Transversely Isotropic 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.

In a transversely isotropic MPM material (or FEA material), one plane is isotropic while the direction normal to that plane defines a unique axis with different properties. Properties in the isotropic plane are subscripted "T" for transverse plane properties and properties along the unique axis are subscripted "A" for axial properties. You can pick from two types of transversely isotropic materials, which differ only by orientation on the unique axes. The only reason two are needed is for 2D modeling Without two variations, 2D calculations could not model both when the isotropic plane as the analysis plane (use Transverse 1) and when the axial direction is in the the analysis plane (use Transverse 2). 3D modeling only needs one (and only Transverse 1 is allowed).

Transverse 1

In this transversely isotropic MPM material (or FEA material), the isotropic plane is the x-y plane, which is the plane for 2D or axisymmetric analyses. The axial direction is along the z axis, which is in the thickness direction for 2D analyses or the hoop direction for axisymmetric anlayses. The stiffness and compliance tensors are:

      [math]\displaystyle{ \mathbf{C}^{-1} = \mathbf{S} = \left(\begin{array}{cccccc} {1\over E_T} & -{\nu_T\over E_T}& -{\nu_A\over E_A} & 0 & 0 & 0 \\ -{\nu_T\over E_T} & {1\over E_T} & -{\nu_A\over E_A} & 0 & 0 & 0 \\ -{\nu_A\over E_A} & -{\nu_A\over E_A} & {1\over E_A} & 0 & 0 & 0 \\ 0 & 0 & 0 & {1\over G_A} & 0 & 0 \\ 0 & 0 & 0 & 0 & {1\over G_A} & 0 \\ 0 & 0 & 0 & 0 & 0 & {1\over G_T} \end{array}\right) }[/math]

where E and G are tensile and shear moduli, ν are Poisson's ratios, and A and T refer to axial and transverse properties. This tensor can be explicitly inverted to:

      [math]\displaystyle{ \mathbf{S}^{-1} = \mathbf{C} = = \left(\begin{array}{cccccc} K_T+G_T & K_T-G_T & 2K_T\nu_A & 0 & 0 & 0 \\ K_T-G_T & K_T+G_T & 2K_T\nu_A & 0 & 0 & 0 \\ 2K_T\nu_A & 2K_T\nu_A & E_A + 4K_T\nu_A^2 & 0 & 0 & 0 \\ 0 & 0 & 0 & G_A & 0 & 0 \\ 0 & 0 & 0 & 0 & G_A & 0 \\ 0 & 0 & 0 & 0 & 0 & G_T \end{array}\right) }[/math]

where [math]\displaystyle{ G_T }[/math] is transverse shear modulus and [math]\displaystyle{ K_T }[/math] is the transverse, plane-strain bulk modulus. These new properties are related to properties in S by:

      [math]\displaystyle{ G_T = {E_T\over 2(1+\nu_T)} \qquad {\rm and} \qquad \frac{1}{K_T} = {2(1-\nu_T)\over E_T} - {4\nu_A^2\over E_A} }[/math]

The thermal and solvent expansion tensors are

      [math]\displaystyle{ \vec\alpha = (\alpha_T, \alpha_T,\alpha_A,0,0,0) }[/math]

      [math]\displaystyle{ \vec\beta = (\beta_T, \beta_T,\beta_A,0,0,0) }[/math]

where again, A and T refer to axial and transverse properties. 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.

This transversely isotropic material is a special case of an orthotropic material where [math]\displaystyle{ E_{x}=E_{y}=E_T }[/math], [math]\displaystyle{ E_{z}=E_A }[/math], [math]\displaystyle{ \nu_{xy}=\nu_{yx}=\nu_T }[/math], [math]\displaystyle{ \nu_{xz}=E_T\nu_A/E_A }[/math], [math]\displaystyle{ \nu_{zx}=\nu_A }[/math], [math]\displaystyle{ \nu_{yz}=E_T\nu_A/E_A }[/math], [math]\displaystyle{ \nu_{zy}=\nu_A }[/math], [math]\displaystyle{ G_{xy}=G_T=E_T/(2(1+\nu_T)) }[/math], [math]\displaystyle{ G_{xz}=G_{yz}=G_A }[/math], [math]\displaystyle{ \alpha_{x}=\alpha_{y}=\alpha_T }[/math], [math]\displaystyle{ \alpha_{z}=\alpha_A }[/math], [math]\displaystyle{ \beta_{x}=\beta_{y}=\beta_T }[/math], and [math]\displaystyle{ \beta_{z}=\beta_A }[/math].

Transverse 2

In this transversely isotropic MPM material (or FEA material), the isotropic plane is the x-z plane. The axial direction is along the y axis, which is in the plane for 2D analyses. For axisymmetric analyses, the isotropic plane is the r-θ plane and the axial direction is along the z axis. The stiffness and compliance tensors are derived from those above by interchanging y (2) and z (3) entries as well as xz (5) and xy (6) entries. The thermal and solvent expansion tensors just interchange y (2) and z (3) entries. All properties are set as explained below.

The material is only allowed for 2D simulations when the axial direction lies in the analysis x-y plane.

Material Properties

The properties for Transverse 1 and Transverse 2 are the same, although their initial orientations are different. The properties are

Property	Description	Units	Default
EA	Axial tensile modulus	pressure units	none
ET	Transverse tensile modulus	pressure units	none
GA	Axial shear modulus	pressure units	none
GT	Transverse shear modulus	pressure units	none
nuA	Axial Poisson's ratio	none	0.33
nuT	Transverse Poisson's ratio	none	none
alphaA	Axial thermal expansion coefficient	ppm/K	40
alphaT	Transverse thermal expansion coefficient	ppm/K	40

In a transversely isotropic material, the three transverse properties (ET, GT, and nuT) are not independent (see above). As a consequence, you can only specify two of these three properties. The Poisson ratios cannot assume any values. They must obey the following relations:

      [math]\displaystyle{ -1\lt \nu_T\lt 1 \qquad -\sqrt{E_A\over E_T} \lt \nu_A \lt \sqrt{E_A\over E_T} \qquad {E_T\nu_A^2\over E_A} \lt {1-\nu_T\over 2} }[/math]

The following properties are only allowed in MPM calculations:

Property	Description	Units	Default
betaA	Axial solvent expansion coefficient	1/(wt fraction)	0
betaT	Transverse solvent expansion coefficient	1/(wt fraction)	0
DA	Axial solvent diffusion constant	diffusion units	0
DT	Transverse solvent diffusion constant	diffusion units	0
kCondA	Axial thermal conductivity	conductivity units	0
kCondT	Transverse thermal conductivity	conductivity units	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 Variables

None

Examples

These commands model carbon fibers as a transversely isotropic material with axial direction in the y direction

Material "carbon","Carbon Fiber","Transverse 2"
  EA 220000
  ET 20000
  GA 18000
  nuT 0.3
  nuA .2
  alphaA -.4
  alphaT 18
  rho 1.76
Done