Category Archives: Materials Science

Classic Laminate Theory: how to derive the reduced stiffness matrix

Those of you who work with composite materials will be familiar with the Classic Laminate Theory. According to this theory, from the properties of the material, the stiffness matrix can be easily obtained. Then, using very tedious expressions, the coefficients of the reduced stiffness matrix can be calculated. The thing is that, based on my own experience, those expressions are introduced without any explanation, so most of the people just use them, ignoring where they come from. So, if you’re curious or you just want to understand a bit more about this theory, keep reading this post!


The first thing that we have to consider is the rotation of our coordinate system . We need to know how to express our new coordinate system in terms of the original one. Following Figure 1 and using basic trigonometry, the relationship can be found.

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Figure 1 Rotation of the original coordinate system

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Carbon Ceramic Brakes

Nowadays, if you are lucky enough to be able to afford a sport car, one decision has to be made with regards to the extras: the brakes. Are the famous carbon ceramic brakes that special? Let´s find out some of their features.


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Carbon ceramic brake (source: Olivier Delorme)

Carbon ceramic brakes consist of carbon fibre reinforced silicon carbide. In this case, the matrix is made of silicon carbide (SiC) and silicon (Si), whereas the reinforcement is made of popular carbon fibre. The matrix provides the hardness to the composite material and the fibres are responsible for the fracture toughness.

The main advantage of this type of brake is its capability to absorb extremely high temperatures. Why is this important? Well, sport cars can go fast… very fast. Therefore, there is also a need for reducing the speed of the vehicle as fast as possible. Since the brakes use friction in order to slow the cars down, heat is generated and it can decrease the efficiency of those particular components. Hence, having brake disks which contain modifications in order to withstand those high temperatures, is a must.

In addition, this composite material reduces the weight of brake disks up to 50% when compared to conventional ones. Furthermore, carbon ceramic brakes do not suffer corrosion, which is a major problem for iron brake disks. Apart from that, other merits include: longer life, less dust (in metal brakes, the dust have magnetic properties due to static electricity, resulting in particles which remain on metal parts around the disk) and less noise.

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About volume and foams

If you had the opportunity to learn the basics of rational mechanics in high school or impact dynamics during your degree, you may be familiar with one specific condition which was specially useful in order to solve problems: the conservation of volume. But, what if I told you that there are certain cases where that particular assumption can be totally wrong?


Let’s start with the so-called “conservation of volume condition”. This condition assumes that when we have a component (e.g. a beam or a bar) and it goes from one state to another (e.g. it is impacted by another body), no mass will be lost. In other terms, it considers that the component will not break into pieces. However, this doesn’t mean that your particular component cannot change its shape. Thus, it is usually taken for granted that even if the shape changes, the volume should remain constant due to the fact that the density (mass over volume) of a certain material should remain the same. Or at least, that’s what we are usually taught…

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Material Science in surfboards

During my recent visit to the American West Coast, I had the opportunity to have a first contact with surfing. Although I didn’t have a totally satisfactory experience, I must say I found this sport quite interesting and, for that reason, I started asking some questions about the materials which are used to manufacture surfboards to my surfing expert brother. In this post I will introduce very briefly the main materials and a bit of comparison between their performance.


To begin with, the surfboards can be classified in two main categories with regards to the materials used in the outer part. For people who, like me, are not familiar with this sport, the way for identifying which of the two types of surfboards is in front of us is simply to look at the external appearance of the board itself. Trust me, anyone can spot the difference!

The first type corresponds to the surfboards made of glass fibre. This boards have been around for over half a century and this fact makes them be considered as the traditional board. The inner part is made of polyurethane (PU), a polymer which stands out for being lightweight and providing a good buoyancy to the final structure. Usually a wooden element is embedded in the core in order to increase the stiffness of the product. Then, the whole thing is surrounded by some layers of glass fibre cloth which provides the strength and resistance to the board. The amount of fibres and their orientation can vary from one surfboard to another, resulting in different strengths and stiffness without changing the weight of the structure too much.

