Chapter 8 Failure.pdf

# Introduction to material failure and fracture mechanics Understanding material failure is crucial for engineers to prevent undesirable consequences such as loss of life, economic losses, and disruption of services. This section introduces the concept of material failure and provides an initial overview of fracture mechanics, focusing on simple fracture, ductile fracture, and brittle fracture [2](#page=2) [3](#page=3). ### 1.1 Importance of studying failure * Failure of engineering materials is almost always undesirable due to its potential to endanger lives, cause economic losses, and disrupt the availability of products and services [2](#page=2). * Common causes of failure include improper material selection and processing, inadequate component design, and misuse of components [2](#page=2). * Damage can also occur during service, necessitating regular inspection, repair, or replacement for safe design [2](#page=2). * Engineers are responsible for anticipating and planning for potential failures, assessing their causes, and implementing preventive measures [2](#page=2). ### 1.2 Overview of fracture mechanics Fracture mechanics is a field that studies the behavior of materials and structures under conditions of crack propagation. A fundamental principle is that applied tensile stress is amplified at the tip of a small incision or notch, which can initiate and drive crack growth [1](#page=1) [2](#page=2). ### 1.3 Simple fracture Simple fracture refers to the separation of a body into two or more pieces under a static or slowly changing stress at temperatures significantly below the material's melting point. This discussion primarily focuses on fractures resulting from uniaxial tensile loads. Fracture can also occur due to fatigue (cyclic stresses) and creep (time-dependent deformation at elevated temperatures), which are discussed in later sections [3](#page=3). > **Tip:** While this section focuses on tensile loads, remember that fracture can also be initiated by compressive, shear, or torsional stresses [3](#page=3). ### 1.4 Modes of fracture: ductile and brittle Two primary modes of fracture exist for metals: ductile and brittle. The classification is based on the material's ability to undergo plastic deformation [3](#page=3). #### 1.4.1 Ductile fracture * Characterized by substantial plastic deformation and high energy absorption before fracture [3](#page=3). * Crack propagation is relatively slow and often occurs after a moderate amount of necking [3](#page=3). * The crack is typically considered stable, meaning it resists further extension unless the applied stress increases [3](#page=3). * Fracture surfaces often exhibit evidence of significant gross deformation, such as twisting and tearing [3](#page=3). * Ductile fracture is generally preferred because the plastic deformation provides a warning that failure is imminent, allowing for preventive actions [3](#page=3). * These materials also require more strain energy to fracture, indicating they are generally tougher [3](#page=3). > **Example:** Photograph (a) in the document illustrates how a small incision in a plastic package makes it easier to tear open, demonstrating the principle of stress amplification at a notch leading to easier crack propagation [1](#page=1). #### 1.4.2 Brittle fracture * Accompanied by little to no plastic deformation and low energy absorption [3](#page=3). * Cracks can spread extremely rapidly, often spontaneously, without a significant increase in applied stress [3](#page=3). * Such cracks are considered unstable [3](#page=3). * Brittle fracture is dangerous because it occurs suddenly and catastrophically without warning [3](#page=3). > **Example:** Photograph (b) depicts an oil tanker that fractured in a brittle manner due to crack propagation around its girth, initiated by a small flaw where stresses were amplified. Photograph (c) shows an aircraft that experienced explosive decompression and structural failure attributed to metal fatigue aggravated by corrosion, leading to crack propagation [1](#page=1). ### 1.5 Factors influencing ductility Ductility is a relative term and depends on several factors [3](#page=3): * **Temperature:** Lower temperatures can cause normally ductile materials to exhibit brittle behavior [3](#page=3). * **Strain rate:** Higher strain rates can also promote brittle fracture [3](#page=3). * **Stress state:** Complex stress states can influence the fracture mode [3](#page=3). > **Tip:** Understanding how temperature, strain rate, and stress state affect ductility is critical for predicting and preventing failure in various service conditions [3](#page=3). ### 1.6 Crack formation and propagation Any fracture process involves two fundamental steps: crack formation and crack propagation, both driven by an imposed stress. The mechanism of crack propagation is key to determining the mode of fracture [3](#page=3). ### 1.7 Fracture mechanics principles * An applied tensile stress is amplified at the tip of a small incision or notch. This stress concentration is a core concept in fracture mechanics and explains why sharp flaws are more dangerous than blunt ones [1](#page=1). * Understanding this amplification is essential for designing components that can resist crack initiation and propagation [1](#page=1). ### 1.8 Key topics in fracture mechanics (as outlined in the document) The broader chapter covers several topics related to material failure, including: * Simple fracture (ductile and brittle modes) [2](#page=2). * Fundamentals