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elastic  function  half space  half  infinite  point  radially inhomogeneous  secondary  seismic  semi infinite  sources  space  surface  tube  wave  waves 
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Preview: The Quarterly Journal of Mechanics and Applied Mathematics - current issue

The Quarterly Journal of Mechanics and Applied Mathematics Current Issue

Published: Thu, 15 Jun 2017 00:00:00 GMT

Last Build Date: Thu, 03 Aug 2017 08:49:35 GMT


Necking in two-dimensional incompressible polyconvex materials: theoretical framework and numerical simulations


We show examples of 2D incompressible isotropic homogeneous hyperelastic materials with a polyconvex stored-energy function that present necking. The construction of the stored-energy function of a material satisfying all those properties requires a fine search. We used the software Algencan to perform numerical experiments and visualise necking for the examples constructed. The algorithm is based on minimisation of the elastic energy under the nonconvex constraint of incompressibility.

Controlling Flexural Waves in Semi-Infinite Platonic Crystals with Resonator-Type Scatterers


We address the scattering and transmission of a plane flexural wave through a semi-infinite array of point scatterers/resonators, which take a variety of physically interesting forms. The mathematical model accounts for several classes of point defects, including mass-spring resonators attached to the top surface of the flexural plate and their limiting case of concentrated point masses. We also analyse the special case of resonators attached to opposite faces of the plate. The problem is reduced to a functional equation of the Wiener–Hopf type, whose kernel varies with the type of scatterer considered. A novel approach, which stems from the direct connection between the kernel function of the semi-infinite system and the quasi-periodic Green's functions for corresponding infinite systems, is used to identify special frequency regimes. We thereby demonstrate dynamically anisotropic wave effects in semi-infinite platonic crystals, with particular attention paid to designing systems that exhibit dynamic neutrality (perfect transmission) and localisation close to the structured interface.

Vortex/Tollmien–Schlichting wave interaction states in the asymptotic suction boundary layer


A self-sustaining interaction between a roll/streak structure and a three-dimensional Tollmien–Schlichting wave is considered at high-Reynolds-number within the asymptotic suction boundary layer. Strongly nonlinear governing equations, taking the form of a vortex–wave interaction (VWI) are derived and solved numerically. Finite amplitude travelling wave states, bifurcating from the lower branch of the linear neutral curve, are obtained. These states exhibit spanwise focusing, developing steep wall-shear gradients at specific spanwise locations as the wave amplitude rises. A spanwise-local analytic analysis reveals explicitly how the solution gradually loses regularity as the nonlinearity of the VWI system is increased.

Elastic waves inside and on the surface of a half-space


The notions of ‘elementary’ seismic sources and ‘elementary’ earthquakes are introduced, as being associated with ‘elementary’ tensorial point forces with a $\delta$-like time dependence (where $\delta$ is the Dirac delta function). The tensorial character of these forces, known in Seismology as the dipole (or double-couple) representation, is given by the tensor of the seismic moment. A regular seismic source and a regular earthquake can be represented as a superposition of elementary sources and, respectively, elementary earthquakes, governed by a space-time structure factor of the seismic focal region. All these are new concepts. Elementary seismic sources are considered here for a homogeneous isotropic elastic half-space bounded by a free plane surface, the sources being located at an inner point in the half-space. A transient regime of generation and propagation of seismic waves is identified, as distinct from the stationary regime of elastic vibrations. This is another new concept. It is shown that elementary seismic sources produce (double-shock) spherical-shell waves (in the wave region), which are the well-known $P$ and $S$ waves associated with the feeble tremor in the recorded seismograms. Their mathematical expression, derived here from the tensorial force, differs from known, particular cases. These waves are called here collectively ‘primary’ waves. It is shown that the primary waves interact with the surface of the half-space, where they give rise to ‘secondary’ wave sources, placed on the surface. The secondary waves generated by the secondary sources (which may be called ‘surface seismic radiation’) are estimated here in a simplified model. It is shown that the secondary waves have a delay time in comparison with the primary waves and give rise to a main shock and a long seismic tail, in qualitative agreement with the seismic records. The secondary wave introduced here is a new concept; the main shock and its long tail derived here are elements of novelty. Similarly, the secondary waves generated by an internal discontinuity in the elastic properties of the half-space (an interface parallel with the free surface) are also estimated; it is shown that the discontinuity reduces appreciably the singular main shock on the free surface of the homogeneous half-space.

Cylindrically Anisotropic and Radially Inhomogeneous Elastic Tube Under Surface Loadings


Explicit solutions of a cylindrically anisotropic and radially inhomogeneous elastic tube subjected to loadings at the inner and outer surfaces of the tube are obtained. Radially inhomogeneity is realized by assuming the elastic stiffnesses exercising a power–law function in the radial direction. For a two-dimensional deformation the result resembles Stroh’s formalism in a rectangular coordinate system. The associated eigenequation is derived and solved analytically. Nonzero total axial force and resultant torque are recognized in general. A cylindrical hole in an infinite space and a solid cylinder (or a cylinder with a pin hole at the centre) are special cases of a tube. Both are also explored and solutions are forthrightly obtained. Conditions of having infinite stresses at the centre of a solid cylinder are discussed. An example of a cylindrically orthotropic and radially inhomogeneous elastic tube subjected to surface loadings is given. The stress amplification and shielding effects are considered. When the power–law index equals zero, homogeneity is resumed and results are analogy to the published literature.