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## AR40316: Continuum mechanics

6
Masters UG & PG (FHEQ level 7)
Semester 1
Semester 2
EX 100% Supplementary Assessment: Like-for-like reassessment (where allowed by programme regulations)
Before taking this unit you must take AR20061 and take AR20082 and take AR20312 Description: Aims:
To introduce continuum mechanics and its application to elasticity, plasticity and fluid mechanics.
To give a brief introduction to curvilinear co-ordinates and tensors which are useful for any advanced work in continuum mechanics, the finite element method, or shell theory.
To stimulate the students interest in physics and its relation to some important areas of current engineering research.

Learning Outcomes:
The successful student will be able to demonstrate an understanding of some of the following techniques, especially in relation to advanced structural analysis:
Equations in three dimensions.
Two dimensional elasticity.
Plasticity.
Navier-Stokes equations, introductory Computational Fluid Dynamics.
Curvilinear co-ordinates.
Symmetric second order tensors
Constitutive equations in elasticity, plasticity and fluid mechanics using curvilinear coordinates.
Geometry of surfaces.
The student will also be able to demonstrate:
* a systematic understanding of this knowledge, and a critical awareness of current problems and/or new insights, much of which is at, or informed by, the forefront of this area of professional practice;
* conceptual understanding that enables the student to evaluate critically current practice and new developments, and propose new solutions;
* an ability to deal with complex issues both systematically and creatively, make sound judgements in the absence of complete data, and communicate their conclusions clearly.

Skills:
An understanding of the theory and curvilinear tensor notation applied to elasticity, plasticity, soil mechanics and fluid mechanics.

Content:
The unit is complementary to other units describing the numerical methods which would be used to solve the equations.
Equations in three dimensions using 'Timoshenko notation'. Stress functions.
Compatability equations.
Two dimensional elasticity: derivation of del4phi=0 and solutions using polynomials.
Reworking of this using cartesian tensor notation to demonstrate its utility.
Plasticity: Tresca and von Mises yield criteria. Application to indentation problems.
Derivation of Navier-Stokes equations in fluid mechanics.
Revision of Navier-Stokes equations and introduction to Computational Fluid Dynamics.
Curvilinear co-ordinates, covariant and contravariant base vectors, metric tensor.
Tensor product. Tensors in curvilinear co-ordinates.
Properties of symmetric second order tensors - principal values and directions,
Mohr's circles in three dimensions. Definition of stress and strain in curvilinear coordinates.
Christoffel symbols and covariant differentiation. Equilibrium equations in curvilinear co-ordinates.
Constitutive equations in elasticity, plasticity and fluid mechanics using curvilinear coordinates.
Geometry of surfaces, metric tensor, second fundamental form, normal curvature and twist, mean and Gaussian curvature. Order of covariant differentiation, Reimann-Christoffel tensor. Gauss's theorem and the Codazzi equations.
Discussion of curvilinear co-ordinates in 4-dimensional space-time, the Bianci relations, the Ricci tensor, the Einstein tensor and the General Theory of Relativity. Programme availability:

#### AR40316 is Compulsory on the following programmes:

Department of Architecture & Civil Engineering
• UEAR-AFM03 : MEng (hons) Civil and Architectural Engineering (Full-time) - Year 4
• UEAR-AFM07 : MEng (hons) Civil Engineering (Full-time) - Year 4

#### AR40316 is Optional on the following programmes:

Department of Computer Science
• RSCM-AFD02 : Doctor of Engineering (EngD) in Digital Media (Full-time) - Year 1

NB. Programmes and units are subject to change at any time, in accordance with normal University procedures.