*Burak Erman and James E. Mark*

- Published in print:
- 1997
- Published Online:
- November 2020
- ISBN:
- 9780195082371
- eISBN:
- 9780197560433
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195082371.003.0007
- Subject:
- Chemistry, Materials Chemistry

In the first section of this chapter, the relationships between the Helmholtz free energy, the stress tensor, and the deformation tensor are given for uniaxial stress. These relations follow from ...
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In the first section of this chapter, the relationships between the Helmholtz free energy, the stress tensor, and the deformation tensor are given for uniaxial stress. These relations follow from the general discussion of stress and strain given in appendix C, and the notation and approach closely follow the classic treatment of Flory. The detailed forms of the stress-strain relations in simple tension (or compression) are given in the remaining sections of the chapter for the (1) phantom network, (2) affine network, (3) constrained-junction model, and (4) slip-link model. Results of theory are then compared with experiment. The effects of swelling on the stress-strain relations are also included in the discussion. It is to be noted that the stress-strain relations in this chapter are obtained by treating the swollen networks as closed systems. The conditions for such systems are fulfilled if solvent does not move in and out of the network during deformation. A network swollen with a nonvolatile solvent and subject to simple tension in air is an example of a closed system. The same network at swelling equilibrium and subjected to compression will exude some of the solvent under increased internal pressure, and is therefore not a closed system. For semiopen systems, such as those under compression, or, in general, networks stressed while immersed in solvent, a more general thermodynamic treatment is required. This situation will be taken up in the following chapter.
Less

In the first section of this chapter, the relationships between the Helmholtz free energy, the stress tensor, and the deformation tensor are given for uniaxial stress. These relations follow from the general discussion of stress and strain given in appendix C, and the notation and approach closely follow the classic treatment of Flory. The detailed forms of the stress-strain relations in simple tension (or compression) are given in the remaining sections of the chapter for the (1) phantom network, (2) affine network, (3) constrained-junction model, and (4) slip-link model. Results of theory are then compared with experiment. The effects of swelling on the stress-strain relations are also included in the discussion. It is to be noted that the stress-strain relations in this chapter are obtained by treating the swollen networks as closed systems. The conditions for such systems are fulfilled if solvent does not move in and out of the network during deformation. A network swollen with a nonvolatile solvent and subject to simple tension in air is an example of a closed system. The same network at swelling equilibrium and subjected to compression will exude some of the solvent under increased internal pressure, and is therefore not a closed system. For semiopen systems, such as those under compression, or, in general, networks stressed while immersed in solvent, a more general thermodynamic treatment is required. This situation will be taken up in the following chapter.

*Burak Erman and James E. Mark*

- Published in print:
- 1997
- Published Online:
- November 2020
- ISBN:
- 9780195082371
- eISBN:
- 9780197560433
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195082371.003.0005
- Subject:
- Chemistry, Materials Chemistry

The classical theories of rubber elasticity presented in chapter 2 are based on a hypothetical chain which may pass freely through its neighbors as well as through itself. In a real chain, however, ...
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The classical theories of rubber elasticity presented in chapter 2 are based on a hypothetical chain which may pass freely through its neighbors as well as through itself. In a real chain, however, the volume of a segment is excluded to other segments belonging either to the same chain or to others in the network. Consequently, the uncrossability of chain contours by those occupying the same volume becomes an important factor. This chapter and the following one describe theoretical models treating departures from phantom-like behavior arising from the effect of entanglements, which result from this uncrossability of network chains. The chains in the un-cross-linked bulk polymer are highly entangled. These entanglements are permanently fixed once the chains are joined during formation of the network. The degree of entanglement, or degree of interpenetration, in a network is proportional to the number of chains sharing the volume occupied by a given chain. This is quite important, since the observed differences between experimental results on real networks and predictions of the phantom network theory may frequently be attributed to the effects of entanglements. The decrease in network modulus with increasing tensile strain or swelling is the best-known effect arising from deformation-dependent contributions from entanglements. The constrained-junction model presented in this chapter and the slip-link model presented in chapter 4 are both based on the postulate that, upon stretching, the space available to a chain along the direction of stretch is increased, thus resulting in an increase in the freedom of the chain to fluctuate. Similarly, swelling with a suitable diluent separates the chains from one another, decreasing their correlations with neighboring chains. Experimental data presented in figure 3.1 show that the modulus of a network does indeed decrease with both swelling and elongation, finally becoming independent of deformation, as should be the case for the modulus of a phantom network. Rigorous derivation of the modulus of a network from the elastic free energy for this case will be given in chapter 5. The starting point of the constrained-junction model presented in this chapter is the elastic free energy.
Less

The classical theories of rubber elasticity presented in chapter 2 are based on a hypothetical chain which may pass freely through its neighbors as well as through itself. In a real chain, however, the volume of a segment is excluded to other segments belonging either to the same chain or to others in the network. Consequently, the uncrossability of chain contours by those occupying the same volume becomes an important factor. This chapter and the following one describe theoretical models treating departures from phantom-like behavior arising from the effect of entanglements, which result from this uncrossability of network chains. The chains in the un-cross-linked bulk polymer are highly entangled. These entanglements are permanently fixed once the chains are joined during formation of the network. The degree of entanglement, or degree of interpenetration, in a network is proportional to the number of chains sharing the volume occupied by a given chain. This is quite important, since the observed differences between experimental results on real networks and predictions of the phantom network theory may frequently be attributed to the effects of entanglements. The decrease in network modulus with increasing tensile strain or swelling is the best-known effect arising from deformation-dependent contributions from entanglements. The constrained-junction model presented in this chapter and the slip-link model presented in chapter 4 are both based on the postulate that, upon stretching, the space available to a chain along the direction of stretch is increased, thus resulting in an increase in the freedom of the chain to fluctuate. Similarly, swelling with a suitable diluent separates the chains from one another, decreasing their correlations with neighboring chains. Experimental data presented in figure 3.1 show that the modulus of a network does indeed decrease with both swelling and elongation, finally becoming independent of deformation, as should be the case for the modulus of a phantom network. Rigorous derivation of the modulus of a network from the elastic free energy for this case will be given in chapter 5. The starting point of the constrained-junction model presented in this chapter is the elastic free energy.