Finite element simulations of elasto-plastic processes in Nb3Sn strands
Introduction
Prediction of the contribution of the mechanical properties of Nb3Sn strand components to the intrinsic strain in the filaments after heat treatment and cooldown has been the subject of stress analysis since the early 1980s [1], [2], [3]. It is possible with relatively simple models (limited temperature dependence of material properties and bilinear plasticity) to achieve reasonable matches with measured values. However, there has been no systematic match of mechanical measurements, superconducting measurements and proper elasto-plastic stress modelling and all of the existing models fail at some point to match the experimental data.
This paper is the first step in a more comprehensive matching process. Based on the materials measurements over the last 60 years, parametric scalings for stress–strain curves from 4 K to 1000 K for the strand components are proposed. Elasto-plastic simulations on strands are performed and compared with the available database of superconducting and mechanical measurements, covering not only stress–strain curves but factors such as permanent length changes through the heat treatment, as a result of thermal cycling. The analysis is extended to strands with a steel jacket. Agreement is good, which is sufficiently encouraging to propose further mechanical measurements on ‘reference’ strands to allow more quantitative comparisons and if appropriate adjustments to the material properties.
Two major new features of the model are the inclusion of thermal stresses prior to heat treatment and cold work/residual stresses from the strand fabrication process. These features are critical to match the observed effects. It appears from interpretation of the mechanical measurements that the common assumption often used in the past, that the heat treatment period at about 650 °C is sufficient to anneal all components and relax internal stresses, is incorrect. The heat treatment modifies the internal stresses but does not completely relax them or anneal all components. This unfortunately greatly complicates the development of models to predict Nb3Sn filament strain in multistrand cables.
Section snippets
Finite element model
The finite element model contains two nodes, from 5 to 8 elements and has 1 degree of freedom (the linear strain of the strand). The elements used are the Ansys 8 LINK1 struts, connected in parallel with an area corresponding to the total cross-section area in the strand of the different components. The material model is multilinear isotropic hardening (Von Mises yield criteria with isotropic work hardening) for copper and bronze. Niobium, tantalum, steel and Nb3Sn are assumed to remain elastic
Loads
The primary loading is thermal. For simulating mechanical tests, a force is applied to one end of the strand.
The start temperature for the analysis is 293 K, and the residual strains for bronze route strand are assumed to be zero. Processing residual strains for internal tin and PIT strands are induced as described in the previous section, by applying an initial strain to the Nb and Ta components. After this, if appropriate, the steel element is activated. The residual strains do not strongly
Model verification
Model verification relies on deduction of overall strand properties that can be verified by mechanical or electrical measurements. There are four types, as given in the sections below.
Interpretation and discussion of results
The finite element model makes predictions of the internal stress system in the strand through the heat treatment and cooldown process. These are shown for the bronze-route and internal tin strands in Fig. 9, Fig. 10.
The residual stresses in the internal tin wire are clear, about 250 MPa in the Nb and 100 MPa in the copper. During heating to 923 K, the copper and bronze are compressed by the Ta and Nb (which are put into tension).
At the heat treatment temperature, stresses are generally low with
Conclusions
The predictions of the finite element model, where they can be quantitatively verified, appear quite accurate. They are also supported by qualitative observations of strand behaviour in cases where insufficient strand data is available to allow proper modelling. This suggests that the material models are realistic, giving an important basis for more complex modelling of superconducting strands.
The simple 1D material model appears adequate to predict all of the mechanical features of Nb3Sn
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