Material Fatigue in Engineering with Calculations

In mechanical design, material fatigue has a very important place. Because of the fatigue, unexpected cracks and failures can occur with stress levels that are lower than the yield stress. We give a very big amount of research and development activities to mechanical parts to prevent or estimate the fatigue behavior of a part. Here we explain the general aspects of material fatigue and the place of material fatigue in mechanical design.

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What is Material Fatigue? 

Fatigue is the development of internal cracks and defects through time with the application of cyclic stresses which are generally lower than the yield stresses of materials. In the early technologies such as around the industrial revolution, we used different kinds of metals and steel in different structural applications. But in most applications, structural parts deteriorated because of unknown events. A metal bridge suddenly deteriorates by a very low load such as a passing of a small car without giving any warning. 

After these events, material fatigue mechanics started to improve, we developed and different fatigue and crack growth estimations. 

Important Points of Material Fatigue in Mechanical Systems

There are lots of theoretical and experimental applications to estimate the fatigue life of an engineering system. But before starting to understand these fatigue approaches, we need to understand some important phenomena about material fatigue. 

Material Fatigue Theorems

There are different kinds of theorems that we are using to calculate the different material fatigue problems in general.

Goodman Theory

Goodman material fatigue theory.
Goodman diagram(Robert L. Moth: Machine Elements In Mechanical Design).

With Safety Factor



 












Without Safety Factor



 








Goodman theory is a very useful theory that you can use for material fatigue calculation of your ductile material, that undergoes alternating stresses. Goodman equations are very basic but, definitions of variables in that theory are very important and depend on you.

We have also a Von-Mises stress calculator for ductile materials.

You can calculate your design’s safety situation with Goodman theory in two modes;

  • With Safety Factor: In this case, we make the calculation of the safety situation with the safety factor. Definition of safety factor is very important, and changes according to the application and material.
With sfety factor.

The above equation gives the Goodman line that we indicate as ‘Safe stress line’ Goodman diagram above.

Kt(Unitless): This is the magnifying factor that magnifies the absolute maximum value of alternating stress that changes with time sinusoidally.

Sigma A(Unit of Pressure: It is the alternating stress that its value changes with time sinusoidally.

SigmaM(Unit of Pressure): Mean stress value of sinusoidal alternating stress.

Sn(Unit of Pressure): Endurance strength value of your material in terms of stress.

Su(Unit of Pressure): Ultimate tensile stress value of your material.

N(Unitless): Factor of safety.

  • Without Safety Factor: In this calculation option of Goodman theory, we do not consider the factor of safety.
Formula without safety factor.

All the variables are the same with calculation with safety factors.

Goodman Theory for Shear Stresses

Fluctuating Shear Stresses: Goodman Method



 










In some cases of engineering problems, all the stresses can be in the form of shear stresses. And these shear stresses can be in fluctuating form which means the magnitude of shear stress can change in changing time. In this formula, the used fluctuating shear stress is in the form of sinusoidal wave.

We can apply Goodman method on ductile materials. So if there is a sinusoidal shear stress application on ductile materials, we can apply Goodman method to see material fatigue situation of this application.

Goodman material fatigue theory with shear stresses.

The formula of Goodman method for shear stresses is like above. It is actually same with Goodman method for tensile stresses. In here, Kt represents the concentration factor which depends of your application and material. You can define a concentration factor according to critical region that are prone to be failure in your material. Ta is the absolute maximum shear stress in sinusoidal wave form. Tm is the mean shear stress in this sinusoidal wave.

Ssn is the endurance shear strength of your material and Ssu is the ultimate shear strength of your material. In general applications we can consider Ssn as 0.577*Sn and Ssu can be considered as 0.75*Su. Sn and Su values are the tensile strength characteristics of your material.

N is the safety factor which depends on your application again.

Gerber Method for Fluctuating Stress

Gerber Method For Fluctuating Stress



 










Gerber method's graph
Gerber method’s graph(Robert L. Moth: Machine Elements In Mechanical Design).

