Dynamic loading (DMA)
Laplace-plane shear operator
> G[L]:=G[R]+ (G[d]*s)/(s+1/tau[sigma]);
G
d
s
G
L
:= G
R
+
1
s +
τ
σ
Applied strain in time plane,
> unprotect(gamma);gamma(t):=gamma[0]*cos(omega*t);
γ,=()t γ
0
cos( ω t)
Applied strain in laplace plane,
> with(inttrans):gamma(s):=laplace(gamma(t),t,s);
γ
0
s
γ,=()s
s
2
+ ω
2
Dynamic modulus in laplace plane,
> G_bar:=G[L]*gamma(s)/gamma[0];
G
d
s
G
R
+
1
s +
τ
σ
s
G_bar,=
s
2
+ ω
2
Invert for time-plane modulus,
> G_t:=invlaplace(G_bar,s,t);
t
τ
σ
ω
2
τ
σ
2
G
R
ω
2
τ
σ
2
cos( ω t) G
R
cos( ω t)
G
d
e ωτ
σ
G
d
sin( ω t)
+
G
d
cos( ω t)
G_t,= + +
ω
2
τ
σ
2
ω
2
τ
σ
2
ω
2
τ
σ
2
+ 1
ω
2
τ
σ
2 2
1 ω
2
τ
σ
1 + 1 ++ +
Simplifying,
> 'G(t)'=factor(collect((G_t),cos(omega(t))));
t
τ
σ
ω
2
τ
σ
2
G
R
ω
2
τ
σ
2
cos( ω t) G
R
cos( ω t)
G
d
e? ωτ
σ
G
d
sin( ω t) + G
d
cos( ω t)+ +
()G =t
ω
2
τ
σ
2
+ 1
Simplifying further and rearranging manually,
t
22
σ
G
*
=
G
d
e
τ
σ
+
G
R
+
G
d
ωτ
σ
cos ()?
G
d
ωτ?
ωt
2 2
22 22
1+ωτ
σ?
1+ωτ
σ
1+ω
σ?
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