## 1 Introduction  
Automotive, non-ferrous alloys and advanced high-strength steels have been extensively investigated by many researchers owing to their lightweight and excellent mechanical properties. Lightweight alloys, such as aluminum and magnesium alloys, exhibit superior specific strengths to conventional steels. By contrast, the increased strength, which can exceed 1 GPa, enables the advanced high-strength steel (AHSS) sheets to have a thinner gauge, thus, rendering them more competitive compared to low-density, lightweight alloys. The excellent strength and enhanced ductility of the AHSS can be achieved owing to their well-controlled microstructure and deformation mechanisms. For example, dual-phase steels or transformation induced steels introduce a hard martensite phase embedded in the softer ferrite phase or transforming martensite from meta-stable austenite. The strength of lightweight alloys can be achieved through the dispersion of optimally designed second phases, such as precipitations. However, even in the presence of these improved plasticity properties, such as tensile strength and elongation at fracture, the intricate microstructure leads to a further complexity in the mechanical properties, especially when the deformation paths become convoluted. The complex mechanical responses include the Bauschinger effect and transient flow behavior under reversed loading path [1; 2; 3; 4; 5]. These anisotropic hardening behaviors of sheet metals have been reported to be critical factors for the springback of automotive sheet parts [6; 7; 8; 9]. More recently, the anisotropic hardening behavior of advanced sheet metals has been further investigated under more complex loading paths than simple reverse loading. For instance, the deformation of ferritic-phased steels shows significant stress overshooting (or larger stress than the monotonic flow stress) when the loading path changes to being orthogonal to the previous loading path [10; 11; 12; 13; 14; 15]. Interestingly, dual-phase steels with mixed ferrite and martensite phases exhibit a clear softening under the same loading path changes [15; 16; 17; 18]. These plastic behaviors under orthogonal loading path changes are called cross-hardening and cross-softening, respectively. Experimental observations on these complex mechanical behaviors, which cannot be explained through the simple isotropic plasticity model, require the implementation of new anisotropic models in the field of metal forming and plasticity. Indeed, this complex behavior can influence the formability and springback of advanced sheet metals [9]. In the literature, a significant number of works have proposed models for anisotropic hardening behavior. Kinematic hardening is a representative concept, which explains the Bauschinger effect and transient flow hardening at load reversal by introducing yield surface translation during plastic deformation. The kinematic hardening models were pioneered by [PERSON] [19] and [PERSON] [20], and further extended by adding nonlinear terms [21] or by coupling with the isotropic hardening model. The series of isotropic and kinematic hardening models was well implemented into finite element (FE) simulations for sheet metals, especially when metals exhibit the Bauschinger effect, transient behavior, and permanent softening under reversed loading paths [22; 23; 24; 25]. The kinematic hardening--based anisotropic hardening models were further extended by combing the distortional hardening concept to reproduce mechanical responses under rather complex loading conditions, beyond the simple loading-reverse loading path. [PERSON] and [PERSON] [26] introduced the distortion of the yield surface by controlling the size of the effective stress. [PERSON] and [PERSON] [27; 28] employed the fourth-rank anisotropic tensor as a function of plastic deformation, but the fundamental basis they used remained that of isotropic-kinematic hardening.