of fracture mechanics [2](#page=2). * Fracture toughness testing [2](#page=2). * The ductile-to-brittle transition [2](#page=2). * Fatigue [2](#page=2). * Creep [2](#page=2). --- # Ductile and brittle fracture mechanisms and analysis This section details the mechanisms of ductile and brittle fracture, including their characteristics, fractographic studies, and the principles of fracture mechanics like stress concentration and fracture toughness. ### 2.1 Fracture fundamentals Fracture is the separation of a body into two or more pieces due to an imposed static stress at temperatures significantly below the material's melting point. Fatigue and creep fracture, occurring under cyclic or time-dependent deformation respectively, are addressed separately. This discussion focuses on fracture resulting from uniaxial tensile loads [3](#page=3). For metals, two primary fracture modes exist: ductile and brittle, distinguished by their capacity for plastic deformation. Ductile materials exhibit substantial plastic deformation and high energy absorption before fracture, while brittle fracture involves little to no plastic deformation and low energy absorption. Ductility is a relative term influenced by temperature, strain rate, and stress state [3](#page=3). All fracture processes involve crack formation and propagation. Ductile fracture is characterized by extensive plastic deformation near the crack tip, progressing slowly and being considered "stable" as it resists extension without increased stress. Brittle fracture involves rapid crack propagation with minimal plastic deformation, considered "unstable" as it can proceed spontaneously [3](#page=3). Ductile fracture is generally preferred due to two main advantages: * It provides warning signs through plastic deformation before imminent failure, unlike the sudden and catastrophic nature of brittle fracture [3](#page=3). * It requires more strain energy, indicating higher toughness [3](#page=3). Metals often exhibit ductile behavior, ceramics are typically brittle, and polymers can show a range of behaviors [3](#page=3). ### 2.2 Ductile fracture Ductile fracture surfaces display distinct macroscopic and microscopic features [3](#page=3). #### 2.2.1 Macroscopic fracture profiles * **Highly ductile fracture:** In very soft metals (e.g., pure gold, lead) or other materials at elevated temperatures, the specimen may neck down to a point, exhibiting near 100% reduction in area [3](#page=3). * **Moderately ductile fracture (Cup-and-cone):** This is the most common tensile fracture profile in ductile metals. It involves several stages [3](#page=3): 1. **Necking:** The cross-section begins to reduce in area [4](#page=4). 2. **Microvoid formation:** Small cavities form in the interior of the cross-section [4](#page=4). 3. **Coalescence:** These microvoids enlarge and merge to form an elliptical crack, oriented perpendicular to the stress direction [4](#page=4). 4. **Crack propagation:** The crack grows parallel to its major axis via microvoid coalescence [4](#page=4). 5. **Final fracture:** The crack propagates rapidly around the outer perimeter of the neck by shear deformation at approximately 45° to the tensile axis (where shear stress is maximum). This results in one mating surface appearing as a "cup" and the other as a "cone". The central region of the fracture surface is often fibrous, indicating plastic deformation [4](#page=4). #### 2.2.2 Fractographic studies Microscopic examination, typically with a scanning electron microscope (SEM), reveals detailed fracture mechanisms [5](#page=5). * **Dimples:** The fibrous central region of a cup-and-cone fracture surface, when viewed under high magnification, shows numerous spherical "dimples". Each dimple represents half of a microvoid that formed and separated during fracture [5](#page=5). * **Shear dimples:** Dimples on the 45° shear lip of a cup-and-cone fracture are elongated or C-shaped, indicative of shear failure [5](#page=5). Fractographs provide valuable information for analyzing fracture mode, stress state, and crack initiation sites [5](#page=5). ### 2.3 Brittle fracture Brittle fracture occurs rapidly with negligible plastic deformation, and the crack propagates nearly perpendicular to the applied tensile stress, resulting in a relatively flat fracture surface [5](#page=5). #### 2.3.1 Macroscopic fracture features * **Chevron markings:** In some brittle materials like certain steels, V-shaped markings ("chevrons") may be observed pointing back towards the crack origin [5](#page=5). * **Radial ridges:** Other brittle fracture surfaces exhibit lines or ridges radiating from the crack origin in a fan-like pattern [5](#page=5). * **Smooth surfaces:** Very hard, fine-grained metals may show no discernible fracture pattern. Amorphous materials like ceramic glasses typically present a shiny and smooth fracture surface [6](#page=6). #### 2.3.2 Microscopic fracture mechanisms * **Cleavage:** In most brittle crystalline materials, crack propagation involves the repeated breaking of atomic bonds along specific crystallographic planes. This is known as transgranular fracture, as the cracks pass through the grains. Macroscopically, the surface may appear grainy or faceted due to variations in cleavage plane orientation between grains [6](#page=6). * **Intergranular fracture:** In some alloys, cracks propagate along grain boundaries. This occurs when grain boundary regions are weakened or embrittled. The fracture surface can reveal the three-dimensional structure