8.3 Mechanical Testing, Fatigue Failure & Fracture Mechanics

8.3 Mechanical Testing, Fatigue Failure & Fracture Mechanics

PRC MELE Core Focus: Mechanical testing and component failure evaluation are mandatory competence areas for mechanical engineers. Board exam problems test standardized tensile test calculations, hardness scale conversions, Charpy impact dynamics, DBTT behavior, fatigue life modeling using Marin factors, fluctuating stress criteria (Goodman, Soderberg, Gerber, ASME-elliptic), and linear elastic fracture mechanics ($K_I, K_{Ic}$).

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1. Tensile Testing & Property Calculations (ASTM E8)

The standard tensile test measures the deformation response of a material under uniaxial tensile loading.

Key Engineering & True Relationships

1. Engineering Stress ($\sigma$) & Engineering Strain ($\epsilon$): $$\sigma = \frac{F}{A_0}, \quad \epsilon = \frac{\Delta L}{L_0} = \frac{L - L_0}{L_0}$$
2. True Stress ($\sigma_{\text{true}}$) & True Strain ($\epsilon_{\text{true}}$): Assuming constant specimen volume ($A L = A_0 L_0$): $$\sigma_{\text{true}} = \sigma (1 + \epsilon)$$ $$\epsilon_{\text{true}} = \ln\left(\frac{L}{L_0}\right) = \ln(1 + \epsilon)$$
3. Hooke's Law & Modulus of Elasticity ($E$): $$\sigma = E \cdot \epsilon \quad (\text{in elastic region, } \sigma \le S_y)$$
4. Ductility Measures: $$\% \text{EL} = \frac{L_f - L_0}{L_0} \times 100\%, \quad \% \text{RA} = \frac{A_0 - A_f}{A_0} \times 100\%$$
5. Modulus of Resilience ($U_r$) & Modulus of Toughness ($U_t$): $$U_r = \frac{S_y^2}{2E} \quad (\text{Elastic strain energy density to yield})$$ $$U_t \approx \left(\frac{S_y + S_u}{2}\right) \cdot \epsilon_f \quad (\text{Total work per unit volume to fracture})$$

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2. Hardness Testing Standards & Approximations

Hardness measures a material's localized resistance to plastic deformation by indentation.

Ultimate Tensile Strength ($S_u$) Estimation for Carbon Steels

In MELE calculations, empirical equations estimate UTS directly from Brinell Hardness ($HB$):

$$S_u (\text{MPa}) \approx 3.45 \times HB$$ $$S_u (\text{psi}) \approx 500 \times HB$$

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3. Impact Testing & Transition Temperature (DBTT)

Impact testing evaluates material behavior under high strain rates and triaxial stress states.

1. Charpy V-Notch Test (ASTM E23): A standard bar ($10\times10\times55\text{ mm}$) with a $2\text{ mm}$ deep $45^\circ$ V-notch is supported as a simple beam and struck behind the notch by a swinging pendulum mass $m$. Impact energy absorbed is: $$E_{\text{impact}} = m g (h - h') \quad (\text{in Joules or ft-lbf})$$
2. Izod Test: The specimen is clamped vertically as a cantilever beam and struck on the notched face.
3. Ductile-to-Brittle Transition Temperature (DBTT):
   • BCC Metals (e.g., Carbon Steels): Experience a sharp drop in impact energy absorption at lower temperatures (transitioning from ductile shear fracture to brittle cleavage).
   • FCC Metals (e.g., 304 Stainless Steel, Aluminum, Copper): Retain high energy absorption down to cryogenic temperatures (no DBTT).
   • Failure Case: Historical ship hull failures (e.g., Liberty ships in WWII) occurred when operating below the DBTT of structural steel.

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4. Fatigue Testing & S-N Curves

Fatigue causes over 80-90% of operational mechanical component failures under cyclic stress levels well below the static yield strength.

S-N Curve & Endurance Limit

• Rotating Beam Test (ASTM E466): Determines stress amplitude ($S$) vs. number of cycles to failure ($N$).
• Unworked Endurance Limit ($S'_e$): For plain carbon and alloy steels ($N \ge 10^6 \text{ cycles}$): $$S'_e = \begin{cases} 0.50 S_u \& \text{for } S_u \le 1400\text{ MPa } (200\text{ ksi}) \\ 700\text{ MPa } (100\text{ ksi}) \& \text{for } S_u > 1400\text{ MPa } (200\text{ ksi}) \end{cases}$$
• For non-ferrous alloys (aluminum, copper), no true endurance limit exists; fatigue strength $S_N$ is reported at a specified life (e.g., $N = 5 \times 10^8 \text{ cycles}$).

