Variational Problem and Elastic Torques

The minimum of (8.18) corresponds to the equilibrium helical (harmonic) structure with wavevector qo = dq /dz and pitch Pq = 2nlqo. Equation (8.18) is correct as long as Po a where a is molecular size. In the opposite case, local biaxiality becomes important (practically, Eq 1 pm, a k, nm). 

In precisely the same way, a spontaneously splay-deformed structure must correspond to the equilibrium condition with finite coefficient fsTi 7 0 in tensor (8.13). The corresponding term should be added to the splay term with (divn). If the molecules have, e.g., pear shape they can pack as shown in Fig. 8.7b. In this case, the local symmetry is Coov (conical) with a polar rotation axis, which is compatible with existence of the spontaneous polarization. However, such packing is unstable, as seen in sketch (b), and the conventional nematic packing (a) is more probable. The splayed stmcture similar to that pictured in Fig. 8.7b can occur close to the interface with a solid substrate or when an external electric field reduces the overall symmetry (a flexoelectric ejfecf). 

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Variational Problem and Elastic Torques 8.3.1 Euler Equation... 

The problem of definition of modulus applies to all tests. However there is a second problem which applies to those tests where the state of stress (or strain) is not uniform across the material cross-section during the test (i.e. to all beam tests and all torsion tests - except those for thin walled cylinders). In the derivation of the equations to determine moduli it is assumed that the relation between stress and strain is the same everywhere, this is no longer true for a non-linear material. In the beam test one half of the beam is in tension and one half in compression with maximum strains on the surfaces, so that there will be different relations between stress and strain depending on the distance from the neutral plane. For the torsion experiments the strain is zero at the centre of the specimen and increases toward the outside, thus there will be different torque-shear modulus relations for each thin cylindrical shell. Unless the precise variation of all the elastic constants with strain is known it will not be possible to obtain reliable values from beam tests or torsion tests (except for thin walled cylinders). 

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