Hooke's Law

${\vec{F}}_{\text{spring}}=-k\vec{x}$

• ${\vec{F}}_{\text{spring}}$ is the elastic force provided by a spring or elastic component on the object the spring is attached to
• $k$ is the spring constant of the spring
• $\vec{x}$ is the displacement of the object from its equilibrium position

${\vec{F}}_{ij}=-k_{ij}\Delta {\vec{r}}_{ij}$

• $\overrightarrow{F_{ij}}$ is the spring force between two points
• $k_{ij}$ is the spring constant between the points
• $\Delta {\vec{r}}_{ij}$ is the deformation between two points from equilibrium

$k_{ij}=E\frac{A_{ij}}{L_{0ij}}$

• $E$ is Young’s modulus

$$\begin{array}{rl}{\vec{F}}_{ij}& =-k_{ij}\Delta {\vec{r}}_{ij}\\ {\sigma }_{ij}& =E{\epsilon }_{ij}\end{array}$$

${\vec{F}}_{ij}=-k_{ij}\Delta {\vec{r}}_{ij}$

$\overrightarrow{F_{ij}}$ is the spring force between points $i,j$

• $k_{ij}=E\frac{A_{ij}}{r_{0ij}}$ is the spring constant between $i,j$
  • $A_{ij}$ is the area between $i,j$
  • $r_{0ij}$ is the equilibrium distance between $i,j$
• $\Delta {\vec{r}}_{ij}$ is the deformation between $i,j$ from equilibrium

${\sigma }_{ij}=E{\epsilon }_{ij}$

• ${\sigma }_{ij}$ is the stress between points $i,j$
• $E$ is the Young’s modulus of the material
• ${\epsilon }_{ij}=\frac{\Delta r_{ij}}{r_{0ij}}$ is the strain between $i,j$

Continuum Mechanics

Textbook