Fractional Kelvin-Voigt

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

Fract_KelvinVoigt

Model name: fractKV

Free parameters: cₐ, a, cᵦ and β

                ________ ╱╲ ________
               |         ╲╱  cₐ, a  |
           ____|                    |____
               |                    |
               |________ ╱╲ ________|
                         ╲╱  cᵦ, β
                

Constitutive Equation

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

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

Relaxation Modulus

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

Creep Modulus

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

Storage Modulus

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

Loss Modulus

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

Fractional (Spring) Kelvin-Voigt

FractS_KelvinVoigt

Model name: fractSpringKV

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_KelvinVoigt, (cₐ = 1.0, a = alpha, k = 1.0)))

end

plotmodel(models, ymaxG = 5.0)

Fractional (Dashpot) Kelvin-Voigt

FractD_KelvinVoigt

Model name: fractDashpotKV

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_KelvinVoigt, (η = 10, cᵦ= 1.0, β = beta)))

end

plotmodel(models, ymaxG = 3.0)

Kelvin-Voigt

KelvinVoigt

Model name: KelvinVoigt

Free parameters: η and k

                        ___
                _________| |________
               |        _|_| η      |
           ____|                    |____
               |                    |
               |____╱╲  ╱╲  ╱╲  ____|
                      ╲╱  ╲╱  ╲╱
                                k
                

models = Vector{RheoModel}()

# plot moduli for varying k
for η in [0.1, 1.0, 5.0]

    push!(models, RheoModel(KelvinVoigt, (η = η, k = 1)))

end

plotmodel(models)

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