Mechanical Engineering Design · Part 1 — Basics

Materials

How material strength and stiffness are measured and described statistically, how cold work, hardness, impact, and temperature change them, and how processing routes, alloying, and systematic selection charts guide the choice of material for a part.

Learning objectives

Interpret engineering and true stress–strain data without confusing stiffness, yield strength, and ultimate strength.
Select equations whose assumptions match elastic, plastic, impact, thermal, and statistical behavior.
Quantify cold work, resilience, specific properties, and material-selection indices with consistent units.
Connect casting, forming, heat treatment, and inspection choices to geometry, defects, cost, and reliability.
Rank candidate materials using transparent constraints and weighted objectives rather than a single property.
Identify when handbook values are insufficient and certified lot data, standards, supplier limits, or tests are required.

2–1Material Strength and Stiffness

The Standard Tensile Test

Most design strengths come from the standard tensile test: a specimen of recorded original diameter \(d_0\) and gauge length \(l_0\) is pulled slowly while load \(P\) and extension are recorded. Load converts to stress through the original area \(A_0 = \pi d_0^2/4\), and extension converts to normal strain — the results are plotted as a stress–strain diagram.

Engineering Stress

\[ \sigma = \frac{P}{A_0} \]Eq. (2–1)

Engineering (Normal) Strain

\[ \epsilon = \frac{l - l_0}{l_0} \]Eq. (2–2)

FIG. 2.1Engineering stress–strain diagram of a ductile material: proportional/elastic limits (pl, el), offset yield (y), ultimate (u), and fracture (f). (Original diagram, redrawn.)

Key Points on the Curve

Proportional limit (pl) — where the curve first departs from a straight line; unloading from here leaves no permanent set. Elastic limit (el) — beyond it, deformation is plastic and a permanent set remains. Yield point / yield strength \(S_y\) — where strain grows rapidly without added stress; for materials with no obvious yield point, \(S_y\) is defined by the offset method: a line of slope \(E\) drawn from a stated permanent set, usually 0.2% strain. Ultimate (tensile) strength \(S_u\) (or \(S_{ut}\)) — the maximum stress on the diagram. Ductile materials neck and fall to a lower fracture point \(f\); brittle materials (some cast irons, high-strength steels) fracture while the trace is still rising, so \(u\) and \(f\) coincide.

Hooke's Law (linear elastic range)

\[ \sigma = E\,\epsilon \]Eq. (2–3)

Young's Modulus — Stiffness

\(E\), the slope of the linear part of the curve, measures material stiffness and has stress units. For steel \(E \approx 30\) Mpsi (207 GPa) regardless of heat treatment, carbon content, or alloying; stainless steel is about 27.5 Mpsi (190 GPa). Stiffness, unlike strength, cannot be changed by processing the same alloy family.

Engineering vs. True Stress and Strain

Engineering values use the original area and length, so they understate reality once the section shrinks. Beyond the ultimate point a ductile specimen "necks" — the local area drops sharply, so the true stress there is far above the engineering stress. True (logarithmic) strain sums incremental elongations over the current length. On a true stress–strain diagram the stress rises continuously all the way to fracture, so the true fracture stress exceeds the true ultimate stress — opposite to the engineering picture.

True (Logarithmic) Strain

\[ \varepsilon = \int_{l_0}^{l}\frac{dl}{l} = \ln\frac{l}{l_0} \]Eq. (2–4)

Compression and Torsion Tests

Compression tests use different specimen geometry (buckling, uneven stress distribution, and post-yield bulging complicate them). For most ductile metals compressive and tensile strengths are about equal; where they differ substantially (cast irons), both are reported — \(S_{ut}\) and \(S_{uc}\) (the latter as a positive number). Torsion tests twist solid round bars and record torque vs. twist angle; shear stress varies linearly from zero at the center to a maximum at the outer radius.

Maximum Shear Stress from the Twist Angle

\[ \tau_{\max} = \frac{G\,r}{l_0}\,\theta \]Eq. (2–5)

Maximum Shear Stress from the Torque

\[ \tau_{\max} = \frac{T\,r}{J}, \qquad J = \tfrac{1}{2}\pi r^4 \]Eq. (2–6)

Modulus of Rupture (torsion)

\[ S_{su} = \frac{T_u\, r}{J} \]Eq. (2–7)

Modulus of Rupture ≠ Ultimate Torsional Strength

\(G\) is the shear modulus (modulus of rigidity), found from the linear part of the torque–twist diagram along with the torsional yield strength \(S_{sy}\). At the peak torque \(T_u\) the outer region of the bar is already plastic and the stress distribution is no longer linear — so Eq. (2–7) defines a modulus of rupture, not a true ultimate shear strength.

Resilience and Toughness

Because the stress–strain diagram couples load with deflection, its area measures absorbed energy per unit volume. Resilience is the capacity to absorb energy elastically (no permanent set); toughness is the capacity to absorb energy without fracture. Units are energy per volume (lbf·in/in³ or J/m³ — numerically psi or Pa). For two materials of equal yield strength, the less stiff one (lower \(E\)) is the more resilient. These definitions assume slow strain rates; for fast loading see impact properties (Sec. 2–5).

