Hysteresis

Lore & Background

Early work on hysteresis in mechanical systems was performed by James Clerk Maxwell. Later, significant contributions came from Ferenc Preisach (Preisach model), Louis Néel, and Douglas Hugh Everett in connection with magnetism and absorption. A more formal mathematical theory was developed in the 1970s by a group of Russian mathematicians led by Mark Krasnosel'skii.

Reader's Guide

Hysteresis describes how a system's state depends on its history, creating loops in plots of variables such as magnetic moment versus field. This history dependence enables durable memory in hard disk drives and the preservation of past magnetic field records in rocks. Hysteresis occurs in ferromagnetic and ferroelectric materials, rubber band deformation, shape-memory alloys, and many other natural phenomena, often associated with irreversible thermodynamic change and internal friction. In engineering, hysteresis is intentionally added to control systems like thermostats and Schmitt triggers to prevent unwanted rapid switching. Systems with hysteresis are nonlinear and mathematically challenging to model; models such as the Preisach, Bouc–Wen, and Jiles–Atherton models attempt to capture general or specific features. The precise definition of hysteresis remains debated, with Isaak D. Mayergoyz noting that its meaning varies across fields and authors.

Did You Know?

• Hysteresis is the basis of memory in hard disk drives and the remanence that retains a record of Earth's magnetic field magnitude.
• Rate-independent hysteresis, such as magnetic hysteresis loops, makes durable memory possible.
• Hysteresis is intentionally added to thermostats and Schmitt triggers to prevent unwanted frequent switching.

The Physics of the Hysteresis Loop

In viscoelastic solids, the response to an applied load is not simply reversible. When a stress is imposed on such a material and then released, the material does not retrace the same path along its stress-strain curve. Instead, the loading and unloading paths diverge, enclosing a region known as a hysteresis loop. The physical meaning of this loop is that mechanical work performed during the loading cycle is not fully recovered; a portion of it is converted into heat within the material because work must be done against internal damping effects. This energy dissipation is what gives viscoelastic behavior its characteristic time-dependence, distinguishing it from the idealized case of linear elasticity, where a material returns exactly to its original undeformed shape once the load is removed. The hysteresis loop thus serves as a diagnostic signature in the stress-strain curve, signaling that the material's response cannot be captured by a single-valued function of stress alone. Understanding this dissipation mechanism is essential whenever a solid component is subjected to repeated cyclic loading, because the accumulated heat generated within the material can influence its subsequent mechanical behavior and the overall performance of the structure in which it is embedded.

Hysteresis Among Competing Response Models

The hysteresis loop occupies a specific niche within the broader family of constitutive models that solid mechanics uses to describe how a material responds to applied stress. At one extreme, pure elasticity assumes that removing the load restores the material to its undeformed state, a behavior captured by Hooke's law and the linear elasticity equations. At the other extreme, plasticity describes the regime beyond a material's yield stress, where deformation becomes permanent and the solid no longer returns to its previous configuration. Viscoplasticity merges these two ideas and finds application in materials such as gels and mud, while thermoelasticity addresses the coupling between mechanical and thermal responses in solids that are neither isothermal nor adiabatic. Viscoelasticity, the model in which the hysteresis loop naturally emerges, sits between these extremes: the material behaves elastically in the sense that it can recover, yet the damping it exhibits means that energy is lost on each cycle. Analysts most often reach for linear material models because they are computationally tractable, but as newer materials are introduced and existing ones are pushed closer to their limits, non-linear models that capture hysteresis and other path-dependent effects are becoming increasingly common in practice.

Engineering and Practical Relevance

The study of hysteresis and viscoelastic response is embedded within solid mechanics, a discipline that underpins an extraordinary breadth of engineering and scientific practice. Civil, aerospace, nuclear, biomedical, and mechanical engineers all rely on the principles of solid mechanics to predict how structures and components will deform, vibrate, and ultimately fail under load. Beyond engineering, the field feeds directly into geology, materials science, and branches of physics and chemistry. In the biomedical realm, understanding how biological materials such as muscles, bones, and heart tissue respond to stress is critical for designing dental prostheses and surgical implants, and for interpreting the anatomy of living organisms. The diversity of materials that solid mechanics must accommodate—steel, wood, concrete, textiles, geological materials like ice, soil, and rock, and various plastics—means that hysteresis behavior must be characterized across an enormous range of compositions and microstructures. Even the Euler-Bernoulli beam equation, one of the most widely used practical tools in the field, rests on assumptions about material response that become more complex when viscoelastic damping and the accompanying hysteresis loop are taken into account.

Mathematical and Computational Treatment

Capturing the hysteresis loop and the time-dependent viscoelastic response it represents requires a mathematical apparatus more sophisticated than a simple proportionality between stress and strain. Solid mechanics extensively employs tensors to describe stresses, strains, and the constitutive relationships linking them, providing a coordinate-free language in which the anisotropic and path-dependent features of viscoelastic materials can be expressed rigorously. For problems where the hysteresis loop is too complex to solve analytically, variational formulations and computational mechanics step in, offering numerical solutions through techniques such as the finite element method. These computational tools allow engineers to model the energy dissipation and heat generation that accompany each loading-unloading cycle in a realistic geometry. Complementing the numerical work, experimental mechanics provides the design and analysis of laboratory methods used to measure the actual stress-strain behavior of solid materials, including the identification of damping characteristics and the shape of the hysteresis loop. Together, the tensor-based continuum description, the finite element discretization, and the experimental validation form a closed loop of understanding that makes it possible to predict how a viscoelastic component will behave under the cyclic, time-dependent loading it will encounter in service.