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Simulation of Elastomers · Filled Elasto-mers · Stress Softening · Implementati-on of material models

The complex mechanical behaviour of filled elastomers comprises several nonlinear and dissipative effects, one of them is stress softening. Previous model approaches to stress softening show difficulties both in their represen-tation of the stress behaviour of filled elastomers and their complexity with respect to an FE implementation. These problems have been treated in a novel model proposed in [1] at DIK, Hannover. Here, we demonstrate a full 3D imple-mentation of that model within the FEM system ABAQUS. The applicability of the description for technical elasto-mer-components within industrial usa-ge is shown by examples.

FEM-Implementierung der Spannungserweichung von Elastomeren Simulation von Elastomeren · Gefüllte Elastomere · Spannungserweichung · Implementierung von Materialmodellen

Das komplexe mechanische Verhalten gefüllter Elastomere weist einige nicht-lineare und dissipative Effekte auf, un-ter anderem Spannungserweichung. Bisherige Ansätze zum Modellieren spannungserweichenden Materialver-haltens wiesen Schwierigkeiten sowohl in der Wiedergabe der Spannungsant-wort gefüllter Elastomere als auch in ihrer Komplexität bezüglich einer FE-Implementierung auf. Diese Defizite wurden in einem neuartigen Modell, welches vom DIK, Hannover, in [1] vor-geschlagen wurde, behandelt. In dieser Arbeit wird dieses Modell in einer voll-ständigen 3D-Formulierung in das FEM-Paket ABAQUS implementiert. Die An-wendbarkeit zur Beschreibung techni-scher gefüllter Elastomer-Komponenten für den industriellen Einsatz wird an Beispielen gezeigt.

Figures and Tables: By a kind approval of the authors.

Motivation & Aim In order to adapt the behaviour of elasto-meric materials to industrial requirements, fillers – e.g. carbon black or silica – are added to the rubber matrix. These fillers improve the mechanical behaviour of the material and therefore expand their field of use within industrial applications, main-ly by adjusting the stiffness and abrasion resistance and therefore extending the life cycle of these components.

A side effect of filling elastomers is the appearance of several nonlinear and dissipative material properties, which are still not understood in all details and are often modelled by terms of pheno-menology. With increasing requirements on the accuracy of the simulation of elastomeric components, the represen-tation of these effects plays a crucial role in predicting their mechanical behaviour. Since the standard procedure of mode-ling the rubber mechanics by the concept of hyperelasticity can’t reproduce these effects, extensions on the standard mod-els are inevitable.

One of the effects occurring at filled elastomers is stress softening – also known as Mullins effect. In the treatment of homogenous stress states – such as uniaxial or equibiaxial tension tests – the influence of stress softening can be avoid-ed by an adequate pre-conditioning of the specimen, which leads to the risk of ne-glecting that effect within experiments for parameter calibrations – and later on in the description of inhomogeneous states within a technical rubber compo-nent. When considering the general case of arbitrary loaded components with het-erogenous stress states, stress softening can play a crucial role in the global me-chanical behaviour due to a heterogenous softening within the component.

With the models proposed in [1] and [2] at DIK, Hannover, a new physical moti-vated approach to the description of stress softening has been made. In [1] softening is modeled as an instantaneous effect only dependent on the previous reached maximum value of deformation. The model [2] is a further development of model [1], which expands it by continu-ous damage effects. The model proposed in [1] will be presented and implemented, as it gives an easier access to the topic

while containing the main ideas of the model. The main difference between the models is the functional relationship be-tween deformation and softening, thus it was possible to use several equations of the model in [2] which provide a better computational efficiency.

The aim of this investigation is the implementation of a new model for the description of filled elastomers with stress softening proposed in [1] at DIK, Hannover, as well as an examination of the model with respect to the applicabil-ity for industrial use. Exemplary, we use the experimental data setup for EPDM as depicted in Fig. 0 resulting from an uni-axial tensil test in quasi-static mode.

Model of spontaneous rubber softeningBy adding fillers such as carbon black or silica to elastomers, both stiffness and strength of the material are usually in-creased. This increase is due to the higher strength of the filler compared to the sur-rounding rubber as well as to the reduced flexibility of the polymer chains. When the material experiences deformation – and thus stresses – the materials struc-ture undergoes reformation mechanisms, which leads to a reduction of both stiff-ness and strength compared to the initial reinforcement, as well as to a softening hysteresis, see [1]. This effect is called stress softening. In general, stress soften-ing is a continuous and anisotropic effect dependent on e.g. temperature and time.

