2.2 Physical Properties, Temperature Effects, Hardness, Impact Toughness & Creep

2.2 Temperature Effects, Hardness, Impact Toughness & Creep

Quick Summary: Component performance in operational environments depends on temperature-dependent and time-dependent material degradation mechanisms. This section covers standard hardness testing methodologies (Brinell, Rockwell, Vickers, Knoop) and their empirical correlation to tensile strength in structural steels, analyzes dynamic impact toughness and crystal lattice effects on ductile-to-brittle transitions, and formulates high-temperature creep modeling using the Larson-Miller parameter alongside thermal shock resistance.

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1. Hardness Testing Methods & Empirical Correlations

Hardness measures a material's resistance to localized permanent plastic deformation, penetration, abrasion, or scratching. In mechanical engineering, indentation hardness testing provides a rapid, non-destructive means of assessing material strength, quality of heat treatment, and surface wear resistance.

Brinell Hardness Test (ASTM E10)

The Brinell test presses a spherical tungsten carbide ball indenter ($D = 10\text{ mm}$) into the specimen surface under a standard static test load $P$:

• Standard load: $3000\text{ kgf}$ ($29.42\text{ kN}$) for structural steels and cast irons.
• Intermediate load: $1500\text{ kgf}$ for copper alloys and aluminum forgings.
• Light load: $500\text{ kgf}$ for soft aluminum alloys and bearing metals.

The load is sustained for $10\text{--}15\text{ seconds}$, and the resulting spherical indentation diameter ($d$) is measured along two orthogonal axes using an optical microscope:

$HBW = \frac{2P}{\pi D \left(D - \sqrt{D^2 - d^2}\right)}$

where $P$ is the applied load in $\text{kgf}$, $D$ is the indenter diameter in $\text{mm}$, and $d$ is the mean indentation diameter in $\text{mm}$.

• Engineering Advantage: The large $10\text{ mm}$ indenter creates an impression spanning multiple grains, effectively averaging local microstructural variations. This makes Brinell testing the preferred method for cast irons, forgings, hot-rolled structural steel shapes, and coarse-grained weldments.
• Designation: E.g., $250\text{ HBW 10/3000}$ denotes a Brinell hardness of $250$ determined with a $10\text{ mm}$ tungsten carbide ball under a $3000\text{ kgf}$ load held for $10\text{--}15\text{ seconds}$.

Rockwell Hardness Test (ASTM E18)

The Rockwell test measures the net permanent increase in indentation depth under a major load relative to an initial minor preload. Unlike the Brinell optical measurement, Rockwell testers provide a direct dial or digital readout:

1. Preliminary Minor Load ($10\text{ kgf}$): Applied to seat the indenter through surface scale, oxidation, and surface roughness, establishing a zero-depth datum ($h_0$).
2. Additional Major Load: Applied and held for the short dwell time specified in ASTM E18, producing plastic penetration.
3. Major Load Removal: The major load is removed while retaining the minor load. The final permanent indentation depth $e$ relative to the datum determines the hardness number: $HR = E_0 - e$.

Primary Rockwell Scales:

• Rockwell C (HRC): $120^\circ$ spheroconical diamond Brale indenter, $150\text{ kgf}$ total load ($10\text{ kgf}$ minor + $140\text{ kgf}$ major). Used for hardened alloy steels, tool steels, quenched and tempered components, and carburized cases exceeding $20\text{ HRC}$ up to $\approx 70\text{ HRC}$.
• Rockwell B (HRB): $1/16\text{ in}$ ($1.588\text{ mm}$) diameter hardened tungsten carbide ball, $100\text{ kgf}$ total load ($10\text{ kgf}$ minor + $90\text{ kgf}$ major). Used for soft and medium-strength materials: annealed carbon steels, structural aluminum alloys, brass, and bronze ($0\text{ to }100\text{ HRB}$).
• Rockwell A (HRA): $120^\circ$ diamond Brale indenter, $60\text{ kgf}$ total load. Used for extremely thin case-hardened surfaces, razor blades, and cemented tungsten carbides to avoid chipping or punching through the thin substrate.