Modulus of Resilience

\[ u_R \approx \int_0^{\epsilon_y} \sigma \, d\epsilon \]Eq. (2–8)

Resilience for a Linear Curve to Yield

\[ u_R \approx \tfrac{1}{2} S_y \epsilon_y = \frac{S_y^{\,2}}{2E} \]Eq. (2–9)

Modulus of Toughness

\[ u_T = \int_0^{\epsilon_f} \sigma \, d\epsilon \]Eq. (2–10)

Toughness — Rough Approximation

\[ u_T \approx \left(\frac{S_y + S_{ut}}{2}\right) \epsilon_f \]Eq. (2–11)

2–2The Statistical Significance of Material Properties

Strength Is a Distribution, Not a Number

A single tensile test describes one specimen. A thousand nominally identical specimens would return strengths spread between a minimum and a maximum — so a material "property" is inherently distributional and statistical, described by a mean, a standard deviation, and a distribution type. Tables that print a single value may be showing the mean, the minimum, or some percentile: always read the footnotes, and treat unqualified single-entry tables with suspicion. When ordering property tests, ask for enough data to identify the statistical parameters and the distribution itself.

EXAMPLE 1 Interpreting a tensile-test data point

A tensile coupon has original diameter \(d_0=10.00\ \text{mm}\) and gauge length \(l_0=50.00\ \text{mm}\). At load \(P=15.0\ \text{kN}\), its gauge length is \(50.060\ \text{mm}\). Find engineering stress, engineering strain, and secant modulus, then assess consistency with a supplied elastic modulus \(E=160\ \text{GPa}\).

Recognition strategy

Original dimensions are explicitly supplied, so this is an engineering stress–strain calculation. The modulus check applies only if the measured point is within the linear elastic range.

Given information

\(d_0=10.00\ \text{mm}\), \(l_0=50.00\ \text{mm}\), \(l=50.060\ \text{mm}\), \(P=15.0\ \text{kN}\), and supplied \(E=160\ \text{GPa}\).

Model and interpretation

The coupon is under quasi-static uniaxial tension, tension is positive, and deformation is uniform over the gauge length. Grip compliance, necking, temperature change, and area reduction are neglected.

Step-by-step solution

1. Establish original area. Engineering stress uses the undeformed section: \[A_0=\frac{\pi d_0^2}{4} =\frac{\pi(10.00\ \text{mm})^2}{4}=78.54\ \text{mm}^2.\]

2. Calculate engineering stress. \[\sigma=\frac{P}{A_0}=\frac{15000\ \text{N}}{78.54\ \text{mm}^2} =191.0\ \text{MPa}.\]

3. Calculate engineering strain. \[\epsilon=\frac{l-l_0}{l_0}=\frac{50.060-50.00}{50.00}=0.001200.\]

4. Infer the secant modulus. \[E_{\mathrm{sec}}=\frac{\sigma}{\epsilon} =159.2\ \text{GPa},\] which differs from the supplied \(160\ \text{GPa}\) by \(0.5\%\).

Verification

Units: Newtons per square millimetre give megapascals; strain is a length ratio, and stress divided by strain retains stress units.

Signs and direction: Load and elongation are tensile and positive under the stated convention, so both engineering stress and strain are positive.

Plausibility: A strain of \(0.0012\) multiplied by \(160\ \text{GPa}\) predicts \(192\ \text{MPa}\), independently matching the measured stress.

Final result: \(\boxed{\sigma=191.0\ \text{MPa},\quad \epsilon=0.001200,\quad E_{\mathrm{sec}}=159.2\ \text{GPa}}\)

Engineering interpretation

The point is consistent with the supplied elastic modulus within ordinary measurement scatter. It does not establish yield or ultimate strength; those require the full test curve and the governing test standard.

2–3Strength and Cold Work

Cold Working and Strain Hardening

Cold working is plastic straining below the recrystallization temperature. Stress a material past yield to a point \(i\) in the plastic region and unload: a permanent strain \(\epsilon_p\) remains, and reloading travels a straight line parallel to the original elastic line, with elastic component \(\epsilon_e = \sigma_i/E\) and total \(\epsilon = \epsilon_p + \epsilon_e\). The material now yields at a higher stress, is less ductile, and is said to be strain-hardened; pushed far enough it becomes brittle.

FIG. 2.2Unloading from point i in the plastic region leaves permanent strain ε_p; reloading is elastic over ε_e along a line parallel to the original elastic slope. (Original diagram, redrawn.)

Reduction in Area (ductility measure)

\[ R = \frac{A_0 - A_f}{A_0} = 1 - \frac{A_f}{A_0} \]Eq. (2–12)

Ductility

The reduction in area at fracture, usually quoted in percent, measures ductility — the ability to absorb overloads and to be cold-worked. Bending, drawing, heading, and stretch forming all demand ductile stock.

Cold-Work Factor

\[ W = \frac{A_0 - A_i'}{A_0} \approx \frac{A_0 - A_i}{A_0} \]Eq. (2–13)

Area After Cold Work

\[ A_i' = A_0\,(1 - W) \]Eq. (2–14)

True Stress–Strain Power Law (plastic region)

\[ \sigma = \sigma_0\, \varepsilon^{\,m} \]Eq. (2–15)

Strain-Strengthening Exponent at the Ultimate Point

\[ m = \varepsilon_u \]Eq. (2–16)

True Strain from Area Ratio (constant volume)

\[ \varepsilon = \ln\frac{l}{l_0} = \ln\frac{A_0}{A} \]Eq. (2–17)

The Power-Law Model

The plastic region of the true stress–true strain curve fits \(\sigma = \sigma_0 \varepsilon^m\), with strength coefficient \(\sigma_0\) and strain-strengthening exponent \(m\); when the load–deformation curve has a stationary point, \(m\) equals the true strain at ultimate, \(\varepsilon_u\). Because necking makes gauge-length strain nonuniform, true strain is better computed from the neck area via the constant-volume relation \(Al = A_0 l_0\).

