/FAIL/HC

The failure strain is described by the Hosford-Coulomb function (refer to /FAIL/EMC for solids). Works only with elasto-plastic material number > 28. This failure criteria was developed by Keunhwan Pack (Massachusetts Institute of Technology MIT) and Dirk Mohr (Swiss Federal Institute of Technology ETH Zurich). ¹

Format

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
/FAIL/HC_DSSE/`mat_ID`/`unit_ID`
`I`_{fail_sh}	`P_thick`_fail		`I-Flag`

If I-Flag = 0; Card 2 - Damage accumulation parameters

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`a`		`b`		`c`		`d`		$n_{f}$ $n_{f}$

If I-Flag = 1; Card 2 - Failure test value input

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`c2`		`c3`		`c4`		`Inst_str`		$n_{f}$ $n_{f}$

Optional line

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`fail_ID`

Definitions

Field	Contents	SI Unit Example
`mat_ID`	Material identifier (Integer, maximum 10 digits)
`unit_ID`	Unit Identifier (Integer, maximum 10 digits)
`I`_{fail_sh}	Shell failure flag. = 1 (Default) Shell is deleted, if the damage criterion is satisfied in one integration point or layer. = 2 Shell is deleted, if the damage criterion is satisfied in all, or defined in `P_thick`_fail, shell layers. For a single layer shell, the stress tensor is set to zero. (Integer)
`P_thick`_fail	Percentage of through thickness integration points that must fail before the element is deleted. (shells only). Default = 1.0 (Real)
`I-Flag`	Input type flag. = 0 (Default) Parameter input using `a`, `b`, `c`, `d`, $n_{f}$ $n_{f}$ . = 1 Failure test input using `c1`, `c2`, `c3`, `c4` 1. (Integer)
`a`	Failure model parameter `a`. (Real)
`b`	Failure model parameter `b`. (Real)
`c`	Failure model parameter `c`. (Real)
`d`	Failure model parameter `d`. (Real)
`c2`	Failure strain in pure shear (triaxiality $η = 0$ $η = 0$ ). (Real)
`c3`	Failure strain in uniaxial tension (triaxiality $η = \frac{1}{3}$ $η = \frac{1}{3}$ ). (Real)
`c4`	Failure strain in plane strain tension (triaxiality = $η = \frac{1}{\sqrt{3}}$ $η = \frac{1}{\sqrt{3}}$ ). (Real)
`Inst_str`	Instability strain in plane strain tension for localized necking. (Real)
$n_{f}$ $n_{f}$	Parameter $n_{f}$ $n_{f}$ . Default = 0.1 (Real)
`fail_ID`	Failure criteria identifier. (Integer, maximum 10 digits)

Example

This example uses the HC-DSSE parameter input (a, b, c, d parameters with I-Flag = 0).

$\begin{matrix} I - F l a g = 0 a = 1.742 b = 0.7 c = 0.029 d = 1.6 \end{matrix}$ $\begin{array}{l} I - F l a g = 0 \\ a = 1.742 \\ b = 0.7 \\ c = 0.029 \\ d = 1.6 \end{array}$ ⇔ $\begin{matrix} I - F l a g = 1 c 2 = 0.8 \begin{matrix} \end{matrix} (P u r e - S h e a r) c 3 = 0.7 \begin{matrix} \end{matrix} (U n i a x i a l - T e n s i o n) c 4 = 0.58 \begin{matrix} \end{matrix} (P l a n e - S t r a i n - T e n s i o n) \end{matrix}$ $\begin{array}{l} I - F l a g = 1 \\ c 2 = 0.8 \begin{matrix} \end{matrix} (P u r e - S h e a r) \\ c 3 = 0.7 \begin{matrix} \end{matrix} (U n i a x i a l - T e n s i o n) \\ c 4 = 0.58 \begin{matrix} \end{matrix} (P l a n e - S t r a i n - T e n s i o n) \end{array}$

and one use physical input (failure strain input with I-Flag=1). HC-DSSE parameters a, b, c, d will be calculated using a curve fit by the Radioss Starter.

Failure definition in these two examples are equivalent.

