Glass Structures & Engineering

, Volume 3, Issue 2, pp 237–256 | Cite as

On cavitation in transparent structural silicone adhesive: TSSA

  • Michael Drass
  • Vladimir A. Kolupaev
  • Jens Schneider
  • Stefan Kolling
SI: Challenging Glass paper

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

A general application form of transparent structural silicones are laminated connections (Santarsiero et al. 2016a, b). The laminated connection is achieved by directly bonding glass to metal using adhesives like TSSA, SentryGlas or PVB (see Fig. 1). To reduce application effort, which is obviously with regard on classical connection systems, like articulated and undercut point fixing courtesies (Overend 2005), the same production process as for laminated glass components is utilized in laminated connections. Regarding the numerical simulation of glass, laminated glass and laminated connections, in general different kind of approaches exist in literature (Pourmoghaddam and Schneider 2017; Alter et al. 2017; Rühl et al. 2017; Drass et al. 2017a).
Fig. 1

Point fixing connected to a glass pane with TSSA.

Permission of Dow Corning Europe (2017)

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.

However, regarding rubber-like materials, which are provided by several fillers like carbon black or nano-silica particles, void fraction increases due to vacuoles at the matrix-filler interaction (Sarkawi et al. 2015). Hence, cavitation criteria accounting for finite void fraction are demanded, where \({p_{\mathrm{cr,HT}}}\) must be smaller than the classical solution of
$$\begin{aligned} {p_{\mathrm{cr,HT}}} = {5/2}\,\mu . \end{aligned}$$
(1)
This was proved by Drass et al. (2017a), where the critical hydrostatic stress is illustrated for finite void fractions. To study the cavitation effect numerically, the above-mentioned cavitation criteria are coupled with a continuum damage formulation of a classical Neo-Hookean material. The damage formulation reduces the initially constant bulk modulus after satisfying a cavitation criterion. Since the classical cavitation criteria account for void growth after reaching a critical stress level, i.e. \({p_{\mathrm{cr,HT}}} = {5/2}\,\mu \) under homogeneous hydrostatic loading, an abrupt or discontinuous damage initiation is considered in the numerical studies. To enable a discontinuous damage formulation, the continuous damage formulation of Drass et al. (2017a) was generalized and extended by adding a critical relative volume \({J_{\mathrm{cr}} }\). The continuously damage formulation describes void growth and therefore cavitation even under smallest volumetric deformations. Hence, for the continuously damage approach cavitation ensues significantly earlier than in comparison to the abrupt or discontinuously approach.

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

To account for nonlinear elasticity in the context of finite theory, large deformations and strain measures must be applied in numerical calculations. Deforming a body \({\mathcal {B}_0}\) to \({\mathcal {B}}\), the material points \({{P}_0}\) of the reference configuration are mapped to the current configuration with 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
$$\begin{aligned} d{{\varvec{x}}} = \mathbf F d{{\varvec{X}}};\;\;\;\;\;\mathbf F = {\nabla _0}{{\varvec{x}}}, \end{aligned}$$
(2)
and
$$\begin{aligned} dv = JdV;\;\;\;\;\;J = \det \left( {{\mathbf{\nabla } _0}{{\varvec{x}}}} \right) =\det \mathbf F . \end{aligned}$$
(3)
The gradient with respect to the material coordinates is characterized by \({\nabla _0}: = {\partial \over {\partial {{\varvec{X}}}}}\). Considering incompressible hyperelastic materials, the determinant of \(\mathbf F \) must satisfy \(\det \mathbf F = J = 1\). Utilizing the polar decomposition, the deformation gradient can be written as \(\mathbf F = RU = VR \), where the left and right Cauchy–Green stretch tensors can be derived by \(\mathbf b = \mathbf{V ^2} = \mathbf F \mathbf{F ^\mathrm{T}}\) and \(\mathbf C = \mathbf{U ^2} = \mathbf{F ^\mathrm{T}}{} \mathbf F \).
Fig. 2

Continuum deformation at finite strains

The constitutive description of incompressible hyperelasticity can be postulated by a Helmholtz free energy function of
$$\begin{aligned} \varPsi = \varPsi \left( \mathbf F \right) - p\left( {J}-1 \right) \end{aligned}$$
(4)
with the restrictions of \(\det \mathbf F = J = 1\). The Helmholtz free energy function \(\varPsi \) depends on the deformation gradient \(\mathbf F \) and the relative volume 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

