mechtest_ufmg: Mechanical Testing Analysis

This library is being developed to assist the students of the course of Mechanical and microstructural characterization of metals. Its content is prepared to guide throughout the lectures and to help the students in the learning process.

The course is offered as an elective discipline for the Metallurgical and Materials Engineering students, and is taught at the School of Engineering of the Federal University of Minas Gerais.

A brief description

The functions provided by this package are intended to calculate the mechanical properties of most of the engineering materials that can be tested in tension. The functions that are already implemented can calculte properties and plot graphs of the data.

Mechanical properties that can be calculated

• Apparent modulus of elasticity ($E$)
• Yield strength ($\sigma_y$)
• Ultimate tensile strength (UTS or $\sigma_R$)
• Uniform elongation ($\varepsilon_u$)
• Resilience ($U_r$): approximation and numerically integrated
• Toughness ($U_t$): approximation and numerically integrated
• Hollomon's equation parameters: resistance coefficient ($K$) and strain hardening exponent ($n$)

Plots obtained as outputs from some functions

• Engineering stress/strain curve
• Plot highlighting the linear portion used to calculate $E$
• Plot highlighting the determination of $\sigma_y$
• Flow stress curve
• True stress flow curve
• Hollomon's equation fitting to the data

Main dependencies

The main dependencies to use mechtest_ufmg are the following.

• os
• matplotlib
• numpy
• scipy in scipy.optimize
• pandas (for data importing)

The first one is a standard library, the others can be installed via pip install package_name. If you are using Anaconda and/or Google Colab it all may work right out of the box.

### Example of installing modules

# !pip install scipy
# !pip install pandas
# !pip install matplotlib

##### Importing the necessary packages

from mechtest_ufmg.Properties import *  # Importing all functions at once
import pandas as pd                     # Importing pandas to read the data

Importing the data to a dataframe with pandas

data = pd.read_csv('data/astm1055.tsv', sep = '\t', decimal = ',')

If you want to check if your data was imported and parsed correctly you can use data.head().

##### Just checking....
# data.head()

Extracting the dataframe columns into series with our data

strain = data['Strain']
stress = data['Stress']

Using plot_eng_SSC() to plot the engineering stress/strain curve

%matplotlib inline
plot_eng_SSC(strain, stress, fig_label = 'ASTM 1055', 
             show_plot = True, save = True, name = 'eng_SSC_1055')

Using young_modulus() to determine the elasticity modulus

E, b, R2 = young_modulus(strain, stress, fig_label = 'ASTM 1055', 
                      show_plot = True, save = True, name = 'elasticity_1055')

Using sigma_y() to calculate the yield strength $\sigma_y$

sig_y = sigma_y(strain, stress, E, b, fig_label = 'ASTM 1055', 
                show_plot = True, save = True, name = 'yield_strength_1055')

The yield strength is 566 MPa.

Using UTS() to calculate the ultimate tensile strength $UTS$ or $\sigma_R$

uts = UTS(strain, stress, show = True)

The ultimate tensile strength is 940 MPa. 

Using uniform_elong() to calculate the uniform elongation.

uniform_e = uniform_elong(strain, stress, show = True)

The uniform elongation is 0.1086.

Calculating the resilience $U_r$ using an approximate mathematical formula and numerical integration

Using aprox_resilience() $U_r$ is calculated with the formula: $$ U_r = \frac{\sigma_y^2}{2E}$$

aprox_res = aprox_resilience(E, sig_y, show = True)

The resilience calculated by the formula yield²/2E is 1.06 MJ/m³. 

Using resilience() $U_r$ is calculated as: $$ U_r = \int_{0}^{\varepsilon_y} \sigma ,d \varepsilon $$

res = resilience(strain, stress, sig_y)

The resilience calculated by trapezoidal integration with dx = 1.0 is 1.15 MJ/m³. 

Calculating the toughness $U_t$ using an approximate mathematical formula and numerical integration

Using aprox_toughness() $U_r$ is calculated with the formula: $$ U_r = \frac{\sigma_y + \sigma_R}{2} \times \varepsilon_f$$

aprox_tough = aprox_toughness(strain, sig_y, uts, show = True)

The material toughness is approximately 112 MJ/m³.

Using toughness() $U_r$ is calculated as: $$ U_r = \int_{0}^{\varepsilon_f} \sigma ,d \varepsilon $$

tough = toughness(strain, stress, show = True)

The material toughness computed by Trapezoidal method is 124 MJ/m³. 

Plotting the flow stress curve

plot_flow_curve(strain, stress, sig_y, uts, fig_label = 'ASTM 1055', 
                show_plot = False, save = True, name = 'flow_curve_1055')

Plotting the true stress/strain curve

plot_true_SSC(strain, stress, sig_y, uts, fig_label = 'ASTM 1055', 
              show_plot = False, save = True, name = 'true_curve_1055')

Plotting the flow model curve with Hollomon's equatiion

n, K = flow_model(strain, stress, sig_y, uts, show_plot = False, save = True, name = 'Hollomon_1055')

The resistance modulus is 1685 MPa and the strain-hardening exponent is 0.2.