Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 132

1

0.75

0.2

0.5

0.25 0.15

0

0 0.1

0.2

0.4

0.05

0.6

0.8

0

1

Fig. 6.1 The temporal evolution of the pressure distribution in a layer of rigid porous material.
The vertical scale is the dimensionless pressure P with a range of 0 to 1. The front horizontal scale
is the dimensionless coordinate X , also with a range of 0 to 1; it traverses the porous layer.
The dimensionless time constant T is plotted from 0 to 0.2 in the third direction. The purpose of the
plot is to illustrate the evolution of the steady state linear distribution of pressure across the layer of
rigid porous material

X in
the notation of the dimensionless variables. The solution to this dimensionless
differential equation for t

Note that the steady-state solution p ( x 1 ) ¼ r gh ( x 1 / L ) þ p o becomes P ¼

0 subject to these boundary and initial conditions is

X

n¼1

2
p

1
n ð

p

n 1 e n 2

2 T sin

P ¼ X 1 Þ ð n p X Þ:

n¼ 1

Substitution of this solution back into the differential equation above may be used
to verify that the solution is a solution to the differential equation. Note that the
steady-state solution P X is recovered as t tends to infinity. The temporal evolu-

tion of the steady-state linear distribution of pore fluid pressure across the layer of
rigid porous material is illustrated in Fig. 6.1 . The vertical scale is the dimensionless
pressure P with a range of 0 to 1. The front horizontal scale is the dimensionless
coordinate X , also with a range of 0 to 1; it traverses the porous layer. The
dimensionless time constant T is plotted from 0 to 0.2 in the third direction.

¼

Problems

6.2.1 Show that the pore fluid density r f satisfies the same differential equation for

diffusion, equation ( 6.11 ), as the pore fluid pressure p ( 6.8 ).
6.2.2 Verify that the form of the rescaled equations ( 6.10 ) and ( 6.17 ) follow from
( 6.13 ) and ( 6.14 ). Describe the shape of a homogeneous orthotropic material
object O that is in the form of a cube after it is rescaled and distorted so that
the differential equation is isotropic.