## 1 Introduction  
Disk springs and slotted disk springs have the advantages of a compact configuration, nonlinear characteristics, easy maintenance, dissipation energy capacity, etc. The static behavior of disk springs and slotted disk springs is a subject of long-standing concern in engineering with both practical importance and scientific fascination [1; 2; 3; 4]. The most common applications for disk springs are clutches [5; 6; 7], sliding hinge joints [8; 9; 10], vibration absorbers [11; 12; 13], energy harvesters [14; 15], and rocking bridge piers [16]. Compared with conical disk springs, slotted disk springs, also known as diaphragm springs, have a larger deflection and a lower stiffness given their regularly arranged slots [17; 18]. A quasi-zero stiffness isolator was developed by [PERSON] [19], in which a slotted disk spring was used to provide negative stiffness. Slotted disk springs have a promising prospect in reducing the damage caused by exceptional or destructive seismic events. The establishment of a mechanical model for slotted disk springs, which can properly describe the load-deflection relationships, has great significance for the application of these springs. A mechanical model for slotted disk springs was developed by incorporating the influences of slots based on those of conical disk springs. [PERSON] and [PERSON] [20] are pioneers of research on disk springs, and they performed seminal work by supplementing [PERSON]'s hypothesis and proposing a load-deflection formula for uniform cross-sectional disk springs. Subsequently, a series of studies were conducted to characterize the nonlinear characteristics of disk springs [2; 3; 21; 22; 23; 24; 25]. To improve the load bearing capacity and optimize the stress distribution of disk springs, research on nonuniform disk springs was carried out. For example, introducing a dimensionless parameter to characterize the thickness features, [PERSON] et al. [26] proposed a new mathematical model for radially tapered disk springs with a linear variable thickness. [PERSON] et al. [27] proposed novel parabolically varying thickness disk springs and derived a new analytical model to predict their load-deflection relationships. [PERSON] et al. [28] developed an analytical solution to characterize the load-deflection characteristics of stepped disk springs, and the finite element method was used to verify the proposed model. Moreover, explorations of the static behaviors of nonuniform disk springs were conducted by [PERSON] [29] and [PERSON] [30] using the finite element method. The deflections of slotted disk springs consist of a rigid deflection and a bending deflection. To clarify the relationship between the load and total deflection of the slotted disk spring, [PERSON] [17] treated the slotted disk spring as a combination of the coned disk spring and a number of cantilever beams with variable widths and performed a systematic theoretical analysis. A novel mathematical model for slotted disk springs was developed by [PERSON] and [PERSON] [31] in which the finite rotation and large deflection theories of a beam and a conical shell were used. [PERSON] et al. [32] proposed an energy method to characterize the mechanical characteristics of bent slotted disk springs, and an inverse algorithm was developed to design slotted disk springs with the desired load-deflection characteristics [33]. It should be mentioned that the above mathematical models developed to characterize the load-deflection characteristics were only employed for slotted disk springs with uniform thickness. Furthermore, the hysteresis caused by friction, which has a significant influence on the load-deflection relationships during loading and unloading, was neglected. [PERSON] and [PERSON] [34] developed a new load-deflection relationship for uniform slotted disk springs by considering the friction occurring due to the contact between the edges of disk springs and supports. However, this relationship is incapable of predicting the load-deflection characteristics of nonuniform slotted disk springs.