Abstract:Nb ₃Sn superconducting magnets can produce high magnetic fields during operation, and the Nb ₃Sn superconducting coils subjected to strong electromagnetic force can result in great mechanical strain. The strain sensitivity of Nb ₃Sn superconducting material will degrade the critical performance of the Nb ₃Sn magnet coil, which has a significant impact on the safety and stability of magnets. Therefore, it is of great scientific significance to accurately calculate the mechanical behavior of superconducting magnets under electromagnetic force. Nb ₃Sn superconducting magnets are mainly made of superconducting wires wound into coil structure and solidified by epoxy resin. Nb ₃Sn superconducting wire is a composite structure mainly made of multiple microfilament embedded in a copper matrix. Therefore, the size from superconducting filaments to superconducting magnets spans several orders of magnitude, which brings challenges for accurate analysis of the mechanical deformation of superconducting coils. Firstly, the representative element (RVE) homogenization method is used to analyze the equivalent mechanical parameters of the whole coil. By comparing the results of the equivalent homogenization model and the actual structure of the coil, it is found that there are significant errors in the equivalent homogenization model. Therefore, we propose a bidirectional homogenization analysis method with high accuracy and low computational cost to study the stress-strain distribution of each component material (Nb ₃Sn filament, copper, and epoxy resin) in the superconducting coil. This method does not require large-scale numerical modeling and the results of this method are in good agreement with that of the actual composite coils, which verifies the effectiveness and accuracy of this method. Finally, based on the proposed multi-scale method, we discuss the stress-strain variations of each layer of the Nb ₃Sn superconducting coil versus turns and layers under the electromagnetic force in detail.