New Yield Strength After Cold Work (\(\varepsilon_i \le \varepsilon_u\))

\[ S_y' = \frac{P_i}{A_i'} = \sigma_0\, \varepsilon_i^{\,m} \]Eq. (2–18)

New Ultimate Strength After Cold Work (\(\varepsilon_i \le \varepsilon_u\))

\[ S_u' = \frac{S_u}{1 - W} \]Eq. (2–19)

Heavily Cold-Worked Case (\(\varepsilon_i > \varepsilon_u\))

\[ S_u' \approx S_y' \approx \sigma_0\, \varepsilon_i^{\,m} \]Eq. (2–20)

Why the New Strengths Matter

A bar strain-strengthened in tension carries the same ultimate load as before — the gain is in the strength values: fatigue strength correlates with local ultimate strength, so fatigue performance improves, and the raised yield strength widens the sustainable elastic load range.

EXAMPLE 2 Planning a cold-drawing area reduction

A round bar is cold-drawn from \(d_0=14.0\ \text{mm}\) to \(d_f=12.0\ \text{mm}\). For a supplied plastic-flow fit \(\sigma=850\epsilon^{0.22}\ \text{MPa}\), determine percent cold work, true plastic strain, and the model-predicted flow stress after drawing. State whether this is a stiffness change.

Recognition strategy

Diameter reduction determines area reduction and, under plastic incompressibility, true strain. The power law predicts local flow stress; it does not change the elastic modulus.

Given information

\(d_0=14.0\ \text{mm}\), \(d_f=12.0\ \text{mm}\), strength coefficient \(850\ \text{MPa}\), exponent \(m=0.22\), room-temperature cold drawing.

Model and interpretation

Use circular, uniform sections and constant plastic volume. The positive strain measure is accumulated tensile true strain; die friction, redundant work, residual stress, and anisotropy are excluded.

Step-by-step solution

1. Express the area ratio. For circular sections, \(A_f/A_0=(d_f/d_0)^2=(12.0/14.0)^2=0.734694\).

2. Compute cold work. \[W=1-\frac{A_f}{A_0}=1-0.734694=0.2653,\] so the nominal area reduction is \(26.53\%\).

3. Compute true strain. Constant volume gives \[\epsilon=\ln\frac{A_0}{A_f}=\ln\frac{1}{0.734694}=0.3083.\]

4. Apply the supplied fit. \[\sigma=850(0.3083)^{0.22}=656.3\ \text{MPa}.\] This is a model flow stress, not Young's modulus.

Verification

Units: Area ratios, cold work, and logarithmic strain are dimensionless; multiplying by the \(850\ \text{MPa}\) coefficient returns megapascals.

Signs and direction: Because final area is smaller, \(A_0/A_f>1\), giving positive accumulated true strain and positive cold work.

Plausibility: A \(26.5\%\) area reduction should give \(\ln[1/(1-0.265)]\approx0.31\); the flow stress remains below the \(850\ \text{MPa}\) coefficient.

Final result: \(\boxed{W=26.53\%,\quad \epsilon=0.3083,\quad \sigma_{\mathrm{flow}}=656.3\ \text{MPa}}\)

Engineering interpretation

Cold work raises flow and yield resistance while reducing remaining ductility, but it does not materially raise Young's modulus. Drawing-force, die-life, residual-stress, and intermediate-anneal decisions require process data and lot-specific material limits.

2–4Hardness

Hardness and the Two Main Tests

Hardness is a material's resistance to penetration by a pointed tool. Rockwell tests (ASTM E-18) are fast and reproducible, reading directly from a dial; scales A, B, C use specified indenters and loads — \(R_B\) is a 100-kg load on a 1/16-in ball, \(R_C\) a 150-kg load on a diamond cone. Rockwell numbers are relative: a value has meaning only against another number on the same scale. Brinell hardness \(H_B\) is the applied load divided by the spherical indentation area, so its units are stress units (rarely written). Both tests are usually nondestructive and correlate directly with ultimate strength — parts could even be strength-checked one by one in production.

Ultimate Strength from Brinell Hardness — Steels (rough)

\[ S_u \approx \begin{cases} 0.5\,H_B & \text{kpsi} \\ 3.4\,H_B & \text{MPa} \end{cases} \]Eq. (2–21)

Minimum Ultimate Strength — Cast Iron (ASTM data)

\[ S_u = \begin{cases} 0.23\,H_B - 12.5 & \text{kpsi} \\ 1.58\,H_B - 86 & \text{MPa} \end{cases} \]Eq. (2–22)

Using the Correlations

For many materials the minimum ultimate strength is roughly linear in Brinell hardness, with a proportionality constant that varies by material class and test load, and with wide data scatter — treat Eqs. (2–21) and (2–22) as rough estimates. SAE minimum-strength charts are more conservative than Eq. (2–22).

2–5Impact Properties

Impact Load

A force is an impact load when its application time is less than one-third of the part's lowest natural period of vibration; otherwise it is a static load.

Charpy / Izod Tests and the Ductile–Brittle Transition

The Charpy (common) and Izod (rare) notched-bar tests strike a standard specimen with a pendulum from a fixed height; the energy absorbed (the impact value) follows from the follow-through swing. Some materials show a ductile–brittle transition: below a narrow critical-temperature band fractures are brittle and shattering, above it tough and tearing, with the impact value climbing steeply through the band. The transition temperature depends on both the material and the notch geometry — so designers should not lean too hard on notched-bar numbers; notch sensitivity, fracture toughness, and fracture mechanics (Chs. 5–6) are sounder tools.

