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# Compression of an elastic body

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

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.

## 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

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

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