Abstract
We theoretically investigate the mechanical response of a transient network, which is characterised by dynamically breaking and re-forming crosslinks, and accounts for the finite chain extensibility (thus permitting the large deformations to be described). We build the general theory that incorporates the widely accepted empirical model of hyper-elasticity at large deformations (the Gent model) and naturally includes the microscopic behavior of transient crosslinks under the local tension applied to them. The full analytical expression for the elastic energy, or equivalently, the constitutive relation for arbitrary deformation is derived, and then the example of uniaxial tensile strain is focused on. In this case, we show that the mechanical response depends on the ratio of the imposed strain rate and the breakage rate of the crosslink: the system flows plastically (over a yield point) when the strain rate is much smaller than the breakage rate, while it remains elastic when the strain rate is much larger than the breakage rate. There is a broad range of this transition when the elastic and plastic regions of the sample coexist, and a resulting necking instability occurs. As a generalisation, we also consider a dual transient network, with two components penetrating each other, each having its own microscopic crosslink dynamics. The two networks add their local forces and share the deformation; we find that the network with a lower breakage rate determines the global deformation of the system.
Highlights
When a transient or dynamically crosslinked gel is uniaxially stretched as schematically shown in Figure 1, it can behave as a plastically flowing melt upon a slow rate of deformation, as an elastic rubber under a fast rate of deformation—or it may enter an intermediate regime of coexistence when a neck is formed, connecting a highly and a weakly deformed region in the sample
We will briefly describe how the crosslinks behave in a deformed polymer network, mainly following the ideas developed by Tanaka and Edwards [31]
We developed a consistent theory for handling a transient network undergoing a finite/large deformation, where the network strands could be stretched to their limit, and where the microscopic dynamics of crosslinks under tensions and the macroscopic deformation of the network are bridged in a unified framework
Summary
When a transient or dynamically crosslinked gel is uniaxially stretched as schematically shown in Figure 1, it can behave as a plastically flowing melt upon a slow rate of deformation, as an elastic rubber under a fast rate of deformation—or it may enter an intermediate regime of coexistence when a neck is formed, connecting a highly and a weakly deformed region in the sample. A transient network under small deformations is well studied theoretically, from either a microscopic [20,21,22,23,24] or a macroscopic viewpoint [25,26], where the crosslinks are dynamically broken and reformed all the time Such theories are valid when the material undergoes a sufficiently small deformation, so the constituting chains remain Gaussian, but cannot be accurate for large deformations when the network chains may be stretched significantly past the Gaussian limit (exploring their finite extensibility). The breakage and the reforming of the crosslinks depend on the local forces exerted on the polymer chains, which can be obtained from the continuum energy density By applying this general model, we will discuss how a transient network deforms when undergoing a uniaxial. A dual transient network will be studied at the end of this paper
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