Last time, I summarized the one-shot destruction techniques.
This time, we will explain fatigue failure.
このThe material mechanics explained in a way that even beginners can understand is an extremely important topic; it's no exaggeration to say that everything we've covered up to now has been done in order to explain fatigue fracture.
The durability of virtually every product in the world is determined by taking fatigue failure into consideration.Products that have not undergone fatigue failure testing, similar to single-shot failure testing,It shouldn't exist.
Conversely, even if you understand drafting, mechanical dynamics, heat, and fluid dynamics, it's no exaggeration to say that you absolutely cannot design machinery if you don't understand fatigue failure, so make sure you understand it thoroughly.
Now, let's begin the explanation of fatigue failure.
If fatigue failure and single-shot failure are broadly categorized, they can be seen in the diagram below.
The basic classification is the same as for one-shot destruction, and the way of thinking is very similar, so those who have forgotten should review it.
Rather than explaining fatigue failure from a materials mechanics perspective, the following link provides a simpler explanation of "what fatigue failure conceptually is," so please take a look if you're interested.
What is fatigue failure?
The deformation and sudden failure described so far have been phenomena that occur when a constant force is acting on the object. Such a load is called a static load.
However, structures rarely experience a constant force; the forces they are subjected to are constantly fluctuating.
For example, buildings and bridges are constantly subjected to fluctuating forces, such as when people are present or absent, when wind blows, or when earthquakes occur.
A familiar example of this is a coil spring; each time the spring expands and contracts, the coil twists and returns to its original shape.
like thisLoads are called fluctuating loads, and stresses are called fluctuating stresses. Generally, they are often represented by a graph with time on the horizontal axis and force on the vertical axis.
A load that fluctuates regularly is called a cyclic load or cyclic stress.
Let's consider which of the two, a static tensile load or a cyclic load, will cause failure at a weaker load, as shown in the diagram.
It breaks quickly when subjected to repeated stress in everyday life. It's like how a wire breaks if you bend it many times.
This means that even with a load below the yield point, within the elastic region of the stress-strain diagram, repeated application will cause it to break.
How muchEven within the elastic range, applying a load causes the internal structure to deform and become damaged.This will be explained in more detail in the section on metal materials.
In other words, Even if a material is designed to be below its yield point, it will eventually break if subjected to fluctuating loads.
This is simply the natural order of things.
The objectSince breaking down is inevitable, all products and structures have a lifespan (number of uses, service life, etc.), or rather, they are designed with a predetermined lifespan.
The reason why the instruction manual doesn't mention the lifespan is because it's designed so that it won't reach the end of its lifespan under normal use.
Products that experience significant fluctuations in load, such as automobiles, are managed by manufacturers through strict maintenance cycles and legally mandated vehicle inspections.
The reason I don't usually worry about it is that I take it to the dealer regularly, and any parts that are nearing the end of their lifespan are replaced.
I will explain how to determine the lifespan of a fatigue failure.
Fatigue limit
The term "fatigue limit" came up suddenly, but let me explain it in more detail.
To manage failure due to fatigue, it's not enough to just look at the load, stress, or strength at which failure occurs; the number of times the force is applied must also be considered.
To explain how this works, a constant load is repeatedly applied to a material until it breaks, and the stress inside the material and the number of repetitions are graphed. Not just one type of load, but several types are applied, and the number of repetitions until each type breaks is recorded.
The direction of stress can also vary, with various types of loads being applied, including not only tension and compression, but also torsion, bending, and rotation of the plate.
In the actual test, various loads are repeatedly applied and broken, and the number of repetitions is checked, such as breaking after 10 repetitions with load A, breaking after 100 repetitions with load B, and breaking after 1000 repetitions with load B.
このThe test is called a fatigue test or exhaustion test.
It's an extremely difficult test, but any material that doesn't meet this requirement is too dangerous to use in design.
So, what are the results? They are shown in the following diagram.
This diagram S-This is called an N-diagram and is a very important graph (SN stands for Stress - Number of Repetitions).By the way, the graph is a logarithmic graph.
At this timeThe stress that can be withstood 10^7 times is called the fatigue limit σw, τw (w from the German word Wechselfestigkeit, meaning fatigue strength).
Furthermore, just like a single-shot destruction...Fatigue limits due to tensile and compressive forces, fatigue limits due to shear forces, and fatigue limits due to bending forces.Each of these exists.
