Toughness of natural rubber compounds under biaxial loading

TL;DR: In this paper, the effect of strain induced molecular orientation on the fracture toughness of natural rubber based compounds was studied under biaxial loading conditions, using non-linear elastic fracture mechanics.

About: This article is published in Engineering Fracture Mechanics.The article was published on 2015-11-01 and is currently open access. It has received 10 citations till now. The article focuses on the topics: Fracture mechanics & Crack growth resistance curve.

Summary

1. Introduction

• It is well known that crystallising rubbers develop a strong strength anisotropy upon stretching: broadly speaking, if stretched along some direction they can become easier to break upon subsequent stretching perpendicularly to this direction [1–3].
• Under certain conditions rubbers may exhibit a crack branching phenomenon that has been variously termed in literature as “sideways crack propagation” [3] or “knotty tearing” [2, 4, 5].
• After sideways crack propagation, if the load is further increased, propagation of a crack along the initial notch direction plane takes place; this subsequent fracture event will be termed “forward propagation”, following the nomenclature proposed in [3]; often a very significant increment is needed.
• The results of this analysis are presented in Section 3.1.

2.1. Materials

• Natural rubber (NR0) and two carbon black filled natural rubber compounds (NR25 and NR50) were considered in this study.
• The filled compounds contain 25 and 50 phr of N330 carbon black.
• Rubber sheets were compression moulded at 160 ◦C and 8 MPa for 15 min so as to assure complete sulfur vulcanisation.
• Uniaxial tensile test-pieces were cut using a die from flat sheets, while biaxial and pure shear specimens (see section 2.2) were compression moulded in specially devised moulds.
• The nominal thickness of the sheets was 1mm.

2.2. Material behaviour characterisation

• It was assumed that the natural rubber compounds used in this study could be described by Ogden’s hyperelastic law for incompressibile materials [14].
• The number of terms in the sum, n, may be regarded as a material parameter as well, however, it is common practice to fix it a priori ; in this work n was arbitrarily fixed to 3.
• At the end of each cut a circular hole was cut in the specimen to reduce the stress concentration.
• The tests were run on a custom-built dynamometer; the experimental setup can be seen in Fig. 1(b).
• To identify the model parameters, data from uniaxial, pure-shear and equibiaxial tests were simultaneously fitted; the fitting procedure and its results are discussed in [21] where it is also shown, through a series of validation checks, that by using the identified material constants the mechanical behaviour of the materials is adequately described.

2.3. Fracture Tests

• To study the effect of orientation on toughness, two fracture tests were considered in this work.
• To keep the description of the phenomenon simple, in this work the nominal stretch ratio along the x direction was chosen as an overall measure of the material orientation induced by the initial pre-stretch.
• Fracture tests were also run on notched square shaped samples (Fig.1(b)).
• The step size chosen for the correlation analysis was about 0.15mm, with subset size 0.6mm.

2.4. Approximate determination of the sideways cracks shape

• Among the materials tested, NR25 and NR50 showed sideways crack propagation before propagation of the main crack, if the orientation was not enough to suppress it.
• In a very schematic way the undeformed configuration they would assume if the samples were unloaded is shown in Fig.4(b): they tend to grow backwards (consistently with the numerical results in [23]) with respect to the initial notch direction.
• Although in brittle materials branches tend to partially follow the main crack path (they grow “forward”) it was here assumed that a power law could be used as a very rough approximation to describe their shape in order to introduce them into a finite element model.
• In Fig.4(c) the measured sideways crack tip positions are shown (the reference system is given in Fig.4(b)).

2.5. Finite element simulations

• To calculate the J-integral FE models were developed using the commercial FE package ABAQUS© [27]; they are shown in Fig.5.
• For the cross shaped specimen a simple 2D non-linear elastic model was used exploiting specimen symmetry (Fig.5(a)).
• Eight-noded plane-stress rectangular elements were used (CPS8 in ABAQUS).
• The elements used were the same as in the cross shaped specimen model.
• Additional models with sideways cracks included were developed to study their effect on the elastostatic fields in the crack region.

3.1. Effect of orientation on toughness

• The dependence of Jc on the applied orientation level λx is shown in Fig.6 for all the three materials.
• There is also a rough agreement between the different test geometries considered.
• With respect to the initial crack dimensions, NR50 (Fig.6(c)) showed deep sideways cracks.
• Looking at the biaxial tests, the results are qualitatively similar to those of the other materials, with a very significant decrease from a value of about 20Nmm−1 for λx = 1 to about 1Nmm−1 for large values of λx.

