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Compression of an elastic body

In this exercise the compression of an elastic body under different external constraints is considered.

Task

An elastic body is embedded in a rigid die without clearance and without friction. The body is loaded with the force F via a rigid plunger.

Given parameters:

Dimensions a, b, c, Young’s modulus E, Poisson’s ratio ν, strain θz

To be determined:

a) Compression force F to compress the body in the z-direction by θz.

b) Pressure on the surfaces of the die.

c) Change of the contour length under the force F when the die is removed.

d) Compression force F * with a compression by θz without a die.

Compression of an elastic body
Compression of an elastic body

Solution

Hooke’s law is:

\[ \tag{1} \epsilon_x = \frac{1}{E} \left[ \sigma_x -- \nu \left( \sigma_y + \sigma_z \right) \right] \]

\[ \tag{2} \epsilon_y = \frac{1}{E} \left[ \sigma_y -- \nu \left( \sigma_x + \sigma_z \right) \right] \]

\[ \tag{3} \epsilon_z = \frac{1}{E} \left[ \sigma_z -- \nu \left( \sigma_x + \sigma_y \right) \right] \]

reg. a)

The strains in direction x and y are Zero.

\[ \tag{4} \epsilon_x = 0 \]

\[ \tag{5} \epsilon_y = 0 \]

The strain in z-direction is

\[ \tag{5} \epsilon_z = -- \frac{\theta_z}{c} \]

The tension in z-direction is

\[ \tag{6} \sigma_z = -\frac{F}{ab} \]

With this Hooke’s law leads to

\[ \tag{7} F = \frac{\theta_z a b E}{c \left( 1 -- \frac{2 \nu^2}{1-\nu} \right)} \]

reg. b)

The pressure at the surfaces of the die is

\[ \tag{8} \sigma_x = \sigma_y = -\frac{F \nu}{ab \left( 1 -- \nu \right)} \]

reg. c)

When the die is removed, the tensions are

\[ \tag{9} \sigma_x = 0 \]

\[ \tag{10} \sigma_y = 0 \]

\[ \tag{11} \sigma_z = -\frac{F}{ab} \]

from which the following changes in the contour lengths result

\[ \tag{12} \Delta b = \epsilon_x b = \frac{\nu F}{Ea} \]

\[ \tag{13} \Delta a = \epsilon_y a = \frac{\nu F}{Eb} \]

\[ \tag{14} \Delta c = \epsilon_z c = -\frac{F c}{abE} \]

reg. d)

The stresses σx and σy are zero. The strain is

\[ \tag{15} \epsilon_z = -\frac{\theta_z}{c} = \frac{\sigma_z}{E} = -- \frac{F^*}{abE} \]

\[ \tag{16} F^* = \frac{\theta_z E a b}{c} \]