Example (HC-DSSE Parameter Input)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
#              MUNIT               LUNIT               TUNIT
                  kg                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  1. MATERIALS:
/MAT/PLAS_TAB/2/1
DP600 from SSAB Homepage
#              RHO_I
              7.8E-6                   0
#                  E                  Nu           Eps_p_max               Eps_t               Eps_m
                 210                  .3                   0                   0                   0
#  N_funct  F_smooth              C_hard               F_cut               Eps_f
         1         0                   0                   0                   0
#  fct_IDp              Fscale   Fct_IDE                EInf                  CE
         0                   0         0                   0                   0
# func_ID1  func_ID2  func_ID3  func_ID4  func_ID5
        14
#           Fscale_1            Fscale_2            Fscale_3            Fscale_4            Fscale_5
                   1
#          Eps_dot_1           Eps_dot_2           Eps_dot_3           Eps_dot_4           Eps_dot_5
                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FAIL/HC_DSSE/2/1
#   Ishell        P_thickfail     I_Flag
         1                 .5          0
#                  a                   b                   c                   d                 n_f
               1.742                 0.7               0.029                 1.6                 0.1
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/14
Mat_Curev Quasi-static DOCOL DP 600 (Material from SSAB Homepage 2010)
#                  X                   Y
                   0                .306                                                            
              .00112                .415                                                            
              .00218                .445                                                            
                .003                .461                                                            
              .00404                .474                                                            
              .00517                .489                                                            
              .00613                .498                                                            
               .0071                .505                                                            
              .00806                .512                                                            
              .00901                .522                                                            
               .0102                 .53                                                            
               .0121                .543                                                            
                .013                 .55                                                            
                .014                .555                                                            
                .015                .561                                                            
               .0159                .567                                                            
               .0171                .572                                                            
               .0181                .577                                                            
               .0204                .592                                                            
               .0303                .632                                                            
               .0405                .663                                                            
               .0502                .687                                                            
                 .06                .706                                                            
               .0702                .722                                                            
               .0807                .737                                                            
                 .09                .749                                                            
               .0997                .758                                                            
                .101                .759                                                            
                 .11                .768                                                            
           .15000001                .805                                                            
                  .2                 .84                                                            
           .30000001                  .9                                                            
                  .5                   1                                                            
                   1                1.21                                                            
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Example (Physical Input)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
#              MUNIT               LUNIT               TUNIT
                  kg                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  1. MATERIALS:
/MAT/PLAS_TAB/2/1
DP600 from SSAB Homepage
#              RHO_I
              7.8E-6                   0
#                  E                  Nu           Eps_p_max               Eps_t               Eps_m
                 210                  .3                   0                   0                   0
#  N_funct  F_smooth              C_hard               F_cut               Eps_f
         1         0                   0                   0                   0
#  fct_IDp              Fscale   Fct_IDE                EInf                  CE
         0                   0         0                   0                   0
# func_ID1  func_ID2  func_ID3  func_ID4  func_ID5
        14
#           Fscale_1            Fscale_2            Fscale_3            Fscale_4            Fscale_5
                   1
#          Eps_dot_1           Eps_dot_2           Eps_dot_3           Eps_dot_4           Eps_dot_5
                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FAIL/HC_DSSE/2/1
#   Ishell        P_thickfail     I_Flag
         1                 .5          1
#                 c2                  c3                  c4            Inst_str                 n_f
                 0.8                 0.7                0.58                 0.1                 0.1    
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/14
Mat_Curev Quasi-static DOCOL DP 600 (Material from SSAB Homepage 2010)
#                  X                   Y
                   0                .306                                                            
              .00112                .415                                                            
              .00218                .445                                                            
                .003                .461                                                            
              .00404                .474                                                            
              .00517                .489                                                            
              .00613                .498                                                            
               .0071                .505                                                            
              .00806                .512                                                            
              .00901                .522                                                            
               .0102                 .53                                                            
               .0121                .543                                                            
                .013                 .55                                                            
                .014                .555                                                            
                .015                .561                                                            
               .0159                .567                                                            
               .0171                .572                                                            
               .0181                .577                                                            
               .0204                .592                                                            
               .0303                .632                                                            
               .0405                .663                                                            
               .0502                .687                                                            
                 .06                .706                                                            
               .0702                .722                                                            
               .0807                .737                                                            
                 .09                .749                                                            
               .0997                .758                                                            
                .101                .759                                                            
                 .11                .768                                                            
           .15000001                .805                                                            
                  .2                 .84                                                            
           .30000001                  .9                                                            
                  .5                   1                                                            
                   1                1.21                                                            
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

This failure criterion is defined in the space of equivalent plastic strain at failure versus stress triaxiality (state of stress) for plane stress condition. This allows for definition of failure strain that varies with different loading conditions. It consists of two curves. One is the fracture locus (that is, below the red curve) and the other is the localized necking locus (blue curve).

Figure 2.

