Chapter 3: Particle and Laminate Composites

3.1 Fundamentals of Particle-Reinforced Composites

3.1.1 Classification of Particle Reinforcement

Particle-reinforced composites are materials with particle-shaped reinforcements dispersed in a matrix. They are classified according to strengthening mechanisms:

3.1.2 MMC (Metal Matrix Composites)

Metal matrix composites are characterized by light weight, high strength, and high heat resistance.

3.1.3 CMC (Ceramic Matrix Composites)

Ceramic matrix composites are materials where fibers or particles are incorporated into a ceramic matrix to improve brittleness.

3.2 Mechanical Models for Particle Strengthening

3.2.1 Elastic Modulus Prediction

For spherical particle reinforcement, the Hashin-Shtrikman upper and lower bound models are commonly used. For isotropic materials:

Bulk Modulus

Shear Modulus

Young's modulus and Poisson's ratio are calculated from:

# 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 hashin_shtrikman_moduli(K_m, G_m, K_p, G_p, V_p):
    """
    Composite material elastic modulus calculation using Hashin-Shtrikman model

    Parameters:
    -----------
    K_m, G_m : float
        Bulk and shear modulus of matrix [GPa]
    K_p, G_p : float
        Bulk and shear modulus of particle [GPa]
    V_p : float or array
        Particle volume fraction

    Returns:
    --------
    E_c, nu_c : float or array
        Young's modulus and Poisson's ratio of composite
    """
    # Bulk modulus
    K_c = K_m + V_p / (1/(K_p - K_m) + 3*(1 - V_p)/(3*K_m + 4*G_m))

    # Shear modulus
    G_c = G_m + V_p / (1/(G_p - G_m) + 6*(K_m + 2*G_m)*(1 - V_p)/(5*G_m*(3*K_m + 4*G_m)))

    # Young's modulus and Poisson's ratio
    E_c = 9 * K_c * G_c / (3 * K_c + G_c)
    nu_c = (3 * K_c - 2 * G_c) / (2 * (3 * K_c + G_c))

    return E_c, nu_c

def E_nu_to_K_G(E, nu):
    """Convert Young's modulus and Poisson's ratio to bulk and shear modulus"""
    K = E / (3 * (1 - 2 * nu))
    G = E / (2 * (1 + nu))
    return K, G

# Al alloy matrix properties
E_m = 70.0   # GPa
nu_m = 0.33
K_m, G_m = E_nu_to_K_G(E_m, nu_m)

# SiC particle properties
E_p = 450.0  # GPa
nu_p = 0.17
K_p, G_p = E_nu_to_K_G(E_p, nu_p)

# Volume fraction range
V_p_range = np.linspace(0, 0.5, 100)

# Elastic modulus calculation
E_c, nu_c = hashin_shtrikman_moduli(K_m, G_m, K_p, G_p, V_p_range)

# Comparison with rule of mixtures (upper and lower bounds)
E_voigt = E_m * (1 - V_p_range) + E_p * V_p_range  # Upper bound
E_reuss = 1 / ((1 - V_p_range)/E_m + V_p_range/E_p)  # Lower bound

# Visualization
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

# Young's modulus
ax1.plot(V_p_range, E_c, 'b-', linewidth=2, label='Hashin-Shtrikman')
ax1.plot(V_p_range, E_voigt, 'r--', linewidth=1.5, label='Voigt (upper bound)')
ax1.plot(V_p_range, E_reuss, 'g--', linewidth=1.5, label='Reuss (lower bound)')
ax1.fill_between(V_p_range, E_reuss, E_voigt, alpha=0.2, color='gray',
                  label='Rule of mixtures range')
ax1.set_xlabel('SiC Volume Fraction')
ax1.set_ylabel('Young\'s Modulus [GPa]')
ax1.set_title('Young\'s Modulus of SiC/Al Composite')
ax1.grid(True, alpha=0.3)
ax1.legend()

