Low-cycle fatigue

Last updated April 27, 2023

Low cycle fatigue (LCF) has two fundamental characteristics: plastic deformation in each cycle; and low cycle phenomenon, in which the materials have finite endurance for this type of load. The term cycle refers to repeated applications of stress that lead to eventual fatigue and failure; low-cycle pertains to a long period between applications.

Study in fatigue has been focusing on mainly two fields: size design in aeronautics and energy production using advanced calculation methods. The LCF result allows us to study the behavior of the material in greater depth to better understand the complex mechanical and metallurgical phenomena (crack propagation, work softening, strain concentration, work hardening, etc.).^[1]

History

Common factors that have been attributed to low-cycle fatigue (LCF) are high stress levels and a low number of cycles to failure. Many studies have been carried out, particularly in the last 50 years on metals and the relationship between temperature, stress, and number of cycles to failure. Tests are used to plot an S-N curve, and it has been shown that the number of cycles to failure decreased with increasing temperature. However, extensive testing would have been too costly so researchers mainly resorted to using finite element analysis using computer software.^[2]

Through many experiments, it has been found that characteristics of a material can change as a result of LCF. Fracture ductility tends to decrease, with the magnitude depending on the presence of small cracks to begin with. To perform these tests, an electro-hydraulic servo-controlled testing machine was generally used, as it is capable of not changing the stress amplitude. It was also discovered that performing low-cycle fatigue tests on specimens with holes already drilled in them were more susceptible to crack propagation, and hence a greater decrease in fracture ductility. This was true despite the small hole sizes, ranging from 40 to 200 μm.^[3]

Characteristics

When a component is subject to low cycle fatigue, it is repeatedly plastically deformed. For example, if a part were to be loaded in tension until it was permanently deformed (plastically deformed), that would be considered one half cycle of low cycle fatigue, or LCF. In order to complete a full cycle the part would need to be deformed back into its original shape. The number of LCF cycles that a part can withstand before failing is much lower than that of regular fatigue.^[4]

This condition of high cyclic strain is often the result of extreme operating conditions, such as high changes in temperature. Thermal stresses originating from an expansion or contraction of materials can exacerbate the loading conditions on a part and LCF characteristics can come into play.

Mechanics

A commonly used equation that describes the behavior of low-cycle fatigue is the Coffin-Manson relation (published by L. F. Coffin in 1954 and S. S. Manson in 1953):

{\frac {\Delta \varepsilon _{p}}{2}}=\varepsilon _{f}'(2N)^{c}+{\frac {\varsigma _{f}'(2N)^{b}}{E}}

where,

Δε_p /2 is the plastic strain amplitude;
ε_f' is an empirical constant known as the fatigue ductility coefficient defined by the strain intercept at 2N =1;
2N is the number of reversals to failure (N cycles);
c is an empirical constant known as the fatigue ductility exponent, commonly ranging from -0.5 to -0.7. Small c results in long fatigue life.
b is an empirical constant known as the fatigue brittleness exponent.

The first half of the equation indicates the Plastic region and the second half of the equation indicates elastic region. ^[5]

Morrow Approximation

In the above given Coffin-Manson relation the constant values (b and c) is determined by the given equations:

$c={\frac {-1}{1+5{\acute {n}}}}$

$b={\frac {-{\acute {n}}}{1+5{\acute {n}}}}$

Notable failures

One noteworthy event in which the failure was a result of LCF was the 1994 Northridge earthquake. Many buildings and bridges collapsed, and as a result over 9,000 people were injured.^[6] Researchers at the University of Southern California analyzed the main areas of a ten-story building that were subjected to low-cycle fatigue. Unfortunately, there was limited experimental data available to directly construct a S-N curve for low-cycle fatigue, so most of the analysis consisted of plotting the high-cycle fatigue behavior on a S-N curve and extending the line for that graph to create the portion of the low-cycle fatigue curve using the Palmgren-Miner method. Ultimately, this data was used to more accurately predict and analyze similar types of damage that the ten-story steel building in Northridge faced.^[7]

Another more recent event was the 2010 Chile earthquake, in which several researchers from the University of Chile made reports of multiple reinforced concrete structures damaged throughout the country by the seismic event. Many structural elements such as beams, walls and columns failed due to fatigue, exposing the steel reinforcements used in the design with clear signs of longitudinal buckling.^[9]^[10] This event caused Chilean seismic design standards to be updated based on observations on damaged structures caused by the earthquake.^[11]

