Composite materials — fiber-matrix, ply properties, laminate theory

1. Fiber-Matrix Fundamentals

Why This Combination Works

The load-sharing mechanism:

1. Fibers are strong in tension along their axis (carbon fiber: $E_f \sim 230$ GPa) but britle and need support
2. Matrix is weak (epoxy: $E_m \sim 3$ GPa) but tough, holds fibers in place, distributes loads

When you pull a composite, the stiff fibers carry most of the load:

$\sigma_{\text{composite}} = \sigma_f V_f + \sigma_m V_m$

where $V_f + V_m = 1$ (volume fractions). Since $E_f \gg E_m$, and strain is equal ($\epsilon_f = \epsilon_m = \epsilon$):

$\sigma_f = E_f \epsilon, \quad \sigma_m = E_m \epsilon$

The effective modulus along fiber direction (longitudinal):

$E_1 = E_f V_f + E_m V_m$

Why this step? The fibers and matrix are bonded together (perfect adhesion assumed), so they stretch the same amount. Each contributes stress proportional to its stiffness and volume. This is the Rule of Mixtures.

Transverse and Shear Properties

Perpendicular to fibers (transverse direction, subscript 2), the load must pass through the weak matrix:

$\frac{1}{E_2} = \frac{V_f}{E} + \frac{V_m}{E_m}$

This is an inverse Rule of Mixtures (series springs).

Derivation from first principles:

Consider a unit cube of composite loaded perpendicular to fibers. The fiber and matrix sections are in series (load flows through both). For series elements:

• Same stress: $\sigma_f = \sigma_m = \sigma$
• Total strain: $\epsilon_{total} = \epsilon_f + \epsilon_m$

If the cube has thickness1, the fiber portion has thickness $V_f$, matrix portion $V_m$:

$\epsilon_{total} = \epsilon_f V_f + \epsilon_m V_m = \sigma\left( \frac{V_f}{E_f} + \frac{V_m}{E_m} \right)$

By definition $E_2 = \sigma/\epsilon_{total}$:

$\frac{1}{E_2} = \frac{V_f}{E_f} + \frac{V_m}{E_m}$

Why this matters: $E_2 \ll E_1$. For our carbon-epoxy example:

$\frac{1}{E_2} = \frac{0.6}{230} + \frac{0.4}{3} = 0.00261 + 0.133 = 0.1356$

$E_2 = 7.4 \text{ GPa}$

The composite is anisotropic: $E_1/E_2 = 139.2/7.4 \approx 19$. This is both a feature (tailorability) and a challenge (must design carefully).

Shear modulus $G_{12}$ (in-plane) follows a similar inverse rule:

$\frac{1}{G_{12}} = \frac{V_f}{G_f} + \frac{V_m}{G_m}$

2. Single Ply (Lamina) Properties

A ply or lamina is a thin layer (~0.125 mm) of unidirectional fibers in matrix. Its properties in the material coordinate system (1 = fiber direction, 2 = transverse) are:

• $E_1, E_2$ (Young's moduli)
• $G_{12}$ (shear modulus)
• $\nu_{12}$ (major Poisson's ratio)
• $\nu_{21} = \nu_{12} E_2/E_1$ (minor Poisson's ratio, from symmetry)

Constitutive Relations (Stress-Strain)

For a ply under plane stress ($\sigma_3 = 0$), the stiffness matrix $[Q]$ relates stress to strain in material axes:

$\begin{Bmatrix} \sigma_1 \ \sigma_2 \\ \tau_{12} \end{Bmatrix} = \begin{bmatrix} Q_{11} & Q_{12} & 0 \\ Q_{12} & Q_{22} & 0 \\ 0 & Q_{66} \end{bmatrix} \begin{Bmatrix} \epsilon_1 \\ \epsilon_2 \\ \gamma_{12} \end{Bmatrix}$

Deriving the $Q$ terms:

Start with the compliance form (strain as a function of stress):

