6.1 High-Cycle Fatigue, S-N Curves & Marin Modifying Factors

6.1 High-Cycle Fatigue, S-N Curves & Marin Modifying Factors

Dynamic and cyclic loadings represent the primary cause of catastrophic structural and mechanical failures across engineering hardware. While static failure theories govern components subjected to invariant loads, dynamic failure can occur at stress levels drastically below the material's yield strength ($S_y$) or ultimate tensile strength ($S_{ut}$). Understanding high-cycle fatigue (HCF), stress-life ($S$-$N$) relationships, Marin modifying factors, and notch sensitivity is essential for machine design and failure prevention.

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The Three Physical Stages of Fatigue Failure

Fatigue is a localized, progressive, and permanent structural damage process that occurs when materials are subjected to cyclic strains and stresses. Even when macroscopic stresses remain within the elastic regime, microscopic plastic deformation occurs at localized stress risers, slip planes, or material inhomogeneities. Fatigue failure unfolds across three distinct chronological stages:

   +-------------------------------------------------------------------------+
   | Stage I: Microcrack Initiation                                          |
   | - Localized plastic shear along crystallographic slip bands (45° planes)|
   | - Microscopic slip bands form extrusions and intrusions on free surface |
   | - Typically consumes 80% to 90% of total life in smooth, pristine parts |
   +------------------------------------+------------------------------------+
                                        |
                                        v
   +------------------------------------+------------------------------------+
   | Stage II: Stable Crack Propagation                                      |
   | - Crack turns 90° to propagate in Mode I (opening mode)                 |
   | - Driven by alternating tensile principal stresses                      |
   | - Produces macroscopic beach marks and microscopic striations           |
   +------------------------------------+------------------------------------+
                                        |
                                        v
   +------------------------------------+------------------------------------+
   | Stage III: Catastrophic Fast Fracture                                   |
   | - Remaining uncracked net ligament cross-section is overstressed       |
   | - Stress intensity reaches fracture toughness: K_I >= K_Ic              |
   | - Sudden, unstable ductile shear or brittle cleavage rupture            |
   +-------------------------------------------------------------------------+

1. Stage I: Microcrack Initiation

Under cyclic shear stresses, dislocation movements cause slip along favorable crystallographic slip planes oriented at roughly 45° to the tensile stress axis. With repeated cycling, persistent slip bands (PSBs) emerge, producing microscopic valleys (intrusions) and peaks (extrusions) on the free surface. Intrusions act as severe stress micro-notches, concentrating cyclic strain until microcracks nucleate. In smooth, unnotched specimens operating in high-cycle fatigue, crack initiation consumes the vast majority (80% to 90%) of the component's total fatigue life.

2. Stage II: Stable Crack Propagation

Once a microcrack reaches a depth spanning several material grains (typically $0.1\text{ to }0.2\text{ mm}$), the stress field transitions from localized shear dominance to macroscopic tensile dominance. The crack changes direction, propagating perpendicular to the maximum alternating principal tensile stress in Mode I (opening mode). As the crack tip sharpens during tensile loading and blunts during compressive loading, it leaves distinct topographical features:

• Microscopic Striations: Minute parallel ridges created during each individual cycle of loading. Striations are visible only under scanning electron microscopy (SEM) and directly map cycle-by-cycle crack advance.
• Macroscopic Beach Marks (Progression Marks / Clamshell Marks): Visible rings radiating outward from the initiation site, formed by changes in cyclic loading magnitude, machine stop-start intervals, or environmental oxidation during prolonged operation. Beach marks always point back toward the initiation site.

3. Stage III: Catastrophic Fast Fracture

As the stable crack reduces the uncracked cross-sectional area, the effective stress on the remaining ligament rises exponentially. When the crack reaches a critical flaw length $a_c$ such that the stress intensity factor equals or exceeds the material's plane-strain fracture toughness ($K_I \ge K_{Ic}$), the crack propagates unstably at high speed. The final failure surface exhibits ductile shear lips or brittle crystalline cleavage, representing an immediate, catastrophic rupture without warning.

