Does zero strain always imply zero stress? In solid mechanics, can I always assume that if an object undergoes no strain, then no stress is applied to it?  I think it's true only because I can't seem to find a counter-example.
 A: Zero strain does not always imply zero stress and visa versa. There are matterials that display stress-strain, $\sigma-\epsilon,$ hysteresis behaviour. In matterials like this, when you start loading them, they behave normally, i.e increasing the stress increases the strain. However, when you start to unload them (remove the load), instead of the stress becoming zero when the strain becomes zero, the matterial has some residual stress applied to it! Similary, if you repeat the cycle, although the stress becomes zero the strain retains a permanent value, i.e. the matterial remains permanently deformed! These are very interesting elastic properties of such matterials. Stress-Strain hysteresis phenomena are very well known and are discussed extensively in literature. 
A: Examples:


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*stress is zero but strain is present= when component is loaded beyond the elastic limit it shows plastic deformation which can not be regained. after unloading the specimen in plastic deformation zone material will follow slope similar to the elastic slope and will come back to zero stress (as load is removed now). but during this process it has already been plastically deformed which can not be restored only elastic deformation can be restored. so now there is some strain in material but the stress is zero.

*Strain is zero but stress is present= consider bar fixed at two ends and there is NO yielding/movement in supports. now when temperature is increased the bar will experience thermal stresses as there is resistance to axial expansion. but strain will be zero as change in length in axial direction is zero.
A: Zero strain implies zero internal stress but you can still have external stress or volume forces applied on it.
The equation of motion of continuum mechanics are:
$$\rho\partial_t^2u_i = C_{ijkl}\nabla_j\nabla_ku_{l} + f_i$$
with $u(\vec{x},t)$ the displacement to the equilibrium position, $C_{ijkl}$ the stiffness matrix, $\rho$ the mass density, $f_i = \nabla_j\tau_{ij}$ the volume forces derived from an applied external (as in not a consequence of Hooke's law and the natural elasticity of your solid) stress $\tau_{ij}$.
A: *

*Strain is present but stress is zero.
Consider an example of a rod  subjected to a rise of temperature, condition is that the rod is not fix at its ends, when the temperature increases rod will experience thermal stresses but when you stop to increase temperature then at the end stresses will become zero but the thermal strain will remain as the rod length has increased
2) Stress is present but strain is zero.
Now consider the same rod fixed at its end, when you increase temperature, stress will induce in it but because of fix condition rod will not show any deflection in longitudinal direction and hence the strain will be zero. But in this case there may be chances of transverse strain developed in rod.

