Estimate the Young's modulus of a composite by the rule of mixtures from the Young's moduli of the fiber and matrix and the fiber volume fraction. Along the fibers the isostrain form (springs in parallel) is used, and across them the isostress form (springs in series). The defaults are a unidirectional composite of carbon fiber (230 GPa, 1.80 g/cm³) at a volume fraction of 60 % in epoxy (3.5 GPa, 1.20 g/cm³): about 139 GPa along the fibers and about 8.6 GPa across them.
Your inputs are kept in this page's URL, so a bookmark or a shared link opens the same calculation. Favorites and recently used calculators are stored only in this browser (localStorage).
Results
Fiber volume fraction Vf
–
Fiber mass fraction Wf
–
Density of composite ρ
–
Young's modulus (rule of mixtures)
Along the fibers E₁ (isostrain, Voigt, upper bound)
–
Across the fibers E₂ (isostress, Reuss, lower bound)
–
Anisotropy E₁/E₂
–
Share of the load along the fibers carried by the fibers
–
Specific stiffness
Along the fibers E₁/ρ
–
Across the fibers E₂/ρ
–
Index for a light, stiff beam √E₁/ρ (GPa^½/(g/cm³))
–
The isostrain and isostress forms are rough upper and lower bounds on Young's modulus. The actual transverse modulus is often a little higher than the isostress value and is corrected with the Halpin–Tsai equations or similar. Short-fiber or particle-reinforced materials and materials with voids depart from these values.
See it as a graph
Other inputs stay at their current values while the chosen input varies from 1/4 to 4 times its current value. The red dot is the current value.
The isostrain form (parallel, Voigt)
When pulled parallel to the fibers, the fiber and matrix stretch by the same amount, like springs connected in parallel. Because the strains are equal, the overall stress is the volume-fraction-weighted average, giving E₁ = VfEf + VmEm. With the defaults, 0.60 × 230 + 0.40 × 3.5 ≈ 139 GPa. The share of the load carried by the fibers is VfEf/E₁, about 99 % with the defaults. Even with only about half the volume as fiber, stiffness and strength along the fibers are set almost entirely by the fibers.
The isostress form (series, Reuss)
When pulled perpendicular to the fibers, the fiber and matrix carry the same stress, like springs connected in series, and their strains add in proportion to volume fraction. This gives 1/E₂ = Vf/Ef + Vm/Em, in which the soft matrix dominates. With the defaults, 1/(0.60/230 + 0.40/3.5) ≈ 8.6 GPa, about one sixteenth of the value along the fibers. At the same volume fraction, the isostrain form gives an upper bound and the isostress form a lower bound (the Voigt and Reuss bounds).
Volume fraction and mass fraction
The rule of mixtures uses volume fractions. In manufacturing, fiber content is often controlled by mass fraction, so convert using the densities: Vf = (Wf/ρf)/(Wf/ρf + (1 − Wf)/ρm). For carbon fiber (1.80 g/cm³) and epoxy (1.20 g/cm³), a mass fraction of 70 % is a volume fraction of about 61 %. When the fiber is denser than the matrix, the volume fraction is smaller than the mass fraction. Without voids, the density of the composite is ρ = Vfρf + Vmρm.
Specific stiffness and materials selection
How stiff a material is for its weight is compared by the specific stiffness E/ρ. Steel (210 GPa, 7.9 g/cm³) is about 27 GPa/(g/cm³), and aluminum and titanium alloys are about the same. The default CFRP is about 89 GPa/(g/cm³) along the fibers, more than three times higher, but across them it is about 5.5, lower than steel. The index for choosing the lightest beam of given bending stiffness is √E/ρ: about 1.8 for steel, against about 7.6 for the default composite along the fibers. In real parts, layers with different fiber directions are stacked to make up for the directional difference, so the advantage in specific stiffness is smaller than this.
