Duhem-Type Viscoelastoplastic Hysteresis

• Duhem-type viscoelastoplastic hysteretic element is a mathematical model that combines elastic, viscous, and plastic behaviors to capture rate-dependent memory effects and hysteresis loops.
• It employs coupled differential equations with internal states and Duhem operators to ensure well-posedness and unique global solutions under locally Lipschitz conditions.
• The model accurately represents asymmetric loading and unloading branches, enabling practical analysis of energy dissipation, the Bauschinger effect, and cyclic plastic yield.

A Duhem-type viscoelastoplastic hysteretic element is a mathematical and physical model that encapsulates the combined elastic, viscous, and plastic hysteretic behaviors observed in many mechanical systems. It forms the basis for modeling rate-dependent and history-dependent restoring forces, prominently featuring in unforced oscillators where it characterizes the memory effects resulting from plastic yield and hysteresis. The defining feature is the use of Duhem operators to realize a restoring force whose relationship to displacement and velocity generates a characteristic hysteresis loop, which captures cyclic energy dissipation, plastic deformation, and differential response during loading and unloading.

1. Mathematical Formulation of the Duhem Model

The prototypical Duhem model, in its minimal single-input, single-output setting, introduces two internal coordinates: a kinematic state $x$ and a hysteretic state $z$. The model equations are: $$\boxed{ \begin{aligned} \dot x &= u,\ \dot z &= u + f_1(z) g_1(u) + f_2(z) g_2(u),\ y &= -[h_1(x) + h_2(z)], \end{aligned} }$$ where $u$ is the input (typically relative velocity), $y$ is the output (restoring force), and $f_1, f_2, g_1, g_2:\mathbb{R}\to\mathbb{R}$ are locally Lipschitz functions. The terms $h_1, h_2$ are strictly increasing, locally Lipschitz homeomorphisms (vanishing at zero), ensuring well-posedness and invertibility. The structure of the $f_i, g_i$ pairs creates direction-dependent response: different "slopes" or correction terms become active as the velocity and internal state cross zero, prerequisites for generating a hysteresis loop with memory effects (Milehins et al., 5 Jan 2026).

2. Coupling into Mechanical Oscillator Dynamics

A Duhem hysteretic element is typically incorporated within an unforced mass–spring–dashpot system, leading to a coupled set of ordinary differential equations governing displacement $x$, hysteretic state $z$, and velocity $v$: $$\boxed{ \begin{aligned} \dot x &= v,\ \dot z &= v + f_1(z) g_1(v) + f_2(z) g_2(v),\ \dot v &= -[h_1(x) + h_2(z)] - c(x,z,v), \end{aligned} }$$ with initial conditions $(x_0, z_0, v_0)$. The elastic spring (often $h_1(x) = k_1 x$), the Duhem element modeling viscoelastoplasticity via $h_2(z) = k_2 z$, and an optional viscous damper $c(x,z,v)$ (usually $c_0 v$) combine to produce a three-dimensional dynamical system. The functions $f_i, g_i$ prescribe different corrections for positive and negative velocity, thus encoding distinct loading and unloading branches, while $c$ must be locally Lipschitz and dissipative [$v\,c(x,z,v)\ge0$]. This structure ensures that the system can model elastic response, yield, reversible and irreversible energy dissipation in a unified ODE framework (Milehins et al., 5 Jan 2026).

3. Regularity Conditions and Sector Properties

Well-posedness of the Duhem oscillator relies on several key mathematical properties:

• All functions $f_1, f_2, g_1, g_2, h_1, h_2,$ and $c$ are locally Lipschitz.
• Zero equilibrium structure: $f_i(0) = g_i(0) = h_i(0) = 0$.
• Strictly increasing $h_1, h_2$ guarantee sector bounds and invertibility.
• Sign constraints on the hysteretic correction functions to encode yield characteristics:

| Regime | Monotonicity/Activation | |----------------|---------------------------------| | $z < 0$ | $0< f_1(z),\ f_2(z)<0$ | | $z > 0$ | $f_1(z)<0,\ 0<f_2(z)$ | | $v > 0$ | $0<g_1(v),\ g_2(v)=0$ | | $v < 0$ | $g_1(v)=0,\ g_2(v)<0$ |

• The damper $c$ must satisfy dissipativity: $v\,c(x,z,v)\ge0$, $c(x,z,0)=0$.

