Isotropic Material

Constitutive Law

This material is a small strain, linear elastic material. The components of stress are related to components of strain by

[math]\displaystyle{ \sigma_{ij} = \bigl(\lambda\varepsilon_{ii} - 3K(\alpha \Delta T+\beta c)\bigr)\delta_{ij} + 2G\varepsilon_{ij} }[/math]

where λ is the Lame coefficient, K is bulk modulus, α is thermal expansion coefficient, ΔT is temperature difference, β is solvent expansion coefficient, c is solvent concentration, and G is shear modulus. Two other isotropic material properties are modulus, E, and Poisson's ratio, ν.

Material Properties

Although deformation properties of an isotropic material can be defined by any two of λ, K, G, E, and ν, this material's properties can only be defined by specifying any two (and exactly two) of E, G, and ν. Those three and other properties for isotropic materials are:

Property	Description	Units	Default
E	Tensile modulus	MPa	none
G	Shear modulus	MPa	none
nu	Poisson's ratio	none	none
alpha	Thermal expansion coefficient	ppm/M	40

The following properties are only allowed in MPM calculations:

Property	Description	Units	Default
beta	Solvent expansion coefficient	1/(wt fraction)	0
D	Solvent diffusion constant	mm2/sec	0
kCond	Thermal conductivity	W/(m-K)	0
(other)	Properties common all materials	varies	varies

If you know K or λ instead of E, G, and ν, they are easily converted to E and ν. Given K and G:

[math]\displaystyle{ E = {9K \over 1+{3K\over G}} \qquad {\rm and} \qquad G = G }[/math]

or given λ and G:

[math]\displaystyle{ E = G\left({2 + {2G\over \lambda} \over 1 + {G\over \lambda}}\right) \qquad {\rm and} \qquad G = G }[/math]

or given K and ν:

[math]\displaystyle{ E = 3K(1-2\nu) \qquad {\rm and} \qquad \nu = \nu }[/math]

Other combinations are easily derived, but the above examples are the most common.

History Variables

None