So far, we've covered most of the basic deformations of objects.
Up until now, we have explained stress and deformation in uniform (same-shaped) cross-sections, but this time we will explain the generation of stress when the shape of the cross-section changes abruptly.
While there may be a theory behind this, we'll mostly be dealing with formulas derived through experimentation. Therefore, there's almost no explanation of the proof or meaning of the formulas. We'll simply introduce the formulas.
What's important here is developing an intuition for how much stress increases depending on the shape change.
Therefore, I want you to understand the relationship between shape and stress.
Before we begin the material mechanics explanation, I'd like to give you a conceptual explanation of "what stress concentration is" in the following link, so please take a look if you're interested.
Now, let's explain stress concentration in the mechanics of materials.
応力集中
Up until now, we've considered simple round bars or beams with a continuous square cross-section, but this time, let's consider cases where the cross-sectional shape changes midway.
Typical examples include rods and plates. Rods may have grooves for inserting O-rings, while plates may have holes or elongated slots for weight reduction or for tightening screws.
We will consider how the stress changes when there is such a change in cross-sectional shape.
応力集中
Now, let's consider an example problem as usual.
It's hard to imagine, but imagine a plate with infinite width and length, with a hole of radius 'a' in the center.
In the diagram, the vertical axis is the y-axis and the horizontal axis is the x-axis, with the center being 0. A uniform stress σ0 is generated in the vertical direction on the plate.
When such a hole is present, the stress at the cross-section where the y-axis is 0 can be expressed by the following equation. (Almost an empirical formula)
$ σy=σ0(1+\frac{a^2}{2x^2}+\frac{3a^4}{2x^4}) $
Here, σy is maximized when x = a, and its value is...
$ σy=3σ0 $
This is the result. The stress is highest at the edge of the circle, and the stress approaches σ0 as you move away from the circle.
in this wayThe phenomenon in which stress increases due to shape is called stress concentration.
To show the degree of stress concentration, we define the reference stress as σn (stress without change in cross-sectional shape) and the maximum stress as σmax, and express it by the following equation.
$ α＝\frac{σmax}{σn} $
This α is called the stress concentration factor or shape factor.
The method for determining the reference stress σn is actually difficult, but in this example, if we take σ0 as the reference...
$ α＝\frac{3σ0}{σ0}=3 $
Becomes
hereA stress concentration factor of 3 is a number that serves as an indicator of stress concentration due to other cross-sectional changes, so please remember it.
It seems like the shape factor is often 3, perhaps by chance.
Therefore, when doing a rough assessment, I sometimes use a 3x multiplier.
If the weight-reducing hole is oval
Next, let's consider the case where the hole in the circle is replaced with an ellipse, as in the previous example.
The ellipse is defined as having a horizontal length of 2a, a vertical length of 2b, and a radius of curvature of ρ (the degree of curvature of a circle, $ρ=\frac{b^2}{a}$).
In this case, the stress at the cross-section at coordinate y=0 is expressed by the following equation.
$ σmax=(1+2\sqrt{\frac{a}{ρ} })σ0 $
If we take σ0 as the reference stress here, the shape factor α is
$ α=1+2\sqrt{\frac{a}{ρ} } $
What's important here isThe smaller the radius of curvature ρ and the sharper the curvature, the higher the stress becomes.
Conversely, if the curvature is gentle, the stress decreases. If the radius of curvature is 'a', it will be 3, the same as a circle.
This is important because the more abrupt the shape change, the more stress concentrates.
Therefore, it is desirable to design the shape to change gradually.
Also, in durability tests, etc.When a test sample breaks, the point of failure is often a place where the cross-section has changed abruptly like this.So be sure to look carefully when searching for the fragments.
Next, I will introduce some relationships between changes in cross-sectional shape and shape factors that are frequently encountered in actual design.
Typical cross-sectional changes and stress concentration factors
If there is a hole in the center of the board
A plate with a width of 2b and a thickness of t has a hole of radius a, and the length of the plate does not need to be greater than the length of the hole. When a tensile load P is applied to the plate, the stress concentration factor is as follows:
Just like before, the smaller the circle, the larger the shape factor, and vice versa.
Here's something to be aware of:WhenEven if the circle is very large, the shape factor will always be greater than 2.
In other words, no matter how smooth the shape is, if the cross-section changes, stress concentration is unavoidable.
