Up until now, in this series on mechanics of materials, we have explained various internal forces and deformations.
These are all important topics related to destruction, and can be considered a stepping stone to understanding the destruction of objects that we will be starting to explore.
Not only mechanical designs, but all structures are calculated, tested, and implemented based on the principles of material mechanics and fracture.
The same calculations are applied to everything from tiny screws to massive structures like the Tokyo Skytree.
このIt's safe to say that objects that haven't been considered for destruction basically don't exist in this world; that's how important it is.
In other words, it is desirable, or rather essential, that everyone involved in manufacturing understands destruction.
I'll begin the explanation here, but it's important, so it will be a bit long. Please try your best to understand it.
Theory of Destruction
While there are theories that exist regarding fracture, there is no general theory that can cover all types of fracture. Therefore, there are theories that apply to each type of fracture, and these are selected and applied according to the material's properties and the nature of the fracture.
Also thisTo explain why fracture theory is important, nowadays, when performing strength calculations using computer simulations (CAE), which are the mainstream method of calculation, a human needs to select and judge which theory should be used to calculate the results and display them.
To determine how to display the results, it is necessary to understand the basics of the theory.
If this is done incorrectly, even if the CAE performs accurate and precise calculations, the result display will be flawed, leading to serious problems.
This article is solely for design purposes, so I will only provide an overview and name without going into too much detail.
Maximum Principal Stress Theory
First, there is a theory called the maximum principal stress theory.
In essence, this involves taking a tiny portion of a structure subjected to external forces (loads or bending) and considering the stress acting on that tiny portion.
For example, if a small shape is cut out from a round bar subjected to a tensile load P, stress will be generated on each surface of that small shape.
そのThis theory states that if the maximum stress within a tiny portion of a material exceeds its tensile or compressive yield point, it will fracture.
This theory generally works well for brittle materials (hard and brittle materials like concrete), but it often doesn't apply to ductile materials (tough materials like Japanese swords or brass) or fracture due to compression. This is because it doesn't take shear stress into account.
absolutelyAvoid using this method for analyzing ductile materials or structures subjected to compression.
またIn CAE results, the stress displayed as the maximum principal stress has both direction and quantity (vector).Be careful.
This detailed explanation requires an introduction to triaxial stress and partial derivatives, so it will be explained after it is covered in the section on industrial mathematics.
最大せん断応力説
The maximum shear stress theory is a shear force version of the maximum principal stress theory.
In essence, this involves taking a tiny portion of a structure subjected to external forces (loads or bending) and considering the shear force acting on that tiny portion.
For example, if a small shape is cut from a round bar subjected to a tensile load P, shear force is generated on each face of that small shape.
そのThis theory states that failure occurs when the maximum shear force within a small area exceeds the shear yield point of the material.
Suitable for ductile materials (thin materials, like Japanese swords or brass) and fracture by compression.This is often the case.
Therefore, this theory's result display is frequently used in CAE.
またIn CAE results, the stress displayed as the maximum shear stress has both direction and quantity (vector).Be careful.
This detailed explanation requires an introduction to triaxial stress and partial derivatives, so it will be covered after it is introduced in the field of industrial mathematics.
Maximum principal strain theory
The maximum shear stress theory is a strain-based version of the maximum principal stress theory.
In essence, this involves taking a tiny portion of a structure subjected to external forces (loads or bending) and considering the strain acting on that tiny portion.
For example, if a small shape is cut from a round bar subjected to a tensile load P, strain will occur on each surface of that small shape.
そのWhen the maximum strain within a tiny portion of a material exceeds its strain limit, it will fracture.
This is often used when comparing with actual test results. When measuring stress in a test, a strain sensor is used, and the raw data is strain, so there is the advantage that it does not need to be converted to stress each time.
Therefore, this theory's results display is often used when comparing tests and CAE.
またIn CAE results, the stress displayed for the maximum principal strain has both direction and quantity (vector).Be careful.
This detailed explanation requires an introduction to triaxial stress and partial derivatives, so it will be explained after it is covered in the section on industrial mathematics.
Total strain energy theory
A new term, "strain energy," has emerged, but it's a fairly simple and useful concept, so I'd like to introduce it.
Stress and strain can be expressed under the name of strain energy, just as the extension (x) and spring properties (spring constant k) of a spring, or the weight (m) and velocity (v) of an object are expressed as kinetic energy ($U= \frac{1}{2}kx^2. U= \frac{1}{2}mv^2$), as learned in physics.
hereIf the strain energy is U, then the following equation holds:
$ U=\frac{1}{2}Eε^2=\frac{σ^2}{2E}, (ε=\frac{σ}{E}) E: Longitudinal elastic modulus $
Shear force is similar.
$ U=\frac{1}{2}Gε^2=\frac{τ^2}{2E}、(ε=\frac{τ}{G}) G: shear modulus$
This strain energy can be explained by the maximum principal stress, and the energy of the three stresses acting on a tiny portion can be calculated and added together. The total amount can then be considered as the total strain energy in that small section (shear force is omitted in the figure).
The diagram shows strain energy only for stress, but of course, shear force also has strain energy.
In other words, The sum of the tiny amounts of strain energy is
Total strain energy = Strain energy due to stress on each surface + Strain energy due to shear force on each surface
Becomes
This concept of strain energy is very convenient because, ignoring the sign (direction) of compression and tension, and the sign (direction) of deformation, if we know the amount of strain and the elastic modulus of the material, we can determine the value of the stress.(This allows us to determine the stress by differentiating the strain energy with respect to the amount of strain and deformation.).
Therefore, it is extremely useful for determining how much stress is being generated during tests using strain sensors.
