MA4012: Finite Elasticity

School Cardiff School of Mathematics
Department Code MATHS
Module Code MA4012
External Subject Code 100400
Number of Credits 20
Level L7
Language of Delivery English
Module Leader Professor Angela Mihai
Semester Spring Semester
Academic Year 2015/6

Outline Description of Module

The main objective of finite elasticity is to predict changes in the geometry of solid bodies. When these changes are very small, i.e. the deformation under working loads are not detectable by the human eye, their configuration can be considered as fixed, and any changes in their geometry can be neglected. This so-called `small strain' assumption is at the basis of the classical theory of linear elasticity, which is successfully used in structural mechanics and many other engineering applications. However, many modern applications (involving soft solids, inflatable structures, polymers and synthetic rubbers) and biological structures (such as plants and vital organs) involve large deformations. In the framework of large deformations, finite elasticity covers the simplest case where internal forces (stresses) only depend on the present deformation of the body and not on its history (i.e. it excludes plasticity, viscosity, and damage). Because of the large deformations involved, the mathematical models used in finite elasticity are always nonlinear, and the numerical solution of the resulting mathematical equations requires a careful approximation strategy and powerful algorithms. The purpose of this course is to give a general description of three-dimensional finite elasticity and its approximation in the setting of quasi-static theory. First the physical models which are used in this field are described, then their mathematical structure is explained, both by considering particular solutions to specific problems and by studying the existence and non-uniqueness of solutions. Finally, the main strategies which can be used for their numerical treatment are presented. Since much of the relevant mathematical theory depends on the use of tensors, a summary of tensor algebra and analysis is given in the introduction. Specific references are indicated throughout the course for background reading and further developments.

Recommended Modules: MA0332 Fluid Dynamics

On completion of the module a student should be able to

  • Describe the general model of nonlinear elastic deformation in the reference (Lagrangian) and current (Eulerian) system of references.
  • Formulate the basic boundary value problem of finite elasticity and solve the governing equations for a selection of problems for (internally) constrained and unconstrained isotropic materials.
  • Reduce the equilibrium equation to variational form by elimination of stresses by constitutive law and discretise the variational problem by isoparametric finite elements.
  • Employ suitable algorithms which are now available for the numerical approximation of the deformation and stresses around a given state.
  • Solve analytically or numerically specific problems, including the Rivlin cube, torsion of a bar, extension and inflation of a cylindrical tube, inflation of a spherical shell, buckling of a rod, elastic cavitation, frictionless contact and self-contact.

How the module will be delivered

33 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 120 hours private study including preparation of worked solutions for problem classes.

Skills that will be practised and developed

Ability to relate mathematical theory to specific computations by employing basic tools of elementary differential geometry, partial differential equations, variational calculus, finite element analysis, linear algebra, and optimization.

How the module will be assessed

Formative assessment is carried out in the problem classes. Feedback to students on their solutions and their progress towards learning outcomes is provided during these classes.

The summative assessment is the written examination at the end of the module. This gives students the opportunity to demonstrate their overall achievement of learning outcomes. It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of four from five equally weighted questions.

Assessment Breakdown

Type % Title Duration(hrs)
Exam - Spring Semester 100 Finite Elasticity 3

Syllabus content

  • Tensor Theory Prerequisites
    • Euclidean vector space
    • Tensor algebra
    • Tensor fields
  • Deformation and Motion
    • Kinematics
    • Deformation and strain
    • Equilibrium equation
    • Material models
      • Hyperelastic compressible (internally unconstrained) material
      • Hyperelastic incompressible (internally constrained) material
    • Linear elastic approximation
  • Boundary Value Problems
    • Formulation of boundary value problems
      • Problems for compressible materials
      • Problems for incompressible materials
    • Variational formulation of equilibrium problem
      • Variational problem for compressible material
      • Variational problem for incompressible material
    • Existence results
  • Numerical Solution Techniques
    • Finite element approximation
      • Compressible case
      • Incompressible case
      • Nearly incompressible case
    • Algebraic solution
  • Contact Problems
    • Equilibrium problem with frictionless contact or self-contact
    • Finite element descretisation
    • Mathematical programming approach

Essential Reading and Resource List

J. M. Ball, Convexity conditions and existence theorems in non-linear elasticity. Arch.Rat. Mech. Anal. 63 (1977) 337-403.

A. E. Green, J. E. Adkins, Large Elastic Deformations (and Non-linear Continuum Mechanics), Oxford University Press, 1970, Second Edition.

P. Le Tallec, Numerical methods for three-dimensional elasticity, in Handbook of Numerical Analysis, v. III, P. G. Ciarlet and J. L. Lions eds., North-Holland, 1994, pp. 465-624.

J. T. Oden, Finite Elements of Nonlinear Continua, Dover, 2006, Second Edition.

R. W. Ogden, Non-Linear Elastic Deformations, Dover, 1997, Second Edition.

R. T. Shield, Deformations possible in every compressible, isotropic, perfectly elastic material, Journal of Elasticity 1 (1971) 91-92.

C. Truesdell, W. Noll, The Non-Linear Field Theories of Mechanics, Springer-Verlag,2004, Third Edition.

Background Reading and Resource List

Not applicable.


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