Microhardness: Vickers (HV) & Knoop (HK) (ASTM E384)

When evaluating thin coatings, decarburized surface layers, individual microscopic phases, or heat-affected zones (HAZ) adjacent to welds, microhardness indentations with microscopic depths are required:

• Vickers Hardness (HV): Uses a square-based diamond pyramid indenter with an apex face angle of $136^\circ$. Loads range from $1\text{ gf}$ to $120\text{ kgf}$. The mean diagonal length ($d$) of the square impression is measured optically: $HV = \frac{2P \sin(136^\circ/2)}{d^2} \approx 1.8544 \frac{P}{d^2}$ The Vickers scale is continuous from soft pure metals ($15\text{ HV}$) to nitrided steels ($1000\text{ HV}$) and diamond-like coatings ($2500\text{ HV}$).
• Knoop Hardness (HK): Uses an elongated rhombic diamond pyramid indenter with a length-to-width ratio of approximately $7.11:1$. It produces a shallow, elongated impression that minimizes subsurface cracking in brittle materials (such as glass, ceramics, and intermetallic compounds) and allows precise hardness profiling across narrow case depths.

Empirical Hardness-Strength Correlation for Structural Steels

Because plastic indentation involves localized compressive yielding and shear deformation, hardness correlates directly with ultimate tensile strength for structural steels. Widely used textbook correlations (for example, Shigley gives $S_{ut} \approx 0.5\,\text{HB}$ kpsi $\approx 3.4\,\text{HB}$ MPa) are:

$S_{ut} \approx 500 \times \text{HB} \quad (\text{psi})$

$S_{ut} \approx 0.50 \times \text{HB} \quad (\text{ksi})$

$S_{ut} \approx 3.45 \times \text{HB} \quad (\text{MPa})$

Examination Constraints: These linear approximations are estimates for plain carbon and low-alloy steels; they lose accuracy at very low and very high hardness, so treat them as screening values rather than certified strengths. They do not apply to work-hardening austenitic stainless steels, cast irons, or non-ferrous alloys (aluminum, brass, titanium), which have different strain-hardening characteristics.

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2. Dynamic Impact Toughness & Fracture Transitions

Under high strain rates, multiaxial constraints, and sub-ambient temperatures, materials that appear ductile in a slow uniaxial tensile test can fail by catastrophic, low-energy brittle fracture without plastic deformation.

Impact Testing: Charpy V-Notch vs. Izod (ASTM E23)

Impact tests measure the mechanical energy absorbed in fracturing a notched specimen struck by a swinging pendulum hammer:

• Charpy V-Notch (CVN): The specimen is a $10\text{ mm} \times 10\text{ mm} \times 55\text{ mm}$ bar containing a central $45^\circ$, $2\text{ mm}$ deep notch with a $0.25\text{ mm}$ root radius. The specimen is supported horizontally at both ends as a simple beam and struck on the face directly opposite the notch.
• Izod Test: The specimen is clamped vertically at one end as a cantilever beam and struck above the notch on the same face containing the notch.

The absorbed impact energy ($E_{\text{abs}}$) is computed from the initial release height ($h_0$) and the maximum swing-through height ($h_f$):

$E_{\text{abs}} = mg(h_0 - h_f)$

Recorded in Joules ($\text{J}$) in SI or foot-pounds ($\text{ft}\cdot\text{lbf}$) in USCS ($1\text{ ft}\cdot\text{lbf} = 1.356\text{ J}$).