$\epsilon_1 = \frac{\sigma_1}{E_1} - \nu_{12} \frac{\sigma_2}{E_1}, \quad \epsilon_2 = -\nu_{21} \frac{\sigma_1}{E_2} + \frac{\sigma_2}{E_2}, \quad \gamma_{12} = \frac{\tau_{12}}{G_{12}}$

In matrix form:

$\begin{Bmatrix} \epsilon_1 \\ \epsilon_2 \\ \gamma_{12} \end{Bmatrix} = \begin{bmatrix} 1/E_1 & -\nu_{12}/E_1 & 0 \\ -\nu_{21}/E_2 & 1/E_2 & 0 \\ 0 & 0 & 1/G_{12} \end{bmatrix} \begin{Bmatrix} \sigma_1 \\ \sigma_2 \ \tau_{12} \end{Bmatrix}$

This is $[\epsilon] = [S][\sigma]$, where $[S]$ is compliance. The stiffness $[Q] = [S]^{-1}$. For a 2×2 block:

$\begin{bmatrix} 1/E_1 & -\nu_{12}/E_1 \\ -\nu_{21}/E_2 & 1/E_2 \end{bmatrix}^{-1} = \frac{1}{(1/E_1)(1/E_2) - (\nu_{12}/E_1)(\nu_{21}/E_2)} \begin{bmatrix} 1/E_2 & \nu_{12}/E_1 \\ \nu_{21}/E_2 & 1/E_1 \end{bmatrix}$

The determinant simplifies using $\nu_{21}E_1 = \nu_{12}E_2$:

$\det = \frac{1 - \nu_{12}\nu_{21}}{E_1 E_2}$

Therefore:

$Q_{11} = \frac{E_1}{1 - \nu_{12}\nu_{21}}, \quad Q_{22} = \frac{E_2}{1 - \nu_{12}\nu_{21}}, \quad Q_{12} = \frac{\nu_{12}E_2}{1 - \nu_{12}\nu_{21}} = \frac{\nu_{21}E_1}{1 - \nu_{12}\nu_{21}}$

$Q_{66} = G_{12}$

Why this step? We inverted the compliance to get stiffness because structural analysis works with applying strains and calculating stresses (or vice versa). The $1 - \nu_{12}\nu_{21}$ term ensures energy compatibility (symmetry of stiffness matrix).

Off-Axis Loading: Transformation

Real structures have plies at angles $\theta$ to the loading direction (e.g., 0°, ±45°, 90° layups). We need the transformed stiffness $[\bar{Q}]$ in the laminate coordinate system (x-y):

$[\bar{Q}] = [T]^{-1} [Q] [T]^{-T}$

where $[T]$ is the transformation matrix. For rotation by angle $\theta$:

$[T] = \begin{bmatrix} \cos^2\theta & \sin^2\theta & 2\sin\theta\cos\theta \\ \sin^2\theta & \cos^2\theta & -2\sin\theta\cos\theta \\ -\sin\theta\cos\theta & \sin\theta\cos\theta & \cos^2\theta - \sin^2\theta \end{bmatrix}$

Physical meaning: Aply at $\theta = 45°$ has fibers at 45° to the x-axis. Pulling in the x-direction now:

• Loads both fiber and transverse directions equally
• Induces significant shear in material axes
• Couples extension and shear: pulling causes shearing

The transformed stiffness has all6 terms ($\bar{Q}_{16}, \bar{Q}_{26}$ coupling terms appear):

$\begin{Bmatrix} \sigma_x \\ \sigma_y \\ \tau_{xy} \end{Bmatrix} = \begin{bmatrix} \bar{Q}_{11} & \bar{Q}_{12} & \bar{Q}_{16} \ \bar{Q}_{12} & \bar{Q}_{22} & \bar{Q}_{26} \\ \bar{Q}_{16} & \bar{Q}_{26} & \bar{Q}_{66} \end{bmatrix} \begin{Bmatrix} \epsilon_x \\ \epsilon_y \\ \gamma_{xy} \end{Bmatrix}$

3. Laminate Theory (Classical Laminated Plate Theory, CLPT)

A laminate is a stack of plies, each potentially at different orientations. Example: [0/45/-45/90]s means symmetric layup with 0°, +45°, -45°, 90° plies, then mirrored.

