Modeling of Materials for Sports Equipment 1 introduction

1.2 PROPERTIES OF METALLIC ALLOYS
Table 1.1summarizes the range of mechanical properties of metallic systems commonly encountered in golf equipment and is typical of the data available in materials handbooks (e.g.,Boyer, Welsch, & Collings, 1994) or online sources such as www.matweb.com. In general, the density values of the alloys do not vary much, usually because the amounts of alloying elements present are limited. An exception is when the atoms are similar in size (and hence mass) as for Fe and Cr in stainless steels. Titanium-based systems are also an exception due to the greater solubility of elements in titanium, with the three alloys inTable 1.1showing a 10% variation in density. Of the other alloy systems, only the addition of Li to Al results in a decrease in density (up to 7% reduction), which also increases the Young’s modulus (by up to 10%). The beneficial improvements in density and modulus are accompanied by strength increases, but at the expense of reduced formability, toughness, and easier crack formation.
Stiffness for efficient energy transfer is important in many sporting applications but, asTable 1.1indicates, the variation in Young’s modulus (and, hence, also in shear and bulk modulus) for the alloys is limited; for
example, the Young’s modulus of the steels falls below 200 GPa for 316
stainless steel, which has a change in structure from bcc (ferrite) to fcc
(austenite) achieved by the addition of more than 25 at.% alloying element. As for density, the major exception is in the Ti-based system where the low temperature hcp (α) phase has a significantly higher modulus (125 GPa) than the higher temperature bcc (β) phase (80 GPa). Therefore using composition and heat treatment, the mix ofαandβcan be altered to control modulus. On the elastic loading of a Ti-based alloy containing a volume fraction, the interfaces between the two phases remain intact, the overall stress, σ,is the same in both phases while the strain, ε, is partitioned between bothphases so that the overall strain is the sum of that inαand inβ.:The Young’s modulus of the alloy (Et) can be estimated from the phasebalance and the properties (in this case moduli) of the individual phases.The principle of additivity of properties based on a particular boundary condition (usually constant stress or strain) is the“rule of mixtures”and is widely used in materials science (particularly composites) for estimating properties and designing alloy structures.
Phase balance is also one of the principal strengthening mechanisms in metallic alloys and can contribute to the range of yield stress and tensile strength values in Table 1.1; for the titanium-based alloy examples, the strength of αis greater than that for βso that, as the modulus is increased through formation of greater amounts ofα, the strength levels also increase.
The same effects of phase balance are seen for mixtures of phases (ferrite, austenite, bainite, and martensite) in steels with reduced options for shaping and reduced toughness. The rule of mixturesbased approaches are used in these cases, with a number of empirically determined equations relating mechanical properties to microstructural parameters in the literature (Llewellyn, 1992). Phase balance is only one of the five main strengthening mechanisms active in most metallic alloys. The others are: 1. Solid solution strengthening: The substitution of solute atoms for solvent in the crystalline matrix results in lattice strains which increase the yield stress of the alloy. Strength levels depend on the amount of the element in solution (Xi) and the mismatch in atom size between solute and solvent (ei ) represented as strengthening coefficients, Ki. The strengthening (Δτss) due to increased solute content of an alloy is estimated fromEq. (1.4):Δτss 5XiKiXninB1 (1.4)
2. Precipitation strengthening: As solute levels rise, the solubility product is exceeded (at lower solute levels as mismatch increases) and fine secondary phases precipitate, (Fig. 1.1). These either introduce elastic strains or block slip paths in the matrix, both of which increase strength. The strength level increases with increasing volume fractions
(Vf) of smaller precipitates (radius r) with the strengthening increment (Δτppt) being given by:Δτppt5Gb2π3 Vf12r(1.5)
whereGis the shear modulus and bis the Burgers vector.
3. Reduced grain size: Grain boundaries (Fig. 1.1) can also act to block slip paths so that yield stress strengthening (Δτgb) is inversely proportionalto the square root of the grain size:Δτgb5kyd212(1.6) whereky is the HallPetch parameter.
4. Work hardening (cold work): Plastic deformation increases the number of dislocations present, which impedes the progress of other dislocations and so raises the strength (although at the expense of toughness and ductility). Strengthening (Δτd) by this effect is given by:Δτd5α1Gb ρdp(1.7)
wherea1is the constant and ρd is the number of dislocations/unit area.
