Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 159

(d) What is the stress on the surface x 3 ¼ h ? Please describe these stresses

in prose.
(e) What is the stress on the surface x 3 ¼ 0? Please describe these stresses in

prose.
(f) In the portion of the figure containing the words “Shear stress” there is a
line drawn across the flow domain. Please sketch in the shear stress
distribution across the flow field using this line as the zero shear
stress reference.
(g) In the portion of the figure containing the word “Velocity” there is a line
drawn across the flow domain. Please sketch in the velocity distribution
across the flow field using this line as the zero velocity reference.

6.5 The Theory of Viscoelastic Materials

An algebraic overview of the theory of linear viscoelastic materials can be obtained
by considering it as a system of fifteen equations in fifteen unknown functions of time.
The fifteen unknown functions of time are the six components of the stress tensor T ,
the six components of the strain tensor E , and the three components of the displace-
ment vector u . The parameter functions of a viscoelasticity problem are the tensor of
viscoelastic coefficients G ( s )

and the action-at-a-distance force d ( s ), which are assumed to be known. The system
of fifteen equations consists of viscoelastic stress-strain relations,

ð s Þ , call the tensor of relaxation functions , the density r

Z s¼t

T G

ÞD

ð x ; t Þ¼ ð t s ð x ; s Þ d s

s ¼1

ð 5 : 36V Þ repeated

Z s¼t

DE
Ds ð

G

¼ ð t s Þ x ; s Þ d s ;

s ¼1

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 ; ð 6 : 18 Þ repeated

Interchanging the roles of stress and strain the current strain is expressed as a
function of the past history of stress by

Z s¼t

DT
Ds ð

E J

ð x ; t Þ¼ ð t s Þ x ; s Þ

d s

;

(6.41)

s¼1

where J

ð s Þ is called the tensor of creep function.