Chapter 1: Tensile Testing Fundamentals

Stress-Strain Behavior and Mechanical Properties

📖 Reading Time: 20-25 minutes 📊 Difficulty: Beginner 💻 Code Examples: 0 📝 Exercises: 0

🌐 EN | 日本語 (準備中) Last sync: 2025-11-16

Materials Science Dojo > Mechanical Testing Introduction > Chapter 1

1.1 Introduction to Tensile Testing

Tensile testing is the most fundamental mechanical test used to determine material properties including strength, ductility, and elastic modulus. A specimen is subjected to controlled tension until failure while measuring force and elongation.

=Ğ Definition: Engineering Stress and Strain
Engineering stress: $$\sigma = \frac{F}{A_0}$$ where $F$ is applied force and $A_0$ is original cross-sectional area.

Engineering strain: $$\epsilon = \frac{\Delta L}{L_0} = \frac{L - L_0}{L_0}$$ where $L_0$ is original length and $L$ is current length.

=» Code Example 1: Stress-Strain Curve Generation

Python Implementation: Engineering Stress-Strain Analysis

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import numpy as np
import matplotlib.pyplot as plt

def generate_stress_strain_curve(material='steel'):
    """Generate engineering stress-strain curve"""
    materials = {
        'steel': {'E': 200e3, 'yield': 250, 'uts': 400, 'fracture_strain': 0.25},
        'aluminum': {'E': 70e3, 'yield': 100, 'uts': 200, 'fracture_strain': 0.15},
        'copper': {'E': 120e3, 'yield': 70, 'uts': 220, 'fracture_strain': 0.45}
    }
    props = materials[material]

    # Elastic region
    strain_elastic = np.linspace(0, props['yield']/props['E'], 100)
    stress_elastic = props['E'] * strain_elastic

    # Plastic region
    strain_plastic = np.linspace(props['yield']/props['E'], props['fracture_strain'], 200)
    K = props['uts'] * 1.1
    n = 0.2
    stress_plastic = K * strain_plastic**n

    strain = np.concatenate([strain_elastic, strain_plastic])
    stress = np.concatenate([stress_elastic, stress_plastic])
    return strain, stress, props

# Visualize different materials
fig, ax = plt.subplots(figsize=(10, 6))
for material in ['steel', 'aluminum', 'copper']:
    strain, stress, props = generate_stress_strain_curve(material)
    ax.plot(strain * 100, stress, linewidth=2, label=material.capitalize())
    yield_idx = np.argmin(np.abs(stress - props['yield']))
    ax.plot(strain[yield_idx] * 100, stress[yield_idx], 'o', markersize=8)

ax.set_xlabel('Engineering Strain (%)', fontsize=12)
ax.set_ylabel('Engineering Stress (MPa)', fontsize=12)
ax.set_title('Engineering Stress-Strain Curves', fontsize=14, fontweight='bold')
ax.legend()
ax.grid(True, alpha=0.3)
plt.show()

1.2 True Stress and True Strain

Engineering stress and strain assume constant dimensions, but materials deform during testing. True stress and strain account for instantaneous dimensions.

=Ğ Definition: True Stress and True Strain
True stress: $\sigma_T = \frac{F}{A} = \sigma(1 + \epsilon)$
True strain: $\epsilon_T = \ln\left(\frac{L}{L_0}\right) = \ln(1 + \epsilon)$

=» Code Example 2: True vs Engineering Conversion

Python Implementation: Stress-Strain Conversion

def engineering_to_true(eng_stress, eng_strain):
    """Convert engineering to true stress-strain"""
    true_stress = eng_stress * (1 + eng_strain)
    true_strain = np.log(1 + eng_strain)
    return true_stress, true_strain

strain_eng, stress_eng, _ = generate_stress_strain_curve('steel')
stress_true, strain_true = engineering_to_true(stress_eng, strain_eng)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))
ax1.plot(strain_eng * 100, stress_eng, 'b-', linewidth=2, label='Engineering')
ax1.set_xlabel('Engineering Strain (%)')
ax1.set_ylabel('Engineering Stress (MPa)')
ax1.set_title('Engineering Stress-Strain')
ax1.grid(True, alpha=0.3)
ax1.legend()

ax2.plot(strain_true * 100, stress_true, 'r-', linewidth=2, label='True')
ax2.set_xlabel('True Strain (%)')
ax2.set_ylabel('True Stress (MPa)')
ax2.set_title('True Stress-Strain')
ax2.grid(True, alpha=0.3)
ax2.legend()
plt.tight_layout()
plt.show()

1.3 Mechanical Properties Extraction

Tensile testing provides critical mechanical properties: elastic modulus, yield strength, ultimate tensile strength, and ductility.

=Ğ Key Mechanical Properties

=» Code Example 3: Mechanical Property Calculation

# Requirements:
# - Python 3.9+
# - scipy>=1.11.0

from scipy import integrate
from scipy.stats import linregress

def calculate_mechanical_properties(strain, stress):
    """Extract mechanical properties from stress-strain curve"""
    properties = {}

    # Elastic Modulus
    elastic_idx = int(len(strain) * 0.05)
    slope, intercept, r_value, _, _ = linregress(strain[:elastic_idx], stress[:elastic_idx])
    properties['elastic_modulus'] = slope

    # Yield Strength (0.2% offset)
    offset_strain = 0.002
    offset_line = slope * (strain - offset_strain) + intercept
    diff = np.abs(stress - offset_line)
    yield_idx = elastic_idx + np.argmin(diff[elastic_idx:])
    properties['yield_strength'] = stress[yield_idx]

    # Ultimate Tensile Strength
    uts_idx = np.argmax(stress)
    properties['ultimate_tensile_strength'] = stress[uts_idx]
    properties['uniform_elongation'] = strain[uts_idx]

