The combination of material properties to be maximized in materials selection: E/ρ for a stiffness-limited tie, √E/ρ for a beam, ∛E/ρ for a plate, and σf/ρ for a strength-limited tie.
Japanese name: 材料選択の性能指数
Material index (performance index) appears in 6 practice questions across 1 topic, including Materials selection and design.
A performance index emerges when the free variables (such as section dimensions) are eliminated, using the constraints (specified stiffness or strength), from the equation expressing the objective (such as mass). For a tie of specified stiffness, the extension condition fixes the cross-sectional area as A ∝ 1/E, and the mass is m = ρAL ∝ ρ/E. So the larger E/ρ, the lighter the tie can be made. In this way, performance separates into the product of three parts, the functional requirements such as load and length, a factor for the shape of the cross section, and the material properties, and the material part alone is the performance index. Even after the shape and dimensions are optimized, the ranking of materials is determined by this index.
For a beam of square section whose dimensions can be varied freely, bending stiffness is proportional to the fourth power of the dimension, so mass is proportional to ρ/√E and the performance index is √E/ρ. For a plate of fixed width whose thickness alone can be varied, stiffness is proportional to the cube of the thickness, giving ∛E/ρ. When strength is the constraint, the indices become σf/ρ for a tie, ∛(σf²)/ρ for a beam and √σf/ρ for a plate. The deeper the root in the index, the relatively greater the influence of density, so low-density materials such as wood, magnesium alloys, CFRP and foams are favored for beams and plates in bending.
When the objective is material cost instead of mass, density multiplied by the price per kg Cm, ρCm, is used, and for a stiffness-limited tie, E/(ρCm) is maximized. When the objective is the energy or CO₂ emissions of production, the per-kg value is multiplied in the same way. In thermal design, indices combining thermal conductivity and thermal diffusivity are used. Note, however, that performance indices are derived on assumptions such as keeping the shape geometrically similar, so they may not be directly comparable when the cross-sectional shape differs greatly between materials or when there is a lower limit on dimensions.
Answer: A. σf/ρ
The condition for not yielding is F/A ≤ σf, so the required area is A = F/σf. The mass of the tie is m = ρAL = FL(ρ/σf), and since F and L are fixed, minimizing mass means choosing the material with the largest σf/ρ (specific strength). E/ρ (specific modulus) is the index to maximize for a tie whose constraint is not strength but the amount of extension, that is, stiffness. ∛(σf²)/ρ is the index for a beam in bending whose cross-section size is varied to avoid yielding; it comes from the bending stress varying inversely with the cube of the section dimension. The key point is that even for “light and strong,” the index changes with how the load is applied.
Answer: B. √E/ρ
The deflection of a beam is inversely proportional to the flexural rigidity EI, and the second moment of area of a square section of side b is I = b⁴/12. To meet the specified stiffness, Eb⁴ must be constant, so b² ∝ 1/√E. The mass of the beam is m = ρb²L ∝ ρ/√E, so minimizing mass means choosing the material with the largest √E/ρ. E/ρ is the index for a tie limited by stiffness, used when the cross-sectional area enters directly. ∛E/ρ is for the bending stiffness of a plate (panel) of fixed width whose thickness alone is varied, coming from the stiffness being proportional to the cube of the thickness. In bending, stiffness increases rapidly as the section grows, so low-density materials are favored, which is why wood and CFRP are strong performers as beams.
No. The performance index for a beam is √E/ρ, so even with the same E/ρ, the lower-density material is better as a beam. Steel and aluminum alloys have almost the same E/ρ, but as beams the aluminum alloy is lighter.
Yes. For example, the unit of √E/ρ is something like √GPa/(Mg/m³). Indices with the same units can be compared in magnitude, but the value of an index itself has no particular physical meaning.
Calculate the required dimensions and mass for each constraint, and for each material compare using the larger mass (the more restrictive constraint). Which constraint is active varies from material to material.
“Make the mass as small as possible” is a quantity to minimize, so it is the objective, and the performance index used to rank materials is derived from it.
In Ashby's method of materials selection, the design requir…When cost is the objective, the performance index becomes E/(Cmρ) (Cm is the price per kg), and CFRP, advantageous in weight, is far worse than steel in material cost.
A tie of fixed length and tensile stiffness is made of stee…So when fracture must be avoided, materials are selected not by strength alone but with K_IC/σy as the performance index, compared against the crack sizes that inspection can detect.
For a component used at a stress equal to its yield stress …Young's modulus CFRP (carbon-fiber-reinforced plastic) Ashby chart (materials selection chart) Yield before fracture Life cycle assessment Pareto optimality Specific stiffness Tramp elements