In the new model proposed in [1] at DIK, Hannover, stress softening of filled elastomers is modelled as an isotropic effect dependent on the maximum value

FEM Implementation of elastomeric stress softening

AuthorsDominik Klein & Herbert Baaser, Univ. of Applied Sciences, Bingen, Germany

Corresponding Author:Prof. Dr.-Ing. Herbert BaaserUniversity of Applied Sciences Bingen55411 Bingen, GermanyE-Mail: h.baaser@th-bingen.de

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of the first invariant I1 of deformation. A change in stiffness and strength is only observed when reaching a new maxi-mum value of the first invariant. By this simplification the model provides an easy access to the simulation of stress

Fig. 1: Von Mises equivalent stress for equibiaxial tension.

1

Fig. 2: Crack tip model.

Fig. 0: Experimental observation on quasi-static uniaxial ten-sion test of a typical EPDM mit 50 phr carbon black (N339). We refer to this mul-ti-hysteresis data setup.

2

0

softening, while maintaining the possi-bility of extensions like continuous dam-age effects, see [2].

The reinforcement sustained by the filler is modelled with an amplification factor (denoted with X), which is a quan-

titative measure for the reinforcement. Regarding the heterogeneity of the ma-terials reinforcement on a microscopic level, the distribution of the amplifica-tion factor on a microscopic level is also heterogenous. The change in the macro-scopic material properties is calculated by an amplified strain energy function W which is, together with the distribution function PX, integrated over the range of amplification factors occurring on micro-scopic level, see [1]:

𝑊𝑊𝑋𝑋(𝐼𝐼,𝑋𝑋𝑚𝑚𝑚𝑚𝑚𝑚 ,𝑋𝑋𝑚𝑚𝑚𝑚𝑚𝑚 ) = � 𝑑𝑑𝑋𝑋𝑋𝑋𝑚𝑚𝑚𝑚𝑚𝑚

𝑋𝑋𝑚𝑚𝑚𝑚𝑚𝑚∙ 𝑃𝑃𝑋𝑋(𝑋𝑋) ∙ 𝑊𝑊(𝑋𝑋 ∙ 𝐼𝐼)

𝑊𝑊𝑋𝑋(𝐼𝐼,𝑋𝑋𝑚𝑚𝑚𝑚𝑚𝑚 ,𝑋𝑋𝑚𝑚𝑚𝑚𝑚𝑚 ) = � 𝑑𝑑𝑋𝑋

𝑋𝑋𝑚𝑚𝑚𝑚𝑚𝑚

𝑋𝑋𝑚𝑚𝑚𝑚𝑚𝑚∙ 𝑃𝑃𝑋𝑋(𝑋𝑋) ∙ 𝑊𝑊(𝑋𝑋 ∙ 𝐼𝐼) (1)

This offers a good adaption of the mate-rials behaviour, while providing a com-paratively simple way of implementa-tion. While previous approaches to sof-tening behaviour, see [3], used an inter-nal average 𝑊𝑊𝑋𝑋 ~ 𝑊𝑊( ⟨𝑋𝑋⟩ ) to consider the microscopic heterogeneity of the rein-forcement, the new model uses an exter-nal average 𝑊𝑊𝑋𝑋 ~ ⟨ 𝑊𝑊( 𝑋𝑋 ) ⟩ . The usage of the external average allows a far better representation of the materials stress response, e.g. the upturn of the stress with increasing deformation, and is also the fundamental difference compared to previous approaches, see [1].

Softening of the material is taken into account by a reduction of the maximum amplification factor. As a quantitative measure of the remaining amplification, the average value of the amplification factors on microscopic level is calculated (“Average Amplification Factor”, AAF). The AAF is dependent on the distribution of the occurring amplification factors, see [4]:

𝑋𝑋� = � 𝑑𝑑𝑋𝑋

𝑋𝑋𝑚𝑚𝑚𝑚𝑚𝑚

𝑋𝑋𝑚𝑚𝑚𝑚𝑚𝑚∙ 𝑃𝑃𝑋𝑋(𝑋𝑋) ∙ 𝑋𝑋 (2)

By the way, this approach can be adapted to any hyperelastic model. Due to its physical motivated background and its good representation of stress for differ-ent deformation states, the non-affine tube model was chosen here, see [1]. To improve the efficiency of the model while maintaining the good characteris-tics of the tube model, several simplifica-tions were carried out, see [2], leading to the adapted tube model𝑊𝑊 =

𝐺𝐺𝑐𝑐2

𝐼𝐼1̅

1 − 1𝑚𝑚 𝐼𝐼1̅

+ 2𝐺𝐺𝑒𝑒𝐼𝐼∗̅ (3)

with Gc and Ge as shear modulus contri-butions due to crosslinking respectively the tube constraints of the polymer

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chains. The parameter n is a measure of the chain length between two network junctions, and 1 respectively * are adapted invariants, see [1] and [5].

