Mechanical Behavior · Elastic anisotropy

Elastic anisotropy: why Young's modulus depends on crystal direction

Pick copper and drag the shape below. Its long arms point along the <111> directions, where a copper crystal is almost three times stiffer than along the cube edges <100>.

How stiff is one crystal in each direction?

Distance from the centre = Young's modulus E along that direction. Drag to turn it. A sphere would mean the same E in every direction. Dashed circle: the polycrystal (Hill) value.

Crystal and elastic constants

Look along

Try it: switch from copper to tungsten. The spiky shape becomes a sphere: tungsten has almost the same E (about 410 GPa) in every direction. Then raise C44 of tungsten and watch the arms grow along <111>.

What is Young's modulus in one crystal plane?

E for every direction lying in the chosen plane. Distance from the centre is E (rings in GPa). Click the plot to pick a direction. Dashed circle: the polycrystal (Hill) value.

Plane and direction

A direction and its opposite have the same E, so 0 to 180 degrees covers the plane. Angles are measured from the first direction listed in the readout.

Try it: with copper, choose the (111) plane. The curve is a perfect circle: every direction in a cubic {111} plane has the same E. Now choose (110): E swings from the lowest value along [001] to the highest along [1-11].

Why do grains of the same metal give different moduli?

Grains seen from above. Colour: E along each grain's surface normal (the indenting direction). Click a grain.
Standard triangle. Every direction folded into one triangle, as on an EBSD inverse pole figure. Dots: the grains.
Single crystal against polycrystal. Pale bar: every value one crystal can show. Ticks: the grains on the map. V, R, H: Voigt (upper bound), Reuss (lower bound) and Hill (their mean) for a random polycrystal.

Grains and averages

Colour grains by
Grain set

An indenter pushes on many directions under the tip at once, so the measured indentation modulus changes less from grain to grain than E along the normal does. The order is the same: in copper, grains with a normal near <111> are stiffest.

Try it: with nickel, click the yellow grains and then the dark blue ones: E along the normal runs from 139 to 292 GPa across the first set of grains, while the Hill value for the whole sample is 223 GPa. Switch to "EBSD IPF colour": red grains (normal near [001]) are the soft ones.

What to remember

One number is not enoughA copper crystal at room temperature has E = 191 GPa along <111> and 67 GPa along <100>, a factor of 2.9.
Two shear constants decide the shapeAlong <100> the stretch is resisted by C' = (C11 - C12)/2, along <111> by C44. Their ratio is the Zener ratio A. A = 1 gives a sphere.
A polycrystal averagesA random polycrystal sits between the Reuss and Voigt bounds. For copper the Hill mean is 127 GPa, the value found in handbooks.
Single grains see their orientationNanoindentation, micropillars and in situ TEM tests probe one grain. Know its orientation from EBSD before you compare moduli.
More detail: the equations and the limits of the model

From stiffness to compliance

The elastic constants Cij (stiffness, Voigt notation) give stress from strain. The page inverts the 6 by 6 matrix to get the compliances Sij, which give strain from stress. For a cubic crystal:

S11 = (C11 + C12) / [(C11 - C12)(C11 + 2 C12)], S12 = -C12 / [(C11 - C12)(C11 + 2 C12)], S44 = 1 / C44

Young's modulus along a direction

Pull along a unit vector with direction cosines l1, l2, l3 to the cube axes (Nye 1957):

1/E = S11 - 2 (S11 - S12 - S44/2)(l1² l2² + l2² l3² + l3² l1²)

The bracket is 0 along <100>, 1/4 along <110> and 1/3 along <111>, so E along <100> and <111> are the two extremes. A short way to see which is larger, with bulk modulus K = (C11 + 2C12)/3:

1/E<100> = 1/(9K) + 1/(3C'), 1/E<111> = 1/(9K) + 1/(3C44)

If C44 > C', that is A = 2C44/(C11 - C12) > 1, the crystal is stiffest along <111>. This holds for Cu, Ni, alpha-Fe, Al and Si. For a hexagonal crystal E depends only on the angle θ from the c axis:

