11.2 Mechanical Properties, Fatigue, Fracture, and Corrosion

11.2 Mechanical Properties, Fatigue, Fracture, and Corrosion

Exam Focus: Mechanical behavior and degradation topics on the FE exam require mastering engineering vs. true stress-strain relationships, elastic constants ((E, G, \nu)), fracture mechanics stress intensity ((K_I = Y \sigma \sqrt{\pi a})), fatigue design with mean stress corrections (Goodman equation), high-temperature creep modeling via the Larson-Miller parameter, and electrochemical corrosion mechanisms.

Engineering vs. True Stress-Strain Relationships and Elastic Constants

Standard tension testing measures the response of a specimen under uniaxial loading. Uniaxial stress and strain are classified as either engineering or true parameters.

Metric Definitions

• Engineering Stress ((\sigma_e)): Force divided by original cross-sectional area: $\sigma_e = \frac{F}{A_0}$
• Engineering Strain ((\epsilon_e)): Change in length divided by original gage length: $\epsilon_e = \frac{\Delta L}{L_0} = \frac{L_i - L_0}{L_0}$
• True Stress ((\sigma_t)): Force divided by instantaneous cross-sectional area (A_i): $\sigma_t = \frac{F}{A_i}$
• True Strain ((\epsilon_t)): Instantaneous elongation strain integrated over gage length: $\epsilon_t = \int_{L_0}^{L_i} \frac{dL}{L} = \ln\left(\frac{L_i}{L_0}\right)$

Mathematical Conversions (Prior to Necking)

Assuming constant plastic volume ((A_0 L_0 = A_i L_i)), the relationship between true and engineering metrics prior to the onset of necking is:

$\frac{L_i}{L_0} = 1 + \epsilon_e \implies A_i = \frac{A_0}{1 + \epsilon_e}$

$\sigma_t = \frac{F}{A_i} = \sigma_e (1 + \epsilon_e)$

$\epsilon_t = \ln(1 + \epsilon_e)$

Elastic Constants and Hooke's Law

Within the linear elastic region:

• Hooke's Law: (\sigma = E \epsilon)
• Poisson's Ratio ((\nu)): Ratio of lateral contracting strain to axial elongating strain: $\nu = -\frac{\epsilon_{\text{lateral}}}{\epsilon_{\text{axial}}}$
• Shear Modulus ((G)): Governs shear stress-strain response ((\tau = G \gamma)) and is related to (E) and (\nu) by: $G = \frac{E}{2(1 + \nu)}$

Key Stress-Strain Curve Parameters

Worked Example: True Stress and Strain Calculation

Problem: A cylindrical titanium alloy specimen with initial diameter (d_0 = 12.8 \text{ mm}) ((A_0 = 128.68 \text{ mm}^2)) and gage length (L_0 = 50.00 \text{ mm}) is loaded in tension. Under a load of (F = 110.0 \text{ kN}) prior to necking, the specimen length is measured as (54.20 \text{ mm}). Calculate the engineering stress, engineering strain, true stress, and true strain.

Solution:

1. Engineering Stress ((\sigma_e)): $\sigma_e = \frac{110,000 \text{ N}}{1.2868 \times 10^{-4} \text{ m}^2} = 8.548 \times 10^8 \text{ Pa} = 854.8 \text{ MPa}$
2. Engineering Strain ((\epsilon_e)): $\epsilon_e = \frac{54.20 - 50.00}{50.00} = \frac{4.20}{50.00} = 0.0840$
3. True Stress ((\sigma_t)): $\sigma_t = \sigma_e (1 + \epsilon_e) = 854.8 \text{ MPa} \times (1 + 0.0840) = 854.8 \times 1.0840 \approx 926.6 \text{ MPa}$
4. True Strain ((\epsilon_t)): $$$$

\approx 0.0807$$

Hardness Testing, Charpy Impact, and Ductile-to-Brittle Transition

Hardness Testing Methods

Hardness measures a material's resistance to localized surface indentation or plastic deformation:

• Rockwell (HRB, HRC): Measures depth of penetration under fixed minor/major loads using a 1/16-inch steel ball (HRB) or a Brale diamond cone penetrator (HRC).
• Brinell (HB): Uses a 10 mm hardened steel or tungsten carbide sphere pressed into the surface under a 3000 kgf load, measuring indentation diameter.
• Empirical Tensile Strength Correlation for Steels: $\text{UTS (MPa)} \approx 3.45 \times \text{HB}$ $\text{UTS (psi)} \approx 500 \times \text{HB}$

Charpy Impact Energy and DBTT

Impact testing measures dynamic energy absorption under high strain rates using a swinging pendulum striker:

• Charpy V-Notch (CVN) Test: Specimen supported as a simple beam with a central V-notch; energy absorbed during fracture is recorded in Joules ((\text{J})) or foot-pounds ((\text{ft}\cdot\text{lbf})).
• Ductile-to-Brittle Transition Temperature (DBTT): Body-Centered Cubic (BCC) and Hexagonal Close-Packed (HCP) metals exhibit a sharp reduction in impact energy absorption as temperature drops below DBTT. Failure transitions from ductile dimpled shear fracture to brittle cleavage fracture along crystallographic planes.
• FCC Metals: Face-Centered Cubic (FCC) metals (such as aluminum, copper, and austenitic stainless steels) retain high toughness down to cryogenic temperatures and do not exhibit a DBTT.

