Mechanics 101, part 4

In my last installment of posts on the basics of mechanics, we discussed some basic principles of dynamics and applied them to a falling ball colliding with a table. Prior to that, we discussed statics. At the end of both posts, I talked about how the deformation of real systems becomes important in creating physically realistic models for events.

In this post, we'll cover some basics of Strength of Materials and talk a little about how we can analyze deforming systems in the context of solid mechanics.

Update 26 Nov:  I added a little bit more to the talk of the Young's Modulus, specifically about how it doesn't generally change much for a given metal alloy, except as you go to high temperatures.
Update 25 Nov: I totally forgot to mention the Poisson effect during the loading experiment. This has been rectified. Please don't hate me.

In the first post in the mechanics 101 series, I briefly defined the concepts of stress and strain. These concepts, and their relationship for a given material is at the core of all solid mechanics. So let's take a deeper look at them, one at a time.

Stress

In part 1, I defined stress as follows:

Stress: stress is a measure of the forces exerted between neighboring particles in a continuous body. It is defined as the force being exerted by neighboring particles per unit area separating them. There are two major types: Normal (force perpendicular to the area) and shear (force parallel to the area). For example, the normal stress, \(\sigma\), on a bar of cross sectional area, A, with an applied normal load, P, is \(\sigma = P/A\). Units are of the form force per unit area, such as N/m\(^2\), which is equivalent to the Pascal (Pa).

Since a picture says a thousand words, here are some pictures illustrating this concept. First, Let's consider some general body with applied external forces and moments. We already know from our discussion of static equilibrium in part 2 that we can slice a section through the body and expect to see the corresponding reaction forces.

Generic body with applied loads and reactions at an arbitrary section

We can then resolve the reaction forces and moments into different components:

Resolved components of the internal reactions

The internal reaction components are defined as follows:

• Normal Force (\(F_n\)): The internal reaction force component acting perpendicular to the section.
• Shear Force (\(F_s\)): The internal reaction force component acting parallel to the section.
• Bending Moment (\(M_B\)): The internal reaction moment component acting to bend the section with respect to the rest of the body.
• Torsional Moment (\(M_T\)): The internal reaction moment component acting to twist the section with respect to the rest of the body.

But we still want to look at the distribution of force and moments within the body. So let's go back to our general body with a generic distribution of forces on the section. If we consider the typical force, \(\Delta F\) acting on any given area of the section, \(\Delta A\), we arrive at a diagram like the following:

Typical force acting on an arbitrary portion of the section area.

We can also resolve the typical force into its shear and normal components. This resolution actually includes the influence of any applied moments as well, which is why you don't see any moments in the right image or at the section on the left. For example, a bending moment on our section would just produce a variation in normal force around the bending axis:

Moments are just distributed forces.

 We can now write the normal and shear components of the force per unit area based on our normal and shear force components:
\[\bar{\sigma_x} = \frac{\Delta F_n}{\Delta A}\]
\[\bar{\tau_{xy}} = \frac{\Delta F_s}{\Delta A}\]
Where \(\bar{\sigma_x}\) is our average normal stress (x indicates the force is acting in the x-direction) and \(\bar{\tau_{xy}}\) is our average shear stress. The subscript for the shear stress (xy) indicates that the shear force is acting parallel to the plane with a normal vector in the x direction, and the y indicates the shear force is acting in the y direction.

But \(\Delta A\) is a macroscopic area, not a point. So how can we find the stress at a point based on these average values? Mathematics, that's how. We can look at the force per unit area as we reduce the area to an infinitesimally small value using the idea of limits:
\[\sigma_x = \lim_{\Delta A \rightarrow 0}\frac{\Delta F_n}{\Delta A} = \frac{\delta F_n}{\delta A}\]
\[\tau_{xy} = \lim_{\Delta A \rightarrow 0}\frac{\Delta F_s}{\Delta A} = \frac{\delta F_s}{\delta A}\]
In other words, the stress is the derivative of the distributed internal reaction force with respect to area. If the normal force tends to act to elongate the body, we refer to that as a tensile stress, whereas if the action tends to compress the body, we call that a compressive stress. Finally, these stresses exist on every surface of an arbitrary volume within our body, and are said to represent the state of stress for that volume.