# Poisson's ratio
ax2.plot(V_p_range, nu_c, 'b-', linewidth=2, label='Hashin-Shtrikman')
ax2.axhline(y=nu_m, color='r', linestyle='--', label=f'Al matrix ({nu_m:.2f})')
ax2.axhline(y=nu_p, color='g', linestyle='--', label=f'SiC particle ({nu_p:.2f})')
ax2.set_xlabel('SiC Volume Fraction')
ax2.set_ylabel('Poisson\'s Ratio')
ax2.set_title('Poisson\'s Ratio of SiC/Al Composite')
ax2.grid(True, alpha=0.3)
ax2.legend()

plt.tight_layout()
plt.savefig('particle_composite_modulus.png', dpi=300, bbox_inches='tight')
plt.close()

# Values at practical particle fractions
V_p_practical = np.array([0.10, 0.15, 0.20, 0.25, 0.30])
E_c_practical, nu_c_practical = hashin_shtrikman_moduli(K_m, G_m, K_p, G_p, V_p_practical)

print("Elastic Properties of SiC/Al Composite:")
print("="*60)
print(f"{'V_p':>6} {'E_c [GPa]':>12} {'Increase[%]':>12} {'Poisson\'s Ratio':>12}")
print("-"*60)
for vp, ec, nuc in zip(V_p_practical, E_c_practical, nu_c_practical):
    increase = (ec / E_m - 1) * 100
    print(f"{vp:6.2f} {ec:12.1f} {increase:12.1f} {nuc:12.3f}")

3.2.2 Strength Prediction

The strength of particle-reinforced composites is determined by the combined effects of the following factors:

• Load transfer effect: Particles bear the load
• Dislocation strengthening: Increased dislocation density around particles
• Orowan mechanism: Dislocations bypass particles
• Thermal expansion mismatch: Residual stress from cooling

# 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 particle_strengthening(sigma_m, V_p, d_p, b, G_m):
    """
    Yield strength prediction for particle-reinforced composites

    Parameters:
    -----------
    sigma_m : float
        Matrix yield strength [MPa]
    V_p : float
        Particle volume fraction
    d_p : float
        Particle diameter [μm]
    b : float
        Burgers vector [nm]
    G_m : float
        Matrix shear modulus [GPa]

    Returns:
    --------
    sigma_c : float
        Composite yield strength [MPa]
    """
    # Load transfer term (simplified model)
    sigma_load = sigma_m * (1 + 0.5 * V_p)

    # Orowan strengthening term
    # Estimation of interparticle spacing
    lambda_p = d_p * (np.sqrt(np.pi / (4 * V_p)) - 1)  # [μm]

    # Orowan stress [MPa]
    G_m_MPa = G_m * 1000  # GPa → MPa
    b_m = b * 1e-9  # nm → m
    lambda_p_m = lambda_p * 1e-6  # μm → m

    sigma_orowan = 0.4 * G_m_MPa * b_m / lambda_p_m / 1e6  # MPa

    # Total strength (simplified addition)
    sigma_c = sigma_load + sigma_orowan

    return sigma_c

# Al alloy matrix
sigma_m = 100  # MPa (annealed)
b = 0.286      # nm (Al Burgers vector)
G_m = 26       # GPa

# Effect of SiC particle size
d_p_range = np.logspace(-1, 1.5, 50)  # 0.1-30 μm
V_p_values = [0.10, 0.15, 0.20, 0.25]

plt.figure(figsize=(10, 6))

for V_p in V_p_values:
    sigma_c = []
    for d_p in d_p_range:
        s_c = particle_strengthening(sigma_m, V_p, d_p, b, G_m)
        sigma_c.append(s_c)

    plt.plot(d_p_range, sigma_c, linewidth=2, label=f'V_p = {V_p:.2f}')

plt.xscale('log')
plt.xlabel('Particle Diameter [μm]')
plt.ylabel('Composite Yield Strength [MPa]')
plt.title('Relationship between Particle Size and Yield Strength (SiC/Al)')
plt.grid(True, alpha=0.3, which='both')
plt.legend()
plt.tight_layout()
plt.savefig('particle_size_strengthening.png', dpi=300, bbox_inches='tight')
plt.close()