In contrast to an incompressible hyperelastic material, compressible hyperelastic materials can undergo finite strains under small volume changes. This class of materials often make use of the volumetric-isochoric split of the deformation gradient \({\mathbf{F}}\) respectively the Helmholtz free energy function \(\varPsi \left( \mathbf C \right) \) or \(\varPsi \left( \mathbf b \right) \), which was originally proposed by Flory (1961). The volumetric-isochoric split is reasonable when materials have large differences between the shear and bulk modulus (see Fig. 2). Following this, the deformation gradient can be rewritten by
$$\begin{aligned} \mathbf F = \left( {{J^{{1 \over 3}}}{} \mathbf I } \right) {\overline{\mathbf{F}} }= {J^{{1 \over 3}}}{\overline{\mathbf{F}} }, \end{aligned}$$
(5)
where \({\mathbf{I}}\) represents a second order identity tensor. Regarding the right Cauchy–Green stretch tensor, it can be formulated by
$$\begin{aligned} \mathbf C = \left( {{J^{{2 \over 3}}}{} \mathbf I } \right) {\overline{\mathbf{C}} }= {J^{{2 \over 3}}}{\overline{\mathbf{C}} }, \end{aligned}$$
(6)
whereas the left Cauchy–Green stretch tensor reads
$$\begin{aligned} \mathbf b = \left( {{J^{{2 \over 3}}}{} \mathbf I } \right) {\overline{\mathbf{b}} }= {J^{{2 \over 3}}}{\overline{\mathbf{b}} }. \end{aligned}$$
(7)
The overline \(\left( {\bar{\bullet } } \right) \) denotes the isochoric part of its argument. Hyperelastic constitutive equations are often formulated based on strain invariants of b, which lead to the strain invariants
$$\begin{aligned} {I_{{1,\mathbf{b}}}}= & {} {b_{ii}} = \lambda _{\mathrm{1}}^2 + \lambda _2^2 + \lambda _3^2, \end{aligned}$$
(8)
$$\begin{aligned} {I_{{2,\mathbf{b}}}}= & {} {1 \over 2}\left( {{b_{ii}}{b_{jj}} - {b_{ji}}{b_{ij}}} \right) = \lambda _{\mathrm{1}}^2\lambda _2^2 + \lambda _2^2\lambda _3^2 + \lambda _{\mathrm{1}}^2\lambda _3^2,\nonumber \\ \end{aligned}$$
(9)
$$\begin{aligned} {I_{{3,\mathbf{b}}}}= & {} \det {b_{ii}}= \lambda _{\mathrm{1}}^2\lambda _2^2\lambda _3^2. \end{aligned}$$
(10)
The isochoric strain invariants can be calculated by
$$\begin{aligned} {I_{{1,{{\bar{\mathbf{b}}}}}}}= & {} {{{I_{1,\mathbf b }}} \over {\root 3 \of {I_{3,\mathbf b }} }} = {J^{ - {2 \over 3}}}{I_{1,\mathbf b }} = {I_{{1,{{\bar{\mathbf{c}}}}}}} \end{aligned}$$
(11)
$$\begin{aligned} I_{2,{{{{\bar{\mathbf{b}}}}}}}= & {} {{I_{2,\mathbf b }} \over {\root 3 \of {I_{3,\mathbf b }^2} }} = {J^{ - {4 \over 3}}}I_{2,\mathbf b } = I_{2,{{{{\bar{\mathbf{c}}}}}}} \end{aligned}$$
(12)
$$\begin{aligned} I_{3,{{{{\bar{\mathbf{b}}}}}}}= & {} I_{3,{{{{\bar{\mathbf{c}}}}}}}=1. \end{aligned}$$
(13)
The strain invariants are often formulated dependent on the principal stretches \(\lambda _i\) with \(i \in \left[ {1,2,3} \right] \). The stretch (or the stretch ratio) is defined as the ratio of the length of a deformed line element to the length of the corresponding undeformed line element, which reads
$$\begin{aligned} \lambda _i = {{\left| x_i \right| } \over {\left| X_i \right| }}. \end{aligned}$$
(14)
Following this, a general compressible hyperelastic material law can be formulated by
$$\begin{aligned} \varPsi ={\varPsi _{{\mathrm{iso}}}}\left( {I_{{1,{{\bar{\mathbf{b}}}}}}},{I_{{2,{{\bar{\mathbf{b}}}}}}} \right) +{\varPsi _{{\mathrm{vol}}}}\left( J \right) , \end{aligned}$$
(15)
which describes the isochoric (volume preserving) and volumetric (volume change) elastic response. This special form of constitutive characterization has been established in literature because it is convenient for numerical implementation (Chaves 2013). An overview for different volumetric Helmholtz free energy functions was given by Hartmann and Neff (2003). However, none of the classical approaches are able to account either for large volume strains or stress softening effect caused by hydrostatic stresses in terms of the cavitation. Following Li et al. (2007), they state that volumetric Helmholtz free energy functions have physically not been well investigated since stress softening effects due to cavitation are disregarded.
Fig. 3

Generic representation of structural behaviour of a \({\varPsi _{{\mathrm{vol,classic}}}}\left( J \right) \), \({p _{{\mathrm{vol,classic}}}}\left( J \right) \) and \({K _{{\mathrm{vol,classic}}}}\left( J \right) \); b and \({\varPsi _{{\mathrm{vol,ND}}}}\left( J \right) \), \({p _{{\mathrm{vol,ND}}}}\left( J \right) \) and \({K _{{\mathrm{vol,ND}}}}\left( J \right) \) utilizing material parameters typical for structural silicones

3 Constitutive modelling of hyperelastic materials

3.1 Compressible Neo-Hookean material model at small volume strains

To model the isochoric deformation behaviour of a Neo-Hookean material, the elementary statistical theory founded by Kuhn (1934, 1936a, b), Kuhn and Grün (1942) and James and Guth (1943) can be utilized. In this theory rubber-like materials are treated as materials exhibiting long, flexible chain molecules, weak intermolecular forces and partial cross links between molecules, which are forming networks. During the deformation process the entropy decreases in accordance to statistical thermodynamics. Hence, it can be postulated that the change in entropy is directly dependent on the work done by temperature. Thus, based on the proposed micro-mechanical background the macroscopic Neo-Hookean free energy can be formulated as
$$\begin{aligned} \varPsi = {\mu \over 2}{I_{{1,{{\bar{\mathbf{b}}}}}}}\;\;\;{\mathrm{with}}\;\;\;\mu \equiv nkT, \end{aligned}$$
(16)
where 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
$$\begin{aligned} {{\varPsi _{{\mathrm{iso,NH}}}} = {\mu \over 2}\left( {{I_{1,{{\bar{\mathbf{b}}}}}}- 3} \right) .} \end{aligned}$$
(17)
Since this constitutive formulation only accounts for isochoric deformations, a volumetric energy term must be included in the material model. This can be achieved by adding the classical, volumetric Helmholtz free energy
$$\begin{aligned} {\varPsi _{{\mathrm{vol,classic}}}} = {K \over 2}{\left( {J - 1} \right) ^2}={1 \over 2\kappa _2}{\left( {J - 1} \right) ^2}, \end{aligned}$$
(18)
which represents a standard formulation in commercial FE packages. The presented volumetric deformation is only dependent on 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
$$\begin{aligned} p = K\left( {J - 1} \right) = {1 \over \kappa _2}{\left( {J - 1} \right) }. \end{aligned}$$
(19)
By two-fold differentiation of Eq. (18) with respect to the relative volume 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

