DAMAGE#

Description#

This object class permits addition of damage mechanisms to a behavior assembly of type gen_evp. These models are of the type “continuum damage mechanics” (CDM), and thus provide interaction through alteration of the elasticity modulus with damage and calculation of plasticity with the use of an effective stress.

Damage models may be used in the gen_evp behavior with and without inelastic deformation potentials. The coupling with inelastic deformations and their hardening variables is discussed below.

Syntax#

The damage mechanisms are added through the use of a token **damage input after the **elasticity declaration. The general syntax is:

**damage dama_type [ *elastic ] [ *plastic ] [ *creep ] [ *coupling type ]

There is a restriction currently that the *damage statement must come after the *elasticity statement in the behavior.

The number of coefficients for each option depends of course on  on the particular damage model selected. The currently implemented types are summarized below:

CODE

DESCRIPTION

*elastic

elastic damage

*anisotropic_elastic

anisotropic elastic damage with scalar and tensorial variables

*plastic

plasticity damage which depends on the rate of inelastic deformation

*creep

time dependent damage

fatigue

cyclic damage
*elastic

The coefficients here are: B0 et alpha. The damage is calculated directly at a given time as:

(378)#\[\begin{split}\begin{aligned} &\bar{Y} = \frac{1}{2}\ten{\varepsilon}_{el}:\tenf D_{el}:\ten{\varepsilon}_{el} \\ &Y_{max}^{t+\Delta t} = \max(\bar{Y}, Y_{max}^t) \\ &d = \begin{array}{ll} \alpha~(Y_{max}^{1/2} - Y_o^{1/2}) & Y_{max}>Y_o \\ 0 & Y_{max}\leq Y_o \end{array} \\ \end{aligned}\end{split}\]
*plastic

This model integrates the damage as a function of the inelastic strain equivalent in the following manner:

(379)#\[\dot{d} = \dot{p}_i\left(Y/{\tt S0}\right)^{\tt s0}\]

For the \(i\)-th component of inelastic deformation. Note that \(\dot{p}\) is calculated as:

(380)#\[\dot{p}_i = \left[ \dfrac{2}{3}\dot{\boldsymbol{ \varepsilon}}_i:deps_i\right]^{1/2}\]
*creep

Classical viscoplastic damage using the Hayhurst stress function. The damage is calculated as:

(381)#\[\chi(\ten \sigma) = \alpha J_0 + \beta J_1 + (1-\alpha-\beta)J_2\]

The rate of damage production will be calculated as:

(382)#\[\dot{d} = \left<\frac{\chi(\ten \sigma)}{A}\right>^r \left(1 - d\right)^{-k}\]

with \(J_0\) the maximum principle stress, \(J_1\) the trace of the stress tensor, and \(J_2\) the second invariant of the deviator \((\frac{3}{2}\ten s:\ten s)^{1/2}\).

Coupling with plasticity#

Applying simply the damage mechanisms to a gen_evp plasticity model will only couple the stress calculation in the potentials and modify the elastic modulus. For the hardening mechanisms to be coupled to the damage rate, several additional changes must be made to the syntax.

Addition of one or more plasticity potentials to a behavior with damage causes the following equations to be used for the inelastic strain rate:

(383)#\[\begin{split}\begin{aligned} &\tilde{\ten \sigma} = \tenf D_{el}:\ten \varepsilon_{el} \\ &\ten \sigma = (1-d) \tilde{\ten \sigma}\\ &\dot{\boldsymbol{\varepsilon}}_{to} = \dot{\boldsymbol{\varepsilon}}_{el} + (1-d)^{-1}\dot{\lambda}\ten n \\ &\dot{\bf h} = \dot{\lambda}\bf m - \dot{\bf \omega} \\ \end{aligned}\end{split}\]

The default situation interfacing damage to the potential only involves use of the effective stress in the place of the actual (weakened) stress. This leads to the following set of evolution equations:

(384)#\[\begin{split}\begin{aligned} &f = f(\tilde{\ten \sigma}, \bf H, p) \\ &\dot{\lambda} = \dot{\lambda}(f,p) \\ &\ten n = \pder{f}{\ten \sigma} = \pder{f}{\tilde{\ten \sigma}}\pder{\tilde{\ten \sigma}}{\ten \sigma} \\ &\bf m = \pder{\phi(\tilde{\ten \sigma}, \bf H, p)}{\bf H} \\ \end{aligned}\end{split}\]

from which it is noted that the hardening forces \(\bf H\) are related to the internal (strain analogue) variables \(\bf h\) by the same linear relationship:

(385)#\[\bf H_{i} = \bf M_{ij}\bf h_j\]

Example#

***behavior gen_evp
 **elasticity isotropic
      young 260000.
      poisson 0.3
 **damage isotropic
  *creep
      alpha  0.75
      beta   0.0
      A      3000.
      r      5.3
      k      15.
  *coupling damage_hardening
 **potential gen_evp ev
  *var_coefs
  *flow alt_norton
      n      7.0
      K      2070.
      K2     1600.0
      K3     19.0
  *kinematic nonlinear
      C      inv_one_minus dv 15000.0
      D      300.0
  *kinematic nonlinear
      C      inv_one_minus dv 6000.0
      D      100.0
  *isotropic constant
      R0     130.
***return