Visco-Energetic solutions to one-dimensional rate-independent problems

Figure 1.  The double-well potential $W$ with its convex envelope in bold (left picture) and an energetic solution $u$ in the case of a strictly increasing load $\ell$ (right picture).

Figure 2.  BV solution for a double-well energy $W$ with an increasing load $\ell$. The blue line denotes the path described by the optimal transition $\vartheta$ solving (4).

Figure 3.  Visco-Energetic solutions for a double-well energy $W$ with an increasing load $\ell$. When $\mu>-\min W''$ (left picture) the solution jumps when it reach the maximum of $W'$ and the transition is the ''double chain'' obtained by solving the Incremental Minimization Scheme with frozen time $t$. When $\mu$ is small (right picture) the optimal transition $\vartheta$ makes a first jump connecting $\mathit{u}_{\tiny \mathsf L}(t)$ with $u_+$ according to the modified Maxwell rule (9): $\mathit{u}_{\tiny \mathsf L}(t)$ and $u_+$ corresponds to the intersection of $W'$ with the red line, whose slope is $-\mu$.

Figure 4.  The one-sided slopes and the stability region when $W$ is a double-well potential. For some suitable choices of $\delta$, $\mathit W_{\mathsf{i}\mathsf{r},{\delta}}'$ is intermediate between $W'_{\mathsf i\mathsf r}$ and $W'$.

Figure 5.  $\mathsf D$-Maxwell's rule for a double well potential: when $\delta=\frac{\mu}{2}|\cdot|^2$, the ''last point'' where $W'$ and $\mathit W_{\mathsf{i}\mathsf{r},{\delta}}'$ coincide is such that the total area between the graph $W'$ and the line whose slope is $-\mu$ is zero.

Figure 6.  Visco-Energetic solution of a double-well potential energy with an oscillating external loading and a quadratic viscous-correction $\delta(u,v)$, turned by a parameter $\mu>-\min W''$.

Figure 7.  Visco-Energetic solutions of a nonconvex energy and an increasing loading. The optimal transition is a combination of sliding and viscous parts.