Related Research Articles

Ductility is a mechanical property commonly described as a material's amenability to drawing. In materials science, ductility is defined by the degree to which a material can sustain plastic deformation under tensile stress before failure. Ductility is an important consideration in engineering and manufacturing. It defines a material's suitability for certain manufacturing operations and its capacity to absorb mechanical overload. Some metals that are generally described as ductile include gold and copper, while platinum is the most ductile of all metals in pure form. However, not all metals experience ductile failure as some can be characterized with brittle failure like cast iron. Polymers generally can be viewed as ductile materials as they typically allow for plastic deformation.

In engineering, deformation refers to the change in size or shape of an object. Displacements are the absolute change in position of a point on the object. Deflection is the relative change in external displacements on an object. Strain is the relative internal change in shape of an infinitesimally small cube of material and can be expressed as a non-dimensional change in length or angle of distortion of the cube. Strains are related to the forces acting on the cube, which are known as stress, by a stress-strain curve. The relationship between stress and strain is generally linear and reversible up until the yield point and the deformation is elastic. The linear relationship for a material is known as Young's modulus. Above the yield point, some degree of permanent distortion remains after unloading and is termed plastic deformation. The determination of the stress and strain throughout a solid object is given by the field of strength of materials and for a structure by structural analysis.

In engineering and materials science, a stress–strain curve for a material gives the relationship between stress and strain. It is obtained by gradually applying load to a test coupon and measuring the deformation, from which the stress and strain can be determined. These curves reveal many of the properties of a material, such as the Young's modulus, the yield strength and the ultimate tensile strength.

<span class="mw-page-title-main">Fracture</span> Split of materials or structures under stress

Fracture is the separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually occurs due to the development of certain displacement discontinuity surfaces within the solid. If a displacement develops perpendicular to the surface, it is called a normal tensile crack or simply a crack; if a displacement develops tangentially, it is called a shear crack, slip band or dislocation.

The field of strength of materials typically refers to various methods of calculating the stresses and strains in structural members, such as beams, columns, and shafts. The methods employed to predict the response of a structure under loading and its susceptibility to various failure modes takes into account the properties of the materials such as its yield strength, ultimate strength, Young's modulus, and Poisson's ratio. In addition, the mechanical element's macroscopic properties such as its length, width, thickness, boundary constraints and abrupt changes in geometry such as holes are considered.

In materials science, fatigue is the initiation and propagation of cracks in a material due to cyclic loading. Once a fatigue crack has initiated, it grows a small amount with each loading cycle, typically producing striations on some parts of the fracture surface. The crack will continue to grow until it reaches a critical size, which occurs when the stress intensity factor of the crack exceeds the fracture toughness of the material, producing rapid propagation and typically complete fracture of the structure.

Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture.

In materials science, work hardening, also known as strain hardening, is the strengthening of a metal or polymer by plastic deformation. Work hardening may be desirable, undesirable, or inconsequential, depending on the context.

In materials science, fracture toughness is the critical stress intensity factor of a sharp crack where propagation of the crack suddenly becomes rapid and unlimited. A component's thickness affects the constraint conditions at the tip of a crack with thin components having plane stress conditions and thick components having plane strain conditions. Plane strain conditions give the lowest fracture toughness value which is a material property. The critical value of stress intensity factor in mode I loading measured under plane strain conditions is known as the plane strain fracture toughness, denoted $. When a test fails to meet the thickness and other test requirements that are in place to ensure plane strain conditions, the fracture toughness value produced is given the designation . Fracture toughness is a quantitative way of expressing a material's resistance to crack propagation and standard values for a given material are generally available.$

In materials science the flow stress, typically denoted as Y_f, is defined as the instantaneous value of stress required to continue plastically deforming a material - to keep it flowing. It is most commonly, though not exclusively, used in reference to metals. On a stress-strain curve, the flow stress can be found anywhere within the plastic regime; more explicitly, a flow stress can be found for any value of strain between and including yield point and excluding fracture : $.$

A deformation mechanism, in geology, is a process occurring at a microscopic scale that is responsible for changes in a material's internal structure, shape and volume. The process involves planar discontinuity and/or displacement of atoms from their original position within a crystal lattice structure. These small changes are preserved in various microstructures of materials such as rocks, metals and plastics, and can be studied in depth using optical or digital microscopy.