Analysing the volumetric behaviour of TSSA under large volume strains, the nano-structure must be investigated to characterize void fraction. Since the nano-structure of TSSA consists of polydimethylsiloxane polymers, a high amount of nano-silica particles (ca. 20–30%) forming aggregates and nano-cavities, special constitutive models must be applied within numerical analyses (Drass et al. 2017a, b). Thus, Drass et al. (2017b) proposed a novel volumetric Helmholtz free energy function, which accounts for isotropic cavitation at finite strains. The constitutive model of Drass et al. (2017b) is based on an inverse polynomial of the form
$$\begin{aligned} {x \over y} = {\beta _0} + {\beta _1}x + {\beta _2}{x^2}, \end{aligned}$$
(20)
which was utilized to obtain an energy-homogenized volumetric Helmholtz free energy function. Based on homogenization schemes in accordance to Hohe and Becker (2001, 2003a, b), the homogenized volumetric strain energy function \({\varPsi _{\mathrm{vol,ND}}}\left( J \right) \) reads
$$\begin{aligned} \begin{aligned} {\varPsi _{\mathrm{vol,ND}}} =&{1 \over 2}{{\ln \left[ {{\kappa _2}{{\left( {J - 1} \right) }^2} + {\kappa _1}\left( {J - 1} \right) + {\kappa _0}} \right] } \over {{\kappa _2}}} \\&- {{{\kappa _1}\arctan \left[ {{{2{\kappa _2}\left( {J - 1} \right) + {\kappa _1}} \over \omega }} \right] } \over {{\kappa _2}\omega }} \\&- {1 \over 2}{{\ln \left[ {{\kappa _0}} \right] } \over {{\kappa _2}}} + {{{\kappa _1}\arctan \left[ {{{{\kappa _1}} \over \omega }} \right] } \over {{\kappa _2}\omega }} \end{aligned}, \end{aligned}$$
(21)
with \(\omega = \sqrt{4{\kappa _0}{\kappa _2} - \kappa _1^2} \). This constitutive approach is able to represent an initial high stiffness, which is equivalent to an initial high bulk modulus corresponding to the natural behaviour of rubber-like materials. The initially high stiffness is followed by a strong degradation of the bulk modulus accordingly the cavitation effect, where the bulk modulus decreases due to excessive void growth. Figure 3b shows the effect of an initially high stiffness till reaching a bifurcation point at which a strong degradation of the bulk modulus takes place.
Fig. 4

a Generic structural response of a cube under hydrostatic tension and b evolution of internal scalar damage variable \({\xi }_{\mathrm{mod} } \) dependent on the chosen critical damage threshold value \({J_{\mathrm{cr}} }\) respectively \(\varepsilon _{\mathrm{cr,vol}}\)

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

In order to describe stress softening effects of hyperelastic materials due to cavitation, a volumetric Helmholtz free energy function coupled with a damage evolution law was developed. The damage model accounts for the onset of cavitation respectively void growth due to high hydrostatic stresses. In correspondence to Timmel et al. (2007) and Timmel et al. (2009), who proposed a micro-mechanical approach to simulate rubber-like materials with damage, in the present work, the undamaged volumetric Helmholtz free energy function is coupled with an internal scalar damage variable \(\xi \in \left[ {0,1} \right] \) following Kachanov (1958) with the result of
$$\begin{aligned} {\varPsi _{{\mathrm{vol}}{\mathrm{}}}} = \left( {1 - \xi } \right) {\varPsi _{{\mathrm{vol,classic}}}}. \end{aligned}$$
(22)
Following Drass et al. (2017a), where a micro-mechanically motivated volumetric damage model for poro-hyperelastic materials was developed, the internal scalar damage variable reads
$$\begin{aligned} \xi = 1 - {{{\varPsi _{{\mathrm{vol}},{\mathrm{ND}}}}} \over {{\varPsi _{{\mathrm{vol}},{\mathrm{classic}}}}}}. \end{aligned}$$
(23)
Inserting the nomenclature of Eqs. (18) and (21) into the formulation of the internal scalar damage variable, it express itself as
$$\begin{aligned} {\xi } = {{\left( {J - 1} \right) \left( {{\kappa _4}\left( {J - 1} \right) + {\kappa _3}} \right) } \over {{\kappa _4}{{\left( {J - 1} \right) }^2} + {\kappa _3}\left( {J - 1} \right) + {\kappa _2}}}. \end{aligned}$$
(24)
Since this formulation is only able to account for continuous damage, the originally proposed internal scalar damage variable accordingly to Drass et al. (2017a) was extended by a critical damage threshold value
$$\begin{aligned} {J_{\mathrm{cr}} }=\varepsilon _{\mathrm{cr,vol}}+1 \end{aligned}$$
(25)
to control the start of damage initiation. \({J_{\mathrm{cr}} }\) defines the critical relative volume at which cavitation initiates, whereas \(\varepsilon _{\mathrm{cr,vol}}\) characterizes the critical volume strain. Therefore, the modified internal scalar damage variable reads
$$\begin{aligned} {\xi }_{\mathrm{mod} }= {{\left( {J - J_{\mathrm{cr}}} \right) \left( {{\kappa _4}\left( {J - J_{\mathrm{cr}}} \right) + {\kappa _3}} \right) } \over {{\kappa _4}{{\left( {J - J_{\mathrm{cr}}} \right) }^2} + {\kappa _3}\left( {J - J_{\mathrm{cr}}} \right) + {\kappa _2}}}. \end{aligned}$$
(26)
To decide whether a void is growing or not and therefore accounting for the stress softening effect due to cavitation, a cavitation criterion must be satisfied. When the cavitation criterion is fulfilled the critical relative volume \({J_{\mathrm{cr}} }\) is also reached and the internal scalar damage variable approaches \({\xi }_{\mathrm{mod} } \rightarrow 1\). Thus, by satisfying the cavitation criterion, the damage threshold is hold constant during numerical analysis. In the case that the cavitation criterion is not fulfilled, no damage evolution takes place and therefore the internal scalar damage variable equals \({\xi }_{\mathrm{mod} }= 0\). As a generic example of the continuum damage formulation of the volumetric part of a classical Helmholtz free energy function, the hydrostatic pressure 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.
In summary, all following numerical analyses are based on a hyperelastic continuum damage formulation, which reads
$$\begin{aligned} {\varPsi } = {\varPsi _{{\mathrm{iso,NH}}}} + \left( {1 - {\xi }_{\mathrm{mod} }} \right) {\varPsi _{{\mathrm{vol,classic}}}}, \end{aligned}$$
(27)
respectively
$$\begin{aligned} {\varPsi } = {\mu \over 2}\left( {{I_{1,{{\bar{\mathbf{b}}}}}}- 3} \right) + \left( {1 - {\xi }_{\mathrm{mod} }} \right) {K \over 2}{\left( {J - 1} \right) ^2}. \end{aligned}$$
(28)
The onset of cavitation is governed by the chosen cavitation criterion.
Fig. 5