In mechanical design, it is generally considered good enough if a part can withstand more than 10^7 cycles (and is therefore considered to be practically permanent).
Furthermore, the point where the line in the S/N diagram becomes horizontal is called the critical number of cycles, and the stress at that point is called the time intensity.
Generally, materials with high durability over time are considered high-grade materials.
Now that we know the stress and fatigue limit that the device can withstand 10^7 cycles, does that mean we can simply design it? The reality is, it's not that easy.
In reality, fatigue tests simply apply the same load in tension, compression, torque, and bending; actual structures are subjected to much more complex and fluctuating loads.
Now, let's explain how to think about these complex loads.
Approach to complex and fluctuating loads and fatigue limits
Before discussing complex and fluctuating loads, we will introduce the basic types of cyclic loads.
Double-handed load, double-handed fatigue limit
The most basic method is to determine the fatigue limit using a fluctuating load. If we let σa be the upper limit stress and -σa be the lower limit stress, we get the following graph (where a is the amplitude).
The horizontal axis represents time, and the vertical axis represents stress in this graph.
In this pattern, the average stress σm (mean mean m) cancels out due to tension and compression, resulting in a stress of 0. The stress generation pattern is the same as in the fatigue test described above.
This is called bidirectional stress, and at this timeThe fatigue limit is where σa becomes σw in the S/N diagram, and this is called the double-swing fatigue limit σw (essentially the fatigue limit).
Unidirectional load, unidirectional fatigue limit
Let's look at the following patterns.
A cyclic stress with a lower limit of 0 stress and an upper limit of 2σa stress, generated by a certain fluctuating load. The average stress is σa.
The horizontal axis represents time, and the vertical axis represents stress in this graph.
This type of fluctuating stress is called unidirectional stress.
このThe fatigue limit stress 2σa for fluctuating stress is called the unidirectional fatigue limit and is represented by σu (I don't know why it's u).
Numerous experiments have shown that the following relationship holds true.
- Fatigue limit due to tensile and compressive forces: Unidirectional fatigue limit σu ≈ 1.60 × Bidirectional fatigue limit σw
・Fatigue limit due to twisting: Unidirectional fatigue limit σu ≈ 1.94 × Bidirectional fatigue limit σ
- Fatigue limit due to bending: Unidirectional fatigue limit σu ≈ 1.66 × Bidirectional fatigue limit σw
In the case of complex fluctuating loads
Finally, I will explain what to do in the case of fluctuating loads that appear to have no discernible pattern.
The upper limit of stress generated by complex fluctuating loads is σmax, and the lower limit is σmin. This can be represented by the following graph.
The vertical axis represents stress, and the horizontal axis represents time.
In situations like this, in the automotive industry, based on my experience,Because it concerns human life, the mean stress σm is taken as the midpoint between the fluctuating upper and lower stress limits and converted into the following graph.
In some industries, more statistical processing might be used to calculate the most efficient mean stress σm and amplitude stress σa, but for products that involve human lives, this method is preferable. If you know Fourier series expansion, you can divide it into even finer and more regular components (I will explain the basics of Fourier series expansion later in industrial mathematics).
This is a slight digression, but there is also the concept of Minor's law, which involves meticulously tracking minute stress fluctuations to examine failure, but let's set that aside for now.
This allows even irregular fluctuating loads to be treated as regular fluctuating loads.
HoweverWhen generating arbitrary mean stress σm and amplitude stress σa, the material's lifespan cannot be directly determined from the fatigue limit σw of the SN diagram.
Therefore, some modifications are necessary, which will be explained in the next section.
We have considered three types of fluctuating loads so far, so let's summarize them in the following diagram.
Fatigue limit diagram
Let's explain what we use to determine the lifespan of a material for any given mean stress σm and amplitude stress σa.
Basically, we use the fatigue limit σw for both swings and the fatigue reduction σu for one swing, as explained up to this point.
Let's consider the basic case of tension and compression.
On a coordinate system with the mean stress σm on the horizontal axis and the amplitude stress σa on the vertical axis, we perform the following manipulation using two numbers (σw, σu).
thisThis is called a fatigue limit diagram. If the stress generated by fluctuating loads falls within this red line, it can withstand $10^7$ cycles, meaning that from a mechanical design perspective, it can be considered to have sufficient lifespan.
