Stress and strain, and Poisson's ratio (tension, compression, shear) explained for beginners.

Starting this time, we're launching a new series: "Materials Mechanics for Beginners" (was it a series before?).

First, I will explain stress, which is the very basics of mechanics of materials.

Many of you may already know this, but I will explain it again as a review, as it is unavoidable when explaining mechanical element design (screws, gears, pins, linkage mechanisms, etc.) that I would like to discuss in the future.

Also, this time there will be a lot of explanations of rules and regulations, and I won't be able to include many practical tips, so please bear with me.

definition of stress

First, I will explain the difference between stress and force, which I myself got confused about decades ago.

The definition is as follows:

This is extremely important when performing strength calculations for machinery, so be sure to remember it.

To explain with a diagram (using a cylinder as an example):

Here, P is the external force, and the stress σ generated at cross-section A of the object is expressed by the following equation.

$ σ(stress) =\frac{P(external force)}{A(area)} $

This isStress is defined as the force or load P divided by the area A.

The important thing isThe units for external force P are Newtons (N), area A is square meters (m²), and stress σ is Pascals (Pa).

Here's a little tip: the units used in design are almost always millimeters (mm).

The unit of Pa is $ \frac{N}{m^2}$, which can seem difficult to handle at first glance.

However, the area is in mm², so converting it to meters gives us 10⁻⁶ mm².

If we take this as the denominator and look at Pa again, we get $ \frac{10^6N}{m^2} $.

Rewriting this unit gives us megapascals (MPa), which makes it easier to understand.

SoIt's helpful to remember that dividing a force (N) by an area ($mm^2$) directly gives you a number equivalent to megapascals (MPa).

Incidentally, $10^3$ is kilo (K), $10^6$ is mega (M), and $10^9$ is giga (G).

It's just like with computers, so you should definitely remember it.

Types of stress

Now that we understand the definition of stress, let's move on to explaining the types of stress.

First, let's explain tensile stress. (We'll use a cylinder as an example.)

The tensile stress σ in section A is given by \frac{P(external force)}{A(area)}. The unit is Pascals (Pa) and the sign is +.

The key here is the correspondence.

Let's explain compressive stress. (Using a cylinder as an example.)

The compressive stress σ in section A is given by \frac{P(external force)}{A(area)}. The unit is Pascals (Pa) and the sign is negative.

I think you'll understand the importance of correspondence here.

I will explain this later.The failure modes of materials are completely different under tension and compression, and the allowable stresses are also completely different, so it is extremely important to be able to distinguish between tension and compression.

Next, we will explain shear stress, which is fundamental to stress.

For this next example, imagine two boards with cylindrical pins inserted between them, creating a situation where the boards are being pulled apart.

It's hard to see like this, so let's remove the pin.

Imagine a pin being pulled from side to side.

$Shear stress τ in section A = \frac{P(external force)}{A(area)}$The unit is Pascals (Pa) and the sign is +.

Shear stress is represented by τ (tau) and has a positive sign.

It might be due to my lack of experience, but I've never seen a negative value for shear stress (excluding the shear force of a beam).

later,Torsional stress exists, but it's a bit complicated, so I'll explain it later.

These are all the types of stress.

Next, I will explain an important concept in mechanics of materials.

Distortion

Let's explain distortion.

This is an extremely important concept.

Let's take a cylinder as an example again. First, an external force P is applied to the cylinder, and a stress σ is generated.

When stress is generated, an object deforms.

Naturally, if you pull on an object, it will stretch.

Here, we define distortion.

The strain ε = \frac{L + ΔL}{L} has no units (dimensionless quantity).

The distortion is represented by ε (epsilon) and is simply calculated by dividing the extended length (L + ΔL) (Δdelta) by the original length L..

Remember this, as it will be extremely important later on.

またThe units are also important, and I know I'm being repetitive, but tension is positive and compression is negative.

Next, let's discuss shear strain, which is a bit more complex.

Let's consider this using a familiar diagram.

When you take out the pin

Considering deformation (when looking at the pin from the front)

So it takes on a shape like a parallelogram. If we define shear strain here then

$ Shear strain γ = \frac{λ (amount of deformation)}{L (original length)} = tan α (angle) $ Unitless (dimensionless quantity) Sign is +

Shear strain is represented by γ (gamma) and is calculated by dividing the deformation λ (lambda) by the original length L. The sign is +.

Perhaps it's due to the author's lack of experience, but I've never seen that before.

Now we have defined the distortion.

This strain is used to explain some extremely important material properties (characteristics).

Let's consider the state where the cylinder is being stretched again.

When you pull on an object, it stretches.

As you may have noticedWhen you pull an object, it not only stretches but also contracts radially.

hereThe direction of stretching is called vertical distortion, and the direction of contraction is called horizontal distortion.And

In the definition, there is a rule that the direction of the load is vertical, and the direction perpendicular to the direction of the load is horizontal.

From the previous equation, you can see that the longitudinal strain ε is $ \frac{L+ΔL}{L}$ and the transverse strain ε' is $ \frac{D−Δd}{D}$.

This defines an important value.

$ Poisson's ratio ν = \frac{|ε'|(lateral strain)}{|ε|(longitudinal strain)} = \frac{1}{m} $

Poisson's ratio ν (nu) is the value obtained by dividing the transverse strain ε' by the longitudinal strain ε when an external force is applied to an object.Therefore, let Poisson's ratio be $ \frac{1}{m}$.When expressed in this way, m is called Poisson's number.

Incidentally, the | symbol | in an expression is called the absolute value and represents the value obtained by removing the sign from the value inside (absolute value).

Poisson's ratio is extremely important, and in mechanical design, it's used in strength calculations for all sorts of mechanical elements, such as press-fits and gears, so be sure to remember it.

Summary

It might be difficult to remember at first, but these are the basic rules, so I really hope you'll learn them.

またMechanical designers aren't academics, so they don't necessarily need to memorize everything, but it would be good to know where to find the information and what to look for.

In other words, it's important to develop proper search skills.

Finally, and I'll repeat this important point,

During my student days, I only understood the difference between + and - as simply a difference in direction on a coordinate system, which caused me a lot of trouble (I stumbled right from the start).

In mechanics,Be careful, as these coincidences often hold significant meaning.

The next installment of "Materials Mechanics for Beginners" will cover the following topics:This section explains elasticity and stress-strain diagrams.

I want to read it together

Elasticity and stress-strain diagrams (plasticity, material properties) that even beginners can understand Continuing from last time, we'll explain the very important concepts of elasticity and the basic properties of materials. This explanation is also extremely important in mechanical design, so please try your best to follow along. Even beginners can understand...

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.

Basic Mechanics of Materials (Revised Edition)

created by Rinker

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