In some engineering problems, some fluctuating stresses or alternating stresses can occur in time. These alternating stresses are generally below the yield strength of material. But application of these alternating stresses can be problematic for part or material. Generally failure occurs in a certain time of application of these alternating stresses. We call this failure as material fatigue failure. We use the Gerber method generally for ductile materials to predict the fatigue failure in certain force or stress application.

Gerber material fatigue formula.

This is the formula of Gerber method to predict material fatigue failure of a certain application. In here, SigmaA is the absolute maximum value of fluctuating or alternating stress. Su is the ultime tensile strength of material that you are using in your application. SigmaM is the mean stress of maximum and minimum peak points of sinusoidal alternating stress. Sn(prime) is the estimated endurance strength of your material.

If the calculated value from Gerber method is bigger than 1, material fatigue failure from this application is possible. If the calculated value with this formula is smaller than 1, there is no place for fatigue failure with this application.

Another alternative of Gerber method is also Goodman method to calculate fatigue situation for ductile materials.

Soderberg Method

Gerber Method For Fluctuating Stress



 










In some engineering problems, some fluctuating stresses or alternating stresses can occur in time. These alternating stresses are generally below the yield strength of material. But application of these alternating stresses can be problematic for part or material. Generally failure occurs in a certain time of application of these alternating stresses. We call this failure as material fatigue failure. We use Gerber method generally for ductile materials to predict the fatigue failure in certain force or stress application.

Gerber method.

This is the formula of Gerber method to predict material fatigue failure of a certain application. In here, SigmaA is the absolute maximum value of fluctuating or alternating stress. Su is the ultime tensile strength of material that you are using in your application. SigmaM is the mean stress of maximum and minimum peak points of sinusoidal alternating stress. Sn(prime) is the estimated endurance strength of your material.

If the calculated value from Gerber method is bigger than 1, material fatigue failure from this application is possible. If the calculated value with this formula is smaller than 1, there is no place for fatigue failure with this application.

Another alternative of Gerber method is also Goodman method to calculate fatigue situation for ductile materials.

Material Fatigue in Ductile and Brittle Materials

Fatigue failures are much more important for the brittle materials than the ductile materials. Because fatigue failure takes place abruptly for brittle materials without giving any warnings. But for ductile materials, there are some warning signs that fatigue would occur. Preliminary deformations and cracks are showing the imminence of the material fatigue growth in ductile materials. 

Material Fatigue for Different Materials

For different materials, the fatigue design approach differs. There are experimental methods developed for cardol steel materials but these experimental methods do not apply to other materials. 

Stress Cycles

We classify the stress cycles in fatigue calculations like below.

Low Cycle Material Fatigue

Firstly, low cycle fatigue is regime where stress cycles are below 1000. There are lots of systems that show low-cycle behavior such as set screws on shafts, truck wheel studs, and automotive glove compartment latches. 

At the low cycle stress, the device will survive 1000 cycles. In this case, designers tend to design the parts in an infinite life approach. In the infinite life approach, stress levels are not high as any fatigue will occur. And the stress levels at low-cycle are much higher than the high-cycle. 

You Can Use the Static Design Approach

Furthermore, in the design of low-cycle fatigue, we can use a static load design approach. Static load design calculations are made by the consideration of the yield strength of the material where the maximum stress is observed without any elastic deformation. Also, safety factors are used in the static design of the components which will probably eliminate the low-cycle fatigue problem. 

High-Cycle Limits for Steel

Moreover, we defined the fatigue limits for steels well according to their maximum allowable stress values. We defined the limit between the low-cycle and high-cycle fatigue below.

The low-cycle fatigue limit for general types of steel is approximately 0.9 times of ultimate fracture strength in bending loading conditions. 

In the axial loading condition of the steel, the limit is defined as 0.75 times of ultimate fracture strength of the steel. 

This value is 0.70 times the ultimate fracture strength of steel in torsional loading. 

