Hooke's Law for Isotropic Materials
engineering fundamentals Hooke's Law: Isotropic Materials
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Isotropic Definition
Most metallic alloys and thermoset polymers are considered isotropic, where by definition the material properties are independent of direction. Such materials have only 2 independent variables (i.e. elastic constants) in their stiffness and compliance matrices, as opposed to the 21 elastic constants in the general anisotropic case.

The two elastic constants are usually expressed as the Young's modulus E and the Poisson's ratio n. However, the alternative elastic constants K (bulk modulus) and/or G (shear modulus) can also be used. For isotropic materials, G and K can be found from E and n by a set of equations, and vice-versa.

Hooke's Law in Compliance Form
Hooke's law for isotropic materials in compliance matrix form is given by,

Some literatures may have a factor 2 multiplying the shear modulii in the compliance matrix resulting from the difference between shear strain and engineering shear strain, where , etc.

Hooke's Law in Stiffness Form
The stiffness matrix is equal to the inverse of the compliance matrix, and is given by,

Some literatures may have a factor 1/2 multiplying the shear modulii in the stiffness matrix resulting from the difference between shear strain and engineering shear strain, where , etc.

Visit the elastic constant calculator to see the interplay amongst the 4 elastic constants (E, n, G, K).

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