Orthotropic Material

Constitutive Law

This anisotropic MPM material (or FEA material) is a small strain, linear elastic material. The stress (σ) and strain (ε) are related by:

      [math]\displaystyle{ \vec\varepsilon = \mathbf{S}\vec\sigma + \vec\alpha\Delta T + \vec\beta c }[/math]

      [math]\displaystyle{ \vec\sigma = \mathbf{C}\vec\varepsilon + \vec M\Delta T + \vec M_\beta c }[/math]

where S and C are the compliance and stiffness tensors, [math]\displaystyle{ \vec\alpha }[/math] and [math]\displaystyle{ \vec\beta }[/math] are the thermal and solvent expansion tensors, and [math]\displaystyle{ \vec M }[/math] and [math]\displaystyle{ \vec M_\beta }[/math] are the stress-temperature and stress-concentraion tensors. ΔT is difference between current temperature and the stress free temperature and c is the weight fracture solvent concentration. These equations use contracted notation where stress and strain tensors contract to vectors:

      [math]\displaystyle{ \vec\varepsilon = (\varepsilon_{xx},\varepsilon_{yy},\varepsilon_{zz},\varepsilon_{yz},\varepsilon_{xz},\varepsilon_{xy}) }[/math]

      [math]\displaystyle{ \vec\sigma = (\sigma_{xx},\sigma_{yy},\sigma_{zz},\sigma_{yz},\sigma_{xz},\sigma_{xy}) }[/math]

and the order of the shear terms is by the standard convention. The stiffness and compliance tensors contract to 6X6 matrices while all thermal and moisture expansion tensors contract to a vector. When used as an FEA material, the solvent expansion and solvent concentration terms are not used.

Material Matrices

For an orthotropic material, the stiffness and compliance tensors are:

      [math]\displaystyle{ \mathbf{C}^{-1} = \mathbf{S} = \left(\begin{array}{cccccc} {1\over E_x} & -{\nu_{xy}\over E_x}& -{\nu_{xy}\over E_x} & 0 & 0 & 0 \\ -{\nu_{yx}\over E_y} & {1\over E_y} & -{\nu_{yz}\over E_y} & 0 & 0 & 0 \\ -{\nu_{zx}\over E_z} & -{\nu_{zy}\over E_z} & {1\over E_z} & 0 & 0 & 0 \\ 0 & 0 & 0 & {1\over G_{xz}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {1\over G_{yz}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {1\over G_{xy}} \end{array}\right) }[/math]

where E and G are tensile and shear moduli, ν are Poisson's ratios, and x, y, and z refer to orthogonal axes of the material. The thermal and solvent expansion tensors are

      [math]\displaystyle{ \vec\alpha = (\alpha_x, \alpha_y,\alpha_z,0,0,0) }[/math]

      [math]\displaystyle{ \vec\beta = (\beta_x, \beta_y,\beta_z,0,0,0) }[/math]

where again, x, y, and z refer to orthogonal axes of the material. The stress-temperature and stress-concentration tensors are found from

      [math]\displaystyle{ \vec M = -\mathbf{C}\vec\alpha \quad{\rm and}\quad \vec M_\beta = -\mathbf{C}\vec\beta }[/math]

All these properties are set as explained below. The solvent expansion terms are for MPM only.

Material Properties

The properties are

Property	Description	Units	Default
Ex (or ER)	x-direction modulus (or R if axiysmmetric)	pressure units	none
Ey (or EZ)	y-direction modulus (or Z if axisymmetric)	pressure units	none
Ez (or ET)	z-direction modulus (or θ if axisymmetric)	pressure units	none
Gxy, Gyx (or GRZ,GZR)	x-y plane shear modulus (or R-Z if asymmetric)	pressure units	none
Gxz, Gxz (or GRT,GTR)	x-z plane shear modulus (or R-θ if asymmetric)	pressure units	none
Gyz, Gzy (or GZT,GTZ)	y-z plane shear modulus (or Z-θ if asymmetric)	pressure units	none
nuxy (or nuRZ)	x-y Poisson's ratio (or R-Z if asymmetric)	none	none
nuyx (or nuZR)	y-x Poisson's ratio (or Z-R if asymmetric)	none	none
nuxz (or nuRT)	x-z Poisson's ratio (or R-θ if asymmetric)	none	none
nuzx (or nuTR)	z-x Poisson's ratio (or θ-R if asymmetric)	none	none
nuyz (or nuZT)	y-z Poisson's ratio (or Z-θ if asymmetric)	none	none
nuzy (or nuTZ)	z-y Poisson's ratio (or θ-Z if asymmetric)	none	none
alphax (or alphaR)	x-direction thermal expansion coefficient (or R if axisymmetric)	ppm/K	none
alphay (or alphaZ)	y-direction thermal expansion coefficient (or Z if axisymmetric)	ppm/K	none
alphaz (or alphaT)	z-direction thermal expansion coefficient (or θ if axisymmetric)	ppm/K	none

You should only set one for each pair of Poisson's ratios (e.g., one of nuxy and nuyx). Note that to define a valid materials, the Poisson's ratios must satisfy:

      [math]\displaystyle{ |\nu_{ij}|\lt \sqrt{\frac{E_{ii}}{E_{jj}} \qquad {\rm and} \qquad 2\nu_{xy}\nu_{yz}\nu_{zx} \lt 1-\nu_{xy}\nu_{yx}-\nu_{yz}\nu_{zy}-\nu_{xz}\nu_{zx} }[/math]

The following properties are only allowed in MPM calculations:

Property	Description	Units	Default
betax (or betaR)	x-direction solvent expansion coefficient (or R if axisymmetric)	1/(wt fraction)	0
betay (or betaZ)	y-direction solvent expansion coefficient (or Z if axisymmetric)	1/(wt fraction)	0
betaz (or betaT)	z-direction solvent expansion coefficient (or θ if axisymmetric)	1/(wt fraction)	0
Dx (or DR)	x-direction solvent diffusion constant (or R if axisymmetric)	diffusion units	0
Dy (or DZ)	y-direction solvent diffusion constant (or Z if axisymmetric)	diffusion units	0
Dz (or DT)	z-direction solvent diffusion constant (or θ if axisymmetric)	diffusion units	0
kCondx (or kCondR)	x-direction thermal conductivity (or R if axisymmetric)	conductivity units	0
kCondy (or kCondZ)	x-direction thermal conductivity (or Z if axisymmetric)	conductivity units	0
kCondz (or kCondT)	x-direction thermal conductivity (or θ if axisymmetric)	conductivity units	0
(other)	Properties common to all materials	varies	varies

History Data

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