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Programme & Unit Catalogues

## MA40049: Elasticity

2019/0 Owning Department/School: Department of Mathematical Sciences
6      [equivalent to 12 CATS credits] Notional Study Hours: 120
Masters UG & PG (FHEQ level 7)
Semester 2
EX-TH 100%*
• Open book examination with a duration of 24 hours* (EX-TH 100%)

*Assessment updated due to Covid-19 disruptions Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Before taking this module you must take MA30253 Description: Aims:
To provide a unified introduction to the mathematical modelling of elastic materials.

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.

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. Programme availability:

#### MA40049 is Compulsory on the following programmes:

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 programmes:

Department of Mathematical Sciences
• RSMA-AFM16 : Integrated PhD Statistical Applied Mathematics
• TSMA-AFM17 : MRes Statistical Applied Mathematics
• TSMA-AFM08 : MSc Modern Applications of Mathematics
• TSMA-AFM16 : MSc Statistical Applied Mathematics
• 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-AFB05 : BSc(Hons) Statistics (Year 3)
• USMA-AAB06 : BSc(Hons) Statistics with Study year abroad (Year 4)
• USMA-AKB06 : BSc(Hons) 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 2019/20 academic year only. Students continuing their studies into 2020/21 and beyond should not assume that this unit will be available in future years in the format displayed here for 2019/20. Programmes 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. Undergraduates: Find out more about these and other important University terms and conditions here. Postgraduates: Find out more about these and other important University terms and conditions here.