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The Rotating-Beam Test & Uncorrected Endurance Limit ($S_e'$)

Laboratory baseline fatigue properties are established using the standardized R.R. Moore rotating-beam test machine:

       Pure Bending Moment (No Shear Force)           Fully Reversed Bending
       ====================================           ======================
                 F/2             F/2                  Stress Ratio: R = -1
                  |               |                   Mean Stress:  sigma_m = 0
            +-----v---------------+-----+
      Motor |  =====[Specimen Neck]=====  | Bearing   Cycle Period: 1,800 rpm
            +---------------------------+             Surface:      Polished mirror

In this apparatus, a polished cylindrical specimen having a minimum neck diameter of $d = 0.30\text{ in}$ ($7.62\text{ mm}$) is supported in bearings and loaded with pure bending moments. As the motor rotates the specimen at high speed (typically $1,800\text{ rpm}$), every point on the neck surface undergoes fully reversed sinusoidal tension and compression ($R = \sigma_{\min}/\sigma_{\max} = -1$) with zero mean stress ($\sigma_m = 0$).

Endurance Limit Estimation for Steels

When test results for polished carbon and alloy steels are plotted on semi-logarithmic or log-log axes ($\%$ of $S_{ut}$ vs cycles $N$), the curve flattens into a horizontal plateau at approximately $10^6$ cycles. This threshold stress is the uncorrected endurance limit ($S_e'$), below which the specimen can endure an infinite number of load cycles without fracturing:

$$S_e' = \begin{cases} 0.50 S_{ut} & \text{for } S_{ut} \le 200\text{ kpsi } (1400\text{ MPa}) \\ 100\text{ kpsi } (700\text{ MPa}) & \text{for } S_{ut} > 200\text{ kpsi } (1400\text{ MPa}) \end{cases}$$

Nonferrous Alloys vs. Steels

Ferrous metals, titanium alloys, and certain polymers display a true endurance limit plateau. In contrast, nonferrous engineering alloys (including wrought and cast aluminum, magnesium, brass, and copper alloys) do not possess a true endurance limit. Their $S$-$N$ curves continue to decrease continuously with increasing cycles. For nonferrous alloys, engineers must design for a finite service life using a fatigue strength ($S_{f,N}$) defined at a designated cycle count—typically $S_{f,5\times 10^8} \approx 0.30\text{ to }0.40 S_{ut}$ at $N = 5 \times 10^8$ cycles.

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Finite-Life Fatigue Strength & Basquin's Equation

In many applications (e.g., aerospace frames, automotive suspension arms, pressure relief cycles), designing for infinite life results in excessive weight and cost. Instead, components are designed for a specified finite service life ($N$ cycles) in the high-cycle fatigue regime ($10^3 \le N \le 10^6$ cycles).

  Log(Stress) ^
       S_ut   +----------------------+
              | Low-Cycle Fatigue    |
              | (Plastic Strain Dom) |
    f * S_ut  + - - - - - - - - - - - + - - - - - - - - - - - - - - - - - +
              |                       \    High-Cycle Fatigue             |
              |                        \   (Basquin Power-Law Regime)     |
         S_e  + - - - - - - - - - - - - - - - - - - - - - - - +===========+
              |                                               | Infinite  |
              |                                               | Life Zone |
              +-----------------------+-----------------------+-----------+--->
                                    10^3                    10^6         Log(N)

Basquin's Power-Law Formulation

On a log-log coordinate grid, the high-cycle fatigue curve between $10^3$ cycles and $10^6$ cycles is idealized as a straight line described by Basquin's equation:

$$S_f = a N^b$$

where:

• $S_f$ is the fatigue strength at $N$ cycles.
• $a$ is the fatigue strength coefficient.
• $b$ is the fatigue strength exponent (Basquin's slope).
• $f$ is the fatigue strength fraction at $10^3$ cycles ($S_{10^3} = f S_{ut}$).

For typical steels in bending, $f$ ranges from $0.75$ to $0.90$, commonly approximated as $f = 0.90$ for $S_{ut} \le 70\text{ kpsi}$ and decreasing toward $f = 0.80\text{--}0.85$ for high-strength steels.

