Tests of the Affinity Assumption in Phantomlike Elastomer Networks
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Tests of the Affinity Assumption in Phantomlike Elastomer Networks Misty Davies and Adrian Lew Mechanical Engineering, Stanford University, Stanford, CA 94305, U.S.A ABSTRACT Phantomlike elastomer simulations do not always deform globally affinely in the way that classical theory predicts. Assuming that each crosslink will deform affinely with its topological neighbors gives much better results, and creates a way to isolate crosslinks with unpredictable deformation properties. The correlation of non-affinities and network properties depends on the constitutive model and boundary condition used. We always find a correlation between local density of crosslinks and degree of non-affinity. INTRODUCTION The assumption that elastomers will deform affinely everywhere if the boundary deforms affinely (“global affinity”) is often made [11, 9]. Recent experiments [4, 23, 22, 21, 28] have shown that these networks do not deform globally affinely at scales below a chain length. This difference between the deformation at the largest scales and at the smallest is considered by J.E. Mark [18] to be “one of the central problems in rubberlike elasticity”. Non-affinity for some materials can be partially explained by variations in chain stiffness [12]. Other proposed causes include crosslink density [16, 3, 4, 23, 7, 21, 25, 26, 17, 6, 27, 10, 6], chain reorientation [20, 8], finite extensibility [22,2,1], strain-induced crystallization [14,2,1], and interchain excluded volume terms [25, 19, 26, 27]. It is likely is that there are many different simultaneous causes. A hypothesis proposed here is that these networks will deform topologically locally affinely (“locally affinely”). This means connected crosslinks deform affinely together, while crosslinks that are in the near neighborhood and unconnected may not. Given a crosslink in an elastomer network, there will be many more unconnected crosslinks than connected in the geometrically local vicinity. Affinity and network property correlations of affinity are studied here for models with and without intrachain excluded volume terms. We also test models with and without finite extensibility. All models tested here neglect interchain effects. THEORY We use simulations with five different energy models and two different boundary conditions (10 total cases) focused at the scale of the individual crosslink. Our network is randomly generated with 1000 crosslinks and 1998 chains within a unit cube—crosslinks have functionality 3 or 4. The simulation method is incapable of capturing interchain interactions. The Fixed boundary condition constrains all of the crosslinks within a small distance from the outside edge; the Free boundary condition constrains crosslinks at the bottom and top of the cube, but leaves the crosslinks along the sides free and does not preserve incompressibility—this creates a network that looks like it is experiencing necking. (See Figure 1.) For each of the constitutive model equations below, A is the Helmholtz free energy and r is the length of an individu
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