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_z} & 0 & 0 & 0 \\ -{\nu_{zx}\over E_x} & -{\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)	MPa	none
Ey (or EZ)	y-direction modulus (or Z if axisymmetric)	MPa	none
Ez (or ET)	z-direction modulus (or θ if axisymmetric)	MPa	none
Gxy, Gyx (or GRZ,GZR)	x-y plane shear modulus (or R-Z if asymmetric)	MPa	none
Gxz, Gxz (or GRT,GTR)	x-z plane shear modulus (or R-θ if asymmetric)	MPa	none
Gyz, Gzy (or GZT,GTZ)	y-z plane shear modulus (or Z-θ if asymmetric)	MPa	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/M	none
alphay (or alphaZ)	y-direction thermal expansion coefficient (or Z if axisymmetric)	ppm/M	none
alphaz (or alphaT)	z-direction thermal expansion coefficient (or θ if axisymmetric)	ppm/M	none

You should only set one for each pair of Poisson's ratios (e.g., one of nuxy and nuyx).

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)	mm2/sec	0
Dy (or DZ)	y-direction solvent diffusion constant (or Z if axisymmetric)	mm2/sec	0
Dz (or DT)	z-direction solvent diffusion constant (or θ if axisymmetric)	mm2/sec	0
kCondx (or kCondR)	x-direction thermal conductivity (or R if axisymmetric)	W/(m-K)	0
kCondy (or kCondZ)	x-direction thermal conductivity (or Z if axisymmetric)	W/(m-K)	0
kCondz (or kCondT)	x-direction thermal conductivity (or θ if axisymmetric)	W/(m-K)	0
(other)	Properties common to all materials	varies	varies

Transverse 1 and Transverse 2 give identical materials but with different initial orientations. You can change to any other orientation when defining material points (in MPM) or elements (in FEA) by selecting rotations angles for particles or elements. In 2D, the only rotation angle that is allowed is about the z axis. Because there is only one rotation angle, you need both material types to allow you to specify all possible orientations for 2D analysis of transversely isotropic materials. In other words, for 2D analysis, the axial direction has to be either in the x-y plane (Transverse 2) or normal to the x-y plane (Transverse 1). Any other orientation would induce out-of-plane shear that is not allowed in 2D calculations. To handle such material orientations, you need to use 3D calculations (MPM only).

History Data

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