Isotropic Material

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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

Examples