Assumptions (Kirchoff-Love)

1. Plies are perfectly bonded (no slip)
2. Plane sections remain plane (no shear deformation through thickness)
3. Thin plate: thickness $h \ll$ lateral dimensions
4. Linear elastic materials, small deformations

Force and Moment Resultants

Instead of tracking stress at every point through thickness, we integrate to get force per unit width $[N]$ and moment per unit width $[M]$:

$N_x = \int_{-h/2}^{h/2} \sigma_x \, dz, \quad M_x = \int_{-h/2}^{h/2} \sigma_x \, z \, dz$

(similarly for $y$ and $xy$ components)

Kinematic relations: Under Kirchhoff assumptions, strain varies linearly through thickness:

$\epsilon_x(z) = \epsilon_x^0 + z \kappa_x$

where $\epsilon_x^0$ is midplane strain, $\kappa_x$ is curvature (bending strain gradient).

In matrix form:

$\begin{Bmatrix} \epsilon_x \ \epsilon_y \\ \gamma_{xy} \end{Bmatrix} = \begin{Bmatrix} \epsilon_x^0 \ \epsilon_y^0 \\ \gamma_{xy}^0 \end{Bmatrix} + z \begin{Bmatrix} \kappa_x \\ \kappa_y \\ \kappa_{xy} \end{Bmatrix}$

Why this step? Plane-sections-remain-plane means the deformation is described by the midplane (stretching) and its curvature (bending). The $z$ distance from midplane scales the bending strain.

ABD Matrix

Substitute stress-strain relation $\sigma = \bar{Q}\epsilon$ and integrate:

$N_i = \int_{-h/2}^{h/2} \sum_j \bar{Q}_{ij} (\epsilon_j^0 + z\kappa_j) \, dz = \sum_j \left[ \int \bar{Q}_{ij} \, dz \right] \epsilon_j^0 + \sum_j \left[ \int \bar{Q}_{ij} z \, dz \right] \kappa_j$

Define:

$A_{ij} = \int_{-h/2}^{h/2} \bar{Q}_{ij} \, dz = \sum_{k=1}^{n} (\bar{Q}_{ij})_k (z_{k} - z_{k-1})$

$B_{ij} = \int_{-h/2}^{h/2} \bar{Q}_{ij} \, z \, dz = \frac{1}{2} \sum_{k=1}^{n} (\bar{Q}_{ij})_k (z_k^2 - z_{k-1}^2)$

$D_{ij} = \int_{-h/2}^{h/2} \bar{Q}_{ij} \, z^2 \, dz = \frac{1}{3} \sum_{k=1}^{n} (\bar{Q}_{ij})_k (z_k^3 - z_{k-1}^3)$

where $k$ indexes plies, $z_k$ is the z-coordinate of ply interfaces.

The ABD matrix couples forces/moments to strains/curvatures:

$\begin{Bmatrix} N_x \\ N_y \ N_{xy} \\ M_x \\ M_y \\ M_{xy} \end{Bmatrix} = \begin{bmatrix} A_{11} & A_{12} & A_{16} & B_{11} & B_{12} & B_{16} \\ A_{12} & A_{22} & A_{26} & B_{12} & B_{22} & B_{26} \\ A_{16} & A_{26} & A_{66} & B_{16} & B_{26} & B_{66} \\ B_{11} & B_{12} & B_{16} & D_{11} & D_{12} & D_{16} \\ B_{12} & B_{22} & B_{26} & D_{12} & D_{22} & D_{26} \ B_{16} & B_{26} & B_{66} & D_{16} & D_{26} & D_{66} \end{bmatrix} \begin{Bmatrix} \epsilon_x^0 \\ \epsilon_y^0 \\ \gamma_{xy}^0 \\ \kappa_x \\ \kappa_y \\ \kappa_{xy} \end{Bmatrix}$

