T8 · CompositesPolymer Composites & Fiber EngineeringGATE 2027 Syllabus (XE5)Shop-Floor Standard

Voigt Longitudinal Rule of Mixtures

Predicts the longitudinal elastic modulus (E11 in GPa) of unidirectional fiber reinforced polymer (FRP) composites parallel to fiber axis.

Mathematical Canonical Relation

E_{11} = V_f \cdot E_f + (1 - V_f) \cdot E_m

Plain Formula: E11 = Vf * Ef + (1 - Vf) * Em

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Fiber Volume Fraction (Vf) (fraction)

Range: 0.1 – 0.75 fraction

Fiber Modulus Ef (GPa) (GPa)

Range: 40 – 600 GPa

Matrix Modulus Em (GPa) (GPa)

Range: 1 – 10 GPa

Calculated Engineering Result:

139.4 GPa

Engineering Verification:Calculated using Voigt longitudinal micromechanics model.

When & Where to Apply

Use when designing continuous fiber reinforced structural parts (CFRP, GFRP) loaded along fiber direction.

Common Engineering Pitfalls

Applying Voigt equation to transverse loading (transverse requires Reuss inverse model)

Physical Assumptions & Validity Limits

Continuous unidirectional parallel fibers
Perfect fiber-matrix interfacial bonding
Isostrain deformation along fiber direction

Variable Symbols, SI Units & Dimensional Specifications

Symbol	Variable Meaning	SI Unit	Typical Industrial Range
E_{11}	Longitudinal composite modulus	GPa	—
V_f	Fiber volume fraction	fraction	0.60
E_f	Fiber tensile modulus	GPa	230
E_m	Matrix resin modulus	GPa	3.5

Worked Industrial Numerical Example: Aerospace Carbon/Epoxy Laminate Longitudinal Stiffness

Problem Statement:

Calculate longitudinal modulus E11 for an aerospace grade unidirectional Carbon/Epoxy prepreg laminate with 60 vol% carbon fiber (Ef = 230 GPa) in epoxy matrix (Em = 3.5 GPa).

Step-by-Step Numerical Solution:

Apply Voigt Model: E11 = (0.60 * 230) + (1 - 0.60) * 3.5
Fiber contribution = 138 GPa
Matrix contribution = 1.4 GPa
Sum contributions: E11 = 138 + 1.4 = 139.4 GPa

Final Answer: E11 = 139.4 GPa

💡 Engineering Interpretation: Longitudinal modulus is 139.4 GPa. Carbon fibers carry 99% of the tensile load, providing high specific stiffness at 1/5th steel density.

Authoritative Technical Reference & Source Provenance

Analysis and Performance of Fiber Composites by B. D. Agarwal & L. J. Broutman (Wiley, 2006), 3rd Edition, Page 75-82.

Connected Curriculum Lesson

Micromechanics & Rule of Mixtures for Composites

Study Full Lesson