Transversely Isotropic Viscoelastic Material

Constitutive Law

(This material is available only in OSParticulas because it is still in development)

This anisotropic MPM material is a small strain, linear viscoelastic material that extends the Viscoelastic Material to model anisotropic viscoelasticity.

The stress (σ) and strain (ε) are related by:

      [math]\displaystyle{ \sigma(t) = \mathbf{C}(t) * \varepsilon(t) }[/math]

Here [math]\displaystyle{ * }[/math] indicates convolution (or Boltzman's superposition) between time-dependent stiffness tensor ([math]\displaystyle{ \mathbf{C}(t) }[/math]) and strain tensor. In Voight-notation with unique axis in the z direction, the time-dependent stiffness tensor is

      [math]\displaystyle{ \mathbf{C}(t) = \left[\begin{array}{cccccc} K_T(t)+G_T(t) & K_T(t)-G_T(t) & \ell(t) & 0 & 0 & 0 \\ K_T(t)-G_T(t) & K_T+G_T & \ell(t) & 0 & 0 & 0 \\ \ell(t) & \ell(t) & n(t) & 0 & 0 & 0 \\ 0 & 0 & 0 & G_A(t) & 0 & 0 \\ 0 & 0 & 0 & 0 & G_A(t) & 0 \\ 0 & 0 & 0 & 0 & 0 & G_T(t) \end{array}\right] }[/math]

Here [math]\displaystyle{ K_T(t) }[/math] is the plane strain, bulk modulus, [math]\displaystyle{ G_T(t) }[/math] is the transverse shear modulus, [math]\displaystyle{ G_A(t) }[/math] is the axial shear modulus, and [math]\displaystyle{ n(t) }[/math] and [math]\displaystyle{ \ell(t) }[/math] give time-dependence of the C₁₁ and C₁₂=C₁₃ elements of the stiffness matrix (as defined here). The time dependence of each property is modeled with a sum of exponentials:

      [math]\displaystyle{ K_T(t) = K_{T0} + \sum_{k=1}^{N_{KT}} K_{Tk} e^{-t/\tau_{KT,k}} }[/math]

      [math]\displaystyle{ G_T(t) = G_{T0} + \sum_{k=1}^{N_{GT}} G_{Tk} e^{-t/\tau_{GT,k}} }[/math]

      [math]\displaystyle{ G_A(t) = G_{A0} + \sum_{k=1}^{N_{GA}} G_{Ak} e^{-t/\tau_{GA,k}} }[/math]

      [math]\displaystyle{ n(t) = n_0 + \sum_{k=1}^{N_n} n_k e^{-t/\tau_{n,k}} }[/math]

      [math]\displaystyle{ \ell(t) = \ell_0 + \sum_{k=1}^{N_\ell} \ell_k e^{-t/\tau_{\ell,k}} }[/math]

Elastic Fiber Direction

Some materials, such as unidirectional composite materials or wood, can be modeled with the fiber direction as the axial direction. Because this direction is typically much stiffer than the isotropic plane direction, it might be expected to have little or now viscoelasticity compared to other directions. If one assumes that axial modulus ([math]\displaystyle{ E_A }[/math]) and Poisson's ratio ([math]\displaystyle{ \nu_A }[/math]) re independent of time, then [math]\displaystyle{ n(t) }[/math] and [math]\displaystyle{ \ell(t) }[/math] are determined by [math]\displaystyle{ K_T(t) }[/math] by setting:

      [math]\displaystyle{ N_n=N_\ell=N_{KT}, \quad \tau_{n,k}=\tau_{\ell,k}=\tau_{KT,k}, \quad n_0 = E_A + 4K_{T0}\nu_A^2, \quad n_k = 4K_{Tk}\nu_A^2, \quad \ell_0 = 2K_{T0}\nu_A, \quad \ell_k = 2K_{Tk}\nu_A }[/math]