Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 162

Fig. 6.7 The relaxation
function G ( t ). The relaxation
function is the stress
response to a step input of
strain at t ¼ 0

Z s¼t

d E 1
d s

T 1 ð G 11 ð

t Þ¼ t s Þ d s : (6.46)

s¼1

The result ( 6.46 ) is a special case of (5.36V). The physical significance of the
relaxation function G 11 ð is the following: it is the uniaxial normal stress T 1 ð

vs.
time response of a specimen subjected to a step increase in the normal strain in the
same direction, E 1 ¼

s Þ t Þ

.
The creep and relaxation functions, J

E o h ð t Þ

G

, are hypothesized to have a
property called fading memory . The fading memory hypothesis for Jð

ð s Þ and ð s Þ

Þð Gð ÞÞ is

that the strain (stress) depends more strongly upon the recent history than it does
upon the distant history of the value of stress (strain). Mathematically this is the
requirement that the functions J

s s

and G are continuously decreasing functions

of the backward running time parameter s . This then decreases the influence of the
more distant events. For each component of the viscoelastic strain-stress relations
to possess this fading memory behavior, it is sufficient that the magnitude of the
slope of each component of the creep function tensor be a continuously decreasing
function of time, thus

ð s Þ ð s Þ

s¼s 1 < s¼s 2

DJ ij ð DJ ij ð s Þ s Þ

for all i ; j ¼ 1 ; 2 ; ... 6 ; and for s 1 > s 2 > 0 : (6.47)

Ds Ds

For each component of the viscoelastic stress-strain relations a similar fading
memory hypothesis holds, thus