Continuing from last time , I will explain the very important concepts of elasticity and the basic properties of materials.
This explanation is extremely important in mechanical design, so please try your best to follow along.
Also, for those who want to learn about mechanics of materials from the beginning in a way that even beginners can understand, please refer to this index → Mechanics of Materials Series for Beginners
There are many rules, but as I mentioned at the end of the last post, memorization isn't the most important thing for a mechanical designer ; what's important is the ability to search correctly. So please make sure to remember where everything is written.
What is elasticity?
First, let's explain the concept of elasticity, which is extremely important in mechanics of materials.
Elasticity is simple: just like a spring, when an object is stretched, a force acts to return it to its original shape. An elastic body is one in which the relationship $constant = \frac{stress}{strain}$ holds true.
Furthermore, the constant defined here , denoted as E, is called the elastic modulus and is an important physical property.
This type of relationship is called Hooke's Law, but the name doesn't really matter.
Let's stop with the formal language here and explain it concretely with diagrams.
First, imagine a simple spring.
Well, obviously, if you pull on a spring, it will try to return to its original position, so I think you can see that P = kx.
In fact, this relationship holds true for many objects as well.
To put it simply, without fear of misunderstanding, an object is like a spring.
However, here's the key point
Just as the force kx (spring constant × distance) of a spring is determined by the amount x it stretches, the stress Eε (elastic modulus × strain) of an object is determined by the amount ε it deforms and the stress it returns to.
$ σ (stress) = E (elastic modulus) × ε (strain) $
The elastic modulus E is very similar to the spring constant K of a spring. The unit of E (elastic modulus) is Pascals (Pa).
The elastic modulus E mentioned here has units of Pa, is a constant, and is determined by the material.
If you've done some mechanical design work, you tend to remember that steel has a load capacity of around 210 GPa, and duralumin has a load capacity of around 70 GPa.
This law also applies to shear forces.
Please recall the previous example of shearing .
Please pay attention to the yellow arrow on the right side of the diagram.
It tries to return to its original shape, just like when it is pulled.
$ τ (shear stress) = G (shear modulus) × λ (strain) $
The unit of G (shear modulus) is Pascal (Pa).
Similar to the elastic modulus E, the shear modulus G has units in pascals (Pa), and its value is determined for each material.
In the explanations so far, we 've described objects as being similar to springs, generating a stress that tries to return them to their original shape, and we've called this relationship elasticity.
However, as the amount of deformation of an object increases, this law no longer holds true.
To see this relationship, we use a graph called a stress-strain diagram .
Stress-strain diagram
You might be wondering, "What is a stress-strain diagram?" I'll explain it carefully.
Tensile test
First, a machine pulls a constricted rod whose shape and dimensions are determined by JIS (Japanese Industrial Standards).
Well, this is called a standard test specimen, but the name doesn't really matter.
The key point is that the shape is fixed . (If you want more details, please look up the JIS standard yourself.)
In this test , the stress and strain generated in cross-section A when the object is pulled are measured, and a graph is created with strain on the horizontal axis and stress on the vertical axis.
This is called a stress-strain diagram. The name itself is important.
The basic properties of an object are as shown in the following diagram.
This is quite important, so please be sure to remember it.
Stress-strain diagram
From here, we will explain in detail how to interpret stress-strain diagrams.
First, let's explain the straight line portion starting from the origin of the coordinate axes, 0.
Elastic range
I apologize for jumping straight to the point, but I'm going to state something important.
The straight line from the origin o of the coordinate system is called the elastic region, and it is the range in which Hooke's Law holds true.
As some of you may have noticed, this is the region where the object I've described has spring-like properties.
What's really important here is that when you pull it, it returns completely to its original position .
In a stress-strain diagram, the stress at the end of the straight line is called the proportional limit (σa) and the elastic limit (σe ).
I know I'm being persistent, but in this area, if you pull it, it will completely return to its original shape.
Next, I'll explain the regions with jagged characteristics. This part is extremely important.
Lyudas Domain
First, let's define the terms.
The stress at the peak just before the jagged knee on the graph is called the upper yield point (①σsu in the figure).
The jagged parts of the graph are called the ludus region.
The stress at the end of the jagged part of the graph is called the yield point (②σsl in the figure).
First, please look at the diagram so that you can remember the name.
Let's start with ①.
Stress and strain increase proportionally from coordinate 0, and the point where this proportionality ends is called the upper yield point.
Up to this point, the object is in the elastic range where it returns to its original shape after being stretched.
Next, let's look at ②.
Section ② is called the yield point, which is the point where the jaggedness ends and a strange curve begins, and the stress at that point is the yield point.
If the distortion increases any further , it will naturally lose the ability to completely return to its original state.
The lowest point of stress in the jagged section of the following ③ is called the yield point.
What's happening here is that the object slips at a 45-degree angle to the direction of the pull (this will be explained in detail in the destruction section, so you can skip it here).
If you simply consider the diameter of the shaft, you would think that it would decrease when pulled (Poisson's ratio), but in reality, it slides diagonally, so the diameter actually increases.
Well, what you should remember here is that while the strain increases, there is a region where the stress remains constant (a jagged pattern) .
This is a critical point where, as the distortion increases, the object loses its property of trying to return to its original shape completely.
It simply cannot be reversed.
Next, I will explain the remaining parts.
plastic region
Let's move on to the next part of the graph.
