Inhomogeneous deformations

All other methods start with a director deformation the relaxation of which takes place in a plane perpendicular to the boundaries. In this case standard methods for the determination of the director orientation, which are also applied to the determination of elastic constants, can be used (e.g. light intensity measurements behind crossed polarizers or measurement of the dielectric constant). Usually a cell with planar surface orientation is used. By means of a magnetic [65] or electric [66] field perpendicular to the plates and somewhat larger than the critical value, a small splay deformation of the director is produced. The relaxation to the planar equilibrium orientation after switching off the field is followed by the methods discussed above. In contrast to the twist deformation, inhomogeneous rotation of the director causes a shear flow in the cell, which is called backflow. The observed relaxation time for a small deformation will be shorter than the analogue to Eq. (40)... 

Nevertheless, as response data have accumulated and the nature of the porous deformation problems has crystallized, it has become apparent that the study of such solids has forced overt attention to issues such as lack of thermodynamic equilibrium, heterogeneous deformation, anisotrophic deformation, and inhomogeneous composition—all processes that are present in micromechanical effects in solid density samples but are submerged due to continuum approaches to mechanical deformation models. 

Inelastic deformation of any solid material is heterogeneous. That is, it always involves the propagation of localized (inhomogeneous) shear. The elements of this localized shear do not occur at random places but are correlated in a solid. This means that the shears are associated with lines rather than points. The lines may delineate linear shear (dislocation lines), or they may delineate rotational shear (disclination lines). The existence of correlation means that when shear occurs between a pair of atoms, the probability is high that an additional shear event will occur adjacent to the initial pair because stress concentrations will lie adjacent to it. This is not the case in a liquid where the two shear events are likely to be uncorrelated. 

Correlated plastic deformation in polymers is very evident in the necking of polymeric rods or filaments. This is one form of inhomogeneous deformation. Experiments with Nylon filaments, for example, have shown that necks in them behave quite similarly to the Lueders bands observed in steel (Dey, 1967). This strongly suggests that plastic deformation in Nylon is associated with the motion of dislocations. 

Note 2 An inhomogeneous deformation is one in which the incremental changes in the undeformed and deformed co-ordinates are related by... 

Note l For an inhomogeneous deformation, the material derivative has to be used to find time derivatives of strain. 

Rivlin-Ericksen tensor of order n, for a viscoelastic liquid or solid in homogeneous deformation, is the nth time derivative of the Cauchy strain tensor at reference time, t. Note 1 For an inhomogeneous deformation the material derivatives have to be used. 

Strain rate, test temperature and the thermal history of the specimen all affect the appearance of shear bands in a particular glassy polymer [119]. The differences in morphology of shear bands was proposed to be due to different rates of strain softening and the rate sensitivity of the yield stress. Microshear bands tend to develop in polymers with a small deformation rate sensitivity of Oy and when relatively large inhomogeneities exist in the specimen before loading. This is sometimes characterized by a factor e j, introduced by Bowden in the form [119] ... 

---