Finite Rotation Shells Basic Equations and Finite Elements for Reiss
This book covers theoretical and computational aspects of non-linear shells. Several advanced topics of shell equations and finite elements, not included in standard textbooks on finite elements, are addressed. Key features include: several sets of 3D equ
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Finite Rotation Shells Basic Equations and Finite Elements for Reissner Kinematics February 25, 2010
Springer
V
To my wife Ewa, my parents, and my children.
Contents
I
PRELIMINARIES
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Subject of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 2 4
2
Operations on tensors and their representations . . . . . . . . 2.1 Cartesian bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Normal bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Gradients and derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 6 10 19
II
SHELL EQUATIONS
3
Rotations for 3D Cauchy continuum . . . . . . . . . . . . . . . . . . . 3.1 Polar decomposition of deformation gradient . . . . . . . . . . . . . 3.2 Rotation Constraint equation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Interpretation of rotation Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Rate form of RC equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Rotations calculated from the RC equation . . . . . . . . . . . . . .
22 22 24 26 28 29
4
3D 4.1 4.2 4.3 4.4 4.5 4.6
formulations with rotations . . . . . . . . . . . . . . . . . . . . . . . . . Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-F formulation for nominal stress . . . . . . . . . . . . . . . . . . . . . . 3-F formulation for nominal stress . . . . . . . . . . . . . . . . . . . . . . 3-F and 2-F formulations for Biot stress . . . . . . . . . . . . . . . . . 3-F and 2-F formulations for second Piola–Kirchhoff stress 2-F formulation with unconstrained rotations . . . . . . . . . . . .
30 31 37 40 42 45 48
Contents
VII
5
Basic geometric definitions for shells . . . . . . . . . . . . . . . . . . . 5.1 Coordinates and position vector . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Basic geometric definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Example: Geometrical description of cylinder . . . . . . . . . . . .
49 49 52 57
6
Shells with Reissner kinematics and drilling rotation . . . 6.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Rotation Constraint for shells . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Shell strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Non-symmetric relaxed right stretch strain . . . . . . . . . 6.3.2 Symmetric relaxed right stretch strain . . . . . . . . . . . . . 6.3.3 Green strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Transverse shear strains satisfying RC. Kirchhoff kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Rotation as an intermediate variable sy
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