ALoading parallel to the fibers is approximated by the isostrain form and loading perpendicular to them by the isostress form
BLoading parallel to the fibers is approximated by the isostress form and loading perpendicular to them by the isostrain form
CThe isostrain form holds in both directions, and Young's modulus does not depend on the fiber orientation
Show answer and explanation
Answer: A. Loading parallel to the fibers is approximated by the isostrain form and loading perpendicular to them by the isostress form
When loaded parallel to the fibers, fiber and matrix stretch by the same amount like springs connected in parallel, so the isostrain form E₁ = VfEf + VmEm applies. When loaded perpendicular to the fibers, fiber and matrix carry the same stress like springs in series, so the isostress form 1/E₂ = Vf/Ef + Vm/Em is used as an approximation. At the same volume fraction, the isostrain form gives an upper bound on Young's modulus and the isostress form a lower bound (the Voigt and Reuss bounds). The choice applying the isostress form to parallel loading and the isostrain form to perpendicular loading swaps parallel and perpendicular. The modulus of unidirectional material differs greatly between the fiber and transverse directions, so saying it does not depend on direction is also wrong. The actual transverse modulus is often somewhat higher than the isostress form predicts, and corrections such as the Halpin–Tsai equations are used.
Perpendicular to the fibers, fiber and resin are treated as lying in series and carrying the same stress, so 1/E₂ = Vf/Ef + Vm/Em is used. 1/E₂ = 0.50/72 + 0.50/3.0 = 0.00694 + 0.1667 = 0.1736 GPa⁻¹, so E₂ = 1/0.1736 ≈ 5.76 GPa, about 5.8 GPa. About 37.5 GPa is the modulus in the fiber direction from the isostrain form 0.50 × 72 + 0.50 × 3.0. About 0.17 GPa forgets to take the reciprocal. The transverse modulus stays close to that of the compliant resin, only about 15% (about 1/6.5) of the modulus in the fiber direction. This is why unidirectional material is not used in parts loaded perpendicular to the fibers.
The specific modulus is 140/1.6 = 87.5 for CFRP and 210/7.9 ≈ 26.6 for steel (both in GPa/(g/cm³)), so the ratio is 87.5/26.6 ≈ 3.3. CFRP has the lower Young's modulus itself, but its density is about one fifth, so at equal weight it is about 3.3 times as stiff. About 0.67 compares only the moduli (140/210), and about 4.9 looks only at the density ratio (7.9/1.6). Note that this value applies only in the fiber direction; perpendicular to the fibers, the specific modulus is far lower than that of steel. Real structures compensate for direction by laminating, so the specific-stiffness advantage is smaller than this value.
FAQ
Which form should I use?
For a unidirectional continuous-fiber composite, approximate loading parallel to the fibers with the isostrain form and loading perpendicular to them with the isostress form. The Young's modulus of particle-dispersed or short-fiber materials usually lies between the two.
Can the rule of mixtures be used for strength too?
The strength along the fibers is sometimes estimated as σc ≈ Vfσf + Vmσm′, using the matrix stress at the fiber failure strain. However, strength is strongly affected by defects, interfacial bonding and fiber length (the critical fiber length), and does not agree as well as Young's modulus does.
Why is the Voigt form an upper bound?
For a given average strain, the strain distribution inside the material is the one that minimizes the elastic strain energy (the principle of minimum potential energy). A uniform strain (isostrain) is just one possible distribution, so the energy calculated from it, and hence the isostrain Young's modulus, is at or above the true value. In reality the soft matrix strains more, which lowers the energy. Likewise, assuming a uniform stress gives a complementary energy at or above the true value, so the isostress form is a lower bound. These two forms are strict bounds on Young's modulus when the two phases have the same Poisson's ratio; if they differ, the modulus of a unidirectional composite along the fibers can slightly exceed the isostrain value. Along the fibers of a unidirectional composite the actual strain is nearly uniform, so the isostrain form agrees well.