Strain-Rate Effects

Stress–strain diagrams are taken at roughly 0.001 s⁻¹ or slower. Raising the strain rate raises both strengths — at very high rates the yield strength approaches the ultimate strength — while total elongation (ductility) changes little. A mild steel can therefore behave elastically over nearly its whole strength range under impact.

2–6Temperature Effects

Strength vs. Operating Temperature

For steels, ultimate strength holds nearly constant as temperature rises, then falls off rapidly past a threshold; yield strength decreases continuously with temperature, while ductility grows. Below the ductile–brittle transition temperature, brittle fracture becomes a real failure possibility (Ch. 5). Welding heat (and clamping during welding) also alters properties and leaves residual stresses — hardness tests can reveal strength changes but not residual stresses.

Creep

Creep is continuous deformation under constant load at elevated temperature — it occurs even at stresses below the short-time yield strength measured at that temperature. The classic constant-load, constant-temperature creep curve has three stages: (1) elastic-plus-plastic deformation with decreasing creep rate (strain hardening), (2) a constant minimum creep rate (annealing effect), and (3) accelerating creep with area reduction and rising true stress, ending in fracture.

FIG. 2.3Typical creep–time curve at constant load and temperature, showing the three characteristic stages. (Original diagram, redrawn.)

2–7Numbering Systems

UNS — Unified Numbering System

SAE pioneered steel numbering; AISI followed; and in 1975 SAE published the UNS, which cross-references the other specification systems. A letter prefix identifies the material family: G carbon and alloy steels, A aluminum alloys, C copper-base alloys, S stainless steels.

Reading a Steel Designation

For G-prefix steels, the first two digits give the alloy composition class — e.g., G10 plain carbon, G11 free-cutting (resulfurized), G13 manganese, G41 chromium-molybdenum, G43 and G86/G87 nickel-chromium-molybdenum, G61 chromium-vanadium, G92 manganese-silicon. The next two digits are the approximate carbon content in hundredths of a percent: G10400 is plain carbon steel with nominally 0.40% C (0.37–0.44%). The fifth digit handles special cases (the old AISI 52100 chromium bearing steel is UNS G52986).

Stainless, Aluminum, and Cast-Iron Designations

Stainless (S-prefix) reuses the old three-digit AISI numbers: leading 2 = chromium-nickel-manganese, 3 = chromium-nickel, 4 = straight chromium; S30200 is the familiar "18-8" (18% Cr, 8% Ni). Aluminum (A-prefix): the first digit gives processing (A9 wrought, A0 cast); the second the main alloy group — 1 = ≥99% pure, 2 = copper, 3 = manganese, 4 = silicon, 5 = magnesium, 6 = magnesium-silicon, 7 = zinc. Cast iron uses the ASTM class system based on tensile strength: class 30 has a minimum \(S_{ut}\) of 30 kpsi, though typical values run higher — always state whether the minimum or typical value entered your factor-of-safety calculation.

2–8Sand Casting

Sand Casting

A basic low-cost process suited to large quantities with virtually no limit on size, shape, or complexity. Molten metal is poured into a sand mold formed around a wood or metal pattern; sand cores create holes and recesses. The designer should mentally place pattern and casting in the mold to anticipate core setting, pattern removal, draft, and solidification. Steel is the hardest metal to cast (highest pouring temperature). Design rules: keep section thickness uniform; make necessary section changes gradual; use generous fillets between adjoining sections; split a complicated part into simple castings joined by fasteners or welds. Typical minimum wall: about 5 mm for steel, gray iron, brass, bronze, and aluminum.

2–9Shell Molding

Shell Molding

A heated metal pattern contacts a dry sand–thermosetting-resin mix, fusing a 5–10-mm shell that is baked, stripped, and stored. Shell halves are clamped or pasted together, backed with a permeable material such as steel shot, and poured. The thin shell conducts heat away for rapid solidification, the resin bond burns out, and gases escape through the backing — yielding fine-grained, stress-free castings with smooth surfaces, minimal draft, and close tolerances. Sand-casting design rules generally apply.

2–10Investment Casting

Investment (Lost-Pattern) Casting

A wax or plastic pattern is invested in mold material, then melted out; the mold (plaster or ceramic, depending on the metal's melting point) is fired and poured hot. Because each pattern is consumed, mechanized pattern production is required. For larger quantities, permanent metal molds (die castings, centrifugal castings) give smooth, bright, accurate surfaces needing little or no machining.

2–11Powder-Metallurgy Process

Powder Metallurgy

Powders of one or more metals (and sometimes nonmetals) are blended, compacted in dies at high pressure, and sintered below the major constituent's melting point, uniting the particles into a strong part. Advantages: no scrap, no machining, low mass-production unit cost, exact composition control. Disadvantages: costly dies, lower physical properties, higher material cost, design limitations, limited material range. Typical products: oil-impregnated bearings, lamp filaments, carbide tool tips, permanent magnets — and some parts (e.g., surgical implants) that can be made no other way.

2–12Hot-Working Processes

Hot Working

Rolling, forging, hot extrusion, and hot pressing all deform metal above its recrystallization temperature. Hot rolling produces standard bar shapes (round, square, flat, hexagon, half-oval) and structural shapes (wide flange, channel, angle, tee, zee) in steel, aluminum, magnesium, and copper alloys. Tubing is rolled from strip with butt- or lap-welded seams, or made seamless by roll-piercing a heated rod over a mandrel. Extrusion forces a heated billet through a restricted orifice — most common for low-melting metals (Al, Cu, Mg, Pb, Sn, Zn). Forging by hammer, press, or machine refines the grain structure, raising strength and ductility; forgings beat castings on strength-per-weight and on accuracy, at the price of more expensive dies.