There is a caveat to all of this: our general body must be continuous, in other words, we smear out the atomic structure, voids, etc. and assume that at any given point in our general body there is matter. In practice, this isn't a terribly constraining requirement and indeed it is possible to account for material discontinuity by constructing equivalent material models. Fun fact: this is why the concept of stress doesn't really exist in the continuum mechanics sense when you talk about bonds in atomic structures (despite the construction of an approximate stress function called the virial stress). Because who doesn't talk about bonding in atomic structures, right?

Strain

Compared to the concept of stress, strain is remarkably straight-forward. In part 1, I defined stress as follows:

Strain: strain is a measure of the deformation of a body relative to some reference. For example, the change in length of a stretched bar vs its original length: \(\epsilon = \Delta L/L\). This measure is unitless.

There are several reasons we use a reference length to normalize the deformation:

• Depending on the size of the structure, the magnitude of the deformation that is important may differ. For example, a long bar can deform a great deal under the same load that would not visibly deform a shorter bar.
• The amount of deformation and the direction of the deformation occuring may differ through a structure. For example, a bar being bent upwards contracts on the upper surface while elongating on the lower surface.

There are two particularly useful strain types:

• Normal Strain (\(\epsilon\)): the change in length of an arbitrary line relative to its original length. Also sometimes called the Cauchy strain or engineering strain.
  \[\epsilon = \frac{L_{final} - L_{initial}}{L}\]
• Shear Strain (\(\tau\)): the change in angle between two lines from the undeformed configuration to the final configuration (in radians):
  \[\tau = \Theta_i - \Theta_f\]

Both of these strains can also be described as derivatives with respect to position of the components of the deformation, u, in a given direction:
\[\epsilon_x = \frac{\delta u_x}{\delta x}\]
\[\tau_{xy} = \frac{\delta u_y}{\delta x} + \frac{\delta u_x}{\delta y}\]

To put it graphically (showing positive strains in both cases):

Schematic of normal and shear strains. Positive strains shown.

Material Stress-Strain Relationships

Now that we know what stress and strain are, let's talk about how they are related in actual materials. This topic is amazingly vast, so we'll only be able to hit a few high points to help understand things better. I find these relationships, called constitutive laws, very interesting, and absolutely recommend looking up some more if you are interested. This is the meat and potatoes of continuum mechanics, elasticity, plasticity, fracture mechanics and a number of other areas of mechanics.

Let's do a mental experiment: we take a material like some common mild steel alloy (which are generally ductile without being excessively so), prepare a sample with a uniform cross-section and put it into a machine that exerts an increasing tensile force and measures the resulting deformation (a tensile test machine):

Tensile test sample ('dog bone').

The region noted by the dotted lines is the so-called gauge length, \(L_g\), and acts as the reference length for our strain calculation as well as the area we measure to determine the strain.  As we pull on this sample, the shape is intended to force the deformation to occur in the narrow section so that we know where to expect any failures to occur. As we apply an ever increasing tensile load, we can calculate our stress and strain using the initial configuration as our reference:
\[\sigma_{nom} = \frac{F}{A_0}\]
\[\epsilon = \frac{L-L_g}{L_g}\]

Plotting the stress (so-called nominal stress) as a function of the strain results in a graph like the following:

Schematic of a typical ductile nominal stress/strain curve. Not to scale.

There are two distinct regions in this curve. The first is called the elastic region. Deformation in this region is reversible. Remove the load, and the system will return to its original configuration. For common steels, this region corresponds to a strain of about 0.002, or a deformation of 0.2 percent of the initial length and a yield stress (\(\sigma_y\)) of about 250 MPa. The proportional limit, \(\sigma_{pl}\) corresponds to the stress at which the roughly linear stress-strain behavior of the material stops. Many metals behave approximately linearly for small stresses and strains. The slope of the linear elastic portion of the stress-strain curve is called the Elastic Modulus or the Young's Modulus, and is a parameter we'll see a lot. This property, at least in most metals of engineering interest, is kind of neat in that it doesn't vary a whole lot for a given alloy regardless of processing. It does, however, vary with temperature. Generally the Young's modulus will decrease as you increase temperature, though the amount may vary with temperature. In principle, it can also vary in different directions within a material, and can change if there is a significant atomic-level structure change.

