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# Fracture of Rubber. Lecture 2

**Fracture mechanics without invoking any field theory. **In Lecture 1 on Fracture of Rubber, we considered the extension of a crack in an elastic body subject to a load. Following Rivlin and Thomas (1953), we regarded the elastic energy stored in the body as a function of two independent variables: the displacement of the load, and the area of the crack. The partial derivative of the elastic energy with respect to the area of the crack defined the energy release rate.

This definition of the energy release rate invokes no field theory. Indeed, the energy release rate can be determined experimentally without measuring any field. The energy release rate has been used as a loading parameter to study diverse phenomena related to the extension of a crack. Examples include fracture energy, *R-*curve, fatigue, and stress corrosion. See Lake (2003) for a review of these studies for rubber.

All these applications of fracture mechanics require no field theory. Fracture mechanics applies whether the material is glass or rubber, and whether deformation is small or large. Rivlin and Thomas (1953) attributed this approach to Griffith (1921).

**The condition of small-scale fracture process zone**. In defining the energy release rate, we have assumed that the body is elastic. In reality, the extension of a crack in the body is always an inelastic process, involving breaking atomic bonds, growing cavities, etc. For a material such as glass or rubber, our daily experience suggests that the inelastic process of fracture is mostly confined in zone around the front of the crack.

So long as this fracture process zone is small compared to the size of a specimen (e.g., the length of the crack, the length of the ligament), the shape of the specimen and the distribution of the load should not affect the fracture process. The magnitude of the load is represented by the energy release rate *G*, which is the only parameter that links the external mechanical boundary conditions and the fracture process.

**What do we gain by specifying a field theory in fracture mechanics?** Most engineering materials, however, can be modeled by field theories of one kind or another. For example, Griffith (1921) modeled glass using the linear elastic theory. What do we gain by introducing such a field theory into fracture mechanics? Here are two basic consequences:

- The energy release rate
*G*is quadratic in applied load. For a given cracked body,*G*can be determined by solving a boundary-value problem. - The crack-tip field is square-root singular. The intensity of the field is specified by
*G*.

Consequence 1 is of immediate practical value. Compared to determining *G* by experimental measurement, it is often convenient to calculate *G* using the finite element method. Consequence 2 helps to quantify the condition of small-scale fracture process zone.

Both consequences have analogs when a material is modeled with the nonlinear elastic theory. This lecture reviews the nonlinear elastic theory, and describes its consequences for fracture of rubber.

These notes belong to a course on fracture mechanics

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Fracture of rubber Lecture 2 2010 03 11.pdf | 220.45 KB |

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## Comments

## Rivlin and Thomas

Dear Zhigang,

Interestingly, Rivlin and Thomas considered the same mathematical expression, (2.2), as Griffith. However, they realized and emphasized that their "characteristic energy" was different from the Griffith "surface energy". That was a great step forward. The next step was due to Barenblatt (1959) who introduced the cohesive zone at the tip of the crack - the predecessor of the modern cohesive zone models.

Kosta

## Re: Rivlin and Thomas

The Rivlin-Thomas (1953) paper really is remarkable. Starting with this paper, fracture mechanics was developed for rubbers. This development paralleled that for metals.

In the early papers on metals and rubbers, there were little overlap between the two extensions. But Rivlin's autobiographic notes suggested interactions among the principals.

I've also learned recently that Thomas (1958) used G to study the extension of a crack in rubber under a cyclic load.

A.G. Thomas, Rupture of rubber. V. Cut growth in natural rubber vulcanizates. Journal of Polymer Science 31, 467-480 (1958).

This paper was published several years before Paris's paper.