# Examination of optimal particle size
V_p_opt = 0.20
d_p_test = np.array([0.5, 1.0, 2.0, 5.0, 10.0])

print("Particle Size and Strengthening Effect:")
print("="*60)
print(f"{'Particle Diameter [μm]':>15} {'Yield Strength [MPa]':>18} {'Strengthening [%]':>15}")
print("-"*60)

for d_p in d_p_test:
    sigma_c = particle_strengthening(sigma_m, V_p_opt, d_p, b, G_m)
    strengthening = (sigma_c / sigma_m - 1) * 100
    print(f"{d_p:15.1f} {sigma_c:18.1f} {strengthening:15.1f}")

3.3 Orowan Mechanism

3.3.1 Dislocation-Particle Interactions

The Orowan mechanism is a strengthening mechanism that occurs when dislocations cannot cut through particles, but instead bypass between particles.

where \(G\): shear modulus, \(b\): Burgers vector, \(\lambda\): interparticle spacing

Interparticle spacing can be estimated from particle size and volume fraction:

3.3.2 Design of Optimal Particle Size and Fraction

To maximize Orowan strengthening, the interparticle spacing must be minimized. However, the following trade-offs exist:

• Small particle size → Large strengthening effect (λ decreases) but particles tend to agglomerate
• Large volume fraction → Large strengthening effect (λ decreases) but ductility decreases
• Optimization of particle size/fraction is important

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

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

def orowan_stress(d_p, V_p, G_m, b):
    """
    Orowan stress calculation

    Parameters:
    -----------
    d_p : float
        Particle diameter [μm]
    V_p : float
        Particle volume fraction
    G_m : float
        Matrix shear modulus [GPa]
    b : float
        Burgers vector [nm]

    Returns:
    --------
    sigma_orowan : float
        Orowan stress [MPa]
    """
    # Interparticle spacing [m]
    lambda_p = d_p * 1e-6 * (np.sqrt(np.pi / (4 * V_p)) - 1)

    # Calculate only when interparticle spacing is positive
    if lambda_p <= 0:
        return 0

    # Orowan stress [MPa]
    G_m_Pa = G_m * 1e9  # GPa → Pa
    b_m = b * 1e-9      # nm → m

    sigma_orowan = 0.4 * G_m_Pa * b_m / lambda_p / 1e6  # MPa

    return sigma_orowan

def ductility_reduction_factor(V_p):
    """
    Ductility reduction factor estimation (empirical model)

    Ductility decreases as V_p increases
    """
    return np.exp(-3 * V_p)

# Al alloy parameters
G_m = 26  # GPa
b = 0.286  # nm

# Parameter ranges
d_p_range = np.logspace(-1, 1.2, 40)  # 0.1-16 μm
V_p_range = np.linspace(0.05, 0.40, 40)

# Create mesh grid
D_p, V_p_grid = np.meshgrid(d_p_range, V_p_range)

# Calculate Orowan stress
sigma_orowan_grid = np.zeros_like(D_p)
performance_index = np.zeros_like(D_p)

for i in range(len(V_p_range)):
    for j in range(len(d_p_range)):
        sigma_o = orowan_stress(D_p[i,j], V_p_grid[i,j], G_m, b)
        sigma_orowan_grid[i,j] = sigma_o

        # Performance index: strength × ductility factor
        ductility = ductility_reduction_factor(V_p_grid[i,j])
        performance_index[i,j] = sigma_o * ductility

# 3D plot
fig = plt.figure(figsize=(16, 6))

# Orowan stress
ax1 = fig.add_subplot(131, projection='3d')
surf1 = ax1.plot_surface(np.log10(D_p), V_p_grid, sigma_orowan_grid,
                          cmap='viridis', alpha=0.8)
ax1.set_xlabel('log₁₀(Particle Diameter) [μm]')
ax1.set_ylabel('Volume Fraction')
ax1.set_zlabel('Orowan Stress [MPa]')
ax1.set_title('Orowan Strengthening Effect')
fig.colorbar(surf1, ax=ax1, shrink=0.5)