2–13Cold-Working Processes

Cold Working and Cold-Finished Bars

Cold working forms metal at (usually) room temperature, giving a bright finish, better accuracy, and less machining than hot work. Most cold-finished bars are made by cold drawing (often miscalled "cold-rolled"): descaled hot-rolled bars are pulled through a die reducing the size by roughly 1/32 to 1/16 in — no material is removed, the size is drawn down. Cold rolling now serves mainly wide flats and sheets; both processes affect properties identically. Cold work distorts (does not refine) the grain: yield strength rises sharply, ultimate strength and hardness rise, ductility falls.

Other Cold Operations

Heading upsets (gathers) metal to form screw and rivet heads. Roll threading squeezes a blank between serrated dies to form threads. Spinning works sheet around a rotating form into circular shapes. Stamping covers punch-press work: blanking, coining, forming, shallow drawing.

2–14The Heat Treatment of Steel

Heat Treatment

Time- and temperature-controlled processes that relieve residual stresses and/or modify hardness (strength), ductility, and toughness. The common operations: annealing, quenching, tempering, and case hardening.

Annealing and Normalizing

Full annealing heats the steel to ~100 °F above the critical temperature, holds until carbon dissolves and diffuses, then cools slowly (in the furnace) so the material transforms per the equilibrium diagram — softening it, restoring ductility, relieving residual stresses, and refining grain. Normalizing heats slightly higher and cools in still air: faster cooling leaves the steel harder than a full anneal and gives a coarser grain that machines more easily in low-carbon steels. It is often the final treatment.

Quenching — Martensite and Bainite

Hardness is governed by cooling rate. Austenite needs time to transform to pearlite, and the transformation ceases near 800 °F — so cooling fast enough to dodge it produces, below ~400 °F, martensite: a supersaturated solution of carbon in ferrite, the hardest and strongest form of steel. Rapid cooling to a hold between 400 and 800 °F instead yields bainite, a family of structures between pearlite and martensite — choosing the hold temperature dials in nearly any structure from coarse pearlite to fine martensite. Media: still air (mild — i.e., normalizing), oil (slow, avoids quench cracks), water (for carbon and medium-carbon low-alloy steels).

Tempering (Drawing)

Fully hardened steel is very hard, brittle, unstable, and full of residual stress. Reheating below the critical temperature for a set time, then air cooling, relieves stresses (stress relieving) and softens (tempering): carbon trapped in the martensite is released as carbide crystals, leaving tempered martensite — a superfine dispersion of iron carbide in fine-grained ferrite. The tempering temperature sets the hardness/toughness trade: as it rises, tensile and yield strength and hardness fall while elongation and reduction in area climb.

Case Hardening

Produces a hard, high-carbon outer case on a low-carbon steel while keeping a ductile, tough core, by enriching surface carbon from solid, liquid, or gaseous carburizing media for a stated time and temperature (which set case depth). The part may be quenched straight from carburizing and tempered, or double heat-treated to condition both case and core. Methods include pack and gas carburizing, nitriding, cyaniding, and — without adding carbon, on medium-carbon steels — induction and flame hardening.

Predicting Heat-Treated Properties

Quantitative prediction from the Jominy end-quench test uses the Crafts–Lamont addition method for plain carbon steels and the Grossman–Field multiplication method for low-alloy steels; both are detailed in standard machine-design handbooks, which can also predict the Jominy curve from a ladle analysis and grain size.

2–15Alloy Steels

What Makes an Alloy Steel

Plain carbon steel is already iron + carbon with traces of Mn, Si, S, and P; it becomes an alloy steel when other elements are added in quantities sufficient to modify its properties substantially — improving physical properties and widening the heat-treatment window.

Effects of the Common Alloying Elements

Chromium — forms very hard carbides yet leaves the steel more ductile than equal hardness from carbon alone; refines grain; net gain in both toughness and hardness. Nickel — dissolves in ferrite (no carbides/oxides), raising strength without losing ductility; improves case-hardened cores; often paired with chromium (Ni toughness + Cr wear resistance). Manganese — deoxidizer/desulfurizer in all steels; above 1% Mn (with low S) it is an alloying element that slows transformation enough to make oil quenching practical. Silicon — deoxidizer; with very low carbon gives magnetically soft, low-hysteresis material; mainly used with Mn, Cr, V to stabilize carbides. Molybdenum — adds hardness and toughness; strongly lowers the transformation point, enabling oil- and air-hardening; greatest hardening effect after carbon, with fine grain retained. Vanadium — strong carbide former and deoxidizer used in small amounts; fine grain, toughening, wide hardening range, resists softening on tempering — hence tool steels. Tungsten — keeps tools hard at red heat; fine dense structure; Mo-like effects but needed in larger amounts.

2–16Corrosion-Resistant Steels

Stainless Steels

Iron-base alloys with at least 12% chromium are stainless steels, resistant to many — not all — corrosive environments. Four types: ferritic chromium, austenitic chromium-nickel, martensitic, and precipitation-hardenable.

