A general formulation of two-dimensional elastic-perfectly-plastic anti-plane straining is presented for materials with arbitrary anisotropic convex yield surfaces. Stress and strain distributions in plastic regions adjoining portions of the boundary are obtained directly in terms of the yield surface geometry. When specialized to the classical torsion problem, results lead directly to a generalization of the well-known plastic roof construction for limit loads. Examples of the determination of fully plastic stress distributions and corresponding limit torques are given for circular and rectangular shafts with various yield conditions. Another specialization is made to the contained plastic deformation created by longitudinal shearing of a body containing a sharp edge notch. Here the determination of the elastic-plastic boundary and strain distribution is reduced to a potential theory problem for a region in the stress plane bounded by straight line segments and a portion of the yield surface, and a membrane analogy is presented which allows effective visualization of the solution. A solution valid for small scale yielding near a crack is given in terms of a conformal transformation of the yield surface to a unit circle, and some specific examples are worked. Particular attention is given to single crystal type yield surfaces made up of straight line segments corresponding to discrete slip planes.
This communication demonstrates a simple, soft lithographic approach to the replication and metrology of nanoscale vertical displacements. We patterned test structures with regular patterns that minimize artifacts in measurements by atomic force microscopy. A composite stamp of poly(dimethylsiloxane) (PDMS) molded against the original test structure served as a template to generate polyurethane replicas. We replicated vertical displacements down to ∼1.5 nm. This replication demonstrates the capability of soft lithography to reproduce features with dimensions similar to those of large molecules.
Heterogeneous, patterned surfaces comprising well-defined hydrophobic and hydrophilic regions and having micrometer-scale periodicities were prepared by patterning the adsorption of omega-functionalized alkanethiolates in self-assembled monolayers (SAMs) on gold. Condensation of water on such surfaces resulted in drops that followed the patterns in the SAMs. These patterned condensation figures (CFs) acted as optical diffraction gratings for reflected (or transmitted) light from a helium-neon laser (wavelength of 632.8 nanometers). Under an atmosphere of constant relative humidity, the development of the condensation figure was monitored quantitatively, as the temperature of the surface was lowered, by following the change in intensity of a first-order diffraction spot. This experimental technique may be useful in the development of new types of optical sensors that respond to their environment by changing the reflectivity of patterned regions and for studying phenomena such as drop nucleation, contact angle hysteresis, and spontaneous dewetting and break-up of thin liquid films.
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