Isotropic Material

Constitutive Law

This MPM material (or FEA 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 (MPM only), c is solvent concentration (MPM only), 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 MPM material (or FEA 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 MPM material (or FEA material) 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

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 = {9KG \over G+3K} , \qquad G = G \qquad {\rm and} , \qquad \nu = {3K-2G\over 6K+2G} }[/math]

or given λ and G:

      [math]\displaystyle{ E = G\left({3\lambda + 2G \over \lambda + G}\right) \qquad {\rm and} \qquad G = G }[/math]

or given K and ν:

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

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

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 to all materials	varies	varies

History Variables

None

Examples

These commands model copper as an isotropic, elastic material (using scripted or XML commands):

Material "copper","Copper","Isotropic"
  E 120000
  nu .34
  alpha 16.5
  rho 8.96
  kCond 401
  Cv 386
Done
 
<Material Type="1" Name="Copper">
  <rho>8.96</rho>
  <Cv>386</Cv>
  <E>0.12</E>
  <nu>0.34</nu>
  <alpha>16.5</alpha>
  <kCond>401</kCond>
</Material>