MATERIALS SELECTION IN MECHANICAL DESIGN, Lecture notes of Design

Download MATERIALS SELECTION IN MECHANICAL DESIGN and more Lecture notes Design in PDF only on Docsity! Troisieme Conference Europeenne sur les Materiaux et les Procedes Avances Euromat ‘93, Paris, June 8-10 1993 MATERIALS SELECTION IN MECHANICAL DESIGN M.F. Ashby and D. Cebon Engineering Department, Trumpington Street, Cambridge CB2 1PZ, UK ABSTRACT A novel materials–selection procedure has been developed and implemented in software. The procedure makes use of Materials Selection Charts: a new way of displaying material property data; and performance indices: combinations of material properties which govern performance. Optimisation methods are employed for simultaneous selection of both material and shape. INTRODUCTION The performance of an engineering component is limited by the properties of the material of which it is made, and by the shapes to which this material can be formed. Under some circumstances a material can be selected satisfactorily by specifying ranges for individual properties. More often, however, performance depends on a combination of properties, and then the best material is selected by maximising one or more ‘performance indices’. An example is the specific stiffness E/ρ (E is Young’s modulus and ρ is the density). Performance indices are governed by the design objectives. One is derived later in this paper and many others are tabulated elsewhere [1, 2]. Component shape is also an important consideration. Hollow tubular beams are lighter than solid ones for the same bending stiffness and I–section beams may be better still. Information about section shape can be included in the performance index to enable simultaneous selection of material and shape. THE PROCEDURE Performance Indices A performance index is a group of material properties which governs some aspect of the performance of a component [1, 2]. They are derived from simple models of the function of the component, as illustrated by the following example. A material is required for a light, stiff beam. The aim is to achieve a specified bending stiffness at minimum weight. The beam has a length L and a square, solid, cross–section as shown in Figure 1a. The mass of the beam is m AL= ρ , (1) where A is the area of the cross–section and ρ is the density of the material of which the beam is made. The stiffness S of a simply–supported beam with modulus E, second moment of area I, central load F, and central deflection δ, is S F C E I L = = δ 1 3 . (2) with C1 = 48 for 3-point bending. Other supports, or other distributions of load, change C1, but nothing else. Assume that the beam has a square section, of side b. The second moment of area is I b A= =4 212 12 (3) The stiffness S and the length L are constrained by the design. The area A is a ‘free’ variable that we wish to choose so as to minimise the mass, while meeting the constraints. Fig. 1 (a) A square–section beam loaded in bending (b) A beam of more complex cross section. Substituting for I in equation (2) and eliminating A between this and (1) gives m S C L E =     ( )     12 1 1 2 5 2 1 2 ρ (4) The mass of the beam can be minimised (and performance maximised) by seeking the material with the largest value of the performance index M E 1 1 2 = ρ (5) The same performance index holds for square–section beams with any value of the design stiffness S, any boundary conditions and distributions of load (defined by C1), and any length L. The cross–section shape of the beam (like the I–section shown in Figure 1b) can be included in the performance index by introducing a dimensionless shape factor φ, defined [3] by φ π= 4 2 I A (6) The value of φ measures the bending efficiency of the section shape. For the solid section of Figure 1(a), φ ≅ 1; that for the I-section of Figure 1(b) is about 5. Real I-sections have efficiencies, φ , as high as 40. The maximum value of φ is limited by manufacturing constraints or by local buckling of the component, and, for this reason, it can be considered to be a material property. Shape factors can also be defined for design against yield or fracture, and for shafts as well as beams. Using equation (6) in place of equation (3) to eliminate A in equation (1) gives the new index: M E 2 1 2 = ( )φ ρ (7) For a constant shape ( φ constant) the criterion reduces to the earlier one; the best selection is then the material with the largest value of M1 (equation (5)). In comparing materials with different shapes, the best choice is that with the greatest value of M2 (equation (7)). Fig. 3 An oar, showing the components and the method of measuring the stiffness. Figure 3 shows an oar: a blade or "spoon" is bonded to a shaft or "loom" which carries a sleeve and collar to give positive location in