1/E = S11 sin⁴θ + S33 cos⁴θ + (2 S13 + S44) sin²θ cos²θ

Voigt, Reuss and Hill

Voigt (1928) assumes every grain has the same strain; this gives an upper bound. Reuss (1929) assumes every grain carries the same stress; this gives a lower bound. Hill (1952) showed that the true value of a random, untextured polycrystal lies between them and proposed their mean. For any crystal (cubic or hexagonal):

9 K_V = (C11 + C22 + C33) + 2 (C12 + C23 + C31)
15 G_V = (C11 + C22 + C33) - (C12 + C23 + C31) + 3 (C44 + C55 + C66)
1/K_R = (S11 + S22 + S33) + 2 (S12 + S23 + S31)
15/G_R = 4 (S11 + S22 + S33) - 4 (S12 + S23 + S31) + 3 (S44 + S55 + S66)

Then E = 9KG/(3K + G) for each average. For a cubic crystal KV = KR, so only the shear modulus differs between the bounds.

Limits of this page

The constants are adiabatic values from ultrasonic tests at room temperature (about 295 to 300 K). All of them fall as temperature rises, and anisotropy usually changes too. For Young's and shear moduli of metals at room temperature, isothermal (static) values differ from these by less than about 1 percent. The grain map assumes a random texture; a rolled or drawn metal has texture and a different average. The shear range shown for a cubic crystal lies between C' and C44; for hexagonal crystals the page searches all planes and directions. The indentation modulus needs the full anisotropic contact solution (Vlassak and Nix 1994); the page does not compute it and shows E along the normal instead.

On this site: How a hardness number gets made (where the indentation modulus comes from) · When your hardness number lies · Reading an IPF map (the colours in the grain map) · Stereographic projections lab (the standard triangle) · Two-beam defect simulator (it uses isotropic elasticity for the strain field; this page shows how far that is from real copper). In this chapter: Dislocations and Burgers vectors · Schmid factor and slip systems · Reading a tensile curve · Strengthening mechanisms · Fracture toughness.

Questions people ask

What is elastic anisotropy?

It means the stiffness of a crystal depends on the direction you load it. A copper crystal stretched along a cube edge <100> has E = 67 GPa; along the body diagonal <111> it has 191 GPa. Glass and a random polycrystal have the same E in every direction.

What is the Zener anisotropy ratio?

A = 2C44/(C11 - C12), the ratio of the two shear stiffnesses of a cubic crystal. A = 1 means isotropic. At room temperature A is 1.01 for tungsten, 1.22 for aluminium, 2.4 for alpha-iron, 2.5 for nickel and 3.2 for copper.

Why is copper stiffest along <111>?

Stretching along <100> is resisted by the shear constant C' = (C11 - C12)/2 = 23.5 GPa. Stretching along <111> is resisted by C44 = 75.4 GPa. C44 is larger, so <111> is stiffer.

Why does nanoindentation give a different modulus in different grains?

Each grain presents a different crystal direction to the tip. The indenter loads many directions at once, so the spread in indentation modulus is smaller than the spread in E along the normal, but it is real. Map the grains with EBSD, then compare moduli grain by grain.

What are the Voigt, Reuss and Hill averages?

They estimate the modulus of a random polycrystal from single-crystal constants. Voigt assumes equal strain in all grains (upper bound), Reuss equal stress (lower bound), and Hill takes their mean. For copper the page gives E = 145, 109 and 127 GPa.

Which metals are elastically isotropic?

Tungsten at room temperature is the classic case: A = 1.01 and E is within 1 percent of 410 GPa in every direction. Aluminium is close (E from 63 to 76 GPa). Magnesium is the nearly isotropic hexagonal metal (43 to 51 GPa).

Does Young's modulus of a hexagonal metal depend on direction?

Yes, but only on the angle from the c axis. Zinc is the extreme case: 36 GPa along c and 124 GPa in the basal plane. Titanium is stiffer along c (143 GPa) than in the basal plane (104 GPa).

Why does a textured sheet have a different modulus in each direction?

