2D mechanical representation of superconducting magnets: A comparative study of plane stress and plane strain

Abstract

When approaching the mechanical design of a superconducting magnet, whenever possible the starting model is a 2D approximation. If rotational symmetry (solenoid-like winding) is present, the 2D representation is unique and contains no approximations. If, on the other hand, a non-axisymmetric system is opted for, the 2D representation is not unique and there are main options available, as plane stress, plane stress with thickness, plane strain and generalized plane strain. Considering z as the direction normal to the 2D plane, the plane stress option defines a stress state in which no normal or shear stresses perpendicular to the xy plane can occur (${\sigma }_z={\sigma }_{xz}={\sigma }_{yz}=0$). In this option, deformation can occur in the thickness direction of the element, which will become thinner when stretched and thicker when compressed; it is generally used for objects with limited depth (thin objects).

In contrast, plane strain refers to the fact that deformation can only occur in plane, which means that no out-of-plane deformation will occur (${ϵ}_z={ϵ}_{xz}={ϵ}_{yz}=0$). The plane strain option is generally appropriate for structures of nearly infinite length, relative to their cross section, that exhibit negligible length changes under load. The “generalized plane strain” option imposes the axial strain equal to a constant value; this condition does not reproduce the real operating condition of the magnet and is therefore excluded. The “plane stress with thickness” uses the same equations as the plane stress option but, output quantities are given per unit length defined with thickness; therefore, it is again excluded from the comparison. Superconducting magnets, such as dipoles, do not fit neatly into any of the above options: they are far from thin but deform longitudinally under load. This work reports a comparative study of plane stress and plane strain in the specific case study of a dipole magnet.