Estimate the stiffness and density of a fiber-reinforced composite from the properties of its constituents. The calculator applies the Voigt (iso-strain) upper bound and the Reuss (iso-stress) lower bound on modulus, plus the linear rule of mixtures for density — the foundation of every composite-design first estimate.
A fiber-reinforced composite combines a stiff, strong fiber with a lighter, more compliant matrix, and its overall properties depend on how much of each is present and how they share the load. The rule of mixtures is the simplest and most widely used way to estimate composite stiffness and density from the constituent properties. This calculator gives both the Voigt upper bound (load along the fibers) and the Reuss lower bound (load across the fibers), bracketing where the real modulus lies, and computes the composite density directly.
When a unidirectional composite is loaded along the fiber direction, the fibers and matrix stretch by the same amount, so they share a common strain (iso-strain). The stresses add in proportion to volume fraction, giving the Voigt rule of mixtures: E_upper = Vf·Ef + Vm·Em, where Vm = 1 − Vf. This is the upper bound on modulus and describes the longitudinal stiffness of aligned fibers — the direction in which a composite is at its strongest and stiffest.
When the same composite is loaded across (transverse to) the fibers, both phases carry the same stress (iso-stress) and their strains add, so the compliances combine: E_lower = 1 / (Vf/Ef + Vm/Em). This harmonic-style average is the Reuss lower bound and represents the transverse stiffness, which is much closer to the soft matrix modulus. The wide gap between the Voigt and Reuss values is the origin of a composite's anisotropy.
The Voigt and Reuss expressions are rigorous upper and lower bounds for any two-phase mixture. Real composites have fibers that are not perfectly aligned, imperfect fiber-matrix bonding, and complex stress transfer near fiber ends, so the measured modulus falls somewhere inside the bounds. For load parallel to well-aligned continuous fibers the Voigt bound is nearly exact; for transverse or randomly oriented short-fiber composites the modulus lies further toward the Reuss bound, and semi-empirical models (such as Halpin-Tsai) refine the estimate.
Density follows a simple linear rule of mixtures with no upper/lower distinction: ρc = Vf·ρf + Vm·ρm, since volume and mass are conserved. Combining stiffness and density gives the specific stiffness (E/ρ), which is why carbon-fiber composites are prized in aerospace: a high longitudinal modulus paired with very low density produces a specific stiffness far above that of steel or aluminium, allowing lighter structures at equal rigidity.
The Voigt (iso-strain) bound assumes the fiber and matrix experience the same strain and gives the upper-bound modulus E = Vf·Ef + Vm·Em — appropriate for loading along the fibers. The Reuss (iso-stress) bound assumes the same stress in both phases and gives the lower-bound modulus 1/(Vf/Ef + Vm/Em) — appropriate for loading across the fibers. The true modulus lies between them.
For unidirectional continuous fibers loaded along their length, use the Voigt (upper) bound — it closely matches reality. For transverse loading or randomly oriented short fibers, the modulus is near the Reuss (lower) bound. When in doubt, the bounds bracket the answer, and refined models like Halpin-Tsai give a better single estimate.
The fiber volume fraction Vf is the fraction of the composite's total volume occupied by fibers, a number between 0 and 1 (for example 0.6 means 60% fibers by volume). The matrix volume fraction is Vm = 1 − Vf. Note this is a volume fraction, not a weight fraction; the two differ when the fiber and matrix densities differ.
Mass and volume are conserved when phases are mixed, so total mass is just the sum of each phase's mass and the density is a true volume-weighted average. Stiffness depends on how the phases share load, which differs between parallel (iso-strain) and series (iso-stress) arrangements, so modulus has distinct upper and lower bounds rather than a single linear average.
Specific stiffness is the modulus divided by density (E/ρ). It measures how much rigidity you get per unit mass. Composites can have a far higher specific stiffness than metals, so they let engineers build structures that are equally stiff but much lighter — critical in aircraft, spacecraft, and high-performance sporting goods.