At the two boundary conditions:

1. At $N = 10^3$ cycles: $S_f = f S_{ut} = a (10^3)^b$
2. At $N = 10^6$ cycles: $S_f = S_e = a (10^6)^b$

Solving these two simultaneous logarithmic equations yields the fundamental Basquin constants:

$$a = \frac{(f S_{ut})^2}{S_e}$$ $$b = -\frac{1}{3} \log_{10}\left(\frac{f S_{ut}}{S_e}\right)$$

Calculating Cycles to Failure

When a machine element is subjected to a completely reversed alternating stress amplitude $\sigma_{\text{rev}}$ that exceeds the endurance limit ($S_e < \sigma_{\text{rev}} < f S_{ut}$), its predicted fatigue life $N$ until failure is calculated by inverting Basquin's equation:

$$N = \left(\frac{\sigma_{\text{rev}}}{a}\right)^{1/b}$$

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Marin Endurance Modifying Factors ($S_e = k_a k_b k_c k_d k_e k_f S_e'$)

The laboratory uncorrected endurance limit $S_e'$ applies strictly to mirror-polished, small ($0.30\text{ in}$), unnotched specimens tested in room-temperature dry air under pure rotating bending. Real-world mechanical parts differ in surface texture, cross-sectional geometry, loading mode, operating temperature, and environmental exposure. Joseph Marin formulated an empirical modifying framework to derive the working endurance limit ($S_e$):

$$S_e = k_a k_b k_c k_d k_e k_f S_e'$$

1. Surface Condition Factor ($k_a$)

Fatigue cracks almost invariably initiate at the outer free surface where alternating tensile stresses peak and surface micro-irregularities act as stress risers. The surface factor is computed using the power-law relation:

$$k_a = a S_{ut}^b$$

Note: These are the widely used Shigley coefficients; other references and later editions publish somewhat different fits, so use the table given in the problem or reference handbook. High-strength steels suffer much more severe percentage reductions in $k_a$ under rough finishes than softer, ductile steels because of their elevated notch sensitivity.

2. Size Factor ($k_b$)

Larger components exhibit lower fatigue limits due to two mechanisms: (1) statistical volume effect (a larger volume of highly stressed material has a higher statistical probability of containing a critical flaw or inclusion), and (2) stress gradient effect (larger diameters experience shallower stress gradients, leaving a thicker sub-surface layer subjected to near-peak stresses).

For rotating cylindrical shafts in bending and torsion:

$$k_b = \begin{cases} (d / 0.3)^{-0.107} = 0.879 d^{-0.107} & \text{for } 0.11\text{ in} \le d \le 2.0\text{ in} \\ 0.91 d^{-0.157} & \text{for } 2.0\text{ in} < d \le 10\text{ in} \end{cases}$$

Some references simply set $k_b = 1$ for $d \le 0.30\text{ in}$ ($7.6\text{ mm}$, the rotating-beam specimen size). In metric units ($d$ in mm):

$$k_b = \begin{cases} 1.24 d^{-0.107} & \text{for } 2.79\text{ mm} \le d \le 51\text{ mm} \\ 1.51 d^{-0.157} & \text{for } 51\text{ mm} < d \le 254\text{ mm} \end{cases}$$

For non-rotating bending or non-circular sections, an equivalent diameter $d_e$ is calculated by equating the volume of material stressed above 95% of maximum stress ($A_{95}$) to that of a rotating round beam ($A_{95} = 0.0766 d^2$):

• Rectangular section ($b \times h$, bending about horizontal axis): $d_e = 0.808 \sqrt{b h}$
• Non-rotating round bar (diameter $d$, bending): $d_e = 0.370 d$
• Pure Axial Loading: In pure axial tension-compression, the stress is uniformly distributed across the entire cross-section (zero stress gradient). Therefore, by definition:

$$k_b = 1.0 \quad \text{(Pure Axial Loading)}$$

3. Loading Factor ($k_c$)

Accounts for the stress state and cross-sectional gradient compared to pure rotating bending ($k_c = 1.0$):

$$k_c = \begin{cases} 1.00 & \text{for Bending} \\ 0.85 & \text{for Pure Axial Loading} \\ 0.59 & \text{for Pure Torsion / Shear} \end{cases}$$

