Continuum Mechanics of Duhem’s Approach to Hysteresis

Although the Duhem model of hysteresis was introduced in the late nineteenth century, it has not received particular attention for nearly a century. Only in the late twentieth century did researchers begin to cite and attribute this model to Pierre Duhem, employing it as a “black-box model” in various applications. The model has been widely used to describe hysteretic behaviours, particularly those observed in piezoelectric materials. In this work, we explore the thermomechanical basis of the Duhem model. In a conser-vative system, the time rate of the dependent variable (e.g., stress) is related to the time rate of the independent variable (e.g., strain) through the second derivative of the Helmholtz free energy. To account for hysteresis, Duhem added a term featuring a continuous function representing dissipation and leading to permanent changes in the state variables. We call this Duhem’s irreversibility function or simply Duhem’s function. Based on the properties of Duhem’s function, the model describes a region of equilibrium states of the system called the natural state surface, for which Duhem’s function vanishes and no irreversible transfor-mations occur. Duhem studied the isothermal case (constant temperature) and the isobaric case (constant pressure/stress). As an example of application, we show how the isothermal Duhem model is equivalent to classical elastoplasticity with isotropic and kinematic hardening, with a judicious choice of Duhem’s function. To illustrate this example, we numerically simulate cyclic loading for materials with both linear and non-linear elastic behaviour, and linear hardening. This work shows how, after more than a century from its conception and without knowledge of the specific system (e.g., the decomposition into elastic and plastic strain), the Duhem model constitutes a viable phenomenological approach to the modelling of hysteresis.

Name thesis:degree_name

Master of Science (MSc)

Discipline thesis:degree_discipline

Engineering – Mechanical & Manufacturing

Grantor dc:publisher.institution

Schulich School of Engineering

Year dc:date.issued

2025

Author dc:creator

• Nfaileh, Najla

Advisors dc:contributor.advisor

• Federico, Salvatore
• Sun, Qiao

Committee members dc:contributor.committeemember

• Feder, David
• Ghasemloonia, Ahmad

• Duhem Model
• Hysteresis
• Elastoplasticity
• Rate-Independent

Statement dc:rights

• University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.

Language dc:language.iso

en

DOI

https://dx.doi.org/10.11575/PRISM/49853

OAI identifier oai:identifier

oai:ucalgary.scholaris.ca:1880/122261