becker_needleman

Description

This model is a direct implementation of the viscoplastic porous damage model given by R. Becker and A. Needleman “Effect of Yield Surface Curvature on Necking and Failure in Porous Plastic Solids,” J. Appl. Mech. v53, 491-498 (1986).

(259)\[\ten B = \ten \sigma - \ten X \qquad \qquad \ten B' = \tenf U:\ten B\]

(260)\[\phi = \frac{3}{2}\frac{\ten B':\ten B'}{\sigma_F^2} + 2q_1f^*\text{cosh}\left[\frac{q_2\ten B:\ten 1}{2\sigma_F}\right] - 1 - q_1 f^{*^2} \qquad \ten n=\pder{\phi}{\ten \sigma}\]

Where \(f^*\) is a modification of the porosity \(f\) ^{1 1}In the porous_plastic behavior \(f^*\) is calculated from \(f\) using the coefficient mechanism so that if \(f<f_c\) \(f^*=f\) and otherwise \(f^* = f_c + ((1/q_1-f_c)/(f_f-f_c))(f-f_c)\)

(261)\[\Delta\lambda = \frac{(1-f)\sigma_F\Delta p}{\ten B:\ten n}\]

(262)\[\rho = \frac{(1-b)}{1-b+b g(p)}\left[\ten B:\ten n\right]^{-1} \left[ \frac{g'(p)(\ten B:\ten n)^2}{(1-f)\sigma_F} + (1-f)(1-g(p))\pder{\phi}{f}(\ten 1:\ten n) \right]\]

Rate equations

(263)\[\dot{\ten \varepsilon}_{el} = \ten \varepsilon - \dot{\lambda}\ten n - \ten \varepsilon_{th}\]

(264)\[\dot{p} = \epsilon_0 \left[\frac{\sigma_F}{(1-b)\sigma_o + b\sigma_o g(p)}\right]^{1/m}\]

(265)\[\dot{f} = \dot{\lambda}(1-f)(\ten 1:\ten n)\]

(266)\[\dot{\ten \alpha} = \dot{\lambda}\rho\ten B\]

Syntax

***behavior becker_needleman modifier \(~\,\) **elasticity <ELASTICITY> [ **thermal_strain <THERMAL_STRAIN> \(~\,\) **model_coef \(~\,~\,~\,\) coefs

q1, q2, f_c, f_f, e_dot0, b, sig0, m, eps0, C

Example

Here is an example using coefficients from the original paper.

***behavior becker_needleman
 **elasticity isotropic
      young   1865.67
      poisson 0.3
 **model_coef
      q1      2.38
      q2      0.748
      eps0    0.0125
      e_dot0  1.e-3  % same as loading rate
      sig0    1.0
      b       1.0    % 1==isotropic, 0==kinematic
      f_c     0.15
      f_f     0.25
      m       0.02 % 0.002
      C       1.
***return