Cavitation criterion accordingly to Gent and Lindley (1959) in three-dimensional principal Cauchy stress space

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

The first important cavitation criterion was proposed by Gent and Lindley (1959) and is presented by
$$\begin{aligned} \frac{1}{3}\,\left( \sigma _\mathrm {1}+\sigma _\mathrm {2} +\sigma _\mathrm {3}\right) -\frac{5}{2}\,\mu =0 \end{aligned}$$
(29)
or
$$\begin{aligned} I_{1,\sigma }-\frac{15}{2}\,\mu =0, \end{aligned}$$
(30)
with the first stress invariant \( I_{1,\sigma }=\mathrm{tr}(\varvec{\sigma })=\sigma _\mathrm {1}+\sigma _\mathrm {2}+\sigma _\mathrm {3}\). It is defined in three-dimensional principal Cauchy stress space, which is also known as Haigh and Westergaard space, see Fig. 5. Since the cavitation criterion is formulated in Haigh and Westergaard space, the principal Cauchy stress tensor reads
$$\begin{aligned} \varvec{\sigma }_{\mathrm{prin}}=\begin{bmatrix} \sigma _\mathrm {1}&\quad 0&\quad 0 \\ 0&\quad \sigma _\mathrm {2}&\quad 0 \\ 0&\quad 0&\quad \sigma _\mathrm {3} \end{bmatrix} . \end{aligned}$$
(31)
Following Kolling et al. (2007), the eigenvalues of a general Cauchy stress tensor \(\varvec{\sigma }\) can be obtained by multiplication with an orthogonal matrix \(\varvec{\Upphi }\) based on the eigenvectors of the right stretch tensor U, which reads
$$\begin{aligned} \varvec{\sigma }_{\mathrm{prin}}=\varvec{\Upphi }^{\mathrm{T}}\varvec{\sigma }\;\varvec{\Upphi }. \end{aligned}$$
(32)
Returning to the developed criterion of Gent and Lindley (1959), it gives a relationship between the initiation of cavitation dependent on the stress state. Regarding homogeneous hydrostatic loading (\({\sigma _\mathrm {1}}={\sigma _\mathrm {2}} = {\sigma _\mathrm {3}} = \sigma \)), it is immediate that the critical homogeneous hydrostatic load corresponds to the well-known result of \({p_{\mathrm{cr,HT}}} = {5/2}\,\mu \) (Ball 1982). Considering different stress states, like uniaxial tension (UT) with \({\sigma _\mathrm {1}} = \sigma \) and \({\sigma _\mathrm {2}} = {\sigma _\mathrm {3}} = 0\), one obtains the solution that cavitation occurs while reaching \({p_{\mathrm{cr,UT}}} = {15/2}\,\mu \). Regarding the case of biaxial tension (BT) with \({\sigma _\mathrm {1}}={\sigma _\mathrm {2}} = \sigma \) and \({\sigma _\mathrm {3}} =0\), the critical load amounts to \({p_{\mathrm{cr,BT}}} = {15/4}\,\mu \), whereby the relationship of \({p_{\mathrm{cr,UT}}} = {1 /2}\,{p_{\mathrm{cr,BT}}}\) can be derived. Since cavitation is a phenomenon under pure homogeneous and inhomogeneous triaxial stresses, the above-mentioned criterion is not physical, since it permits cavitation even under isochoric (volume-preserving) deformations.

