Two-scale geometric modelling for defective media: An approach using fibre bundles

A new geometrically exact micro-structured model is constructed in order to propose a generalisation of the notion of Riemann–Cartan manifolds. The construction is done using the theory of affine bundles, Ehresmann connections, and solder forms. This model is based on the concept of two different length scales: a macroscopic scale—a manifold <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="double-struck">B</mml:mi> </mml:mrow> </mml:math> of dimension 1, 2, or 3—and a microscopic scale—an affine space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="script">B</mml:mi> </mml:mrow> <mml:mrow> <mml:mover accent="false"> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> of dimension 3 at each <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mover accent="false"> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>∈</mml:mo> <mml:mi mathvariant="double-struck">B</mml:mi> </mml:mrow> </mml:math> . As they interact with each other, they produce emergent behaviours such as dislocations (torsion) and disclinations (curvature). A first-order placement map <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="bold">F</mml:mi> <mml:mo>:</mml:mo> <mml:mi mathvariant="normal">T</mml:mi> <mml:mi mathvariant="script">B</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">T</mml:mi> <mml:mi mathvariant="script">E</mml:mi> </mml:mrow> </mml:math> between a micro-structured body <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="script">B</mml:mi> </mml:mrow> </mml:math> and the micro-structured ambient space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="script">E</mml:mi> </mml:mrow> </mml:math> is constructed, allowing to pull back the ambient geometry onto the body <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="script">B</mml:mi> </mml:mrow> </mml:math> . In order to allow for curvature to arise, F is, in general, not required to be a gradient. Central to this model is the new notion of compatible pseudo-metric, providing, in addition to a macroscopic metric (the usual Cauchy–Green tensor) and a microscopic metric, a notion of coupling between the microscopic and macroscopic realms. A notion of frame indifference is formalised and invariants are computed. In the case of a micro-linear structure, it is shown that the data of these invariants are equivalent to the data of the pseudo-metric.