Applied Mathematics
MP410: Nonlinear Elasticity (5 ECTS)
(This course will be run every other year.) Description: This course is concerned with continuum mechanics applied to the behaviour of elastic solids. Topics covered include Tensor algebra, Kinematics of continuum deformation and motion, Balance laws and equations of motion, Constitutive equations for soft elastic materials.
Taught in Semester(s) I. Examined in Semester(s) I.
Workload: 36 hours (24 Lecture hours, 12 Tutorial hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- go from a field in material (reference) form to its expression in spatial (current) form, and vice-versa;
- compute the principal stretch ratios and principal strain directions for a homogeneous deformation;
- be familiar with some simple constitutive equations used for soft solids and their corresponding stress-strain relations;
- solve some simple boundary value problems corresponding to simple states of stress;
- have a good grasp of how tensor and linear algebra, and partial differential equations can be applied to the modelling of soft solids;
- construct mathematical models to describe the mechanical behaviour of soft solids encountered in Nature, Science and Engineering;
- interpret the results of mechanical testing protocols for soft materials.
Indicative Content
This course is concerned with continuum mechanics applied to the behaviour of elastic solids.
Topics covered include:
- Tensor algebra: Trace, determinant, orthogonal tensors, gradient, curl, divergence, Cayley-Hamilton theorem, eigenvalues and eigenvectors;
- Kinematics of continuum deformation and motion: Bodies, configurations, motions, material time derivative, deformation gradient, deformation of line, area and volume elements, polar decomposition, analysis of deformation, homogeneous deformations, analysis of motion, transport formulas;
- Balance laws and equations of motion: Mass conservation, forces, momentsmeasures of stress and strain;
- Constitutive equations for soft elastic materials: Hyperelastic materials, objectivity, isotropy, incompressibility, stress-strain representations, application to homogeneous deformations, experimental determination of material parameters.
Module Resources
Introduction to the Theory of Elasticity by R.J. Atkins and N. Fox (2005)
Continuum Mechanics by P. Chadwick (1999).
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