This fatigue limit diagram can be drawn in Excel or similar software if you know the fatigue limit σw, the unidirectional fatigue σu, and the yield point σs, so I encourage you to try drawing it yourself. Initially, you can ignore the compression side and focus only on the tension side.
To make it easier to write in Excel, I'll show the coordinates of the ends of the tensioned line.
Even with the same material, the fatigue limits for bending, torsion (both double-handed and single-handed) differ from those for tensile and compressive forces, so don't get them confused.Let's try to do that (the reason is the same as for one-shot destruction, so if you're curious, check out the previous page).
Note that, as shown in the diagram, the average stress σm at the point where the lines intersect the blue lines at 45° angles for tension, compression, bending, and torsion is 0.83σw for bending and 0.97σw for torsion.
Note that in the bending fatigue limit diagram, the bending yield point is the same as the tensile yield point and the compression yield point.
Once we've reached this point, the strength analysis of the components is almost complete.
Fracture surface when fatigue failure occurs
Up to this point, we have explained what considerations should be taken to determine the lifespan in relation to fatigue failure. Now, let's explain what happens to a component that has undergone fatigue failure.
As explained earlier, fatigue failure occurs when damage accumulates due to repeated loading.
In fact, fatigue-induced failure leaves fatigue marks on the fracture surface of a component, as shown in the following diagram.
そのThe marks left by fatigue and damage resemble the patterns that form on a seashell as it grows.
ItThese are called beach marks (seashell patterns) and are a very important part of the appearance.
When a component undergoes fatigue failure, the effective cross-section that can withstand stress is gradually reduced due to damage caused by fatigue.
When the reduced cross-sectional area becomes sufficiently small in relation to the applied load, it reaches fracture stress and a sudden failure occurs.
In other words, destruction isIt can only be destroyed in a single hit; fatigue destruction does not occur on its own, but only in combination with other factors.
ThereforeEach fracture surface always leaves traces of the individual fractures.So be careful and look closely.
There are two modes of failure in fatigue failure, and each of themThese are called low-cycle fatigue failure and high-cycle fatigue failure.
The number of beach marks varies greatly in each case. When there are very few beach marks, it is called low-cycle fatigue failure, and when there are many beach marks, it is called high-cycle fatigue failure.
While this difference cannot be clearly distinguished by the number of beach marks, a smaller number of beach marks, which are easily recognizable to anyone, indicates low-cycle fatigue failure.
This low-cycle fatigue failure occurs when a load, not as large as a single-shot failure but clearly exceeding the fatigue limit, is repeatedly applied.
While this is often due to insufficient consideration, if low-cycle fatigue failure occurs after thorough life calculations and testing, it means that a large load that was not anticipated during the initial planning stage is at work, so we should carefully consider countermeasures.
The destruction is occurring in a complex way.The basic mechanism of fatigue failure is,
become.
Examining these fracture surfaces is extremely important. When I break an engine during durability testing, I don't clean it up at all. Instead, I gather all the scattered parts and fragments, assemble them like a puzzle to recreate the original shape, and check every single fracture surface.
At that time, the secondary damage was divided into three categories: those that suffered a single-shot failure, those that suffered low-cycle fatigue failure, and those that suffered high-cycle fatigue failure.
In that context, we carefully examine the cross-section of the component that has undergone high-cycle fatigue failure to find the starting point.
Based on the analysis results, we will create a story to reveal the true nature of the engine failure, explaining what forces were at work inside it and what kind of destruction mode caused the damage.
In this way, we always investigate how the destruction occurred and try to find out who the culprit was.
Unless we do this, we cannot take effective countermeasures.
As an aside, if the fracture surface of a destroyed part is crushed and difficult to see due to secondary damage, you can lightly polish the suspicious surface to reveal the traces of the destruction.
These tasks are analyzed by bringing together the combined efforts of departments such as design, testing, and materials.
またEven if no clearly damaged parts are found after durability testing, cracks may still be present, so be sure to inspect carefully (crack check).
When cracks or Luders wires like these appear, the component will soon break down, so in most cases, countermeasures are necessary.
As such, calculating and testing the lifespan of components is extremely important, and every company conducts and verifies these tests during product development, even though they are not inexpensive.
またThe basic idea behind durability testing is to set the load so that each component is subjected to $10^7$ cycles using statistical methods. (There are various approaches.)
As you can see, product lifecycle involves many people and departments in product development, including not only design but also testing and quality control, so it's important to understand it well.
