Bilinear Elasto-Plastic Oscillator

• Bilinear elasto-plastic oscillator is a canonical model for systems experiencing both elastic and irreversible plastic deformations under random forcing.
• It utilizes a bilinear force–displacement law and stochastic variational inequalities to capture hysteresis and non-smooth phase transitions.
• Small-jump regularization and deterministic PDE methods enable convergence analysis and efficient computation of long-time statistical behaviors.

The bilinear elasto-plastic oscillator is a canonical model for mechanical and structural systems exhibiting both elastic and plastic (irreversible) deformations under stochastic forcing. It is defined by a piecewise linear (bilinear) force-displacement relationship with a sharp yield threshold, and the evolution is governed by coupled stochastic differential equations (SDEs) or stochastic variational inequalities (SVIs). This oscillator serves as a mathematically rigorous prototype for hysteretic phenomena encountered in engineering and physics, supporting both analytical tractability and high-fidelity numerical analysis in the presence of noise, with particular application to reliability and serviceability assessment in structural engineering (Bensoussan et al., 2011, Guo et al., 11 Jan 2026).

1. Mathematical Formulation and State Variables

The bilinear elasto-plastic oscillator is formulated in both SDE and SVI frameworks, with the latter offering a precise treatment of the nonsmooth switching between elastic and plastic regimes.

The principal state variables (notation as in (Bensoussan et al., 2011, Guo et al., 11 Jan 2026)):

• $x(t)$ or $X(t)$: total displacement.
• $y(t) = \dot{x}(t)$ or $Y(t)$: velocity.
• $z(t)$ or $Z(t)$: elastic (reversible) deformation, confined to $[-Y, Y]$ or $[-b, b]$.
• $\Delta(t) = x(t) - z(t)$ or $\Delta(t) = X(t) - Z(t)$: plastic (irreversible) deformation.

Physical parameters include the spring stiffness $k > 0$, viscous damping $c_0 > 0$, yield threshold $Y > 0$ or $b > 0$, bilinearity parameter $\alpha \in [0,1]$ (with $\alpha = 0$ corresponding to an elasto-perfectly-plastic case), and stochastic forcing via a standard Wiener process $w(t)$ or $W(t)$, with intensity $\sigma > 0$.

The governing SDE-SVI system (see (Bensoussan et al., 2011, Guo et al., 11 Jan 2026)) is: $$\begin{cases} dx(t) = y(t)\,dt, \ dy(t) = -\big(c_0 y(t) + k\,z(t)\big)\,dt + dw(t), \end{cases} \quad |z(t)| \leq Y,$$ with the plastic (internal) variable evolving via the variational inequality: $$(\dot{z}(t) - y(t))(\phi - z(t)) \geq 0, \quad \forall \phi \in [-Y, Y].$$

For the general bilinear case with hysteresis (Guo et al., 11 Jan 2026): $$\begin{cases} dX = Y\,dt, \ dY = \big[\mathfrak{f}(X, Y) - k(1-\alpha) Z - k\alpha X\big]\,dt + \sigma dW, \ (dZ - Y\,dt)(\xi - Z) \geq 0, \quad \forall |\xi| \leq b, |Z| \leq b, \end{cases}$$ where $\mathfrak{f}$ represents additional deterministic forces.

2. Bilinear Force–Displacement Law and Hysteresis

The restoring force in the oscillator is bilinear, i.e., it is linear in the elastic regime and saturates on a plastic plateau once the yield threshold is reached. Explicitly,

$F(t) = k\,z(t)$

with the force-displacement relationship: $$f(x) = \begin{cases} k\,x, & |x| \leq Y, \ k\,Y\,\operatorname{sign}(x), & |x| > Y. \end{cases}$$ This law produces characteristic hysteretic behavior (Bauschinger effect) and nontrivial energy dissipation, fundamental in the modeling of metallic structures and other hysteretic systems (Guo et al., 11 Jan 2026).

3. Stochastic Variational Inequality and Phase Identification

The SVI formalism replaces explicit phase tracking by a constraint-based evolution: $$(dZ - Y\,dt)\,(\xi - Z) \geq 0, \quad \forall |\xi| \leq b, |Z| \leq b.$$ This allows the model to resolve elastic excursions and plastic flow without enumerating switching points. Importantly, the process may exhibit infinitely many micro-elastic excursions under pure white noise, making phase counting intractable using discontinuous phase indicators. The SVI approach preserves a mathematically rigorous and compact trajectory description (Bensoussan et al., 2011). However, when a sharp demarcation between phases is desired (e.g., for simulation purposes), a small-jump regularization can be imposed, as described below.

