Table 1.

Constitutive laws. The generalized curvature ${\mathit{\kappa }}_{\mathcal{L}}$ is associated with the bending κ₁, κ₂ about the principal directions (d₁, d₂) and the twist κ₃ about the longitudinal one (d₃), while ${\mathit{\sigma }}_{\mathcal{L}}=\mathbf{\text{Q}}(e\mathbf{\text{t}}-{\mathbf{\text{d}}}_3)$ is associated with the shears σ₁, σ₂ along the principal directions (d₁, d₂) and the axial extensional or compression σ₃ along the longitudinal one (d₃). The material properties of the rod are captured through the Young’s (E) and shear (G) moduli, while its geometric properties are accounted for via the cross-sectional area A, the second moment of inertia I and the constant ${\alpha }_c=\frac{4}{3}$ for circular cross sections [58]. The diagonal entries of the bending/twist $\mathbf{\text{B}}\in {\mathbb{R}}^{3\times 3}$ and shear/stretch $\mathbf{\text{S}}\in {\mathbb{R}}^{3\times 3}$ matrices are, respectively, (B₁, B₂, B₃) and (S₁, S₂, S₃). Pre-strains are modelled via the intrinsic curvature/twist ${\mathit{\kappa }}_{\mathcal{L}}^o$ and shear/stretch ${\mathit{\sigma }}_{\mathcal{L}}^o$.

deformation modes	strains	rigidities	loads
bending about d1	κ1	B1=EI1	${\tau }_1=B_1({\kappa }_1-{\kappa }_1^o)$
bending about d2	κ2	B2=EI2	${\tau }_2=B_2({\kappa }_2-{\kappa }_2^o)$
twist about d3	κ3	B3=GI3	${\tau }_3=B_3({\kappa }_3-{\kappa }_3^o)$
shear along d1	σ1	S1=αcGA	$n_1=S_1({\sigma }_1-{\sigma }_1^o)$
shear along d2	σ2	S2=αcGA	$n_2=S_2({\sigma }_2-{\sigma }_2^o)$
stretch along d3	σ3	S3=EA	$n_3=S_3({\sigma }_3-{\sigma }_3^o)$