4. Temperature Factor ($k_d$)

At moderate operating temperatures ($T \le 450^\circ\text{F} / 230^\circ\text{C}$), fatigue strength tracks tensile strength. At elevated temperatures ($T > 450^\circ\text{F}$), oxidation accelerates and thermal creep mechanisms interact with cyclic strain, degrading fatigue resistance:

$$k_d = \frac{S_T}{S_{RT}} = 0.975 + 0.432\times 10^{-3} T - 0.115\times 10^{-5} T^2 + 0.104\times 10^{-8} T^3 - 0.595\times 10^{-12} T^4$$

(with $T$ in $^\circ\text{F}$). For temperatures from $70^\circ\text{F}$ to $400^\circ\text{F}$, $k_d \approx 1.00$.

5. Reliability Factor ($k_e$)

Standard fatigue endurance limits represent a $50\%$ survival probability (the mean of experimental scatter, with typical coefficient of variation $C_v \approx 0.08$). For mission-critical mechanical systems, higher reliability is enforced through $k_e = 1 - 0.08 z_a$:

6. Miscellaneous Effects Factor ($k_f$)

All other operating and environmental factors are grouped into $k_f$:

• Residual Stresses: Compressive surface stresses introduced by shot peening, roller burnishing, or cold-working inhibit crack opening and can boost fatigue life significantly ($k_f > 1.0$). In contrast, tensile residual stresses from unannealed grinding or welding degrade life.
• Corrosion Fatigue: In corrosive environments (e.g., saltwater), pitting acts as continuous sharp notch generation. The $S$-$N$ curve loses its endurance limit entirely; $k_f$ can fall below $0.30$.
• Electroplating: Hard chrome plating or nickel plating introduces high tensile residual stresses and hydrogen embrittlement, reducing the endurance limit by $20\%\text{ to }40\%$ ($k_f \approx 0.60\text{--}0.80$).
• Fretting Corrosion: Microscopic oscillating motion between tightly clamped contact surfaces produces micro-galling and debris pits, drastically lowering fatigue strength ($k_f \approx 0.25\text{--}0.60$).

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Fatigue Stress Concentration Factor ($K_f$) & Notch Sensitivity ($q$)

Geometric stress concentrations (fillets, keyways, transverse holes, retaining ring grooves) concentrate stress elastically by a factor $K_t = \sigma_{\max} / \sigma_0$. However, under dynamic cyclic loading, materials do not experience the full geometric penalty $K_t$. Instead, they exhibit a smaller effective fatigue stress concentration factor ($K_f$):

$$K_f = \frac{\text{Unnotched Endurance Limit}}{\text{Notched Endurance Limit}} = \frac{S_e}{S_{e,\text{notched}}}$$

Notch Sensitivity Index ($q$)

The notch sensitivity $q$ is defined as the fraction of the geometric penalty realized by the material:

$$q = \frac{K_f - 1}{K_t - 1} \iff K_f = 1 + q(K_t - 1)$$

where $0 \le q \le 1$:

• If $q = 0$: $K_f = 1.0$ (material is completely insensitive to the notch, typical of grey cast iron where internal graphite flakes dwarf external notches).
• If $q = 1$: $K_f = K_t$ (material is fully sensitive to the notch, typical of ultra-high-strength brittle alloys).

Neuber's Constant Formulation

Notch sensitivity is quantified by Neuber's equation as a function of notch root radius $r$ (or $\rho$) and the Neuber material constant $\sqrt{a}$:

$$q = \frac{1}{1 + \frac{\sqrt{a}}{\sqrt{r}}}$$

For steels under bending or axial fatigue loading, Neuber's constant $\sqrt{a}$ (in $\sqrt{\text{in}}$) correlates directly with ultimate tensile strength:

$$\sqrt{a} = 0.246 - 3.08\times 10^{-3} S_{ut} + 1.51\times 10^{-5} S_{ut}^2 - 2.67\times 10^{-8} S_{ut}^3 \quad (S_{ut} \text{ in kpsi})$$