Ferritic Chromium Steels

12–27% Cr; corrosion resistance scales with chromium content (below 12% some resistance remains but rusting is possible). Quench-hardenability depends on Cr and C: very high-carbon grades harden up to ~18% Cr, low-carbon grades only to ~13% (a little nickel extends this to ~20%). Above 18% Cr they become hard to weld, and at very high Cr the hardness demands careful attention to service conditions. Since chromium is expensive, choose the lowest content the corrosive environment allows.

Austenitic Chromium-Nickel Steels

Retain the austenitic structure at room temperature, so they cannot be hardened by heat treatment — but cold working strengthens them greatly (and makes them magnetic and hard to machine). All are weldable, and they out-resist plain chromium grades in corrosion; adding more Cr for still better resistance requires more Ni to keep the structure austenitic.

2–17Casting Materials

Gray Cast Iron

The most widely used cast material: very cheap, easy to cast in quantity, easy to machine — but brittle and weak in tension. High carbon (>1.7%, usually >2%) plus high silicon yields pearlite, ferrite, and graphite; the graphite's thin evenly distributed flakes darken the fracture, hence "gray." Welding tends to crack it (careful preheat helps). Tensile strength runs 100–400 MPa (15–60 kpsi); compressive strength is 3–4× tensile; \(E\) varies widely, 75–150 GPa, and the stress–strain curve is not Hookean — the modulus falls steadily as stress rises. A mild anneal relieves cooling stresses and improves machinability.

Ductile (Nodular) Cast Iron

Made by adding MgFeSi to the melt, ductile iron carries its graphite as spheroids, combining malleable-iron ductility with gray-iron castability and machinability — in the as-cast state (a 1-h anneal and slow cool adds even more ductility). High modulus (~172 GPa / 25 Mpsi) with a genuinely linear elastic portion; compressive strength still exceeds tensile, but by less than in gray iron.

White and Malleable Cast Iron

White iron carries all carbon as cementite and pearlite (no graphite), made by low C–Si composition or by casting gray-iron composition against chills for rapid cooling: very brittle, hard to machine, extremely wear-resistant. Chills are often used locally to put a hard wear surface on an otherwise gray casting. Malleable iron is white iron annealed (up to 6 days for heavy castings) so the carbon frees as nodular "temper carbon": tensile strength can top 350 MPa (50 kpsi) with up to 18% elongation — versus barely 1% for gray iron — at higher cost due to the long anneal.

Alloy Cast Irons and Cast Steels

Nickel (≤5%) raises strength, density, wear resistance, and machinability; at 10–18% Ni an austenitic, heat- and corrosion-resistant structure forms. Chromium raises hardness and wear resistance and promotes white iron with chills; Ni + Cr together improve hardness and strength without hurting machinability. Molybdenum (≤1.25%) raises stiffness, hardness, tensile strength, and impact resistance. Cast steels suit complex high-strength parts cheaper than fabrication; their higher melting temperatures demand close attention to cores, sections, fillets, and cooling — but they accept the same alloying as wrought steels, can be heat-treated, and unlike cast irons can be welded.

2–18Nonferrous Metals

Aluminum

Prized for strength-to-weight, corrosion resistance, and thermal/electrical conductivity. Density ≈ 2770 kg/m³ vs. 7750 for steel; \(E = 71.7\) GPa (10.4 Mpsi) — about one-third the stiffness of steel — for pure metal and alloys alike. Pure Al is weak (~90 MPa tensile) but gains greatly from cold work and alloying (Cu, Si, Mn, Mg, Zn). It casts (sand, die, permanent mold; melts at 660 °C), extrudes, machines, press-works, and joins by soldering/brazing/welding with alloy-specific precautions. Corrosion protection comes from a spontaneous oxide film — anodizing grows an extra-heavy, color-controllable film electrolytically. Heat-treatable alloys dissolve an alloying element at temperature, quench, and then age-harden (duraluminum 2017 hardens in ~4 days at room temperature, so it is refrigerated or formed immediately after quench; slower-aging alloys are furnace-aged after forming). Non-heat-treatable alloys harden only by cold work; annealing erases either kind of hardening.

Magnesium and Titanium

Magnesium — lightest commercial metal (~1800 kg/m³: ⅔ of aluminum, ¼ of steel); modest absolute strength but competitive strength-to-weight; \(E = 45\) GPa (cold work, curiously, lowers it); loses yield strength at boiling-water temperatures; main markets aircraft and automotive. Titanium — strength like moderate-strength steel at half the weight; excellent corrosion resistance, low thermal conductivity, nonmagnetic, strong at high temperature; \(E = 114\) GPa (16.5 Mpsi), between aluminum and steel; used in aerospace, marine hardware, chemical processing, fluid handling, and implants — its drawbacks are cost and difficult machining.

Brasses (Copper–Zinc)

Copper + zinc = brass. 5–15% Zn (gilding 5%, commercial bronze 10%, red brass 15%): ductile, easy to cold work, corrosion-resistant, poor machining; red brass serves hot-water tubing (radiators, condensers). 20–36% Zn (low brass 20%, cartridge 30%, yellow 35%): cheaper, stronger, more machinable, but weaker corrosion resistance and prone to stress cracking; cartridge brass has the best ductility–strength combination (named for the draw-anneal-draw cartridge process). Lead additions (low-leaded, high-leaded/"engraver's", free-cutting brass) greatly improve machinability at the cost of cold-working and welding ability. Admiralty metal (28% Zn + 1% Sn) and aluminum brass (22% Zn + 2% Al) add saltwater and erosion resistance for power-plant and chemical service. 36–40% Zn (Muntz metal 40%, naval brass +0.75% Sn): less ductile, usually hot-worked or extruded.

