Transversely Isotropic Material
Constitutive Law
This anisotropic 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 is the compliance and stiffness tensors, [math]\displaystyle{ \vec\alpha }[/math] and [math]\displaystyle{ \vec\beta }[/math] are the thermal and moisture 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.
In a transversely isotropic 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.
Transverse 1
In this transversely isotropic material, the isotropic plane in the x-y plane, which is the plane for 2D analyses. The axial direction is along the z axis, which is in the thickness direction for 2D analyses. 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. The thermal and moisture 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.
Transverse 2
In this transversely isotropic material, the isotropic plane in the x-z plane. The axial direction is along the y axis, which is in the plane for 2D analyses. The stiffness and compliance tensors are:
[math]\displaystyle{ \mathbf{C}^{-1} = \mathbf{S} = \left(\begin{array}{cccccc} {1\over E_T} & -{\nu_A\over E_A} & -{\nu_T\over E_T} & 0 & 0 & 0 \\ -{\nu_A\over E_A} & {1\over E_A} & -{\nu_A\over E_A} & 0 & 0 & 0 \\ -{\nu_T\over E_T} & -{\nu_A\over E_A} & {1\over E_T} & 0 & 0 & 0 \\ 0 & 0 & 0 & {1\over G_A} & 0 & 0 \\ 0 & 0 & 0 & 0 & {1\over G_T} & 0 \\ 0 & 0 & 0 & 0 & 0 & {1\over G_A} \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. The thermal and moisture expansion tensors are
[math]\displaystyle{ \vec\alpha = (\alpha_T, \alpha_A,\alpha_T,0,0,0) }[/math]
[math]\displaystyle{ \vec\beta = (\beta_T, \beta_A,\beta_T,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.
Material Properties
Can rotate to any other orientation
The only reason two are needed is support all options in 2D calculations.
History Variables
None