of the grains [7](#page=7). ### 2.4 Principles of fracture mechanics Fracture mechanics quantifies the relationship between material properties, stress, crack-producing flaws, and crack propagation [7](#page=7). #### 2.4.1 Stress concentration Theoretical calculations of fracture strength are often higher than measured values due to the presence of microscopic flaws and cracks. These flaws amplify applied stress at their tips, a phenomenon known as stress concentration. The localized stress ($\sigma_m$) at a crack tip can be approximated by [8](#page=8): $$ \sigma_m = 2\sigma_0\left(\frac{a}{\rho_t}\right)^{1/2} \quad (8.1) $$ where $\sigma_0$ is the nominal applied tensile stress, $\rho_t$ is the radius of curvature of the crack tip, and $a$ is the crack length (or half-length for an internal crack) [8](#page=8). The stress concentration factor ($K_t$) quantifies this amplification: $$ K_t = \frac{\sigma_m}{\sigma_0} = 2\left(\frac{a}{\rho_t}\right)^{1/2} \quad (8.2) $$ Stress concentration also occurs at macroscopic discontinuities like voids, inclusions, sharp corners, and scratches. The effect is more pronounced in brittle materials, as ductile materials can undergo plastic deformation to redistribute stress [9](#page=9). #### 2.4.2 Critical stress for crack propagation In a brittle material, the critical stress ($\sigma_c$) required for crack propagation can be expressed as: $$ \sigma_c = \left(\frac{2E\gamma_s}{\pi a}\right)^{1/2} \quad (8.3) $$ where $E$ is the modulus of elasticity and $\gamma_s$ is the specific surface energy. When the stress at a flaw tip exceeds this critical value, a crack forms and propagates, leading to fracture [10](#page=10). > **Tip:** The concept of critical stress explains why materials with tiny flaws have significantly lower fracture strengths than theoretically predicted based on atomic bonding energies. **Example Problem 8.1: Maximum Flaw Length Computation** A glass plate subjected to a tensile stress of 40 MPa has a specific surface energy of 0.3 J/m$^2$ and a modulus of elasticity of 69 GPa. To find the maximum flaw length possible without fracture, we rearrange Equation 8.3: $$ a = \frac{2E\gamma_s}{\pi \sigma_c^2} $$ Substituting the given values: $$ a = \frac{ (69 \times 10^9 \, \text{N/m}^2)(0.3 \, \text{J/m}^2)}{\pi(40 \times 10^6 \, \text{N/m}^2)^2} = 8.2 \times 10^{-6} \, \text{m} = 8.2 \, \mu\text{m} $$ [2](#page=2). Thus, the maximum length of a surface flaw that can exist without causing fracture is 8.2 micrometers [10](#page=10). #### 2.4.3 Fracture toughness Fracture toughness ($K_c$) quantifies a material's resistance to brittle fracture in the presence of a crack. It relates critical stress for crack propagation ($\sigma_c$) and crack length ($a$) through the equation [10](#page=10): $$ K_c = Y\sigma_c\sqrt{\pi a} \quad (8.4) $$ $K_c$ has units of MPa$\sqrt{\text{m}}$ or psi$\sqrt{\text{in.}}$. $Y$ is a dimensionless parameter dependent on crack and specimen geometry, and loading conditions [10](#page=10). For thin specimens, $K_c$ depends on thickness. However, when specimen thickness significantly exceeds crack dimensions, $K_c$ becomes independent of thickness, defining a condition of **plane strain**. Under plane strain conditions, the fracture toughness is denoted as the **plane strain fracture toughness**, $K_{Ic}$. This is defined for mode I (opening or tensile) crack displacement [11](#page=11): $$ K_{Ic} = Y\sigma\sqrt{\pi a} \quad (8.5) $$ $K_{Ic}$ is the fracture toughness typically cited for most situations. Materials with low $K_{Ic}$ values are susceptible to catastrophic failure, while ductile materials have higher $K_{Ic}$ values [11](#page=11). > **Tip:** Plane strain fracture toughness ($K_{Ic}$) is a critical material property for designing against brittle fracture, especially in structures where significant flaws might be present. The plane strain fracture toughness is influenced by temperature, strain rate, and microstructure. $K_{Ic}$ generally decreases with increasing strain rate and decreasing temperature. It typically increases with decreasing grain size and can decrease with additions that enhance yield strength [12](#page=12). #### 2.4.4 Design using fracture mechanics Designing with fracture mechanics involves considering fracture toughness ($K_c$ or $K_{Ic}$), applied stress ($\sigma$), and flaw size ($a$) [12](#page=12). * **Design stress calculation:** If $K_{Ic}$ and allowable flaw size ($a$) are known, the critical design stress ($\sigma_c$) can be calculated: $$ \sigma_c = \frac{K_{Ic}}{Y\sqrt{\pi a}} \quad (8.6) $$ * **Maximum allowable flaw size calculation:** If stress ($\sigma$) and fracture toughness ($K_{Ic}$) are fixed, the maximum allowable flaw size ($a_c$) can be determined: $$ a_c = \frac{1}{\pi}\left(\frac{K_{Ic}}{\sigma Y}\right)^2 \quad (8.7) $$ Nondestructive testing (NDT) techniques are employed to detect and measure flaws, aiding in quality control and in-service inspection [13](#page=13). **Design Example 8.1: Material Specification for a Pressurized Cylindrical Tank** A thin-walled cylindrical tank (radius 0.5 m, thickness 8.0 mm) contains a fluid at 2.0 MPa pressure. The hoop stress ($\sigma_h$) is calculated as: $$ \sigma_h = \frac{pr}{t} = \frac{(2.0 \, \text{MPa})(0.5 \, \text{m})}{8.0 \times 10^{-3} \, \text{m}} = 125 \, \text{MPa} $$ The design considers two failure scenarios: 1. **Leak-before-break (LBB):** The crack penetrates the wall before catastrophic failure, allowing detection by leakage. This occurs when the critical crack length ($c_c$) is greater than or equal to the wall thickness ($t$). The critical crack length is given by [14](#page=14): $$ c_c = \frac{1}{\pi N^2}\left(\frac{K_{Ic}}{\sigma_h}\right)^2 \quad (8.10) $$ where $N$ is a factor of safety (given as 3.0). 