麻省理工学院：《polymer engineering》lec 16

scs oC4 GCL-SAY 58z 七 cC公区c△ c=f≤cte出c 七

Derive complex modulus for Zener solid using superposition Relaxation modulus >G: =t ->G R+(G U -G R)*exp(-t/tau)i G:=t→GR+(GU-GR)e Sinusoidal strain input(unit magnitude) unprotect(gamma): gamma: =t - sin(omega*t)i Y:=t→sin(ot) Superposition integral for stress output >G star: =int(G(t-xi)*diff (gamma(xi),xi),xi=0. t)i Simplify a bit collect(factor(G star), sin(omega*t))i (-oGUT-GR)Sin(t o) 1+0-τ OG Ute -OGUtcos( o)+og coS(Io-ate Gr 1+0^τ

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 + ω2 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 + ω2 τ 2 Page 1