Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 133

2 P

6.2.3 Show that the solution to the dimensionless differential equation @P
@ T ¼ @

@X 2 and

the prescribed boundary and initial conditions that p ¼ p o everywhere in the

medium and on its boundaries for t < 0, that p ¼ p o at x ¼ 0 for for

t 2 [ 1 , 1 ] and p ¼ p 1 at L ¼ h for t 2 [0, 1 ], is given by

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:

n ¼ 1

6.2.4 Record the explicit matrix form for the constitutive relation for the Darcy
medium (5.6D).
6.2.5 Record the explicit matrix form for the constitutive relation for a
Darcy porous medium in an inhomogeneous transversely isotropic material.
6.2.6 Record the explicit matrix form for the constitutive relation for Darcy's law
in a homogeneous isotropic material.

6.3 The Theory of Elastic Solids

An overview of the theory of linear elastic solids can be obtained by considering it
as a system of fifteen equations in fifteen scalar unknowns. The fifteen scalar
unknowns are the six components of the stress tensor T , the six components
of the strain tensor E , and the three components of the displacement vector u .
The parameters of an elasticity problem are the tensor of elastic coefficients C , the
density , and the action-at-a-distance force d , which are assumed to be known. The

system of fifteen equations consists of stress-strain relations from the anisotropic
Hooke's law,

r

T ¼ C E; where C ¼ C T

:ð 5 : 7H Þ and ð 5 : 24H Þ repeated

the six strain-displacement relations,

T

E ¼ð 1 = 2 Þððr u Þ þr u Þ; ð 2 : 49 Þ repeated

and the three stress equations of motion,

T T

r€ u ¼r T þ r d ; T ¼ : (6.18)

This form of the stress equations of motion differs from (3.38) only in notation:
the acceleration is here represented by

x , a result that follows from
(2.20) by taking the time derivative twice and assuming that the material and spatial
reference frames are not accelerating relative to one another. The system of fifteen
equations in fifteen scalar unknowns may be reduced to a system of three equations
in three scalar unknowns by accomplishing the following algebraic steps: (1) sub-
stitute the strain-displacement relations (2.49) into the stress-strain relations

€ u rather than €