4. Small-Jump Regularization and Convergence

To enforce clear phase separation, a regularized jump process is defined, where at each detected transition to the plastic regime ($|z|=Y$ and $y=0$), the state is reset via a small jump: $$z^\varepsilon(\tau_n^\varepsilon) = \mathrm{sign}\big(z^\varepsilon(\tau_n^\varepsilon-)\big)(Y-\varepsilon),\quad y^\varepsilon(\tau_n^\varepsilon) = 0.$$ The continuous trajectory is then reconstructed via the SVI until the next jump. This regularization introduces artificial but vanishingly small modifications to the original dynamics.

The main convergence theorem asserts that, assuming a technical restriction on the stiffness for well-posedness,

$$k> X_+(c_0) := \frac{1}{2} \left(-\frac{c_0}{3} + c_0 \sqrt{\frac{1}{9} + 4\,\frac{c_0}{6}}\right),$$

one has (for any finite horizon $T>0$): $\lim_{\varepsilon \to 0} \frac{1}{\varepsilon} \E\left[ \sup_{0 \leq t \leq T} \big(|y(t) - y^\varepsilon(t)|^2 + k |z(t) - z^\varepsilon(t)|^2\big)\right] = 0,$ i.e., uniform convergence in mean square at rate $o(\varepsilon)$ (Bensoussan et al., 2011).

5. Invariant Measures and Stationary Analysis via PDE

For the white-noise-driven bilinear elasto-plastic oscillator with hysteresis (BEPO), the existence of an invariant probability measure $\mu$ is established by construction of a Lyapunov function: $$V(x, y) = \left(k \alpha + \tfrac{c_0 d_0}{2} - c_1\right)x^2 + y^2 + c_0 x y$$ with specified conditions ensuring negative drift outside compact sets ((Guo et al., 11 Jan 2026), Theorem 3.2). Tightness and positive recurrence then follow, implying the existence of $\mu$ via standard Krylov–Bogoliouboff arguments.

The invariant measure $\mu(x, y, z)$ solves the stationary Fokker–Planck (Kolmogorov forward) PDE on the state space $\mathbb{R}^2 \times [-b, b]$ with nontrivial divergence-form structure, incorporating the elastic and plastic regime transitions through interface and no-flux boundary conditions.

Key equations in the interior ($|z|<b$): $$-\partial_x(y\mu) - \partial_y(\beta(x, y, z)\mu) - \partial_z(y\mu) + \frac{\sigma^2}{2} \partial_y^2 \mu = 0$$ with interface corrections and matching/no-flux conditions at $z = \pm b$.

6. Deterministic PDE Methods for Long-Time Statistics

The deterministic PDE approach permits the direct computation of long-term time-averaged statistics without Monte Carlo simulation. Observables are computed as stationary averages: $\lim_{t \to \infty} \E\,g(X(t), Y(t), Z(t)) = \int g\,d\mu$ by solving the resolvent PDE: $\lambda u_\lambda - A u_\lambda = g$ with appropriate interface and normalization conditions, then taking the limit $\lambda \to 0$.

Applications include:

• Threshold-crossing frequency:

$$\nu(a_1) = \int_{\bar D} |y|\,\delta(x-a_1) \,\mu(dx\,dy\,dz)$$

provides an alternative to Rice’s formula for level crossings.

• Probability of serviceability:

$$P(a_2) = \int_{\bar D} 1_{|x-z| \leq a_2}\,\mu(dx\,dy\,dz)$$

quantifies the likelihood of displacement remaining within service limits.

Numerically, the system is discretized on a truncated domain using second-order upwind schemes for transport and centered differences for diffusion, with sparse linear solver techniques (GMRES with ILU preconditioning) (Guo et al., 11 Jan 2026). Upwind schemes are adapted near interfaces to enforce correct physical fluxes.

7. Implications and Extensions

The bilinear elasto-plastic oscillator, rigorously formulated in the SVI-PDE framework, models a range of highly nonlinear and hysteretic phenomena in engineering contexts. Its inclusion of stochastic forcing captures microscale elastic excursions and noise-induced plastic flow, essential for reliability and serviceability analysis. The convergence guarantees associated with small-jump regularization solidify its use in both simulation and analytical regimes.

Recent extensions to higher-dimensional oscillators with richer hysteretic laws—enabled by Lyapunov-based existence proofs and deterministic Fokker–Planck solvers—generalize the oscillator's applicability beyond classical elasto-perfectly-plastic domains to systems manifesting the Bauschinger effect and more sophisticated rate-independent memory (Guo et al., 11 Jan 2026). Deterministic PDE methods make previously intractable long-time statistics computationally accessible, providing an alternative to Monte Carlo with advantageous convergence properties.

A plausible implication is the deployment of deterministic PDE solvers in the design and reliability codes of structural engineering, where standard Monte Carlo schemes are computationally burdensome for low-probability rare events. The oscillator's mathematical structure remains a paradigm for analyzing and simulating non-smooth, noise-driven dynamical systems with path-dependent memory.