# What is the difference between stress and pressure?

What is the difference between stress and pressure? Are there any intuitive examples that explain the difference between the two? How about an example of when pressure and stress are not equal?

• As you can see from the answers it is hard to assume what "intuitive" is for you and what level of expectations you have without being more specific. Apr 11, 2014 at 1:34
• The question came about when reading about Overburden Pressure (stress), en.wikipedia.org/wiki/Overburden_pressure. The linked article defines it as both. I had hoped to learn how to distinguish the two using an explanation a high school student or an engineering undergraduate would understand. Apr 11, 2014 at 13:39
• Stress is valence 2 tensor (represented by a matrix). "Pressure" is a special case: a stress tensor that can be written as a "scalar" quantity (i.e. of the form $p\,\mathrm{id}_3$, where $\mathrm{id}_3$ matrix is the $3\times3$ identity and $p$ the pressure scalar). Mar 28, 2015 at 3:22

Pressure is defined as force per unit area applied to an object in a direction perpendicular to the surface. And naturally pressure can cause stress inside an object. Whereas stress is the property of the body under load and is related to the internal forces. It is defined as a reaction produced by the molecules of the body under some action which may produce some deformation. The intensity of these additional forces produced per unit area is known as stress (pretty picture from wikipedia):

Overburden Pressure or lithostatic pressure is a case where the gravity force of the object's own mass creates pressure and results in stress on the soil or rock column. This stress increases as the mass (or depth) increases. This type of stress is uniform because the gravity force is uniform.

http://commons.wvc.edu/rdawes/G101OCL/Basics/earthquakes.html

Included in lithostatic pressure are the weight of the atmosphere and, if beneath an ocean or lake, the weight of the column of water above that point in the earth. However, compared to the pressure caused by the weight of rocks above, the amount of pressure due to the weight of water and air above a rock is negligible, except at the earth's surface. The only way for lithostatic pressure on a rock to change is for the rock's depth within the earth to change.

Since this is a uniform force applied throughout the substance due to mostly to the substance itself, the terms pressure and stress are somewhat interchangeable because pressure can be viewed as both an external and internal force.

For a case where they are not equal, just look that the image of the ruler. If pressure is applied at the far end (top of image) it creates unequal stress inside the ruler, especially where the internal stress is high at the corners.

• Regarding fluids and pressure, the fluid within the interstitial space or pore space of the rock is typically referred to having pore pressure or fluid pressure. Would it be incorrect to say pore stress or fluid stress? If I understand it correctly, from the other answers to the original question and from an explanation found here: en.wikipedia.org/wiki/Fluid#Physics it seems that pressure is the normal component of stress. So if all the other components are zero, you are left with just pressure making up the stress. Thus, you can say pore stress? Apr 11, 2014 at 22:16
• @jakemcgregor I'm not sure. I don't think it would be called stress because it is a force generated by pockets which can be non-uniform and vary throughout the substance. But I'm not a geologist, perhaps they mix the terms. Apr 12, 2014 at 0:15
• You say nothing about the tensor nature of stress as opposed to the scalar nature of pressure. Also, gas pressure, for example, is related to the internal forces of the gas molecules. Aug 5, 2015 at 20:02

Given a stress tensor $\mathbf{\sigma}$, which has 9 components in general, the pressure (in continuum mechanics at least) is defined as $P = 1/3 tr(\mathbf{\sigma})$.

So the pressure at a point in the continuum is the average of the three normal stresses at the point. The off-diagonal terms manifest as shear stress.

It's hard to say "stress" without being more specific in your question because stress is not a scalar. Pressure is always different from stress, but the two are related.

• Pressure is always different from stress because pressure is a scalar. As far as I know, pressure is defined as compressive isotropic normal stress. By this definition pressure has direction sign (positive for compressive?) and a magnitude? Also, by this definition, it seems that pressure is defined as a special case of stress (simple stress situation). If this is true, can fluid pressure be called fluid stress? My source: en.wikipedia.org/wiki/Stress_%28mechanics%29#Simple_stresses Apr 12, 2014 at 1:27

The difference between stress and pressure has to do with the difference between isotropic and anisotropic force. There's a Wikipedia section on the decomposition of the Cauchy stress $\boldsymbol{\sigma}$ into "hydrostatic" and "deviatoric" components, $$\boldsymbol{\sigma}=\mathbf{s}+p\mathbf{I}$$ where the pressure $p$ is $$p=\frac{1}{3}\text{tr}(\boldsymbol{\sigma})$$ where $\mathbf{I}$ is the $3\times 3$ identity matrix, and where $\mathbf{s}$ is the traceless component of $\boldsymbol{\sigma}$.

