Difference between revisions of "Transversely Isotropic Viscoelastic Material"
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The stress (σ) and strain (ε) are related by: | The stress (σ) and strain (ε) are related by: | ||
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<math>\sigma(t) = \mathbf{C}(t) * \varepsilon(t)</math> | <math>\sigma(t) = \mathbf{C}(t) * \varepsilon(t)</math> | ||
Here <math>*</math> indicates convolution (or Boltzman's superposition) between time-dependent stiffness tensor (<math>\mathbf{C}(t)</math>) and strain tensor. | Here <math>*</math> indicates convolution (or Boltzman's superposition) between time-dependent stiffness tensor (<math>\mathbf{C}(t)</math>) and strain tensor. In Voight-notation with unique axis in the ''z'' direction, the time-dependent stiffness tensor is | ||
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<math>\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 & 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 | |||
Revision as of 15:46, 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 & 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