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T8 · CompositesPolymer Composites & Fiber EngineeringGATE 2027 Syllabus (XE5)
Reuss Transverse Rule of Mixtures Modulus (Inverse / Series Model)
Calculates the lower-bound transverse elastic modulus (E2) perpendicular to fiber alignment in continuous unidirectional fiber-reinforced composites under isostress assumption.
Mathematical Canonical Relation
Plain Formula: 1 / E2 = (Vf / Ef) + ((1 - Vf) / Em)
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V_fib
Range: 0.1 – 0.75 fraction
E_fib
Range: 30 – 500 GPa
E_mat
Range: 1 – 8 GPa
Calculated Engineering Result:
E2 = 8.6 GPa (vs E1 = 139.4 GPa)
Engineering Verification:Anisotropy ratio E1/E2 is 16.3x.
When & Where to Apply
Use when calculating laminate ABD stiffness matrices, finite element composite ply orientations, and assessing matrix-dominated transverse stiffness.
Common Engineering Pitfalls
- Using Voigt parallel rule of mixtures for transverse loading direction
- Forgetting to invert the result (E2 = 1 / [Vf/Ef + Vm/Em])
Physical Assumptions & Validity Limits
- Constant stress across fiber and matrix (isostress / series model)
- Perfect fiber-matrix interfacial bonding without voids
- Linear elastic isotropic matrix and transversely isotropic fibers
Variable Symbols, SI Units & Dimensional Specifications
| Symbol | Variable Meaning | SI Unit | Typical Industrial Range |
|---|---|---|---|
| E_2 | Transverse composite Young’s modulus | GPa | — |
| V_f | Fiber volume fraction (0.1 to 0.7) | fraction | 0.60 |
| E_f | Fiber tensile modulus | GPa | 230.0 |
| E_m | Polymer matrix tensile modulus | GPa | 3.5 |
Worked Industrial Numerical Example: Carbon Fiber / Epoxy Prepreg Transverse Modulus E2
Problem Statement:
A unidirectional carbon fiber epoxy aerospace prepreg has fiber volume fraction Vf = 0.60. Carbon fiber modulus Ef = 230 GPa, epoxy matrix modulus Em = 3.5 GPa. Calculate longitudinal E1 (Voigt) and transverse E2 (Reuss).
Step-by-Step Numerical Solution:
- Calculate Voigt E1: 0.60 * 230 + 0.40 * 3.5 = 138 + 1.4 = 139.4 GPa
- Calculate Reuss 1/E2: (0.60 / 230) + (0.40 / 3.5)
- Term 1: 0.60 / 230 = 0.00261
- Term 2: 0.40 / 3.5 = 0.11429
- Sum = 0.11690 => Invert E2 = 1 / 0.11690 = 8.55 GPa
Final Answer: E2 = 8.55 GPa (compared to E1 = 139.4 GPa)
💡 Engineering Interpretation: While longitudinal modulus E1 reaches 139.4 GPa due to high-stiffness carbon fibers, transverse modulus E2 is only 8.55 GPa (matrix dominated), illustrating extreme mechanical anisotropy and necessitating cross-ply [0/90] or quasi-isotropic [0/±45/90] layup sequences.
Authoritative Technical Reference & Source Provenance
Mechanics of Composite Materials by Robert M. Jones (CRC Press, 1998), 2nd Edition, Page 92-99.
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