5.2 “Mathematical tools” in biomechanics
5.2.1 Vectors and scalars
5.2.1.1 Vectors
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Vectors are quantities that have an associated magnitude and direction.\(^{**}\)
Vector have components
- \(\vec{F}_x = {F}_x i\)
- \(\vec{F}_y = {F}_y j\)
- \(\vec{F} = \vec{F}_x + \vec{F}_y = {F}_x i +{F}_y j\)
Examples of vectors
- Forces (Weight)
- Moments (Torques)
- Velocities
5.2.1.2 Vector sums
- Vector addition is component by component
- Proper (2D) summation results in a parallelogram
5.2.1.3 Scalars
- Scalars have magnitude without direction.
- Examples
- Levels of gray in a image
- Patient mass
- Patient bone density
- Concentration of a drug per unit volume
- Time
- Altitude
5.2.2 Relationship between vectors and scalars
- Scalars and vectors can be linked mathematically
Example: diffusion of nutrients into intervertebral disk driven by gradient of glucose
5.2.2.1 Gradient of a scalar
\[\nabla f = \left(\frac{\partial f}{\partial x_1 }, \dots, \frac{\partial f}{\partial x_n } \right).\]
Gradient of a vector becomes a 2nd rank tensor \[\mathbf{f}=({{f}_{1}},{{f}_{2}},{{f}_{3}})\] \[\nabla \mathbf{f}=\frac{\partial {{f}_{j}}}{\partial {{x}_{i}}}{{\mathbf{e}}_{i}}{{\mathbf{e}}_{j}}\]
Example: strain is gradient of displacement vector
- \[\vec{u} = u {{e_x}}+ v {{e_y}}+ w {{e_z}}\]
- \[= {\varepsilon_{ij}}= \nabla \vec{u} = \left[ \begin{array}{ccc} {\varepsilon_{xx}}& {\varepsilon_{xy}}& {\varepsilon_{xz}}\\ {\varepsilon_{yx}}& {\varepsilon_{yy}}& {\varepsilon_{yz}}\\ {\varepsilon_{zx}}& {\varepsilon_{yz}}& {\varepsilon_{zz}}\\ \end{array} \right]\]
5.2.2.2 Divergence
Similarly, the divergence operator (\(\nabla \cdot ()\)) reduces the
rank of a tensor- i.e., makes a scalar out of a vector
Example: the divergence calculates the strength
of a source or sink of a velocity field in fluid flow
5.2.3 Rigid body and flexible body assumptions
- All bodies are flexible, meaning, all bodies deform when loaded
- For convenience, we often assume that a body is rigid
- (ie we assume that it does not deform when loaded.)
- With this assumption, the mechanics/mathematics is simplified