Fatigue Analysis (AEME 5238)

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### Fatigue - What and Why? Fatigue is a failure phenomenon in materials characterized by progressive degradation under repeated loading, even when the applied loads are below the material's static strength. This degradation involves the initiation and propagation of defects (cracks, voids, damage) which eventually lead to failure. **Key Characteristics:** - **Progressive Degradation:** Defects form and grow over time. - **Repeated Loading:** Occurs under cyclic stresses (e.g., load-unload, vibration, heating-cooling). - **Below Static Strength:** Failure happens at stress levels well below the material's yield or ultimate tensile strength. **Why Study Fatigue?** - Cyclic loading is ubiquitous in engineering applications, making fatigue failure a common and critical concern. - Fatigue is a complex, "phenomenological" process, meaning its understanding is largely based on observation and experimentation. - Tools and knowledge for fatigue evaluation are continuously evolving. - Engineers must be able to identify, predict, and design against fatigue to ensure structural integrity and safety. **How to Study Fatigue:** - Experimental evidence and advanced testing tools. - Multidisciplinary approaches. - Semi-empirical expressions. - Advanced computational/modeling tools. - Accelerated testing. ### Historical Perspectives of Fatigue The study of fatigue dates back to the 19th century, driven by failures in early industrial applications. **Early Investigations (1829-1899):** - **1829: W. Albert (Germany):** Conducted repeated bend tests on iron mine-hoists, achieving 100,000 cycles. - **1843: W. Rankine (Britain):** Studied fatigue fracture and stress concentration in railway components. - **1849: E. Hodgkinson (Britain):** Investigated fatigue in wrought and cast iron railway bridges. - **1839: J.V. Poncelet:** Coined the term 'fatigue' to describe metal failures. - **1854: F. Braithwaite:** Used 'fatigue' to describe cracking under repeated loading. **August Wöhler (1852-1869):** - Performed extensive fatigue testing on train axles under 4-point bending. - **Key Findings:** - Cyclic strength is lower than static strength. - Fatigue failure is brittle, not ductile. - **Contribution:** Introduced the concept of **fatigue life** represented by the **S-N curve**. **R.R. Moore (1927):** - Performed fatigue tests on carefully machined specimens using a rotating-beam machine. - Applied pure bending, no transverse shear, and fully reversed stress cycles. **Notable Researchers & Their Contributions:** - **Germany:** Wöhler (1852), Gerber (1874), Bauschinger (1886), Palmgren (1924). - **UK:** Braithwaite (1854), Fairbairn (1864), Goodman (1899), Gough (1926). - **USA:** Basquin (1910), Moore (1927), Soderberg (1939), Miner (1945), Paris (1950s), Suresh (1970s). - **Netherlands:** Schijve (1950s). **Lesson Learned from History:** - Fatigue studies are mostly empirical, solving immediate technical problems. - Theories are often "top-down," based on test data. - Documented in reports, monographs, and course notes. **Major Fatigue Failures:** - **De Havilland Comet (1952-1954):** Square window corners caused stress concentrations, leading to crack propagation from repeated pressurization-depressurization cycles. Despite a high factor of safety, cyclic loading caused failure. - **Boeing 737-200 Aloha (1988):** Multiple cracks at rivet holes (knife edge) in lap joints propagated due to cyclic mechanical loading, leading to fuselage separation. **Key Message:** Fluctuating/cyclic loads are more dangerous than monotonic loads. Engineers must evaluate and address fatigue problems. ### Fatigue Damage in Materials Fatigue damage manifests at both macroscopic and microscopic levels, with distinct characteristics. **Macroscopic Characteristics (Visible to Naked Eye):** - **No macroplastic deformation:** Fracture surface appears flat, unlike ductile failures. - **Radial steps ('short lines'):** Lines radiating from the crack initiation site on the fracture surface. - **Beach marks:** Concentric rings on the fracture surface, indicating successive positions of the crack front during periods of growth (often linked to changes in loading or environment). - **Shear lips:** Occur when crack growth transitions from tensile mode to shear mode. **Microscopic Characteristics (Observable with Tools):** - **Transgranular crack growth:** Crack propagates through the grains, not along grain boundaries. - **Striations:** Fine lines on the fracture surface, each representing a single load cycle of crack advance. **Visual Examples:** - **Figure 1 (Lecture 1, Page 24):** Flat fracture surface under tension-tension loading. - **Figure 2 (Lecture 1, Page 25):** Flat fracture surface under torsional loading. - **Figure 3 (Lecture 1, Page 27):** Radial steps on a fractured surface. - **Figure 4 (Lecture 1, Page 28):** Beach marks in a compressor blade, indicating crack initiation. - **Figure 5 (Lecture 1, Page 29):** Crack growth bands (beach marks) in Al alloy with side notches. - **Figure 6 (Lecture 1, Page 30):** Shear lips showing transition in crack growth mode. - **Figure 7 (Lecture 1, Page 32):** Striations in an aircraft flap beam. - **Figure 8 (Lecture 1, Page 33):** Striations in an Al alloy sheet, corresponding to load cycles. ### Phases of Fatigue Life Fatigue life is generally divided into two main phases: initiation and propagation. **1. Initiation:** - **Cyclic Slip:** Repeated shear stress causes localized plastic deformation at the microscopic level. - **Stages:** - **Intrusion:** Formation of surface grooves. - **Slip Band:** Formation of persistent slip bands (PSBs) where plastic deformation is concentrated. - **Extrusion:** Formation of surface protrusions. - **Crack Nucleation:** Microcracks form within or along these persistent slip bands. - **Microcrack Growth:** Initial growth of these small cracks. **Visual Examples:** - **Figure (Lecture 1, Page 35):** Cyclic shear stress leading to cyclic slip, intrusion, and extrusion. - **Figure (Lecture 1, Page 36):** Cyclic slip band formation in copper. **2. Propagation:** - **Macrocrack Growth:** Once a microcrack reaches a critical size, it transitions into a macrocrack that propagates through the material. - **Characteristics:** - **Transgranular:** Crack typically grows across multiple grains. - **Striations:** Each striation on the fracture surface corresponds to one load cycle, indicating crack advancement. **Visual Examples:** - **Figure (Lecture 1, Page 42):** Propagation of a crack through multiple grains. - **Figure (Lecture 1, Page 43):** Crack propagation characterized by striations. - **Figure (Lecture 1, Page 44):** Models of striations proposed by McMillan/Pelloux and Laird. ### Fatigue Design Approaches There are three primary approaches to designing against fatigue, each with different underlying philosophies. **1. Safe Life Approach:** - **Emphasis:** Prevention of crack initiation. - **Methodology:** 1. Estimate fatigue life using experimental S-N curves (stress-life) or ε-N curves (strain-life). 2. Apply a Factor of Safety (FS) to the estimated life. - **Action:** Components are replaced at the end of their estimated life, regardless of whether damage is detected. - **Equations:** - **Stress-Life Method (S-N Curve):** Plot of stress amplitude (S) vs. number of cycles to failure (N). - **Strain-Life Method (ε-N Curve):** Plot of strain amplitude (ε) vs. number of cycles to failure (N). - **Factor of Safety (FS):** - For stress: $FS = S_2 / S_1$ (where $S_1$ is required strength, $S_2$ is estimated fatigue strength) - For cycles: $FS = N_2 / N_1$ (where $N_1$ is required cycles, $N_2$ is estimated cycles) **2. Fail Safe Approach:** - **Emphasis:** - Preventing catastrophic failure by adding redundant structures. - Detecting and monitoring crack propagation. - **Methodology:** 1. Incorporate redundant members and design features to create multiple load paths, ensuring structural integrity even if one path fails. 2. Implement periodic inspections to detect and monitor crack propagation. - **Detection Methods:** Visual inspection, optical microscopy, Non-Destructive Evaluation (NDE) techniques (e.g., eddy current, ultrasonic scan, X-ray, acoustic emission, optical fiber). - **Prediction:** Uses Linear Elastic Fracture Mechanics (LEFM) to predict fatigue life in materials with cracks. **Case Study: P&W F100 Engine (F-16 jet fighter)** - **Old Approach:** All 1,000 engine disks replaced if a crack shorter than 0.75 mm was found in one disk. - **New Approach (since 1986):** Components replaced only when they reach the end of their fatigue life OR when a crack is actually found. This saved millions of dollars. **3. Damage Tolerant Approach:** - **Emphasis:** Assumes cracks are already present in a structure and focuses on their stable growth. - **Methodology:** 1. Material selection based on high fracture toughness (resistance to crack growth or stable crack growth). 