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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 7,5 c Grading: TH 7,5 c 28 Oct 2014 am M, 03 Jan 2015 pm V, 27 Aug 2015 am M

#### In programs

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

#### Examiner:

Professor  Magnus Ekh

#### Course evaluation:

http://document.chalmers.se/doc/a558b4e1-a33a-44ab-b412-d6bce3781f9b

#### Eligibility:

In order to be eligible for a second cycle course the applicant needs to fulfil the general and specific entry requirements of the programme that owns the course. (If the second cycle course is owned by a first cycle programme, second cycle entry requirements apply.)
Exemption from the eligibility requirement: Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling these requirements.

#### 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 with particular focus on solids. An important part of the course is the derivation 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 elasticity is used for different structural elements such as beams, plates and shells. Energy methods are introduced to show important concepts and phenomena in linear elasticity such as superposition and reciprocity. The relation between energy methods and the finite element method is shown. A short introduction to the finite element method is given in terms of both running a commercial software and of own programming in Matlab.

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

• Manipulate tensor expressions using index notation, and use the divergence theorem and the transport theorem.
• Derive the equations of continuity, momentum and energy for a continuum.
• Extract essential aspects of a given stress state, such as principal values, principal directions, hydrostatic stress, deviatoric stress, stress vector on a plane, 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 equilibrium of a continuum with boundary conditions.
• Derive and utilize Clapeyron's theorem and reciprocity relations.
• Derive the weak form (virtual work formulation) of linear momentum and show how it is used in the finite element method.
• Establish the principle of minimum potential energy for linear elasticity and show the relation to the weak form.
• Derive the plate equation together with proper boundary conditions and specialize to axisymmetry.
• Derive the buckling loads for a rectangular plate.
• Derive equilibrium equations for membrane condition of axisymmetric shells.
• Formulate the kinematic assumption, equilibrium equations and the deflection of cylindrical shells subjected to membrane and bending condition.

#### 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; Potential energy; Virtual work formulation; Finite element method; Plates; Buckling; Shells.

#### 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: Thu 04 Feb 2021.