Summary of fatigue failure
Let's summarize fatigue failure.
- When subjected to repeated loading, materials will break even if the magnitude of the load is below the material's yield point.
The limit of a material's fatigue failure is called the fatigue limit, and it is the stress that it can withstand for more than 10^7 cycles when the same repeated load is applied.
- The fatigue limit is represented by an S/N diagram, which shows the characteristics of the material in relation to repeated stress and number of cycles.
- The fluctuating loads applied to the member include bidirectional stress, where the same load is applied repeatedly with the same period, and unidirectional fatigue stress, where the same load of 0 or greater is applied repeatedly. The fatigue limits for bidirectional fatigue stress and unidirectional fatigue stress are denoted as bidirectional fatigue limit σw and unidirectional fatigue limit σu, respectively.
The fatigue limit in the S/N diagram is the same as the dual fatigue limit σw.
Irregular fluctuating loads are converted into regular fluctuating loads by taking the upper and lower limits of the fluctuating load and determining the mean stress σm and amplitude stress σa.
To determine the fatigue limit for any given mean stress σm and amplitude stress σa, a fatigue limit diagram is required.
The fatigue limit diagram can be created from the material's bidirectional fatigue limit σw, unidirectional fatigue limit σu, yield point σs, and single-shot fracture strength σb.
- Beach marks (seashell patterns) always appear on the cross-section where fatigue failure occurs.
- The number of beach marks indicates whether the failure is low-cycle or high-cycle.
This explanation will cover the most important information yet.
Products that have not been tested for fatigue failure do not exist, and should not exist.
If you haven't done this before, it's likely that this information is already included in the calculation programs your company uses, so you've been doing it without even realizing it (you might regret it later if you continue doing it without knowing).
This content is setIt's not just the calculation department; almost all departments, including testing and quality control, must understand it.
ProductsWhenever something breaks, whether during testing or in the market, we always investigate the point of failure and look for beach marks to determine the cause.
If we can't do this, we won't be able to respond, and we'll either have to abandon development or take back all of the products on the market.
At that timeThe damage can be enormous, and sometimes even a miscalculation of the lifespan of a single component can lead to a management crisis or even bankruptcy for a fairly large company (like Mitsubishi Motors).
In the worst-case scenario, you could be sued by consumers or local governments, and you'll lose the case 99% of the time.
vice versaHaving company-specific fatigue limit data that even material suppliers don't possess provides a tremendous advantage in product development.
Therefore, many companies, including automobile companies and even those that don't seem to have anything to do with materials at first glance, have materials departments.
Also, when designing something, always check the material specifications, and pay particular attention to the fatigue limit.
If the fatigue limit is not listed, you should be able to get it from the material manufacturer (ideally, they should have an S/N diagram).
IfIf the supplier says they don't have it, I might consider stopping using their materials altogether.
Then, based on the fatigue limits you've checked, create your own fatigue limit diagram. Materials suppliers and materials departments generally only create S/N diagrams and not fatigue limit diagrams, so the design team will have to do their best from there.
Finally, for those who are in a hurry and don't have time to check their fatigue limits, I'll share some tips on saving money based on my own experience.
If the tensile strength is σb, then most steel materials are
- Fatigue limit for bending both sides σw ≈ 0.49σb
- Dual-handed tensile fatigue limit σw ≈ 0.73σb
- Dual-handed twisting fatigue limit σw ≈ 0.58σb
- Fatigue limit for bending both sides σw ≈ 0.38σb
- Dual-handed tensile fatigue limit σw ≈ 0.33σb
- Dual-handed twisting fatigue limit σw ≈ 0.25σb
From this, a fatigue limit diagram can be created.
This is just a guideline, and naturally the numbers will vary depending on the type of steel, heat treatment, type of aluminum alloy, and heat treatment, but it will be roughly around this range.
This is a treat for readers who have read this far through this niche and boring blog.
However, this number doesn't guarantee everything will be fine, so be sure to check the fatigue limit of the material.
This concludes my introduction to fatigue failure. It ended up being a bit long, but the content is important, so I thought it would be better to go through it all in one go rather than break it up.
That concludes the basic destruction, but next I will explain creep destruction, which rarely happens but is the worst when it does occur.
To those who found this article helpful in understanding design:
While there is basically no textbook covering this content, and it is my own original work, I will introduce the textbook that I have been using since I was a student.
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