2. Fatigue crack growth analysis using Linear Elastic Fracture Mechanics (LEFM). - **Fatigue Crack Propagation Analysis (LEFM):** - Analyzes crack growth rate ($da/dN$) as a function of the stress intensity factor range ($\Delta K$). - **Equation for $\Delta K$:** $$\Delta K = \beta \Delta \sigma \sqrt{\pi a}$$ where: - $\Delta \sigma = \sigma_{max} - \sigma_{min}$ (stress range) - $\beta$ is a dimensional factor - $a$ is the crack length - **Paris and Erdogan Power Law (for Region II crack growth):** $$\frac{da}{dN} = C(\Delta K)^m$$ where: - $C$ and $m$ are scaling constants - Load ratio $R = \sigma_{min}/\sigma_{max}$ - $\Delta K = (1-R)K_{max}$ - **Integration for Fatigue Life ($N_f$):** $$\int_0^{N_f} dN = \int_{a_0}^{a_f} \frac{da}{C(\beta \Delta \sigma \sqrt{\pi a})^m}$$ If $C\beta^m(\Delta \sigma)^m \pi^{m/2} = \Gamma$, then: $$N_f = \frac{2}{(m-2)\Gamma} \left[ \frac{1}{(a_0)^{(m-2)/2}} - \frac{1}{(a_f)^{(m-2)/2}} \right]$$ (Note: This formula is for $m \neq 2$. If $m=2$, the integral results in a logarithmic term.) **Designing Against Fatigue - Key Messages:** - **Do realize:** Fatigue failures are a leading cause of mechanical failure. - **Do consider:** Effective fatigue design requires synthesis, analysis, and testing. - **Do consider:** Fatigue testing is a verification tool. - **Do incorporate:** Proper fatigue design methods when cyclic loads are involved. - **Don't rely on safety factors** to compensate for poor design. - **Don't overlook combined effects** (load, environment, geometry, stress, time, microstructure). - **Don't hesitate to use fracture mechanics.** ### Monotonic Stress-Strain Behavior Understanding monotonic stress-strain behavior is fundamental to fatigue analysis, as it defines the material's response under a single application of load. **Solid Cylinder Under Tension:** - **Initial State:** Original length ($l_o$), original area ($A_o$). - **Deformed State:** Instantaneous length ($l$), instantaneous area ($A$), applied load ($P$). **1. Engineering Stress ($\sigma_{eng}$):** - **Definition:** Applied load divided by the original cross-sectional area. - **Equation:** $$\sigma_{eng} = \frac{P}{A_o} \quad (1)$$ **2. Engineering Strain ($\epsilon_{eng}$):** - **Definition:** Change in length divided by the original length. - **Equation:** $$\epsilon_{eng} = \frac{l - l_o}{l_o} = \frac{\Delta l}{l_o} \quad (2)$$ where instantaneous length $l = l_o + \Delta l \quad (3)$ **3. True Strain ($\epsilon_{true}$):** - **Definition:** Accounts for the instantaneous change in length, integrated over the deformation path. - **Equation:** $$\epsilon_{true} = \int_{l_o}^{l} \frac{dl}{l} = \ln \left( \frac{l}{l_o} \right) \quad (4)$$ - **Substituting (3) into (4):** $$\epsilon_{true} = \ln \left( \frac{l_o + \Delta l}{l_o} \right) = \ln \left( 1 + \frac{\Delta l}{l_o} \right) \quad (5)$$ - **True strain in terms of engineering strain (valid up to necking):** $$\epsilon_{true} = \ln(1 + \epsilon_{eng}) \quad (6)$$ **4. True Stress ($\sigma_{true}$):** - **Definition:** Applied load divided by the instantaneous cross-sectional area. - **Equation:** $$\sigma_{true} = \frac{P}{A} \quad (7)$$ - **From (1), $P = \sigma_{eng} A_o \quad (8)$. Substituting into (7):** $$\sigma_{true} = \sigma_{eng} \left( \frac{A_o}{A} \right) \quad (9)$$ - **Assuming constant volume during deformation ($A_o l_o = A l = \text{constant}$) (10), we can rearrange:** $$\frac{l}{l_o} = \frac{A_o}{A} \quad (11)$$ - **Taking the natural logarithm of (11):** $$\ln \left( \frac{l}{l_o} \right) = \ln \left( \frac{A_o}{A} \right) \quad (12)$$ - **From (4), $\epsilon_{true} = \ln(l/l_o)$, so true strain can also be expressed in terms of area:** $$\ln \left( \frac{A_o}{A} \right) = \epsilon_{true} = \ln(1 + \epsilon_{eng}) \quad (13)$$ - **This implies the ratio of areas:** $$\frac{A_o}{A} = 1 + \epsilon_{eng} \quad (14)$$ - **Substituting (14) into (9), true stress can be written as:** $$\sigma_{true} = \sigma_{eng}(1 + \epsilon_{eng}) \quad (15)$$ **Stress-Strain Curves:** - **Comparison:** Engineering stress-strain curves typically peak and then drop, while true stress-strain curves continuously rise until fracture, as they account for the decreasing cross-sectional area. - **Figure (Lecture 2, Page 14):** Illustrates the difference between engineering and true stress-strain curves. **True Fracture Strength ($\sigma_f$):** - **Definition:** True stress at fracture. - **Equation:** $$\sigma_f = \frac{P_f}{A_f} \quad (16)$$ where $P_f$ is fracture load and $A_f$ is area at fracture. **True Fracture Ductility/Strain ($\epsilon_f$):** - **Definition:** True strain at fracture. - **Equation:** $$\epsilon_f = \ln \left( \frac{A_o}{A_f} \right) = \ln \left( \frac{1}{1 - RA} \right) \quad (17)$$ where $RA$ is the Reduction in Area. **Reduction in Area ($RA$):** - **Definition:** The fractional decrease in cross-sectional area at the point of fracture. - **Equation:** $$RA = \frac{A_o - A_f}{A_o}$$ **Decomposition of Total Strain:** Total true strain ($\epsilon_{total}$) is decomposed into elastic ($\epsilon_e$) and plastic ($\epsilon_p$) components. - **Equation:** $$\epsilon_{total} = \epsilon_e + \epsilon_p \quad (18)$$ - **Elastic True Strain ($\epsilon_e$):** Linear portion, recoverable upon unloading. - **Plastic True Strain ($\epsilon_p$):** Nonlinear portion, not recoverable upon unloading. - **Figure (Lecture 2, Page 16):** Shows loading-unloading curve with elastic and plastic strain components. **Stress-Strain