Computational Homogenization of Heterogeneous Materials with Finite Elements

By James Lee and Carl Von Linne

Book Overview

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Why computational homogenization? . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Brief historical and recent advances . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Industrial applications and use in commercial softwares . . . . . . . . . . 31.4 Position of the present monograph as compared to available otherbooks on that topic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Overview and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Review of classical FEM formulations and discretizations . . . . . . . . . . . 52.1 Steady-state thermal problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 Strong form of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Weak forms of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 2D FEM discretization with linear elements . . . . . . . . . . . . . . 72.1.4 Assembly of the elementary systems . . . . . . . . . . . . . . . . . . . . 122.1.5 Prescribing Dirichlet conditions . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.1 Strong form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 Weak form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.3 2D discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.4 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Conduction properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1 The notion of RVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Localization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Averaged quantities and Hill-Mandel lemma . . . . . . . . . . . . . . . . . . . . 313.3.1 Averaging theorem: temperature gradient . . . . . . . . . . . . . . . . 313.3.2 Averaging theorem: heat flux . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3.3 Hill-Mandel lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4 Computation of the effective conductivity tensor . . . . . . . . . . . . . . . . 333.4.1 The superposition principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4.2 Definition of the effective conductivity tensor . . . . . . . . . . . . 34vvi Contents3.5 Periodic boundary conditions for the thermal problem: numericalimplementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.6 Numerical calculation of effective conductivity with 2D FEM . . . . . 393.6.1 Transverse effective conductivity . . . . . . . . . . . . . . . . . . . . . . . 393.6.2 Computation of the out-of plane properties using a 2D RVE 413.7 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 Elasticity and thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.1 Localization problem for elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Averaged quantities and Hill-Mandel lemma . . . . . . . . . . . . . . . . . . . . 504.2.1 Averaging theorem: strain .

• Format: Paperback
• Language: English
• ISBN: 3030183858
• ISBN13: 9783030183851
• Release Date: August 2020
• Publisher: Springer
• Length: 223 Pages
• Weight: 0.76 lbs.
• Dimensions: 0.5" x 6.1" x 9.2"