Thermo-mechanical fatigue is the overlay of a cyclical mechanical loading, that leads to fatigue of a material, with a cyclical thermal loading. Thermo-mechanical fatigue is an important point that needs to be considered, when constructing turbine engines or gas turbines.

TRIP steel are a class of high-strength steel alloys typically used in naval and marine applications and in the automotive industry. TRIP stands for "Transformation induced plasticity," which implies a phase transformation in the material, typically when a stress is applied. These alloys are known to possess an outstanding combination of strength and ductility.

In Earth science, ductility refers to the capacity of a rock to deform to large strains without macroscopic fracturing. Such behavior may occur in unlithified or poorly lithified sediments, in weak materials such as halite or at greater depths in all rock types where higher temperatures promote crystal plasticity and higher confining pressures suppress brittle fracture. In addition, when a material is behaving ductilely, it exhibits a linear stress vs strain relationship past the elastic limit.

Polymer fracture is the study of the fracture surface of an already failed material to determine the method of crack formation and extension in polymers both fiber reinforced and otherwise. Failure in polymer components can occur at relatively low stress levels, far below the tensile strength because of four major reasons: long term stress or creep rupture, cyclic stresses or fatigue, the presence of structural flaws and stress-cracking agents. Formations of submicroscopic cracks in polymers under load have been studied by x ray scattering techniques and the main regularities of crack formation under different loading conditions have been analyzed. The low strength of polymers compared to theoretically predicted values are mainly due to the many microscopic imperfections found in the material. These defects namely dislocations, crystalline boundaries, amorphous interlayers and block structure can all lead to the non-uniform distribution of mechanical stress.

Crack tip opening displacement (CTOD) or $is the distance between the opposite faces of a crack tip at the 90° intercept position. The position behind the crack tip at which the distance is measured is arbitrary but commonly used is the point where two 45° lines, starting at the crack tip, intersect the crack faces. The parameter is used in fracture mechanics to characterize the loading on a crack and can be related to other crack tip loading parameters such as the stress intensity factor and the elastic-plastic J-integral.$

Plasticity theory for rocks is concerned with the response of rocks to loads beyond the elastic limit. Historically, conventional wisdom has it that rock is brittle and fails by fracture while plasticity is identified with ductile materials. In field scale rock masses, structural discontinuities exist in the rock indicating that failure has taken place. Since the rock has not fallen apart, contrary to expectation of brittle behavior, clearly elasticity theory is not the last work.

Solder fatigue is the mechanical degradation of solder due to deformation under cyclic loading. This can often occur at stress levels below the yield stress of solder as a result of repeated temperature fluctuations, mechanical vibrations, or mechanical loads. Techniques to evaluate solder fatigue behavior include finite element analysis and semi-analytical closed-form equations.

A crack growth equation is used for calculating the size of a fatigue crack growing from cyclic loads. The growth of fatigue cracks can result in catastrophic failure, particularly in the case of aircraft. A crack growth equation can be used to ensure safety, both in the design phase and during operation, by predicting the size of cracks. In critical structure, loads can be recorded and used to predict the size of cracks to ensure maintenance or retirement occurs prior to any of the cracks failing.

