Fractional Poynting-Thomson

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

Fract_PT

Model name: fPT

Free parameters: cₐ, a, cᵦ, β, cᵧ and γ

                     _________╱╲_________
                    |         ╲╱ cₐ, a   |
                ____|                    |______╱╲____
                    |                    |      ╲╱
                    |_________╱╲_________|        cᵧ, γ
                              ╲╱
                                 cᵦ, β
                                 

Constitutive Equation

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

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

Relaxation Modulus

\[\tilde{G}(s) = \frac{1}{s}\frac{c_\gamma s^\gamma \cdot \left[c_\alpha s^\alpha + c_\beta s^{\beta}\right]}{c_\gamma s^\gamma+c_\alpha s^{\alpha}+c_\beta s^{\beta}}\]

Creep Modulus

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

Storage Modulus

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

Loss Modulus

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

Fractional SLS (PT)

FractSLS_PT

Model name: fSLS_PT

Free parameters: cₐ, a, kᵦ and kᵧ

                     _________╱╲_________
                    |         ╲╱ cₐ, a   |
                ____|                    |______╱╲  ╱╲  ╱╲  ____
                    |                    |        ╲╱  ╲╱  ╲╱
                    |____╱╲  ╱╲  ╱╲  ____|                   kᵧ
                           ╲╱  ╲╱  ╲╱
                                     kᵦ
                                 

models = Vector{RheoModel}()

# plot moduli for varying α
for alpha in [0.1, 0.25, 0.5, 0.74, 0.9]

    push!(models, RheoModel(FractSLS_PT, (cₐ = 1, a = alpha, kᵦ = 1, kᵧ = 1)))

end

plotmodel(models)

Fractional Jeffreys (PT)

FractJeffreys_PT

Model name: fjeff_PT

Free parameters: ηₐ, cᵦ, β and ηᵧ

                                      ___
                              _________| |________
                             |        _|_| ηₐ     |        ___
                         ____|                    |_________| |_____
                             |                    |        _|_| ηᵧ
                             |_________╱╲_________|
                                       ╲╱
                                          cᵦ, β
                                          

models = Vector{RheoModel}()

# plot moduli for varying β
for beta in [0.1, 0.25, 0.5, 0.74, 0.9]

    push!(models, RheoModel(FractJeffreys_PT, (ηₐ = 1, cᵦ = 1, β = beta, ηᵧ = 1)))

end

plotmodel(models, ymaxG = 0.5)

Standard Linear Solid (PT)

SLS_PT

Model name: SLS_PT

Free parameters: η, kᵦ and kᵧ

                             ___
                     _________| |________
                    |        _|_| η      |
                ____|                    |______╱╲  ╱╲  ╱╲  ____
                    |                    |        ╲╱  ╲╱  ╲╱
                    |____╱╲  ╱╲  ╱╲  ____|                   kᵧ
                           ╲╱  ╲╱  ╲╱
                                     kᵦ
                                 

models = Vector{RheoModel}()

# plot moduli for varying kᵦ
for k in [1.0, 3.0, 5.0]

    push!(models, RheoModel(SLS_PT, (η = 1, kᵦ = k, kᵧ = 1)))

end

plotmodel(models)

Jeffreys (Zener)

Jeffreys_PT

Model name: jeffreys_PT

Free parameters: ηₐ, k and ηᵧ

                             ___
                     _________| |________
                    |        _|_| ηₐ     |        ___
                ____|                    |_________| |_____
                    |                    |        _|_| ηᵧ
                    |____╱╲  ╱╲  ╱╲  ____|
                           ╲╱  ╲╱  ╲╱
                                     k
                                 

models = Vector{RheoModel}()

# plot moduli for varying ηₐ
for eta in [1.0, 5.0, 8.0]

    push!(models, RheoModel(Jeffreys_PT, (ηₐ = eta, k = 3, ηᵧ = 1)))

end

plotmodel(models)

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