Physical meaning:

• [A] (extensional stiffness): relates in-plane forces to midplane strains. Units: N/m (force per width strain)
• [B] (coupling stiffness): couples extension to bending. Non-zero for unsymmetric laminates. Units: N (force per width per curvature)
• [D] (bending stiffness): relates moments to curvatures. Units: N·m (moment per width per curvature)

Why the integrals? $A$ sums stiffness contributions of all plies (thicker laminate → more stiffness). $D$ weights by $z^2$ because outer plies contribute more to bending resistance (like a beam's $I = bh^3/12$).

4. Design Implications for Spacecraft

Quasi-Isotropic Laminates

To approximate isotropic behavior in-plane, use quasi-isotropic layups: equal fiber fractions at 0°, ±60° (or 0°, ±45°, 90°). Example: [0/+45/-45/90]s.

This gives:

• $A_{16} = A_{26} = 0$ (no extension-shear coupling)
• $A_{11} \approx A_{22}$ (similar stiffness in x and y)
• Still anisotropic out-of-plane

Why this matters: Early spacecraft used quasi-isotropic laminates because design codes were written for metals. Now we optimizeply angles for specific load paths.

Coefficient of Thermal Expansion (CTE)

Carbon fiber has negative CTE along fiber axis ($\alpha_1 \sim -0.5 \times 10^{-6}$ /°C), epoxy has positive CTE ($\alpha_m \sim 60 \times 10^{-6}$ /°C).

A composite ply's CTE:

$\alpha_1 \approx \frac{E_f V_f \alpha_f + E_m V_m \alpha_m}{E_f V_f + E_m V_m} \approx \alpha_f \quad \text{(fiber-dominated)}$

$\alpha_2 \approx (1+\nu_m) V_m \alpha_m + (1+\nu_f) V_f \alpha_f -_1 \nu_{12}$

Typical carbon-epoxy: $\alpha_1 \sim 0$, $\alpha_2 \sim 30 \times 10^{-6}$ /°C.

Spacecraft application: Dimensional stability for optics and antennas. A symmetric [0/90]s laminate has near-zero CTE in both x and y if balanced. Aluminum's $\alpha \sim 23 \times 10^{-6}$ /°C causes unaceptable distortion over the $\pm 150°$C thermal swing in orbit.

Failure Modes

Composites fail differently than metals:

1. Fiber breakage (tension in fiber direction): catastrophic, little warning
2. Matrix cracking (transverse tension, shear): first damage, may not be critical
3. Delamination (interlaminar shear, impact): plies separate, loses bending stiffness
4. Fiber microbuckling (compression in fiber direction): local fiber kinking

Analysis: Use first-ply failure criteria (Tsai-Wu, Hashin) on each ply in material axes, checking all load cases. Then assess if damage propagates (progressive failure analysis).

Connections

• 3.6.14-material-selection — Why composites beat metals in specific strength/stiffness
• 3.6.16-sandwich-structures — Composites as facesheets over honeycomb cores
• 3.7.1-launch-loadsand-quasi-static — Designing composite structures for launch g-loads
• 4.2-thermal-design — Low CTE critical for thermal stability of instruments
• 5.3-finite-element-analysis — Modeling laminates with shell elements and ABD matrices
• 2.4-mass-budgets — Composites enable mass savings for more payload

#flashcards/physics

What is a composite material and why is it used in spacecraft? :: A composite combines high-strength fibers (carbon, glass) in a matrix (epoxy) to achieve high specific strength (strength/weight ratio) and tailorable anisotropic properties. Spacecraft use them to save mass (5× better specific stiffness than aluminum) while maintaining strength and low thermal expansion.

Concept Map

Hinglish (regional understanding)