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Abstract
Riemann-Cartan geometry is used to model continuum with defects. In order to illustrate the differences induced by two possible definitions for the strain (spatial or material) in this framework, propagation of 3D waves is studied for a simple example of infinite continuum with uniform and stationary defects density. Anisotropy and attenuation are caught by both models even if these effects are quite different. Furthermore the material strain induces chirality and uniform breathing mode.
Highlights
The role played by geometry in physics is commonly acknowledged
In 1909, Cosserat brothers introduce continuum model involving independent field of rotation in addition to the displacement. Inspired by this approach Cartan has developed continuum models based on Riemann-Cartan (RC) manifold, endowed with the metric and an affine connection ∇ which replaces the classical gradient operator ∇ [1]
A uniform distribution of screw dislocation within an infinite continuum is modeled by a constant torsion density, so that Sjik = 0 except S213 = −S213 = S
Summary
In 1909, Cosserat brothers introduce continuum model involving independent field of rotation in addition to the displacement Inspired by this approach Cartan has developed continuum models based on Riemann-Cartan (RC) manifold, endowed with the metric (measuring the shape change) and an affine connection ∇ which replaces the classical gradient operator ∇ [1]. According to Noll’s definition: a continuum is said to be homogeneous if there exists a state of deformation of material manifold in which the mass density is uniform [5] Following this approach, one of the authors proposes a class of non-homogenity allowing discontinuities of scalar fields and vector fields described by torsion and curvature [6]. Application to wave propagation in non-homogeneous continuum is developed in [7] under the hypothesis of covariant spatial strain for continuum
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