/FAIL/TSAIWU

Block Format Keyword Tsai-Wu failure criterion for composite materials failure modeling. This criterion is available for solids and shells.

Format

(1)	(3)	(5)	(7)	(9)	(10)
/FAIL/TSAIWU/`mat_ID`/`unit_ID`
$σ_{1}^{t}$ $σ_{1}^{t}$	$σ_{2}^{t}$ $σ_{2}^{t}$	$σ_{1}^{c}$ $σ_{1}^{c}$	$σ_{2}^{c}$ $σ_{2}^{c}$	${¯ σ}_{12}$ ${\bar{σ}}_{12}$
$α$ $α$	$τ_{max}$ $τ_{\max}$	`F`_cut		`I`_{fail_sh}	`I`_{fail_so}

Optional Line:

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

Definition

Field	Contents	SI Unit Example
`mat_ID`	Material identifier. (Integer, maximum 10 digits)
`unit_ID`	(Optional) Unit identifier. (Integer, maximum 10 digits)
$σ_{1}^{t}$ $σ_{1}^{t}$	Longitudinal tensile strength. Default = 10²⁰ (Real)	$[Pa]$ $[Pa]$
$σ_{2}^{t}$ $σ_{2}^{t}$	Transverse tensile strength. Default = 10²⁰ (Real)	$[Pa]$ $[Pa]$
$σ_{1}^{c}$ $σ_{1}^{c}$	Longitudinal compressive strength. Default = 10²⁰ (Real)	$[Pa]$ $[Pa]$
$σ_{2}^{c}$ $σ_{2}^{c}$	Transverse compressive strength. Default = 10²⁰ (Real)	$[Pa]$ $[Pa]$
${¯ σ}_{12}$ ${\bar{σ}}_{12}$	Shear strength. Default = 10²⁰ (Real)	$[Pa]$ $[Pa]$
$α$ $α$	Interaction parameter 12. Default = 0.0 (Real)
$τ_{max}$ $τ_{\max}$	Dynamic time relaxation. 5 Default = 10²⁰ (Real)	$[s]$ $[s]$
`F`_cut	Stress tensor filtering frequency. Default = 0.0 (Real)	$[\frac{1}{s}]$ $[\frac{1}{s}]$
`I`_{fail_sh}	Shell failure model flag. = 0 (Default) Shell is never deleted and no stress softening. = 1 Shell is deleted, if damage is reached for one layer. = 2 Shell is deleted, if damage is reached for all shell layers. (Integer)
`I`_{fail_so}	Solid failure model flag. = 0 Solid is never deleted and no stress softening. = 1 (Default) Solid is deleted, if damage is reached for one integration point of solid. = 2 Solid is deleted, if damage is reached for all integration points. (Integer)
`fail_ID`	(Optional) Failure criteria identifier. 4 (Integer, maximum 10 digits)

▸Example

/UNIT/1
                  kg                  mm                  ms
/FAIL/TSAIWU/1/1
#           SIGMA_1T            SIGMA_2T            SIGMA_1C            SIGMA_2C            SIGMA_12
                 0.6                0.525                 0.8                0.75                0.075  
#              ALPHA             TAU_MAX                 FCUT                      IFAIL_SH  IFAIL_SO
                 0.9             0.005932                                                  2          2

Comments

This failure model is available for shells and solids. It considers a composite material ply with the fibers oriented in the direction 1 (also denoted m1) and the matrix oriented in transverse direction, that is, in direction 2 (and 3 for solids). Each direction considers a critical strength value in tension and compression as;

Figure 1.

Where, $σ_{1}^{t}$ $σ_{1}^{t}$ , $σ_{2}^{t}$ $σ_{2}^{t}$ , $σ_{1}^{c}$ $σ_{1}^{c}$ , $σ_{2}^{c}$ $σ_{2}^{c}$ , ${¯ σ}_{12}$ ${\bar{σ}}_{12}$ are respectively the critical strength in tension for direction 1, in tension for direction 2, in compression for direction 1, in compression for direction 2 and in shear.
The failure criterion for shells is written as:(1)
$F = F_{11} σ_{1}^{2} + F_{22} σ_{2}^{2} + F_{66} σ_{12}^{2} + 2 F_{12} σ_{1} σ_{2} + F_{1} σ_{1} + F_{2} σ_{2} \leq 1$ $F = F_{11} σ_{1}^{2} + F_{22} σ_{2}^{2} + F_{66} σ_{12}^{2} + 2 F_{12} σ_{1} σ_{2} + F_{1} σ_{1} + F_{2} σ_{2} \leq 1$