The Ductile-to-Brittle Transition Temperature (DBTT)

When Charpy absorbed energy is plotted as a function of test temperature, ferritic and low-alloy structural steels exhibit a characteristic S-shaped curve:

Absorbed Energy
  ^
  |          Upper Shelf (Ductile, fibrous microvoid coalescence)
  |          -----------------\
  |                            \
  |                             \  <-- Transition Region (DBTT)
  |                              \
  |                               \-----------------
  |                                 Lower Shelf (Brittle, cleavage fracture)
  +---------------------------------------------------> Temperature (T)

1. Upper Shelf: At elevated and ambient temperatures, failure occurs by ductile rupture (microvoid nucleation, growth, and coalescence). Fracture surfaces appear dull, fibrous, and display distinct shear lips. Energy absorption is high ($> 50\text{--}150\text{ J}$).
2. Lower Shelf: At sub-ambient temperatures, failure transitions to rapid transgranular cleavage along crystallographic cleavage planes ($\{100\}$ in ferrite). Fracture surfaces appear bright, flat, highly reflective, and crystalline. Energy absorption drops drastically ($< 10\text{--}15\text{ J}$).
3. Transition Criteria: Design standards (e.g., ASME Boiler and Pressure Vessel Code, ASTM A370) define the DBTT using:
   • Specific absorbed energy threshold: typically $20\text{ J}$ ($15\text{ ft}\cdot\text{lbf}$) or $27\text{ J}$ ($20\text{ ft}\cdot\text{lbf}$) for structural steels.
   • Fracture Appearance Transition Temperature (FATT): the temperature at which the fracture surface displays 50% shear (fibrous) and 50% cleavage (crystalline) area.
   • Lateral expansion threshold: minimum transverse expansion at the specimen's compression face (e.g., $\ge 0.38\text{ mm} / 0.015\text{ in}$).

Crystal Structure Influence: BCC vs. FCC vs. HCP

The existence of a DBTT is directly dictated by crystal lattice structure and dislocation mobility:

• Body-Centered Cubic (BCC) Metals: Carbon steels, low-alloy steels, ferritic/martensitic stainless steels, chromium, molybdenum, and tungsten exhibit a pronounced DBTT. In BCC lattices, dislocation motion (plastic slip) requires overcoming a steep lattice resistance barrier known as the Peierls-Nabarro stress. At lower temperatures, thermal activation energy decreases, the Peierls stress rises exponentially, dislocation motion freezes, and the stress required for slip exceeds the cleavage fracture stress. BCC steels must never be specified for service below their DBTT.
• Face-Centered Cubic (FCC) Metals: Austenitic stainless steels (AISI 304, 316), aluminum alloys, copper, nickel, and austenitic nickel-base superalloys do not exhibit a DBTT. FCC crystals possess 12 close-packed slip systems ($\{111\}\langle110\rangle$) with very low Peierls-Nabarro lattice resistance. Dislocation mobility remains active down to cryogenic temperatures ($4\text{ K} / -269^\circ\text{C}$). FCC alloys maintain exceptional impact toughness, ductility, and resistance to brittle shock at cryogenic temperatures (e.g., liquid nitrogen tanks at $-196^\circ\text{C}$, LNG piping at $-162^\circ\text{C}$, and liquid hydrogen aerospace systems at $-253^\circ\text{C}$).
• Hexagonal Close-Packed (HCP) Metals: Titanium, magnesium, and zinc have limited active slip systems at room temperature, leading to moderate toughness and gradual temperature sensitivity without the steep inflection seen in BCC steels.

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3. Elevated Temperature Mechanics: Creep & Stress Rupture

At homologous temperatures exceeding approximately $40\%$ of a metal's absolute melting temperature:

$T_H = \frac{T}{T_m} > 0.40 \quad (\text{with } T, T_m \text{ in Kelvin or Rankine})$

materials undergo creep—continuous, time-dependent, irreversible plastic deformation under sustained mechanical stress, even when the applied stress is significantly below the static yield strength ($S_y$).

