Affiliation:
1. Department of Mathematics, King Fahd University of Petroleum and Minerals , Dhahran 31261 , Saudi Arabia
2. The Interdisciplinary Research Center in Construction and Building Materials, King Fahd University of Petroleum and Minerals , Dhahran 31261 , Saudi Arabia
Abstract
Abstract
In this study, we consider a viscoelastic Shear beam model with no rotary inertia. Specifically, we study
ρ
1
φ
t
t
−
κ
(
φ
x
+
ψ
)
x
+
(
g
∗
φ
x
x
)
(
t
)
=
0
,
−
b
ψ
x
x
+
κ
(
φ
x
+
ψ
)
=
0
,
\begin{array}{rcl}{\rho }_{1}{\varphi }_{tt}-\kappa {\left({\varphi }_{x}+\psi )}_{x}+\left(g\ast {\varphi }_{xx})\left(t)& =& 0,\\ -b{\psi }_{xx}+\kappa \left({\varphi }_{x}+\psi )& =& 0,\end{array}
where the convolution memory function
g
g
belongs to a class of
L
1
(
0
,
∞
)
{L}^{1}\left(0,\infty )
functions that satisfies
g
′
(
t
)
≤
−
ξ
(
t
)
ϒ
(
g
(
t
)
)
,
∀
t
≥
0
,
g^{\prime} \left(t)\le -\xi \left(t)\Upsilon \left(g\left(t)),\hspace{1.0em}\forall t\ge 0,
where
ξ
\xi
is a positive nonincreasing differentiable function and
ϒ
\Upsilon
is an increasing and convex function near the origin. Using just this general assumptions on the behavior of
g
g
at infinity, we provide optimal and explicit general energy decay rates from which we recover the exponential and polynomial rates when
ϒ
(
s
)
=
s
p
\Upsilon \left(s)={s}^{p}
and
p
p
covers the full admissible range
[
1
,
2
)
\left[1,2)
. Given this degree of generality, our results improve some of earlier related results in the literature.