By applying formula one to the adapted tube model, the strain energy function of the reinforced elastomer was calculated and could subsequently be implemented. We want to highlight that the integration in eq. (1) with the strain energy density function in eq. (3) could be solved analyti-cally using only elementary functions, see [2]. This leads to an improvement in both accuracy and efficiency of the calculations required within the FE simulations.

Numerical realizationThe realization is carried out in [4] in a full three-dimensional formulation; thereby arbitrary two- and three-dimen-sional simulations can be calculated. The model is embedded in the UMAT envi-ronment provided by ABAQUS to imple-ment user defined material laws.

The implementation is based on [6], where an automated algorithm for the implementation of material models is presented. By using this automated algo-rithm, the process of implementation can be accomplished very efficiently and new models can be implemented in a compa-ratively short time. This yields a signifi-cant advantage concerning economic and scientific requirements in the treatment and examination of new material models.

The algorithm computes all required derivatives as well as the tensors and their matrix representations required within the FE simulation. As the derivati-ves are calculated analytically, the usage of an automated algorithm is inevitable due to the complex calculations. The for-mulation of the material model takes place in the main axis of deformation, based on a formulation in [7]. By the for-mulation of the material model in prin-ciple axis and by the analytical calculati-

on of the derivatives, the implementati-on has an improved numerical stability and precision. Therefore, even mathema-tical challenging material models and si-mulations can be realized.

Furthermore, future extensions to the material model described in the previous section can easily be added to the exis-ting implementation.

Stress response of the modelAt first, a prototype deformation is simu-lated in order to show the characteristic stress strain curve of the model. When comparing uniaxial and equibiaxial ten-sion with the same value of displace-ment, equibiaxial tension shows a more pronounced stress softening due to the higher values of the first invariant of de-formation. Hence, the characteristic be-haviour of the model is shown with equibiaxial tension. The model is loaded in three sequences (100% / 250% / 150 % tension of its edge length). The simula-tion was carried out with parameters for multi-hysteresis data, see [2]: Gc = 0.21 MPa, Ge = 0.33 MPa, n = 36.8, γ = 1.25, c = 3.7, χ = 3.17.

In fig. 1 the characteristic stress res-ponse of the model is shown. Since du-

ring the first load the material is softe-ned, the stress at relaxation is smaller compared to the initial loading. The stress during the second load follows the relaxation curve of the first load, until a new maximum of deformation is reached – as soon as the deformation reaches a new maximum value, the stress curve follows the virgin curve. In both sequences the material shows a pronounced softening hysteresis due to different mechanical material proper-ties during loading and relaxation. In the third load, no new maximum value of the deformation is reached, therefore neither a change in the materials me-chanical properties nor softening hyste-resis occurs.

Applications of the new ImplementationSubsequently, some simulations are per-formed to investigate the functioning of the model in terms of numerical stability and its softening behaviour. As a meas-ure for the remaining reinforcement of the elastomer (therefore being a meas-ure for the softening) the dimensionless Average Amplification Factor (AAF) is used, see eq. (2).

Fig. 3: Average Amplification Factor of the crack tip model after load.Fig. 3 a: AAF in - Fig. 3 b: AAF in - at the crack tip.

3

Fig. 4: AAF and von Mises equivalent stress at maximum deforma-tion along the symme-try axis.

4

a) b)

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The following simulations were carried out with parameters for multi-hysteresis data, see [2]: Gc = 0.21 MPa, Ge = 0.33 MPa, n = 36.8, γ = 1.25, c = 3.7, χ = 3.17.

Crack tip When a rubber component has sharp edges – its limiting case being a crack tip – the stress in this area is considerably higher than in the rest of the compo-nent, leading to a highly heterogeneous stress. The softening behaviour of the model for a highly varying stress within a component is examined.

The specimen, see fig. 2, is loaded in three sequences in vertical direction (20% / 30% / 10% of its edge length).

