**MSE203 Mechanical Behaviour: Continuum Mechanics**

In second year, I teach a 10-lecture course on stress tensors in the spring term. Here we learn how to deal with stress *states* in materials. So we define stress tensors and examine how to rotate them, both using tensor rotations and Mohr’s circle. We look at how to analyse stresses in simple bodies like shafts and pressure vessels. Then we look at strain, elasticity and how to measure strains by diffraction. We can then look at anisotropic elasticity, particularly in crystalline materials. We finish up by studying yielding and yield criteria.

As with MSE104, the course is taught by Peer Instruction, supported by notes and videos on YouTube.

The notes are here, which are also accompanied by examples here. In the tutorials, we will go through some of these examples (Tutorial 1: Q5,6,8,9 and Tutorial 2: Q20, 23,24,25); the others are discussed in the videos. Solutions are provided with the examples (but don’t look until you’ve tried them :-).

If you like lectures, then they are online on YouTube. The video numbering corresponds to the ‘lecture’ classes in which we will discuss that material. As with MSE104, there is no intention to re-lecture the material in class: instead students should review the material before class in their own time, ready to discuss and explore it in the class session.

After the end of the lectures, there is a short 1-hr test to prepare students for the exam in June.

**Lecture 1.** 1a: Defining Stress States 1b: Rotations in 2D – the inclined plane

**Lecture 2.** 2a: Rotations in 2D – Principal stresses and axes. 2b: Mohr’s circle

**Lecture 3.** 3a: Pressure Vessels 3b: Q1-4 Pressure Vessels and Using Mohr’s Circle

**Lecture 4.** 4a: 3D Stress Tensors. 4Q7 – Q7 hollow rotating power transmission shaft. 4Q10 – biaxial test specimen.

**Lecture 5.** 5a: More 3D Stress Tensors. 5b Q11-12 A Mohr’s circle and some eigenvalues.

**Lecture 6.** 6a: Strain and 6b Isotropic Elasticity. Also Q15, Q16 and Q27.

**Lecture 7.** 7a: Anisotropic Elasticity

**Lecture 8.** 8a: Strain Measurement

**Lecture 9.** 9a: Yield Surfaces and Criteria

**Additional Resources**

Dieter (Ch2) is really great for this course, and is probably the core reference. But, the main drawback of Dieter is that he soft-pedals the tensor mathematics.

To go further, particularly for anisotropic elasticity, the Nye’s book is the core reference. But, its quite dense and written in the style of the time.

If you are struggling to conceive of what a tensor is as a mathematical object, then I quite like this video.

Hello Sir.

May i know the complete name of the Nye’s book

JF Nye, Physical Properties of Crystals, Oxford University Press, 1957. It’s in the reading list on p1 of the notes.

thanks for the answer

Hello Sir

I am an engineering student.I really want to know some practical applications of these concepts,or in other sense, why these concepts are developed?

(i) Real materials are subjected to mutliaxial stress situations – and you want to know how they will respond (elastically strain, yield, flow), and therefore if your structure will respond to the loading in a safe manner.

(ii) Real materials are anistropic, and often if you want to measure these properties, you need to understand how to manipulate stress and strain states.

(iii) You can’t always measure the loads applied, but its relatively easy to apply strain gauges to a surface and measure the strains – which you will then have to convert into the stress state somehow.

(iv) Most materials are made up of many individual single crystals, so measurements and modelling of how these respond is very important to developing better materials – which means understanding and working with stress and strain states.

–> All these considerations mean that its very difficult to work with stresses and strains meaningfully without treating them properly as tensors. Sure, you can work on idealised pin-jointed beams, but surely we want to do more than that?

Hello sir,

First of all I would like to sincerely thank you for your videos. Your way of explaining a concept is really commendable. After watching them I really have started to develop an appreciation for the subject. I was wondering whether you would consider uploading lectures on Shear force and Bending moment diagrams as well in the near future.

Sorry, but I don’t lecture these subjects; colleagues in my Department do. One day, maybe!

Dear Professor Dye. First of all thanks a lot for putting up the course material for this course. This is a very important course for people interested in mechanical metallurgy, espescially because in conventional materials science and metallurgy curriculum, not much of the emphasis is given to mechanics or the mathematics of it. I believe in order to grow further in this field knowledge of Linear Algebra becomes very important.

There is one doubt that I would like to clarify from you. In a uniaxial tensile test cylindrical specimen, at any point inside the cylinder, is it theoretically possible to achieve a uniaxial state of stress? Or, is our notion of uniaxial stress in the specimen just an assumption based on the fact that the length of the specimen is very big in comparison to its diameter (cross-section) which would lead to stress state closer to uniaxial one?

