<ANISOTROPIC_DAMAGE>#

Description#

These behavior objects are specific to the elastic and viscoplastic anisotropic damage models. They allow different types of damage variable to be used easily, and in combination. All the coefficients and functions entered here are local to the each damage object, and must therefore be input for all applicable objects even if the coefficients have the same meaning.

Syntax#

**damage [ type ] [ *use_e_bar ] [ *dont_use_e_bar ] [ *eta <COEFFICIENT> ] % default 1.0 [ *K_coeffs \(~V_{1}~V_2~V_3~V_4~V_5~V_6~V_7~V_8\) ] [ *g <G_FUNCTION> ]

where type may be from the following types:

CODE

DESCRIPTION

scalar

scalar variables with fixed direction

elastic_tensorial

tensorial variable

rate_tensorial

tensorial variable
*use_e_bar

Makes the calculation of \(Y\) or \(\bf Y\) use the \(\overline{\bf \varepsilon}\) tensor in place of \(\bf eel\).

*dont_use_e_bar

use \(\bf eel\) to calculate \(Y\) or \(\bf Y\) in place of \(\overline{\bf \varepsilon}\). This option is currently the default.

*K_coeffs

coefficients are entered to construct the tensor \(\bf K\) use the compliance matrix (\(\bf S = \tenf D_{el}^{-1}\)) and are as follows:

(322)#\[\begin{split}\bf H_0 = \left[ \begin{matrix} S_{11}V_1 & S_{12}V_4 & S_{13}V_5 & & & \\ & S_{22}V_2 & S_{23}V_6 & & & \\ & & S_{33}V_3 & & & \\ & & & S_{44}V_7 & & \\ & sym & & & S_{55}V_8 & \\ & & & & & S_{66}V_9 \\ \end{matrix} \right]\end{split}\]

note that the tensors are stored as

(323)#\[[S_1 \ldots S_6] = [ S_{11}~~S_{22}~~S_{33}~~S_{12}~~S_{23}~~S_{31} ]\]
(324)#\[\bf K = \tenf D_{el}:\bf H_0:\tenf D_{el}\]

The factors for \(V\) are entered in order after the *K_coeffs keyword is entered. See the behavior descriptions for examples.

*K

matrix form for the \(\bf K\). This input is entered as an ELASTICITY object.

Only one definition for the matrix \(\bf K^0_i\) may be given with the K_coeffs or K. The coefficients correspond directly to the associate terms marked with a subscript \(i\).

<ANISOTROPIC_DAMAGE> scalar#

Description#

Each scalar damage variable is defined using a “damage direction” input by the user, \(\vect{n}_i\), and an effective fourth order “damage effects” tensor \(\tenf K^0_i\). The modulus modification induced by a scalar damage variable \(i\) is the following:

(325)#\[\tenf C^{eff}_i = \delta_i (\tenf K^0_i - \eta_i h(-\bar{\epsilon}_i) \tenf N_i:\tenf K^0_i:\tenf N_i)\]

where \(\delta_i\) is the scalar internal damage variable. The above uses the following terms to measure the degree of opening strain:

(326)#\[\begin{split}\begin{aligned} &\bar{\epsilon}_i = \left(\vect{n}_i\otimes\vect{n}_i \right): (\ten \varepsilon - \ten \varepsilon_c) \\ &\bf N_i = \left(\vect{n}_i\otimes\vect{n}_i \right)\otimes \left(\vect{n}_i\otimes\vect{n}_i \right) \end{aligned}\end{split}\]

The material coefficient \(\eta\) is used to describe the influence of closing on the damage softening.

**damage scalar \(~\,~\,\) std options [ *n <VECTOR> ] % orientation

*n

the directions of the axes of damage for the isotropic variables entered as a vector.

Example#

Please see the examples in the anisotropic behaviors aniso_damage and visco_aniso_damage.

<ANISOTROPIC_DAMAGE> elastic_tensorial#

Description#

This model using a 2nd order tensorial damage variable is calculated as follows:

(327)#\[\tenf C^{eff}_j = \tenf D_j:\tenf K^0_j - \sum_{n=0}^3\eta_j h(-\bar{\epsilon}_n) \tenf{P}_n:(\tenf{D}_j:\bf K^0_j):\tenf{P}_n)\]

where the summation over \(n\) is over the three principal directions of the strain tensor, and \(\tenf D_j\) is a fourth order tensor constructed based on the second order damage variable \(\ten d_j\). This will use the following calculations based on the principal strain eigenvectors \(\vect{p}_n\):

(328)#\[\begin{split}\begin{aligned} &\bar{\epsilon}_n = \left(\vect{p}_n\otimes\vect{p}_n \right): (\ten \varepsilon - \ten \varepsilon_c) \\ &\tenf{P}_n = \left(\vect{p}_n\otimes\vect{p}_n \right)\otimes \left(\vect{p}_n\otimes\vect{p}_n \right) \end{aligned}\end{split}\]

and the fourth order tensors \(\tenf D_j\) are constructed their tensorial variables as:

(329)#\[\tenf D_j = (\alpha - \beta)(\ten 1\otimes\ten d_j)_s + \dfrac{\beta}{2} \left( (\ten 1\overline{\otimes}\ten d)_s + \ten 1\underline{\otimes}\ten d)_s \right)\]