Fractional Maxwell

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

Fract_Maxwell

Model name: fractmaxwell

Free parameters: cₐ, a, cᵦ and β

             ___╱╲__________╱╲____
                ╲╱          ╲╱
                  cₐ,a         cᵦ, β
                 

Constitutive Equation

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

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

Relaxation Modulus

\[G(t) = c_\beta t^{-\beta} E_{\alpha-\beta,1-\beta}\left(-\frac{c_\beta}{c_\alpha} t^{\alpha-\beta}\right)\]

Creep Modulus

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

Storage Modulus

\[G^{\prime}(\omega) = \frac{\left(c_\beta \omega^\beta\right)^2 \cdot c_\alpha \omega^\alpha \cos(\alpha \frac{\pi}{2}) + \left(c_\alpha \omega^\alpha\right)^2 \cdot c_\beta \omega^\beta \cos(\beta \frac{\pi}{2})}{\left(c_\alpha \omega^\alpha\right)^2+\left(c_\beta \omega^\beta\right)^2+2c_\alpha \omega^\alpha \cdot c_\beta \omega^\beta \cos((\alpha-\beta)\frac{\pi}{2})}\]

Loss Modulus

\[G^{\prime\prime}(\omega) = \frac{\left(c_\beta \omega^\beta\right)^2 \cdot c_\alpha \omega^\alpha \sin(\alpha \frac{\pi}{2}) + \left(c_\alpha \omega^\alpha\right)^2 \cdot c_\beta \omega^\beta \sin(\beta \frac{\pi}{2})}{\left(c_\alpha \omega^\alpha\right)^2+\left(c_\beta \omega^\beta\right)^2+2c_\alpha \omega^\alpha \cdot c_\beta \omega^\beta \cos((\alpha-\beta)\frac{\pi}{2})}\]

Fractional (Spring) Maxwell

FractS_Maxwell

Model name: fractmaxwell_spring

Free parameters: cₐ, a and k

           ___╱╲_________╱╲  ╱╲  ╱╲  ________
              ╲╱           ╲╱  ╲╱  ╲╱
                cₐ,a               k
               

models = Vector{RheoModel}()

# plot moduli for varying α
for alpha in [0.1, 0.3, 0.5, 0.7, 0.9]

    push!(models, RheoModel(FractS_Maxwell, (cₐ = 1.0, a = alpha, k = 1.0)))

end

plotmodel(models)

Fraction (Dashpot) Maxwell

FractD_Maxwell

Model name: fractmaxwell_dashpot

Free parameters: η, cᵦ and β

                  ___
              _____| |_________╱╲____
                  _|_|         ╲╱
                    η            cᵦ, β
                 

models = Vector{RheoModel}()

# plot moduli for varying β
for beta in [0.1, 0.3, 0.5, 0.7, 0.9]

    push!(models, RheoModel(FractD_Maxwell, (η = 10, cᵦ= 1.0, β = beta)))

end

plotmodel(models, ymaxG = 2.0)

Maxwell

Maxwell

Model name: maxwell

Free parameters: η and k

                ___
            _____| |________╱╲  ╱╲  ╱╲  ___
                _|_|          ╲╱  ╲╱  ╲╱
                  η                  k
               

models = Vector{RheoModel}()

# plot moduli for varying k
for k in [5.0, 10.0, 20.0]

    push!(models, RheoModel(Maxwell, (η = 10, k = k)))

end

plotmodel(models)

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