LAW12 and LAW14

Describes orthotropic solid material which use the Tsai-Wu formulation. The materials are 3D orthotropic-elastic, before the Tsai-Wu criterion is reached. LAW12 is a generalization and improvement of LAW14.

Elastic Phase

Both material laws require Young's modulus, shear modulus and Poisson ratio (9 parameters) to describe the material orthotropic in elastic phase.
Figure 1.


[ ε 11 ε 22 ε 33 γ 12 γ 23 γ 31 ] = [ 1 E 11 ν 12 E 11 ν 31 E 33 0 0 0 1 E 22 ν 23 E 22 0 0 0 1 E 33 0 0 0 1 2 G 12 0 0 s y m m . 1 2 G 23 0 1 2 G 31 ] [ σ 11 σ 22 σ 33 σ 12 σ 23 σ 31 ]

Stress Damage

Figure 2.


Stress limits σ t 1 ,   σ t 2   and   σ t 3 (in tensile/compression) are requested for damage. These stress limits could be observed from a tensile test in 3 related directions.
Figure 3.


Once stress limit is reached, then damage to material begins (stress reduced with damage parameter δ ). If Damage ( D i = D i +δ ) reaches D=1, then stress is reduced to 0.
Figure 4.


Tsai-Wu Yield Criteria

In LAW12 (3D_COMP), the Tsai-Wu yield criteria is:

F ( σ ) = F 1 σ 1 + F 2 σ 2 + F 3 σ 3 + F 11 σ 1 2 + F 22 σ 2 2 + F 33 σ 3 2 + F 44 σ 12 2 + F 55 σ 23 2 + F 66 σ 31 2 + 2 F 12 σ 1 σ 2 + 2 F 23 σ 2 σ 3 + 2 F 13 σ 1 σ 3

The 12 coefficients of the Tsai-Wu criterion could be determined using the yield stress from the following tests:

Tensile/Compression Tests
  • Longitude tensile/compression (in direction 1):
    Figure 5.


    F 1 = 1 σ 1 y c + 1 σ 1 y t
    F 11 = 1 σ 1 y c σ 1 y t
  • Transverse tensile/compression (in direction 2):
    Figure 6.


    F 2 = 1 σ 2 y c + 1 σ 2 y t
    F 22 = 1 σ 2 y c σ 2 y t
  • Transverse tensile/compression (in direction 3):
    Figure 7.


    F 3 = 1 σ 3 y c + 1 σ 3 y t
    F 33 = 1 σ 3 y c σ 3 y t

Then the interaction coefficients can be calculated as:

F 12 = 1 2 ( F 11 F 22 )
F 23 = 1 2 ( F 22 F 33 )
F 13 = 1 2 ( F 11 F 33 )

Shear Tests
  • Shear in plane 1-2 test:
    Figure 8.


    σ 12 y t and σ 12 y c can result from the sample tests below:
    Figure 9.


    F 44 = 1 σ 12 y c σ 12 y t
  • Shear in plane 1-3
    Figure 10.


    σ 31 y t and σ 31 y c can result from the sample tests below:
    Figure 11.


    F 66 = 1 σ 31 y c σ 31 y t
  • Shear in plane 2-3:
    Figure 12.


    F 55 = 1 σ 23 y c σ 23 y t
The parameters shown below in LAW12 and LAW14 are requested to calculate the Tsai-Wu criteria:
Figure 13.


The yield surface for Tsai-Wu is F ( σ ) = 1 . As long as ( F ( σ ) 1 ) , the material is in the elastic phase. Once ( F ( σ ) > 1 ) , the yield surface is exceeded and the material is in nonlinear phase.

In these two material laws, the following factors could also be considered for the yield surface.
  • Plastic work W p with parameter B and n
  • Strain rate ε ˙ with parameter ε ˙ 0 and c.
    F ( W p , ε ˙ ) = ( 1 + B W p n ) ( 1 + c ln ε . ε . o )
Then the yield surface will be F ( σ ) = F ( W p , ε ˙ ) .
  • Material will be in elastic phase, if F ( σ ) F ( W p , ε ˙ )
  • Material will be in nonlinear phase, if F ( σ ) > F ( W p , ε ˙ )

This yield surface F ( W p , ε ˙ ) will be limited with f max ( F ( W p , ε ˙ ) f max ), where f max is the maximum value of the Tsai-Wu criterion limit.

f max = ( σ max σ y ) 2

Depending on parameter B, n, c and ε ˙ 0 , the yield surface is between 1 and f max .
Figure 14. Tsai-Wu Yield Criteria in 1-2 Plane