I agree with PM-14's approach to the analysis. Curling of the unrestrained alloy should be into a circular form.

Maximum moment whilst gripping the pencil will be at the top, where the lever arm (perpendicular to the gripping force) is longest ($h_{max}$): $$M_{max} = F_{grip}\cdot h_{max}$$

I do not fully comprehend the curling mechanism, but it is very much possible, that the maximum moment is independent of it. So it would depend just on classic mechanical properties of the "beam": $$\sigma = -z\cdot \frac{M}{I_y}$$

where $z$ is maximum distance from the neutral axis in the beam section and $J_y$ is the second moment of area with respect to $y$ axis.

Combining the 2 equations and setting $z$ to half of the beam section height $b/2$ and stress $\sigma$ to maximum allowable stress $f$: $$F_{grip} = 2\frac{f\cdot J_y}{b\cdot h_{max}}$$

Finally, assuming that the original length of the gripper is $L$ (and it does not change when curling) and it will bend to a circle with diameter $h_{max}$ the formula can be expressed as this

$$F_{grip} = 2\pi\frac{f\cdot J_y}{b\cdot L}$$

Simplifications:

• Combination of stiff and extending layers will increase overall length when curled.
• There is most likely some pre-stress after the curling before any external loading is applied.
• I used formula for stress in straight beam, where the axial stress from bending varies linearly across the section, but this is no longer the case for curved beams, where more complicated formulae would be applied.

I agree with PM-14's approach to the analysis. Curling of the unrestrained alloy should be into a circular form.

Maximum moment whilst gripping the pencil will be at the top, where the lever arm (perpendicular to the gripping force) is longest ($h_{max}$): $$M_{max} = F_{grip}\cdot h_{max}$$

I do not fully comprehend the curling mechanism, but it is very much possible, that the maximum moment is independent of it. So it would depend just on classic mechanical properties of the "beam": $$\sigma = -z\cdot \frac{M}{I_y}$$

where $z$ is maximum distance from the neutral axis in the beam section and $I_y$ is the second moment of area with respect to $y$ axis.

Combining the 2 equations and setting $z$ to half of the beam section height $b/2$ and stress $\sigma$ to maximum allowable stress $f$: $$F_{grip} = 2\frac{f\cdot I_y}{b\cdot h_{max}}$$

Finally, assuming that the original length of the gripper is $L$ (and it does not change when curling) and it will bend to a circle with diameter $h_{max}$ the formula can be expressed as this

$$F_{grip} = 2\pi\frac{f\cdot I_y}{b\cdot L}$$

Simplifications:

• Combination of stiff and extending layers will increase overall length when curled.
• There is most likely some pre-stress after the curling before any external loading is applied.
• I used formula for stress in straight beam, where the axial stress from bending varies linearly across the section, but this is no longer the case for curved beams, where more complicated formulae would be applied.

I agree with PM-14's approach to the analysis. Curling of the unrestrained alloy should be into a circular form.

Maximum moment whilst gripping the pencil will be at the top, where the lever arm (perpendicular to the gripping force) is longest ($h_{max}$): $$M_{max} = F_{grip}\cdot h_{max}$$

I do not fully comprehend the curling mechanism, but it is very much possible, that the maximum moment is independent of it. So it would depend just on classic mechanical properties of the "beam": $$\sigma = -z\cdot \frac{M}{I_y}$$

where $z$ is maximum distance from the neutral axis in the beam section and $J_y$ is the second moment of area with respect to $y$ axis.

Combining the 2 equations and setting $z$ to half of the beam section height $b/2$ and stress $\sigma$ to maximum allowable stress $f$: $$F_{grip} = 2\frac{f\cdot J_y}{b\cdot h_{max}}$$

Finally, assuming that the original length of the gripper is $L$ (and it does not change when curling) and it will bend to a circle with diameter $h_{max}$ the formula can be expressed as this

$$F_{grip} = 2\pi\frac{f\cdot J_y}{b\cdot L}$$

I agree with PM-14's approach to the analysis. Curling of the unrestrained alloy should be into a circular form.

Maximum moment whilst gripping the pencil will be at the top, where the lever arm (perpendicular to the gripping force) is longest ($h_{max}$): $$M_{max} = F_{grip}\cdot h_{max}$$

I do not fully comprehend the curling mechanism, but it is very much possible, that the maximum moment is independent of it. So it would depend just on classic mechanical properties of the "beam": $$\sigma = -z\cdot \frac{M}{I_y}$$

where $z$ is maximum distance from the neutral axis in the beam section and $J_y$ is the second moment of area with respect to $y$ axis.

Combining the 2 equations and setting $z$ to half of the beam section height $b/2$ and stress $\sigma$ to maximum allowable stress $f$: $$F_{grip} = 2\frac{f\cdot J_y}{b\cdot h_{max}}$$

Finally, assuming that the original length of the gripper is $L$ (and it does not change when curling) and it will bend to a circle with diameter $h_{max}$ the formula can be expressed as this

$$F_{grip} = 2\pi\frac{f\cdot J_y}{b\cdot L}$$

Simplifications:

• Combination of stiff and extending layers will increase overall length when curled.
• There is most likely some pre-stress after the curling before any external loading is applied.
• I used formula for stress in straight beam, where the axial stress from bending varies linearly across the section, but this is no longer the case for curved beams, where more complicated formulae would be applied.