4.2 Cavitation criterion: Hou and Abeyaratne

A second criterion, which determines the critical load for void growth under arbitrary triaxial loading (\({\sigma _\mathrm {1}} \ne {\sigma _\mathrm {2}} \ne {\sigma _\mathrm {3}} \ge 0 \)), was presented by Hou and Abeyaratne (1992) with
$$\begin{aligned}&(4\,\sigma _\mathrm {1}-\sigma _\mathrm {2}-\sigma _\mathrm {3})\, (4\,\sigma _\mathrm {2}-\sigma _\mathrm {3}-\sigma _\mathrm {1})\, (4\,\sigma _\mathrm {3}-\sigma _\mathrm {1}-\sigma _\mathrm {2})\nonumber \\&\quad -125\,\mu ^3=0 \end{aligned}$$
(33)
or
$$\begin{aligned} \frac{8}{27}\,I_{1,\sigma }^3-\frac{50}{3}\,I_{1,\sigma }\,I_{2,\sigma }' +125\,I_{3,\sigma }'-125\,\mu ^3=0. \end{aligned}$$
(34)
In this context, \(I_{2,\sigma }'\) represents the second invariant of the stress deviator, which reads
$$\begin{aligned} I_{2,\sigma }' = {1 \over 6}\left[ {{{\left( {{\sigma _{\mathrm{1}}} - {\sigma _{{\mathrm{2}}}}} \right) }^2} + {{\left( {{\sigma _{{\mathrm{2}}}} - {\sigma _{{\mathrm{3}}}}} \right) }^2} + {{\left( {{\sigma _{{\mathrm{3}}}} - {\sigma _{\mathrm{1}}}} \right) }^2}} \right] . \end{aligned}$$
(35)
The third invariant of the stress deviator can be expressed as
$$\begin{aligned} I_{3,\sigma }' = \left( {{\sigma _{\mathrm{1}}} - {{{I_{1,\sigma }}} \over 3}} \right) \left( {{\sigma _{{\mathrm{2}}}} - {{{I_{1,\sigma }}} \over 3}} \right) \left( {{\sigma _{{\mathrm{3}}}} - {{{I_{1,\sigma }}} \over 3}} \right) .\nonumber \\ \end{aligned}$$
(36)
The approximate expression to account for cavitation accordingly to Hou and Abeyaratne (1992) was calculated by employing a particular class of kinematically admissible deformations fields, where the following restrictions concerning the Cauchy stresses must be considered:
$$\begin{aligned} \begin{aligned} 4{\sigma _\mathrm {1}} - {\sigma _\mathrm {2}} - {\sigma _\mathrm {3}}> 0,\\ 4{\sigma _\mathrm {2}} - {\sigma _\mathrm {3}} - {\sigma _\mathrm {1}}> 0,\\ 4{\sigma _\mathrm {3}} - {\sigma _\mathrm {1}} - {\sigma _\mathrm {2}} > 0. \end{aligned} \end{aligned}$$
(37)
The formulated restrictions are necessary under a physical viewpoint since disregarding them would lead to cavitation even under isochoric deformations as it is the case for the cavitation criterion proposed by Gent and Lindley (1959). Based on this, an illustration of the proposed criterion is given in Fig. 6 in three-dimensional principal Cauchy stress space under different viewpoints. The proposed cavitation surface represents the onset of cavitation under arbitrary triaxial loading conditions. The colour scale from blue to red represents qualitatively a measure for the homogeneity of the triaxial loading, where blue indicates pure, homogeneous hydrostatic tension. Hence, blue indicates that all principal stresses are equal to each other. While reaching and penetrating into the cavitation surface, voids begin excessively to grow. From Eq. (33) it follows that cavitation occurs for homogeneous hydrostatic loading in accordance to Eq. (1), which corresponds to the well-established solution of Gent and Lindley (1959) and Ball (1982).
Fig. 6

Cavitation criterion accordingly to Hou and Abeyaratne (1992) in three-dimensional principal Cauchy stress space

4.3 Cavitation criterion: Lopez-Pamies

A third closed-form cavitation criterion was developed by Lopez-Pamies et al. (2011b). The proposed closed-form cavitation criterion represents an approximated solution of a more complex cavitation criterion proposed in (Lopez-Pamies et al. 2011a). However, for all practical purposes the approximated solution reads
$$\begin{aligned}&8\,\sigma _\mathrm {1}\,\sigma _\mathrm {2}\,\sigma _\mathrm {3} -12\,\mu \,\left( \sigma _\mathrm {1}\,\sigma _\mathrm {2}+\sigma _\mathrm {2}\,\sigma _\mathrm {3} +\sigma _\mathrm {3}\,\sigma _\mathrm {1}\right) \nonumber \\&\quad +18\,\mu ^2\,(\sigma _\mathrm {1}+\sigma _\mathrm {2}+\sigma _\mathrm {3})-35\,\mu ^3=0 \end{aligned}$$
(38)
or
$$\begin{aligned}&8\,\left( \frac{I_{1,\sigma }^3}{3^3}-\frac{I_{1,\sigma }\,I_{2,\sigma }'}{3}+I_{3,\sigma }'\right) -12\,\mu \,\left( \frac{I_{1,\sigma }^2}{3}-I_{2,\sigma }'\right) \nonumber \\&\quad +18\,\mu ^2\,I_{1,\sigma }-35\,\mu ^3=0 \end{aligned}$$
(39)
In the case of homogeneous hydrostatic loading the critical load for the onset of cavitation is equal to Eq. (1), which corresponds to the well-established solutions for the onset of cavitation. Since, the approximate solution of the criterion was restricted towards \({\sigma _k} > {3 \over 2}\mu \) with \(k \in \left\{ {\mathrm {I},\mathrm {II},\mathrm {III}} \right\} \) (cf. Poulain et al. 2017), no cavitation occurs under isochoric deformations. The above-mentioned three-dimensional plot is illustrated in Fig. 7 under different viewpoints, where one main cavitation surface is illustrated, which result from the approximated solution for the onset of cavitation (Lopez-Pamies et al. 2011b). In contrast to Fig. 6, the failure surface proposed by Lopez-Pamies et al. (2011b) is considerably larger, hence cavitation can occur under more triaxial stress combinations. Regarding a homogeneous hydrostatic loading, both criteria lead to the same result.
Fig. 7

Cavitation criterion accordingly to Lopez-Pamies et al. (2011b) in three-dimensional principal Cauchy stress space

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.

