__Intro:__

"How does this particular material behave?" "Is it brittle? Or is ductile? And to what degree?" "What loads can it withstand before it breaks?" All of these questions and more can be answered by the analysis of the material's Stress/Strain Curve. In the field of materials science, it is a widely used and practical tool that can help you to understand and predict material behavior.

__What to expect from this Instructable:__

This instructable explains the basic steps you would take to analyze different materials' Stress/Strain Curves in order to interpret the results and understand its real-world implications. The final step will ask you to sketch your own Stress/Strain Curves and label all key points discussed. Various regions of the Stress/Strain Curve will have points of interest that include:

-Correctly Labeled Axes (with appropriate units)

-Young's Modulus of Elasticity

-Yield Strength

-Strain Hardening

-Max Tensile Strength

-Necking

-True Stress

-Fracture

-Comparison of Brittle & Ductile materials

__Supplies Needed:__

- Pencil and paper (used for verification of reading comprehension)

__Details about Instructable:__

-For the sake of consistency, all units will be expressed in the form of SI (or metric) units.

-All Stress/Strain Curve Diagrams utilized in this Instructable are typical of most metal materials. Plastic materials, however, can have varying types of curves. For simplicity, plastic materials will be neglected in this Instructable.

-Image references came from various engineering websites, listed here:

http://www.tpub.com/doematerialsci/materialscience59.htmhttp://www.substech.com/dokuwiki/doku.php

http://www.business-analysis-made-easy.com/Create-A-Graph.html

## Step 1: What Is the Stress-Strain Curve?

__What is it?:__

The Stress/Strain Curve is the relationship between the magnitude of applied stress on a material and the resulting strain (or elongation) caused to the material.

__Units and Symbols:__

-Stress is denoted by the lowercase Greek symbol, σ (pronounced 'sigma'). It is an applied force (Newtons, or N) over a given area (millimeters squared, or mm^2). One unit of stress is one Pascal (N/m^2).

-Strain is denoted by the lowercase Greek symbol, ε (pronounced 'epsilon'). Strain is the length of elongation (millimeters, or mm) divided by the original length of the material (mm). It follows that strain is expressed as millimeters/millimeters (mm/mm). Since the units cancel out, strain can be considered unitless. Also, since strain is a ratio of change in length to original length, it can be expressed as a percentage (as seen in the Figure 2).

__How it is Graphed:__

The stress-strain relationship is displayed on an x-y graph, where the y axis (vertical axis) represents stress, and the x axis (horizontal axis) represents strain (as seen in Figure 2). Therefore the stress-strain slope (change in y over change in x) is Stress divided by Strain. For the linear portion of the curve, this slope is known as Young's Modulus, denoted as E (also seen in figure). Before further explaining the Stress/Strain Curve, however, we should take a quick look at how the Stress-Strain Curve is developed.

## Step 2: How Is the Stress-Strain Curve Determined for Any Given Material?

__Intro:__

In order to understand Stress/Strain Curves, it is important to have a general idea of how the data is produced in the first place.

__Testing Procedure:__

A material's Stress/Strain Curve is typically determined by testing a 'dog-bone' specimen of the material (as seen in Figure 3) of the material in question with a Tensile Strength Testing Machine, or TSTM. The dog-bone shaped specimen is gripped at the far ends and a tensile load of a slowly increasing magnitude is applied. The dog-bone shaped specimen is designed to snap somewhere in the middle, which allows for easy evaluation of the elongation that occurs. For continuous elongation data output during the loading process, a strain gauge is attached to the specimen and electronically connected to a DAQ (data acquisition system). The TSTM is also connected to the DAQ. The DAQ then communicates to your computer and plots the Stress-Strain curve until the specimen eventually breaks.

__Can the Data be Trusted?:__

For robustness of results, the test is often repeated with the hope of producing consistent Stress/Strain Curves. If results are inconsistent, there is likely an issue with the testing process and the data output is inconclusive. Even if your results are consistent, you could still have an errant test system. Therefore, it is important that the Stress/Strain Curves you are analyzing have been cross-tested on multiple test systems. Reducing the risk of bad data can also be achieved by ensuring the data was not only coming from different test systems, but also from different groups of researchers. In fact, when comparing tabulated Stress/Strain information on a material from one organization to another, you will often find some variation. These are just important things to keep in mind when looking at a Stress/Strain Curve. Trusting someone else's data always has its risks.

__Moving On:__

Now that we know how Stress/Strain Curves are produced, let's continue to look at the Stress/Strain Curves themselves...

## Step 3: Regions on the Curve: Linear Elastic Portion (Young's Modulus)

__Young's Modulus:__

As previously stated, the slope on the linear portion of the curve (units of Pa/(mm/mm), or the equivalent Pa) is known as Young's Modulus and is denoted as 'E' (seen in figure). Other names for the Young's Modulus you might run into include 'The Modulus of Elasticity', 'Elastic Modulus', and 'Young's Modulus of Elasticity'.

A material's Young's Modulus value is the most concise description available for any material's linear elastic behavior. It tells you how much you can expect the material to elastically stretch when applying a particular tensile load.