Bronzes (Copper + Other Elements)

Silicon bronze (3% Si, 1% Mn) — mild-steel-level mechanical properties plus good corrosion resistance; hot/cold workable, machinable, weldable. Phosphor bronze (≤11% Sn + P) — outstanding fatigue and corrosion resistance, high tensile strength and energy absorption, wear-resistant: a classic spring material. Aluminum bronze (≤12% Al, heat-treatable) — stronger and more corrosion-resistant than brass; with up to 4% Fe gains high endurance limit, shock resistance, and wear resistance. Beryllium bronze (~2% Be, heat-treatable) — very corrosion-resistant, high strength/hardness/wear resistance; expensive, used for fatigue-loaded springs in corrosive service. Most copper alloys are also available cast.

2–19Plastics

Thermoplastics vs. Thermosets

A thermoplastic flows or molds when heated (or under pressure) and can be remolded repeatedly. A thermoset completes its polymerization in the hot mold and cannot be remolded.

Property Ranges and Selection Cautions

Common thermoplastics include the ABS group, acetals, acrylics, fluoroplastics, nylon, phenylene oxide, polycarbonate, polyester, polyimide, polyphenylene sulfide, polystyrene, polysulfone, and PVC; common thermosets include alkyds, allylics, the amino group, epoxies, phenolics, and silicones. Ultimate strengths span roughly 1.5–50 kpsi and moduli about 0.03–1.6 Mpsi, with enormous spreads in elongation, dimensional stability, and heat/chemical resistance — so published property tables are for orientation only, never for a final design decision. Cost, moldability, friction coefficient, weathering, impact strength, and the effect of fillers and reinforcements all enter; manufacturers' catalogs are the working reference.

2–20Composite Materials

Filler + Matrix

Composites combine two or more dissimilar materials that stay distinct at the macroscopic level (unlike alloys). Most engineering composites pair a reinforcement (filler) — continuous straight or woven fibers, short chopped fibers, or particulates — providing stiffness and strength, with a matrix (usually a plastic resin, sometimes metal) that binds the material and transfers load between discontinuous reinforcements.

Anisotropy and Laminates

Metals are isotropic — properties independent of position and direction, fully described by \(E\) and Poisson's ratio \(\nu\). A composite ply is directional: graphite fibers in epoxy give very high strength and stiffness along the fibers and very low values transverse to them. A single ply needs four constants — longitudinal modulus \(E_1\), transverse modulus \(E_2\), major Poisson's ratio \(\nu_{12}\), and shear modulus \(G_{12}\) — with the minor ratio from reciprocity, \(\nu_{21}/E_2 = \nu_{12}/E_1\). Structures are built as laminates of plies at chosen angles, which makes hand analysis impractical and computer tools standard. Payoff: strength-to-weight up to ~5× high-strength steels and stiffness-to-weight up to ~8× structural metals — hence the growth in automotive, marine, aircraft, and spacecraft use.

2–21Materials Selection

List-and-Rank Selection

Selection may be trivial (1020 steel by precedent) or as demanding as any design task. A basic method: list the design-relevant properties (strength, stiffness, cost, …), weight them by importance, rank all candidate materials under each property, take the top performers from each list, keep the materials common to every list, then grade and weight that short list.

Ashby Material-Selection Charts

Ashby's systematic method plots material families (metals, technical and nontechnical ceramics, glasses, polymers, elastomers, composites, foams, natural materials) on log–log property charts. A bar chart of \(E\) shows each material as a range; a "bubble" chart of \(E\) vs. density \(\rho\) shows family clusters and lets guideline lines of constant \(E^{\beta}/\rho\) be drawn. The ratio \(E/\rho\) is the specific modulus (specific stiffness), key when minimizing weight under a deflection, stiffness, or natural-frequency limit; on \(\beta = 1\) lines, \(c = (E/\rho)^{1/2}\) is also the speed of sound in the material. \(\beta\) depends on the load case: linear (\(\beta = 1\)) for axial loading, \(\beta = 1/2\) for bending. A strength–density chart works the same way for specific strength \(S/\rho\) (axial \(\beta = 1\), bending \(\beta = 2/3\)).

Performance Metric and the Material Index

The performance \(P\) of a structural element depends on functional requirements \(F\), geometry \(G\), and material \(M\). When the function separates, the material's contribution \(f_3(M)\) — the material efficiency coefficient — can be maximized or minimized on its own, independent of loading and geometry. Lines of increasing material index \(M\) march up and to the left on the \(E\)–\(\rho\) chart; the search region above the chosen line (further cut by hard limits such as \(E \ge 50\) GPa) contains the candidates.

Performance as a Function of F, G, M

\[ P = f(F,\, G,\, M) \]Eq. (2–23)

Separable Form

\[ P = f_1(F)\cdot f_2(G)\cdot f_3(M) \]Eq. (2–24)

Worked Illustration — a Light, Stiff Cantilever

Minimize the mass of an end-loaded cantilever of circular section with required stiffness \(k = F/\delta\). The end deflection is \(\delta = Fl^3/(3EI)\), so the stiffness and section relations below combine into a mass expression that cleanly separates into \(f_1(F) = 2\sqrt{\pi/3}\,k^{1/2}\) (function), \(f_2(G) = l^{5/2}\) (geometry), and \(f_3(M) = \rho/E^{1/2}\) (material). Minimizing mass means maximizing \(M = E^{1/2}/\rho\) — i.e., \(\beta = 1/2\) — which points to certain woods, composites, and ceramics; imposing an additional limit such as \(E \ge 50\) GPa then eliminates the woods. Real selections add environment, cost, availability, machinability, and process selection — the bookkeeping that selection software exists to manage.