2. **Brittle fracture:** Rapid crack propagation occurs when the crack reaches a critical length shorter than that for LBB [14](#page=14). For steel alloy 4140 tempered at 370°C, with a minimum $K_{Ic}$ of 55 MPa$\sqrt{\text{m}}$, the critical crack length is: $$ c_c = \frac{1}{\pi ^2}\left(\frac{55 \, \text{MPa}\sqrt{\text{m}}}{125 \, \text{MPa}}\right)^2 = 6.8 \times 10^{-3} \, \text{m} = 6.8 \, \text{mm} $$ [3](#page=3). Since $c_c$ (6.8 mm) is less than the wall thickness (8.0 mm), LBB is unlikely for this steel alloy. Brittle fracture is likely for alloys not meeting the LBB criterion [15](#page=15) [16](#page=16). Standardized tests, developed by organizations like ASTM, are used to measure fracture toughness values, typically involving specimens with pre-existing sharp cracks [16](#page=16). --- # Fracture toughness testing and ductile-to-brittle transition This section details methods for assessing fracture toughness and explains the phenomenon of the ductile-to-brittle transition in materials. ### 3.1 Fracture toughness testing Fracture toughness testing aims to determine a material's resistance to fracture under specific conditions, using established criteria before results are considered acceptable. While most tests are for metals, some have been adapted for ceramics, polymers, and composites [17](#page=17). #### 3.1.1 Impact testing techniques Before the development of fracture mechanics, impact testing methods were established to evaluate fracture characteristics at high loading rates, as laboratory tensile tests (at low loading rates) could not reliably predict fracture behavior. These tests simulate severe fracture conditions, including low temperatures, high strain rates, and triaxial stress states, often introduced by notches [17](#page=17). ##### 3.1.1.1 Charpy and Izod tests The Charpy and Izod tests are standardized methods to measure impact energy, sometimes referred to as notch toughness. The Charpy V-notch (CVN) technique is widely used in the United States. Both tests employ specimens with a square cross-section and a machined V-notch [17](#page=17). The testing apparatus involves a pendulum hammer released from a fixed height ($h$) that strikes the notched specimen at its base. The hammer's subsequent swing to a maximum height ($h'$) allows for the calculation of absorbed energy, which is the difference between $h$ and $h'$. The notch serves as a stress concentration point for the high-velocity impact. The primary distinction between Charpy and Izod tests lies in the specimen support method [17](#page=17). > **Tip:** Impact tests are considered qualitative and are primarily useful for relative comparisons and identifying the ductile-to-brittle transition, rather than for precise design calculations. Plane strain fracture toughness tests, though more complex and expensive, provide quantitative fracture properties like $K_{Ic}$ [17](#page=17). ##### 3.1.1.2 Relationship between impact tests and fracture toughness Attempts to correlate plane strain fracture toughness values with Charpy V-notch energies have yielded limited success. While impact tests are simpler and less costly to perform than fracture toughness tests, their results are more qualitative and less directly applicable to design engineering [17](#page=17). ### 3.2 Ductile-to-brittle transition A critical function of the Charpy and Izod tests is to identify if a material undergoes a ductile-to-brittle transition as temperature decreases, and to map the temperature range over which this occurs. This phenomenon can have severe consequences, as exemplified by failures in steel structures like oil tankers [17](#page=17). #### 3.2.1 Characteristics of the transition The ductile-to-brittle transition is characterized by a temperature-dependent change in impact energy absorption. For a typical steel, as temperature decreases, the Charpy V-notch (CVN) impact energy drops significantly over a narrow temperature range, transitioning from a high energy absorption at higher temperatures (ductile fracture) to a low, constant energy absorption at lower temperatures (brittle fracture). This behavior is represented by curve A in Figure 8.14 [18](#page=18) [19](#page=19). > **Example:** The catastrophic splitting of welded transport ships during World War II is a classic example of failure due to the ductile-to-brittle transition. The steel alloy used had adequate toughness in room-temperature tensile tests, but brittle fractures occurred at relatively low ambient temperatures (around 4 °C [40°F]), close to the material's transition temperature [20](#page=20). #### 3.2.2 Fracture surface appearance The appearance of the fracture surface provides insight into the fracture mode and can aid in determining transition temperatures. Ductile fractures exhibit a fibrous or dull appearance, indicative of shear. Conversely, brittle fractures present a granular, shiny texture, characteristic of cleavage. Over the transition range, fracture surfaces will show characteristics of both ductile and brittle modes [19](#page=19). > **Tip:** Plotting the percentage of shear fracture against temperature (curve B in Figure 8.14) provides another perspective on the ductile-to-brittle transition [19](#page=19). #### 3.2.3 Defining the transition temperature