Additional Examples

PyPlot needs to be installed to run these examples and display plots of the data.

using PyPlot

using RHEOS

Example 1

Loading experimental data from a .csv file
Fitting a model
Compare original data with fitted model prediction

# Make sure the examples folder is the current directory
# check by typing "pwd()"

# Import data
data = importcsv("assets/example1_data.csv", t_col = 1, ϵ_col = 2, σ_col = 3)

# Plot data
fig, ax = subplots(1, 1, figsize = (7, 5))
ax.plot(data.t, data.σ, ".", color = "green")
ax.plot(data.t, data.ϵ, "-", color = "blue")
ax.set_ylabel("Strain (blue), Stress (green)")
ax.set_xlabel("Time")

# We now fit a Maxwell model
maxwell_model = modelfit(data, Maxwell, strain_imposed)


Model name: maxwell

Free parameters: 


Fixed parameters: (η = 4.0054017305374146, k = 2.003000020980835)

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

Note that the fitting function requires guidance regarding the type of testing used. It helps optimise the fitting process.

The data in this example is the stress response to a strain ramp followed by plateau. It therefore corresponds to a strain imposed process.

We now want to calculate the stress values predicted by the model given the experimental strain data. Let's create a new data set with the strain profile.

maxwell_predict = onlystrain(data)
# and calculate the stress based on the model
maxwell_predict = modelpredict(maxwell_predict, maxwell_model)
# Now we can plot data and model together for comparison

# Plot data
fig, ax = subplots(1, 1, figsize = (7, 5))
ax.plot(data.t, data.σ, ".", color = "green")
ax.plot(maxwell_predict.t, maxwell_predict.σ, color = "red")
ax.set_xlabel("Time")
ax.set_ylabel("Stress")

This script is a slight modification of Example 1 to present to the user the possibility of creating new RheoModelClass from an existing one with some of the parameters frozen to specific values. As an example, we fix the spring constant of the model above (k) to 2 and we let RHEOS fit the viscosity η.

Maxwell_springFix = freeze_params(Maxwell, k = 2)


Model name: maxwell

Free parameters: η


Fixed parameters: (k = 2.0,)

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

maxwellD_model = modelfit(data, Maxwell_springFix, strain_imposed)
maxwellD_predict = onlystrain(data)
# and calculate the stress based on the model
maxwellD_predict = modelpredict(maxwellD_predict, maxwellD_model)
# Now we can plot data and model together for comparison

# Plot data
fig, ax = subplots(1, 1, figsize = (7, 5))
ax.plot(data.t, data.σ, ".", color = "green")
ax.plot(maxwellD_predict.t, maxwellD_predict.σ, color = "red")
ax.set_xlabel("Time")
ax.set_ylabel("Stress")

Example 3

This script shows how to use RHEOS to explore the behaviour of various models This involves:

Creating a strain function
Defining models based on parameter values

PyPlot needs to be installed to run these examples and display plots of the data.

# Creates a time only dataset
dϵ = timeline(t_end = 10)
# calculates strain data by applying a function of time
dϵ = strainfunction(dϵ, t -> sin(t))

# Plot strain data
fig, ax = subplots(1, 1, figsize = (7, 5))
ax.plot(dϵ.t, dϵ.ϵ, "--b")

# we can now simulate various models based on this strain only dataset
# Let's study the role of the dashpot strength in the MAxwell model
for η in [0.1, 0.3, 1, 3, 10]
    maxwell_model = RheoModel(Maxwell, k = 2., η = η)
    d_maxwell = modelpredict(dϵ, maxwell_model)
    ax.plot(d_maxwell.t, d_maxwell.σ)
end
ax.set_xlabel("Time")
ax.set_ylabel("Stress")
ax.grid("on")

Example 4

This example demonstrates generating a timeline and stress data, fitting multiple models to the data, calling the extractfitdata function, listing the errors, and determining which model fits the best.

Generate Timeline

datat = timeline(t_start = 0, t_end = 20.0, step = 0.02)  # Create a timeline from 0 to 20 seconds with a step size of 0.02 seconds

Generate Stress Data (Ramp & hold)

dramp_stress = stressfunction(datat, ramp(offset = 4.0, gradient = 0.8))  # Generate a ramp stress function with offset 4.0 and gradient 0.8
dhold_stress = dramp_stress - stressfunction(datat, ramp(offset = 5.0, gradient = 0.8))  # Generate a hold stress function by subtracting a shifted ramp

Define the rheological model and predict

model = RheoModel(SLS_Zener, (η = 1, kᵦ = 1, kᵧ = 1))
data = modelpredict(dhold_stress, model)

Fit three models to the data

SLS_Zener_model = modelfit(data, SLS_Zener, stress_imposed)
Maxwell_model = modelfit(data, Maxwell, stress_imposed)
BurgersLiquid_model = modelfit(data, BurgersLiquid, stress_imposed)


Model name: burgerliquid

Free parameters: 


Fixed parameters: (η₁ = 2.2791392395938356e16, k₁ = 1.9214746727288188e16, η₂ = 0.765625, k₂ = 1.03125)


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

Call the extractfitdata function to extract fitting data

extracted_data = extractfitdata(data)

Dict{String, Vector{@NamedTuple{params, info, index::Int64}}} with 3 entries:
  "SLS_Zener"    => [@NamedTuple{params, info, index::Int64}(((η = 1.00165, kᵦ …
  "burgerliquid" => [@NamedTuple{params, info, index::Int64}(((η₁ = 2.27914e16,…
  "maxwell"      => [@NamedTuple{params, info, index::Int64}(((η = 57.064, k = …

Determine which model fits best by comparing errors

best_model = ""
min_error = Inf

for (model_name, fitdata) in extracted_data
    err = fitdata[1].info.error

    println("Model: $model_name, Total Error: $err")

    if err < min_error
        min_error = err
        best_model = model_name
    end
end

println("Best fitting model: $best_model with total error: $min_error")

Create strain-only data for model predictions

stress_only_data = onlystress(data)

Get model predictions for plotting

SLS_Zener_predict = modelpredict(stress_only_data, SLS_Zener_model)
Maxwell_predict = modelpredict(stress_only_data, Maxwell_model)
BurgersLiquid_predict = modelpredict(stress_only_data, BurgersLiquid_model)

Plot data and fitted models

fig, ax = subplots(1, 1, figsize = (7, 5))
ax.plot(data.t, data.ϵ, ".", color = "green", label = "Original Data")
ax.plot(SLS_Zener_predict.t, SLS_Zener_predict.ϵ, "-", color = "red", label = "SLS_Zener Model")
ax.plot(Maxwell_predict.t, Maxwell_predict.ϵ, "--", color = "blue", label = "Maxwell Model")
ax.plot(BurgersLiquid_predict.t, BurgersLiquid_predict.ϵ, ":", color = "purple", label = "BurgersLiquid Model")
ax.set_xlabel("Time")
ax.set_ylabel("Strain")
ax.legend()
ax.grid("on")

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Additional Examples

Example 1

Example 2

Example 3

Example 4