scholarly journals The constitutive tensor of linear elasticity: Its decompositions, Cauchy relations, null Lagrangians, and wave propagation

2013 ◽  
Vol 54 (4) ◽  
pp. 042903 ◽  
Author(s):  
Yakov Itin ◽  
Friedrich W. Hehl
Keyword(s):  
Wave Propagation ◽  
Linear Elasticity ◽  
Constitutive Tensor ◽  
Cauchy Relations

Rayleigh wave equations with couple stress: modeling and dispersion characteristic

Geophysics ◽  
2021 ◽  
pp. 1-64
Author(s):  
Yanqi Wu ◽  
Jianwei Ma
Keyword(s):  
Wave Propagation ◽  
Half Space ◽  
Scale Effect ◽  
Linear Elasticity ◽  
Rayleigh Waves ◽  
Couple Stress ◽  
Characteristic Length ◽  
Couple Stress Theory ◽  
Stress Theory ◽  
Homogeneous Half Space

In elastostatics, the scale effect is a phenomenon in which the elastic parameters of a medium vary with specimen size when the specimen is sufficiently small. Linear elasticity cannot explain the scale effect because it assumes that the medium is a continuum and does not consider microscopic rotational interactions within the medium. In elastodynamics, wave propagation equations are usually based on linear elasticity. Thus, nonlinear elasticity must be introduced to study the scale effect on wave propagation. In this work, we introduce one of the generalized continuum theories—couple stress theory—into solid earth geophysics to build a more practical model of underground medium. The first-order velocity-stress wave equation is derived to simulate the propagation of Rayleigh waves. Body and Rayleigh waves are compared using elastic theory and couple stress theory in homogeneous half- space and layered space. The results show that couple stress causes the dispersion of surface waves and shear waves even in homogeneous half-space. The effect is enhanced by increasing the source frequency and characteristic length, despite its insufficiently clear physical meaning. Rayleigh waves are more sensitive to couple stress effect than body waves. Based on the phase-shifting method, it was determined that Rayleigh waves exhibit different dispersion characteristics in couple stress theory than in conventional elastic theory. For the fundamental mode, the dispersion curves tend to move to a lower frequency with an increase in characteristic length l. For the higher modes, the dispersion curves energy is stronger with a greater characteristic length l.

NULL LAGRANGIANS IN LINEAR ELASTICITY

1995 ◽  
Vol 05 (04) ◽  
pp. 415-427 ◽  
Author(s):  
M.R. LANCIA ◽  
G. VERGARA CAFFARELLI ◽  
P. PODIO-GUIDUGLI
Keyword(s):  
Linear Elasticity ◽  
Energy Functional ◽  
Elasticity Tensor ◽  
Theory Of Elasticity ◽  
Stored Energy ◽  
Cauchy Relations ◽  
Null Lagrangian

The concept of null Lagrangian is exploited in the context of linear elasticity. In particular, it is shown that the stored energy functional can always be split into a null Lagrangian and a remainder; the null Lagrangian vanishes if and only if the elasticity tensor obeys the Cauchy relations, and is therefore determined by only 15 independent moduli (the so-called “rari-constant” theory of elasticity).

Electron Microscopy of Shock Loaded Polycrystalline Beryllium

1974 ◽  
Vol 32 ◽  
pp. 506-507 ◽  
Author(s):  
J. M. Galbraith ◽  
L. E. Murr ◽  
A. L. Stevens
Keyword(s):  
Electron Microscopy ◽  
Wave Propagation ◽  
Structural Change ◽  
Single Crystal ◽  
Diffraction Pattern ◽  
Shock Loading ◽  
Slip Systems ◽  
Compression Tests ◽  
X Ray Diffraction ◽  
X Ray

Uniaxial compression tests and hydrostatic tests at pressures up to 27 kbars have been performed to determine operating slip systems in single crystal and polycrystal1ine beryllium. A recent study has been made of wave propagation in single crystal beryllium by shock loading to selectively activate various slip systems, and this has been followed by a study of wave propagation and spallation in textured, polycrystal1ine beryllium. An alteration in the X-ray diffraction pattern has been noted after shock loading, but this alteration has not yet been correlated with any structural change occurring during shock loading of polycrystal1ine beryllium.This study is being conducted in an effort to characterize the effects of shock loading on textured, polycrystal1ine beryllium. Samples were fabricated from a billet of Kawecki-Berylco hot pressed HP-10 beryllium.

Computing cell tractions from particle displacements on silicone rubber substrata

1994 ◽  
Vol 52 ◽  
pp. 162-163
Author(s):  
Tim Oliver ◽  
Akira Ishihara ◽  
Ken Jacobsen ◽  
Micah Dembo
Keyword(s):  
Plane Stress ◽  
Linear Elasticity ◽  
Silicone Rubber ◽  
Mathematical Analysis ◽  
Elastic Recovery ◽  
Video Images ◽  
Cell Traction Forces ◽  
Latex Beads ◽  
Cell Traction ◽  
Better Than

In order to better understand the distribution of cell traction forces generated by rapidly locomoting cells, we have applied a mathematical analysis to our modified silicone rubber traction assay, based on the plane stress Green’s function of linear elasticity. To achieve this, we made crosslinked silicone rubber films into which we incorporated many more latex beads than previously possible (Figs. 1 and 6), using a modified airbrush. These films could be deformed by fish keratocytes, were virtually drift-free, and showed better than a 90% elastic recovery to micromanipulation (data not shown). Video images of cells locomoting on these films were recorded. From a pair of images representing the undisturbed and stressed states of the film, we recorded the cell’s outline and the associated displacements of bead centroids using Image-1 (Fig. 1). Next, using our own software, a mesh of quadrilaterals was plotted (Fig. 2) to represent the cell outline and to superimpose on the outline a traction density distribution. The net displacement of each bead in the film was calculated from centroid data and displayed with the mesh outline (Fig. 3).

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