Spin–orbital structure of the Timoshenko shear factor: An anisotropic Cosserat reduction

This work presents a one-dimensional Micropolar Mechanics (MM) reduction of Timoshenko beam theory (TBT) in which the cross-section is identified with a rigid microcontinuum endowed with an anisotropic MM constitution. The direct integration of the three-dimensional micropolar balance laws over the section yields a system in which two structurally distinct shear coefficients emerge: κ v in the transverse balance, accounting for the total shear flow through the section, and κ θ in the angular balance, representing the spin–orbital transfer between translational and rotational channels. Both coefficients are independent and parametrised by the anisotropic shear modulus μ ∗ and the anisotropy parameter η . Static consistency of the reduced model determines the micropolar characteristic length and ensures exact recovery of the Euler–Bernoulli bending stiffness for any pair ( κ v , κ θ ) . Classical TBT is recovered structurally at μ ∗ = μ , η = κ s − 2 , where κ s is Cowper’s coefficient (Cowper, 1996). The general case κ v ≠ κ θ provides a structural counterpart to the two-coefficient TBT reported in Franco-Villafañe and Méndez-Sánchez (2014). The isotropic limit η = 0 produces the structural identity κ θ = 2 ( κ v − 1 ) , which illuminates the spin–orbital reading of the two channels.