In real alloy systems, a number of strengthening mechanisms operate,
although one may dominate. The strength level achievable and suitable processing routes for that alloy (which affect the shapes that may be achieved) will depend on the operating strengthening mechanisms which, for simple systems, can be selected on the basis of the phase diagrams inFig. 1.2. Fig. 1.2shows, schematically, the phase diagrams for a binary eutectic alloy, which exhibits three classes of alloy, namely: (A) solid solution strengthening; (B) precipitation hardening; and (C) phase mbalance. For alloys with a composition similar to A, the operative strengthening mechanisms are solid solution strengthening, grain size refinement, and work hardening; this class of alloy would require shaping by cold working, for example, drawing, rolling, or forging at temperatures below one-third of the absolute melting temperature. Alloys of this type, for example, 316 stainless steel and 5xxx series AlMg alloys, are better suited to sheet and wire applications although, from Table 1.1, their overall strengths are limited. Higher strength values are achieved by strains or block slip paths in the matrix, both of which increase strength. The strength level increases with increasing volume fractions(Vf) of smaller precipitates (radius r) with the strengthening increment (Δτppt) being given by:Δτppt5
whereGis the shear modulus and bis the Burgers vector.
3. Reduced grain size: Grain boundaries (Fig. 1.1) can also act to block slippaths so that yield stress strengthening (Δτgb ) is inversely proportional to the square root of the grain size: Δτgb5kyd whereky is the HallPetch parameter.
4. Work hardening (cold work): Plastic deformation increases the number of dislocations present, which impedes the progress of other dislocations and so raises the strength (although at the expense of toughness and ductility). Strengthening (Δτd ) by this effect is given by:Δτd5α1Gb wherea1is the constant and ρd is the number of dislocations/unit area. In real alloy systems, a number of strengthening mechanisms operate, although one may dominate. The strength level achievable and suitable processing routes for that alloy (which affect the shapes that may be achieved) will depend on the operating strengthening mechanisms which, for simple systems, can be selected on the basis of the phase diagrams inFig. 1.2. Fig. 1.2shows schematically, the phase diagrams for a binary eutectic alloy, which exhibits three classes of alloy, namely: (A) solid solution strengthening; (B) precipitation hardening; and (C) phase balance. For alloys with a composition similar to A, the operative strengthening mechanisms are solid solution strengthening, grain size refinement, and work hardening; this class of alloy would require shaping by cold working, for example, drawing, rolling, or forging at temperatures below one-third of the absolute melting temperature. Alloys of this type, for example, 316 stainless steel and 5xxx series AlMg alloys, are better suited to sheet and wire applications although, from Table 1.1, their overall strengths are limited. Higher strength values are achieved byAfter quenching, there is a strong chemical driving force for the excess solute to precipitate, but diffusion is too slow at room temperature for this to happen (except for certain Al-based alloys and, even then, it is still slow). Hence, the alloys are aged or tempered in the two-phase region (120°C175°C for Al-based alloys, 400°C550°C for metastableβ-Ti alloys, and 350°C650°C for steels). These temperatures are much lower than the dissolution temperature so that the chemical driving force (ΔGch) remains high, resulting in a large number of fine and finely spaced precipitates. The number and size of the precipitates depend on their nucleation (Iv ) and growth (Y) rates, which are both time (t) and temperature (T) dependent:Iv5vDC0exp2ΔGkBTexp2ΔGmRT wherevDis the Debye frequency, C0 is the number of nucleation sites per unit volume,ΔG is the activation barrier for nucleation (decreases
with decreasing temperature),ΔGmis the activation barrier for diffusion (increases slightly on decreasing temperature), Dis the diffusivity, Ris the universal gas constant, andkBis the Boltzmann’s constant.
During the initial stages of aging/tempering, nucleation dominates so that the number of precipitates increases causing an increase in strength, but, in the later stages, growth dominates and the precipitates increase in size (reducing in number). This causes a loss in strength and is known as overaging. The effective use of high-strength (HS) metallic alloys requires optimization of this heat treatment through all stages of processing (including machining and joining, see Section 1.4) and in-service. As most sports equipment is used at ambient temperatures, overaging in service is usually only a problem in motorsport applications. Eutectic alloys (C inFig. 1.2) develop at least a two-phase structure directly from the liquid with the mix of phases developing strength (as for αandβphases of titanium above). The eutectics do not readily dissolve without liquation and so these alloys are mostly used in the as-cast state with higher cooling rates during solidification refining the eutectic to increase strength levels, although these are not usually as high as wrought age-hardened alloys