Then, we have the epoxy boards. The first thing we notice about this category is its plastic exterior appearance. In this case, the core is made of polystyrene (PS) foam, and it is worth saying that depending on its type, the foam will absorb a different quantity of water. This material is also the responsible for the buoyancy of the surfboard. Then this core is coated with an epoxy resin, being this the reason behind its name. Another option within this category is the use of an expanded polysterene (EPS) foam core, which is an even lighter material that we can find, for instance, in bicycle helmets. It should be said that the use of this kind surfboards was not very extended until they started appearing in main events where professional surfers had a great performance.

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Surfer in Malibu, California (extracted from my personal Flickr account)

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Split Hopkinson Pressure Bar Test

One of the experimental techniques for characterising materials under the effect of high strain rates is the split Hopkinson pressure bar (SHPB) test. This week a brief introduction to the basics of this technique is covered.


If you have an engineering background, it is likely that you have come across one of the most famous testing methods for characterising materials: the tensile/compression test. In that case, a sample is usually subjected to a controlled displacement (usually in mm/min). The result is the force-displacement curve and from those results and the geometry of the sample, the determination of the stress-strain curve for the material is pretty straight forward. Despite the fact that this test can be performed for a different range of speeds, these velocities are normally quite low. In these terms, there are many cases where engineers are interested in the stress-strain curve for dynamic cases, since some materials can exhibit different behaviour depending on the strain rate (e.g. crushing of an automotive component). For that reason, in order to obtain the desired characteristic curve, other methods have to be used, such as the split Hopkinson pressure bar test.

One of the most interesting things about the SHPB test is that there is no official standard to follow. However, there are some common features with regards to the necessary equipment. In these terms, every Hopkinson bar test should have:

  • Two cylindrical long bars. Their length has to be big enough in order to obtain one-dimensional wave propagation. They are called incident and transmitted bar, respectively.
  • Fixtures to ensure that the bars are perfectly aligned and that they can freely move after an impact occurs.
  • Gas gun. This device is the launches a striker bar which impacts the incident bar. Hence, a controlled compressive pulse is achieved.
  • Two sets of strain gages. Each set should be placed in the middle of both the incident and transmitted bars.
  • Equipment for data acquisition (e.g. amplifiers, oscilloscopes…).
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Figure 1 Schematic of the split Hopkinson pressure bar test [1]

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About Hashin´s failure criteria in FEA

When using Finite Element Analysis (FEA) for studying composite materials, one of the most used failure criterion is the one which was proposed by Hashin in 1980. This theory is included in all the main FEA packages and, probably, you are more than familiar with this particular model. However, what you might not know is that the failure criteria that you are defining is not exactly the Hashin’s one. If you want to know why, this is your place.


Since the available failure criteria at that point presented some inconsistencies, in 1980 Hashin developed a new criteria which differentiated between failure modes. His theory considered four different ways in which the material could fail:

Hasin_original

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Is the Maximum Strain Theory a non-interactive failure criterion?

Failure criteria for composite materials are usually classified in two categories: non-interactive and interactive theories. In literature, you can find that the main non-interactive failure criteria are the Maximum Stress Theory and the Maximum Strain Theory. However, one question arises: is the second one a non-interactive theory in reality? Let’s figure it out.


To begin with, a non-interactive failure criterion is that one which only takes into account the effect of one stress or strain component for each failure condition. In other words, it does not consider any interaction between the different components. For example, the Maximum Stress Theory considers that the material fails when one of the stress components reaches a maximum value. Hence, considering a sample loaded in tension:

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Where subindex 1 refers to the fibre direction and 2 corresponds to the transverse direction. When the stress reaches the limit value (measured experimentally under uniaxial stress conditions), the material fails. It is clear how in that failure criterion only one stress component is considered for each condition.

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