# I have a basic doubt in the theory of stress and elasticity. Please help me

## Main Question or Discussion Point

A Book says, " When a load is acting on a member, it undergoes deformation. An internal resistance is developed against deformation by the member and the intensity of this internal resistance is called stress." I also read, "When a member is in its elastic range, it offers resistance against deformation and when it goes beyond the elastic range, i.e. plastic range, there is no resistance offered by the member against deformation."

But, in a stress-strain curve, the stress keeps increasing even after the elastic range until the ultimate tensile stress which is in the plastic range.
How does stress ( RESISTANCE) increase in the plastic range when it is clearly stated that the member does not actually offer resistance after the elastic range??

## Answers and Replies

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A good question, and I think you've spotted an inconcistency there. Yes, you are absolutely right, the stress often increases in the plastic range (e.g. for steel) until you reach an ultimate stress value. In reality, there are different types of material, steel, soil, etc, and they all have different stess-strain graphs. For some materials the stress rises in the plastic regime and for some materials it may be idealized to stay constant. For example, when you design concrete, you often assume that the stress-strain graph is horizontal in the plastic range, i.e. no additional resistance to stress. I'm not sure where there are any real materials that fully behave like this, but it is used in idealizations for some. Your book may have been referring to those specific materials.

Also, during plastic movement, you usually get elastic movement happening at the same time. Elastic movement is due to lattice distortion, causing atomic bonds to be stretched, thereby leading to stresses. Plastic movement is due to dislocation slip, maybe you can find a bit more info on this online (I am a bit rusty on this, to be honest).

I hope that helps.

I think the glitch in the understanding is not taking the geometrical modification of the member into account once the yield stress level is reached. Imagine pulling a rod in both ends, once the yield stress level is reached the rod will locally undergo a large deformation, i.e the cross section area of the material is decreased, hence the stress in the material is locally increased due to the smaller x-section area until the rod finally breaks...

AlephZero
Homework Helper
A Book says, " When a load is acting on a member, it undergoes deformation. An internal resistance is developed against deformation by the member and the intensity of this internal resistance is called stress." I also read, "When a member is in its elastic range, it offers resistance against deformation and when it goes beyond the elastic range, i.e. plastic range, there is no resistance offered by the member against deformation."
Hm.... IMO, most of that quote is so wrong that it''s hard to know where to start to correct it.

Can you give the title and author of the book? Was it originally in English, or is the translation into English wrong?

Yes Aleph Zero is correct - ditch that book double quick.

Here is a better extract from a better book

Structural Analysis by example - Hambly

When a load is applied to a structure it is resisted by a system of forces which can be visualised as 'flowing' through the structure from the loading to the support reactions. The forces within the components cause them to deform, so that the whole structure 'gives' under load. The flow of forces tends to be concentrated in the stiffer components (ie those providing greater resistance to deformation)

256bits
Gold Member
A Book says, " When a load is acting on a member, it undergoes deformation. An internal resistance is developed against deformation by the member and the intensity of this internal resistance is called stress." I also read, "When a member is in its elastic range, it offers resistance against deformation and when it goes beyond the elastic range, i.e. plastic range, there is no resistance offered by the member against deformation."

But, in a stress-strain curve, the stress keeps increasing even after the elastic range until the ultimate tensile stress which is in the plastic range.
How does stress ( RESISTANCE) increase in the plastic range when it is clearly stated that the member does not actually offer resistance after the elastic range??
Your last sentance , or question, misinterprets the stress - strain diagram. The author is correct.

If you look at the stress strain curve, up to the yield stress, if the stress is relaxed the member will return to its original length.

Into the plastic region, the member will deform permanently. Notice that the author has not stated that the stress has been reduced to zero but that the member offers no resistance to the present stress. If you continue to stress the member at that level where it behaves as a plastic just above the yield point, the strain will increases with no more stress added, up to a level where work hardening ( strain hardening ) will increase the yield strength and more stress is needed to deform the material. Increase the stress again and the material behaves plastically until work hardening increases the yield stress once more, and on and on, until the ultimate stress level is reached.

Strain in the elastic region is mainly due to change in atomic spacing.

In the plastic region, other mechanisms are at play.
This all has to do with movement of dislocations within the crystal structure of the material called slip and twinning, where atoms will move relative to one another. Once all the dislocations within the crystal sructure have been used up, that specific sample of material is as strong as it will ever become. A perfect cystal made of the same material would show no strain hardening.

Not all materials exhibit this behavior. Ductile carbon steel is the usual example.
You can compare the stress strain curve of a brittle material to that of a ductile material as an exercise.

Chestermiller
Mentor
This is another vote in agreement with the recommendation to Studiot and AlephZero. I cringed when I read the original quote.