However, since various cross-sectional shapes are required in mechanical design, the key is how to reduce the shape factor.
Incidentally, be aware that shapes with holes like this are common in mechanical design, such as bolt holes and pinholes.
If the board has a notch
Next, let's consider the case where the same board (width 2b) has semicircular and elongated notches at both ends.
The left side of the diagram shows a semicircular notch with radius a. The right side shows an elongated notch with distance h from the end to the start of the circle, radius r, and the shortest distance between the two notched circles being 2a.
If we set the reference stress in the left figure to $σ0=\frac{P}{2(ba)t}$ and in the right figure to $σ0=\frac{P}{2at}$, the shape factor α changes as shown in the following figure.
This notched shape also exhibits a similar tendency to the plate with a circular hole in the center that we have described.
In a semicircular cutout shapeIf the radius a of the circle is sufficiently small compared to the width 2b of the board, the shape factor will be 3.07 (approximately 3).As the radius 'a' increases, that is, as the shape change becomes smoother, the shape coefficient also decreases.
For elongated notches, the shape factor can reach as high as 6 if the notch depth h is deep and the radius of the circle is small.However, if the depth h is shallow and the reciprocal of the shape change amount b/a is large, the shape coefficient will decrease rapidly.
In other words,It means to make it smooth.
While this shape isn't commonly designed for flat plates, it's frequently used as a groove in shafts and other components. Be careful, as it's used in many applications, such as O-rings and positioning.
If the shaft has a groove
Now let's consider the case where the shaft actually has grooves.
Let's consider a material with a diameter D and a semicircular groove of radius r at an arbitrary position around its axis, where the diameter of the groove's bottom is d. Then, let's try twisting it with a torque T, bending it with a bending moment M, and pulling it with a load P.
The change in the shape factor is similar even on this axis, when there is a notch in the plate.For twisting, the minimum shape factor is 2, and for tensile stress, it is 3.
Naturally, just like with notches in a plate, if the groove has depth, the shape factor will increase progressively depending on the depth and the radius of the circle.
In machinery, shafts require many grooves, so be careful. Basically, design them so that the shape factor is the smallest possible value with a torsional coefficient of 2.
If the shape factor becomes large, the mechanism itself will need to be changed.
For example, if the pressure is too high and a large O-ring is needed for airtightness, it can be replaced with an oil seal, or if the positioning groove becomes too deep, the flange shape can be used for positioning—this is where the skill of the manufacturer comes into play.
Special stress concentrations and techniques to avoid them
Occasionally, even after understanding the explanations up to this point and proceeding with the design, stress concentration can still occur for some reason.
Surface roughness and stress concentration
What's happening is that the surface roughness is too great, causing stress to concentrate along the roughness of the surface. The gray area is the material, and the jagged edges represent the surface shape.
For example, if the surface roughness has a maximum height (Rz) of 100, it is equivalent to having a 100μm notch, or a 0.1mm notch, and cracks may propagate from the valleys in the surface roughness.
In my area of expertise, engines, I've actually seen piston pins and crankshaft pins, which withstand immense forces, break during testing due to cracks caused by surface roughness or shape. Of course, since weight directly impacts performance in these parts, the strength is designed to the absolute limit, so even a slight concentration of stress can cause them to break.
The solution involved polishing the surface to make it smooth and shiny.
in this wayFor parts and areas subjected to large forces, such as impact-like loads, pay attention not only to their shape but also to their surface roughness.
Escape ditch
This article introduces relief grooves, one technique for avoiding stress concentration.
Let's consider an example, as usual. For instance, consider a transmission mechanism where the driving side has a circular concave shape and rotates, while the driven side has a convex shape that receives the force and rotates, or simply a case where a flanged shaft passes through a wall with a hole in it.
At this point, the radius shape (a small piece of a circle) at the base of the convex part on the driven side and the radius shape at the base of the flange on the shaft must be smaller than the radius shapes of the driving side and the wall, respectively, otherwise they will interfere and cannot be assembled.
However, if R is too small, stress concentration will occur and the device will break.
In such cases, a relief groove (undercut) is added.
Adding this relief groove allows for a smoother shape that avoids interference with the radius of the drive side or the wall radius during assembly. Furthermore, since the shaft will be machined on a lathe anyway, this shape will hardly increase costs.
