Mechanical Modeling of Material Behavior - Continuum Mechanics of Anisotropic Materials - page 19

Now from the equation for the sumof the forces in the two branches ( F ¼ F L þ F R )

it follows that F R ¼ F F L and since F L ¼ kx,F R ¼ F kx . Substituting this equation

for F R in the equation involving F R above it follows that

F þð= k R )(d F =

d t

Þ¼ kx þ ð þ k = k R Þ

(d x

=

d t

Þ:

1 (1.7)

The physical implications of the constant parameters characterizing the standard
linear solid are easier to understand if the constants

and k R are redefined in terms
of time constants. To that end ( 1.7 ) is rewritten in the form

þ t x d F þ t F d x

d t

F

d t ¼

kx ; (1.8)

where t x and t F are material time constants defined by

¼ k þ t x

t x ¼ k R t F ¼ k

k
k R

and 1 þ (1.9)

These material time constants will be shown to have interpretations as the
characteristic relaxation times of the load associated with a steady, constant deflec-
tion and the deflection associated with a steady, constant load, respectively. Note
that, from the definitions,

t F > t x since

, k , and k R are positive.

■
The standard linear solid, characterized by the linear differential equation ( 1.8 ),
provides a reasonable first model for the phenomena of creep under constant load
and stress relaxation under constant deflection, phenomena observed in many
materials. The creep function is the increase in time of the deflection x ( t ) when a
unit force is applied to the element at t 0 and held constant forever. The

relaxation function is the decrease in time of the force F ( t ) when a unit deflection
is applied to the element at t

¼

0 and held constant forever. The creep and
relaxation functions for the standard linear solid are obtained from solving the
governing differential equation ( 1.8 ). In Sect. A.16 of the Appendix the Laplace
transform method (Thomson 1960 ) for solving the differential equation ( 1.8 )is
described; this is the simplest method of solution. In Sect. A.17 of the Appendix the
more complicated approach of using direct integration is described. The creep
function is the solution of ( 1.8 ) for x ( t ) when F ( t ) is specified to be the unit step
function h ( t ) and the relaxation function is the solution of ( 1.8 ) for F ( t ) when x ( t )is
specified to be the unit step function h ( t ) (see (A219) in the Appendix for a
definition of the unit step function). The unit step function h ( t ) is therefore
employed in the representation of deflection history x ( t ) to obtain the creep function
c ( t ) as well as in the representation of the force history F ( t ) to obtain the relaxation
function r ( t ). The creep function c ( t ) for the standard linear solid is given by

¼

e t=t F

h ð t Þ

t x
t F

c ð t Þ¼ 1 1 : (1.10)

k