The Three Stages of Creep

Under constant uniaxial tensile load and constant elevated temperature, a creep curve of strain versus time exhibits three characteristic regimes:

Creep Strain (ε)
  ^                                         Rupture (*)
  |                                        /
  |                        Stage II       /  Stage III (Tertiary)
  |                       (Secondary)    /
  |                      /--------------/
  |         Stage I     /
  |        (Primary)   /
  |       /-----------/
  |      /
  |     /
  +----+-----------------------------------------> Time (t)
   Instantaneous
   Elastic Strain (ε_0)

1. Primary (Transient) Creep (Stage I): Immediately follows elastic deflection. The creep strain rate ($\dot{\epsilon} = d\epsilon/dt$) decreases continuously over time as dislocation multiplication and forest entanglement produce strain hardening faster than thermal recovery can relieve it.
2. Secondary (Steady-State) Creep (Stage II): The strain rate reaches a nearly constant minimum value ($\dot{\epsilon}_s$). A dynamic equilibrium is established between strain hardening (dislocation accumulation) and thermal recovery mechanisms (dislocation climb over obstacles, vacancy diffusion, and subgrain boundary migration). Stage II dominates the operational life of engineered components. Steady-state creep rate is modeled by the Norton-Bailey power law: $\dot{\epsilon}_s = A \sigma^n \exp\left(-\frac{Q}{R T}\right)$ where $A$ is a material constant, $\sigma$ is the applied stress, $n$ is the creep stress exponent (typically $3 \le n \le 8$ for dislocation climb), $Q$ is the activation energy for creep diffusion ($\text{J/mol}$), $R$ is the universal gas constant ($8.314\text{ J/(mol}\cdot\text{K)}$), and $T$ is absolute temperature.
3. Tertiary Creep (Stage III): The creep rate accelerates rapidly. Grain boundary sliding initiates micro-void cavitation, intergranular cracks propagate, and cross-sectional necking reduces load-bearing area. This instability terminates in catastrophic stress rupture at time $t_r$.

The Larson-Miller Parameter (LMP)

Long-term creep rupture testing (e.g., $100,000\text{ hours} \approx 11.4\text{ years}$) is impractical during design cycles. The Larson-Miller parameter provides a time-temperature superposition method to extrapolate short-term, accelerated high-temperature test data to predict decades-long operating life at lower service temperatures under identical stress levels:

$LMP = T \left(C + \log_{10} t_r\right) \times 10^{-3}$

• Variables & Units:
  • $T$: Absolute temperature in Rankine ($^\circ\text{R} = ^\circ\text{F} + 459.67$) in USCS practice, or in Kelvin ($\text{K} = ^\circ\text{C} + 273.15$) in SI practice.
  • $t_r$: Time to stress rupture in hours.
  • $C$: Dimensionless material-specific Larson-Miller constant, typically taken as $C = 20$ for structural steels, low-alloy boiler steels, and nickel-base superalloys.
  • The factor $10^{-3}$ is conventionally included to scale $LMP$ into a convenient range ($30\text{ to }60$ in $^\circ\text{R}\cdot\text{hr}$, or $15\text{ to }35$ in $\text{K}\cdot\text{hr}$).
• Equivalence Principle: For a given alloy under a specific stress $\sigma$, $LMP$ is a constant value. Therefore, a component operating at a lower temperature for a long duration accumulates the same creep damage as an accelerated specimen tested at a higher temperature for a short duration: $T_1 \left(C + \log_{10} t_{r1}\right) = T_2 \left(C + \log_{10} t_{r2}\right)$

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4. Thermal Shock Resistance & Constrained Thermal Stresses

When mechanical equipment undergoes rapid heating or quenching cycles (e.g., steam turbine rotors, brake discs, internal combustion engine cylinder heads), sharp transient temperature gradients induce differential thermal strains.