# Performance index
ax2 = fig.add_subplot(132, projection='3d')
surf2 = ax2.plot_surface(np.log10(D_p), V_p_grid, performance_index,
                          cmap='plasma', alpha=0.8)
ax2.set_xlabel('log₁₀(Particle Diameter) [μm]')
ax2.set_ylabel('Volume Fraction')
ax2.set_zlabel('Performance Index [Strength×Ductility]')
ax2.set_title('Overall Performance Index')
fig.colorbar(surf2, ax=ax2, shrink=0.5)

# Contour plot
ax3 = fig.add_subplot(133)
contour = ax3.contourf(np.log10(D_p), V_p_grid, performance_index,
                        levels=20, cmap='plasma')
ax3.set_xlabel('log₁₀(Particle Diameter) [μm]')
ax3.set_ylabel('Volume Fraction')
ax3.set_title('Performance Index Contour')
fig.colorbar(contour, ax=ax3)

# Find optimal point
max_idx = np.unravel_index(np.argmax(performance_index), performance_index.shape)
d_p_opt = D_p[max_idx]
V_p_opt = V_p_grid[max_idx]
sigma_opt = sigma_orowan_grid[max_idx]

ax3.plot(np.log10(d_p_opt), V_p_opt, 'r*', markersize=15,
         label=f'Optimal: d_p={d_p_opt:.2f} μm, V_p={V_p_opt:.2f}')
ax3.legend()

plt.tight_layout()
plt.savefig('orowan_optimization.png', dpi=300, bbox_inches='tight')
plt.close()

print("Optimal Design of Orowan Strengthening:")
print("="*60)
print(f"Optimal particle diameter: {d_p_opt:.2f} μm")
print(f"Optimal volume fraction: {V_p_opt:.2f}")
print(f"Orowan stress: {sigma_opt:.1f} MPa")
print(f"Interparticle spacing: {d_p_opt * (np.sqrt(np.pi/(4*V_p_opt)) - 1):.2f} μm")

3.4 Laminate Composites

3.4.1 Types and Characteristics of Laminates

By laminating different materials in layers, composite materials can be designed to utilize the properties of each layer.

3.4.2 Thermal Stress in Laminates

When materials with different thermal expansion coefficients are laminated, stress develops at the interface due to temperature changes.

where \(E_i\): elastic modulus of each layer, \(\alpha_i\): thermal expansion coefficient, \(t_i\): layer thickness, \(\Delta T\): temperature change

# 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 thermal_stress_bilayer(E1, E2, alpha1, alpha2, t1, t2, delta_T):
    """
    Thermal stress calculation for bilayer laminate

    Parameters:
    -----------
    E1, E2 : float
        Young's modulus of each layer [GPa]
    alpha1, alpha2 : float
        Thermal expansion coefficient of each layer [/°C]
    t1, t2 : float
        Thickness of each layer [mm]
    delta_T : float
        Temperature change [°C]

    Returns:
    --------
    sigma1, sigma2 : float
        Thermal stress in each layer [MPa]
    """
    # Thermal stress (simplified model)
    E1_GPa = E1 * 1000  # GPa → MPa
    E2_GPa = E2 * 1000

    sigma_thermal = (E1_GPa * E2_GPa * (alpha1 - alpha2) * delta_T /
                     (E1_GPa * t2 + E2_GPa * t1))

    # Layer 1 is compression, layer 2 is tension (when α1 > α2)
    sigma1 = -sigma_thermal * t2 / t1
    sigma2 = sigma_thermal

    return sigma1, sigma2

# Al/Ti laminate
E_Al = 70   # GPa
E_Ti = 110  # GPa
alpha_Al = 23e-6  # /°C
alpha_Ti = 9e-6   # /°C

# Varying layer thickness ratio
t_total = 10  # mm (total thickness)
t1_ratio = np.linspace(0.1, 0.9, 50)
t1 = t1_ratio * t_total
t2 = (1 - t1_ratio) * t_total

delta_T = -155  # °C (180°C → 25°C)

sigma_Al = []
sigma_Ti = []

for t1_val, t2_val in zip(t1, t2):
    s_Al, s_Ti = thermal_stress_bilayer(E_Al, E_Ti, alpha_Al, alpha_Ti,
                                         t1_val, t2_val, delta_T)
    sigma_Al.append(s_Al)
    sigma_Ti.append(s_Ti)

sigma_Al = np.array(sigma_Al)
sigma_Ti = np.array(sigma_Ti)

