Low-cycle fatigue

Last updated May 15, 2024

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 quarter 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 _{t}}{2}}={\frac {\Delta \varepsilon _{p}}{2}}+{\frac {\Delta \varepsilon _{e}}{2}}=\varepsilon _{f}'(2N)^{c}+{\frac {\sigma _{f}'(2N)^{b}}{E}}

where,

Δεt /2 is the total strain amplitude;
Δε_p /2 is the plastic strain amplitude;
Δεe /2 is the elastic strain amplitude;
2N is the number of reversals to failure (N cycles);
ε_f' is an empirical constant known as the fatigue ductility coefficient defined by the strain intercept at 2N =1;
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.
ς_f' is a constant known as the fatigue strength coefficient
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 refers to the ability of a material to sustain significant plastic deformation before fracture. Plastic deformation is the permanent distortion of a material under applied stress, as opposed to elastic deformation, which is reversible upon removing the stress. Ductility is a critical mechanical performance indicator, particularly in applications that require materials to bend, stretch, or deform in other ways without breaking. The extent of ductility can be quantitatively assessed using the percent elongation at break, given by the equation:

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 infinitesimal 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. Elasticity in materials occurs when applied stress does not surpass the energy required to break molecular bonds, allowing the material to deform reversibly and return to its original shape once the stress is removed. 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 appearance of a crack or complete 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.

<span class="mw-page-title-main">Compressive strength</span> Capacity of a material or structure to withstand loads tending to reduce size

In mechanics, compressive strength is the capacity of a material or structure to withstand loads tending to reduce size. In other words, compressive strength resists compression, whereas tensile strength resists tension. In the study of strength of materials, tensile strength, compressive strength, and shear strength can be analyzed independently.

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.

$<span class="mw-page-title-main">Toughness</span> Material ability to absorb energy and plastically deform without fracturing$

In materials science and metallurgy, toughness is the ability of a material to absorb energy and plastically deform without fracturing. Toughness is the strength with which the material opposes rupture. One definition of material toughness is the amount of energy per unit volume that a material can absorb before rupturing. This measure of toughness is different from that used for fracture toughness, which describes the capacity of materials to resist fracture. Toughness requires a balance of strength and ductility.

In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.

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 engineering and materials science, necking is a mode of tensile deformation where relatively large amounts of strain localize disproportionately in a small region of the material. The resulting prominent decrease in local cross-sectional area provides the basis for the name "neck". Because the local strains in the neck are large, necking is often closely associated with yielding, a form of plastic deformation associated with ductile materials, often metals or polymers. Once necking has begun, the neck becomes the exclusive location of yielding in the material, as the reduced area gives the neck the largest local stress.

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 : $.$

The Ramberg–Osgood equation was created to describe the non linear relationship between stress and strain—that is, the stress–strain curve—in materials near their yield points. It is especially applicable to metals that harden with plastic deformation, showing a smooth elastic-plastic transition. As it is a phenomenological model, checking the fit of the model with actual experimental data for the particular material of interest is essential.

Viscoplasticity is a theory in continuum mechanics that describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the deformation of the material depends on the rate at which loads are applied. The inelastic behavior that is the subject of viscoplasticity is plastic deformation which means that the material undergoes unrecoverable deformations when a load level is reached. Rate-dependent plasticity is important for transient plasticity calculations. The main difference between rate-independent plastic and viscoplastic material models is that the latter exhibit not only permanent deformations after the application of loads but continue to undergo a creep flow as a function of time under the influence of the applied load.

<span class="mw-page-title-main">Fiber-reinforced composite</span>

A fiber-reinforced composite (FRC) is a composite building material that consists of three components:

the fibers as the discontinuous or dispersed phase,
the matrix as the continuous phase, and
the fine interphase region, also known as the interface.

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.

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.

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 word.

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 a fatigue crack can result in catastrophic failure, particularly in the case of aircraft. When many growing fatigue cracks interact with one another it is known as widespread fatigue damage. 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. Safety factors are used to reduce the predicted fatigue life to a service fatigue life because of the sensitivity of the fatigue life to the size and shape of crack initiating defects and the variability between assumed loading and actual loading experienced by a component.

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. 28. Wallace J, Massone L, Bonelli P, Dragovich J, Lagos R, Lüders C, Moehle J: 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. 28. Wallace J, Massone L, Bonelli P, Dragovich J, Lagos R, Lüders C, Moehle J: 281–299. doi:10.1193/1.4000047. S2CID 110387165.

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