Constrained Thermal Stresses

For a body subjected to a temperature change $\Delta T$, unconstrained thermal strain is $\epsilon_{\text{th}} = \alpha \Delta T$, where $\alpha$ is the coefficient of thermal expansion ($\text{m/(m}\cdot^\circ\text{C)}$ or $\text{in/(in}\cdot^\circ\text{F)}$). If the material surface is constrained by colder interior material (or rigid external boundaries), thermal stresses develop:

• 1D Fully Constrained Bar: $\sigma_{\text{th}} = -E \alpha \Delta T$
• 2D Equibiaxially Constrained Surface (Thermal Shock): $\sigma_{\text{th}} = -\frac{E \alpha \Delta T}{1 - \nu}$

During rapid surface cooling (quenching), the surface contracts against the hotter interior, generating severe surface tensile stresses that can initiate surface cracks (thermal crazing or heat checking).

Thermal Shock Resistance Parameters

A material's ability to resist fracture under rapid thermal cycles is characterized by thermal shock parameters:

1. Severe Thermal Shock (High Biot Number, sudden quench): The surface temperature changes almost instantly, so the allowable temperature jump before surface stress reaches the strength ($\sigma_f$ for brittle materials, $S_y$ for ductile metals) is: $R = \frac{S_y (1 - \nu)}{E \alpha}$
2. Mild Thermal Shock (Low Biot Number): Heat conduction inside the part limits the gradient, so thermal conductivity ($k$) enters the parameter: $R' = \frac{k S_y (1 - \nu)}{E \alpha}$

• Design Strategy: To maximize thermal shock resistance, engineers select materials with:
  • High thermal conductivity ($k$) to dissipate temperature gradients rapidly.
  • High tensile/yield strength ($S_y$) to withstand induced stresses.
  • Low coefficient of thermal expansion ($\alpha$) to minimize thermal strain.
  • Low modulus of elasticity ($E$) to minimize the stress generated per unit thermal strain.

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5. Physical Properties Used in Machine Design

Mechanical properties describe how a material resists load. Physical properties describe how heavy it is, how much it expands, and how it stores and conducts heat and electricity. Exam problems combine the two constantly: a shaft-to-housing fit at operating temperature, the inertia of a rotating disk, the temperature rise of a brake rotor, or the resistance of a sensing element.

How each property enters machine design:

• Density ($\rho$) sets mass and rotating inertia ($J = \tfrac{1}{2} m r^2$ for a solid disk). Specific stiffness $E/\rho$ is nearly identical for steel, aluminum and titanium (about $25\ \text{GPa per g/cm}^3$), so switching metals saves weight only when the design is strength-limited, not stiffness-limited.
• Coefficient of thermal expansion ($\alpha$) gives free growth $\delta = \alpha L \Delta T$. Mismatched pairs (aluminum housings on steel bearings, polymer bushings on steel shafts) change fits and preloads with temperature.
• Thermal conductivity ($k$) and thermal diffusivity $a = k/(\rho c_p)$ control how fast heat spreads, which matters for brake rotors, welding heat-affected zones and thermal-shock resistance.
• Specific heat ($c_p$) sets the temperature rise when energy is absorbed: $\Delta T = E/(m c_p)$.
• Melting point or solidus limits brazing and soldering temperatures, and the absolute melting temperature sets the creep threshold ($T > 0.4 T_m$).
• Electrical resistivity ($\rho_e$) gives conductor resistance $R = \rho_e L / A$ (copper windings, heater elements, strain-gage alloys). Ferritic and martensitic steels are ferromagnetic, while austenitic stainless steels, aluminum and copper are not, which decides whether magnetic particle inspection or magnetic sensors can be used.

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6. Worked Numerical Examples

Example 1: Brinell Hardness & Shaft Strength Estimation

Problem: A transmission intermediate shaft machined from AISI 4140 alloy steel undergoes Brinell quality verification. A $10\text{ mm}$ tungsten carbide ball under a $3000\text{ kgf}$ load produces a spherical indentation with an average diameter $d = 3.80\text{ mm}$.

