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Biomechanics : concepts and computation / Cees Oomens, Marcel Brekelmans, Frank Baaijens. — Cambridge ; New York : Cambridge University Press, 2009. – (58.17111/O59) |
Contents
Contents
1 Vector calculus
1.1 Introduction
1.2 Definition of a vector
1.3 Vector operations
1.4 Decomposition of a vector with respect to a basis
Exercises
2 The concepts of force and moment
2.1 Introduction
2.2 Definition of a force vector
2.3 Newton's Laws
2.4 Vector operations on the force vector
2.5 Force decomposition
2.6 Representation of a vector with respect to a vector basis
2.7 Column notation
2.8 Drawing convention
2.9 The concept of moment
2.10 Definition of the moment vector
2.11 The two-dimensional case
2.12 Drawing convention of moments in three dimensions
Exercises
3 Static equilibrium
3.1 Introduction
3.2 Static equilibrium conditions
3.3 Free body diagram
Exercises
4 The mechanical behaviour of fibres
4.1 Introduction
4.2 Elastic fibres in one dimension
4.3 A simple one-dimensional model of a skeletal muscle
4.4 Elastic fibres in three dimensions
4.5 Small fibre stretches
Exercises
5 Fibres: time-dependent behaviour
5.1 Introduction
5.2 Viscous behaviour
5.3 Linear visco-elastic behaviour
5.4 Harmonic excitation of visco-elastic materials
5.5 Appendix: Laplace and Fourier transforms
Exercises
6 Analysis of a one-dimensional continuous elastic medium
6.1 Introduction
6.2 Equilibrium in a subsection of a slender structure
6.3 Stress and strain
6.4 Elastic stress-strain relation
6.5 Deformation of an inhomogeneous bar
Exercises
7 Biological materials and continuum mechanics
7.1 Introduction
7.2 Orientation in space
7.3 Mass within the volume V
7.4 Scalar fields
7.5 Vector fields
7.6 Rigid body rotation
7.7 Some mathematical preliminaries on second-order tensors
Exercises
8 Stress in three-dimensional continuous media
8.1 Stress vector
8.2 From stress to force
8.3 Equilibrium
8.4 Stress tensor
8.5 Principal stresses and principal stress directions
8.6 Mohr's circles for the stress state
8.7 Hydrostatic pressure and deviatoric stress
8.8 Equivalent stress
Exercises
9 Motion: the time as an extra dimension
9.1 Introduction
9.2 Geometrical description of the material configuration
9.3 Lagrangian and Eulerian description
9.4 The relation between the material and spatial time derivative
9.5 The displacement vector
9.6 The gradient operator
9.7 Extra displacement as a rigid body
9.8 Fluid flow
Exercises
10 Deformation and rotation， deformation rate and spin
10.1 Introduction
10.2 A material line segment in the reference and current configuration
10.3 The stretch ratio and rotation
10.4 Strain measures and strain tensors and matrices
10.5 The volume change factor
10.6 Deformation rate and rotation velocity
Exercises
11 Local balance of mass， momentum and energy
11.l Introduction
11.2 The local balance of mass
11.3 The local balance of momentum
11.4 The local balance of mechanical power
11.5 Lagrangian and Eulerian description of the balance equations
Exercises
12 Constitutive modelling of solids and fluids
12.1 Introduction
12.2 Elastic behaviour at small deformations and rotations
12.3 The stored internal energy
12.4 Elastic behaviour at large deformations and/or large rotations
12.5 Constitutive modelling of viscous fluids
12.6 Newtonian fluids
12.7 Non-Newtonian fluids
12.8 Diffusion and filtration
Exercises
13 Solution strategies for solid and fluid mechanics problems
13.1 Introduction
13.2 Solution strategies for deforming solids
13.3 Solution strategies for viscous fluids
13.4 Diffusion and filtration
Exercises
14 Solution of the one-dimensional diffusion equation by means of the Finite Element Method
14.1 Introduction
14.2 The diffusion equation
14.3 Method of weighted residuals and weak form of the model problem
14.4 Polynomial interpolation
14.5 Galerkin approximation
14.6 Solution of the discrete set of equations
14.7 Isoparametric elements and numerical integration
14.8 Basic structure of a finite element program
14.9 Example
Exercises
15 Solution of the one-dimensional convection-diffusion equation by means of the Finite Element Method
15.1 Introduction
15.2 The convection-diffusion equation
15.3 Temporal discretization
15.4 Spatial discretization
Exercises
16 Solution of the three-dimensional convection-diffusion equation by means of the Finite Element Method
16.1 Introduction
16.2 Diffusion equation
16.3 Divergence theorem and integration by parts
16.4 Weak form
16.5 Galerkin discretization
16.6 Convection-diffusion equation
16.7 Isoparametric elements and numerical integration
16.8 Example
Exercises
17 Shape functions and numerical integration
17.1 Introduction
17.2 Isoparametric， bilinear quadrilateral element
17.3 Linear triangular element
17.4 Lagrangian and Serendipity elements
17.5 Numerical integration
Exercises
18 Infinitesimal strain elasticity problems
18.1 Introduction
18.2 Linear elasticity
18.3 Weak formulation
18.4 Galerkin discretization
18.5 Solution
18.6 Example
Exercises
References
Index