the rowlock. The lower part of the figure shows how the oar stiffness is measured: a 10 kg weight is hung on the oar 2.05 m from the collar and the deflection at this point is measured. A soft oar will deflect nearly 50 mm; a hard one only 30. A rower, when ordering an oar, specifies how hard it should be [5]. The oar must also be light; extra weight increases the wetted area of the hull and the drag that goes with it. So there we have it: an oar is a beam of specified stiffness and minimum weight. The performance index we want was derived in earlier; it is M1 = E 1/2 ρ . What materials make good oars? Figure 2 shows the appropriate chart, with a selection line for the index placed on it. It identifies three classes of material: woods, carbon- and glass-fibre reinforced polymers and certain ceramics (Table 1). Ceramics are brittle; they have low values of toughness; if you dropped a ceramic oar, it would probably shatter. This can be analysed, leading to further performance indices [1], but there is insufficient space to do so here. We simply note that ceramics are eliminated because they are brittle and expensive. The recommendation is clear. Make your oars out of wood or - better - out of CFRP. TABLE 1 MATERIALS FOR OARS MATERIAL M (GPa)1/2/(Mg/m3) COMMENT Woods 5-8 Cheap, traditional, but not easily controlled. CFRP 4-8 As good as wood, more control of properties. GFRP 3.5-5.5 Cheaper than CFRP but lower M. Ceramics 4-8 Good M but brittle and expensive Of what, in reality, are oars made? Racing oars and sculls are made either of wood or of a high performance composite: carbon-fibre reinforced epoxy, CFRP. Wooden oars are made today, as they were 100 years ago, by handcraftsmen who use Sitka spruce from the northern US or Canada, the further north the better because the short growing season gives a finer grain. A spruce oar weighs between 4 and 4.3 kg, and costs (in 1993) about £150 or $250. Composite blades are a little lighter than wood, for the same stiffness. The component parts are fabricated from a mixture of carbon and glass fibres in an epoxy matrix, assembled and glued. The advantage of composites lies partly in the saving of weight (typical weight: 3.9 kg) and partly in the greater control of performance: the shaft is moulded to give the stiffness specified by the purchaser. At a price, of course: a CFRP oar costs about £300 ($450). Materials for Precision Instruments The precision of a measuring device, like a sub-micrometer displacement gauge, is limited by its stiffness, and by the dimensional change caused by temperature gradients. Compensation for elastic deflection can be arranged; and corrections to cope with thermal expansion are possible too - provided the device is at a uniform temperature. Thermal gradients are the real problem: they cause a change of shape - that is, distortion - of the device for which compensation is not possible. Sensitivity to vibration is also a problem: natural excitation introduces noise into the measurement. So - in precision instrument design - it is permissible to allow expansion, provided distortion does not occur [6]. Elastic deflection is allowed, provided natural vibration frequencies are high. What, then, are good materials for precision devices? Figure 4. A precision instrument. It consists of a force loop, an actuator and a sensor Figure 4 shows, schematically, such a device: it consists of a force loop, an actuator and a sensor; we aim to choose a material for the force loop. It will, in general, support heat sources: electrical components which generate heat. The relevant performance index is found by considering the simple case of one-dimensional heat flow through a rod insulated except at its ends, one of which is at ambient and the other connected to the heat source. In the steady state, Fourier’s law is q dT dx = −λ (8) where q is heat input per unit area, λ is the thermal conductivity and dT dx is the resulting temperature gradient. The strain is related to temperature by ε α= −( )T T0 (9) where α is the thermal conductivity and T0 is ambient temperature, from which d dx dT dx q ε α α λ = =     (10) Thus for a given geometry and heat flow, the distortion d dxε is minimised by selecting materials with large values of the index M3 = λ α Figure 5. A Chart of thermal conductivity, λ , and expansion coefficient, α, allowing selection of materials for the force loop of precision instruments. The other problem is vibration. The sensitivity to external excitation is minimised by making the natural frequencies of the device as high as possible. The flexural vibrations have the lowest frequencies, they are proportional, once again, to M E 1 1 2 = ρ A high value of this index will minimise the problem. Finally, of course, the device must not cost too much.