Texture means the grains share a preferred orientation, so the polycrystal inherits part of the single-crystal anisotropy. The Voigt, Reuss and Hill values on this page assume random grains and do not apply to a strongly textured sheet.

References

Show the 13 references and the source of each set of constants
  1. J. F. Nye, Physical Properties of Crystals, Oxford University Press (1957, reissued 1985), chapter VIII (elasticity, the directional Young's modulus).
  2. J. P. Hirth and J. Lothe, Theory of Dislocations, 2nd ed., Wiley (1982), chapter 2 and chapter 13 (anisotropic elasticity, elastic constant tables).
  3. G. Simmons and H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook, 2nd ed., MIT Press (1971). Source of the Cu, Al, Ni and Ti constants used here (room temperature).
  4. R. deWit, Elastic constants and thermal expansion averages of a nontextured polycrystal, Journal of Mechanics of Materials and Structures 3, 195 to 212 (2008), doi:10.2140/jomms.2008.3.195. Tables 1 and 3 list the Simmons and Wang values for Cu, Al and Ti.
  5. H. M. Ledbetter and E. R. Naimon, Elastic properties of metals and alloys. II. Copper, J. Phys. Chem. Ref. Data 3, 897 (1974), doi:10.1063/1.3253150. Recommended room-temperature Cu values (169.1, 122.2, 75.4 GPa) agree with the preset within 0.7 percent.
  6. H. M. Ledbetter and R. P. Reed, Elastic properties of metals and alloys, I. Iron, nickel, and iron-nickel alloys, J. Phys. Chem. Ref. Data 2, 531 (1973), doi:10.1063/1.3253127.
  7. J. A. Rayne and B. S. Chandrasekhar, Elastic constants of iron from 4.2 to 300 K, Physical Review 122, 1714 (1961), doi:10.1103/PhysRev.122.1714. Source of the alpha-Fe constants: their Table I gives C44 = 117.8, (C11 - C12)/2 = 48.83 and (C11 + C12 + 2C44)/2 = 302.1 GPa at 300 K, which give C11 = 233.1 and C12 = 135.4 GPa.
  8. F. H. Featherston and J. R. Neighbours, Elastic constants of tantalum, tungsten, and molybdenum, Physical Review 130, 1324 (1963), doi:10.1103/PhysRev.130.1324. Source of the W constants: 523.3, 204.5 and 160.7 GPa at 300 K (their Table III).
  9. J. J. Hall, Electronic effects in the elastic constants of n-type silicon, Physical Review 161, 756 (1967), doi:10.1103/PhysRev.161.756. Source of the Si constants (room temperature).
  10. D. Tromans, Elastic anisotropy of HCP metal crystals and polycrystals, International Journal of Research and Reviews in Applied Sciences 6(4), 462 to 483 (2011). Table 2 (from Hearmon) is the source of the Mg and Zn constants (room temperature).
  11. R. Hill, The elastic behaviour of a crystalline aggregate, Proceedings of the Physical Society A 65, 349 (1952), doi:10.1088/0370-1298/65/5/307.
  12. C. Zener, Elasticity and Anelasticity of Metals, University of Chicago Press (1948) (the anisotropy ratio A).
  13. J. J. Vlassak and W. D. Nix, Measuring the elastic properties of anisotropic materials by means of indentation experiments, J. Mech. Phys. Solids 42, 1223 (1994), doi:10.1016/0022-5096(94)90033-7.
Cite this page: Tripathy, Manisha. “Elastic Anisotropy: Why Young's Modulus Depends on Crystal Direction.” untethered atom, 2026, https://untetheredatom.com/mechanical-behavior/elastic-anisotropy.
BibTeX 
@misc{tripathy2026elasticanisotropy,
  author = {Tripathy, Manisha},
  title  = {Elastic Anisotropy: Why Young's Modulus Depends on Crystal Direction},
  year   = {2026},
  howpublished = {\url{https://untetheredatom.com/mechanical-behavior/elastic-anisotropy}},
  note   = {Interactive web tool}
}
Last updated 24 September 2026.