1. Calculate the Brinell Hardness Number ($HBW$).
2. Estimate the ultimate tensile strength ($S_{ut}$) in both $\text{ksi}$ and $\text{MPa}$.

Solution:

1. Brinell hardness calculation: $HBW = \frac{2P}{\pi D \left(D - \sqrt{D^2 - d^2}\right)}$ Substitute $P = 3000\text{ kgf}$, $D = 10\text{ mm}$, and $d = 3.80\text{ mm}$: $\sqrt{D^2 - d^2} = \sqrt{10^2 - 3.80^2} = \sqrt{100 - 14.44} = \sqrt{85.56} = 9.2499\text{ mm}$ $D - \sqrt{D^2 - d^2} = 10 - 9.2499 = 0.7501\text{ mm}$ $HBW = \frac{2(3000)}{\pi (10)(0.7501)} = \frac{6000}{23.565} \approx 254.6 \approx 255\text{ HBW}$

2. Tensile strength estimation: Using the empirical correlation for low-alloy structural steels: $S_{ut} \approx 500 \times \text{HB} \text{ psi} = 0.50 \times 255 = 127.5\text{ ksi}$ In SI units: $S_{ut} \approx 3.45 \times \text{HB} \text{ MPa} = 3.45 \times 255 = 879.8\text{ MPa} \approx 880\text{ MPa}$

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Example 2: Boiler Superheater Tube Creep Rupture Life Evaluation

Problem: An alloy steel superheater tube in an industrial boiler operates at a sustained internal pressure generating a hoop stress of $\sigma = 120\text{ MPa}$.

• In an accelerated laboratory creep rupture test conducted at $T_1 = 650^\circ\text{C}$ under the same stress of $120\text{ MPa}$, the specimen ruptured after $t_{r1} = 250\text{ hours}$.
• The material follows the Larson-Miller parameter with $C = 20$, where $T$ is in Kelvin: $LMP = T \left(20 + \log_{10} t_r\right) \times 10^{-3}$

Determine:

1. The Larson-Miller parameter ($LMP$) for this stress level.
2. The expected operational service life ($t_{r2}$) in hours and years if the tube operates at $T_2 = 550^\circ\text{C}$.

Solution:

1. Convert test temperature to Kelvin: $T_1 = 650^\circ\text{C} + 273.15 = 923.15\text{ K}$ Calculate $\log_{10} t_{r1}$: $\log_{10}(250) \approx 2.3979$ Calculate $LMP$: $LMP = 923.15 \times (20 + 2.3979) \times 10^{-3} = 923.15 \times 22.3979 \times 10^{-3} = 20.6767$

2. Calculate service life at $T_2 = 550^\circ\text{C}$: $T_2 = 550^\circ\text{C} + 273.15 = 823.15\text{ K}$ Because the stress level is unchanged ($\sigma = 120\text{ MPa}$), $LMP$ remains invariant ($LMP = 20.6767$): $823.15 \times \left(20 + \log_{10} t_{r2}\right) \times 10^{-3} = 20.6767$ $20 + \log_{10} t_{r2} = \frac{20.6767 \times 10^3}{823.15} = \frac{20676.7}{823.15} \approx 25.1190$ $\log_{10} t_{r2} = 25.1190 - 20 = 5.1190$ $t_{r2} = 10^{5.1190} \approx 131,520\text{ hours}$

   Converting operating hours to years (assuming continuous operation at $8,760\text{ hours/year}$): $\text{Life} = \frac{131,520\text{ hours}}{8,760\text{ hours/year}} \approx 15.0\text{ years}$

   Engineering Takeaway: Reducing the operating temperature by $100^\circ\text{C}$ (from $650^\circ\text{C}$ to $550^\circ\text{C}$) increases component rupture life from approximately $10\text{ days}$ ($250\text{ hours}$) to $15\text{ years}$ ($131,520\text{ hours}$). This highlights the exponential sensitivity of creep rupture life to operating temperature.