High Cycle Material Fatigue

High cycle fatigue is the fatigue stresses of materials between the 10^3 and 10^7 stress cycles. Between these stress cycles, the stress levels lower below the yield strength of the material. 

High cycle is also classified as finite and infinite life of materials. Few materials are showing infinite life endurance limit stresses such as titanium alloys and ferrous materials. Below this endurance limit, the material shows any material fatigue. 

There is no infinite life for any material. The stress levels below the endurance limits lead to failure of the material at gigacycles. Gigacycles are a very high cycle of use and there is no machine work in gigacycles. So, we consider gigacycles to be infinite life of the components and materials.

In the S-N curves of materials, the slope of the low is higher than the high-cycle fatigue

Propagation of Cracks in Material Fatigue Failuıre

Typical material fatigue failure.
Typical fatigue failure(Image Source: Yenaengineering.com).

In general mechanical fatigue incidents, crack propagation starts from the surface of the material. Possible surface defects are the starting points of the cracks and incisions which are the reason for fatigue cracks. So, surface characteristics are very important in fatigue propagation. 

How to Improve Material Fatigue Performance of Materials and Parts?

In the design phase of fatigue, there are different approaches are applied to prolong the fatigue life of the design. These applications are; 

  • Improving the total length of the crack in which material fatigue failure occurs, 
  • Improving the total time that material fatigue crack occurs, 
  • Slowing down the material fatigue crack propagation time, 
  • Improving the surface characteristics of parts to prevent material fatigue failure.

By using different methods, the fatigue performance of the developed parts and systems can be improved. 

Total Lenght of the Crack

The total length of the crack where material fatigue fracture takes place is a very important parameter in the fatigue design. In general, materials that have the best fracture toughness values are the most durable materials to fatigue fracture. If the fracture toughness is high, the feature of withstanding failures when there is a crack improves. So, there is a direct relationship between the total length of the crack and the fracture toughness of materials. 

Crack Occurance Time

To improve the fatigue performance of a part or material, the occurrence time of the fatigue crack is also a very important parameter. If the crack occurrence time is long, the material fatigue performance will be better. To minimize the crack occurrence, compressive surface residual stresses are applied by applying different processes such as shot peening. Small metal beads are thrown to the surfaces of the mechanical parts to impart surface residual stresses to improve the total tensile stresses required to crack propagation.

Crack Propagation Time

This is not about the crack occurrence. This time is about the crack propagation time after the crack occurs. To improve the crack propagation time, surface characteristics are improved. To prevent the propagation of the cracks, the grain boundaries are produced in a parallel direction to the surface of the part. So, grain boundaries are used as barriers to the crack growth paths. 

Surface Characteristics

The surface characteristics of the material have a very important effect on the performance. If the surface characteristics such as smoothness of the surface are improved, the possibility of crack propagation is prevented. Extra processes are applied to improve the surface characteristics such as grinding and polishing. 

Cyclic Stresses in Material Fatigue Calculations

The situation of the cyclic stresses in mechanical one is a very important factor. Different types of stresses can occur and cause fatigue on material or part. These stresses are; axial stresses which are tensile and compressive, bending stresses, and torsional stresses.

To see the cyclic stress situations of a mechanically loaded system, you can analyze the general situation of cyclic stresses by using this cyclic stress calculator.

The cyclic application of these stresses can cause fatigue situations in materials and parts. In different mechanical applications, cyclic applications of these stresses can occur on materials. 

Experimental and Theoretical Methods to Calculate Material Fatigue Strength

There are various theories are developed to calculate the fatigue strength of different materials. But these theories generally can not be generalized to further calculations of different materials with different surface characteristics. Because of this reason, fatigue calculations are depending on much more experiment-based results. Also, the development of possible general material fatigue experiments is a very hard thing. In general engineering applications, engineers are considering the worst-case scenarios and they produce their designs according to these worst-case scenario. But the worst-case scenario in the fatigue is the worst condition where crack occurs and fatigue fracture takes place. So in only one cycle of loading, fatigue fracture takes place. 

