Image Source: https://mechtics.com/design/strength-of-material/stress-strain-curve-for-mild-steel-with-defination-of-stress-and-strain/
I know that A is the proportional limit, B is the elastic limit. Up to B if the loading is removed the material will restore its original dimensions in theory. C is the upper yield point, beyond C plastic deformations occur.
I have referred many books but couldn't find what's happening in the portion BC? the material is behaving elastically or plastically?
One point of note is that the yield region is not as cleanly defined as BCD is in the image (although most books have it that way).
In reality the yield region looks like
The following image is one from many actual measurement that I took years ago (its Force Displacement).
You can see the jagged yield region after the elastic region.
The point is that the yield point is not clearly defined. It is a borderline between elastic and plastic. At the yield point, some parts of the material are still in elastic "mode", while other have entered the plastic "mode" (i.e. the dislocations with that part of the material start to move).
Which parts start to move is stochastic, however as the yield process completes (point D), all of the material is now in plastic "mode".
Point "C" is the "offset yield point" or "proof stress" as described below.
The yield strength or yield stress is a material property and is the stress corresponding to the yield point at which the material begins to deform plastically. The yield strength is often used to determine the maximum allowable load in a mechanical component, since it represents the upper limit to forces that can be applied without producing permanent deformation. In some materials, such as aluminium, there is a gradual onset of non-linear behavior, making the precise yield point difficult to determine. In such a case, the offset yield point (or proof stress) is taken as the stress at which 0.2% plastic deformation occurs. Yielding is a gradual failure mode which is normally not catastrophic, unlike ultimate failure.
Some metals, such as mild steel, reach an upper yield point before dropping rapidly to a lower yield point. The material response is linear up until the upper yield point, but the lower yield point is used in structural engineering as a conservative value.
Yes; from B to C it is behaving elastically AND plastically. A small amount of plastic strain ( less than 0.2%) is mixed with elastic strain.
Summary: There is increasing plastic behavior and decreasing elastic behavior as the sample is strained from B to C. The transition is caused by random grain orientations and variable resolved shear stress.
With respect, the other answers generally have the right idea, but are missing the important meso-scale mechanism for why there is a non-linear shape between points B and C.
Between points B and C on the stress-strain diagram there is a transition from elastic stretch to plastic flow. The transition occurs because bulk metals are typically composed of many randomly-oriented crystal grains. As you may know, metal crystal grains flow by application of shear stress. The resolved shear stresses on the crystal grain slip systems is what matters for initiating plastic flow. Grains at different orientations experience different resolved shear stresses. Grains more "favorably" oriented, with larger resolved stresses, experience plastic flow earlier. Because grain orientations are random, each grain will start to flow at a specific stress level, giving a distribution of stresses from point B to C. At the macro-scale the distribution means that the sample gradually transitions from "no plastic flow" at point B to "all flow" at C.
At least, until unlocking of interstitial atoms trapped at dislocations starts to occur between points C and D. See this QA for reasons why D is at lower stress than C, and for why NMech's answer plot is shaped the way it is.
For a little discussion of resolved shear stresses see this QA.