For this purpose, the criterion of Lopez-Pamies et al. (2011b) was transferred into the context of a void growth criterion, where the second stress invariant of the principal Cauchy stress tensor was neglected. Hence, the new developed void growth criterion is defined as
$$\begin{aligned} \alpha {I_{3,\sigma } } + \gamma {\mu ^2} {I_{1,\sigma } } - \delta {\mu ^3} = 0. \end{aligned}$$
(40)
By neglecting the second invariant, the unknown parameters \(\alpha \) and \(\gamma \) can be motivated physically as stress values for the onset of cavitation in the case of uniaxial respectively hydrostatic deformations. Since the present paper analyses cavitation under hydrostatic loading only, the parameters \(\alpha \) and \(\gamma \) must be chosen such that the criterion is fulfilled for volumetric deformations only. However, regarding isochoric deformations the fulfilment of the criterion must be precluded. Therefore, critical stresses for uniaxial tension and hydrostatic tension are defined, which represent the critical stress for the onset of cavitation. Due to the parametrized formulation of the void growth criterion, a critical stress \({p_{\mathrm{cr,UT}}}\) for uniaxial tension and for hydrostatic tension \({p_{\mathrm{cr,HT}}}\) can be chosen individually, which represent the critical stress value for the onset of cavitation. Thus, the parameters for the void growth criterion can be calculated by
$$\begin{aligned} \alpha = -\delta {{\left( {3{p_{{\mathrm{cr}},{\mathrm{HT}}}} - {p_{{\mathrm{cr,UT}}}}} \right) } \over {{p_{{\mathrm{cr}},{\mathrm{UT}}}}\,{p_{{\mathrm{cr}},{\mathrm{HT}}}}^3}} \end{aligned}$$
(41)
and
$$\begin{aligned} \gamma {\mathrm{= }}{\delta \over {p_{\mathrm{cr,UT}}}}, \end{aligned}$$
(42)
while setting \(\delta =35\), which is exactly the value given in Lopez-Pamies et al. (2011b). Since TSSA shows a temperature-dependent structural behaviour (Santarsiero et al. 2017), the parameters can be extended as function of the actual temperature. Under consideration of \({p_{\mathrm{cr,HT}}} \approx 0\;{{\mathrm{MPa}}} \), one obtains a three-dimensional illustration of this criterion in the Haigh and Westergaard-space, which is depicted under different perspectives in Fig. 8. The formulated criterion exhibits no intersections for uniaxial and biaxial loading conditions, if \({p_{{\text {cr}},{\text {UT}}}} \rightarrow \infty \). Thus, void growth under isochoric deformation modes is precluded. This can be substantiated since the intersection representing the onset of cavitation is related directly to the defined values for \({p_{{\text {cr}},{\text {UT}}}}\) and \({p_{{\text {cr}},{\text {HT}}}}\). Hence, regarding a continuous damage formulation, the value for \({p_{\mathrm{cr,HT}}} \) must approximate \({p_{\mathrm{cr,HT}}} \approx 0\;{{\mathrm{MPa}}} \), whereas the value for \({p_{{\text {cr}},{\text {UT}}}}\) must approximate infinity to preclude cavitation under isochoric deformations. Thus, the presented void growth criterion is substantiated only phenomenologically, where the user has to define the critical stress values for cavitation based on observations during experimental testing.
Fig. 8

Stress based void growth criterion in three-dimensional principal Cauchy stress space under different viewpoints

