MATX42

Format

Example

Definitions

Comments

1. The material identification number must be that of an existing MAT1 Bulk Data Entry. Only one MATXi material extension can be associated with a particular MAT1.
2. MATX42 is only applied in geometric nonlinear analysis subcases which are defined by ANALYSIS = EXPDYN. It is ignored for all other subcases.
3. The recommended Poisson's ratio for incompressible material is NU = 0.495. NU is defined on the corresponding MAT1.
4. The strain energy density $W$ is computed using:
   (1)

   $$W=\sum _p\frac{{\mu }_p}{{\alpha }_p}\left({\overline{\lambda }}_1^{{\alpha }_p}+{\overline{\lambda }}_2^{{\alpha }_p}+{\overline{\lambda }}_3^{{\alpha }_p}-3\right)+\frac{K}{2}{\left(J-1\right)}^2$$

   Where, ${\overline{\lambda }}_i$ is the i^th principal engineering stretch ( ${\overline{\lambda }}_i=1+{\epsilon }_i$ , ${\epsilon }_i$ is the i^th principal engineering strain). The Cauchy stress is computed as:

   (2)

   $${\sigma }_{\text{i}}=\frac{{\lambda }_{\text{i}}}{\text{J}}\cdot \frac{\partial \text{W}}{\partial {\lambda }_{\text{i}}}-\text{P}$$

   with J = $\lambda$ ₁ * $\lambda$ ₂ * $\lambda$ ₃ being the relative volume:

   (3)

   $$\text{J}=\frac{{\rho }_0}{\rho }$$

   The quantity, P is the pressure:

   P = K * FBULK * f (J) * (J - 1)

   The Bulk Modulus, $\mathrm{K}$ is:(4)

   $$\text{K}=\mu \cdot \frac{2(1+\upsilon )}{3\left(1-2\upsilon \right)}$$

   with the ground shear modulus $\mu$ :

5. An incompressible Mooney-Rivlin material is governed by:

   W = C₁₀ (I₁ - 3) + C₀₁ (I₂ - 3)

   Where, I_i is i^th invariant of the right-hand Cauchy-Green Tensor. It can be modeled using the following parameters:

   $\mu$ ₁ = 2 * C₁₀

   $\mu$ ₂ = -2 * C₀₁

   α₁ = 2.0

   α₂ = -2.0

6. Coefficients of the Prony series (G_i, T_i) are used to describe viscous effects using the Maxwell model (which can be described in a simplified manner as a system of n springs with stiffness' G_i and dampers $\eta$ _i):

   
   
   Figure 1.

   The hyperelastic ground shear modulus, $\mu$ is exactly the long-term shear modulus G∞ in the Maxwell model, and T_i is the relaxation time:(5)

   $$\mathit{Ti}=\frac{\eta i}{\mathit{Gi}}$$

   The G_i and T_i values must be positive.

7. Viscous effects are modeled through a convolution integral using Prony series. This is an extension of small strain theory (described in comment 6) to large nonlinear strain case. The Kirchhoff viscous stress is given by:
   (6)

   $${\tau }^v={\sum _{i=1}^M{G}_i\underset{0}{\overset{t}{\int }}e}^{-\frac{t-s}{{\tau }_i}}\frac{d}{ds}\left[dev\left(\overline{F}{\overline{F}}^T\right)\right]ds$$

   with, F being the deformation gradient matrix, $\overline{\text{F}}={\text{J}}^{-\frac{1}{3}}\mathrm{F}$ and $\text{dev}\left(\overline{\mathrm{F}}{\overline{\mathrm{F}}}^{\text{T}}\right)$ denotes the deviatoric part of tensor $\overline{\mathrm{F}}{\overline{\mathrm{F}}}^{\mathrm{T}}$ .

   The viscous-Cauchy stress is written as:(7)

   $${\sigma }^v(t)=\frac{1}{J}{\tau }^v(t)$$
8. This card is represented as a material in HyperMesh.