The former takes the following form:(1)
${¯ ε}_{H C}^{p r} (η, θ) = b {(1 + c)}^{\frac{1}{n_{f}}} {{[\frac{1}{2} ({(f_{1} - f_{2})}^{a} + {(f_{2} - f_{3})}^{a} + {(f_{1} - f_{3})}^{a})]}^{\frac{1}{a}} + c (2 η + f_{1} + f_{3})}^{- \frac{1}{n_{f}}}$ ${\bar{ε}}_{H C}^{p r} (η, θ) = b {(1 + c)}^{\frac{1}{n_{f}}} {{[\frac{1}{2} ({(f_{1} - f_{2})}^{a} + {(f_{2} - f_{3})}^{a} + {(f_{1} - f_{3})}^{a})]}^{\frac{1}{a}} + c (2 η + f_{1} + f_{3})}^{- \frac{1}{n_{f}}}$

$f_{1}$ $f_{1}$ , $f_{2}$ $f_{2}$ and $f_{3}$ $f_{3}$ are functions of the Lode angle $θ$ $θ$ :

$w i t h ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} f_{1} (θ) = \frac{2}{3} cos (\frac{π}{6} (1 - θ)) f_{2} (θ) = \frac{2}{3} cos (\frac{π}{6} (3 + θ)) f_{3} (θ) = - \frac{2}{3} cos (\frac{π}{6} (1 + θ)) \end{matrix}$ $w i t h {\begin{matrix} f_{1} (θ) = \frac{2}{3} \cos (\frac{π}{6} (1 - θ)) \\ f_{2} (θ) = \frac{2}{3} \cos (\frac{π}{6} (3 + θ)) \\ f_{3} (θ) = - \frac{2}{3} \cos (\frac{π}{6} (1 + θ)) \end{matrix}$

$θ = 1 - \frac{2}{π} a r c cos (- \frac{27}{2} η (η^{2} - \frac{1}{3}))$ $θ = 1 - \frac{2}{π} a r c \cos (- \frac{27}{2} η (η^{2} - \frac{1}{3}))$

$d D_{H C} = \frac{d {¯ ε}_{p}}{{¯ ε}_{H C}^{p r} (η)}$ $d D_{H C} = \frac{d {\bar{ε}}_{p}}{{\bar{ε}}_{H C}^{p r} (η)}$

The latter is defined only between uniaxial and eqi-biaxial tension conditions for ( $\frac{1}{3} < η < \frac{2}{3}$ $\frac{1}{3} < η < \frac{2}{3}$ ) in which localized necking can occur.(2)
${¯ ε}_{D S S E}^{p r} (η) = b ⎧ ⎪ ⎨ ⎪ ⎩ {[\frac{1}{2} ({(g_{1} - g_{2})}^{d} + g_{1}^{d} + g_{2}^{d})]}^{\frac{1}{d}}}^{- \frac{1}{p}}$ ${\bar{ε}}_{D S S E}^{p r} (η) = b {{[\frac{1}{2} ({(g_{1} - g_{2})}^{d} + g_{1}^{d} + g_{2}^{d})]}^{\frac{1}{d}}}^{- \frac{1}{p}}$
$w i t h ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} g_{1} (η) = \frac{2}{3} η + \sqrt{\frac{1}{3} - \frac{3}{4} η^{2}} g_{2} (η) = \frac{2}{3} η - \sqrt{\frac{1}{3} - \frac{3}{4} η^{2}} \end{matrix}$ $w i t h {\begin{matrix} g_{1} (η) = \frac{2}{3} η + \sqrt{\frac{1}{3} - \frac{3}{4} η^{2}} \\ g_{2} (η) = \frac{2}{3} η - \sqrt{\frac{1}{3} - \frac{3}{4} η^{2}} \end{matrix}$ (3)
$d D_{D S S E} = \frac{d {¯ ε}_{p}}{{¯ ε}_{D S S E}^{p r} (η)}$ $d D_{D S S E} = \frac{d {\bar{ε}}_{p}}{{\bar{ε}}_{D S S E}^{p r} (η)}$
- The universal exponent of $p$ $p$ =0.01 is used.
- Localized necking is predicted in a non-local manner when all integration points or layers have $d D_{D S S E} > 1$ $d D_{D S S E} > 1$ > 1.
- The failure curve parameters a, b, c, and d, must be calibrated based on test results.
  $η$ $η$ is the triaxiality.
- Alternatively, switching the I-Flag to 1, allows Radioss to do the parameter fit internally.
- No shell fracture in compression area ( $η < - \frac{1}{3}$ $η < - \frac{1}{3}$ ).
- Damage is calculated as:(4)
  $D = \sum \frac{Δ {¯ ε}_{p}}{{¯ ε}_{H C}^{p r} (η)}$ $D = \sum \frac{Δ {\bar{ε}}_{p}}{{\bar{ε}}_{H C}^{p r} (η)}$