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## Syllabus for

TME235 - Mechanics of solids

Owner: MPAME
7,5 Credits
Grading: TH - Five, Four, Three, Not passed
Education cycle: Second-cycle
Major subject: Mechanical Engineering
Department: 42 - APPLIED MECHANICS

Teaching language: English
Open for exchange students
Block schedule: C

 Course module Credit distribution Examination dates Sp1 Sp2 Sp3 Sp4 Summer course No Sp 0111 Examination 7,5c Grading: TH 7,5c 24 Oct 2013 pm M, 16 Jan 2014 pm V, 28 Aug 2014 am M

#### In programs

MPAME APPLIED MECHANICS, MSC PROGR, Year 1 (compulsory)
MPENM ENGINEERING MATHEMATICS AND COMPUTATIONAL SCIENCE, MSC PROGR, Year 1 (compulsory elective)

#### Examiner:

Professor  Magnus Ekh

#### Course evaluation:

http://document.chalmers.se/doc/39d00fff-4d40-4278-a5c9-dafa350684d4

#### Eligibility:

For single subject courses within Chalmers programmes the same eligibility requirements apply, as to the programme(s) that the course is part of.

#### Course specific prerequisites

Linear algebra, Calculus in several variables, Mechanics, Solid mechanics and Fluid mechanics.

#### Aim

The course provides an introduction to the mechanics of continuous media and, in particular, solids. A strong focus is placed on deriving and understanding the general field equations in three dimensions. These equations provide a generic basis for solid mechanics, fluid mechanics and heat transport. To be able to formulate the equations in three dimensions Cartesian tensors and the index notation will be used. The role of constitutive equations in distinguishing different types of problem will be emphasized. In particular, linear elastic solids as well as elastic plates will be studied in more detail. Some special cases when analytical solutions are possible to obtain will be highlighted. These analytical solutions will be compared with solutions obtained from finite elements. Energy methods will be introduced to show important concepts and phenomena in linear elasticity such as superposition and reciprocity.

#### Learning outcomes (after completion of the course the student should be able to)

• Confidently manipulate tensor expressions using index notation, and use the divergence theorem and the transport theorem.
• Derive the equations of continuity, momentum and energy
• Extract essential aspects of a given stress state, such as principal values, principal directions, pressure, maximum values, stress vectors on planes, etc.
• Account for the role of a constitutive equation and determine its nature (e.g. solid/fluid, incompressible etc)
• Formulate linear constitutive equations: Hookean solid, Newtonian fluid, Fourier s law
• Formulate Hooke's law for general three dimensional stress-strain condition with specialization to plane stress and plane strain.
• Formulate the boundary value problem for linear momentum with proper boundary conditions.
• Derive Clapeyron's theorem and reciprocity relations.
• Derive the weak form (virtual work formulation) of linear momentum and show its relation to the finite element method.
• Establish the principle of minimum potential energy for linear elasticity and show the relation to the weak form.
• Show that elastic waves can be decomposed into pressure and shear waves.
• Derive the plate equation together with proper boundary conditions and specialize to axisymmetry.
• Derive the buckling loads for a rectangular plate.

#### Content

Index notation; Tensors; Principal values and directions; Spatial derivatives and divergence theorem; Stress tensor; Eulerian and Lagrangian description of motion; The field equations of continuity, momentum and energy; Constitutive equations: Fourier's law, viscous fluids, elastic solids; Elastic solids; Superposition and reciprocity; Principle of minimum of potential energy; Virtual work formulation; Finite element method; Waves; Elastic plates; Buckling.

#### Organisation

Lectures, tutorials, assignment supervision

Lecture notes.

#### Examination

To pass the course the student must pass three assignments and a written exam. The grade is determined by the written exam.

Page manager Published: Mon 28 Nov 2016.