Marin Modification Factors

The fully corrected endurance limit $S_e$ for a real mechanical component is:

$$S_e = k_a \cdot k_b \cdot k_c \cdot k_d \cdot k_e \cdot k_f \cdot S'_e$$

where:

• $k_a$ (Surface Factor): $k_a = a S_u^b$ (Ground: $a=1.58, b=-0.085$; Machined/Cold-drawn: $a=4.51, b=-0.265$; Hot-rolled: $a=57.7, b=-0.718$; As-forged: $a=272, b=-0.995$ for $S_u$ in MPa).
• $k_b$ (Size Factor): For rotating/bending shafts ($d$ in mm): $$k_b = \begin{cases} 1.24 d^{-0.107} \& \text{for } 2.79 \le d \le 51\text{ mm} \\ 1.51 d^{-0.157} \& \text{for } 51 < d \le 254\text{ mm} \end{cases}$$
• $k_c$ (Load Factor): $k_c = 1.0$ for bending, $k_c = 0.85$ for axial load, $k_c = 0.59$ for pure torsion/shear.
• $k_d$ (Temperature Factor): $k_d = 1.0$ for $T \le 450^\circ\text{C}$.
• $k_e$ (Reliability Factor): $90\% \implies 0.897$; $95\% \implies 0.868$; $99\% \implies 0.814$; $99.9\% \implies 0.753$.
• $k_f$ (Miscellaneous Factor): Accounts for corrosion, residual stress, fretting, plating.

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5. Fatigue Failure Criteria for Fluctuating Stress

When cyclic loading involves a non-zero mean stress ($\sigma_m$):

Fluctuating Stress Components

$$\sigma_a = \left|\frac{\sigma_{\text{max}} - \sigma_{\text{min}}}{2}\right| \quad (\text{Alternating Stress})$$ $$\sigma_m = \frac{\sigma_{\text{max}} + \sigma_{\text{min}}}{2} \quad (\text{Mean Stress})$$ $$\text{Stress Ratio } R = \frac{\sigma_{\text{min}}}{\sigma_{\text{max}}}, \quad \text{Amplitude Ratio } A = \frac{\sigma_a}{\sigma_m}$$

Design Equations (Factor of Safety $n$)

1. Modified Goodman Line: $$\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_u} = \frac{1}{n}$$
2. Soderberg Line (Most Conservative): $$\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_y} = \frac{1}{n}$$
3. Gerber Parabola (Best fit to experimental data): $$\frac{\sigma_a}{S_e} + \left(\frac{\sigma_m}{S_u}\right)^2 = \frac{1}{n}$$
4. ASME Elliptic Criterion: $$\left(\frac{\sigma_a}{S_e}\right)^2 + \left(\frac{\sigma_m}{S_y}\right)^2 = \frac{1}{n^2}$$

Palmgren-Miner Linear Damage Rule

For multi-stress level cumulative fatigue:

$$D = \sum_{i=1}^{k} \frac{n_i}{N_i} = \frac{n_1}{N_1} + \frac{n_2}{N_2} + \dots + \frac{n_k}{N_k} \le 1.0$$

where $n_i$ is applied cycles at stress $\sigma_i$, and $N_i$ is fatigue life at $\sigma_i$. Failure occurs when cumulative damage $D = 1.0$.

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6. Linear Elastic Fracture Mechanics (LEFM)

LEFM analyzes components containing pre-existing microcracks or flaws under load.

Stress Intensity Factor ($K_I$)

Mode I (Opening Mode) stress field near a crack tip:

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

where:

• $K_I$ = Stress intensity factor ($\text{MPa}\cdot\sqrt{\text{m}}$ or $\text{ksi}\cdot\sqrt{\text{in}}$).
• $Y$ = Dimensionless geometry correction factor ($Y = 1.0$ for an internal crack of length $2a$ in an infinite plate; $Y = 1.12$ for an edge crack of length $a$).
• $\sigma$ = Nominal applied tensile stress.
• $a$ = Crack length (for edge crack) or half-length (for internal crack).

