Reason for stress and strain

Question

Do you think that the stress is due to strain or vice versa?

I have this doubt because of two of the following scenarios:

1. Consider the case of the rigidly fixed bar. It is now heated (say be some temperature difference of delta T). In this case, the net strain in the bar would be zero as it is fixed at the ends. In this case, the stress is present independent of the presence of the strain.

2. Consider the modified by the more conventional case of the bar kept on the smooth horizontal surface heated (with the same temp difference). In this case, the stress won't be present though (thermal) strain exits. In this case, the strain is present independent of the presence of the stress.

Can anyone please correct my idea of understanding?

Answer 1

Let's take a step back and look at the more general case, as follows.

We model an elastic solid as if it were a coil spring. Pull on its ends, and it stretches while generating an opposing force to your hands, in proportion to the amount of displacement.

In system dynamics we learn that it is impossible to instantaneously assign a displacement to a spring, a current flow through an inductor, a voltage across a capacitor, or a velocity to a mass. For example, in the case of a mass, it responds to a force by integrating it over time to yield a velocity- and we say that masses possess integral causality when forces are applied: you apply a force as an input, and obtain an output- velocity- as a causal consequence.

In the case of a spring, you apply a velocity difference across the ends and the spring integrates this to yield an opposing force: the reverse of the situation with a mass. Springs have integral causality to velocity differences whereas masses have integral causality to forces.

An elastic solid is a spring, which means that the velocity difference imposed across its ends produces a force. This means that the velocity difference comes first and the stress (caused by deflection of the spring) is the causal result.

If our dynamical system contains for example a voltage source driving a capacitor, the system is unphysical as the current would go to infinity. It exhibits what is called differential causality and when excited, the system "looks for" any stray series inductance (however tiny) it can find which would permit an integral-causality solution.

A similar situation exists for an inductor driven by a current source.

Answer 2

(a) "Do you think that the stress is due to strain or vice versa?" The question is meaningless as a general question about stress and strain. If there is a specific context, then one of these modes of thinking may seem more natural than the other.

(b) The apparent failure of stress/strain to equal a finite, non-zero constant in the two cases you've given is because strain =$\frac{\Delta l}l$ by itself is an inadequate definition. Let us impose the condition that the increase in length be isothermal. That immediately removes the paradox in your case 2: the fractional increase in length due to thermal expansion doesn't count as strain. In case 1 the strain is equal to $(–)\frac{\text{thermal expansion in case 2}}{\text{ original length}}$, because we'd have to reduce the length of the expanded bar at its raised temperature to its original length by an amount equal to the thermal expansion, giving the bar, at its raised temperature, a true compressive strain, with accompanying stress.