For torsional shear loading, the shear notch sensitivity $q_s$ and fatigue shear stress concentration factor $K_{fs}$ use the torsional Neuber fit $\sqrt{a_s} = 0.190 - 2.51\times 10^{-3} S_{ut} + 1.35\times 10^{-5} S_{ut}^2 - 2.67\times 10^{-8} S_{ut}^3$ ($\sqrt{\text{in}}$, with $S_{ut}$ in kpsi):

$$K_{fs} = 1 + q_s(K_{ts} - 1)$$

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Summary Table: Marin Modifying Factors & Relationships

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Realistic Worked Engineering Scenario: Transmission Shaft Step Fillet

                     D = 1.875 in             d = 1.50 in
                +---------------------+        +---------------------+
                |                     |        |                     |
   M <======    |                     +---r----+                     |    ======> M
   (Reversed)   |                         (Fillet: r = 0.150 in)     |    (Reversed)
                |                     +---r----+                     |
                +---------------------+        +---------------------+

Problem Statement

A solid stepped drive shaft for a high-speed centrifugal pump is fabricated from cold-drawn AISI 1045 steel having properties:

• Ultimate tensile strength: $S_{ut} = 91.0\text{ kpsi}$ ($627\text{ MPa}$)
• Yield strength: $S_y = 77.0\text{ kpsi}$ ($531\text{ MPa}$)

Geometric and operational parameters:

• Major diameter: $D = 1.875\text{ in}$
• Minor diameter: $d = 1.500\text{ in}$
• Shoulder fillet radius: $r = 0.150\text{ in}$
• Theoretical stress concentration factor in bending: $K_t = 1.68$
• Surface finish: Machined ($a = 2.70$, $b = -0.265$ for $S_{ut}$ in kpsi)
• Operating environment: Ambient temperature ($70^\circ\text{F}$), dry, no corrosive agents
• Target reliability: $99\%$ ($k_e = 0.814$)
• Applied cyclic load: Completely reversed bending moment $M = 3,500\text{ lbf}\cdot\text{in}$ ($R = -1$)

Calculate:

1. The uncorrected endurance limit $S_e'$ and fully modified endurance limit $S_e$.
2. The fatigue notch sensitivity $q$ and fatigue stress concentration factor $K_f$.
3. The peak alternating bending stress $\sigma_a$ and the factor of safety $n_f$ for infinite life.
4. The predicted fatigue life $N$ (cycles) if an overload moment of $M_{\text{over}} = 6,500\text{ lbf}\cdot\text{in}$ is applied (using $f = 0.875$).

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Step-by-Step Mathematical Solution

Step 1: Uncorrected Endurance Limit ($S_e'$)

Because $S_{ut} = 91.0\text{ kpsi} \le 200\text{ kpsi}$:

$$S_e' = 0.50 S_{ut} = 0.50 \times 91.0\text{ kpsi} = 45.50\text{ kpsi}$$

Step 2: Marin Endurance Modifying Factors

1. Surface factor ($k_a$):

$$k_a = a S_{ut}^b = 2.70 \times (91.0)^{-0.265} = 2.70 \times 0.30259 = 0.8170 \approx 0.817$$

1. Size factor ($k_b$): For rotating bending with $d = 1.50\text{ in}$ ($0.11 \le d \le 2.0\text{ in}$):

$$k_b = 0.879 d^{-0.107} = 0.879 \times (1.50)^{-0.107} = 0.879 \times 0.9575 = 0.842$$

(Equivalently, $(d/0.3)^{-0.107} = 5^{-0.107} = 0.842$.)

1. Load factor ($k_c$): For pure bending loading, $k_c = 1.00$.

2. Temperature factor ($k_d$): At $70^\circ\text{F}$, $k_d = 1.00$.

3. Reliability factor ($k_e$): For $99\%$ reliability, $k_e = 0.814$.

4. Miscellaneous factor ($k_f$): Standard baseline environment, $k_f = 1.00$.

Compute fully corrected endurance limit $S_e$:

$$S_e = k_a k_b k_c k_d k_e k_f S_e' = (0.817) \times (0.842) \times (1.00) \times (1.00) \times (0.814) \times (1.00) \times 45.50\text{ kpsi}$$ $$S_e = 0.5600 \times 45.50\text{ kpsi} = 25.48\text{ kpsi}$$

