Reduction or area deformation meaning?

[ehabmozart]

Hey there!

While reading my mechanical design book, I had hard time to understand a particular paragraph if anyone could help. Attached to this post are two figures. 2-6 a and b. The first one is pretty simple to understand; the engineering normal stress strain curve accompanied with cold working. It is then given as follows in the book for the second picture

" It is possible to construct a similar diagram as in Fig 2-6b where the abcissa is the area deformation and the ordinate is the applied load. The reduction in area corresponding to the load pf at fracture is defines as R= 1- (Af/A0) " ...

My precise question is WHAT IS THE AREA DEFORMATION?? What is the units of it? What does the are under the graph present and how do we calculate the area deformation in the first place?

 

[stockzahn]

If for example a stick is stretched, its length increases, which causes a reduction of its diameter (dependent on the material the ratio of increased length to decreased breadth is given by the poisson's ratio). This decrease of diameter means a decrease of the cross section area - the area is deformed (reduced when stretched and increased when compressed). As there is a fixed ratio (given by poisson's ratio) the strain ε and the decrease of the cross section area are interchangable. Now there is a "problem" with the definition of the stress in the σ,ε-diagram: The stress is always referred to the original cross section area A₀. Otherwise the curve would have to incline in the point "u" instead of declining, due to the smaller cross section area, when stretched. I suppose the stress was exchanged for the load to avoid a technically wrong diagram (as the stress would always be referred to the deformed area and not the original one). Using the load P the curve's shape is the same as in a standard σ,ε-diagram.

 

[ehabmozart]

First of all, stockzahn, I thank you for your effort and time to look at my question and reply. I highly appreciate it. Moving to your reply, I get the first part until the sentence "now there is a problem...". I definitely understand that strain and area deformation are interchangeable due to volume consistency as well. The stress in the diagram is nominal; which is the engineering stress. That's why it is declining after the tensile strength. That's why it must be using the original A0 in calculations. Now, what I am still stuck at is what is the unit of area deformation and what does the are under the second graph show. Moreover, why load is in the ordinate? Why not stress for example? In addition, I assume the A0 is the largest but it seems not to be since it is at the beginning of the x axis... Thanks once more and thanks in advance to whoever spends some time here .

 

[stockzahn]

I'm sorry for my last post: I didn't answer your questions, I only wrote some of my thoughts. Reason for that is, that I'm not used to this kind of diagram, I tried to give some basic information, but I also only can try to find it out.

1) If both, the abscissa and the ordinate, are multiplied by an area, the area under the graph would have the same dimension than in the σ,ε-diagram. As the area under the graph of the σ,ε-diagram is a measure for work, it seems to be the case, that the area under the graph of the P,A-diagram also is an indicator for the work put (and partly stored) in the system. Also the similar shape of curves is a sign for that - or it must be be a property which has the same progress as the work during stretching the material.

2) The formula for the reduction indicates that A₀ is the original area, whereas A_f is the area reached when the sample fails. According to the diagrams you posted, the abscissa shows the area development while increasing the force. In my opinion the reduction or deformation of the area (in a plane rectangular to the direction of the force) can only be connected to the work done by the force with a material specific proportion (poissons's ratio) - please tell me if there is a mistake in this thought. If now the area under the curve is the work input, which seems to be the case due to the curve's shape corrisponding to the curve in the σ,ε-diagram, the dimension of the abscissa must be connected to the displacement of the force by a material or geometrical coefficient. I think the deformation area is used for other purposes, like receiving a possibility to compare different materials regarding their ductility.

3) I suppose the load is taken at the ordinate for two reasons: First of all as already mentioned in 1), abscissa and ordinate are multiplied with a value of the same dimension which gives the area under the curve the same dimension as in the σ,ε-diagram. Secondly, using the stress, it would be confusing that the stress given at the ordinate doesn't actually match with the area at the abscissa, although the area shown at the abscissa should be the actual value corrisponding to it. If made like that, the curve would have the (in my former post) mentioned incline in the point "u". To keep the shape the same, the load was used for the graph.

4) It for sure isn't a commonly used diagram - maybe because there is (almost) no more information to gain from it.

I hope somebody here knows better and is willing to share his knowledge.

 

[ehabmozart]

Thanks a lot stockzahn once more for replying. I just appreciate this long and elaborate reply, which shows your professionalism in helping. This is a virtue. Back to your reply. Well, I gave it a bit of thought and would like to share it with you. Some agree with your points and the other alters by a bit. Eventually I realized that reduction is just the same as area deformation. This being said, keeps area deformation unitless and A0 being the SMALLEST value since the reduction is the ratio of difference between the current area and A0 over the original area. i.e (Ai-A0)/A0 ... Here Ai is any arbitrary area. I think the x-axis here shows the percentage rather than the absolute value of area itself. I didn't get your thought exactly on the second point where you said "can only be connected to the work done by the force with a material specific proportion (poissons's ratio". Anyway, if the reduction is unitless, then the area under the second graph is work per unit length which I still don't get how can we use it. This reminds me with your first point which I think you missed somethings about. The area under the stress strain graph is not just work but work per unit volume. Ultimately, I think that the x-axis is the area and y-axis is the load because to keep the graph the same as the once in figure a, the force is constant in cold working while the ultimate strengths and yield strengths change. These are just my thoughts and I am more than welcome to explain anything subtle in my context I've just written above.