For solids the criterion becomes:(2)
$F = F_{11} σ_{1}^{2} + F_{22} σ_{2}^{2} + F_{22} σ_{3}^{2} + F_{66} σ_{12}^{2} + F_{66} σ_{13}^{2} + 2 F_{12} σ_{1} σ_{2} + 2 F_{12} σ_{1} σ_{3} + F_{1} σ_{1} + F_{2} σ_{2} + F_{2} σ_{3} \leq 1$ $F = F_{11} σ_{1}^{2} + F_{22} σ_{2}^{2} + F_{22} σ_{3}^{2} + F_{66} σ_{12}^{2} + F_{66} σ_{13}^{2} + 2 F_{12} σ_{1} σ_{2} + 2 F_{12} σ_{1} σ_{3} + F_{1} σ_{1} + F_{2} σ_{2} + F_{2} σ_{3} \leq 1$
Where, the $F$ $F$ coefficients are obtained with:
- $\begin{matrix} F_{1} = \frac{1}{σ_{1}^{t}} - \frac{1}{σ_{1}^{c}} F_{2} = \frac{1}{σ_{2}^{t}} - \frac{1}{σ_{2}^{c}} F_{11} = \frac{1}{σ_{1}^{t} \cdot σ_{1}^{c}} F_{22} = \frac{1}{σ_{2}^{t} \cdot σ_{2}^{c}} F_{12} = - \frac{α}{2} \sqrt{F_{11} \cdot F_{22}} F_{66} = \frac{1}{{¯ σ}_{12}} \end{matrix}$ $\begin{array}{l} F_{1} = \frac{1}{σ_{1}^{t}} - \frac{1}{σ_{1}^{c}} \\ F_{2} = \frac{1}{σ_{2}^{t}} - \frac{1}{σ_{2}^{c}} \\ F_{11} = \frac{1}{σ_{1}^{t} \cdot σ_{1}^{c}} \\ F_{22} = \frac{1}{σ_{2}^{t} \cdot σ_{2}^{c}} \\ F_{12} = - \frac{α}{2} \sqrt{F_{11} \cdot F_{22}} \\ F_{66} = \frac{1}{{\bar{σ}}_{12}} \end{array}$
Here, $α$ $α$ is the interaction parameter between direction 1 and 2. The criterion is considered to be reached when $F = 1$ $F = 1$ . In fact, the damage variable corresponds to the criterion itself $D = F$ $D = F$ .
Once the criterion is reached $D = F = 1$ $D = F = 1$ , two behaviors can be set up:
- If I_{fail_sh} = 0 or I_{fail_so} = 0, there is no stress softening and elements are never deleted. In this case, the failure criterion is purely visual using the output of the damage variable.
- If I_{fail_sh} ≠ 0 or I_{fail_so} ≠ 0, a stress relaxation is generated to decrease the load carrying capacity of the element.(3)
  $σ (t) = f (t) \cdot σ_{d} (t_{r})$
  
  With $f (t) = \exp (- \frac{t - t_{r}}{τ_{\max}})$ and $t \geq t_{r}$ .
  
  Where,
  
  $t$
  
  Time.
  
  $t_{r}$
  
  Start time of relaxation when the damage criteria is assumed.
  
  $τ_{\max}$
  
  Time of dynamic relaxation.
  
  $σ_{d} (t_{r})$
  
  Stress tensor when the criterion is reached.
  
  When the stresses reach 1% of the stress value at the beginning of the failure, the element is deleted. This is necessary to avoid instabilities coming from a sudden deletion of an element and a failure “chain reaction” in the neighboring elements. Even if the failure criterion is reached, there will be no element deletion with the default value of $τ_{\max} = 1.0 E 20$ . Therefore, it is recommended to define a value for $τ_{\max}$ 10 times larger than the simulation time step.
To avoid “chain reaction” when deleting elements, you can also define a stress tensor filtering frequency F_cut. Thus, the stress tensor used to calculate the TSAIWU criterion is first be filtered according to:(4)
$σ_{n + 1}^{f i l t} = α σ_{n + 1} + (1 - α) σ_{n}^{f i l t}$

With $α = \frac{2 π \cdot F_{c u t} \cdot Δ t}{2 π \cdot F_{c u t} \cdot Δ t + 1}$

Where, $Δ t$ is the current timestep.

If a filtering frequency is not defined (F_cut= 0.0), the filtering effect is deactivated.
The fail_ID is used with /STATE/BRICK/FAIL and /INIBRI/FAIL. There is no default value. If the line is blank, no value will be output for failure model variables in the /INIBRI/FAIL (written in .sta file with /STATE/BRICK/FAIL option).