There are several fatigue strength tests are developed. One of them is Moore’s machine for the calculation of material fatigue strength. In this machine, a specific geometry of cylindrical test specimen is prepared and loaded with the pure bending condition. For different surface characteristics and loading conditions, lots of kinds of experiments are made to obtain very strict data that shows the material’s specific fatigue strength. By using this data, engineers can design parts or systems that have specific fatigue strength and fatigue life. 

Moore's specimen for fatigue tests.
Moore’s specimen for fatigue tests(Image Source: Steven R. Schmid, Bernard J. Hamrock, Bo O. Jacobson – Fundamentals of Machine Elements, 3rd Ed., Pg:163).

According to the manufacturing techniques that are used to produce a specific part, the average and worst surface characteristics where the crack initiators would occur are used as test surfaces. The different loading conditions are also specified according to the application. 

Material Fatigue Growth at Microstructural Level

At the microstructural level, there are different microstructural fatigue crack growth behaviors that occur in the ongoing time to the fatigue failure. Near the origin of the fatigue crack, fatigue crack occurrence is very slow according to the surface finish on the part. When fatigue cracks are developed over time, the development of cracks are being faster with the ongoing time. Visible striations can be seen in a typical fatigue crack after the fatigue fracture takes place. When the required amount of striations occurs, an abrupt brittle fatigue fracture takes place. 

S-N Curves

S-N curve of materials.
The S-N curves of aluminum alloys(Image Source: Steven R. Schmid, Bernard J. Hamrock, Bo O. Jacobson – Fundamentals of Machine Elements, 3rd Ed., Pg: 166).

Firstly, S-N curves are the plottings where several cycles and the fatigue stress levels that they show. You can find the estimated corresponding number of cycles of the stress levels from these curves of materials.

We also call them Wöhler Diagrams named after a German engineer August Wöhler who published his first material fatigue research in 1870.

How to Obtain These Curves?

And also, they plot the S-N curves with the rotating beam experiments. In rotation beam experiments, we use a standard beam. And we load with pure bending. And then, we rotate the beam under these bending conditions up to failure. This pure bending load is below the yield strength of the material. By the rotation of the test specimen, we obtain cyclic pure bending loading. So, they are logging into the graphics up to which cycles that this test specimen will endure. These graphs are the S-N curves of the specimens. 

Endurance Limits on S-N Curves

In some of these curves of materials, there is a limit which we call an endurance limit. Cyclic stress levels are below the endurance limits of materials, that material will not fail because of material fatigue. So, below this level, we can apply an infinite number of cycles to the part. Furthermore, not all the materials are showing endurance limits on S-N curves. 

Which Materials Have Endurance Limits?

Endurance limits.
You can see the exact endurance limit of steel(Image Source: Wikimedia.).

In general ferrous materials and titanium alloys are showing straight endurance limits. And such materials as aluminum alloys, copper, and magnesium are not showing any straight endurance limits in their S-N curves. 

But the prediction of the endurance limits for materials is not an easy thing. We require lots of kinds of experiments that may predict endurance limits. Once we define the endurance limits of materials, we can produce safe structures in terms of material fatigue. 

Endurance Limits of Steels

There is a prediction formula for the endurance limits of steel. With the different loading conditions, different kinds of endurance limits can be calculated. 

  • In the bending load, the endurance limits of general streels are approximately 0.5xUltimate Strength. 
  • In the axial load, which may occur as compression or tension, the total endurance limit can be calculated as 0.4xUltimate Strength. 
  • In the torsional loading conditions on steels, the approximate value of the endurance limit can be calculated as 0.30x Ultimate Strength. 

For some materials, there are no data to show all the S-N curve diagrams with changing load stresses. Research and development institutions, obtain these curves. And also endurance limits may be specified to use in such applications. 

Conclusion

Also, there are other forms of fatigue failure such as corrosion fatigue and thermal fatigue. But we are dealing with mechanical applications and others are out of our scope here. 

Do not forget to leave your comments and questions below about fatigue failure and fatigue in mechanical design. 

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