4.13 Dynamic loading (DMA) Laplace-plane shear operator> G[L]:=G[R]+ (G[d]*s)/(s+1/tau[sigma]); := GL + GRGds + s1τσApplied strain in time plane:> unprotect(gamma);gamma(t):=gamma[0]*cos(omega*t); := ()γtγ0()cos ωtApplied strain in laplace plane:> with(inttrans):gamma(s):=laplace(gamma(t),t,s); := ()γ sγ0s + s2ω2Dynamic modulus in laplace plane:> G_bar:=G[L]*gamma(s)/gamma[0]; := G_bar⎛⎝⎜⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟⎟ + GRGds + s1τσs + s2ω2Invert for time-plane modulus:> G_t:=invlaplace(G_bar,s,t); := G_t − + + + Gde⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟−tτσ + ω2τσ21ωτσGd()sin ωt + ω2τσ21GRω2τσ2()cos ωt + ω2τσ21GR()cos ωt + ω2τσ21ω2τσ2Gd()cos ωt + ω2τσ21Simplifying:> 'G(t)'=factor(collect((G_t),cos(omega(t)))); = ()G t − + + + Gde⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟−tτσωτσGd()sin ωtGRω2τσ2()cos ωtGR()cos ωtω2τσ2Gd()cos ωt + ω2τσ21Simplifying further and rearranging manually:() ()22*22 22 22cos sin11 1tdd dRGG GGeG t tστσσσσ σωτ ωτωωωτ ωτ ωτ−⎛⎞⎛⎞=++ −⎜⎟⎜⎟++ +⎝⎠⎝⎠The first term is an initial transient that diminishes with time; the second term is varying in phase with the applied strain and is thus the real modulus G', and the third term is varyiing 90 deg out of phase Page 1and is thus the loss modulus G". For the generalized Zener model we sum over the number of Maxwell arms:2222 2211,11mmddRiiGGGG Gσσσσωτωτωτωτ==′′′=+ =++∑∑Page
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