Specifying a single ductile-to-brittle transition temperature can be challenging due to the transition occurring over a temperature range. Common definitions include the temperature at which the CVN energy reaches a specific value (e.g., 20 J or 15 ft-lb$_{f}$) or corresponds to a certain fracture appearance (e.g., 50% fibrous fracture). Different criteria may yield different transition temperatures [19](#page=19). A conservative approach defines the transition temperature as that at which the fracture surface becomes 100% fibrous. For the steel in Figure 8.14, this is approximately 110°C (230°F) [19](#page=19). #### 3.2.4 Factors influencing the transition For low-strength steels with a BCC crystal structure, the transition temperature is sensitive to alloy composition and microstructure [20](#page=20). * **Grain size:** Decreasing the average grain size lowers the transition temperature, thereby strengthening and toughening the steel [20](#page=20). * **Carbon content:** Increasing carbon content raises the CVN transition temperature, even though it increases steel strength [20](#page=20). #### 3.2.5 General types of impact energy-temperature behavior Materials exhibit three general types of impact energy versus temperature behavior [20](#page=20): * **Upper Curve:** Low-strength FCC and HCP metals (e.g., some aluminum and copper alloys) do not typically undergo a ductile-to-brittle transition and maintain high impact energies (toughness) even at low temperatures [20](#page=20). * **Middle Curve:** This represents the characteristic ductile-to-brittle transition observed in low-strength steels with a BCC crystal structure [20](#page=20). * **Lower Curve:** High-strength materials, such as high-strength steels and titanium alloys, have impact energies that are relatively insensitive to temperature. However, these materials are generally very brittle, as indicated by their consistently low impact energies [20](#page=20). #### 3.2.6 Transition in other materials Most ceramics and polymers also experience a ductile-to-brittle transition. For ceramics, this transition typically occurs at very high temperatures, often exceeding 1000°C (1850°F). The transition behavior in polymers is discussed elsewhere [20](#page=20). --- # Fatigue failure analysis and influencing factors Fatigue is a failure mode in materials subjected to cyclic stresses, leading to failure at stress levels significantly lower than static strengths, and typically occurs suddenly and without warning [21](#page=21). ### 4.1 Cyclic stresses Cyclic stresses are fluctuating stresses applied to a material over time. These stresses can be axial, flexural, or torsional. The nature of the stress cycle can be characterized by several parameters [21](#page=21): * **Mean stress ($\sigma_m$)**: The average of the maximum and minimum stresses in a cycle. $$ \sigma_m = \frac{\sigma_{max} + \sigma_{min}}{2} $$ [21](#page=21). * **Stress range ($\sigma_r$)**: The difference between the maximum and minimum stresses. $$ \sigma_r = \sigma_{max} - \sigma_{min} $$ [21](#page=21). * **Stress amplitude ($\sigma_a$)**: Half of the stress range. $$ \sigma_a = \frac{\sigma_r}{2} = \frac{\sigma_{max} - \sigma_{min}}{2} $$ [21](#page=21). * **Stress ratio (R)**: The ratio of the minimum stress to the maximum stress. Tensile stresses are conventionally positive, and compressive stresses are negative. $$ R = \frac{\sigma_{min}}{\sigma_{max}} $$ [21](#page=21). For a reversed stress cycle, where the stress alternates symmetrically about zero, $R = -1$ [21](#page=21). > **Tip:** Increasing the stress ratio $R$ leads to a decrease in stress amplitude $\sigma_a$ [22](#page=22). ### 4.2 The S-N curve The fatigue behavior of materials is typically represented by S-N curves, which plot stress (S) against the logarithm of the number of cycles to failure (N). These curves are generated from laboratory tests, commonly using rotating-bending machines that impose reversed stress cycles ($R = -1$) [23](#page=23). * **Fatigue limit (or endurance limit)**: For some ferrous and titanium alloys, the S-N curve becomes horizontal at high N values, indicating a limiting stress level below which fatigue failure will not occur. This is the largest fluctuating stress that can be sustained for an essentially infinite number of cycles. For many steels, the fatigue limit is between 35% and 60% of the tensile strength [23](#page=23). * **Fatigue strength**: For materials like aluminum and copper alloys, which do not exhibit a fatigue limit, fatigue response is specified as fatigue strength. This is the stress level at which failure occurs for a specific number of cycles, e.g., $10^7$ cycles [24](#page=24). * **Fatigue life ($N_f$)**: The number of cycles to cause failure at a specified stress level [24](#page=24). > **Tip:** Fatigue S-N curves often show considerable scatter in data due to variations in specimen preparation, material properties, alignment, mean stress, and test frequency. The curves typically represent average values [25](#page=25) [26](#page=26). #### 4.2.1 Low-cycle vs. High-cycle fatigue Fatigue behavior can be divided into two domains: * **Low-cycle fatigue**: Occurs at relatively high loads, causing both elastic and plastic deformation in each cycle. This results in shorter fatigue lives, typically less than $10^4$ to $10^5$ cycles [26](#page=26). * **High-cycle fatigue**: Occurs at lower stress levels where deformations are entirely elastic, leading to longer fatigue lives, generally greater than $10^4$ to $10^5$ cycles [26](#page=26). > **Example:** A rotating-bending test on a 15.0 mm diameter 1045 steel bar requires a maximum cyclic load of 1712 N to ensure no fatigue failure, assuming a factor of safety of 2.0 and a fatigue limit of 310 MPa [26](#page=26). > **Example:** To ensure a 70Cu-30Zn brass bar subjected to axial tension-compression with a load amplitude of 10,000 N does not fail at $10^7$ cycles with a factor of safety of 2.5, the minimum diameter must be 16.6 mm, based on a fatigue strength of 115 MPa [27](#page=27). ### 4.3 Crack initiation and propagation Fatigue failure proceeds in three distinct stages: 1. **Crack initiation**: A small crack forms at a point of high stress concentration, often on the surface. Common initiation sites include surface scratches, fillets, threads, and microscopic surface discontinuities from dislocation slip steps [27](#page=27) [28](#page=28). 2. **Crack propagation**: The crack grows incrementally with each stress cycle. This stage is characterized by microscopic features like fatigue striations, where each striation ideally represents one cycle of crack advance. Beachmarks are macroscopic ridges representing periods of crack growth, often seen when the machine operates intermittently. Thousands of striations can exist within a single beachmark [27](#page=27) [28](#page=28). 3. **Final failure**: Occurs rapidly once the crack reaches a critical size. This final fracture region may exhibit ductile or brittle characteristics and does not typically show beachmarks or striations [27](#page=27) [29](#page=29). > **Tip:** The presence of beachmarks and/or striations on a fracture surface strongly suggests fatigue failure. However, their absence does not rule out fatigue, as they may not be observable in all metals or conditions [29](#page=29). ### 4.4 Factors that affect fatigue life Several factors significantly influence a material's fatigue resistance: #### 4.4.1 Mean stress Increasing the mean stress ($\sigma_m$) of a cyclic loading condition generally leads to a decrease in fatigue life, as represented by a downward shift of the S-N curves [30](#page=30). #### 4.4.2 Surface effects Since most fatigue cracks initiate at the surface, the surface condition is critical. Factors that improve fatigue resistance include: * **Design Factors**: Geometric discontinuities (notches, holes, threads) act as stress raisers. Sharp corners and sudden contour changes should be avoided by using rounded fillets with large radii of curvature [30](#page=30). * **Surface Treatments**: * **Polishing**: Improving surface finish by polishing significantly enhances fatigue life by reducing surface irregularities [30](#page=30). * **Shot peening**: This process introduces residual compressive stresses into the surface layer by plastically deforming it with high-velocity shot particles. This compression counteracts applied tensile stresses, reducing the likelihood of crack formation [31](#page=31). * **Case hardening**: Techniques like carburizing or nitriding create a harder, carbon- or nitrogen-rich outer layer. This improves fatigue properties through increased surface hardness and the introduction of beneficial residual compressive stresses [31](#page=31). > **Example:** Shot peening can significantly improve the fatigue life of steel, as shown by a higher S-N curve compared to untreated steel [31](#page=31). #### 4.4.3 Environmental effects External environments can also influence fatigue behavior, leading to phenomena like thermal fatigue and corrosion fatigue [31](#page=31). --- # Creep behavior, its influencing factors, and high-temperature applications Creep is the time-dependent, permanent deformation of materials under constant stress at elevated temperatures, often being the critical factor limiting a component's lifespan [32](#page=32). ### 5.1 Generalized creep behavior Creep is studied through creep tests, where a specimen is subjected to a constant load or stress at a fixed temperature, with deformation measured over time. Most tests use constant load to gather engineering data, while constant stress tests offer deeper insight into creep mechanisms [32](#page=32) [33](#page=33). #### 5.1.1 Typical creep curve A typical constant-load creep curve for metals exhibits three distinct regions after an initial instantaneous elastic deformation [33](#page=33): * **Primary (transient) creep:** Characterized by a continuously decreasing creep rate, indicating increasing creep resistance due to strain hardening [33](#page=33). * **Secondary (steady-state) creep:** Features a constant creep rate, resulting in a linear portion of the strain-time plot. This stage is often the longest and represents a balance between strain hardening and recovery. The slope of this segment, $\Delta \epsilon / \Delta t$, is known as the minimum or steady-state creep rate, $\dot{\epsilon}_s$, and is a crucial engineering design parameter for long-life applications [33](#page=33) [34](#page=34). * **Tertiary creep:** Marked by an accelerating creep rate leading to failure, often termed rupture. This is caused by microstructural changes like grain boundary separation, internal cracks, voids, or necking under tensile loads, which reduce the effective cross-sectional area. The total time to rupture is the rupture lifetime, $t_r$, which is critical for shorter-life applications like turbine blades [33](#page=33) [34](#page=34). > **Tip:** Creep tests conducted to failure are called creep rupture tests, providing the rupture lifetime ($t_r$) as a design consideration [33](#page=33). > **Tip:** For most materials, creep properties are largely independent of the loading direction, though uniaxial compression tests are preferred for brittle materials to avoid stress amplification and crack propagation [33](#page=33). ### 5.2 Stress and temperature effects Both applied stress and temperature significantly influence creep characteristics. Below approximately 0.4$T_m$ (absolute melting temperature), strain is time-independent after initial deformation. Increasing either stress or temperature leads to [33](#page=33): 1. Increased instantaneous strain upon stress application [33](#page=33). 