This is how stress concentration is avoided.
However, if you make the relief groove too large, the wall thickness will become thinner or the shaft diameter will decrease, so do it in moderation.
How to properly use the letter R
Now let's consider the case of pulling a simple I-shaped structure.
Some designers are aware of stress concentration and therefore tend to prefer larger radii (R) sizes. Furthermore, larger radii improve moldability.
For example, suppose we attach a single R to an I-shaped structure as shown in the following diagram.
While this may seem good at first glance, under strong tensile loads, stress concentration at the R1.5 tip can occur, potentially becoming the starting point for fracture.
So, how do we deal with this? Contrary to what we've explained so far, we make R smaller and create a short straight line in the central column of I.
If you do this...The smoother shape helps avoid stress concentration. Furthermore, it allows for a slight reduction in weight, resulting in a better shape.
Furthermore, since parts with this shape are often manufactured using molds, the cost remains unchanged.
The only drawback is that it makes drawing instructions more complicated.
theseA series of small improvements leads to a good product.
I've introduced three techniques, but the important thing is to be flexible and adapt to the situation.
Summary
Let's summarize the stress concentration.
When the cross-sectional shape of a component changes abruptly, the stress increases, and this is called stress concentration.
The degree of stress concentration is defined by the stress concentration factor and the shape factor α, and these values are calculated by dividing the maximum stress that occurs by the reference stress.
Stress concentration occurs in areas with weight-reducing holes or notches, and in most cases, the shape factor is around 3.
- Stress concentration in the shaft groove is minimal, with a shape factor of approximately 3.
- The cross-sectional shape change should be designed so that the shape factor is minimized.
Be aware that a rough surface can cause stress concentration.
- To avoid stress concentration, incorporate relief grooves or devise ways to create radii.
Actual stress values are rarely determined using the shape factors described above. Instead, they are usually calculated using simulations (CAE).
However, it's important to understand this characteristic, as a lack of knowledge will prevent proper consideration during the layout and sketching stages.
Oh dear,It may be a hassle, but it's the accumulation of these small considerations and clever ideas that leads to inexpensive, high-quality products.
Without mastering these fundamentals, it's practically impossible to engage in trendy innovations.
This is something that should never be taken lightly.
Next time, I will explain the theory behind actual damage and strength.
We'll start with a review of stress-strain diagrams and a little extra, but it's very important, so please stick with me.
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.
Comment:
Comment list (23)
According to Dr. Kunichika Kubota (Engineering) of Daicel Innovation Park, a materials informatics expert, explaining deep learning solely through multilayer neural networks with four or more layers is incomplete. He explains that the speed of solving problems is increased by cutting neurons according to a set probability called dropout. Since all programming is fundamentally based on mathematics, for example, the origin of artificial intelligence can be traced back to the perceptron, but if viewed from the perspective of adding functions together (function junction theory), its origins can be traced back to the method of calculating the stress concentration factor of long cracks in materials mechanics. It's quite a profound subject.
To the DX Education Manager
Thank you for your comment. (I apologize for the very late reply.)
I don't have much knowledge about deep learning, but I understand what you're saying.
When considering stress concentration in material mechanics from a macroscopic perspective, I think you should use the shape factor mentioned in the article.
However, with today's advanced computers and simulation software (CAE), it's possible to perform fairly accurate calculations without needing to give special consideration to stress concentration in relation to the shape. The reason this is possible is that the shape is discretized (represented as a point cloud), and all the relationships between these points are solved by connecting them with differential equations.
I think the discretization process and the way the function is constructed to connect the discrete data are probably similar to the artificial intelligence concepts you're describing.
Also, in engineering fields (especially mechanical engineering), few people take mathematics seriously, but personally, I feel that since mathematics is the foundation of many technologies, you can reach greater heights if you approach it seriously.
I hope to eventually offer practical mathematics lessons on this blog.
Dr. Kunichika Kubota (Engineering), who lectures on materials informatics at Daicel, presents a fascinating re-educational approach to mathematical materials physics. While many consider the history of neural networks (deep learning) to begin with the perceptron (equation 2), tracing the history of equation 1 of functional junction theory, which appears within it, leads to the calculation of stress concentration in deep notches. This, apparently, predates the advent of fracture mechanics.
Sustainable Friction
Thank you for your comment. (I apologize for the very late reply.)