# Visualization
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

# Thermal stress
ax1.plot(t1_ratio, sigma_Al, 'b-', linewidth=2, label='Al layer stress')
ax1.plot(t1_ratio, sigma_Ti, 'r-', linewidth=2, label='Ti layer stress')
ax1.axhline(y=0, color='k', linestyle='--', linewidth=0.5)
ax1.set_xlabel('Al Layer Thickness Ratio (t_Al / t_total)')
ax1.set_ylabel('Thermal Stress [MPa]')
ax1.set_title(f'Thermal Stress in Al/Ti Laminate (ΔT = {delta_T}°C)')
ax1.grid(True, alpha=0.3)
ax1.legend()

# Comparison with yield strength
sigma_y_Al = 100  # MPa (annealed Al)
sigma_y_Ti = 350  # MPa (pure Ti)

# Safety factor
SF_Al = np.abs(sigma_y_Al / sigma_Al)
SF_Ti = np.abs(sigma_y_Ti / sigma_Ti)
SF_min = np.minimum(SF_Al, SF_Ti)

ax2.plot(t1_ratio, SF_Al, 'b-', linewidth=2, label='Al layer safety factor')
ax2.plot(t1_ratio, SF_Ti, 'r-', linewidth=2, label='Ti layer safety factor')
ax2.plot(t1_ratio, SF_min, 'k--', linewidth=2, label='Minimum safety factor')
ax2.axhline(y=1.0, color='g', linestyle=':', linewidth=1.5, label='Safety limit')
ax2.set_xlabel('Al Layer Thickness Ratio (t_Al / t_total)')
ax2.set_ylabel('Safety Factor')
ax2.set_title('Safety Factor of Each Layer')
ax2.set_ylim([0, 10])
ax2.grid(True, alpha=0.3)
ax2.legend()

plt.tight_layout()
plt.savefig('laminate_thermal_stress.png', dpi=300, bbox_inches='tight')
plt.close()

# Optimal thickness ratio (maximum safety factor)
opt_idx = np.argmax(SF_min)
t1_opt_ratio = t1_ratio[opt_idx]

print("Thermal Stress Analysis of Al/Ti Laminate:")
print("="*60)
print(f"Temperature change: {delta_T}°C")
print(f"Optimal Al layer thickness ratio: {t1_opt_ratio:.2f}")
print(f"Minimum safety factor: {SF_min[opt_idx]:.2f}")
print(f"\nStresses at thickness ratio {t1_opt_ratio:.2f}:")
print(f"  Al layer stress: {sigma_Al[opt_idx]:.1f} MPa")
print(f"  Ti layer stress: {sigma_Ti[opt_idx]:.1f} MPa")

3.4.3 Functionally Graded Materials (FGM)

Materials with continuously varying composition to mitigate thermal stress. Representative example: Graded material from ZrO₂ (ceramic) → Ni (metal)

# 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 fgm_property_profile(z, n, prop_ceramic, prop_metal):
    """
    FGM property distribution based on power law

    Parameters:
    -----------
    z : array
        Thickness direction coordinate (0: ceramic side, 1: metal side)
    n : float
        Gradient index (n=1: linear, n>1: concentrated on ceramic side)
    prop_ceramic, prop_metal : float
        Property values of ceramic and metal

    Returns:
    --------
    prop : array
        Property value at position z
    """
    V_metal = z**n
    prop = prop_ceramic * (1 - V_metal) + prop_metal * V_metal
    return prop

