On geometric conditions for reduction of the Moreau sweeping process to the Prandtl-Ishlinskii operator

Figure 1.  Hysteresis loops of a Prandtl-Ishlinsii operator with 3 springs. $g$ is the deformation and $R$ is the stress (cf. (25)). Thick (thin) lines correspond to the increasing (decreasing) deformation. The direction of motion along the loops is clockwise. Both the outer hysteresis loop and the inner loop are centrally symmetric (with respect to the corresponding loop center). The ascending branch of the inner loop is congruent to the part $AB$ of the ascending branch of the outer loop. Dashed line $OC$ is the loading curve. The ascending branch $ABC$ of the outer loop is obtained by dilation of the loading curve with the scaling factor 2 and translation which maps the initial point $O$ to $A$

Figure 2.  Connection of 5 nodes with 9 elastic-ideal plastic springs

Figure 3.  A nonlinear spring connecting two nodes in the model of Moreau can be represented as a combination of a dry friction element and an ideally elastic spring obeying the Hooke's law (3). The deformation $e_{ij}$ of the ideal spring is called elastic deformation; $\varepsilon_{ij}$ is the elongation of the distance between the nodes $i$ and $j$ (with respect to the distance $\Delta_{ij} = x_j^*-x_i^*$ at zero configuration). Dry friction between the box $B$ and the surface $S$ produces the friction force, which is opposite to the ideal spring force at all times (the quasistatic model). The magnitude of the friction force is limited by the maximal value $a_{ij}\rho_{ij}$. Therefore, the ideal spring deforms but the box does not move with respect to the surface $S$ as long as $|\sigma_{ij}| < a_{ij}\rho_{ij}$. When $|\sigma_{ij}| = a_{ij}\rho_{ij}$, the deformation $e_{ij}$ of the ideal spring and the force remain constant, while the box moves with respect to the surface in the direction of the spring force

Figure 4.  An example of an input-output trajectory $c_0\, c_1\, \ldots\, c_{15}$ of a nonlinear spring with $c_n = (\varepsilon_{ij}(t_n), e_{ij}(t_n))$ and $t_0 < t_1 < \cdots$. The time series of the deformation $\varepsilon_{ij}$ and the elastic deformation $e_{ij}$ are related by the stop operator $e_{ij} = {\mathcal S}_{\rho_{ij}}[\varepsilon_{ij}]$. Slanted lines have the slope $1$; horizontal lines are $e_{ij} = \pm \rho_{ij}$

Figure 5.  (a) Parallel connection of springs in the Prandtl-Ishlinskii model. (b) A reducible connection of springs

Figure 6.  A "linear" connection of springs, which is equivalent to the Prandtl-Ishlinskii model

Figure 7.  Violations of conditions of Theorem 1. The shaded area represents the polytope $\Pi\cap V$. The vector $f_0$ used in the definition (24) of the Moreau process points in the direction of the vector $B_0B_1$. (a) The prism $\Omega$ does not belong to $\Pi\cap V$. The polyline $B_0B_1B_2C_1C_2C_3$ that contains the polyline $\gamma = B_0B_1B_2$ represents the trajectory of a solution $u$ to (24) for the input $g(t)f_0$ where $g$ increases from zero to a maximum value $g_*$ and then decreases to the minimum value $-g_*$. Formula (36) defines a different polyline $B_0B_1B_2DC_3$ for the same input. (b) Conditions (31), (32) are violated because ${\rm dim}\, F_{1} = {\rm dim}\, F_2$. The polylines $B_0B_1B_2B_3C_1C_2C_3$ and $B_0B_1B_2B_3D_1D_2C_3$ are the trajectory of the inclusion (24) and the curve prescribed by formula (36) in response to the same input as in panel (a).

Figure 8.  A 'minimal' system, which may be not reducible to the PI operator (depending on the parameters of springs)

Figure 9.  A graph $G$ of a general linear connection of $m$ springs (thick lines). Thin dashed lines represent possible additional edges as introduced in Theorem 2