Transversely Isotropic Viscoelastic Material

Constitutive Law

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(t)+G_T(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}} \qquad G_T(t) = G_{T0} + \sum_{k=1}^{N_{GT}} G_{Tk} e^{-t/\tau_{GT,k}} \qquad 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}} \qquad\qquad \ell(t) = \ell_0 + \sum_{k=1}^{N_\ell} \ell_k e^{-t/\tau_{\ell,k}} }[/math]

In terms of axial modulus [math]\displaystyle{ E_A }[/math], and Poisson's ratio, [math]\displaystyle{ \nu_A }[/math], we can write:      

      [math]\displaystyle{ n(t) = E_A(t) + 4K_T(t)\nu_A(t)^2 \qquad\qquad \ell(t) = 4K_T(t)\nu_A(t) }[/math]

This material lumps all these time dependencies into [math]\displaystyle{ n(t) }[/math] and [math]\displaystyle{ \ell(t) }[/math], but note that selection of those properties will determine time dependencies of [math]\displaystyle{ E_A(t) }[/math] and [math]\displaystyle{ \nu_A(t) }[/math]

      [math]\displaystyle{ \nu_A(t) = {\ell(t)\over 2K_T(t)} \qquad {\rm and} \qquad E_A(t) = n(t) - \frac{\ell(t)^2}{K_T(t)} }[/math]

Rotated Material Axes

The initial axial direction is along the z axis (or θ axis for axisymmetric calculations). The axial direction can be changed to any other direction using the same method used to orient transversely isotropic elastic material with the swapz material property.

Isotropic with Time-Dependent Bulk Modulus

The available isotropic viscoelastic material is limited to materials with time-independent bulk modulus because that is a good approximation for most isotropic, viscoelastic materials. These transversely-isotropic materials, however, do not place any restrictions on which properties are time dependent. As result, it can model an isotropic material with a time-dependent bulk modulus as a special case. Imagine an isotropic material with K(t) and G(t) as time-dependent bulk and shear moduli, respectively. To model using a transversely isotropic material, choose G_A(t) = G_T(t) = G(t) along with:

      [math]\displaystyle{ K_T(t) = K(t) + \frac{G_T(t)}{3},\quad n(t)= K(t) + \frac{4G_T(t)}{3},\quad {\rm and}\quad \ell(t) = K(t) - \frac{2G_T(t)}{3} }[/math]

Finally, any other material properties (such as thermal expansion coefficients) should be set to the special cases for an isotropic material.

Fibrous Materials

A potential use for a transversely isotropic viscoelastic material is to model fiber-reinforced composites including wood (although transversely isotropic is not a great model for wood because it cannot represent low transverse shear modulus compared to radial and tangential transverse tensile moduli). If the axial direction is the fiber direction, one might expect that time dependence will be much weaker in the axial direction than in the transverse direction. This behavior could be approximated by setting [math]\displaystyle{ n }[/math] and [math]\displaystyle{ \ell }[/math] independent of time. Because an elastic material has [math]\displaystyle{ n=E_A+4K_T\nu_A^2 }[/math] and [math]\displaystyle{ \ell = 2K_t\nu_A }[/math], this approximation implies [math]\displaystyle{ E_A }[/math], [math]\displaystyle{ K_T }[/math], and [math]\displaystyle{ \nu_A }[/math] are all independent of time. The final result is that only [math]\displaystyle{ G_T(t) }[/math] and [math]\displaystyle{ G_A(t) }[/math] are time dependent. This model is one approximation, but would have zero creep in the axial direction and therefore no use in modeling that response (unless experiments also show zero creep).

A second option might be to let [math]\displaystyle{ K_T(t) }[/math] depend on time while [math]\displaystyle{ E_A }[/math] and [math]\displaystyle{ \nu_A }[/math] remain independent of time. This approach does not work and results in non-physical response to axial loading. The problem appears that it predicts that [math]\displaystyle{ n(t) }[/math] decreases in time. For isotropic materials, [math]\displaystyle{ n(t) }[/math] increases in time and it also likely increases for fibrous materials as well. In brief, to model viscoelastic properties of fibrous material that might include effects in the axial direction, the modeling will need to input all five time-dependent properties allowed by this material. Those properties should be based on experimental observations.

Effective Time Implementation

This material handles variations in temperature and solvent concentration by the same methods used for isotropic viscoelastic materials.

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	The long term (or fully-relaxed) transverse shear modulus	pressure units	none
GA0	The long term (or fully-relaxed) axial shear modulus	pressure units	none
KT0	The long term (or fully-relaxed) plane-strain bulk modulus	pressure units	none
en0	The long term (or fully-relaxed) C33 element of the stiffness tensor	pressure units	none
ell0	The long term (or fully-relaxed) C13=C23 elements of the stiffness tensor	pressure units	none
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 the series for the previous long-term property that was entered. Use multiple Pk values for each term on the series.	pressure units	none
tauk	The next relaxation time in the series for the previous long-term property that was entered. Enter multiple tauk values for each term on the series.	time units	none
Tref	Reference temperature to shift relaxation times using the WLF equation. If Tref<0, then no shifting is done and relaxation times will be independent of temperature.	K	-1
C1	Coefficient in WLF equation used when Tref≥0 to shift relaxation times. If the entered value is negative, the thermal shift switches to Arrhenious activation energy with ΔHa = -RC1,	none	17.44
C2	Coefficient in WLF equation used when Tref≥0 to shift relaxation times (not used if C1 is negative).	none	51.6
mref	Reference concentration to shift relaxation times using the WLF-style equation. If mref<0, then no shifting is done and relaxation times will be independent of concentration.	K	-1
Cm1	Coefficient in WLF-style equation used when mref≥0 to shift relaxation times	none	10
Cm2	Coefficient in WLF-style equation used when mref≥0 to shift relaxation times (must be positive)	none	0.0625
TI Properties	Enter thermal expansion, solvent expansion, and poroelasticity properties as usual for transversely isotropic materials.	varies	varies
swapz	Set to >0 to move axial direction from z axis to the y axis (or from θ axis to Z axis in axisymmetric calculations). This property is only needed for 2D simulations that want axial direction in the analysis plane (it is not allowed in 3D MPM simulations).	none	0
(other)	Properties common to all materials	varies	varies

The material properties need to define the time dependence 5 properties. The process for each one is to enter the long-term value first (GT0, GA0, KT0, en0, ell0) and then to follow each one by ntaus (only needed in XML files) and by one Pk and tauk value for each term in the series. Elastic value, or the property values at time zero, are sums of all Pk terms for that property. For example:

      [math]\displaystyle{ G_T(0) = G_{T0} + \sum_{k=1}^{N_{GT}} G_{Tk} }[/math]

Other elastic properties are given by

      [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]

      [math]\displaystyle{ \frac{1}{E_T(0)} = \frac{1}{4K_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]

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]

These relations must apply for all time, but only the initial values are validated before starting a simulation.

Deprecated Material Properties

Prior to the swapz material property, there were two types on transversely isotropic viscoelastic materials named "TIViscoelastic 1" and "TIViscoelastic 2". Although these can still be used as the material type, they are deprecated. The prior "TIViscoelastic 1" is identical to this material with swapz=0. The prior "TIViscoelastic 2" material is identical to this material with swapz=1.

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.