Does the strain induces stress or does the stress induces strain? Or is it both ways? I have thought intuitively that when a force is applied on a material, the material responds by stress which is the internal force. And then, because of this internal force, the material undergoes some deformation, which is the strain.
So in this way, the stress induces the strain.
Later, I have come across the concept of stiffness tensor (from Hooke's law), the one that relates stress to strain. And it is computed as the partial derivative of stress with respect to strain, from here the stress could vary with respect to change of strain, and thus we get the stiffness tensor.
So this means that both stress and strain could contribute to each others. But how could the strain induces stress? The cause of deformation is stress, so how come stress could be changed due to strain?
Unless, stress induces strain, then while particles are changing their position, they also induce stress??
 A: In terms of cause and effect, an external force (stress) causes deformation in an object (strain) which in turn induces internal forces (stress) which oppose the external force. If the internal forces are strong enough then deformation stops at the point where the internal forces balance the external force and the object is in equilibrium. If the internal forces are not strong enough then deformation is not limited, and the object will eventually break (although a ductile material may undergo very large deformations without breaking).
Mathematically, it does not matter whether we think of stress as a function of strain, or strain as the inverse function of stress. The mathematical formulation is quite separate from the chain of physical cause and effect.
A: 
Does the strain induces stress or does the stress induces strain? Or
is it both ways?

When we are dealing with mechanical stress and strain, which is what we deal with most frequently, it is normally the stress resulting from loading that induces strain.
On the other hand, if we are dealing with thermal stress, strain can induce stress if the body is constrained from expansion or contraction.
For example, take a metal bar and place the ends between two fixed surfaces. Heat the bar. When the bar attempts to elongate it will be constrained from doing so by the fixed surfaces resulting in compressive stress. Or take a heated bar and attach the ends to fixed surfaces. When the bar cools and attempts to shorten the constraints will prevent it from doing so, resulting in tensile stress.
Hope this helps.
A: TL; DR: boundary conditions
Within the material they are determined self-consistently. However, which  one the cause and which is the consequence could be tied to the boundary conditions:

*

*If we are given the boundary conditions in terms of the forces applied to the body (the loads), we first have the stress, and the strain follows.

*if we are given the position of the body boundaries, then we are given strain, and the stress is secondary. Note that this is not simply a question of what we are given mathematically - an object may have strains without any external loads applied to it, as is the case many varieties of steel or concrete (which in many languages is appropriately called by german word beton, literally meaning "stress").

*It could be both.