I'll give "The Rearmament of Materials Physics and Mathematics" a read.
I personally believe that stress concentration and various fields of computer science all have their roots in mathematics, so a strong understanding of mathematics is crucial. I wish Japanese education would put a little more emphasis on mathematics, the source of science.
Did you know that SLD-MAGIC is used in parts for Toyota's hydrogen engine vehicles? Also, it's used in molds for cold-formed skeletal components, and, uh, in robot-related speed reducers, semiconductor vacuum equipment, gear pumps, and basically it's a high-performance special steel with a wide range of applications in cutting-edge fields.
Exoelectron
thank you for your comment.
I had heard rumors about the SLD-MAGIC material itself, but I didn't know it was being used in Toyota's hydrogen engine (thanks for the information).
Before receiving the comments, I thought of SLD-MAGIC as just a high-end version of SKD11 (tool steel), but after researching it further, I realized it's a Nobel Prize-worthy invention.
The reason is that this material was developed using the new tribology theory CCSC (a theory worthy of a Nobel Prize).
For someone like me, a mechanical engineer and engine designer, the theory we primarily use in tribology is EHL (Elastic Fluid Lubrication Theory). This theory largely disregards the chemical properties of materials and is based on the shape of the material, the viscosity of the fluid, and the sliding velocity, resulting in a relatively simple and concise formula.
Therefore, mechanical engineers who are not experts in mathematics or chemistry are dominated by EHL theory (which is why they gravitate towards surface treatments such as DLC and PVD).
On the other hand, the SLD-MAGIC theory, CCSC, incorporates chemical phenomena, so I think it's on a completely different level from conventional theories.
It seems that Hitachi Metals, where the professor worked before him, has now become Proterial. I see, mathematics is important, as stress concentrations in things like ball-on-disk systems are calculated using Hertzian stress.
Biomechanics
thank you for your comment.
It seems like the PhD's move from Hitachi Metals to Proterial was a promotion. Since Proterial is a wholly owned subsidiary of Hitachi, I think it was a move with executive-level treatment.
I agree that mathematics is very important. Not only is it related to the mechanics of materials, but almost all physical phenomena can be expressed using differential equations, so I think understanding mathematics is a prerequisite for setting up equations, solving them, and checking the solutions.
Regarding ball-on-disc contact, since it involves contact between a curved surface and a flat surface, it's a typical pattern where Hertz stress (Hertz surface pressure) is applied. Furthermore, when extending this to heat wear calculations, the PV value is determined by multiplying the surface pressure P (Pa) calculated using Hertz stress by the sliding velocity V (m/s) between the objects.
However, based on my personal experience, while Hertz stress can be used when calculating on a macroscopic level, the values often differ when looking at the deformation of an object or the contact surface on a microscopic level, so careful consideration is important.
Recently, there have been numerous reports of exceeding GPa in cold-formed high-tensile steel forming press technology for automobiles. Looking back, it seems that the emergence of SLD-MAGIC, a high-performance cold-work die steel that reigns supreme among martensitic steels manufactured by Proterial (formerly Hitachi Metals), was the breakthrough. The story of how they successfully used CAE alloy design with artificial intelligence technology (neural networks), which is now commonly heard of, and imparted self-lubricating properties through thermodynamic phase diagram analysis, is well-known in the industry, especially in Nagoya. This is how it relates to the basic friction coefficients of bearings, gears, rolling rolls, reducers, and sliding machine parts, which shows that CAE technology is full of even more possibilities.
Takotsubo Organization Cross-cutting Force
thank you for your comment.
I've never designed an alloy using CAE before, so I don't have much knowledge in that field (does CAE specifically for alloy design even exist?).
However, it seems that general CAE (Computer-Aided Engineering) has made considerable progress in multiphysics (simultaneous calculation of mechanics in multiple fields). Furthermore, AI is being used to optimize parameter settings to obtain the best solution.
In addition to these, if we use deep learning to train an AI with past CAE calculation data (shape, parameter settings, results), I feel that the era is not far off when AI will tell us the optimal shape for everyday mechanical elements without even needing to perform CAE calculations.
As technology advances further, we can envision a future where AI automatically determines whether or not CAE analysis is necessary, and if so, performs CAE under optimal conditions and presents a solution.
