Derive complex modulus for Zener solid using superposition
Relaxation modulus
> G:=t -> G_R+(G_U - G_R)*exp(-t/tau);
t
τ
G,= t → G_R + ( G_U? G_R ) e
Sinusoidal strain input (unit magnitude)
> unprotect(gamma):gamma:=t -> sin(omega*t);
γ,= t → sin( ω t)
Superposition integral for stress output
> G_star:=int(G(t-xi)*diff(gamma(xi),xi),xi=0..t);
Simplify a bit
> collect(factor(G_star),sin(omega*t));
(?ω
2
G_U τ
2
G_R ) sin( t ω)
1 + ω
2
τ
2
ω G_U τ e
t
τ
ω τ e
t
τ
ω G_U τ cos( t ω) ω G_R τ+ cos( t ω)? G_R
1 + ω
2
τ
2
Page 1