If you pull an object beyond the jagged edge (Lydus region), it enters a region where it will not return to its original shape completely.
This is called the plastic region, and the name is important.
Let's explain this in detail with diagrams.
Let's say we pull the specimen further from the lower yield point, and then stop pulling at point P, for example.
In the elastic region, an object returns to its original shape completely, but here we are in the plastic region, so it will not return completely.
The amount of deformation that an object cannot completely return to its original shape is called permanent strain (①εp in the diagram).
However, in reality, there is a slight tendency to return to its original shape, which is called elastic strain (②εe in the diagram).
If we try to stretch it further from here, there is a coefficient that is proportional to the elastic strain εe (though not perfectly proportional), and we return to point P accordingly.
If you pull the test piece even harder, it will begin to deform and constrict.
The stress generated at this time is called the tensile strength (③σf in the figure). The generated stress is the maximum in this tensile test.
Pulling it any further will cause it to break.
What's interesting here is that once the waist starts to constrict, the distortion increases, but the stress doesn't.
This is when the object starts to harden on its own.
Because it hardens itself, its strength increases automatically, and even if the strain increases, the stress generated decreases.
I think the detailed mechanisms behind this are being researched at universities and such.
While there are methods to increase the strength of an object by utilizing this spontaneous hardening property, such as work hardening and age hardening, these will be explained separately.
Furthermore, there's a technique called plastic deformation that utilizes areas that cannot be returned to their original state.
These include forging and press working, which everyone enjoys (I'll explain the details in the processing section).
This concludes the explanation of stress-strain diagrams.
The stress-strain diagram described here is a typical example for mild iron (tough iron).
In fact, this tensile test can reveal the mechanical properties of all materials except resins and rubber.
From here on, we will refer to objects as materials and introduce stress-strain diagrams for various materials.
Stress-strain diagrams for typical materials
Here, we present stress-strain diagrams for some representative materials. These are just general guidelines.
I think you'll be able to visualize comparing the materials.
Here, another important concept emerges.
As you can see from the diagram, it's difficult to determine a clear yield point for materials other than soft iron.
The concept used in this context is yield strength.
The limit of stress in the region where a material exhibits mostly elastic properties but retains only a very small amount of permanent strain when stretched is called yield strength.
In the industrial world, it is acceptable to allow for a permanent strain of 0.2%, and this stress point is called the 0.2% proof stress point. For materials where the boundary between the elastic and plastic regions is unclear, the 0.2% proof stress point can be treated the same as the yield point.
With this explanation, you should now understand the meaning of the material specifications provided by each company.
Practical Application of Stress-Strain Diagrams
When designing machinery, please try to look at stress-strain diagrams if possible.
However, since you'll often be pressed for time, be sure to look at the specifications of the materials you'll be using and visualize their properties in your mind.
When designing machinery, it is essential to select materials that are appropriate for the stresses that inevitably arise.
Therefore, the basic principle is to design within the lower yield point (which is called the yield strength in design) or within the 0.2% proof-of-conductance limit.
However, it is fundamentally impossible for a machine to reach the end of its lifespan after receiving a simple load only once.
Machines are subjected to various loads repeatedly depending on how they are used.
The stress generated by repeated loading is called cyclic stress.
Unfortunately, this explanation does not allow us to determine whether the material can withstand this repeated stress.
Such assessments require fatigue limit diagrams and S/N diagrams in addition to stress-strain diagrams.
I will explain this in detail later, but understanding stress-strain diagrams is essential for understanding fatigue limit diagrams and S/N diagrams, which are extremely important, so please make sure you understand them well.
I've discussed various aspects, but let's start by designing it with the basic principle that the lower yield point equals the yield point.
Conversely, the only situations in which materials in the plastic region can be used are for machines that only need to be held once for testing, or simply when they are continuously supported without any change in stress; in such cases, it is acceptable to use materials in the plastic region and tensile strength.
Furthermore, it is used only for special applications such as plastic tightening or plastic deformation of bolts (it is not generally used for parts that are meant to be reused).
Since we're on the subject, let me also introduce a simple example of material specifications.
I'm sure you now understand what each item means.
We'll gradually power up the table, so please bear with us for now.
Additionally, the physical properties of commonly used materials will be compiled separately.
As a practical technique, let's say your company or a related manufacturer proposes a new material.
In that case, be sure to request the stress-strain diagram, fatigue limit diagram, and S/N diagram.
Without this, it's impossible to create a proper design.
In most cases, they only bring basic specifications for the materials at first.
When you request the above documents, people are often reluctant.
However, any company with real capabilities should have conducted various tests and possessed the necessary data.
Conversely, if they don't provide the material even after you request it, it might be because they're using a material they've heard about secondhand, or they haven't bothered to do the testing because it's too much trouble (the testing is quite difficult).
In such cases, it is the designer's prerogative to recommend against using that material if possible.
Summary
To be honest, this lesson is quite extensive, and there might be a lot to remember, which could make it boring.
However, that's precisely why it's so important, and not only designers but everyone involved in manufacturing should be constantly using stress-strain diagrams in practice.
Otherwise, you're practically an amateur.
Conversely, if you can grasp even a little of what we've discussed, I think you can say you've already taken your first step as an engineer .
With this level of understanding, I believe you can already work as an engineer with confidence.
Next time, I'd like to explain torsional stress .
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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