bonded to a point fixing using TSSA as bonding material. The bonded area was circular with a constant diameter of \(d=50\;{\mathrm{mm}}\) and a nominal bonding thickness of \(t=1\;{\mathrm{mm}}\). To obtain detailed descriptions of the experimental investigations, the reader is referred to Drass et al. (2017b). However, after preparing the special test specimens for the pancake tests, the specimens were applied into the testing device and afterwards axially tensioned. Due to the nearly incompressible material behaviour of TSSA, a hydrostatic stress state occurred in the bulk material. Based on the occurring hydrostatic stresses in the bulk material, inherent voids start growing, which manifest itself through a pronounced stress whitening processes of the initially transparent structural silicone. It was shown by Drass et al. (2017b) that the stress whitening effect is caused by void growth processes and the cavitation effect respectively. The hydrostatic stress state within the pancake test specimen was visualized by Drass et al. (2017a), where the triaxiality proposed by Drass et al. (2017c) was visualized during numerical calculation.

Regarding the numerical model of the pancake test, symmetry boundary conditions were exploited in terms of modelling an angle ratio \(\varphi \) between a full, cylindrical model and a part of it, Fig. 12b in order to reduce computation time. Considering a cylindrical coordinate system, the bottom side of the numerical model was only supported in z-direction, hence the degree’s of freedom \(r \;{\text {and}}\;\varphi \) were free in motion. Regarding the top side of the numerical model, the radial and tangential degree’s of freedom (*r* and \(\varphi \)) were fixed, whereas in vertical z-direction an axial displacement was applied corresponding to the deformation of the pancake test. Hence, the finite element calculation was performed displacement controlled. Corresponding to the three-dimensional model of the single element test, the pancake test was also three-dimensionally modelled with three-dimensional higher order solid elements. An overview on the constitutive parameters of the analysed material models combined with different damage formulations and cavitation criteria are illustrated in the “Appendix” in Table 1.

Figure 14 shows the results of the numerical investigations in comparison with the experimental results. The min./max. values of the pancake tests are illustrated as a grey envelope within the numerical responses should lie ideally. The experimental results were normalized by dividing the force signal with the initial shear modulus of TSSA of \(\mu = 2.532\,{\mathrm{MPa}}\) in accordance to Drass et al. (2017a). The numerical results for the classical cavitation criteria combined with the continuum damage approach are displayed as blue lines. The classical approaches were equipped additionally with numerical damping to obtain convergency in the post-critical behaviour since the strong degradation of the bulk modulus leads to loss of well-posedness, such as loss of ellipticity, of the underlying boundary value problem. Regarding the initial stiffness in comparison to the experimental results, the classical formulations lead to an overestimation of it. Additionally, the critical fracture load is significantly overestimated. Hence, the proposed classical cavitation criteria are not able to represent the structural behaviour of TSSA within the pancake test. This can be explained by the fact that the classical cavitation criteria account for an infinite amount of void fraction in the bulk material. Therefore, it can be postulated that the approach of infinite void fraction for TSSA is insufficient because of the overestimation of the failure load. This stays in agreement with the work of Drass et al. (2017b), where finite void fraction for TSSA was experimentally observed. Thus, the classical cavitation criteria should be reformulated accounting for finite void fraction to represent the correct critical load for the onset of cavitation. Reaching large displacements the classical approaches approximate the experimental solution.

In contrast to the results obtained with the classical cavitation criteria, the present void growth criteria are more suitable to represent the structural behaviour of TSSA within the pancake test since the critical load is not overestimated. Since void growth initiates even for smallest volumetric deformations, the degradation of the bulk modulus is a continuously progress. Hence, the structural responses as well as the critical failure load can be represented adequately through the novel void growth criteria combined with the proposed continuum damage formulation. Through the continuous damage approach no numerical damping must be added within the numerical simulation to obtain convergence. The effect of a strong localised damage zone upon mesh refinement is not as distinct as for classical damage approaches since damage initiation within the pancake tests starts not locally at a particular finite element but more globally. Finally, it is to be noted that the results of the stress and strain based criterion lead to identical results for the pancake test.

## 7 Conclusions

Since cavitation is a common damaging effect of hyperelastic materials, a continuum damage formulation was proposed to describe the strong degradation of the initially high bulk modulus. Cavitation in rubber-like materials is understood commonly as sudden void growth of an initially voided material under hydrostatic tension. To study the cavitation effect numerically, in the present work classical cavitation criteria were coupled with a continuum damage formulation of a Neo-Hookean material. The cavitation criterion defines a failure surface in three-dimensional stress or strain space, which describe the onset of void growth and therefore a strong degradation of the bulk modulus. To study classical cavitation criteria with two novel formulations, which describe cavitation as void growth, general three-dimensional illustrations in stress and strain space were presented. For better comparison, the three-dimensional formulations were transferred into the context of a two-dimensional illustration without any loss of information.

To analyse cavitation numerically a novel damage formulation was coupled with all presented cavitation criteria. As numerical examples a single element test was simulated under hydrostatic tension. Furthermore, the pancake tests were analysed numerically. It was observed that the continuum damage model coupled with the classical cavitation criteria led to retarded but abrupt damage initiation, which is followed by a strong degradation of the bulk modulus. Thus, a degradation of the bulk modulus, loss of well-posedness, such as loss of ellipticity, of the underlying boundary value problem was observed. However, to obtain convergent results numerical damping was added within the numerical calculations. In contrast to the inadequate results obtained with the classical cavitation criteria, the proposed novel void growth criteria are ideally suitable to represent the structural behaviour of TSSA within the pancake test. Since void growth initiates even for smallest volumetric deformations, the degradation of the bulk modulus is a continuous progress. Hence, the structural responses as well as the critical failure load was adequately represented through the present void growth criteria combined with the proposed continuum damage formulation. Here, no numerical damping must be added within the numerical simulation to obtain convergency.

The development of three-dimensional cavitation criteria accounting for finite porosity is topic of future investigations. Furthermore, the continuum damage formulation will be equipped with path tracking algorithms to obtain a stable damage progress in the post-critical behaviour. Additionally, three dimensional calculations of isochoric experiments like the uniaxial tension test or the bulge test will be investigated.

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