Biharmonic Fields Research Articles

Conformal mapping on rough boundaries. II. Applications to biharmonic problems

We use a conformal mapping method introduced in a companion paper [Damien Vandembroucq and St\'ephane Roux, Phys. Rev. E 55, 6171 (1997)] to study the properties of biharmonic fields in the vicinity of rough boundaries. We focus our analysis on two different situations where such biharmonic problems are encountered: a Stokes flow near a rough wall and the stress distribution on the rough interface of a material in uniaxial tension. We perform a complete numerical solution of these two-dimensional problems for any univalued rough surfaces. We present results for a sinusoidal and a self-affine surface whose slope can locally reach 2.5. Beyond the numerical solution we present perturbative solutions of these problems. We show in particular that at first order in roughness amplitude, the surface stress of a material in uniaxial tension can be directly obtained from the Hilbert transform of the local slope. In the case of self-affine surfaces, we show that the stress distribution presents, for large stresses, a power-law tail whose exponent continuously depends on the roughness amplitude.

Representations of ?, biharmonic vector fields, and the equilibrium equation of planar elasticity

The equilibrium equation for an elastic body subjected to surface forces asserts the linear dependence of the Laplacian and the gradient of the divergence of the vector field which gives the displacement at each point. James Clerk Maxwell (1831–1879) was the first to point out that the component functions of such a field are biharmonic, i.e., their Laplacians are harmonic functions. Using only algebraic tools familiar to advanced undergraduates we show that the usual complex variable representation of two-variable biharmonic functions falls naturally out of a power series construction based on matrix representations of ℂ. Under the assumption of linear stress and strain components, this construction is then used to describe the solutions to the planar equilibrium equation in terms of the geometry of the Moebius plane.

Suppression of phase-locking chaos in long Josephson junctions by biharmonic microwave fields.

We consider the problem of phase locking of fluxon oscillations in long Josephson junctions irradiated by an external microwave field consisting of two harmonic signals, one at frequency {omega} and the other at frequency {omega}/2 added together (biharmonic driver). The analysis is performed in terms of a two-dimensional map constructed in terms of the fluxon time of flight and fluxon's energy inside the junction. As a result we find that the second driver enhances the stability of the phase-locking state and can be used to suppress the phase-locking chaos in the middle of the step induced by the first driver.

Periphractic construction of elasticity and plasticity on Gurtin's generalized stress functions and the criterion for yielding

An alternative approach is made to Gurtin's stress functions of non-Beltrami feature to paraphrase its anholonomic periphractic standpoint and to provide a basis to reconstruct the Theory of Yielding. The stress is defined with reference to a three-dimensional version of Einstein's coordinates. The meaning and construction of elasticity is attributed to the aeolotropic periphracticies of material continua. The multidimensional picture for the yielding manifold is reached in terms of average metric involving Killing's field. Account is taken of a microphysical restriction on variational degrees of freedom to assume the form of Klein-Goldon field. The fourth order partial differential equation is so secured for the field equation of yielding. The final forms of the boundary condition equations have to be fixed by taking account of the periphracticy effects, especially of the standpoint of the plasticity tensor B klmn , two components of which are independent for isotropic materials, owing to integration constants. That these equations agree with only a slight modification with the ones that have been proposed many years since is checked with reference to the isotropic uniform material. The modification is concerned also with the microphysical and thermodynamical implications. A spontaneous unification with the postulate of the principle of relativity is included summarizing also the relation between spatially-temporally fluctuating harmonic and biharmonic fields.