2. An increased steady-state creep rate ($\dot{\epsilon}_s$) [33](#page=33). 3. A decreased rupture lifetime ($t_r$) [33](#page=33). #### 5.2.1 Empirical relationships for creep rate Empirical relationships describe the steady-state creep rate ($\dot{\epsilon}_s$) as a function of stress ($\sigma$) and temperature ($T$): * **Stress dependence:** $$ \dot{\epsilon}_s = K_1 \sigma^n $$ where $K_1$ and $n$ are material constants. The stress exponent, $n$, can be determined by plotting $\log(\dot{\epsilon}_s)$ versus $\log(\sigma)$, which yields a straight line with a slope of $n$ [34](#page=34) [35](#page=35). * **Stress and temperature dependence:** $$ \dot{\epsilon}_s = K_2 \sigma^n \exp\left(-\frac{Q_c}{RT}\right) $$ where $K_2$ and $Q_c$ are constants. $Q_c$ is the activation energy for creep, and $R$ is the gas constant ($8.31$ J⋅mol/K) [35](#page=35). > **Example:** Computation of Steady-State Creep Rate for Aluminum at $260$ °C. > Given creep rate data for aluminum at $260$ °C and stresses of $3$ MPa and $25$ MPa, we can compute the steady-state creep rate at $10$ MPa. > The relationship $\ln \dot{\epsilon}_s = \ln K_1 + n \ln \sigma$ is used. > Using the provided data: > $\ln(2.0 \times 10^{-4} \, \text{h}^{-1}) = \ln K_1 + n \ln(3 \, \text{MPa})$ > $\ln(3.65 \, \text{h}^{-1}) = \ln K_1 + n \ln(25 \, \text{MPa})$ > Solving these equations simultaneously yields $n \approx 4.63$ and $K_1 \approx 1.24 \times 10^{-6}$. > Then, at $\sigma = 10 \, \text{MPa}$: > $\dot{\epsilon}_s = (1.24 \times 10^{-6})(10 \, \text{MPa})^{4.63} \approx 5.3 \times 10^{-2} \, \text{h}^{-1}$ [36](#page=36). #### 5.2.2 Creep mechanisms Theoretical mechanisms explaining creep include stress-induced vacancy diffusion, grain boundary diffusion, dislocation motion, and grain boundary sliding. Each mechanism is associated with a specific value of the stress exponent ($n$) [36](#page=36). * **Deformation mechanism maps:** These diagrams visually represent stress-temperature regimes where different creep mechanisms dominate, often including constant-strain-rate contours [36](#page=36). ### 5.3 Data extrapolation methods Extrapolating creep data is essential for predicting behavior over prolonged service times, often achieved by conducting tests at higher temperatures for shorter durations and then extrapolating to the desired in-service condition [36](#page=36). #### 5.3.1 Larson-Miller parameter A commonly used extrapolation method is the Larson-Miller parameter ($m$), defined as: $$ m = T(C + \log t_r) $$ where $T$ is the absolute temperature (Kelvin), $t_r$ is the rupture lifetime (hours), and $C$ is a material constant, typically around $20$. The rupture lifetime of a material at a specific stress level remains constant for a given Larson-Miller parameter value. Data can be plotted as the logarithm of stress versus the Larson-Miller parameter for design purposes [36](#page=36) [37](#page=37). > **Example:** Rupture Lifetime Prediction for S-590 Alloy. > Using Larson-Miller data for S-590 alloy, predict the time to rupture for a component under $140$ MPa stress at $800$ °C ($1073$ K). > From a stress vs. Larson-Miller parameter plot (Figure $8.34$), at $140$ MPa, the Larson-Miller parameter is $24.0 \times 10^3$ (K-h). > $24.0 \times 10^3 = 1073(20 + \log t_r)$ > Solving for $t_r$: > $22.37 = 20 + \log t_r$ > $t_r = 10^{2.37} \approx 233$ hours (approximately $9.7$ days) [37](#page=37). ### 5.4 Factors influencing creep resistance Several factors affect a material's resistance to creep [37](#page=37): * **Melting temperature:** Higher melting temperatures generally correlate with better creep resistance [37](#page=37). * **Elastic modulus:** A higher elastic modulus contributes to improved creep resistance [37](#page=37). * **Grain size:** Larger grain sizes generally enhance creep resistance. Smaller grains allow for more grain boundary sliding, which increases creep rates. This is contrary to low-temperature mechanical behavior where smaller grains increase strength [37](#page=37). ### 5.5 Alloys for high-temperature applications Materials commonly used in high-temperature service applications due to their creep resistance include stainless steels and superalloys [38](#page=38). * **Superalloys:** Their creep resistance is enhanced by solid-solution alloying and the formation of precipitate phases. Advanced processing techniques like directional solidification, producing elongated grains or single-crystal components, further improve creep resistance in components like turbine blades [38](#page=38). --- ## Common mistakes to avoid - Review all topics thoroughly before exams - Pay attention to formulas and key definitions - Practice with examples provided in each section - Don't memorize without understanding the underlying concepts

| Term | Definition | |------|------------| | Fracture | The separation of a body into two or more pieces in response to an imposed