    # Ductility
    properties['elongation_percent'] = strain[-1] * 100

    # Toughness
    properties['toughness'] = integrate.trapz(stress, strain)

    return properties

strain, stress, _ = generate_stress_strain_curve('steel')
props = calculate_mechanical_properties(strain, stress)

print(f"Elastic Modulus: {props['elastic_modulus']/1000:.1f} GPa")
print(f"Yield Strength: {props['yield_strength']:.1f} MPa")
print(f"UTS: {props['ultimate_tensile_strength']:.1f} MPa")
print(f"Elongation: {props['elongation_percent']:.1f}%")
print(f"Toughness: {props['toughness']:.1f} MJ/m³")

1.4 Testing Standards

ASTM E8 and ISO 6892 provide standardized procedures ensuring reproducibility. Key requirements include specimen geometry, strain rate, and environmental conditions.

=» Code Example 4: ASTM E8 Specimen Design

class TensileSpecimen:
    """ASTM E8 tensile specimen calculator"""

    def calculate_dimensions(self, diameter=12.5):
        """Calculate specimen dimensions per ASTM E8"""
        area = np.pi * (diameter/2)**2
        gauge_length = 4 * np.sqrt(area)

        return {
            'diameter': diameter,
            'area': area,
            'gauge_length': gauge_length,
            'total_length': gauge_length * 1.5 + 60
        }

    def calculate_test_speed(self, gauge_length, strain_rate=0.005):
        """Calculate crosshead speed"""
        return strain_rate * gauge_length

specimen = TensileSpecimen()
dims = specimen.calculate_dimensions(diameter=12.5)
speed = specimen.calculate_test_speed(dims['gauge_length'])

print(f"Gauge Length: {dims['gauge_length']:.2f} mm")
print(f"Test Speed: {speed:.3f} mm/min")

1.5 Strain Hardening

During plastic deformation, flow stress increases with strain (strain hardening), described by the Hollomon equation.

=Ğ Hollomon Equation: $\sigma_T = K \epsilon_T^n$
where $K$ is strength coefficient and $n$ is strain hardening exponent.

=» Code Example 5: Hollomon Equation Fitting

def fit_hollomon_equation(true_strain, true_stress):
    """Fit Hollomon equation to data"""
    plastic_idx = 100
    strain_plastic = true_strain[plastic_idx:]
    stress_plastic = true_stress[plastic_idx:]

    log_strain = np.log(strain_plastic)
    log_stress = np.log(stress_plastic)

    n, log_K, r_value, _, _ = linregress(log_strain, log_stress)
    K = np.exp(log_K)

    return K, n, r_value**2

strain_eng, stress_eng, _ = generate_stress_strain_curve('steel')
stress_true, strain_true = engineering_to_true(stress_eng, strain_eng)
K, n, R2 = fit_hollomon_equation(strain_true, stress_true)

print(f"Hollomon: Ã = {K:.1f} * µ^{n:.3f}")
print(f"R² = {R2:.4f}")
print(f"Strain hardening exponent: n = {n:.3f}")

1.6 Temperature Effects

Mechanical properties vary with temperature. Higher temperatures reduce strength and increase ductility.

=» Code Example 6: Temperature Dependence

def temperature_dependent_properties(T, T_ref=293):
    """Calculate temperature-dependent yield strength"""
    R = 8.314
    Q = 50000
    sigma_ref = 250

    sigma_y = sigma_ref * np.exp(Q/R * (1/T - 1/T_ref))
    E = 200e3 * (1 - 0.0005 * (T - T_ref))
    elongation = 25 * (T / T_ref)**0.5

    return sigma_y, E, elongation

temperatures = np.linspace(200, 800, 100)
results = [temperature_dependent_properties(T) for T in temperatures]

plt.figure(figsize=(10, 6))
plt.plot(temperatures - 273, [r[0] for r in results], label='Yield Strength')
plt.xlabel('Temperature (°C)')
plt.ylabel('Yield Strength (MPa)')
plt.grid(True, alpha=0.3)
plt.legend()
plt.show()

1.7 Necking and Instability

Necking occurs at UTS when localized deformation begins. The Considère criterion predicts necking onset.

=Ğ Considère Criterion: Necking when $\frac{d\sigma_T}{d\epsilon_T} = \sigma_T$
For Hollomon equation: $\epsilon_T^{neck} = n$

=» Code Example 7: Necking Prediction

def predict_necking(K, n):
    """Predict necking using Considère criterion"""
    necking_strain_true = n
    necking_stress_true = K * necking_strain_true**n

    necking_strain_eng = np.exp(necking_strain_true) - 1
    necking_stress_eng = necking_stress_true / (1 + necking_strain_eng)

    return {
        'true_strain': necking_strain_true,
        'true_stress': necking_stress_true,
        'eng_strain': necking_strain_eng,
        'eng_stress': necking_stress_eng
    }

K, n = 550, 0.22
necking = predict_necking(K, n)
print(f"Necking at true strain: {necking['true_strain']:.3f}")
print(f"Engineering strain: {necking['eng_strain']*100:.1f}%")

=İ Chapter Exercises

 Exercises

  1. Calculate engineering and true stress for a specimen with original diameter 12.5 mm loaded to 62 kN with diameter 10.8 mm.
  2. Extract elastic modulus, yield strength, and UTS from stress-strain data. Calculate toughness.
  3. Fit Hollomon equation to plastic region and predict necking strain.
  4. Design ASTM E8 rectangular specimen for 2 mm thick sheet.
  5. Analyze property changes from 25°C to 400°C for high-temperature application.

Summary

� Series Overview Chapter 2: Hardness Testing ’

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