5.2 Void growth criterion in strain space

A second void growth criterion is formulated in three-dimensional Hencky strain space, which has the advantage that voids start to grow only if an equivalent volumetric strain \({\varepsilon _{\mathrm{eqn,vol}}}\) is positive. In this context, the equivalent volumetric strain is characterized by \({\varepsilon _{\mathrm{eqn,vol}}} \in \mathbb {R}_0^ + \) representing positive volume changes, whereas \({\varepsilon _{\mathrm{eqn,vol}}} \in \mathbb {R}^ - \) describes isochoric deformations only. In contrast to the proposed stress-based void growth criterion of Sect. 5, which was developed purely phenomenological, the strain-based void growth criterion is physically motivated. Referring to the work of Gosse and Christensen (2001), a strain invariant failure criterion for analysing damage initiation within the polymer matrix of a composite material was developed. In Gosse and Christensen (2001), the total volumetric strain \({\varepsilon _{\mathrm{vol}}}\) was defined as sum of the first three strain invariants of the principal Hencky strain tensor dealing with engineering strains. The proposal of Gosse and Christensen (2001) can be proved by expanding the volume strain with respect to the engineering strains, which reads
$$\begin{aligned} {\varepsilon _{{\mathrm{vol}}}}= & {} {{\varDelta V} \over V_0} = \left( {1 + {\varepsilon _1}} \right) \left( {1 + {\varepsilon _2}} \right) \left( {1 + {\varepsilon _3}} \right) - 1 \nonumber \\= & {} {\varepsilon _1} + {\varepsilon _2} + {\varepsilon _3} + {\varepsilon _1}{\varepsilon _2} + {\varepsilon _2}{\varepsilon _3} + {\varepsilon _1}{\varepsilon _3} + {\varepsilon _1}{\varepsilon _2}{\varepsilon _3}\nonumber \\= & {} I_{1,\varepsilon } + I_{2,\varepsilon } + I_{3,\varepsilon } \end{aligned}$$
(43)
The volumetric strain \({\varepsilon _{\mathrm{vol}}}\) is utilized to define a new equivalent volumetric strain \({\varepsilon _{\mathrm{eqn,vol}}}\), which is defined in Hencky strain space. Utilizing the equivalent volumetric strain \({\varepsilon _{\mathrm{eqn,vol}}}\), the above-mentioned strain based void growth criterion can be calculated by
$$\begin{aligned} \,\underbrace{\alpha {I_{1,\varepsilon } } + \beta {I_{2,\varepsilon } } + \gamma {I_{3,\varepsilon } }}_{ \equiv {\varepsilon _{{\mathrm{eqn,vol}}}}\;{\mathrm{for}}\;\alpha = \beta = \gamma = 1} + \delta = 0 \end{aligned}$$
(44)
with
$$\begin{aligned} {I_{1,\varepsilon } }= & {} {\varepsilon _1} + {\varepsilon _2} + {\varepsilon _3} \end{aligned}$$
(45)
$$\begin{aligned} {I_{2,\varepsilon } }= & {} {\varepsilon _1}\,{\varepsilon _2} + {\varepsilon _2}\,{\varepsilon _3} + {\varepsilon _1}\,{\varepsilon _3} \end{aligned}$$
(46)
$$\begin{aligned} {I_{3,\varepsilon } }= & {} {\varepsilon _1}\,{\varepsilon _2}\,{\varepsilon _3}. \end{aligned}$$
(47)
The postulated void growth criterion accounts for finite strains (volume strains), hence Hencky strains are applied in the criterion. For all following studies, the critical hydrostatic strain at which cavitation initiates was chosen to \({\varepsilon _{\mathrm{cr,vol}}} \approx 0\), which enables void growth even for very small volumetric deformations. The prefactors \(\alpha \), \(\beta \) and \(\gamma \) were constrained to \(\alpha = \beta = \gamma = 1\), whereby a non-weighted void growth criterion arises. With this assumption, the proposed void growth criterion can be motivated physically, since void growth can only occur under volumetric deformations, where the equivalent volumetric strain \({\varepsilon _{\mathrm{eqn,vol}}}\) is \({\varepsilon _{\mathrm{eqn,vol}}} \ge 0\). Since the void growth criterion was defined parametrically with the parameters \(\alpha \), \(\beta \) and \(\gamma \), these parameters can be adjusted based on experimental observation. Especially, regarding the temperature-dependency of TSSA, these parameters must be adjusted to make it possible to represent the experimental results. To obtain a solution for \(\delta \), which is dependent on the prefactors \(\alpha \), \(\beta \) and \(\gamma \) as well as \({\varepsilon _{\mathrm{cr,vol}}}\), Eq. (48) can be adduced:
$$\begin{aligned} \delta = \varepsilon _{\mathrm{cr,vol}}^3 + 3\,\varepsilon _{\mathrm{cr,vol}}^2 + 3{\varepsilon _{\mathrm{cr,vol}}}. \end{aligned}$$
(48)
Fig. 9

Strain based void growth criterion in three-dimensional principal Hencky strain space under different viewpoints

Fig. 10

Haigh and Westergaard-space of arbitrary cavitation criterion in terms of principal Cauchy stresses (\({\sigma _{\mathrm{1}}}, {\sigma _{\mathrm{2}}}, {\sigma _{\mathrm{3}}}\)), the transformed coordinates \({\xi _1},\) \({\xi _2}\) and \({\xi _3}\) and the \(\pi \)-plane at different sectional planes (\(\alpha {I_1}, \beta {I_1}, \gamma {I_1}\))

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

For a better understanding of the presented criteria for cavitation and void growth, all three-dimensional illustrations except of the criterion of Gent and Lindley (1959) were reformulated into a two-dimensional context. The cavitation criterion of Gent and Lindley (1959) was disregarded since it represents just a plane in three-dimensional Haigh and Westergaard space.
Fig. 11

Illustration of the Burzyński-plane as well as \(\pi \)-plane for the cavitation criterion of Hou and Abeyaratne (1992) (ab), Lopez-Pamies et al. (2011b) (cd), stress based void growth criterion (ef) and strain based void growth criterion (gh). a Burzyński-plane for Hou and Abeyaratne (1992). b \(\pi \)-Plane for Hou and Abeyaratne (1992). c Burzyński-plane for Lopez-Pamies et al. (2011b). d \(\pi \)-Plane for Lopez-Pamies et al. (2011b). e Burzyński-plane for stress-based criterion. f \(\pi \)-plane for stress-based criterion. g Burzyński-plane for strain-based criterion. h \(\pi \)-plane for strain-based criterion

Fig. 12

Schematic illustration of: a single element test under hydrostatic loading condition and b pancake test using symmetry boundary conditions for an efficient computation