stress. This can occur through various mechanisms including ductile fracture, brittle fracture, fatigue, and creep. | | Ductile fracture | A mode of fracture characterized by significant plastic deformation and high energy absorption before failure. The process involves crack formation and relatively slow propagation with visible gross deformation at the fracture surfaces. | | Brittle fracture | A mode of fracture that occurs with very little or no plastic deformation and is characterized by rapid crack propagation. The fracture surfaces typically show little gross deformation and can exhibit features like chevron markings or radial patterns. | | Fracture mechanics | A field of study that quantifies the relationships among material properties, stress levels, the presence of flaws, and crack propagation mechanisms to predict and prevent structural failures. | | Stress concentration | The amplification of an applied stress at the tip of a small incision, notch, or flaw. This phenomenon significantly lowers the stress required for fracture initiation and propagation. | | Stress raiser | A term used for flaws or discontinuities that cause stress concentration, thus detrimentally affecting the fracture strength of a material. | | Fracture toughness | A measure of a material's resistance to brittle fracture when a crack is present. It is quantitatively expressed as $K_c$ or $K_{Ic}$ and depends on material properties and crack geometry. | | Plane strain fracture toughness ($K_{Ic}$) | The fracture toughness value measured under plane strain conditions, where the strain component perpendicular to the crack front is negligible. It is a critical parameter for designing against catastrophic failure in thick components. | | Fatigue | A phenomenon where materials fail under dynamic and fluctuating stresses, often at stress levels significantly lower than their static tensile or yield strength. It occurs after a period of repeated stress cycling. | | S-N curve | A plot of stress amplitude (S) versus the logarithm of the number of cycles to failure (N). This curve characterizes the fatigue behavior of a material, showing how stress level affects its fatigue life. | | Fatigue limit (or endurance limit) | The maximum stress level below which a material will not fail from fatigue, regardless of the number of stress cycles. This is observed in some materials, like certain steels and titanium alloys. | | Fatigue strength | The stress level at which a material will fail after a specified number of cycles. This term is used for materials that do not exhibit a distinct fatigue limit, such as many nonferrous alloys. | | Fatigue life ($N_f$) | The number of cycles a material can withstand at a specified stress level before failure occurs, as determined from the S-N curve. | | Creep | Time-dependent, permanent deformation of materials when subjected to a constant load or stress, typically at elevated temperatures. It is a significant factor in the lifetime of components operating under sustained stress at high temperatures. | | Steady-state creep rate | The constant rate of deformation during the secondary stage of creep, represented by the slope of the linear portion of the strain-time creep curve. It is a key parameter for long-life creep applications. | | Rupture lifetime ($t_r$) | The total time elapsed until a material fails by rupture under creep conditions. This is a critical design parameter for short-life creep applications. | | Ductile-to-brittle transition | A phenomenon where a material, typically a steel with a BCC crystal structure, changes from ductile fracture behavior to brittle fracture behavior as the temperature decreases. This transition can lead to catastrophic failure at low temperatures. | | Transgranular fracture | A type of fracture where the crack propagates through the grains of a polycrystalline material, breaking atomic bonds across crystallographic planes within the grains. | | Intergranular fracture | A type of fracture where the crack propagates along the grain boundaries of a polycrystalline material. This often occurs when grain boundary regions are weakened or embrittled. | | Thermal fatigue | Fatigue failure induced by fluctuating thermal stresses, which arise from the restraint of thermal expansion and contraction due to temperature variations. Mechanical stresses are not necessarily present. | | Corrosion fatigue | A type of fatigue failure that occurs due to the simultaneous action of cyclic stress and a corrosive environment. The corrosive environment can initiate pits, act as stress concentrators, and accelerate crack propagation. | | Beachmarks | Macroscopic markings on a fatigue fracture surface that indicate the position of the crack tip at different stages of growth, often representing periods of operation or interruptions in cyclic loading. | | Fatigue striations | Microscopic markings on a fatigue fracture surface, each thought to represent the advance of the crack front during a single stress cycle. Their width depends on the stress range. | | Shot peening | A surface treatment process that involves projecting small, hard particles onto a material surface to induce residual compressive stresses. This enhances fatigue resistance by counteracting applied tensile stresses. | | Case hardening | A surface treatment process, such as carburizing or nitriding, that creates a harder outer layer (case) on steel alloys. This improves surface hardness and fatigue life by increasing hardness and introducing residual compressive stresses. |