When do we say a material is isotropic? When properties such as density, Young's modulus etc. are same in all directions. If these properties are direction-dependent then we can say that the material is anisotropic.

Now, when do we say a material is homogeneous? If I have steel with BCC crystal structure, when do we say that this is homogeneous and non-homogeneous? Can someone give specific examples to explain - especially what a non-homogeneous material would be?

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    $\begingroup$ That was always the prelude to a problem. "Assume a homogeneous and isotropic medium". It is pretty simple. Homogeneous means there is the same stuff everywhere, like hydrogen gas or a block of copper. Isotropic means it has the same properties in all directions. Glass would be isotropic on a macro scale, a crystal would not. $\endgroup$ – C. Towne Springer Dec 13 '14 at 0:41

In short, to my understanding:

homogeneous : the property is not a function of position, i.e. it does not depend on $x$, $y$ or $z$.

isotropic: the property does not depend on a particular direction.

NB: you can have a homogenous property that is not isotropic, i.e. the refractive index of a birefringent material: it is a constant, but this constant has two different values along the two axes of the material.

A non-homogeneous material could be, say, the Earth itself: its density depends on whereabouts you are (which layer, crust, mantle etc.).

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    $\begingroup$ Also, isotropic is always homogeneous but the reverse is not true. And another way to say it all is that an isotropic property is invariant under translation and rotation. $\endgroup$ – tpg2114 Mar 31 '15 at 14:07
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    $\begingroup$ @tpg2114 False: isotropic but non homogeneous patterns are possible. The two properties are independent from each other. See here for example: astro.ucla.edu/~wright/cosmo_01.htm $\endgroup$ – valerio Jan 16 '18 at 12:01
  • $\begingroup$ @SuperCiocia How is it possible for a homogenous property to be not isotropic if it has the same value in every point? $\endgroup$ – Antonios Sarikas Jun 12 '20 at 11:09
  • $\begingroup$ See examples in Valerio’s answer. $\endgroup$ – SuperCiocia Jun 12 '20 at 15:42

Homogeneity = translational invariance

A material is homogeneous with respect to the property $f$ (for example density) if

$$f(\mathbf r) = f (\mathbf r + \mathbf r')$$

i.e. property $f$ does not depend on the spatial position. If you measure property $f$ at point $\mathbf r$ or $\mathbf r+\mathbf r'$, you will find the same result.

Examples: most materials are homogeneous at a large enough scale, but they can reveal inhomogeneities if we look close enough. See the section about scale.

Isotropy = rotational invariance

A material is isotropic with respect to the property $f$ if

$$f(\mathbf r) = f (|\mathbf r|)$$

i.e. property $f$ does not depend on the direction of its argument. If you measure property $f$ along any direction in the material, you will find the same result.

Examples: fluids and amorphous solids are isotropic. Most crystals (with a few exceptions like the cubic crystal system) are not isotropic.

Scale dependence

Notice that both homogeneity and isotropy are scale-dependent quantities: they depend on the spatial scale where we choose to effectuate our measurements.

To give you a specific example, consider steel: steel is an iron-carbon alloy. At a large enough scale (let's say the mm scale), steel is homogeneous. However, if you look at it close enough ($\mu$m scale), this is what you see (source):

enter image description here

Definitely not homogeneous. Another example is granite:

enter image description here

Other examples of materials which are homogenous/isotropic on large scales but inhomogeneous/anisotropic on smaller scales, apart from alloys, are polycrystalline materials.

Also a normal simple cubic crystal (figure below), which is isotropic on large scales, is anisotropic on small scales. To see this, just think about standing in the center of the cube: how many atoms will you encounter if you move towards one of the faces? And how many if you move along one of the diagonals? The answer is different.

enter image description here

To conclude, I will just remark that homogeneity and isotropy are independent from each other. Below you can see an homogeneous but not isotropic pattern on the left and an isotropic but not homogeneous pattern on the right (source).

enter image description here

  • $\begingroup$ You say that most crystals (except the cubic crystal system) are anisotropic, but the link you give states that the cubic crystal system is one of the most commonly found in nature. Anyway, my question is, how come the cubic crystal system is isotropic? If I use your mathematical definition I would get that it is isotropic only in the crystallic principal axis. But what about an arbitrary direction? If I measure the resistivity of say potassium in a non crystallographic direction, can I expect it to be the same as in the a-b plane or the c direction? $\endgroup$ – AccidentalBismuthTransform Aug 28 '19 at 11:59

Further to your example, although a block of steel with BCC crystal structure may be considered homogeneous and isotropic, industrial processing such as heat treatment, annealing, cold rolling and welding can be used to create anisotropic stress-strain relationships. For example, if a steel rod is heated at one end, it would be considered non-homogenous, however, a structural steel section like an I-beam which would be considered a homogeneous material, would also be considered anisotropic as it's stress-strain response is different in different directions.


I think a body is homogeneous when the properties that defines its physical structure are same at all points(or space) while a body is isotropic if the value of properties,that affect some physical phenomenon,is same in all directions

  • $\begingroup$ It's important to note that a body can be inhomogeneous but isotropic or homogeneous but anisotropic. So these terms don't exclude each other. $\endgroup$ – engineer Aug 11 '15 at 8:41
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    $\begingroup$ "according to me" is probably not the ideal opener for a generally accepted concept. $\endgroup$ – engineer Aug 11 '15 at 8:42

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