Based on the work of Altenbach et al. (2014), Altenbach and Kolupaev (2014), Fahlbusch et al. (2016), a general procedure was adopted to illustrate cavitation criteria by means of an orthogonal transformation into two-dimensional space. Here, the principal Cauchy stress tensor is split into volumetric and deviatoric components, whereby a rotated orthogonal coordinate system is introduced with the coordinates \({\xi _1},\) \({\xi _2}\) and \({\xi _3}\) considering the Haigh and Westergaard space, Fig. 10. With the help of the rotated coordinate system, the hydrostatic axis is represented by \({\xi _1}\), whereas \({\xi _2}\) and \({\xi _3}\) define the cross-section of the analysed criterion. Based on this transformation the analysis of the failure surface \(\Phi \) can be easily accomplished because firstly the meridians at constant stress angles \(\theta \) can be displayed in the \({\xi _1}\)-\({\xi _2}\) plane and secondly the \(\pi \)-plane can be described by \({\xi _2}\) and \({\xi _3}\) while \({\xi _1}\) is set to a constant value. In Altenbach et al. (2014), it was recommended to use the Burzyński-plane as a beneficial illustration form, where the failure surface \(\Phi \) respectively a meridian of the failure surface is represented by a single line. Using a scaled illustration of the Burzyński-plane in accordance to Fahlbusch et al. (2016), the abscissa corresponds to the first stress invariant \({I_1}\) and the ordinate is characterized by \(\sqrt{3I_2'} \). An illustration of the Burzyński-plane as well as \(\pi \)-plane is given for all presented cavitation criteria in Fig. 11. Regarding the Burzyński-plane, three different stress angles \(\theta \) are illustrated. With these stress angles, unique meridians can be described:
$$\begin{aligned} \theta= & {} 0 \Rightarrow I_3' = {{2\sqrt{3} } \over {{3^2}}}{\left( I_2' \right) ^{{3 \over 2}}} \end{aligned}$$
(49)
$$\begin{aligned} \theta= & {} {\pi \over 6} \Rightarrow I_3' = 0 \end{aligned}$$
(50)
$$\begin{aligned} \theta= & {} {\pi \over 3} \Rightarrow I_3'= -{{2\sqrt{3} } \over {{3^2}}}{\left( I_2'\right) ^{{3 \over 2}}} \end{aligned}$$
(51)
Returning the Burzyński-planes illustrated in Fig. 11a, c, the cavitation criteria of Hou and Abeyaratne (1992) and Lopez-Pamies et al. (2011b) exhibit a point of intersection at \({I_1 } = 7.5\;{{\mathrm{MPa}}} \). Keeping in mind that the Cauchy stress invariant is defined as
$$\begin{aligned} {I_1 } = {\sigma _1} + {\sigma _2} + {\sigma _3} \buildrel \wedge \over = 3\,{\sigma _1}, \end{aligned}$$
(52)
one obtains the desired solution of \({p_{\mathrm{cr,HT}}} = {5/2}\,\mu \). Hence, inserting \({\sigma _1} = {\sigma _2} = {\sigma _3} = 5/2\mu \;{\mathrm{with}}\;\mu = 1.0\;{\mathrm{MPa}}\) into Eq. (33) respectively Eq. (38), the criterion has a simple root, which represents the point of intersection with the Burzyński-plane.
Fig. 13

Comparison of different continuum damage formulations analysing one finite element under hydrostatic tension. a Damage evolution of unit cell under HT. b Aperture of damage evolution

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 \(\kappa _2 = 0.337875\) .

As can be seen in Fig. 13, the structural behaviour depends on the chosen cavitation criterion. The initial stiffness of all numerical models is equal since the damage variable is exactly zero in the case of the discontinuous damage formulation. Regarding the value of the internal scalar damage variable for the continuous damage approaches, the damage variable approximates
$$\begin{aligned} \mathop {\lim }\limits _{J \rightarrow 1} {\xi _{\bmod }}\left( {{\kappa _0},{\kappa _1},{\kappa _2},{J_{{\mathrm{cr}}}}} \right) = 0. \end{aligned}$$
(53)
However, the main differences between the classical cavitation criteria (blue lines) in comparison to the present void growth criteria (red lines) lie within the point of onset of cavitation. Regarding the classical approaches, cavitation initiates at a critical hydrostatic load of \({p_{\mathrm{cr}}} = {5 /2}\mu \). Reaching this critical load, volumetric damage initiates due to excessive void growth. Thus, the internal scalar damage variable is active with the result of an extreme degradation of the initial bulk modulus. It is interesting to note that in the case of a homogeneous hydrostatic loading, the structural responses of the damage approaches equipped with different classical cavitation criteria lead to identical results. This is obvious, since all analysed models for the discontinuous damage formulation describe the onset of cavitation failure at the same critical load of \({p_{\mathrm{cr}}} = {5/2}\mu \). Reaching the critical load, the internal scalar damage variable begins to grow in the same manner for all models.
In contrast, regarding the novel void growth criteria, which describe the start of void growth even under smallest volumetric deformations, a continuous degradation of the bulk modulus can be observed. Hence, a smooth stress softening effect is observed for the present approaches, whereas the classical cavitation criteria exhibit a more abrupt damage evolution with respect to the bulk modulus. The onset of cavitation for the classical approaches is illustrated by the star symbol Open image in new window , whereas regarding the novel criteria void growth is marked with a triangle symbol \(\varvec{\triangle }\). Note that the present stress and strain based void growth criteria lead to nearly same structural responses under homogeneous hydrostatic loading conditions. The differences between the results of the stress and strain based void growth criteria are very small. Hence, for a practical purpose, both criteria lead for the proposed parameter setting to the same results. However, regarding other parameter sets, the results would be not identical anymore. So in conclusion, with a skilful selection of parameters for the phenomenological void growth criterion, the physically motivated strain based void growth criterion can be included.
Fig. 14

Comparison of different continuum damage formulations analysing the pancake test. a Structural behaviour of pancake test. b Aperture of structural behaviour of pancake test

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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Structural Mechanics and DesignTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Fraunhofer Institute for Structural Durability and System ReliabilityDarmstadtGermany
  3. 3.Institute of Mechanics and MaterialsTechnische Hochschule MittelhessenGiessenGermany

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