Burgers (Liquid)

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

BurgersLiquid

Model name: burgerliquid

Free parameters: η₁, k₁, η₂ and k₂


                                                         ___
                                                 _________| |________
                 ___                            |        _|_| η₂     |
             _____| |________╱╲  ╱╲  ╱╲  _______|                    |____
                 _|_|          ╲╱  ╲╱  ╲╱       |                    |
                    η₁                  k₁      |____╱╲  ╱╲  ╱╲  ____|
                                                       ╲╱  ╲╱  ╲╱
                                                                 k₂

                

Constitutive Equation

\[\sigma(t) + p_1 \frac{d \sigma(t)}{dt} + p_2 \frac{d^2 \sigma(t)}{dt^2} = q_1 \frac{d \epsilon(t)}{dt} + q_2 \frac{d^2 \epsilon(t)}{dt^2}\]

\[\text{where}\; \ p_1 = \frac{\eta_1}{k_1}+\frac{\eta_1}{k_2}+\frac{\eta_1}{k_2}\text{,}\; \ p_2 = \frac{\eta_1 \eta_2}{k_1 k_2}\text{,}\; \ q_1 = \eta_1\; \ \text{and}\; \ q_2 = \frac{\eta_1 \eta_2}{k_2}\]

Relaxation Modulus

\[G(t) = \left[ \left(q_1 - q_2 r_1\right) e^{-r_1 t} - \left(q_1 - q_2 r_2\right) e^{-r_2 t} \right]/A\]

\[\text{where}\; \ r_1 = (p_1-A)/2p_2 \text{,}\; \ r_2 = (p_1 +A) /2p_2 \text{, and}\; \ A = \sqrt{p_1^2 - 4p_2}\]

Creep Modulus

\[J(t) = \frac{1}{k_1} + \frac{t}{\eta_1} + \frac{1}{k_2} \left(1-e^{-k_2 t/\eta_2}\right)\]

Storage Modulus

\[G^{\prime}(\omega) = \frac{p_1 q_1 \omega^2 - q_2 \omega^2 (1-p_2 \omega^2)}{p_1^2 \omega^2 + (1-p_2 \omega^2)^2}\]

Loss Modulus

\[G^{\prime \prime}(\omega) = \frac{p_1 q_2 \omega^3 + q_1 \omega (1-p_2 \omega^2)}{p_1^2 \omega^2 + (1-p_2 \omega^2)^2}\]

Qualitative Behaviours of the Moduli

models = Vector{RheoModel}()

# plot moduli for varying k₁
for k in [0.5, 1.0, 2.0]

    push!(models, RheoModel(BurgersLiquid, (η₁ = 1, k₁ = k, η₂ = 1, k₂ = 1)))

end

plotmodel(models)

Reference: Findley, William N., and Francis A. Davis. Creep and relaxation of nonlinear viscoelastic materials. Courier Corporation, 2013.

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