Lecture Notes: Equilibrium in Multi-Dimensional Systems

Key Concepts

• Equilibrium Equation in 1D Mechanics
  • The equilibrium condition: ( \frac{dR}{dX} + B = 0 )
    • (\frac{dR}{dX}): Rate of change of internal force
    • (B): Body force
• Alternate version involving stress and cross-sectional area:
  • ( \frac{d(A \sigma)}{dx} + B = 0 )

Transition to Multi-Dimensional Systems

• Focus on the 2D case, with 3D extension being straightforward.
• Aim: Expression for equilibrium at each point in terms of stresses.

Approach for 2D Equilibrium

• Consider an arbitrary body and examine a specific point.
• Cut out a square of material with sides (\Delta X) and (\Delta Y).
• Analyze stresses and forces acting on this element.

Stress Analysis

• Forces in the X-direction:
  • Stress (\sigma_{xx}) on the left side.
  • On the right side, stress (\sigma_{xx} + \Delta \sigma_{xx}).
  • On the bottom and top, forces (\sigma_{yx}) and (\sigma_{yx} + \Delta \sigma_{yx}) respectively.
  • Multiply by area for force calculation.

Force Equilibrium Equation

• Sum of forces in the X-direction:
  • Initially includes terms that cancel each other out.
  • Simplified expression: ( \Delta \sigma_{xx}/\Delta X + \Delta \sigma_{yx}/\Delta Y + B_X = 0 )
  • Take the limit as (\Delta X, \Delta Y \to 0) to derive:
    • (\frac{\partial \sigma_{xx}}{\partial X} + \frac{\partial \sigma_{xy}}{\partial Y} + B_X = 0)
  • Represents force equilibrium in the X-direction.

Y-Direction Force Equilibrium and Shear Stresses

• Similar process for Y-direction:
  • Equilibrium equation: (\frac{\partial \sigma_{xy}}{\partial X} + \frac{\partial \sigma_{yy}}{\partial Y} + B_Y = 0)
• Moment Equilibrium:
  • Sum of moments about the Z-axis implies (\sigma_{xy} = \sigma_{yx}).
  • Ensures symmetry of the stress tensor.

Summary of Equations

• 2D Equilibrium Equations
  • Two PDEs (X and Y directions) and one algebraic equation (moment equilibrium).
  • Total: 3 equilibrium equations.
• Variables and Unknowns
  • 4 stress components: (\sigma_{xx}, \sigma_{yy}, \sigma_{xy}, \sigma_{yx}).
  • Statically indeterminate: Always one less equation than unknowns.

Conclusion

• Stress determination in multi-dimensional mechanics is inherently statically indeterminate.
• This feature highlights the complexity of determining pointwise stresses in bodies.