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Since the units are the same ($\frac{N}{m^2}$), what's the difference between pressure and stress?

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Pressure is a force applied against the surface of the material in question. It is divided by area because it describes distributed forces (eg. force from a compressed gas or liquid, or stacked/piled solids.)

Stress is a force distributed through the thickness of the material in question. It is divided by area because force gets shared (though not always evenly) by the cross section of the material. For example if you have a solid block of material supporting a weight, the force from the weight, divided by the width and depth of that bock, gives you the stress.

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    $\begingroup$ I feel like this is an overly simplistic answer that gives the impression that stress is only something that happens to solids. Stresses do indeed exist in fluids. The distinction is that pressure is a scalar quantity; it is isotropic -- the same in every direction. Stress, on the other hand, is a tensor quantity, it is directional, but it follows certain rules of frame invariance. $\endgroup$
    – Tristan
    Commented Jan 27, 2015 at 16:42
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    $\begingroup$ OK. That's fair. It was not clear to me how formal of an answer was expected. I was just trying to communicate the broad concept in a clear way. Obviously the person who asked the question can select a different answer if it more clearly resolves their question. $\endgroup$
    – Ethan48
    Commented Jan 27, 2015 at 18:00
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While some of these answers are close, they are (at the time this answer is written) all incorrect to some degree.

Pressure and stress are very closely related -- in fact, one could argue that pressure is, in a sense, a subset of stress. To be specific, the pressure in a material is the isotropic part of the total stress in a material. Pressure is a scalar quantity -- the same in every direction, while stress is a tensor quantity that captures all deforming forces.

Pressure and stress are related as follows: if the components of the stress tensor are given by $\sigma_{ij}$, then the pressure is (using Einstein notation)

$$p = -\frac{1}{3}\sigma_{ii}$$

That is to say, the pressure is the opposite of the average of the diagonal elements of the stress tensor.

When speaking more specifically in terms of a boundary condition or an applied load for a structural analysis problem, it refers specifically to an applied normal stress over a given area.

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Pressure and stress are both forces distributed on a surface, but are in essence two quite different concepts. The main difference between them is that pressure is external and stress is internal.

When you have an object, pressure is the surface-force perpendicular on the 'skin' of this object.

To define stress it is useful to imagine a solid object with a set of external forces (actions and reactions) working on its surface. Because of these forces the object gets deformed, until it is in a state of equilibrium. When you would make a cut through this object and remove a part of it, forces on the surface exposed by the cut would be needed to keep the object in the same deformed state and to keep it in equilibrum. These internal surface-forces are called stresses.

While pressure is defined to be perpendicular on the surface of the object, this restriction does not apply to stresses. Stresses can be applied in any direction on the internal surface. This is another difference between pressure and stress. Stresses perpendicular to the internal surface are called 'normal stresses' (compression or tension). Stresses parallel to the internal surface are called 'shear stresses'.

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One could say they are closely related, but while pressure is more generic, omnidirectional (like in gas), stress is defined in a solid, and is a tensor - with factors responsible for displacement force in 3 dimensions plus twisting force in 3 axis.

With pressure, you take an imaginary piston in cyllinder with vacuum, with a dynamometer attached to the piston, and measure what force the medium exerts on that wall, dividing it by the piston surface. No matter how you turn it, the value is the same.

Now take a bunch of strain guages:

enter image description here

and cover them with concrete, forming a concrete beam. At first they will all show the same, pressure of liquid concrete. But as the concrete solidifies, the readouts will change. Some will show negative values as the beam bends and strains along the outer side. Others will show lateral pressure of the beam exerting its own weight perpendicular to its length. If you compress the beam, you'll get quite extreme values length-wise, but tiny negatives outwards from the axis as the compressed material expands to the sides. If you try to bend the beam, you'll get some small negatives on the outer side of the bend, some small positives on the inner side, and then the beam will snap; it's way weaker against negative forces (pulling it apart) and these are exerted on the outer side of the bend.

So, when using the 'stress' value, unless you give the full tensor, it's always essential to write which direction of the stress you're describing - just putting it down like pressure isn't all that helpful.

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    $\begingroup$ One correction -- it is incorrect to say that stress happens in a solid, while pressure happens in a gas. Both happen in both -- pressure is related to the first invariant of the total stress tensor. Stress indeed happens in fluids -- look at Couette flow for a trivially easy example. $\endgroup$
    – Tristan
    Commented Jan 27, 2015 at 16:44
  • $\begingroup$ @Tristan: Yes, in moving liquids and gasses, where forces of viscosity replace structural bonds. If they reach equilibrium, it quickly gets levelled out. OTOH, it may remain in solids - even without external forces; latent stresses are an important engineering problem. See Prince Rupert's drop, where a minimal damage to the structure of the drop causes the whole thing to explode, accumulated latent stress leading to violent destruction of the drop. $\endgroup$
    – SF.
    Commented Jan 27, 2015 at 18:21
  • $\begingroup$ (well, at least in perfect liquids; the surface tension effects like the meniscus or the capillary action are very much a stress-related effects. But if you take a bulk of immobile liquid, the directional factors become negligible.) $\endgroup$
    – SF.
    Commented Jan 27, 2015 at 18:47
  • $\begingroup$ Considering that most engineering problems involving fluids involve them, well, flowing, I think the distinction is rather moot. Stress is a continuum mechanics concept; it does not care what makes up the continuum -- that's what constitutive equations are for. $\endgroup$
    – Tristan
    Commented Jan 27, 2015 at 18:50
  • $\begingroup$ @Tristan: Let me partially disagree. Most engineering problems involving liquids neglect the tension factors of the liquid dynamics. Sure there are domains (like marine engineering) where they are critical, but in machinery, industrial chemistry, civil engineering, and most branches that deal with bulk amounts of liquids moving at moderate pace or at high pressure, usually it's the pressure that really matters, and the rest is often treated as "let's give it enough surplus pressure never to bother with it." $\endgroup$
    – SF.
    Commented Jan 27, 2015 at 21:06
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Pressure is force applied per unit area. It arises due to external forces on the surface of an object.

When external forces are applied, in order to avoid deformation internal forces are generated which are called Stresses. Both pressure and stress have the same unit.

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