Step 3: Notch Sensitivity $q$ and Fatigue Concentration $K_f$

1. Determine Neuber constant $\sqrt{a}$ for $S_{ut} = 91\text{ kpsi}$ (bending):

$$\sqrt{a} = 0.246 - 3.08\times 10^{-3}(91) + 1.51\times 10^{-5}(91)^2 - 2.67\times 10^{-8}(91)^3 = 0.246 - 0.2803 + 0.1250 - 0.0201 = 0.0706\,\sqrt{\text{in}}$$

1. Compute notch sensitivity $q$ with fillet radius $r = 0.150\text{ in}$:

$$q = \frac{1}{1 + \frac{\sqrt{a}}{\sqrt{r}}} = \frac{1}{1 + \frac{0.0706}{\sqrt{0.150}}} = \frac{1}{1 + \frac{0.0706}{0.3873}} = \frac{1}{1 + 0.1824} = 0.8457 \approx 0.846$$

1. Compute fatigue stress concentration factor $K_f$:

$$K_f = 1 + q(K_t - 1) = 1 + 0.846(1.68 - 1) = 1 + 0.846(0.68) = 1 + 0.575 = 1.575$$

Step 4: Nominal Stress, Peak Stress & Factor of Safety

1. Elastic section modulus for solid circular cross-section at minor diameter $d = 1.50\text{ in}$:

$$Z = \frac{\pi d^3}{32} = \frac{\pi (1.50)^3}{32} = \frac{3.375 \pi}{32} = 0.33134\text{ in}^3$$

1. Nominal alternating bending stress $\sigma_0$:

$$\sigma_0 = \frac{M}{Z} = \frac{3{,}500\text{ lbf}\cdot\text{in}}{0.33134\text{ in}^3} = 10{,}563\text{ psi} = 10.56\text{ kpsi}$$

1. Actual peak alternating stress $\sigma_a$ applying $K_f$:

$$\sigma_a = K_f \sigma_0 = 1.575 \times 10.563\text{ kpsi} = 16.64\text{ kpsi}$$

1. Factor of safety for infinite life ($n_f$):

$$n_f = \frac{S_e}{\sigma_a} = \frac{25.48\text{ kpsi}}{16.64\text{ kpsi}} = 1.53$$

Because $n_f = 1.53 > 1.0$, the shaft operates safely in the infinite-life regime under the $3{,}500\text{ lbf}\cdot\text{in}$ moment.

Step 5: Finite-Life Evaluation Under Overload Moment

When the bending moment increases to $M_{\text{over}} = 6{,}500\text{ lbf}\cdot\text{in}$:

$$\sigma_{0,\text{over}} = \frac{6{,}500}{0.33134} = 19{,}617\text{ psi} = 19.62\text{ kpsi}$$ $$\sigma_{a,\text{over}} = K_f \sigma_{0,\text{over}} = 1.575 \times 19.617\text{ kpsi} = 30.90\text{ kpsi}$$

Because $\sigma_{a,\text{over}} = 30.90\text{ kpsi} > S_e = 25.48\text{ kpsi}$, the shaft has a finite fatigue life.

Compute Basquin constants ($f = 0.875$):

$$f S_{ut} = 0.875 \times 91.0\text{ kpsi} = 79.625\text{ kpsi}$$ $$a = \frac{(f S_{ut})^2}{S_e} = \frac{(79.625)^2}{25.48} = \frac{6340.14}{25.48} = 248.83\text{ kpsi}$$ $$b = -\frac{1}{3} \log_{10}\left(\frac{f S_{ut}}{S_e}\right) = -\frac{1}{3} \log_{10}\left(\frac{79.625}{25.48}\right) = -\frac{1}{3} \log_{10}(3.1250) = -\frac{1}{3}(0.49485) = -0.16495$$

Calculate cycles to failure $N$:

$$N = \left(\frac{\sigma_{a,\text{over}}}{a}\right)^{1/b} = \left(\frac{30.90}{248.83}\right)^{-1/0.16495} = (0.12418)^{-6.0624} \approx 311{,}000\text{ cycles}$$

The stepped shaft is expected to survive approximately $3.1 \times 10^5$ cycles at the overload moment before fatigue fracture.