Expressions that estimate the properties of a composite from the properties and volume fractions of its constituents. Along the fibers, the isostrain form E = VfEf + VmEm applies; perpendicular to them, the isostress form is used as an approximation.
Japanese name: 複合則
Rule of mixtures appears in 3 practice questions across 2 topics, including Composite materials, Polymers and composites.
Field: Polymers and composites
When a material with unidirectionally aligned fibers is pulled along the fibers, the fibers and matrix stretch by the same amount, so the force each carries is proportional to the product of its volume fraction and Young's modulus. Adding the forces gives E = VfEf + VmEm (equal strain, the Voigt equation). When pulled perpendicular to the fibers, the two are in series and carry the same stress, so adding the strains gives 1/E = Vf/Ef + Vm/Em (equal stress, the Reuss equation). Both are derived from an assumption about how each phase carries the force.
The tensile strength along the fibers can also be estimated as σc ≈ Vfσf + Vmσm′, where σm′ is the matrix stress when the fibers reach their failure strain. With too few fibers, however, the matrix can carry the load after the fibers break, so the strength can even fall below that of the matrix alone. For fibers to strengthen the material, their volume fraction must exceed a minimum value and a critical value needed to surpass the matrix strength. For short fibers, being longer than the critical fiber length is also a condition.
Density is simply additive by mass, so ρc = Vfρf + Vmρm holds exactly. Thermal and electrical conductivity also take the same form of equations for parallel and series arrangements. The thermal expansion coefficient, on the other hand, needs an equation weighted by Young's modulus as well as volume fraction, because the stiffer phase constrains the deformation. The Young's modulus of particle-reinforced materials falls between the equal-strain and equal-stress equations, so these serve as upper and lower bounds, and the Hashin–Shtrikman bounds, which give a narrower range, are also used.
For short fibers or fibers with scattered orientations, the estimate is multiplied by an orientation efficiency factor. With Krenchel's factors, it is 1 for unidirectional fibers, 3/8 for random in-plane orientation and 1/5 for random three-dimensional orientation. For example, a resin (3 GPa) with 20% by volume of glass fibers (72 GPa) randomly oriented in-plane gives E ≈ 3/8 × 0.2 × 72 + 0.8 × 3 ≈ 7.8 GPa, less than half the unidirectional upper bound of 16.8 GPa. In actual molded parts, fiber-length effects and voids can lower it further.
Answer: C. About 139 GPa
When load is applied along the fibers, fiber and resin strain by the same amount (isostrain), so the modulus is the volume-fraction-weighted average E = VfEf + VmEm = 0.6 × 230 + 0.4 × 3.5 = 138 + 1.4 ≈ 139 GPa. About 94 GPa swaps the volume fractions of fiber and resin. About 8.6 GPa comes from the isostress expression 1/E = Vf/Ef + Vm/Em, which estimates the modulus perpendicular to the fibers. Because unidirectional material differs in stiffness by more than a factor of ten depending on direction, real components stack plies with different fiber orientations.
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.
Use volume fraction, because the fraction of cross-sectional area carrying the load corresponds to the volume fraction. If the formulation was specified by mass fraction, divide by the density of each component to convert to volume fraction first.
Errors become large when the fiber–matrix interface is debonded, when fibers are too short, when there are many voids, or in strongly nonlinear ranges of stress. Properties determined by local defects, such as fracture, also cannot be expressed by simple addition.
The condition that the whole material strains equally is an assumption that makes the stiffer phase carry more load than it really does. Conversely, the equal-stress equation assumes the maximum deformation of the softer phase and gives a lower bound, and the actual value lies between the two.
Young's modulus Crystallinity Glass transition temperature Thermosetting resin Viscoelasticity