The linked article actually gives a pretty good intuitive explanation of $p\mathbf{I}$:

(From article) A mean hydrostatic stress tensor $p\mathbf{I}$, which tends to change the volume of the stressed body.

This follows since the surface force experienced by a plane with normal vector $\mathbf{n}$ is given by $$\mathbf{T}^{(\mathbf{n})}=\mathbf{n}\cdot\boldsymbol{\sigma}$$ which for a purely hydrostatic stress becomes $$\mathbf{T}^{(\mathbf{n})}=\mathbf{n}\cdot p\mathbf{I}=p\mathbf{n}$$ which points in the same direction as the normal to the plane. This basically means that a cube of material will want to expand like a ballon if $p>0$, and contract if $p<0$.

Meanwhile, the deviatoric component means that there are forces at play which don't just tend to expand or contract things, such as shear forces.

How about an example of when pressure and stress are not equal?

In a solid, pure shear waves can exist. Unlike in acoustic pressure waves, shear waves have constant pressure; the forces that propagate the wave are not due to pressure, but are due to shear strain.

• So if you only have compressive isotropic normal stress, it is called hydrostatic pressure or just pressure. So pressure is the normal components acting in compression that make up stress? I can say I have fluid stress of +100 psi and be equally correct as saying I have fluid pressure of 100 psi? Apr 12, 2014 at 0:41
• @jakemcgregor: Yeah, as far as I know pressure is the normal component of stress. In an isotropic fluid, if you try to shear it, there is no restoring force (because it's a fluid, and it can move), so it's not really possible to have deviatoric (non-normal) stresses, so stress and pressure are sort of interchangeable as far as wording goes. But if you've got rocks and stuff mixed in, it might be more complicated. Apr 12, 2014 at 0:52
• You certainly can have deviatoric stresses in a fluid. Without them, the fluid would not flow! In a Newtonian fluid, for example, the deviatoric stress is proportional to the "strain rate" tensor via viscosity. May 5, 2014 at 18:51
• @TylerOlsen see my comment above to DumpsterDoofus and my comment to tpg2114's answer (below). What are your thoughts? Jun 19, 2014 at 18:44

Pressure is perpendicular to the object, it is an external force only. Pressure causes stress inside of the object, so stress is an internal force.

• You might want to include some of the requested examples that show one versus the other and what happens when they're not equal. Mar 17, 2015 at 12:39

Pressure is an external force, when applied on another body, the effect is easily seen on the outer part of body and it first affected the outer area of the body. In the case of stress, the molecular deformation is developed internal of the body and stress is generated slowly slowly in the internal part of any object due to load. And simply pressure affected the outer area of the body and stress affected the body internally.Stress is observed due to the load applied,whereas pressure is a sort of load on a body.

The pore pressure of a fluid in an underground reservoir is not normally related to the overburden or lithostatic pressure. There are exceptions, known as overpressured reservoirs. Typically the pore pressure at depth is equivalent to the pressure caused by a column of salt water. The vertical stress in the rock is typically a function of the column of rock to the surface. The two principle horizontal stresses are normally unequal and lower than the vertical stress. Rocks behave in a plastic fashion and the difference in horizontal stresses are, inter alliance, caused by plate tectonics.

One can get stressed by pressure. Either pulling at you or pushing at you. Pressure comes before the stress and can be seen as a reaction to pressure. Though they have the same unit, there is a temporal asymmetry. Pressure comes prior to stress.

whenever external force is applied on the object automatically a restoring force is developed inside the object to restrict the deformation of the object.The ratio of restoring force perpendicular to the surface to the area is known as stress.The ratio of external force perpendicular to the surface to the area is known as pressure.

for example if you press a ball u r applying pressure and what the ball apply to u you is stress and if both r not equal than one dominates over other pressure is an external force and stress is an internal force

• What about the requested examples? Mar 28, 2015 at 3:22