In the theory of destructionThe strain energy at tensile strength is calculated from the stress-strain diagram, and fracture occurs when a small fraction of the total strain energy reaches that strain energy (when converted to stress, it is almost the same as the tensile strength).
This theory fits well with ductile materials.
このThe strain energy theory is very easy to calculate and useful, but it has a major drawback: the stress displayed in CAE results is only a quantity.Thing.
In other words, we know the amount of stress, but we don't know its direction, so we can't tell if it's tensile or compressive.
このThe difference between tensile and compressive force is very important and will be explained in detail later, but the characteristics of the strain when a load is applied are completely different, and the strength against each force is also completely different.They're completely different things now.
Therefore, be very careful when using this theory.
A detailed explanation of strain energy requires mathematical knowledge such as partial differentiation, so it will be explained after introducing industrial mathematics.
Shear strain energy theory
This shear strain energy theory is very simple and is derived from the total strain energy theory mentioned earlier, focusing only on the shear force.
It has the same characteristics as the total strain energy theory and fits well with the fracture of ductile materials.
このThe stress determined based on the shear strain energy theory is called the von Mises stress (by differentiating the strain energy with respect to the amount of strain and deformation).
このStress is one of the most frequently used results in CAE (Computer-Aided Engineering).
The reason is that, firstly, the calculation is performed using only the energy due to shear force, so the amount of computation required is overwhelmingly less compared to other methods.
In other words, a smaller computational load reduces errors, and the faster physical calculation speed allows for quicker results, making it a frequently used method.
However,Similar to the total strain energy theory, the direction of stress cannot be determined from the quantity of stress alone.
In other words, it's impossible to tell from this result alone whether it's tension or compression.
This orientation is often overlooked, so be careful.
The state of atoms when a material is destroyed
Next, I will explain the basics of the ultra-microscopic atomic movements that occur when matter is destroyed.
The details will be explained in the section on metal materials, so only an overview will be provided here.
Understanding this analogy of atomic movement will help you understand the relationship between the crystal structure and composition of a material and its strength and toughness (ductility), so be sure to grasp it.
First, a quick review of high school chemistry: matter exists as molecules, formed by many atoms bonding together. Metals, in particular, which are often used in machinery, are composed of metallic bonds unique to metal atoms.
In metallic bonding, metal atoms are bonded together by sharing extra electrons throughout the material. This bond is formed by the negative charge of these shared electrons ($e^-$) and the Coulomb force acting between the protons in the atomic nucleus.
This is how metallic materials are formed.
Now let's apply a shear force to the atom in the first row. The bonds between the atoms will hold together to some extent, but they will eventually break.
When intermolecular bonds are broken, the following occurs.
Interestingly, when an intermolecular bond breaks, the atom recombines with its neighbors, atoms 2, 3, and 4, and sticks together. The image is like the atom sliding sideways.
This is called a dislocation.If we consider this phenomenon from the perspective of materials mechanics, we can see that it is quite similar to what happens when a shear force is applied to a material.
In other words, since material fracture is basically caused by shear force, the fracture theories commonly used in the explanation above are those that utilize shear force.
The emergence of this concept led to a rapid advancement in materials development.
If these intermolecular bonds are viscous, the material becomes tough (extensible), but if they are not, it becomes brittle.
So, does being brittle mean it's completely unusable? Not at all. On the contrary, because it's prone to dislocations, it becomes a material with excellent machinability. It's especially well-suited for cutting and pressing processes.
Materials with excellent toughness have superior properties such as strength as components, but they are difficult to process because they do not easily undergo dislocation. Processing them would typically involve forging.
In shortIt's a matter of balancing performance and cost, and finding the right application.
Furthermore, many materials such as iron and aluminum can be processed into materials with various properties by adding various additives (such as molybdenum, carbon, and silicon) or by applying and cooling heat to alter their crystal structure and intermolecular bonds.
This concept is very important, so make sure you understand it.
Regarding tension, it's simply a matter of applying a force that overcomes the bonding force of the atoms to break the bond.
If you'd like to learn more about dislocations, please take a look here.
Summary
To summarize the destruction
There are mainly five theories of fracture: the maximum principal stress theory, the maximum shear stress theory, the maximum principal strain theory, the total strain energy theory, and the shear strain energy theory.
The maximum principal stress theory is a theory that works well for brittle materials. It should not be used for ductile materials.
The maximum shear stress theory is a theory that fits well with ductile materials.
The maximum principal distortion theory is useful for comparing with actual equipment tests because it is based on the amount of distortion.
• The total strain energy theory requires less computation and is easier to calculate, but note that it does not reveal the direction of the calculated stress.
The shear-strain energy theory requires the least amount of computation and is therefore frequently used.
• The stress calculated using the shear strain energy theory is called the von Mises stress and is frequently used, but note that its direction is unknown.
When a substance is destroyed, dislocations, which are lateral slips between atoms, occur.
become.
This is a bit more information than usual, but it's very important, so please understand it well.
Especially these days, manual calculations are rarely done; simulations (CAE) are the mainstream method.
However,CAE simply performs calculations; it's humans who decide which fracture theory to apply and whether it matches the physical phenomenon..
To make the right decision, you need to understand the underlying principles; otherwise, you might make a mistake, and when you actually use it, it breaks easily—this happens all the time.
It's fine if it can be fixed with a few minor adjustments, but if you make a mistake in the foundation, you have to start all over again, which is a huge problem.
Furthermore, when looking at the results of calculations performed by others, it's impossible to even look at them or make a judgment without first understanding which theory was used for the calculations.
A mistake in the applied theory is far more frightening than a minor calculation error.'S.
Now that AI and simulations are developing at such an accelerating pace, I want you to take the time, even if it's a bit tedious, to thoroughly understand the theory.
Next time, I will introduce a more specific concept of destruction.
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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