Question

Harmonic response of the Maxwell element The Maxwell element for modeling a viscoelastic system treats it as a spring and a viscous dashpot connected in series as shown in the diagram below. The red arrows indicate the direction that the stress is applied. E and η are the Young's modulus of the spring and the viscosity of the dashpot component, respectively. Both parameters are real constants and characterize the model completely. dashpot spring (a) Show that the time rate of change of strain is governed by -+-The stress σ is the dt E dt n same in both components, while the overall strain ε isasum ofthe stransofthe spring anddash ot (b) Consider an oscillatory stress σ = ơoexp (wt) and strain ε = Eoexp (dot), where σ° and εο are complex constants. Define the complex modulus G*-G' + iG", so that σο-G'εο. Define the relaxation time τ = η/E. By substituting σ and ε in the differential equation derived in (a), prove that and G" 12T (c) Plot GYE and G"/E in the same graph with ωτ as the abscissa in the range 0 Comment on the behavior of G" as ωτ--o and G, as ωτ-+ oo. ωτ 10.

Answer 1