Mechanical Modeling of Material Behavior - Continuum Mechanics of Anisotropic Materials - page 15

difference between the fluid chamber and the outside air is p, thus q ¼ k ( p / L ) ¼

k ( p / L ) where k is the permeability constant.

∂ ∂

Example 1.8.1
Show that, from the viewpoint of the relationship between applied force F and time
rate of change of the deflection (d x /d t ), the permeability element is equivalent to a
dashpot element characterized by the constant . An equivalent statement of the

problem would be to show that the dashpot constant Z of the permeability element
(Fig. 1.6c ) is related to the permeability

A 2
A o

where A is the area of the
piston in the permeability element, L is the thickness of the piston, and A o is the
cross-sectional area of the orifice in the piston.
Solution : To show this, first note that a force balance applied to a free object
diagram of the piston of the permeability element (Fig. 1.6c ) shows that F

k ¼

L
k

by

pA
where A is the cross-sectional area of the piston. If d x /d t denotes the time rate of
changes of the downward movement of the piston and q the volume flow rate
through the orifice in the piston, then three different representations of the time rate
of change of the fluid volume in the cylinder chamber are given by

¼

d V
d t ¼

A d x

d t ¼

qA o

where A o is the cross-sectional area of the piston orifice. It follows that q , the
volume flow rate per unit area through the orifice, may be expressed as

A
A o

d x
d t :

q ¼

Combining this result with F ¼ pA and q ¼ k ( p / L ), a constitutive relation for

the dashpot of the form ,F ¼ (d x /d t ), is again obtained but with

A 2
A o

L
k :

¼

This result identifies the source of the dashpot-like viscous loss in the perme-
ability element as fluid movement. The velocity of the fluid movement and the fluid
pressure are the primary parameters in the permeability element; the applied force
and the piston deflection associated with the dashpot element are secondary. It will
come as no surprise that the permeability element will often behave just like a
dashpot element. However the permeability element is not the same as the dashpot
element because the source of the viscous loss is identified as fluid movement in the
permeability element and it is unspecified in the dashpot element.