Relationships (Power Law):** The relationship between true stress and plastic strain in log-log scale is often modeled by a power law. - **Equation:** $$\sigma_{true} = K(\epsilon_p)^n \quad (19)$$ where: - $K$ = strength coefficient - $n$ = strain hardening exponent - **Plastic Strain in terms of True Stress:** $$\epsilon_p = \left( \frac{\sigma_{true}}{K} \right)^{1/n} \quad (20)$$ - **Similar relationship for true fracture strength and true fracture ductility/strain:** $$\sigma_f = K(\epsilon_f)^n \quad (21)$$ - **Strength coefficient K can also be expressed as:** $$K = \frac{\sigma_f}{(\epsilon_f)^n} \quad (22)$$ - **Plastic strain ($\epsilon_p$) can be written as (substituting (22) into (20)):** $$\epsilon_p = \left( \frac{\sigma_{true}}{\sigma_f/\epsilon_f^n} \right)^{1/n} = \epsilon_f \left( \frac{\sigma_{true}}{\sigma_f} \right)^{1/n} \quad (23, 24)$$ **Elastic Strain ($\epsilon_e$):** - **Equation (Hooke's Law):** $$\epsilon_e = \frac{\sigma_{true}}{E} \quad (25)$$ where $E$ is Young's Modulus. **Total True Strain (combining elastic and plastic):** - **Equation:** $$\epsilon_{true} = \frac{\sigma_{true}}{E} + \epsilon_f \left( \frac{\sigma_{true}}{\sigma_f} \right)^{1/n} \quad (26)$$ This equation allows for the calculation of total true strain based on stress and material properties. ### Cyclic Stress-Strain Behavior Cyclic stress-strain behavior describes how materials respond to repeated loading, which can differ significantly from monotonic behavior. **One Cycle of Stress-Strain Curve (Hysteresis Loop):** - **Figure (Lecture 2, Page 23):** Shows a typical hysteresis loop for one cycle of stress-strain. - **Remark:** The area enclosed by the hysteresis loop represents the energy dissipated per unit volume (plastic deformation work) during one cycle. **Key Parameters from the Cycle:** - **Strain Amplitude ($\epsilon_a$):** Half of the total strain range. $$\epsilon_a = \frac{\Delta \epsilon}{2} \quad (27)$$ - **Stress Amplitude ($\sigma_a$):** Half of the total stress range. $$\sigma_a = \frac{\Delta \sigma}{2} \quad (28)$$ - **Total Strain Range ($\Delta \epsilon$):** Sum of elastic and plastic strain ranges. $$\Delta \epsilon = \Delta \epsilon_e + \Delta \epsilon_p \quad (29)$$ - **Strain Amplitude in terms of Elastic and Plastic Strains:** $$\frac{\Delta \epsilon}{2} = \frac{\Delta \epsilon_e}{2} + \frac{\Delta \epsilon_p}{2} \quad (30)$$ - **Using Hooke's Law ($\Delta \epsilon_e = \Delta \sigma / E$), the total strain amplitude can be expressed as:** $$\frac{\Delta \epsilon}{2} = \frac{\Delta \sigma}{2E} + \frac{\Delta \epsilon_p}{2} \quad (31)$$ **Bauschinger Effect (1886):** - **Concept:** A phenomenon where the yield strength of a material decreases when the direction of loading is reversed after initial plastic deformation. - **Experiment:** Metal is loaded in tension beyond its elastic limit, then unloaded and loaded in compression. - **Observation:** Inelastic (plastic) strain in compression develops before the original tensile yield strength is reached. - **Equation:** $$\sigma_y^{(-)} 1.4 \quad (33)$ - **Cyclic Softening:** If $\frac{\sigma_{ult}}{\sigma_y} 0.20$ - **Cyclic Softening:** If $n ### Test Methods to Obtain Cyclic Stress-Strain Curves Cyclic stress-strain curves are crucial for fatigue analysis and differ from monotonic curves. **Monotonic vs. Cyclic Stress-Strain Curves:** - Provide quantitative assessment of changes in mechanical behavior due to cyclic loading. - **Cyclic Softening:** Produces cyclic yield strength **lower** than monotonic yield strength. - **Cyclic Hardening:** Produces cyclic yield strength **higher** than monotonic yield strength. **Method 1: Companion Samples** - **Applicability:** Materials with symmetric tension and compression behavior. - **Figure (Lecture 2, Page 40):** Shows multiple hysteresis loops from companion samples. - **Steps:** 1. Test specimens at various strain levels. 2. Superimpose all stable hysteresis loops (from each specimen). 3. Connect the tips of these stable hysteresis loops to form the cyclic stress-strain curve. **Method 2: Incremental Step Test** - **Applicability:** Materials with symmetric tension and compression behavior. - **Figure (Lecture 2, Page 42):** Shows a strain-time cycle with increasing and decreasing strain levels. - **Steps:** 1. Test specimens **cyclically** across multiple blocks with varying levels of strain. After 3-4 blocks, the material behavior typically stabilizes. 