CONTINUUM MECHANICS for ENGINEERS

9.13 For the hereditary integral, Eq 9.5-2
assume σ(t) = est where s is a constant. Let T = t – t′ be the “elapsed
time” of the load application and show that σ(t) = sest where
is the Laplace transform of J(t).
9.14 Using σ(t) = est as in Problem 9.13, together with the hereditary integral
Eq 9.5-5a
Js () Js ()
2
and the result of Problem 9.13 show that GsJs ()()= 1/
s where Gs
is the Laplace transform of G(t). Assume s is real.
9.15 Taking the hereditary integrals for viscoelastic behavior in the form
Eq 9.5-4
and Eq 9.5-5c
show that for the stress loading σ(t) = e st and with T = t – t′, the
expression
results, where here
and
∫
t
( ) ′
( ) ( ′ ) ′
γ()= t J t − t′ dσ t / dt dt
∫
−∞
t
( ) ′
( ) ( ′ ) ′
σ()= t G t − t′ dγ t / dt dt
−∞
()= ()+ ( ′ ) ( − ′ ) ( − ′ )
γ t J σ t σ t dJ t t / d t t dt
o
t
∫ 0
[ ] ′
()= ()+ ( ′ ) ( − ′ ) ( − ′ )
σ t G γ t γ t dG t t / d t t dt
o
t
∫ 0
[ ] ′
()+ ()+ () ()= 0
GAs JBs AsBs
o o
∫ 0
∞
()= ( )
−sT
As e dJ/ dTdT
∫
∞
()= ( )
−sT
Bs e dG/ dTdT
0
()