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 for for MPM only.

History Data

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