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 C33 and C13=C23 elements of the stiffness tensor (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]
TIViscoelastic 1 and 2
TIViscoelastic 1 and TIViscoelastic 2 give identical materials but with different initial orientations. TIViscoelastic 1 has the unrotated axial direction along the z (or θ if axisymmetric) axis (see above tensors) while TIViscoelastic 2 has unrotated axial direction along the y (or Z if axisymmetric) axis (exchange y and z directions in above tensors). You can change the unrotated direction to any other orientation when defining material points by selecting rotation angles. For 2D analyses, the two options are needed to allo for axial direction in the x-y (or R-Z if axisymmetric) analysis plane (TIViscoelastic 2) or normal to that plane (TIViscoelastic 1). For 3D analyses, only TIViscoelastic 1 is allowed (and it in the only one needed).
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 terms to define the exponential series used for up to five material properties.
| 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 |
| TI Properties | Enter thermal expansion, solvent expansion, and poroelasticity properties as usual for transversely isotropic materials. When fibrous is 1, also enter EA and nuA. | varies | varies |
| (other) | Properties common to all materials | varies | varies |
The material properties need to define time dependence of 3 properties (when fibrous is 1) or 5 properties (when fibrous is 0). The process for each one is to enter the long-term value first (GT0, A0, KT0, en0, ell0) and to follow each one by ntaus (only needed in XML files) and by one Pk and tauk for each term in the series.
When fibrous is 1, you enter EA and nuA. When fibrous is 0, their initial values are:
[math]\displaystyle{ \nu_A(0) = {\ell(0)\over 2K_T(0)} \qquad {\rm and} \qquad E_A(0) = n(0) - 4K_T(0)\nu_A^2 = n(0) - \frac{\ell(0)^2}{K_T(0)} }[/math]
The remaining, initial transverse properties are:
[math]\displaystyle{ \frac{1}{E_T(0)} = \frac{1}{2K_T(0)} +\frac{1}{4G_T(0)} + {\nu_A(0)^2\over E_A(0)} \qquad {\rm and} \qquad \nu_T(0) = \frac{E_T(0)}{2G_T(0)}-1 }[/math]
The initial values, when all exponentials in the series are 1, is sum of all terms fro that property. For example
[math]\displaystyle{ G_T(0) = G_{T0} + \sum_{k=1}^{N_{GT}} G_{Tk} }[/math]
For valid modeling, the initial Poisson's ratios must satisfy
[math]\displaystyle{ -1\lt \nu_T(0)\lt 1 \qquad -\sqrt{E_A(0)\over E_T(0)} \lt \nu_A(0) \lt \sqrt{E_A(0)\over E_T(0)} \qquad {E_T(0)\nu_A(0)^2\over E_A(0)} \lt {1-\nu_T(0)\over 2} }[/math]
The relations must apply for all time, but on the initial values are validate before starting a simulation.
History Variables
This material tracks internal history variables (one for each relaxation time and each component of stress) for implementation of linear viscoelastic properties, but currently none of these internal variables are available for archiving.