A stress analysis of superconducting solenoids is presented which includes a generalized plane strain (GPS) condition for the axial strain. The GPS condition is introduced on the assumption that the deformation of a solenoid from a right circular cylinder is small. The GPS assumption results in an analytic solution for all three components of stress and strain in a solenoid. The work is presented in the context of the historical development of stress analysis for solenoids. The general stress equations for a magnetic solenoid are formulated. The relationship between a right cylinder deformation and the generalized plane strain condition is examined for the physical conditions in the central region of a solenoid magnet. The general analytic solutions of the stress equations are given for the cases of magnetic and thermal loading. The constant coefficients are determined for cases of common interest in solenoid magnet design. The analytic results are compared with numerical analysis results for an example solenoid consisting of a single coil with external reinforcement. In particular, the degree to which the axial strain is a constant and satisfies the GPS assumption is examined for the example solenoid. The analysis reveals features of the axial stress in solenoids, including the Poisson’s ratio induced axial stress and the axial stress distribution between coil and reinforcement during cooldown and operation. The strong agreement between the GPS and numerical analysis results shows that the assumptions contained in the GPS analysis accurately represent the conditions in the central region of a solenoid magnet.
The plastic behavior of a superconducting material is investigated and the corresponding elastoplastic formulation for the distribution of stress and strain in a superconducting solenoid magnet is presented. The analysis calculates stress and strain at the midsection, where tangential stress exhibits its maximum value and shear stress is negligible. The prediction of stress and strain is essential for both the mechanical and electrical design of high-field superconducting magnets containing Nb 3 Sn superconductor. The concept of plasticity is introduced for the first time in the context of magnet design for Nb 3 Sn conductor and compared to alternative approaches using conventional elasticity theory. Individual coil sections of a superconducting magnet can be reinforced by an outer section of structural material, the effect of which is included in this formulation. The results show that the elasticity approach using the ''secant modulus method'' does not fully predict the strain distribution; however, it can be used to approximate the stresses. It is shown that for an accurate strain prediction the true nonlinear elastoplastic nature of the superconducting materials should be considered and proper yield criteria should be used. The inaccurate prediction of strains ͑tangential or radial͒ can affect critical current density and the evaluation of the reinforcements.
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