References

↑ Pineau, Andre (2013). "Low-Cycle Fatigue". Fatigue of Materials and Structures: Fundamentals: 113–177. doi:10.1002/9781118623435.ch4. ISBN 9781848210516.
↑ Agrawal, Richa (July 2014). "Low Cycle Fatigue Life Prediction" (PDF). Ijeert. Richa Agrawal. Retrieved 2016-02-18.
↑ Murakami, Y.; Miller, K. J. (2005-08-01). "What is fatigue damage? A view point from the observation of low cycle fatigue process". International Journal of Fatigue. Cumulative Fatigue Damage Conference - University of Seville 2003 Cumulative Fatigue Damage Conference. 27 (8): 991–1005. doi:10.1016/j.ijfatigue.2004.10.009.
↑ "Understanding Fatigue" (PDF). ASME. D.P DeLuca.
↑ O'Donnell, W.J. and B.F. Langer. Nuclear Science and Engineering, Vol 20, pp. 1-12, 1964.
↑ Taylor, Alan. "The Northridge Earthquake: 20 Years Ago Today". The Atlantic. Retrieved 2016-02-18.
↑ Nastar, Navid (2008). "Effects of Low-Cycle Fatigue on a Ten-Story Steel Building" (PDF). Archived from the original (PDF) on 2016-10-20. Retrieved 2016-02-18.
↑ Rojas, F.; et al. (2011). "Performance of tall buildings in Concepción during the 27 February 2010 moment magnitude 8.8 offshore Maule, Chile earthquake". The Structural Design of Tall and Special Buildings. 20 (37–64): 37–64. doi:10.1002/tal.674. S2CID 109286598.
↑ Egger, J. E.; Rojas, F. R.; Massone, L. M. (2021-09-24). "High-Strength Reinforcing Steel Bars: Low Cycle Fatigue Behavior Using RGB Methodology". International Journal of Concrete Structures and Materials. 15 (38). doi: 10.1186/s40069-021-00474-9 . S2CID 237629712.
↑ Massone, L. M.; Herrera, P.A. (2019-05-22). "Experimental study of the residual fatigue life of reinforcement bars damaged by an earthquake". Materials and Structures. 52 (61). doi:10.1617/s11527-019-1361-x. S2CID 182197597.
↑ Wallace, John W.; Massone, Leonardo M.; Bonelli, Patricio; Dragovich, Jeff; Lagos, René; Lüders, Carl; Moehle, Jack (2012). "Damage and implications for seismic design of RC structural wall buildings". Earthquake Spectra. Wallace J, Massone L, Bonelli P, Dragovich J, Lagos R, Lüders C, Moehle J. 28: 281–299. doi:10.1193/1.4000047. S2CID 110387165.

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[1] Pineau, Andre (2013). "Low-Cycle Fatigue". Fatigue of Materials and Structures: Fundamentals: 113–177. doi:10.1002/9781118623435.ch4. ISBN 9781848210516.

[2] Agrawal, Richa (July 2014). "Low Cycle Fatigue Life Prediction" (PDF). Ijeert. Richa Agrawal. Retrieved 2016-02-18.

[3] Murakami, Y.; Miller, K. J. (2005-08-01). "What is fatigue damage? A view point from the observation of low cycle fatigue process". International Journal of Fatigue. Cumulative Fatigue Damage Conference - University of Seville 2003 Cumulative Fatigue Damage Conference. 27 (8): 991–1005. doi:10.1016/j.ijfatigue.2004.10.009.

[4] "Understanding Fatigue" (PDF). ASME. D.P DeLuca.

[5] O'Donnell, W.J. and B.F. Langer. Nuclear Science and Engineering, Vol 20, pp. 1-12, 1964.

[6] Taylor, Alan. "The Northridge Earthquake: 20 Years Ago Today". The Atlantic. Retrieved 2016-02-18.

[7] Nastar, Navid (2008). "Effects of Low-Cycle Fatigue on a Ten-Story Steel Building" (PDF). Archived from the original (PDF) on 2016-10-20. Retrieved 2016-02-18.

[8] Rojas, F.; et al. (2011). "Performance of tall buildings in Concepción during the 27 February 2010 moment magnitude 8.8 offshore Maule, Chile earthquake". The Structural Design of Tall and Special Buildings. 20 (37–64): 37–64. doi:10.1002/tal.674. S2CID 109286598.

[9] Egger, J. E.; Rojas, F. R.; Massone, L. M. (2021-09-24). "High-Strength Reinforcing Steel Bars: Low Cycle Fatigue Behavior Using RGB Methodology". International Journal of Concrete Structures and Materials. 15 (38). doi: 10.1186/s40069-021-00474-9 . S2CID 237629712.

[10] Massone, L. M.; Herrera, P.A. (2019-05-22). "Experimental study of the residual fatigue life of reinforcement bars damaged by an earthquake". Materials and Structures. 52 (61). doi:10.1617/s11527-019-1361-x. S2CID 182197597.

[11] Wallace, John W.; Massone, Leonardo M.; Bonelli, Patricio; Dragovich, Jeff; Lagos, René; Lüders, Carl; Moehle, Jack (2012). "Damage and implications for seismic design of RC structural wall buildings". Earthquake Spectra. Wallace J, Massone L, Bonelli P, Dragovich J, Lagos R, Lüders C, Moehle J. 28: 281–299. doi:10.1193/1.4000047. S2CID 110387165.

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