3.2. Analysis of sideways cracks

• All the results that will be shown henceforth were calculated at the experimental displacement values corresponding to forward propagation, i.e. at the initiation of the main crack, after sideways propagation took place.
• When there are no sideways cracks, the strain energy density is singular and approaches the tip following a 1/x power law, as indicated by the corresponding dashed line.
• Similar results were obtained for the cross shaped specimen, although the singularity seems to be slightly stronger.
• This result is similar to what one would expect by analogy with the linear elastic wedge case; while it was somewhat expected, such a result is definitely not obvious, as the only theoretical results on the asymptotic behaviour of the elastostatic fields at wedges and notches in hyperelastic materials are for other models of the strain energy function [13, 28].
• Since the J-integral vanishes, it can no longer be used to study the onset of the main crack after sideways crack propagation has taken place, at least without having to resort to some ad-hoc additional criterion, as for example the maximum opening stress criterion or the so called “stress at a distance criteria”.

4. Summary and concluding remarks

• And paired with some new results on a different test set-up, to study how the strain induced strength anisotropy influences toughness.
• For natural rubber filled with 50 phr of carbon black the effect is even stronger.
• As shown by the FEM analysis the J-integral vanishes at the tip of a crack which exhibits sideways propagation and therefore the results can no longer be analysed in the framework of non-linear fracture mechanics.
• For filled compounds, when the orientation exceeds some threshold, which depends on the carbon black content and above which sideways crack propagation becomes suppressed, Jc quickly falls to a value which is about 1Nmm−1, irrespective of the carbon black content.

Figures

Figure 4: Sideways crack shape

Figure 6: Fracture toughness vs. orientation along the notch direction.

Figure 7: Sideways cracks: FEM results.

Figure 3: Schematics of the cross shaped specimen. Dimensions in mm

Table 1: Ogden’s model identified parameters for the three compounds.

Figure 1: Biaxial tensile test

Figure 2: Uniaxial mechanical beahviour of the three compounds. Experimental data (symbols) and Ogden’s model fit (solid lines).

Figure 5: FE models of fracture specimens. Dimensions in mm

Frequently asked questions

Q1. What have the authors contributed in "Toughness of natural rubber compounds under biaxial loading" ?

Strain induced molecular orientation effect on the fracture toughness of natural rubber based compounds was studied under biaxial loading conditions, using non-linear elastic fracture mechanics. This is the post-print author version of the work. Publisher 's version of the paper is available at http: //dx. doi. org/10. 2015. 08. 003 This work is licensed under Creative Commons Attribution Non-Commercial No Derivatives License 3. 

Q2. What future works have the authors mentioned in the paper "Toughness of natural rubber compounds under biaxial loading" ?

In this work previous results relevant to fracture tests under biaxial loading conditions were analysed anew in the framework of non-linear elastic fracture mechanics, and paired with some new results on a different test set-up, to study how the strain induced strength anisotropy influences toughness. 

Q3. What is the main reason why rubbers are hard?

It is well known that crystallising rubbers develop a strong strength anisotropy upon stretching: broadly speaking, if stretched along some direction they can become easier to break upon subsequent stretching perpendicularly to this direction [1–3]. 

Q4. What was the effect of stretching on the rubber?

Gent and Kim [2] tested notched rubber strips that were stretched, before testing, in a direction perpendicular to the testing one; their grip system allowed them to keep the pre-stretching level fixed during testing. 

Q5. What is the effect of the cuts on the material behaviour?

Nevertheless it is to be noted these cuts acts as stress raiser, reducing the range of strains for which the biaxial material behaviour can be measured, as they cause premature failure at the edges. 

Q6. What was the power law used to describe the shape of the cracks?

Although in brittle materials branches tend to partially follow the main crack path (they grow “forward”) it was here assumed that a power law could be used as a very rough approximation to describe their shape in order to introduce them into a finite element model. 

Q7. What is the strain energy density of the crack tip?

When there are no sideways cracks, the strain energy density is singular and approaches the tip following a 1/x power law, as indicated by the corresponding dashed line. 

Q8. What is the reason for the weak planes?

The very same phenomenon that has been conjectured to be source of sideways crack propagation [4], i.e. the development of weak planes due to the orientation and crystallisation of rubber molecules, may as well be cause of the strong toughness decrease with applied pre-stretch. 

Q9. What is the effect of sideways cracks on the material resistance?

In this sense the development of the sideways cracks completely shields the crack tip, strongly reducing the stress intensity near the original crack tip and providing a huge improvement in the apparent material resistance to crack propagation. 

Q10. How was the vanishing of the J-integral verified?

The vanishing of the J-integral was also verified by the compliance method [30], i.e. by extending the main crack tip along the x direction at fixed boundary displacement and studying the variation of the total strain energy U as a function of the crack length a. 

Q11. What was the difference between the two tests?

For both the tests the simulations were run for a time corresponding to the fracture time, defined as the initiation of propagation of the main in crack in the direction parallel to the initial notch, as determined from the video-recordings.