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MA40049: Elasticity

[Page last updated: 09 August 2024]

Academic Year: 2024/25
Owning Department/School: Department of Mathematical Sciences
Credits: 6 [equivalent to 12 CATS credits]
Notional Study Hours: 120
Level: Masters UG & PG (FHEQ level 7)
Period:
Semester 2
Assessment Summary: EX 100%
Assessment Detail:
  • Examination (EX 100%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module you must take MA30253
Learning Outcomes: Students should be able to derive the governing equations of the theory of elasticity from basic principles and be able to solve simple problems.


Aims: To provide a unified introduction to the mathematical modelling of elastic materials.

Skills: Problem Solving T/F A
Written and Spoken Communication F (solutions to exercise sheets, problem classes)

Content: Revision: Kinematics of deformation, balance laws, Cauchy stress tensor, boundary conditions.
Analysis of strain: Polar decomposition theorem, stretch tensors, principal stretches, left and right Cauchy-Green tensors, principal invariants, homogeneous deformations.
Nonlinear Elasticity: Response functions for the Cauchy and 1st Piola-Kirchhoff stress tensors. Frame-indifference, material symmetries, Isotropy. Hyperelastic materials, the stored energy function, equilibrium equations and simple solutions. Examples of non-uniqueness of equilibrium solutions.
Linear Elasticity: Derivation of field equations of linear elasticity from the nonlinear theory. Linear elasticity tensor and its symmetries. Lame moduli;
Young's modulus, Poisson's ratio. Strain energy function for linear elasticity; uniqueness of equilibrium solutions. Simple problems of elastostatics: including expansion of a spherical shell, extension of a cylinder.
Further topics may be chosen from the following:
Variational approaches, symmetries and conservation laws. Constitutive assumptions and inequalities. Applications to modelling phase transitions, composite materials. Incompressible hyperelastic materials.
Linear Elastodynamics: Basic equations and general solutions; plane waves in unbounded media, simple reflection problems; surface waves.

Course availability:

MA40049 is Compulsory on the following courses:

Department of Physics
  • USXX-AFM01 : MSci(Hons) Mathematics and Physics (Year 4)
  • USXX-AAM01 : MSci(Hons) Mathematics and Physics with Study year abroad (Year 5)
  • USXX-AKM01 : MSci(Hons) Mathematics and Physics with Year long work placement (Year 5)

MA40049 is Optional on the following courses:

Department of Mathematical Sciences
  • TSMA-AFM19 : MSc Mathematics with Data Science for Industry
  • TSMA-AWM19 : MSc Mathematics with Data Science for Industry
  • USMA-AFB15 : BSc(Hons) Mathematical Sciences (Year 3)
  • USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
  • USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
  • USMA-AFB13 : BSc(Hons) Mathematics (Year 3)
  • USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
  • USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
  • USMA-AFB01 : BSc(Hons) Mathematics and Statistics (Year 3)
  • USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
  • USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
  • USMA-AFM14 : MMath(Hons) Mathematics (Year 3)
  • USMA-AFM14 : MMath(Hons) Mathematics (Year 4)
  • USMA-AAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
  • USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
  • USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)
Department of Physics
  • USXX-AFB03 : BSc(Hons) Mathematics and Physics (Year 3)
  • USXX-AAB04 : BSc(Hons) Mathematics and Physics with Study year abroad (Year 4)
  • USXX-AKB04 : BSc(Hons) Mathematics and Physics with Year long work placement (Year 4)

Notes:

  • This unit catalogue is applicable for the 2024/25 academic year only. Students continuing their studies into 2025/26 and beyond should not assume that this unit will be available in future years in the format displayed here for 2024/25.
  • Courses and units are subject to change in accordance with normal University procedures.
  • Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.
  • Find out more about these and other important University terms and conditions here.