/FAIL/TSAIHILL
Block Format Keyword TsaiHill failure criterion for composite materials failure modeling. This criterion is available for solids and shells.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/FAIL/TSAIHILL/mat_ID/unit_ID  
${X}_{11}$  ${X}_{22}$  ${S}_{12}$  I_{fail_sh}  I_{fail_so}  
${\tau}_{\mathrm{max}}$  F_{cut} 
(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) 

${X}_{11}$  Longitudinal critical
strength. Default = 10^{20} (Real) 
$\left[\text{Pa}\right]$ 
${X}_{22}$  Transverse critical
strength. Default = 10^{20} (Real) 
$\left[\text{Pa}\right]$ 
${S}_{12}$  Shear critical strength. Default = 10^{20} (Real) 
$\left[\text{Pa}\right]$ 
I_{fail_sh}  Shell failure model flag.
(Integer) 

I_{fail_so}  Solid failure model flag.
(Integer) 

${\tau}_{\mathrm{max}}$  Dynamic time relaxation. 5 Default = 10^{20} (Real) 
$\left[\text{s}\right]$ 
F_{cut}  Stress tensor filtering
frequency. Default = 0.0 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
fail_ID  (Optional) Failure criteria
identifier. 4 (Integer, maximum 10 digits) 
Example
/UNIT/1
Mg mm s
/FAIL/TSAIHILL/1/1
# X11 X22 S12 IFAIL_SH IFAIL_SO
520. 316. 407.5 1 1
# TAU_MAX FCUT
1.0E4 100.0
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 directions 2 (and 3 for
solids). Each direction considers a critical strength value valid for both
tension and compression.
Where, ${X}_{11}$ , ${X}_{22}$ , ${S}_{12}$ are respectively the critical strength in direction 1, critical strength in direction 2 and in shear.
 The failure criterion
for shells is written as:$$F=\frac{{\sigma}_{1}^{2}}{{X}_{11}^{2}}\frac{{\sigma}_{1}{\sigma}_{2}}{{X}_{11}^{2}}+\frac{{\sigma}_{2}^{2}}{{X}_{22}^{2}}+\frac{{\sigma}_{12}^{2}}{{S}_{12}^{2}}\le 1$$
For solids the criterion becomes:
$$F=\frac{{\sigma}_{1}^{2}}{{X}_{11}^{2}}\frac{{\sigma}_{1}{\sigma}_{2}}{{X}_{11}^{2}}\frac{{\sigma}_{1}{\sigma}_{3}}{{X}_{11}^{2}}+\frac{{\sigma}_{2}^{2}}{{X}_{22}^{2}}+\frac{{\sigma}_{3}^{2}}{{X}_{22}^{2}}+\frac{{\sigma}_{12}^{2}}{{S}_{12}^{2}}+\frac{{\sigma}_{31}^{2}}{{S}_{12}^{2}}\le 1$$The criterion is considered to be reached when $F=1$ . In fact, the damage variable corresponds to the criterion itself $D=F$ .
 Once the criterion is
reached
$D=F=1$
, 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.$$\sigma (t)=\mathrm{f}(t)\cdot {\sigma}_{d}({t}_{r})$$
With $\mathrm{f}(t)=\mathrm{exp}\left(\frac{t{t}_{r}}{{\tau}_{\mathrm{max}}}\right)\text{\hspace{0.05em}}\text{\hspace{0.05em}}$ and $t\ge {t}_{r}$ .
Where, $t$
 Time.
 ${t}_{r}$
 Start time of relaxation when the damage criteria is assumed.
 ${\tau}_{\mathrm{max}}$
 Time of dynamic relaxation.
 ${\sigma}_{d}\left({t}_{r}\right)$
 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 ${\tau}_{\mathrm{max}}=1.0E20$ . Therefore, it is recommended to define a value for ${\tau}_{\mathrm{max}}$ 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 TSAIHILL criterion is
first be filtered according to:$${\sigma}_{n+1}^{filt}=\alpha {\sigma}_{n+1}+\left(1\alpha \right){\sigma}_{n}^{filt}$$
With $\alpha =\frac{2\pi \cdot {F}_{cut}\cdot \text{\Delta}t}{2\pi \cdot {F}_{cut}\cdot \text{\Delta}t+1}$
Where, $\text{\Delta}t$ 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).