Difference between revisions of "Transversely Isotropic Viscoelastic Material"
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\end{array}\right]</math> | \end{array}\right]</math> | ||
Here <math>K_T(t)</math> is the plane strain, bulk modulus, <math>G_T(t)</math> is the transverse shear modulus, <math>G_A(t)</math> is the axial shear modulus, and <math>n(t)</math> and <math>\ell(t)</math> give time-dependence of the ''C<sub>11</sub>'' and ''C<sub>12</sub>=C<sub>13</sub>'' elements of the stiffness matrix (as [[Transversely Isotropic Material#Transverse 1|defined here]]). | Here <math>K_T(t)</math> is the plane strain, bulk modulus, <math>G_T(t)</math> is the transverse shear modulus, <math>G_A(t)</math> is the axial shear modulus, and <math>n(t)</math> and <math>\ell(t)</math> give time-dependence of the ''C<sub>11</sub>'' and ''C<sub>12</sub>=C<sub>13</sub>'' elements of the stiffness matrix (as [[Transversely Isotropic Material#Transverse 1|defined here]]). The time dependence of each property is modeled with a sum of exponentials: | ||
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<math>K_T(t) = K_{T0} + \sum_{k=1}^{N_{KT}} K_{Tk} e^{-t/\tau_{KT,k}}</math> | |||
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<math>G_T(t) = G_{T0} + \sum_{k=1}^{N_{GT}} G_{Tk} e^{-t/\tau_{GT,k}}</math> | |||
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<math>G_A(t) = G_{A0} + \sum_{k=1}^{N_{GA}} G_{Ak} e^{-t/\tau_{GA,k}}</math> | |||
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<math>n(t) = n_0 + \sum_{k=1}^{N_n} n_k e^{-t/\tau_{n,k}}</math> | |||
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<math>\ell(t) = \ell_0 + \sum_{k=1}^{N_\ell} \ell_k e^{-t/\tau_{\ell,k}}</math> | |||
Revision as of 16:42, 7 January 2021
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]