# T-Splines for Isogeometric Analysis of the Large Deformation of Elastoplastic Kirchhoff-Love Shells
## 1 Introduction  
Thin-walled structures are widely used in various engineering fields, e.g., aerospace, vehicular, and machinery, due to their specific advantages such as easy fabrication, efficient material use, and excellent mechanical properties. In the fabrication process or in practical operations, thin-walled structures comprising metallic materials may undergo large displacements, rotations, and strains under extreme loading conditions. Two classical situations are the sheet metal forming process and automotive collision. How to efficiently and accurately capture the deformation behaviors of thin-walled structures under strong nonlinearities has long been a hot topic, and presents many challenges. A series of traditional finite element methods have been proposed to solve geometrical or material nonlinearity in various types of shell structures [1; 2; 3; 4; 5; 6; 7]. There are two dominant shell assumptions that are widely used in shell analysis. One is the Reissner-Mindlin (RM) theory for thick shell analysis, which requires only C0 continuity across element boundaries. Another is the Kirchhoff-Love (KL) theory for thin shell analysis, for which a C1 discretization is necessary. Research on the latter has been popular in the last few decades with the development of isogeometric analysis (IGA) [8; 9]. The requirement of C1 continuity across element boundaries, which is a primary barrier to the development of KL shell theory in finite element analysis, has been naturally satisfied in IGA. What is more, the Kirchhoff-Love shell theory is applicable to thin shell structures with \(R/t>20\), where \(R\) is the radius of curvature of the surface and \(t\) is the thickness of the shell. However, for the analysis of thick shells, RM shell theory was found to be more accurate than KL theory because RM shell theory considers the transverse shear deformation, which is negligible in case of KL shell theory. The concept of isogeometric analysis (IGA) was first proposed in 2005 [9]. After decades of development, IGA has gradually demonstrated its inherent advantages in the seamless integration of computer-aided design (CAD) and computer-aided engineering (CAE) for many application fields, including thin-walled structures. In IGA, the spline basis functions to express geometric models in CAD can be directly applied to the analysis as the shape functions without need for conversion. The reusing of geometric models helps researchers avoid the tedious manipulation of mesh generation and avert the error caused by discretization in the traditional finite element method. Furthermore, higher-order and smooth basis functions of IGA could improve the accuracy and robustness of the numerical analysis [8; 10; 11]. Since nearly all CAD software takes nonuniform rational B-splines (NURBS) as the basis function in representing geometric models, in the thin-walled structure analysis, the NURBS-based IGA has been extensively investigated and successfully applied to solid shells [12; 13], hierarchic shells [14; 15], RM shells [16; 17; 18; 19; 20; 21], KL shells [22; 23; 24; 25; 26; 27], and blended shells [28]. From the perspective of plastic material modeling, the crucial problem is the derivation of the plastic constitutive equations for isogeometric analysis. A lot of work has been devoted to the research of this topic. Three-dimensional elastoplastic formulations have been applied to isogeometric analysis based on Bezier extraction of NURBS [29; 30], which can be embedded in the existing FEA program. In references [31; 32; 33], IGA was used for three-dimensional and two-dimensional nearly incompressible large deformation plastic problems. The implementation of plastic constitutive equations of shell elements for isogeometric analysis is a more challenging topic.