Compatibility conditions of continua using Riemann-Cartan geometry

Abstract

The compatibility conditions for generalised continua are studied in the framework of differential geometry, in particular Riemann-Cartan geometry. We show that Vallée's compatibility condition in linear elasticity theory is equivalent to the vanishing of the three dimensional Einstein tensor. Moreover, we show that the compatibility condition satisfied by Nye's tensor also arises from the three dimensional Einstein tensor which appears to play a pivotal role in continuum mechanics not mentioned before. We discuss further compatibility conditions which can be obtained using our geometrical approach and apply it to the micro-continuum theories.

arXiv comment19 pages; v2 extended references and minor corrections

References in corpus (14)

• Torsion Gravity: a Reappraisal
• Dual gauge field theory of quantum liquid crystals in two dimensions
• Emergent spacetime and gravitational Nieh-Yan anomaly in chiral $p+ip$ Weyl superfluids and superconductors
• Elasticity tetrads, mixed axial-gravitational anomalies, and 3+1d quantum Hall effect
• World Nematic Crystal Model of Gravity Explaining the Absence of Torsion
• Cartan's spiral staircase in physics and, in particular, in the gauge theory of dislocations
• Tetrads in solids: from elasticity theory to topological quantum Hall systems and Weyl fermions
• Continuum mechanics with torsion
• Torsional Monopoles and Torqued Geometries in Gravity and Condensed Matter
• A gauge theoretic approach to elasticity with microrotations
• Rotational elasticity
• Rotational elasticity and couplings to linear elasticity
• From continuum mechanics to general relativity
• New Gauge Symmetry in Gravity and the Evanescent Role of Torsion

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• A geometric formulation of Schaefer's theory of Cosserat solids
• First-order hyperbolic formulation of the pure tetrad teleparallel gravity theory
• Micropolar continua as projective space of Skyrmions