# ZrO2/Ni FGM
E_ZrO2 = 200   # GPa
E_Ni = 210     # GPa
alpha_ZrO2 = 10e-6   # /°C
alpha_Ni = 13e-6     # /°C

# Thickness direction coordinate
z = np.linspace(0, 1, 100)

# Effect of gradient index
n_values = [0.5, 1.0, 2.0, 5.0]

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

for n in n_values:
    E_profile = fgm_property_profile(z, n, E_ZrO2, E_Ni)
    alpha_profile = fgm_property_profile(z, n, alpha_ZrO2, alpha_Ni)

    ax1.plot(z, E_profile, linewidth=2, label=f'n = {n}')
    ax2.plot(z, alpha_profile * 1e6, linewidth=2, label=f'n = {n}')

ax1.set_xlabel('Thickness Direction Coordinate z (0: ZrO₂, 1: Ni)')
ax1.set_ylabel('Young\'s Modulus [GPa]')
ax1.set_title('Young\'s Modulus Distribution in FGM')
ax1.grid(True, alpha=0.3)
ax1.legend()

ax2.set_xlabel('Thickness Direction Coordinate z (0: ZrO₂, 1: Ni)')
ax2.set_ylabel('Thermal Expansion Coefficient [×10⁻⁶ /°C]')
ax2.set_title('Thermal Expansion Coefficient Distribution in FGM')
ax2.grid(True, alpha=0.3)
ax2.legend()

plt.tight_layout()
plt.savefig('fgm_property_profile.png', dpi=300, bbox_inches='tight')
plt.close()

print("Property Distribution of Functionally Graded Material (FGM):")
print("="*60)
print(f"{'Gradient Index n':>12} {'Center E [GPa]':>18} {'Center α [10⁻⁶/°C]':>25}")
print("-"*60)

for n in n_values:
    E_mid = fgm_property_profile(0.5, n, E_ZrO2, E_Ni)
    alpha_mid = fgm_property_profile(0.5, n, alpha_ZrO2, alpha_Ni)
    print(f"{n:12.1f} {E_mid:18.1f} {alpha_mid*1e6:25.2f}")

3.5 Summary

In this chapter, we learned about particle-reinforced and laminate composites:

• Classification of particle-reinforced composites (dispersion strengthening, particle strengthening, fillers)
• Types and manufacturing methods of MMC/CMC
• Elastic modulus prediction using Hashin-Shtrikman model
• Orowan mechanism and optimal particle design
• Thermal stress in laminates and functionally graded materials (FGM)

In the next chapter, we will learn about mechanical evaluation methods for composite materials (tensile testing, bending testing, impact testing) and nondestructive inspection (ultrasonic, X-ray CT, thermography).

Exercises

References

1. Chawla, N. and Chawla, K. K., "Metal Matrix Composites", 2nd ed., Springer, 2013, pp. 89-156, 234-278
2. Clyne, T. W. and Withers, P. J., "An Introduction to Metal Matrix Composites", Cambridge University Press, 1993, pp. 67-112
3. Kainer, K. U., "Metal Matrix Composites: Custom-made Materials for Automotive and Aerospace Engineering", Wiley-VCH, 2006, pp. 45-89
4. Courtney, T. H., "Mechanical Behavior of Materials", 2nd ed., Waveland Press, 2005, pp. 389-445
5. Hashin, Z. and Shtrikman, S., "A Variational Approach to the Theory of the Elastic Behaviour of Multiphase Materials", Journal of the Mechanics and Physics of Solids, Vol. 11, 1963, pp. 127-140
6. Koizumi, M., "FGM Activities in Japan", Composites Part B, Vol. 28, 1997, pp. 1-4
7. Suresh, S. and Mortensen, A., "Fundamentals of Functionally Graded Materials", IOM Communications, 1998, pp. 23-67, 134-189
8. Naebe, M. and Shirvanimoghaddam, K., "Functionally Graded Materials: A Review of Fabrication and Properties", Applied Materials Today, Vol. 5, 2016, pp. 223-245