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Research Papers

# Fracture of Elastomeric Materials by Crosslink Failure

[+] Author and Article Information
Yunwei Mao

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139

Lallit Anand

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: anand@mit.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received January 26, 2018; final manuscript received April 27, 2018; published online June 4, 2018. Assoc. Editor: Thomas Siegmund.

J. Appl. Mech 85(8), 081008 (Jun 04, 2018) (14 pages) Paper No: JAM-18-1056; doi: 10.1115/1.4040100 History: Received January 26, 2018; Revised April 27, 2018

## Abstract

If an elastomeric material is subjected to sufficiently large deformations, it eventually fractures. There are two typical micromechanisms of failure in such materials: chain scission and crosslink failure. The chain scission failure mode is mainly observed in polymers with strong covalent crosslinks, while the crosslink failure mode is observed in polymers with weak crosslinks. In two recent papers, we have proposed a theory for progressive damage and rupture of polymers with strong covalent crosslinks. In this paper, we extend our previous framework and formulate a theory for modeling failure of elastomeric materials with weak crosslinks. We first introduce a model for the deformation of a single chain with weak crosslinks at each of its two ends using statistical mechanics arguments, and then upscale the model from a single chain to the continuum level for a polymer network. Finally, we introduce a damage variable to describe the progressive damage and failure of polymer networks. A central feature of our theory is the recognition that the free energy of elastomers is not entirely entropic in nature; there is also an energetic contribution from the deformation of the backbone bonds in a chain and/or the crosslinks. For polymers with weak crosslinks, this energetic contribution is mainly from the deformation of the crosslinks. It is this energetic part of the free energy which is the driving force for progressive damage and fracture of elastomeric materials. Moreover, we show that for elastomeric materials in which fracture occurs by crosslink stretching and scission, the classical Lake–Thomas scaling—that the toughness Gc of an elastomeric material is proportional to $1/G0$, with $G0=NkBϑ$ the ground-state shear modulus of the material—does not hold. A new scaling is proposed, and some important consequences of this scaling are remarked upon.

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## References

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## Figures

Fig. 1

Schematic of a single chain with weak crosslinks at each end: (a) rest state and (b) stretched state. The Kuhn segments are assumed to be rigid while the crosslinks are assumed to be deformable.

Fig. 2

(a) Schematic of the single-edge-notch specimen geometry; all dimensions are in mm. The thickness of the sample is 1 mm; the notch length is denoted by c; and ρ=0.1 mm is the notch-root radius. (b) Calculated force–displacement curves for c=3, 6, 9 mm. Contour plots for the damage variable d at points (a)–(h) on the load displacement curve for a specimen with c=9 mm are shown in Fig. 3.

Fig. 3

Images of the deformed geometry with contour plots of the damage variable d. To aid visualization of the damage, elements with an average value of d>0.99 are removed from the plots. Since the length scale ℓ=100 μm is very small when compared with the overall dimension of the specimen (∼20 mm), the damage zone is barely visibly in this this sequence of contour plots for d.

Fig. 4

Contours of crosslink stretch λc during the fracture process in a specimen with c=9 mm. The crosslink stretch is appreciable only in a small zone near the crack tip. The contours are plotted on the reference configuration. Elements with d>0.99 are removed from the visualization.

Fig. 5

(a) Schematic of the asymmetric-double-edge-notch specimen geometry; all dimensions are in mm. The thickness of the sample is 1 mm; the notches are of length c = 3 mm; and ρ=0.1 mm is the notch-root radius. (b) Calculated force–displacement curve. The contour plots for the damage variable d at points (a)–(j) on the load displacement curve are shown in Fig. 6.

Fig. 6

The deformed geometry with contour plots of the damage variable d. To aid visualization of the damage, elements with an average value of d>0.99 are removed from the plots. Since the length scale ℓ=100 μ m is very small compared with the overall dimension of the specimen (∼20 mm), the damage zone is barely visibly in this sequence of plots.

Fig. 7

Schematic of the geometry of specimen with several circular and elliptical holes; all dimensions are in mm. The thickness of the sample is 1 mm.

Fig. 8

(a) Calculated force–displacement curve. There are four different stages within the curve: (i) The blue line indicates the first stage. (ii) The dashed-black line indicates the second stage. (iii) The solid-yellow line indicates the third stage. And (iv) the dotted-pink line indicates the fourth stage. (b)–(d) show zoom-in figures for the corresponding force drop stages. The contour plots for the damage variable d at points (a)–(h) on the load displacement curve are shown in Figs. 9 and 10.

Fig. 11

(a) Contour plots with severely damaged elements. (b) The same geometry without the severely damaged elements.

Fig. 10

Images of the deformed geometry with contour plots of the damage variable d for third and fourth stages. To aid visualization of the damage, elements with an average value of d>0.99 are removed from the plots. The arrows indicate the ligaments in which damage and rupture occurs.

Fig. 9

Images of the deformed geometry with contour plots of the damage variable d for first and second stages. To aid visualization of the damage, elements with an average value of d>0.99 are removed from the plots. The arrows indicate the ligaments in which damage and rupture occurs.

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