In this era, I think the skills engineers need are not just detailed specification review and technical calculations, but also the ability to consider what kind of machine and what kind of concept will be accepted by the market (product design and evaluation can be done by AI).
That's right. That's where the CCSC model comes into play. This boundary lubrication theory presents the idea that each friction sliding mode, or even each combination of sliding materials, has its own unique properties. Even extreme pressure additives are included. I think one of the reasons it's been accepted in the market is that it presents a cross-cutting force for practical performance in a world that was previously compartmentalized by part type and material type, using ball-on-disc lubrication.
I found a copy of "Fundamentals of Solid Mechanics" by Takeshi Kunio at the library, which contains the process by which the PhD used to explain on social media how to derive the stress intensity factor from stress concentration using polar coordinate transformation. I haven't yet obtained GRIrwin's paper, though.
This is the cutting edge of tool steel development, specifically multiscale alloy design for advanced materials used in molds.
No matter how much you think about it in quality engineering, you won't find the answer. How to reduce ball-on-disc variation. Who would have thought it would be the precise adsorption of a small amount of paraffin oil (poor lubrication)?
This is about the CCSC model for organic analysis using Raman spectroscopy.
The latest episodes on martensite and tribology are very informative.
Tribofilm Fullmetal Alchemist
thank you for your comment.
We will be explaining the crystal structures of steel, such as martensite, on our website soon, so please wait for further information.
Martensite has a BCT structure, where the cubic BCC is slightly elongated into a rectangle, right? The strain in that structure determines the substructure of the martensite microstructure, such as lath, block, and packet, and it's the subject of various studies using orientation analysis, such as vane relationships. Also, when the amount of dissolved carbon exceeds 0.6 mass%, it forms a lens-like structure, which makes it brittle and increases retained austenite (while retained austenite normally improves toughness, the embrittlement caused by the lens-like structure outweighs this). There are also things like epsilon martensite and butterfly martensite, but they are not commonly used in practical steels.
Global Journal Bearings
Thank you for your comment, and I apologize for the late reply.
As you mentioned, martensite has a long, needle-like crystalline structure. It's extremely hard, but that also makes it brittle and prone to distortion, so it's difficult to use effectively without annealing.
I've seen advanced crystals like epsilon martensite and butterfly martensite at academic conferences, but I feel that mass production and practical applications are still a long way off.
I found a doctoral paper on materials mechanics. It provides a clear explanation of the calculation of the second moment of area in the stress analysis of a Japanese sword. I'm sharing it as an aside.
http://www.nihontomessageboard.com/articles/Study_of_Japanese_sword_from_a_viewpoint_of_steel_strength.pdf
For those involved in heat treatment
Thank you for your comment. I apologize for the late reply.
I think it's wonderful that it includes not only stress analysis and the second moment of area, but also the crystal structure of metallic materials (mainly carbon steel).
I think the photographs of the martensite-martensite single crystal, pearlite eutectic, and ferrite eutectic are particularly wonderful.
Recently, the spread of artificial intelligence, such as ChatGPT and generative AI, has become a boom, hailed as the impact of an algorithmic revolution. It's said to be breaking away from the theory-driven approach of Newton and Einstein and opening up a data-driven world. Naturally, since this algorithm simulates human thought, it inevitably influences philosophy and even ideologies like China's Cultural Revolution. Furthermore, artificial intelligence faces the black box problem, where even mathematical decomposition doesn't reveal why it behaves the way it does. Amidst this, "re-arming of materials physics mathematics," which claims that simple problems can be decomposed, has been gaining attention. This takes a broad view of how to shape nonlinear functions, for example, the invisible hand of God as argued in economics' "The Wealth of Nations" is the act of combining two functions, known as function confluence theory, and its higher-order states are said to be connected to the forefront of AI research, including neural networks. This function confluence theory is also called the KPI competition model in management studies, and its ideas are spreading to various fields. This stirring of a new philosophy, true to its name as "philosophy," is beginning to shake the very foundations of everything. Isn't this also a polytheistic way of thinking?
Japanese Original Liberal Arts Philosophy
Thank you for your comment, and I apologize for the late reply.
I don't understand philosophy, but I'm certain that the future will be an era where the main approach to theory building will be to find relationships and construct solid theories based on statistical processing of vast amounts of data, rather than relying on the experience, intuition, and talent of past researchers.