Cantilever Stiffness

\[ k = \frac{F}{\delta} = \frac{3EI}{l^3} \]Eq. (2–25)

Second Moment of a Circular Section

\[ I = \frac{\pi D^4}{64} = \frac{A^2}{4\pi} \]Eq. (2–26)

Required Area for Stiffness k

\[ A = \left(\frac{4\pi k\, l^3}{3E}\right)^{1/2} \]Eq. (2–27)

Beam Mass

\[ m = A\, l\, \rho \]Eq. (2–28)

Mass in Separated Form

\[ m = 2\sqrt{\frac{\pi}{3}}\;\bigl(k^{1/2}\bigr)\bigl(l^{5/2}\bigr)\left(\frac{\rho}{E^{1/2}}\right) \]Eq. (2–29)

Material Efficiency Coefficient (minimize)

\[ f_3(M) = \frac{\rho}{E^{1/2}} \]Eq. (2–30)

Material Index (maximize)

\[ M = \frac{E^{1/2}}{\rho} \]Eq. (2–31)

Integrated material-selection application

Physical meaning

Material properties describe responses under specified test conditions, not an abstract ranking of “best” materials. Strength limits irreversible failure, stiffness controls elastic deformation, toughness measures energy to fracture, and processing determines the microstructure and defects actually present in the delivered part.

Problem-solving strategy

Recognize the governing failure and service limits: static or fatigue loading, deflection, temperature, impact, corrosion, wear, and life.
Translate each limit into a property requirement or material index, keeping local stress distinct from tabulated strength.
Screen by nonnegotiable constraints before ranking survivors by mass, cost, manufacturability, availability, and environmental burden.
Select a material-process-heat-treatment combination, then iterate geometry because stiffness, tolerances, finish, and strength change together.
Verify values against named standards, certified lot data, supplier processing limits, and tests representative of section size and temperature.

EXAMPLE 3 Screening materials for a stiffness-limited instrument arm

A geometrically scalable, circular instrument arm is governed by bending stiffness and minimum mass, so rank candidates by \(M=E^{1/2}/\rho\). Supplied screening data are: aluminum \(E=69\ \text{GPa}\), \(\rho=2.70\ \text{Mg}/\text{m}^3\); steel \(E=200\ \text{GPa}\), \(\rho=7.85\ \text{Mg}/\text{m}^3\); and glass-fiber composite \(E=25\ \text{GPa}\), \(\rho=1.90\ \text{Mg}/\text{m}^3\). Calculate each index and the aluminum advantage over steel.

Recognition strategy

The geometry may scale and elastic deflection governs, so the bending stiffness material index applies. Yield strength would be the wrong first ranking property for this stated objective.

Given information

The three supplied \(E,\rho\) pairs, common circular geometry family, small elastic deflection, room temperature, and objective of minimum mass at fixed stiffness.

Model and interpretation

Use consistent GPa and \(\text{Mg}/\text{m}^3\) values for comparative indices. Composite modulus is assumed along the arm axis; joints, transverse properties, cost, damage tolerance, and processing are not in the index.

Step-by-step solution

1. Confirm the index. For a stiffness-limited beam whose section may scale, minimizing mass is equivalent to maximizing \(M=E^{1/2}/\rho\).

2. Evaluate aluminum and steel. \[M_{\mathrm{Al}}=\frac{\sqrt{69}}{2.70}=3.076,\qquad M_{\mathrm{steel}}=\frac{\sqrt{200}}{7.85}=1.802.\]

3. Evaluate the composite. \[M_{\mathrm{GFRP}}=\frac{\sqrt{25}}{1.90}=2.632,\] so the screening order is aluminum, glass-fiber composite, then steel.

4. Quantify the comparison. The index ratio \(M_{\mathrm{Al}}/M_{\mathrm{steel}}=3.076/1.802=1.707\), a \(70.7\%\) advantage on this objective.

Verification

Units: Every candidate uses the same GPa and \(\text{Mg}/\text{m}^3\) basis, so index ratios are dimensionless and ranking is unaffected by a common conversion.

Signs and direction: Modulus and density are positive; because the objective is maximization, a larger index correctly indicates lower achievable mass.

Plausibility: Steel's larger modulus is offset by much higher density, while the composite's low density cannot fully offset its low supplied axial modulus.

Final result: \(\boxed{M_{\mathrm{Al}}=3.076>M_{\mathrm{GFRP}}=2.632>M_{\mathrm{steel}}=1.802;\ \text{Al is }70.7\%\text{ above steel}}\)

Engineering interpretation

Aluminum wins the stated first-pass index, not the complete design. Yield, fatigue, buckling, joints, corrosion, thermal drift, stock form, tolerances, cost, composite layup variability, and certified temperature-dependent data must be screened before release.

Chapter recap

Stress–strain curves separate elastic stiffness, yielding, strain hardening, ultimate load, necking, and fracture.
Properties are distributions tied to specimen geometry, temperature, rate, orientation, processing, and test method.
Manufacturing and heat treatment create the delivered microstructure, residual stress, surface condition, tolerances, and defects.
Selection first screens hard constraints, then ranks survivors with the index and trade-offs appropriate to the load case.

Before moving on

You should be able to compute engineering stress and strain, distinguish strength from stiffness, quantify cold work, explain when hardness or impact data are only proxies, select an appropriate material index, and state which property values require standards, certificates, or testing.