Non-Linear Continuum Mechanics

Code:

M.EM019

Acronym:

MNLMC

Keywords
Classification	Keyword
OFICIAL	Applied Mechanics

Instance: 2023/2024 - 2S

Active?	Yes
Responsible unit:	Applied Mechanics Section
Course/CS Responsible:	Master in Mechanical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
M.EM	33	Syllabus	1	-	4,5	39	121,5

Teaching language

English

Objectives

• 
  
  Identify the different sources of non-linearity.
  
  


• 
  
  Consolidate and extend the knowledge in tensor algebra, linear operators, linearization and directional derivative.
  
  


• 
  
  Generalization of the concepts of deformation state, stress state and their relations considering large displacements, rotations and deformations.
  
  


• 
  
  Understand and model the behavior of the deformation of materials considering the thermodynamics of the irreversible processes.
  
  


• 
  
  Formulate, through the finite element method, an engineering problem of non-linear nature and derive an algorithm to solve it.
  
  


• 
  
  Know how to use numerical simulation programs, developing technical skills, critical thinking and analysis skills.
  
  

Learning outcomes and competences

Working method

Presencial

Program

• Tensors: algebra, linear operators, calculation;


• Differentiation: operators of first, second, third and fourth-order;


• Kinematics: movement, spatial and material configuration, deformation gradient, polar decomposition, volume variation;


• Kinematics: deformation measurements, strain rates, rate of change in volume;


• Balance equations: mass, momentum, energy, entropy;


• Stress: Cauchy's theorem and alternative stress measurements and their relations;


• Stress: deviating and isochoric component, stress rates;


• Principle of Virtual Work;


• Invariance: observer, rigid body movements;


• Constitutive laws: elastic, visco-elastic, hyperplastic;


• Finite element formulation;


• Derivation of the nonlinear finite element method;


• Iterative solution of a system of nonlinear equations - Newton Raphson and Arc-length method;


• Integration algorithms of constitutive laws - Euler integration schemes;


• Consistent tangent matrix calculation.

Mandatory literature

Javier Bonet; Nonlinear continuum mechanics for finite element analysis. ISBN: 0-521-57272-X

Complementary Bibliography

Gerhard A. Holzapfel; Nonlinear solid mechanics. ISBN: 0-471-82304-X
de Borst, R., Crisfield, M. A., Remmers, J. J. C., Verhoosel, C. V. ; Nonlinear Finite Element Analysis of Solids and Structures, John Wiley & Sons, 2012
Belytschko, T., Liu, W.K. & Moran, B.; Nonlinear Finite Elements for Continua and Structures, Wiley

Teaching methods and learning activities

Theoretical-Practical classes. In the Classes, the exposition of several theoretical subjects will be made as well as the supervision of individual assignments that will be performed in the period of classes, in the classes and at home.

Software

Matlab

Evaluation Type

Evaluation with final exam

Assessment Components

Designation	Weight (%)
Exame	60,00
Trabalho prático ou de projeto	40,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	54,50
Frequência das aulas	39,00
Trabalho laboratorial	28,00
Total:	121,50

Eligibility for exams

The final classification is the average of the classifications obtained in the component related to the assignments (40%) and in the exam (60%). For each of the components, it is necessary to obtain a minimum grade of 7 Values. Failure to obtain at least 7 values in the assignment inhibits the possibility to do any of the subsequent examinations.

Calculation formula of final grade

Examinations or Special Assignments

An assignment on Finite Element analysis/programming.

Special assessment (TE, DA, ...)

A written exam. The computer assignment has to be done during the first semester.

Classification improvement

Students can improve their grade by attending resit exam.