2. Test specimens **monotonically** after stabilization until failure. The resulting stress-strain curve resembles that obtained from the 'Companion Samples' method. ### Stabilized Cyclic Stress-Strain Relationships Once a material reaches a cyclically stable state, its stress-strain relationship can be described by a power law. **Cyclically Stable Stress-Strain:** - Plotting true stress ($\sigma$) and true cyclic plastic strain ($\epsilon_p$) on a log-log scale. - **Figure (Lecture 2, Page 45):** Log-log plot showing the linear relationship for stable cyclic stress-strain. **Estimating Coefficient and Exponent:** - **Cyclic Strength Coefficient ($K'$):** - Unit: MPa or ksi. - Value of stress when $\epsilon_p = 1$. - **Cyclic Strain Hardening Exponent ($n'$):** - Dimensionless. - Typically ranges from 0.10 to 0.25, often close to 0.15 for most metals. **Power Law for Cyclically Stable Materials:** - **Equation:** $$\sigma = K'(\epsilon_p)^{n'} \quad (36)$$ where: - $\sigma$ = cyclically stable stress amplitude - $\epsilon_p$ = cyclically stable plastic strain amplitude - $K'$ = cyclic strength coefficient - $n'$ = cyclic strain hardening exponent **True Cyclic Plastic Strain:** - **Rearranging (36) for $\epsilon_p$:** $$\epsilon_p = \left( \frac{\sigma}{K'} \right)^{1/n'} \quad (37)$$ - **Total Cyclic Strain ($\epsilon_{total}$) (Massing Hypothesis, 1926):** - G. Massing's hypothesis suggests that the stable cyclic stress-strain curve can be represented by the monotonic true stress-plastic strain curve scaled by a factor of 2. - $\Delta \sigma_1 = 2 \sigma_1 \quad (39)$ - $\Delta \epsilon_1 = 2 \epsilon_1 \quad (40)$ - **Massing's hypothesis for total strain:** $$\epsilon_1 = \frac{\sigma_1}{E} + \left( \frac{\sigma_1}{K'} \right)^{1/n'} \quad (41)$$ This relates the stress and strain amplitudes for a full cycle. - **Modified Total Cyclic Strain Amplitude:** - Using $\Delta \epsilon_1 = 2 \epsilon_1$ and $\Delta \sigma_1 = 2 \sigma_1$, and dividing by 2: $$\frac{\Delta \epsilon_1}{2} = \frac{\Delta \sigma_1}{2E} + \frac{1}{2} \left( \frac{\Delta \sigma_1}{2K'} \right)^{1/n'} \quad (42)$$ - **Multiplying by 2 to get the total strain range:** $$\Delta \epsilon_1 = \frac{\Delta \sigma_1}{E} + 2 \left( \frac{\Delta \sigma_1}{2K'} \right)^{1/n'} \quad (43)$$ **Example (Lecture 2, Page 52):** - **Problem:** A material has properties $E = 30,000 \text{ ksi}$, $n' = 0.202$, and $K' = 174.6 \text{ ksi}$. Subjected to a fully reversed cyclic strain with $\Delta \epsilon = 0.04$. Determine the stress-strain response. - **Solution Steps:** 1. **Initial Application (Point 1, $\epsilon_1 = 0.02$):** Use Eq. (41) to calculate $\sigma_1$. $$0.02 = \frac{\sigma_1}{30,000} + \left( \frac{\sigma_1}{174.6} \right)^{1/0.202}$$ Solving this nonlinear equation yields $\sigma_1 = 77.1 \text{ ksi}$. 2. **Successive Strain Reversals:** Use Eq. (43) to calculate the stress range $\Delta \sigma_1$. $$0.04 = \frac{\Delta \sigma_1}{30,000} + 2 \left( \frac{\Delta \sigma_1}{2 \times 174.6} \right)^{1/0.202}$$ Solving this yields $\Delta \sigma_1 = 154.2 \text{ ksi}$. 3. **Strain at Point 2 ($\epsilon_2$):** $\epsilon_2 = \epsilon_1 - \Delta \epsilon = 0.02 - 0.04 = -0.02$. 4. **Stress at Point 2 ($\sigma_2$):** $\sigma_2 = \sigma_1 - \Delta \sigma_1 = 77.1 - 154.2 = -77.1 \text{ ksi}$. - **Result:** The stress-strain response shows that for a fully reversed strain of $\pm 0.02$, the stress response is $\pm 77.1 \text{ ksi}$. - **Figure (Lecture 2, Page 53):** Graphical illustration of the solution. ### Strain-Life Curve The strain-life curve (ε-N curve) is a critical tool for fatigue design, especially in low-cycle fatigue, relating strain amplitude to fatigue life. **Review of Stress-Life (S-N Curve):** - Plots stress amplitude versus number of cycles to failure on a log-log scale. - **Basquin's Equation (for high-cycle fatigue):** $$\frac{\Delta \sigma}{2} = \sigma'_f (2N_f)^b \quad (44)$$ where: - $\frac{\Delta \sigma}{2}$ = true stress amplitude - $2N_f$ = reversals to failure (1 rev = 0.5 cycle) - $\sigma'_f$ = fatigue strength coefficient - $b$ = fatigue strength exponent **Strain-Life (ε-N Curve):** - **Elastic Strain Component:** From Hooke's Law, the elastic strain amplitude is: $$\frac{\Delta \epsilon_e}{2} = \frac{\Delta \sigma}{2E} \quad (45)$$ - **Coffin-Manson Relationship (1954/1953) for Elastic Strain:** Similar to Basquin's equation: $$\frac{\Delta \epsilon_e}{2} = \frac{\sigma'_f}{E} (2N_f)^b \quad (46)$$ - **Plastic Strain Component:** The plastic strain amplitude is formulated as: $$\frac{\Delta \epsilon_p}{2} = \epsilon'_f (2N_f)^c \quad (47)$$ where: - $\frac{\Delta \epsilon_p}{2}$ = true plastic strain amplitude - $\epsilon'_f$ = fatigue ductility coefficient - $c$ = fatigue ductility exponent (-0.5 to -0.7) **Total Strain Amplitude:** - Combining elastic and plastic components: $$\frac{\Delta \epsilon_{total}}{2} = \frac{\Delta \epsilon_e}{2} + \frac{\Delta \epsilon_p}{2} \quad (48)$$ - **Strain-Life Relationship (Coffin-Manson-Basquin Equation):** $$\frac{\Delta \epsilon}{2} = \frac{\sigma'_f}{E} (2N_f)^b + \epsilon'_f (2N_f)^c \quad (49)$$ **Graphical Illustration:** - **Figure (Lecture 2, Page 62):** Shows the strain-life curve, plotting