Fracture Mechanics and Stress Intensity

Conventional stress analysis assumes flaw-free materials. Linear Elastic Fracture Mechanics (LEFM) evaluates component safety in the presence of pre-existing microcracks or defects.

Stress Concentration Factor ((K_t))

Geometric discontinuities (notches, holes, fillets) intensify local stresses:

$\sigma_{\text{max}} = K_t \sigma_0$

For an internal elliptical crack of length (2a) with crack tip radius (\rho_t):

$\sigma_{\text{max}} = \sigma_0 \left(1 + 2\sqrt{\frac{a}{\rho_t}}\right)$

Stress Intensity Factor ((K_I)) and Fracture Toughness ((K_{Ic}))

Under Mode I (tensile opening) loading, the stress field at a crack tip is defined by the Stress Intensity Factor ((K_I)):

$K_I = Y \sigma \sqrt{\pi a}$

where:

• (Y) = dimensionless geometry factor (typically (Y \approx 1.0 - 1.12))
• (\sigma) = nominal applied tensile stress ((\text{MPa}))
• (a) = crack depth for an edge crack, or half-length for an internal crack ((\text{m}))

Catastrophic fast brittle fracture occurs when (K_I) reaches the critical material property known as Plane-Strain Fracture Toughness ((K_{Ic})):

$K_I \ge K_{Ic}$

Critical Stress and Critical Flaw Size Equations

$\sigma_c = \frac{K_{Ic}}{Y \sqrt{\pi a}}$

$a_c = \frac{1}{\pi} \left(\frac{K_{Ic}}{Y \sigma}\right)^2$

Worked Example: Allowable Tensile Stress via Fracture Mechanics

Problem: A structural steel plate containing a through-thickness internal crack of total length (2a = 16.0 \text{ mm}) ((a = 8.0 \text{ mm} = 0.0080 \text{ m})) is subjected to a uniform tensile stress. The material has a plane-strain fracture toughness (K_{Ic} = 45.0 \text{ MPa}\cdot\text{m}^{1/2}) and a yield strength (\sigma_y = 550 \text{ MPa}). Assuming (Y = 1.0):

1. Calculate the critical tensile stress (\sigma_c) that causes brittle fracture.
2. Determine whether the plate fails by yielding or brittle fracture first.

Solution:

1. Calculate Critical Stress (\sigma_c): $\sigma_c = \frac{K_{Ic}}{Y \sqrt{\pi a}} = \frac{45.0 \text{ MPa}\cdot\text{m}^{1/2}}{1.0 \times \sqrt{\pi \times 0.0080 \text{ m}}} = \frac{45.0}{\sqrt{0.025133}} = \frac{45.0}{0.15853} \approx 283.9 \text{ MPa}$
2. Evaluate Failure Mode: Since (\sigma_c = 283.9 \text{ MPa}) is substantially less than the yield strength (\sigma_y = 550 \text{ MPa}), the component will fail by sudden brittle fracture before yielding occurs.

Fatigue Mechanics, Goodman Diagram, and Creep Mechanics

Cyclic Fatigue Parameters

Fatigue represents progressive structural damage under fluctuating or cyclic stress state loading:

• Mean Stress ((\sigma_m)): (\sigma_m = \frac{\sigma_{\text{max}} + \sigma_{\text{min}}}{2})
• Stress Range ((\Delta \sigma)): (\Delta \sigma = \sigma_{\text{max}} - \sigma_{\text{min}})
• Alternating Stress Amplitude ((\sigma_a)): (\sigma_a = \frac{\sigma_{\text{max}} - \sigma_{\text{min}}}{2})
• Stress Ratio ((R)): (R = \frac{\sigma_{\text{min}}}{\sigma_{\text{max}}})

S-N Curves and Endurance Limit

Ferrous metals (steels) exhibit an Endurance Limit ((S_e)) (typically (S_e \approx 0.5 S_{ut}) up to (100 \text{ ksi})), below which the component can endure an infinite number of cycles without fatigue failure. Non-ferrous metals (e.g., aluminum) do not possess a true endurance limit; their fatigue strength continuously decreases with cycle count.

Mean Stress Corrections (Goodman Equation)

When cyclic stress is accompanied by a positive mean tensile stress ((\sigma_m > 0)), the allowable alternating stress is reduced according to the Modified Goodman Equation:

$\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = \frac{1}{n_f}$

where (S_e) is the uncorrected endurance limit, (S_{ut}) is ultimate tensile strength, and (n_f) is the fatigue factor of safety.