## Step 4: Regions on the Curve: Linear Elastic Portion (Yield Strength)

__Yield Strength:__

This linear portion of the Stress/Strain Curve indicates for what range of stress values that the material stretches elastically. If the testing stops before breaching this linear elastic limit, the material will return to its original length. But if the load magnitude on the material continues to increase, the linear elastic limit will be breached and the material will begin to plastically deform, or 'yield'. The loading stress value that causes this yielding to occur is commonly referred to as the 'Yield Strength' (as seen in Figure 5).

__Why Does it Matter?__

Engineers are mainly concerned with this linear region, because it is where material behavior is more easily predicted. Failure of structural design often occurs when a structural material experiences a value larger than its yield strength.

__Moving On:__

Now we know how to interpret the linear portion of the Stress/Strain Curve. So let's look past the Yield Strength, where plastic deformation begins to occur...

## Step 5: Regions on the Curve: Nonlinear Deformation With Elastic Recovery (.2% Offset Method)

__.2% Offset Method:__

Before discussing what happens after Yield Strength is breached, there is actually one caveat about Yield Strength that should be mentioned:

One would expect Yield Strength to be *exactly *at the point where the Stress/Strain Curve ceases to be linear. But in fact, due to the nature of molecular bonds that is beyond the scope of this instructable, the material can return to its original length even after a short amount of nonlinear strain. Conveniently, nearly all materials have the same general position on the Stress/Strain Curve where this true elastic yield occurs; It has a become a good rule of thumb to approximate the yield stress at a .2% offset from the linear region (as seen in Figure 6). That is to say, at a strain value of 0.002, you would place a line parallel to the linear portion of your curve. Where this new line intersects with your Stress/Strain Curve is where you would expect to have your Yield Stress.

## Step 6: Regions on the Curve: Plastic Deformation (Strength Hardening & Ultimate Tensile Strength)

__Intro to Plastic Deformation:__

Plastic Deformation, by definition, is deformation that is permanent after the release of an applied load. From this point on, only the Plastic Deformation portion of the Stress/Strain Curve will be discussed.

__Strain Hardening:__

As you continue to increase the load, the material will undergo a process called Strain Hardening (as seen in Figure 7). The material, while still experiencing some deformation at this point, is still able to receive more tensile stress without weakening. Strain Hardening can have other names, which include Work Hardening, Strength Hardening and Cold Working. These names are adopted when the material is decidedly placed under load cycles designed to increase the strength of the material (as seen in Figure 8). This is a common and practical procedure, because it can significantly increase the load-carrying capacity of the material. Metals such as steel, for example, are very commonly cold-worked because they are not particularly brittle. (Note: Details about the differences between brittle and ductile materials will be outlined in a later step.)

__Ultimate Tensile Strength:__

This is the point where the material cannot handle any more increase of tensile stress (labeled in Figure 7).

## Step 7: Regions on the Curve: Plastic Deformation (Necking, True Stress, and Fracture)

__Necking:__

Necking occurs after the material has experienced Maximum Tensile Stress and is depicted as the downward sloped portion of the Stress-Strain Curve (as labeled in Figure 9). It is called the 'Necking' portion of the Curve because this is where the specimen's cross-sectional Area begins to shrink significantly.

__True Stress:__

But if the cross-sectional Area is decreasing, doesn't that mean the Stress should be increasing? Remember that Stress = Force/Area. A smaller Area means a larger Stress. But most Stress-Strain graphs show a decrease in Stress at the Necking portion of the Curve! This is because the change in Cross-Sectional Area of the dog-boned specimen is not being taken into account. A correcting calculation could be made to indicate what the True Stress would be at this point in the tensile loading process and would be given a new Stress/Strain Curve (as seen in Figure 10).

__Fracture:__

This is the point on the curve where the material finally snaps. It is indicated by the "X" in Figure 9.

## Step 8: Regions on the Curve: Plastic Deformation (Comparison of Brittle & Ductile Materials)

__Brittle vs. Ductile:__

How would you expect a brittle material's Stress-Strain Curve to differ from that of a ductile material? The figure above should come as no surprise. The brittle material in comparison experiences very little plastic deformation before fracture. On the other hand, the ductile material experiences significant deformation before it eventually breaks. Also note that brittle materials typically have higher yield strengths than ductile materials.

## Step 9: Quiz What You Know!

__Conclusion:__

Now we have walked through, step-by-step, every portion of a Stress/Strain Curve. At this point, you should be able to make some sketches of your own! Complete the following Quiz to assure yourself that you have mastered the Stress/Strain Curve.

__Quiz:__

With a pencil and paper, complete the following:

1. Sketch a Stress/Strain Curve on an x-y coordinate system and label all key points:

-Stress and Strain axes (include units)

-Young's Modulus

-Yield Stress (2% offset)

-Strength Hardening

-Ultimate Tensile Strength

-Necking

-Fracture

-True Stress

2. Sketch two Stress/Strain Curves on the same graph, making one Curve for a brittle material and one Curve for a ductile material. Below the graph, briefly mention in your own words what the key difference is between the two types of material.