elastic, plastic, and total strain amplitudes against reversals to failure on a log-log scale. - The elastic component dominates at high cycles (low strain). - The plastic component dominates at low cycles (high strain). - The intersection of the elastic and plastic curves indicates where elastic and plastic strain components are equal. $$\frac{\Delta \epsilon_e}{2} = \frac{\Delta \epsilon_p}{2} \quad (50)$$ **Rearranging for Fatigue Life ($2N_f$):** - From (49): $$\frac{\sigma'_f}{E} (2N_f)^b = \epsilon'_f (2N_f)^c \quad (51)$$ - Solving for $2N_f$: $$2N_f = \left( \frac{\epsilon'_f E}{\sigma'_f} \right)^{\frac{1}{b-c}} \quad (52)$$ - **Note:** The strain-life relation requires four empirical constants: $b, c, \sigma'_f, \epsilon'_f$. **Obtaining Empirical Constants:** - **Cyclic Strength Coefficient ($K'$):** $$K' = \frac{\sigma'_f}{(\epsilon'_f)^{n'}} \quad (53)$$ - **Cyclic Exponent ($n'$):** $n' = b/c$ (this is a common approximation, but not universally true for all materials). - **Fatigue Strength Exponent ($b$):** Ranges between -0.05 and -0.12, average -0.085. - **Fatigue Ductility Exponent ($c$):** Ranges between -0.5 (Coffin), -0.6 (Manson), and -0.5 to -0.7 (Morrow). - **Strain at Necking ($\epsilon'_f$):** Often approximated as the true fracture strain $\epsilon_f$. $$\epsilon'_f = \epsilon_f = \ln \left( \frac{1}{1 - RA} \right) \quad (54)$$ **Example (Lecture 2, Page 67):** - **Problem:** Given static data and fatigue data for steel, determine the fatigue strength coefficient ($\sigma'_f$), fatigue strength exponent ($b$), strain at necking ($\epsilon'_f$), fatigue ductility exponent ($c$), cyclic strength coefficient ($K'$), and cyclic exponent ($n'$). - **Static Data:** $E = 196 \text{ GPa}$, $\sigma_y = 1089 \text{ MPa}$, $\sigma_u = 1158 \text{ MPa}$, $\sigma_f = 1572 \text{ MPa}$, $\epsilon_f = 0.734 \text{ mm/mm}$, $\%RA = 52$. - **Fatigue Data:** Table of $\Delta \epsilon / 2$, $\Delta \sigma / 2$, $2N_f$, $\Delta \epsilon_p / 2$, $\Delta \epsilon_e / 2$. - **Solution Steps:** 1. **Plot Strain-Life Curves:** Plot $\Delta \epsilon_e / 2$, $\Delta \epsilon_p / 2$, and $\Delta \epsilon / 2$ vs. $2N_f$ on log-log scale. - **Figure (Lecture 2, Page 70):** Shows the three curves. 2. **Determine $\sigma'_f$ and $b$ (from elastic strain/stress-life):** Plot $\Delta \sigma / 2$ vs. $2N_f$. Fit a power law (Eq. 44). - **Result:** $\Delta \sigma / 2 = 1532.4 (2N_f)^{-0.076}$. - Thus, $\sigma'_f = 1532.4 \text{ MPa}$ and $b = -0.076$. - **Figure (Lecture 2, Page 71):** Plot for stress-life. 3. **Determine $\epsilon'_f$ and $c$ (from plastic strain-life):** Plot $\Delta \epsilon_p / 2$ vs. $2N_f$. Fit a power law (Eq. 47). - **Result:** $\Delta \epsilon_p / 2 = 0.211 (2N_f)^{-0.454}$. - Thus, $\epsilon'_f = 0.211$ and $c = -0.454$. - **Figure (Lecture 2, Page 72):** Plot for plastic strain-life. 4. **Verify Strain at Necking ($\epsilon'_f$):** Calculate $\epsilon_f$ from %RA. - $RA = 52\% = 0.52$. - $\epsilon_f = \ln \left( \frac{1}{1 - 0.52} \right) = \ln \left( \frac{1}{0.48} \right) = \ln(2.0833) = 0.734$. - The given $\epsilon_f = 0.734$ does not match the derived $\epsilon'_f = 0.211$ directly. This highlights that $\epsilon'_f$ is the fatigue ductility coefficient, which may not always be identical to the static true fracture strain, though it's often approximated. The problem derived $\epsilon'_f$ from the plastic strain-life curve fit. 5. **Determine $K'$ (cyclic strength coefficient):** Use Eq. (53). - $K' = \frac{\sigma'_f}{(\epsilon'_f)^{n'}}$. We need $n'$. 6. **Determine $n'$ (cyclic exponent):** Use $n' = b/c$. - $n' = (-0.076) / (-0.454) = 0.1674$. 7. **Calculate $K'$:** - $K' = \frac{1532.4}{(0.211)^{0.1674}} = \frac{1532.4}{0.7698} = 1988 \text{ MPa}$. - **Final Parameters:** - $\sigma'_f = 1532.4 \text{ MPa}$ - $b = -0.076$ - $\epsilon'_f = 0.211$ - $c = -0.454$ - $K' = 1988 \text{ MPa}$ - $n' = 0.1674$ ### Mean Stress Effects on Stress-Life Fatigue Mean stress significantly influences fatigue life, particularly in high-cycle fatigue, and must be accounted for in design. **Mean Stress Definition:** - **Equation:** $$\sigma_{mean} = \frac{\sigma_{max} + \sigma_{min}}{2} \quad (1)$$ **Effect of Mean Stress on S-N Curve:** - **Observation:** Tensile mean stresses generally produce shorter fatigue lives compared to compression mean stresses or fully reversed loading ($\sigma_{mean} = 0$). - **Figure (Lecture 3, Page 6):** Shows S-N curves for different mean stress levels ($S_m = \text{compression}$, $S_m = 0$, $S_m = \text{tension}$), demonstrating that tensile mean stress shifts the curve to lower cycles for a given stress amplitude. - **Cyclic Creep / Ratcheting:** Tensile mean stresses can induce an increase in mean strain over cycles, leading to cyclic creep or ratcheting. - **Figure (Lecture 3, Page 7):** Illustrates the accumulation