Worked Example: Goodman Fatigue Safety Factor

Problem: A steel shaft with (S_{ut} = 700 \text{ MPa}) and (S_e = 280 \text{ MPa}) experiences cyclic tensile loading between (\sigma_{\text{max}} = 400 \text{ MPa}) and (\sigma_{\text{min}} = 100 \text{ MPa}). Calculate the factor of safety (n_f) using the Goodman relation.

Solution:

1. Calculate Mean and Alternating Stresses: $\sigma_m = \frac{400 + 100}{2} = 250 \text{ MPa}$ $\sigma_a = \frac{400 - 100}{2} = 150 \text{ MPa}$
2. Apply Goodman Equation: $\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = \frac{150}{280} + \frac{250}{700} = 0.5357 + 0.3571 = 0.8928$ $n_f = \frac{1}{0.8928} \approx 1.12$

Cumulative Damage: Miner's Rule

Under multi-amplitude cyclic loading, fatigue damage accumulates linearly according to Miner's Rule:

$\sum_{i=1}^{k} \frac{n_i}{N_i} = C$

where (n_i) is the number of applied cycles at stress level (\sigma_i), and (N_i) is the fatigue life at stress level (\sigma_i). Fatigue failure is predicted when (C \ge 1.0).

Creep Mechanics and Larson-Miller Parameter

Creep is time-dependent plastic deformation occurring under constant load at elevated temperatures (typically (T > 0.4 T_m) in Kelvin).

• Creep Curve Stages:
  1. Primary (Transient) Creep: Strain rate decreases over time due to work hardening.
  2. Secondary (Steady-State) Creep: Minimum constant creep rate ((\dot{\epsilon}_s)) established by balance between work hardening and thermal recovery.
  3. Tertiary Creep: Strain rate accelerates rapidly due to internal necking and micro-void formation, leading to rupture.
• Larson-Miller Parameter (LMP): Predicts rupture time ((t_r) in hours) at absolute temperature (T): $\text{LMP} = T [C + \log_{10}(t_r)]$ where (T) is temperature in Kelvin ((\text{K})) or Rankine ((\text{R})), and (C) is a material constant (typically (C \approx 20) for steels).

Electrochemical Corrosion Mechanisms and Mitigation

Corrosion is the destructive electrochemical attack of a material by reaction with its surrounding environment.

Electrochemical Cell Components

An electrochemical corrosion cell requires four essential elements:

1. Anode: The site of oxidation where metal atoms dissolve and release electrons.
2. Cathode: The site of reduction where electrons are consumed.
3. Electrolyte: An aqueous solution conducting ionic current.
4. Electrical Connection: Metallic path conducting electron flow from anode to cathode.

Electrochemical Half-Cell Reactions

• Anodic Oxidation Reaction (Metal Dissolution): $M \rightarrow M^{n+} + n e^-$
• Cathodic Reduction Reactions:
  • Acidic solutions (Hydrogen Evolution): (2H^+ + 2e^- \rightarrow H_2 (g))
  • Neutral/Basic aerated solutions (Oxygen Reduction): (O_2 + 2H_2O + 4e^- \rightarrow 4OH^-)

Faraday's Law for Corrosion Mass Loss

$m = \frac{I \cdot t \cdot M}{n \cdot F}$

where (I) is corrosion current ((\text{A})), (t) is time ((\text{s})), (M) is molar mass ((\text{g/mol})), (n) is valence state, and (F = 96,485 \text{ C/mol}).

Forms of Corrosion

• Galvanic Corrosion: Occurs when two dissimilar metals are electrically coupled in an electrolyte. The metal higher in the galvanic series (more active / anodic) corrodes rapidly to protect the cathodic metal.
• Crevice Corrosion: Localized corrosion occurring in narrow gaps or stagnant fluid pockets driven by differential oxygen concentration cells.
• Pitting Corrosion: Extremely localized breakdown of passive oxide films, creating deep pits in stainless steels exposed to chloride ions ((\text{Cl}^-)).
• Intergranular Corrosion: Sensitization of austenitic stainless steels during welding between (500^\circ\text{C}) and (800^\circ\text{C}), causing chromium carbide ((\text{Cr}_{23}\text{C}_6)) precipitation along grain boundaries and depleting local chromium below (12 \text{ wt% Cr}).
• Stress Corrosion Cracking (SCC): Rapid brittle cracking resulting from simultaneous tensile stress and a specific corrosive environment.

Corrosion Control Methods

• Sacrificial Anode Cathodic Protection: Attaching sacrificial anodes (e.g., zinc or magnesium) to protect steel structures.
• Impressed Current Cathodic Protection (ICCP): Applying external DC voltage to drive the structure cathodic.
• Passivation: Maintaining a self-healing protective oxide film (e.g., (\text{Cr}_2\text{O}_3) on stainless steel).