Hi,

1) The RHS of the equation quantifies the deformation caused by movement of dislocations in a crystal. Since, dislocations can only propagate in specific planes and directions in a crystal and is only caused by shear stresses on those particular planes. This relations assumes deformation by only dislocation slip , and further relates the shear rates to slip resistances associated with those slip planes, hardening and so on..

2) We are equating the plastic deformation of the body to aggregated shear deformation caused by dislocation slip, assuming the body before and after deformation is still continuous. Now, there are two classes of constitutive equations that follow, phenomenological which I mentioned earlier, and physics based dislocation density models.

Hope this helps.

Thanks Aritra for your answers !

May be, my questions weren't clear enough. let me try to put in this way..

I would like to get a good understanding of the product of gamma dot  and schmid  tensor .

My understanding of the schmid tensor roughly is, it signifies the plane bounded by the slip normal and slip direction(not so sure ). Now, you multiply the gamma_dot (which is in certain discrete directions ) with the schmid tensor - What do you get and what does it signify ?

Say, If I do a double dot product of cauchy tensor to the schmid tensor, i clearly see that I am obtaining the component of applied stress on a particular slip system (RSS), like this what does the above signify geometrically ?

Then , in the framework of continuum crystal plasticity, the body is always continuous having one to one deformation mapping - this is always true! there are no discontinuities.

If you take a closer look of the velocity gradient, it gives an idea of how the velocity varies spatially.

say in a 1-D flow, L = v (x+dx) - v(x) / dx with the limit dx->0 !

Lp is a part of L , i dont know what it means physically as much as i understand L!

Now you equate a continuum quantity to the RHS term, where the length scale is atleast 3 orders less than continuum. My question is, how can you equate quantities defined at two different length scales.

as I mentioned earlier, it assumes that deformation of the body is through slip occurring in discrete planes represented by the slip direction and slip plane normal. you need to implement a constitutive equation to find a relation between gamma dot and slip for example through the phenopwer law - gamma_dot = gamma_ref*|RSS/CRSS|^n, where RSS is resolved shear stress; CRSS being critical resolved shear stress, which are characteristic of particular slip systems. so gamma dot depends on the macroscopic stress. Lp is that part of L which is associated with the strain in the material, similar to rate of deformation tensors, but not same. 

1) Consider a volume element with a single slip system. The slip system is defined with a slip direction ms and a slip plane normal ns. For convenience, a third unitary vector ks, such that ms x ks = ns, is introduced. Because the slip system produces a homogeneous shear transformation, the velocity vector v is given by:

vm=z dgam/dt

vk=0

vn=0

In the above relations, we used the coordinate system defined by (ms,ks,ns). The sole non-zero component of the velocy gradient tensor l is thus lmn=vm,n whose expresion is:

lmn=vm,n=dvm/dz=dgam/dt

If you want to express the velocity gradient tensor in a arbitrary coordinate system, you obtain:

lij=msi  nsj dgam/dt

If we now want to generalize to a multiple slip system configuration, we just need to add the different contributions:

lij=sums(msi nsj dgam/dt)

2) To derive the above relations, we used continuum mechanics (standard kinematics). We just need the volume element to be large enough so that we do not have to employ a detailed description of crystallographic defects (e.g. dislocations).