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.