While we load the bar, there's interesting phenomenon that occurs: it doesn't just elongate. the cross section begins to narrow as well. If you've stretched out a rubber band and watched it narrow, you've seen this effect: it's called the Poisson Effect. This effect is essentially a consequence of the conservation of energy during deformation. The amount of lateral strain that occurs is related to the axial strain as follows:
\[\nu = - \frac{\epsilon_{lat}}{\epsilon_{axial}}\]
Where \(\nu\) is called Poisson's ratio. For the vast majority of materials in the elastic regime, it has a value between 0 and 0.5 (0.5 being the theoretical maximum), with many metals falling around 0.3. Theoretically, values as low as -1 are possible though few real materials have exhibited negative Poisson's ratios once measurement error is taken into account. The value of Poisson's ratio need not stay constant, and in most materials it changes once you enter into the second region of the above plot.

The second region of the stress-strain curve is the plastic region. Deformation in this region is at least partially irreversible (completely irreversible in the case of fracture). The first part is the yielding region. In this region, we've passed the elastic limit of the material and deformation occurs without any increase in the supported load. The reason for this in ductile materials like metals is essentially that the microstructure begins to migrate and relax to a lower energy state. At some point, the microstructural relaxation ends and the supported load begins to increase again as the deformation increases, a phenomenon called strain hardening. The strain hardening effect is actually used heavily by metalworkers such as blacksmiths and is sometimes referred to as work hardening. At some point, however, the hardening stops and the supported stress begins to decrease as the deformation increases. The maximum stress supported is called the ultimate tensile stress, \(\sigma_u\). In our test specimen, what you'd see is that some region in the center will dramatically reduce in width and deformation will localize there, like pulling taffy apart. This phenomenon is called necking (yes, I made the joke too), and is the precursor to fracture of the specimen. The reason the stress decreases is because the reduced area can't support as high a tensile force, but we continue to use the initial area in our stress calculation (force goes down, area stays the same: stress goes down). This convention is mainly done out of convenience, but it is strictly possible to update the cross-section as you perform the test, obtaining a quantity sometimes called the true stress. Finally, after necking, fracture occurs. The stress at which the specimen finally fractures is called the fracture stress, \(\sigma_{fr}\). 

I implied that some of the deformation in the plastic regime is reversible. Recall that in the elastic region, that deformation is completely reversible. Also, to get into the plastic regime we must pass through the elastic regime, so it would make some sense if we had to pass through an elastic regime during unloading as well. As it ends up, if you load into the plastic regime, and then release the load you end up with a stress-strain curve like the following:

Relaxation in the plastic regime

Note that the slope of the unloading curve and the linear elastic curve are the same. What this means is that a portion of the deformation (\(\epsilon_{max} - \epsilon_{final}\)) is recovered, or reversed, just by unloading the sample. The strain \(\epsilon_{final}\) is the permanent deformation or set that developed as a result of the loading. In general, there's a bit more going on here and repeated loading and unloading will have varying levels of recovery. This variation in recovery is known as hysteresis.

The above discussion on stress-strain relationships made use of the tensile loading of a simple ductile material to illustrate the main points. But in reality, there are a wide variety of materials and behaviors. One other general type is a brittle material, like a ceramic (though some metals can behave in a brittle manner as well). The main characteristic of a brittle material is that it exhibits very little yielding and plastic deformation before fracture. Some materials also have a strong dependance on loading rate, and their mechanical behavior can change dramatically even for modest changes in loading rate, even in the elastic regime. Materials with this behavior are referred to as visco-elastic, examples of which include some polymers and even some conventional materials at high temperatures. For our discussion on stress-strain relationships, we've been assuming that our loading rates are very slow, allowing plenty of time for the material to relax and for any propagating waves to dissipate. This assumption is called the quasi-static assumption.

In the next segment of this series, we'll continue in the mechanics of materials trend and talk about using these principles to analyze some problems.