Springpot

using RHEOS
# include a helper function for plotting
include("assets/plothelper.jl");

By typing the name of the model, it is possible to visualise its graphical representation and its parameters.

Springpot

Model name: springpot

Free parameters: cᵦ and β

                ____ ╱╲ ____
                     ╲╱  cᵦ, β
                

Constitutive Equation

\[\sigma(t) = c_{\beta} \frac{d^\beta \epsilon(t)}{dt^\beta}\]

\[\text{for}\; \ 0 \leq \beta \leq 1\]

Relaxation Modulus

\[G(t) = \frac{c_{\beta} }{\Gamma(1-\beta)} t^{-\beta}\]

Creep Modulus

\[J(t) = \frac{1}{c_\beta \Gamma(1+\beta)}t^\beta\]

Storage Modulus

\[G^{\prime}(\omega) = c_\beta \omega^\beta \cos(\frac{\pi}{2}\beta)\]

Loss Modulus

\[G^{\prime\prime}(\omega) = c_\beta \omega^\beta \sin(\frac{\pi}{2}\beta)\]

Spring

When β = 0 the springpot specialises to a spring.

Spring

Model name: spring

Free parameters: k

                ___╱╲  ╱╲  ╱╲  ________
                     ╲╱  ╲╱  ╲╱  k
                

Dashpot

When β = 1 the springpot specialises to a dashpot.

Dashpot

Model name: dashpot

Free parameters: η

                 ___
             _____| |_____
                 _|_|
                     η
                

Qualitative Behaviours of the Moduli

models = Vector{RheoModel}()

# Spring
push!(models, RheoModel(Spring, k = 1.0))

# plot moduli for varying β
for beta in [0.2, 0.5, 0.8]

    push!(models, RheoModel(Springpot, cᵦ = 1.0, β = beta))

end

# Dashpot
push!(models, RheoModel(Dashpot, η = 1.0))

plotmodel(models, ymaxG = 2.0)

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