This tutorial is for curve fitting in hyperelastic models.

Hint

The package supports

neo-Hookean model
Mooney-Rivlin model
Ogden model
Yeoh models model

Note

The package also supports the Drucker stability criterion.

In order to perform curve fitting, experimental data are required. Some experimental data are available in HyperData.jl repository.

The experimental data can be Strain or Stretch for x axis and the nominal stress for yaxis.

Steps TODO Curve Fitting

Load Package
Read Experimental Data
Curve Fitting Inputs
Call curve fitting function
Check the accuracy of the obtained material constant(s).

Load Package

Load the required packages for curve fitting as

using Revise          # to automatically reload changed code
using FerriteHyperelastic
using GLMakie
using HyperData

Read Experimental Data

Read the experimental data as

λ_exp, P_exp = read_data!(Treloar_1944, uniaxial)

Curve Fitting Inputs

Set inputs of curve fitting function

Warning

The input MUST be Strain and the Nominal Stress, for this reasion if the data is different user must convert them to starin and nominal stress

# Input parameters
inputDataType = "lambda"
data_type = "uniaxial"
modelType = "neo-hookean"

Sexp = P_exp
strainExp = λ_exp .- 1

Note

Please use lowercase letter for all model types (this example is "neo-hookean")

Call curve fitting function

The curve-fitting function is now called. For neo-Hookean, the material constant is C1.

# Get material constants by fitting (assumed function)
mat_cons_solver = solver_constants_hyper(data_type, modelType, strainExp, Sexp)
C1 = mat_cons_solver[1]

Check the accuracy of the obtained material constant(s)

# Generate strain values for smooth curve
ϵ = collect(LinRange(strainExp[1], strainExp[end], 100))

# Calculate stretches for incompressible uniaxial tension
λ1 = @. ϵ + 1
λ = λ1
λ2 = @. 1 / sqrt(λ)
λ3 = λ2   

# neo-Hookean stress (uniaxial nominal stress)
P_model = @. 2 * C1 * (λ - 1 / λ^2)

# Plot results
GLMakie.closeall()

fig = Figure(size=(800, 600), fontsize=26)
ax = Axis(fig[1, 1], xlabel= L"\mathscr{ε}", ylabel=L"P",  xgridvisible=false, ygridvisible=false)

lines!(ax, ϵ, P_model, color = :black, label="Fit, neo-Hookean")
scatter!(ax, strainExp, P_exp, marker=:circle, color=:red, label="Uniaxial experiment")

axislegend(ax, position=:lt, backgroundcolor=(:white, 0.7), framecolor=:gray)
display(fig)

Plain program

The version of the code without comments.

using Revise
using FerriteHyperelastic
using GLMakie
using HyperData


λ_exp, P_exp = read_data!(Treloar_1944, uniaxial)


inputDataType = "lambda"
data_type = "uniaxial"
modelType = "neo-hookean"

Sexp = P_exp
strainExp = λ_exp .- 1


mat_cons_solver = solver_constants_hyper(data_type, modelType, strainExp, Sexp)
C1 = mat_cons_solver[1]


ϵ = collect(LinRange(strainExp[1], strainExp[end], 100))


λ1 = @. ϵ + 1
λ = λ1
λ2 = @. 1 / sqrt(λ)
λ3 = λ2   


P_model = @. 2 * C1 * (λ - 1 / λ^2)


GLMakie.closeall()

fig = Figure(size=(800, 600), fontsize=26)
ax = Axis(fig[1, 1], xlabel= L"\mathscr{ε}", ylabel=L"P",  xgridvisible=false, ygridvisible=false)

lines!(ax, ϵ, P_model, color = :black, label="Fit, neo-Hookean")
scatter!(ax, strainExp, P_exp, marker=:circle, color=:red, label="Uniaxial experiment")

axislegend(ax, position=:lt, backgroundcolor=(:white, 0.7), framecolor=:gray)
display(fig)