Strain Measures and Energies for Crimped Fibres and Novel Analytical Expressions for Fibre Populations: Ingredients for Structural Fibre Network Models

Abstract

This paper deals with the structural modelling of fibre networks with a focus on the description of populations of initially crimped fibres. It presents a systematic approach of introducing appropriate strain measures for single fibres based on a deformation decomposition and by transferring knowledge from the field of elastoplasticity. On this basis, for example, the often used Biot-type fibre strain measures lambda-lambda_{mathrm{w}} and lambda /lambda _{mathrm{w}}-1, with stretch lambda and waviness lambda _{mathrm{w}}, are consistently assigned to different classes of material strain (“additive”) or intermediate strain (“multiplicative”) descriptions, respectively. We review in this work different fibre strain energies based on the different stain measures and present extensive comparisons on the physical implications and the results on the fibre population and network scale. These investigations also include formulations with a Hencky-type energy based on a logarithmic strain. Furthermore, we present novel analytical expressions for fibre populations that make the evaluation of integral expression superfluous and thus lead to a significant reduction in computational time.