Regarding the symmetry of the model, only the upper half is simulated. After the loading and reloading cycles the ave-rage amplification factor visualizes which domains were highly stressed res-pectively nearly unstressed, see fig. 3, and therefore represents the softening of the material. Near the crack tip the decrease of the amplification factor is most distinct because of the high stress – in general, the heterogeneity of the amplification factor due to the heteroge-nous stress stands out.

In figure 4 the course of amplification factor and stress along the symmetry axis (which extends horizontally through the tip of the crack, see green line in fig.

2) at maximum deformation is pictured. With increasing distance to the crack tip the stress decreases and therefore the average amplification factor increases. Since a new maximum of deformation is reached, the current value of the amplifi-cation factor is dependent on the current value of strain respectively stress. Thus figure 4 also pictures the relationship between softening and stress.

Grosch wheelThe Grosch wheel is a suitable test for both numerical stability and the stability of the softening behaviour of the model, see [8]. The simulation of the wheel was carried out two dimensional.

A quite simplified model of the arrange-ment is given in fig. 5. The filled elastomer wheel (shaded) is connected to a rigid axis on its inner circle and rests on a rigid plane. The radius of the inner circle is 0.6 mm, the radius of the outer circle is 1 mm. In the first step the wheel is loaded with a force of F = 4 N charging on the rigid axis, after-wards the wheel is rotated several times.

Once the initial force is applied on the wheel, the axis is displaced vertically, leading to a deformation of the wheel and thus to stress, see fig. 6.

Because of the stress near the contact surface, the material in this area is softe-ned. When the wheel is rotated it is gradu-ally softened all around. At the second ro-tation the material is already softened due to the first rotation, therefore the vertical displacement increases. This effect also occurs at the next rotations – after several rotations, vertical displacement and softe-ning should reach a stable limit. Thereby the stability of the softening is examined.

The progress of the softening within the first rotation is shown in fig. 7. A comparison between the stress, see fig. 6, and the softening due to the initial

Fig. 5: Grosch wheel.

5 Fig. 6: Von Mises equivalent stress in MPa distribution due to load axis and con-tact establishment.

6

Fig. 7: Evolution of the AAF over the first rotation. Fig. 7 a-d: AAF in.

7a)

c)

b)

d)

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Fig. 8: Concentric distribution of the AAF.Fig. 8 a: AAF in -, after first rotation; Fig. 8 b: AAF in -, after third rotation.

8force, see fig. 7a, shows the correlation between both quantities. As soon as the already softened material is near the contact area, the softening progresses, see fig. 7d.

The vertical displacement increases re-markably from the first to the second ro-tation, see fig 9. After the first rotation, the softening is nearly completed, see fig. 8. Between the second and third rotation, both softening and displacement virtually no longer change. Therefore, it can be as-sumed that after three rotations a stable limit of softening has been reached.Discussion and OutlookThe simulations show the numerical sta-bility of the implemented model as well as its stable softening behaviour. Even for highly heterogeneous stress states and multiple loading cycles, the model represents pronounced softening behav-iour. Furthermore, the representation of the stress response of filled elastomers is accurately reproduced.

Although the modelling of stress softe-ning as an isotropic, instantaneous effect is a simplification of the physical reality, the model is capable of simulating a cru-cial property of filled elastomers within industrial use: the heterogeneous distri-bution of both stiffness and strength within a component, which results from the generally occurring heterogeneous stress state within rubber components. The heterogeneity of the mechanical ma-terial properties occurring in filled elasto-mers has a significant influence on the components behaviour, therefore the mo-del is a reasonable extension on the stan-dard hyperelastic material models.

In fact, softening is an anisotropic, re-versible and continuous phenomenon which takes place over several loading cycles. The extension to the model pre-sented in [2], which includes continuous damage effects, is the next step in the description and understanding of stress softening.

With this work we want to focus on the effect of stress softening, and we al-so want to take a next step in establi-shing the model within industrial usage.

AcknowledgementHereby, the authors would like to thank the German Rubber Society (DKG) for the financial support carrying out this pro-ject in short time successfully. The au-thors also want to thank Jan Plagge, DIK Hannover, for his support throughout the project as well as the provision of measuring data.

AuthorsB. Eng. Dominik Klein worked on the pro-ject during his bachelor thesis at the University of Applied Sciences Bingen. Prof. Dr.-Ing. Herbert Baaser is professor for mechanical engineering at the Uni-versity of Applied Sciences Bingen.

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[9] H. Baaser, Ch. Hopmann and A. Schobel: Re-formulation of strain invariants at incom-pressibility. In: Archive of Applied Mechanics 83.2 (2013).

Fig. 9: Vertical displacement of the axis center.

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a) b)