These regularity and monotonicity conditions ensure (i) local Lipschitz continuity of the vector field so that solutions exist and are unique, and (ii) global a priori bounds via Lyapunov methods, excluding finite-time blow-up (Milehins et al., 5 Jan 2026).

4. Asymptotic Analysis and Lyapunov Convergence

For the unforced Duhem oscillator, the equilibrium set is

$$\mathcal{E} = \{(x,z,v): h_1(x) + h_2(z) = 0,\ v=0\}.$$

The central asymptotic result establishes that for any initial condition, the solution is global, bounded, and converges to a unique point in $\mathcal{E}$. The proof employs a LaSalle-type invariance argument using the Lyapunov candidate: $$V(x,z,v) = \int_0^x h_1(s) ds + \int_0^z h_2(s) ds + \frac{1}{2}v^2,$$ which is positive definite and radially unbounded. Its time derivative along trajectories

$$\dot V = h_2(z)[f_1(z)g_1(v) + f_2(z)g_2(v)] - v\,c(x,z,v) \le 0,$$

is nonpositive by the sign assumptions. The only invariant subset where $\dot V = 0$ is the equilibrium manifold $\mathcal{E}$, implying every trajectory converges to a static equilibrium by LaSalle's invariance principle (Milehins et al., 5 Jan 2026).

5. Physical Interpretation and Energy Dissipation

• $x$: Mass displacement (external coordinate).
• $v = \dot x$: Velocity.
• $z$: Hysteretic internal coordinate corresponding to plastic or frictional "memory."
• $h_1(x)$: Elastic restoring force of the spring.
• $h_2(z)$: Yield-type restoring force of the Duhem element.
• $f_1(z),g_1(v);\, f_2(z),g_2(v)$: Correction terms—$f_1(z)g_1(v)$ (active for $v>0$) models reloading, $f_2(z)g_2(v)$ (active for $v<0$) models unloading, capturing asymmetric slopes and yielding phenomena.
• $c(x,z,v)$: Viscous damping, always dissipative if $v\,c\ge0$.

Cyclic driving of the input (velocity $v$) leads to a plot of restoring force $y$ against displacement $x$ that forms a closed loop—the area within the loop quantifies energy lost per cycle. The flexibility to set $f_i$ and $g_i$ enables accurate representation of kinematic hardening, Bauschinger effect, and rate-dependent yielding (Milehins et al., 5 Jan 2026).

6. Hysteresis Loop Structure and Operational Behavior

The Duhem element's constitutive law can be written as: $$\frac{d}{dt}[x, z, y] = \left[ v,\, v + f_1(z)g_1(v) + f_2(z)g_2(v),\, -[h_1(x)+h_2(z)] \right].$$ Switching of branches occurs at sign changes in $v$:

• For $v > 0$, $f_1(z)g_1(v)$ dictates the "loading" slope.
• For $v < 0$, $f_2(z)g_2(v)$ governs the "unloading" slope.

When subjected to slow periodic motion, the locus of $(x,y)$ or $(v,y)$ traces a counterclockwise loop characterized by two dominant slopes (loading/unloading) joined by smooth transitions—the hallmark of rate-dependent viscoelastoplastic hysteresis. The monotonic decrease of total mechanical energy (kinetic + stored elastic + stored hysteretic) is mathematically equivalent to the area inside this loop, which does not increase in unforced dynamics (Milehins et al., 5 Jan 2026).

7. Well-Posedness and Practical Implications

Under the aforementioned Lipschitz and sign-definite conditions, the 3-D dynamical system comprising the mass, the Duhem-type viscoelastoplastic element, and the damper has unique, global-in-time solutions for all initial conditions and physical parameter regimes. In the absence of external forcing, all trajectories converge to static equilibria determined by the intersection of spring and hysteretic branch forces. The Duhem-type element thus rigorously models fundamental aspects of mechanical hysteresis, memory, and plastic yield in engineering systems, with proofs of convergence and stability relying on classical energy methods and invariance principles (Milehins et al., 5 Jan 2026).