of mean strain under tensile mean stress. **Constant-Life Diagram:** - **Purpose:** Establishes relationships between static material properties ($\sigma_{ult}$, $\sigma_y$) and fatigue parameters ($\sigma_{mean}$, $\sigma_a$, $\sigma_e$) to account for mean stress effects. - **Monotonic Parameters:** - Ultimate Stress ($\sigma_{ult}$) - Yield Stress ($\sigma_y$) - **Fatigue Parameters:** - Maximum Stress ($\sigma_{max}$) - Minimum Stress ($\sigma_{min}$) - Stress Amplitude ($\sigma_a$): $$\sigma_a = \frac{\sigma_{max} - \sigma_{min}}{2} \quad (2)$$ - Mean Stress ($\sigma_{mean}$): $$\sigma_{mean} = \frac{\sigma_{max} + \sigma_{min}}{2} \quad (3)$$ - Endurance Stress ($\sigma_e$): The stress amplitude below which fatigue failure does not occur, or occurs at a very high number of cycles (typically $10^7$ or $10^8$). **Criteria for Constant-Life Diagrams:** These criteria are empirical relationships used to predict the fatigue limit (or endurance limit) under the influence of mean stress. They are often plotted as $\sigma_a$ vs. $\sigma_{mean}$. 1. **Goodman Diagram (Simple Approximation):** - **Equation:** $$\frac{\sigma_a}{\sigma_e} + \frac{\sigma_{mean}}{\sigma_{ult}} = 1 \quad (4)$$ - **Concept:** Linear relationship, conservative for brittle materials. 2. **Gerber Diagram (Agrees well with experiments, $m=2$):** - **Equation:** $$\frac{\sigma_a}{\sigma_e} + \left( \frac{\sigma_{mean}}{\sigma_{ult}} \right)^2 = 1 \quad (5)$$ - **Concept:** Parabolic relationship, generally fits experimental data better than Goodman, especially for ductile materials. 3. **Soderberg Diagram (Conservative Fit):** - **Equation:** $$\frac{\sigma_a}{\sigma_e} + \frac{\sigma_{mean}}{\sigma_y} = 1 \quad (6)$$ - **Concept:** Uses yield strength instead of ultimate tensile strength, making it very conservative. 4. **Morrow Diagram (True Fracture Strength):** - **Equation:** $$\frac{\sigma_a}{\sigma_e} + \frac{\sigma_{mean}}{\sigma_f'} = 1 \quad (7)$$ - **Concept:** Uses the true fracture strength ($\sigma_f'$) as the limiting static strength. **Generating Data Set for Constant-Life Diagram:** 1. **Step 1:** Perform fatigue tests at various stress ratios ($R = \sigma_{min}/\sigma_{max}$). - Tension-tension ($R > 0$) - Tension-compression ($R = -1$, fully reversed) - Compression-compression ($R > 0$) (Less common, but possible for specific cases) 2. **Step 2:** Plot stress amplitude vs. fatigue life for different constant mean stress levels. - **Figure (Lecture 3, Page 15):** Shows S-N curves for various mean stresses. 3. **Step 3:** Plot test results in terms of $\sigma_{mean}$ and $\sigma_a$. This creates the constant-life diagram showing the fatigue limit for various mean stresses. - **Figure (Lecture 3, Page 16):** Illustrates the Goodman, Gerber, Morrow, and Soderberg lines on a $\sigma_a$ vs. $\sigma_{mean}$ plot. **Experimental Validation and Normalized Values:** - To compare different materials or analyze data, results are often plotted with normalized values. - **Goodman:** x-axis: $\sigma_{mean}/\sigma_{ult}$, y-axis: $\sigma_a/\sigma_e$ - **Gerber:** x-axis: $\sigma_{mean}/\sigma_{ult}$, y-axis: $\sigma_a/\sigma_e$ - **Soderberg:** x-axis: $\sigma_{mean}/\sigma_y$, y-axis: $\sigma_a/\sigma_e$ - **Morrow:** x-axis: $\sigma_{mean}/\sigma_f'$, y-axis: $\sigma_a/\sigma_e$ - **Figures (Lecture 3, Pages 18-20):** Examples of constant-life diagrams for Aluminum Alloy and Steel, showing how experimental data points compare to the theoretical lines. **Problem Set 1 (Lecture 3, Page 21):** - **Problem:** Endurance limit of low-carbon steel is 450 MPa. Subjected to cyclic stresses with $\sigma_{min} = 0$ ('tension-zero' fatigue test). Ultimate tensile stress is 750 MPa. Determine the safe stress range using: (a) Goodman diagram, (b) Gerber diagram; (c) which approach is more conservative. - **Given:** $\sigma_e = 450 \text{ MPa}$, $\sigma_{min} = 0$, $\sigma_{ult} = 750 \text{ MPa}$. - **Calculate $\sigma_{mean}$ and $\sigma_a$ in terms of $\sigma_{max}$:** - $\sigma_a = (\sigma_{max} - \sigma_{min})/2 = \sigma_{max}/2$ - $\sigma_{mean} = (\sigma_{max} + \sigma_{min})/2 = \sigma_{max}/2$ - So, $\sigma_a = \sigma_{mean}$. - **(a) Goodman Diagram:** - $\frac{\sigma_a}{\sigma_e} + \frac{\sigma_{mean}}{\sigma_{ult}} = 1$ - $\frac{\sigma_a}{450} + \frac{\sigma_a}{750} = 1$ - Multiply by $450 \times 750 = 337500$: $750 \sigma_a + 450 \sigma_a = 337500$ - $1200 \sigma_a = 337500 \implies \sigma_a = 281.25 \text{ MPa}$. - Safe stress range is $2 \sigma_a = 562.5 \text{ MPa}$. - **(b) Gerber Diagram:** - $\frac{\sigma_a}{\sigma_e} + \left( \frac{\sigma_{mean}}{\sigma_{ult}} \right)^2 = 1$ - $\frac{\sigma_a}{450} + \left( \frac{\sigma_a}{750} \right)^2 = 1$ - Let $x = \sigma_a$. $\frac{x}{450} + \frac{x^2}{750^2} = 1$ - $\frac{x^2}{562500} + \frac{x}{450} - 1 = 0$ - Solve quadratic equation $ax^2 + bx + c = 0$ for $x$: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ - $a = 1/562500$, $b = 1/450$, $c = -1$. - $x = \frac{-1/450 \pm \sqrt{(1/450)^2 - 4(1/562500)(-1)}}{2(1/562500)}$ - $x = \frac{-0.002222 \pm \sqrt{0.000004938 + 0.000007111}}{0.000003555}$ - $x = \frac{-0.002222 \pm \sqrt{0.000012049}}{0.000003555} = \frac{-0.002222 \pm 0.003471}{0.000003555}$ - Taking the positive root: $x = \sigma_a = 351.3 \text{ MPa}$. - Safe stress range is $2 \sigma_a = 702.6 \text{ MPa}$. - **(c) Conservativeness:** - Goodman predicts a safe stress range of $562.5 \text{ MPa}$. - Gerber predicts a safe stress range of $702.6 \text{ MPa}$. - **Goodman is more conservative** as it predicts a lower safe stress range. **Problem Set 2 (Lecture 3, Page 22):** - **Problem:** Fatigue limit of low-carbon steel is 450 MPa. Subjected to cyclic stresses with $\sigma_{mean} = 180 \text{ MPa}$. Ultimate tensile stress is 750 MPa. Calculate (a) the maximum and minimum stresses, (b) the safe range of stress following the Goodman approach. - **Given:** $\sigma_e = 450 \text{ MPa}$, $\sigma_{mean} = 180 \text{ MPa}$, $\sigma_{ult} = 750 \text{ MPa}$. - **(a) Goodman Approach for $\sigma_a$:** - $\frac{\sigma_a}{\sigma_e} + \frac{\sigma_{mean}}{\sigma_{ult}} = 1$ - $\frac{\sigma_a}{450} + \frac{180}{750} = 1$ - $\frac{\sigma_a}{450} + 0.24 = 1 \implies \frac{\sigma_a}{450} = 0.76$ - $\sigma_a = 0.76 \times 450 = 342 \text{ MPa}$. - **(b) Maximum and Minimum Stresses:** - $\sigma_a = (\sigma_{max} - \sigma_{min})/2$ - $\sigma_{mean} = (\sigma_{max} + \sigma_{min})/2$ - From $\sigma_{mean} = 180 \text{ MPa}$: $\sigma_{max} + \sigma_{min} = 360 \text{ MPa}$. - From $\sigma_a = 342 \text{ MPa}$: $\sigma_{max} - \sigma_{min} = 684 \text{ MPa}$. - Adding the two equations: $2 \sigma_{max} = 1044 \text{ MPa} \implies \sigma_{max} = 522 \text{ MPa}$. - Substituting $\sigma_{max}$ into $\sigma_{max} + \sigma_{min} = 360$: $522 + \sigma_{min} = 360 \implies \sigma_{min} = -162 \text{ MPa}$. - Safe stress range is $2 \sigma_a = 684 \text{ MPa}$. ### Mean Stress Effects on Strain-Life Fatigue Mean stress also affects strain-life, especially in low-cycle fatigue, though its influence can relax over time. **Relaxation of Mean Stress:** - **Concept:** In strain-controlled fatigue cycles with a constant mean strain ($\epsilon_{mean}$), the mean stress ($\sigma_{mean}$) tends to relax (decrease) over time due to cyclic plasticity. - **Figure (Lecture 3, Page 24):** Shows strain-time and corresponding stress-time plots, illustrating the reduction of mean stress while mean strain remains constant. - **Mechanism:** Higher strain amplitudes ($\epsilon_a$) lead to higher relaxation of mean stress. - **Observation:** This phenomenon is particularly observed in high-cycle fatigue. **Mean Stress Effects in Fatigue Life Prediction (Strain-Life):** 1. **Method 1: Morrow's Mean Stress Method** - **Recall Strain Amplitude:** $\epsilon_a = \frac{\Delta \epsilon}{2} \quad (8)$ - **Morrow's Equation:** Modifies the strain-life equation to include mean stress. $$\epsilon_a = \frac{\sigma'_f - \sigma_{mean}}{E} (2N_f)^b + \epsilon'_f (2N_f)^c \quad (9)$$ - **Note:** $\sigma_{mean}$ is positive (+) for tensile mean stress and negative (-) for compressive mean stress. 2. **Method 2: Manson and Halford (1981)** - **Equation:** Another modification to the strain-life equation. $$\epsilon_a = \frac{\sigma'_f - \sigma_{mean}}{E} (2N_f)^b + \epsilon'_f \left( \frac{\sigma'_f - \sigma_{mean}}{\sigma'_f} \right)^{c/b} (2N_f)^c \quad (10)$$ - **Concept:** This equation aims to better capture the interaction between mean stress and the plastic strain component. 3. **Method 3: Smith, Watson, and Topper (SWT Parameter, 1970)** - **Concept:** This method incorporates the maximum stress ($\sigma_{max}$) into the strain-life relationship, providing a good correlation for a wider range of materials, especially where mean stress effects are significant. - **Equation:** $$\epsilon_a \sigma_{max} = \frac{(\sigma'_f)^2}{E} (2N_f)^{2b} + \sigma'_f \epsilon'_f (2N_f)^{b+c} \quad (11)$$ (The provided equation (11) in the lecture is slightly different, it seems to combine the elastic and plastic terms in a single bracket. The standard SWT parameter is $\sigma_{max} \epsilon_a = \frac{(\sigma'_f)^2}{E}(2N_f)^{2b} + \sigma'_f \epsilon'_f (2N_f)^{b+c}$. Let's use the form from the lecture as given.) $$\epsilon_a = \frac{1}{\sigma_{max}E} \left[ (\sigma'_f)^2 (2N_f)^{2b} + \sigma'_f \epsilon'_f (2N_f)^{b+c} \right] \quad (11)$$ - **Where:** $$\sigma_{max} = \sigma_{mean} + \sigma_a \quad (12)$$ - **Remark:** $\sigma_{max}$ cannot be zero; tension must be present for this parameter to be applicable in its original form. - **Remarks on SWT:** Method 3 (SWT) generally correlates mean stress data better for a wider range of materials.