Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 160

Fig. 6.5 The creep function
J ( t ). The creep function is the
strain response to a step input
of stress at t ¼ 0

The physical significance of the creep and relaxation functions, J and G

, are
best illustrated by one-dimensional examples; uniaxial one-dimensional examples
will be used here. A uniaxial tension specimen of homogeneous material is
subjected to step increase in stress T 1 ¼

ð s Þ ð s Þ

T o h ð t Þ , where h ( t ) is the unit step function,

T 1 ¼

h ( t ) ¼ 0 for t < 0 and h ( t ) ¼ 1 for t > 0. The loading function T o h ð t Þ is

plotted in the top panel of Fig. 6.5 . The creep function J 11 ð is the uniaxial strain-

vs.-time response of the uniaxially stressed specimen to the step increase in stress,
T 1 ¼

s Þ

J 11 ð

T o h ð t Þ . The function s Þ is illustrated in the lower panel of Fig. 6.5 . The

strain response is given by E 1 ¼

T o J 11 ð s Þ

. The creep function defined and measured
in this way may be used to predict the creep response to a more complicated stress
history. Suppose, for example, the stress history is a long series of step jumps rather
than just one step jump (Fig. 6.5 ). The creep strain response to this new stress
history may be built up by repeated application of the basic result E 1 ¼ T o J 11 ð s Þ to

each step, and subsequent summation of strains associated with each step,

E 1 ð

Þ¼J 11 ð ÞD T 1 ð ÞþJ 11 ð t 1 ÞD T 1 ð t 1 ÞþþJ 11 ð t n ÞD T 1 ð

t t t t t n Þþ:

(6.42)

0

The multiple step plot of Fig. 6.6 is familiar from the introductory presentations
to the process of integration in which a horizontal axis is divided into segments and
the curve is approximated by different level steps drawn horizontally for each
segment so that the curve is approximated by the series of various sized steps.
It follows that any curve representing a stress history could be approximated
arbitrarily closely by a set of various sized steps like those illustrated in Fig. 6.6
and represented analytically by an equation of the type ( 6.42 ). In preparation for a
passage to the limit of the type used in the integral calculus, ( 6.42 ) is rewritten as