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https://hdl.handle.net/2142/87772
Description
Title
On the Continuum-Mechanics of Nematic Elastomers
Author(s)
Anderson, David Robert
Issue Date
1999
Doctoral Committee Chair(s)
Eliot Fried
Carlson, Donald E.
Department of Study
Theoretical and Applied Mechanics
Discipline
Theoretical and Applied Mechanics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Applied Mechanics
Language
eng
Abstract
A nematic elastomer is a rubber-like solid formed by the cross-linking of a polymeric fluid that includes nematic liquid crystalline molecules. We develop a continuum theory with the basic kinematic ingredients being the deformation of material particles and the orientation of the nematic microstructure. The kinetics of our theory consist of separate momentum balances for forces acting conjugate to each kinematic variable. We restrict our attention to a purely mechanical setting, so the thermodynamic structure rests on an energy imbalance that serves in lieu of the first and second laws of thermodynamics. We consider only nematic elastomers that are incompressible and microstructurally inextensible. In our treatment of these material constraints, we start with a mathematical decomposition of the dependent fields based on the geometry of the constraint manifold. This naturally gives rise to active and reactive components, where only the former enter into the energy imbalance because the latter automatically expend zero power in processes consistent with the constraints. The reactive components are scaled by constitutively indeterminate multipliers. We introduce constitutive equations for the active components, and the requirement that these equations be consistent with the energy imbalance in all processes leads to the active components being determined by an energy density response function depending on the deformation gradient, the orientation, and the orientation gradient. After enforcing the requirements of observer independence and material symmetry for such a function, the representation is expressed in terms of 40 scalar invariants. To obtain a reduced form of the energy response function, we use a polynomial approximation, in terms of the invariants in which we view the principal extension ratios and the magnitude of the nondimensionalized orientation gradient as small quantities. Our energy density encompasses the successful Mooney-Rivlin description of rubber and the Oseen-Zocher-Frank description of nematic fluids as specializations. To investigate the nature of solutions, we derive a necessary and sufficient condition for ellipticity of the equilibrated governing system of equations. In situations where the ellipticity condition fails, we examine the existence of solutions that allow for surface defects. Several example energy densities are considered to study the implications of these conditions.
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