For a uniaxial testing machine that is perfectly aligned, then for a well behaved material the stress state will be uniaxial. Real machines and load trains aren’t perfectly aligned or perfectly rigid, so avoiding bending moments can be tricky. The ASTM HCF and LCF test standards contain good advice.

Dear Professor Dye,

Is it possible to get access to the notes “A Mostofi, MSE201 notes on Tensors”, which are mentioned in your notes?

Thanks!

Here you are!

If you are in Imperial, they are on blackboard. Otherwise, they aren’t publicly downloadable that I know of.

Hello Mr. David,

I am glad for all your lectures. I am attending “Stress Tensors” lectures of yours. However, I would like to ask “where can I find derivation for direction cosines of maximum shear stresses as mentioned in Dieter?”

it is like:

l1 0 +- 1/(2)^(1/2) +- 1/(2)^(1/2)

same for l2 and l3

In anticipation of your urgent response.

If you mean equation 2-20 in Dieter, then this is found by doing a 45 degree rotation between each of the pairs of principal axes. Probably the easiest way to prove it is to simply do the tensor rotation, or by sketching it out.

The state of stress at a point is given as ( all in Mpa ) sigma xx= 40 .sigma yy= -40 sigma zz= 60 . Tau xx=20 taught yz=20 taught yz=25 tau xz =15 .find the resultant stress on an oblique plane equally inclined to the three axes also.find the normal shear stress component.

Imagine we have these Principal stresses: S1=1179.7, S2=411, S3=164.2.

According to Mohr circle (Tresca) the maximum shear available in the system is (S1-S3)/2=507.75

While the Von Misus Stress is larger than that Svm=sqrt(0.5((S1-S3)^2+(S1-S2)^2+(S2-S3)^2))=917.3

Would you please elaborate on this? Why the difference is that much?

My area is soil-structure interaction which one is more appropriate to be used in my research?

Well yes, exactly. They are very different in certain circumstances, such as this one. Which is appropriate depends on how the material actually behaves physically. I don’t know much about soils, except that they are complicated!

Hi sir,

I enjoy your lectures, they are very helpful, thank you. I am currently trying to analyse samples for residual stresses. However I am using a low penetration XRD machine. I am interested in learning how to plot the “d vs sin2ψ” graph(Not sure where the x and y data are obtained). Should you have any examples or source recommendation that may help me archive this, I would really appreciate it. Thank you.

Try ASTM standard E2860, or this UK National standards institution good practice guide. http://www.npl.co.uk/upload/pdf/Determination_of_Residual_Stresses_by_X-ray_Diffraction_-_Issue_2.pdf

Thank you, I will go through them.

Hello Prof. Dye,

In my search of fully understanding eigenvectors & eigenvalues (applied to mechanics, data analysis, linear algebra) I’m stucked with a detail on the 4a chapter’s video. I do not understand why Mohr’s angle is different from the obtained with eigenvectors; actually, the value is 18.4º yet, one is measured from the right of the Y axis, and the other from the left of Y axis, then it is not the same angle (I am confident about the fact I am missing some detail which is key to understand why it is somehow the same result).

Finally, I’d like to ask why we multiply R’*S*R when we want to rotate stress tensor, instead of doing S*R directly.

In any case, I really appreciate your time and effort, your lessons are outstanding, so are you.

Regards.

On tensor rotation (Rt*S*R) you need to look at Ch8 and 26 of Hobson Riley and Bence – one way to think about it is that we need to rotate both the forces and the planes, so we have to do it twice. Another is to compare to the inclined plane and notice that it gives the same answer as the rotation using tensors. The tricky part is getting it right – we are doing a rotation of the axes (passive) not the objects (active).

Mohr’s angle must be the same as eignevectors, if you are doing the correct dot product of the correct eigenvector. I’ve checked this on problems, including this one!

Hello, Teacher. Could you help me in a exercise? I can not attach it here, but i can send you by email if you permit.

Thanks

Sorry, without knowing your context, I don’t think I should…

Hello Dr.

First, I would like express my gratitude for your huge effort to explain such awesome but hard subject in such a simple way. I am a mechanical engineer who is interested also in anistropic materials behavior, specifically piezoelectric materials. As you know Dr., piezoelectric materials are orthotropic materials, and the voltage output depends on the strain developed due to loading condition, and their constitutive equations depends on using not only the matrices of constants associated with mechanical and electrical fields, but also on a coupling coefficient matrices. As I seek to better understand this materials, I surfed the net and could not find a clear and simple numerical examples to illustrate how to use the constitutive equations, rather, I found a lot of papers that solve it using finite element analysis. with regards to the above, I wonder if you could provide some lectures that simplify piezoelectric materials behavior, accompanied by some simple numerical examples that give a glimpse to how coupled fields can treated.

Thank you

Piezoelectric materials and in general, the other Tensor properties of crystals, are fascinating. Nye’s book is probably still the best, referred to in the references list. Good luck!