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 no 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]

Material Properties

The unusual task for this material is to use multiple Gk and tauk properties (all with the same property name) to enter a material with multiple relaxation times.

Property	Description	Units	Default
GT0 (or G0)	The long term (or fully-relaxed) transverse shear modulus (can be entered as G0 to be compatible with isotropic viscoelastic material properties)	pressure units	0
GA0	The long term (or fully-relaxed) axial shear modulus	pressure units	0
KT0	The long term (or fully-relaxed) plane-strain bulk modulus	pressure units	0
en0	The long term (or fully-relaxed) C11 element of the stiffness tensor	pressure units	0
ell0	The long term (or fully-relaxed) C12=C13 elements of the stiffness tensor	pressure units	0
ntaus	The number of relaxation times of the previous long-term property that was entered. This property is only needed in XML files and must come before any subsequent Pk or tauk properties. In scripted files, the number is automatically determined from the number of relaxation times you provide.	none	none
Pk	The next property in series for the previous long-term property that was entered. Use multiple Pk for each term on the series.	pressure units	none
tauk	The next relaxation time. Enter multiple tauk properties for each term on the series.	time units	none
fibrous	Enter 1 to indicate fiber direction is independent of time (or [math]\displaystyle{ E_A }[/math] and [math]\displaystyle{ \nu_A }[/math] are independent of time). Enter 0 to allow [math]\displaystyle{ n(t) }[/math] and [math]\displaystyle{ \ell(t) }[/math] to be entered and to depend on time.	dimensionless	1
(other)	Properties common to all materials	varies	varies