Fracture Criterion & Plane Strain Fracture Toughness ($K_{Ic}$)

Brittle catastrophic failure occurs when:

$$K_I \ge K_{Ic}$$

where $K_{Ic}$ is the critical plane strain fracture toughness (a fundamental material property measured per ASTM E399).

Critical Crack Size ($a_c$) & Allowable Stress ($\sigma_c$)

$$a_c = \frac{1}{\pi} \left(\frac{K_{Ic}}{Y \cdot \sigma}\right)^2, \quad \sigma_c = \frac{K_{Ic}}{Y \sqrt{\pi a}}$$

Griffith Theory of Brittle Fracture

For ideal brittle solids (glass, ceramics), crack propagation stress is:

$$\sigma_f = \sqrt{\frac{2 E \gamma_s}{\pi a}}$$

where $\gamma_s$ is specific surface energy. Orowan modified this for metals by incorporating plastic work energy $\gamma_p$:

$$\sigma_f = \sqrt{\frac{2 E (\gamma_s + \gamma_p)}{\pi a}}$$

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7. Worked Engineering Calculations

Calculation 1: Fatigue Safety Factor (Goodman vs. Soderberg)

Problem: A machine shaft manufactured from AISI 1045 cold-drawn steel ($S_u = 600\text{ MPa}, S_y = 420\text{ MPa}$) operates under a fluctuating axial load producing $\sigma_{\text{max}} = 250\text{ MPa}$ and $\sigma_{\text{min}} = -50\text{ MPa}$. The fully corrected endurance limit is $S_e = 200\text{ MPa}$. Calculate the factor of safety $n$ using:

1. The Modified Goodman criterion.
2. The Soderberg criterion.

Solution:

1. Compute mean stress ($\sigma_m$) and alternating stress ($\sigma_a$):

   $$\sigma_m = \frac{\sigma_{\text{max}} + \sigma_{\text{min}}}{2} = \frac{250 + (-50)}{2} = 100\text{ MPa}$$ $$\sigma_a = \frac{\sigma_{\text{max}} - \sigma_{\text{min}}}{2} = \frac{250 - (-50)}{2} = 150\text{ MPa}$$

2. Modified Goodman Factor of Safety:

   $$\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_u} = \frac{1}{n_G} \implies \frac{150}{200} + \frac{100}{600} = \frac{1}{n_G}$$ $$0.750 + 0.1667 = 0.9167 = \frac{1}{n_G} \implies n_G = \frac{1}{0.9167} = 1.0909 \approx 1.09$$

3. Soderberg Factor of Safety:

   $$\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_y} = \frac{1}{n_S} \implies \frac{150}{200} + \frac{100}{420} = \frac{1}{n_S}$$ $$0.750 + 0.2381 = 0.9881 = \frac{1}{n_S} \implies n_S = \frac{1}{0.9881} = 1.0120 \approx 1.01$$

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Calculation 2: Critical Crack Size Determination

Problem: An alloy steel pressure vessel wall ($K_{Ic} = 65\text{ MPa}\cdot\sqrt{\text{m}}$) is subjected to a nominal hoop tensile stress $\sigma = 320\text{ MPa}$. Non-destructive ultrasonic inspection can reliably detect internal cracks with total length $2a \ge 8.0\text{ mm}$. Assuming $Y = 1.0$:

1. Calculate the critical internal crack half-length $a_c$ and total length $2a_c$.
2. Determine if the ultrasonic NDT system is adequate to prevent catastrophic fracture.

Solution:

1. Apply LEFM fracture formula:

   $$K_{Ic} = Y \sigma \sqrt{\pi a_c} \implies 65 = (1.0)(320) \sqrt{\pi a_c}$$ $$\sqrt{\pi a_c} = \frac{65}{320} = 0.203125$$ $$\pi a_c = (0.203125)^2 = 0.041260\text{ m}$$ $$a_c = \frac{0.041260}{\pi} = 0.013133\text{ m} = 13.13\text{ mm}$$

   Total critical internal crack length $2a_c = 2 \times 13.13\text{ mm} = 26.26\text{ mm}$.

2. Since the NDT resolution ($8.0\text{ mm}$) is well below the critical crack length ($26.26\text{ mm}$), the inspection system will reliably detect dangerous flaws before catastrophic fracture occurs ($8.0\text{ mm} < 26.26\text{ mm}$).

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8. MELE Exam Tips