# What is isotropic vs homogeneous in structural engineering

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

Now, when perform we say a material is homogeneous? If I have steel with BCC crystal structure, once carry out we say that this is homogeneous and also non-homogeneous? Can someone provide certain examples to explain - especially what a non-homogeneous material would certainly be?

In brief, to my understanding:

homogeneous : the property is not a role of place, i.e. it does not depfinish on \$x\$, \$y\$ or \$z\$.

isotropic: the home does not depfinish on a details direction.

NB: you can have a homogenous residential or commercial property that is not isotropic, i.e. the refrenergetic index of a birefringent material: it is a consistent, yet this constant has two different worths alengthy the 2 axes of the material.

A non-homogeneous product might be, say, the Planet itself: its thickness counts on whereabouts you are (which layer, crust, mantle etc.).

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answered Dec 13 "14 at 1:17

SuperCiocia♦SuperCiocia
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## Homogeneity = translational invariance

A material is homogeneous through respect to the residential or commercial property \$f\$ (for instance density) if

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

i.e. home \$f\$ does not depfinish on the spatial position. If you measure property \$f\$ at point \$mathbf r\$ or \$mathbf r+mathbf r"\$, you will uncover the same result.

Examples: most products are homogeneous at a big enough range, but they can expose inhomogeneities if we look cshed enough. See the section about scale.

## Isotropy = rotational invariance

A material is isotropic through respect to the residential or commercial property \$f\$ if

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

i.e. residential or commercial property \$f\$ does not depend on the direction of its discussion. If you measure property \$f\$ alengthy any type of direction in the product, you will discover the very same result.

Examples: fluids and also amorphous solids are isotropic. Many crystals (through a couple of exceptions like the cubic crystal system) are not isotropic.

## Scale dependence

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

To give you a specific instance, think about steel: steel is an iron-carbon alloy. At a large sufficient scale (let"s say the mm scale), steel is homogeneous. However before, if you look at it close sufficient (\$mu\$m scale), this is what you watch (source):

Definitely not homogeneous. Anvarious other example is granite:

Other examples of products which are homogenous/isotropic on huge scales but inhomogeneous/anisotropic on smaller sized scales, acomponent from alloys, are polycrystalline products.

Also a normal basic cubic crystal (number below), which is isotropic on large scales, is anisotropic on little scales. To watch this, simply think around standing in the center of the cube: just how many type of atoms will certainly you encounter if you relocate towards among the faces? And just how many if you move along one of the diagonals? The answer is various.

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