COUPLED PROBLEMS IN FUNCTIONAL MATERIALS

This course is for undergraduate and graduate students.

Introduction to Tensor notation. Basics of continuum (solid) mechanics. Kinematics of finite deformations. Constitutive laws for nonlinear active materials. Basics of nonlinear electro-elasticity. Maxwell stress. Nonlinear coupling in active materials. Plane waves in active nonlinear composites.

SEMESTER START DATE: October 29, 2025

Contact Hours per Week: 3

Day & Time: Tuesday, via Zoom from 14:30-17:30 (Israeli Time which is one hour ahead of Germany, France, Denmark, Switzerland, Czech Republic, and the Netherlands; same time zone as Estonia)

Textbooks and References:

1. Nonlinear Theory of Electroelastic and Magnetoelastic Interactions. Springer US, 2014 Authors: Dorfmann, A. Luis and Ogden, Ray W. ISBN-13: 978-1-4614-9596-3

2. Continuum Mechanics of Electromagnetic Solids. Elsevier: North-Holland, Amsterdam, 1988 Author: Maugin, G.A. ISBN: 9780444703996

3. Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Wiley, 2000 Author: Holzapfel, Gerhard A. ISBN: 978-0-471-82319-3

4. Mechanics of Soft Materials. Springer Singapore, 2016 Author: Volokh, Konstantin ISBN: 978-981-10-1598-4

5. The Non-Linear Field Theories of Mechanics. Springer Book Archive, 1965 Author: Truesdell, Clifford and Noll, Walter. ISBN: 978-3-662-10388-3

At the completion of this course, students should:

• be confident with tensor analysis;

• be able to correctly apply nonlinear elasticity theories in modeling of active materials

• be able to read and understand advanced scientific publications on nonlinear active material models

• be able to apply and evaluate various stress, strain and deformation measures

• be able to model and predict the behavior of active materials

• be familiar with nonlinear multiphysics modeling approaches

• be able to estimate the magneto- or electro-mechanical response of the active materials

• understand the physical meaning of instabilities in materials with coupled physics

• be familiar with the mathematical formulation of stability analysis in materials with coupled physics

6 Homework assignments 18%

3 Reading assignments 12%

Online Exam or Project 70%

Course Outline:

Week 1

• Introduction to Tensors: Scalars, vectors, second-order tensors, and high order tensors; their physical counterparts (energy, traction, stress, tensor of elastic moduli). Examples and Exercises.

Week 2

• Introduction to Tensors: Algebraic and differential operations on tensors. Examples and Exercises.

Week 3

• Kinematics: concept of current and reference configurations; Lagrangian and Eulerian formulations; displacement and velocity fields, material and spatial derivatives. Concept of strain; relations between linear and nonlinear models. Examples and Exercises.

Week 4

• Kinematics: deformation gradient, different measures of strain; strain tensors and stretch-rotation decomposition; examples of homogenous deformations.

Week 5

• Balance laws and stress: tractions and stresses, Cauchy (true) and nominal (referential) stress tensors, conversation of mass, momentum balance, equation of motion and equilibrium.

Week 6

• Constitutive laws for purely elastic materials: isotropic hyperelastic materials, incompressible hyperelastic materials, compressible hyperelastic materials, strain-energy functions. On material isotropy.

Week 7

• Incremental equations: "small on large" motions, tensor of elastic moduli. Plane waves; acoustic tensor, strong ellipticity condition.

Week 8

• Electroelasticity: Maxwell Equations for electrostatics; Maxwell stress tensor, total stress tensor; electroelastic isotropic materials; structure of energy functions for materials with coupled physics, coupled invariants. Dielectric actuator.

Week 9

• Plane waves in electroelastic materials subject to electromechanical loading: Incremental equations. Derivation of generalized acoustic tensor. Dependence of elastic wave velocities on an electric field.

Week 10

• Magnetoelasticity: Maxwell Equations for magnetostatics; magnetic Maxwell stress tensor, magnetic field intensity, boundary conditions for magnetic fields.

Week 11

• Nonlinear magnetoelasticity: Quasi-magnetic approximation; structure of energy functions for magnetoelastic isotropic materials. Incremental equations. Example of magnetodefomation of a bi-laminate.

Week 12

• Student project presentations and preparation for the final exam.

Introduction to Theory of Elasticity (00840515) or equivalents

Lectures & Exercises