Deformable Bodies

Connecting to Deformable Bodies

Deformable bodies are like pistons

• when supporting a mass, they deform until the ‘reactive force’ equals the weight of the object
• harder objects have large K, soft objects lower K

Until the deformation required to counteract the mass is too large, and then it breaks.

Stress-Strain Relationship

Why?

• Choosing materials for robot components
• Choosing gripper material
• Task dependent: go from A to B, apply force F with sufficient positional tolerance
• We want lightweight robots
  • Low energy, cheap, safe
• We want robust robots
  • 20,000+ hours of service
• We want precise robots
  • No (or well modeled) deformation

Stress

Stress: imagine that for a rod made of a certain material that we are considering for our robot pulling force (tensile) is applied from both sides. The mechanical stress is defined as the force divided by the cross-section area: $\sigma =\frac{F}{A_0}$.

Strain

Imagine that with this pulling force, our rod stretches a bit ($\Delta l$)

• Strain: the ratio of change in length by the length when no force is applied $ϵ=\frac{\Delta l}{l_0}$

The Relationship

At low pulling forces, stress-strain have a linear relationship for both.

Hook’s law: $\sigma =Eϵ$

The slope is called the Youngs modulus ($E$)

• Units of pressure
  • Pa
  • Pounds per square inch (PSI)

The linear portion of the stress-strain curve is also called the elastic regime

• The material will recover its original shape if the load is removed

The second portion is the plastic regime

• More load leads to yield permanent change of shape

Shear Force and Bending Moment

When a beam is loaded, there are internal forces that appear to maintain equilibrium.

• Forces parallel to the beam section
• Forces perpendicular to the beam section (along the beam)

Shear force is the force parallel to the cross section of the beam. The bending moment is caused by unequal forces along the beam, creating a torque/moment.