Tsai–Wu failure criterion explained

The Tsai–Wu failure criterion is a phenomenological material failure theory which is widely used for anisotropic composite materials which have different strengths in tension and compression. The Tsai-Wu criterion predicts failure when the failure index in a laminate reaches 1. This failure criterion is a specialization of the general quadratic failure criterion proposed by Gol'denblat and Kopnov and can be expressed in the form

Fi~\sigmai+Fij~\sigmai~\sigmaj\le1

where

ij=1...6

and repeated indices indicate summation, and

Fi,Fij

are experimentally determined material strength parameters. The stresses

\sigmai

are expressed in Voigt notation. If the failure surface is to be closed and convex, the interaction terms

Fij

must satisfy

FiiFjj-

2
F
ij

\ge0

which implies that all the

Fii

terms must be positive.

Tsai–Wu failure criterion for orthotropic materials

For orthotropic materials with three planes of symmetry oriented with the coordinate directions, if we assume that

Fij=Fji

and that there is no coupling between the normal and shear stress terms (and between the shear terms), the general form of the Tsai–Wu failure criterion reduces to

\begin{align} F1\sigma1+&F2\sigma2+F3\sigma3+F4\sigma4+F5\sigma5+F6\sigma6\\ &+F11

2
\sigma
1

+F22

2
\sigma
2

+F33

2
\sigma
3

+F44

2
\sigma
4

+F55

2
\sigma
5

+F66

2
\sigma
6

\\ &    +2F12\sigma1\sigma2+2F13\sigma1\sigma3+2F23\sigma2\sigma3\le1 \end{align}

Let the failure strength in uniaxial tension and compression in the three directions of anisotropy be

\sigma1t,\sigma1c,\sigma2t,\sigma2c,\sigma3t,\sigma3c

. Also, let us assume that the shear strengths in the three planes of symmetry are

\tau23,\tau12,\tau31

(and have the same magnitude on a plane even if the signs are different). Then the coefficients of the orthotropic Tsai–Wu failure criterion are

\begin{align} F1=&\cfrac{1}{\sigma1t

}-\cfrac ~;~~ F_2 = \cfrac-\cfrac ~;~~ F_3 = \cfrac-\cfrac ~;~~ F_4 = F_5 = F_6 = 0 \\ F_ = & \cfrac ~;~~ F_ = \cfrac ~;~~ F_ = \cfrac ~;~~ F_ = \cfrac ~;~~ F_ = \cfrac ~;~~ F_ = \cfrac \\ \end The coefficients

F12,F13,F23

can be determined using equibiaxial tests. If the failure strengths in equibiaxial tension are

\sigma1=\sigma2=\sigmab12,\sigma1=\sigma3=\sigmab13,\sigma2=\sigma3=\sigmab23

then

\begin{align} F12&=

2}\left[1-\sigma
\cfrac{1}{2\sigma
b12

(F1+F2)-\sigma

2(F
11

+F22)\right]\\ F13&=

2}\left[1-\sigma
\cfrac{1}{2\sigma
b13

(F1+F3)-\sigma

2(F
11

+F33)\right]\\ F23&=

2}\left[1-\sigma
\cfrac{1}{2\sigma
b23

(F2+F3)-\sigma

2(F
22

+F33)\right]\end{align}

The near impossibility of performing these equibiaxial tests has led to there being a severe lack of experimental data on the parameters

F12,F13,F23

.

It can be shown that the Tsai-Wu criterion is a particular case of the generalized Hill yield criterion.

Tsai-Wu failure criterion for transversely isotropic materials

For a transversely isotropic material, if the plane of isotropy is 1–2, then

F1=F2~;~~F4=F5=F6=0~;~~F11=F22~;~~F44=F55~;~~F13=F23~.

Then the Tsai–Wu failure criterion reduces to

\begin{align} F2(\sigma1+\sigma2)&+F3\sigma3+F22

2
(\sigma
1

+

2)
\sigma
2

+F33

2
\sigma
3

+F44

2
(\sigma
4

+

2)
\sigma
5

+F66

2
\sigma
6

\\ &    +2F12\sigma1\sigma2+2F23(\sigma1+\sigma2)\sigma3\le1 \end{align}

where

F66=2(F11-F12)

. This theory is applicable to a unidirectional composite lamina where the fiber direction is in the '3'-direction.

In order to maintain closed and ellipsoidal failure surfaces for all stress states, Tsai and Wu also proposed stability conditions which take the following form for transversely isotropic materials

F22~F33-

2
F
23

\ge0~;~~

2
F
12

\ge0~.

Tsai–Wu failure criterion in plane stress

For the case of plane stress with

\sigma1=\sigma5=\sigma6=0

, the Tsai–Wu failure criterion reduces to

F2\sigma2+F3\sigma3+F22

2
\sigma
2

+F33

2
\sigma
3

+F44

2
\sigma
4

+2F23\sigma2\sigma3\le1

The strengths in the expressions for

Fi,Fij

may be interpreted, in the case of a lamina, as

\sigma1c

= transverse compressive strength,

\sigma1t

= transverse tensile strength,

\sigma3c

= longitudinal compressive strength,

\sigma3t

= longitudinal strength,

\tau23

= longitudinal shear strength,

\tau12

= transverse shear strength.

Tsai–Wu criterion for foams

The Tsai–Wu criterion for closed cell PVC foams under plane strain conditions may be expressed as

F2\sigma2+F3\sigma3+F22

2
\sigma
2

+F33

2
\sigma
3

+2F23\sigma2\sigma3=1-k2

where

F23=-\cfrac{1}{2}\sqrt{F22F33

} ~;~~ k = \cfrac ~. For DIAB Divinycell H250 PVC foam (density 250 kg/cu.m.), the values of the strengths are

\sigma2c=4.6

MPa,

\sigma2t=7.3

MPa,

\sigma3c=6.3

MPa,

\sigma3t=10

MPa.

For aluminum foams in plane stress, a simplified form of the Tsai–Wu criterion may be used if we assume that the tensile and compressive failure strengths are the same and that there are no shear effects on the failure strength. This criterion may be written as

3~\tilde{J}2+(η2-

2
1)~\tilde{I}
1

=η2

where

\tilde{J}2:=

2}
\tfrac{1}{3}\left(\cfrac{\sigma
1c

-\cfrac{\sigma1\sigma2}{\sigma1c\sigma2c

} + \cfrac\right) ~;~~ \tilde_1 := \cfrac + \cfrac

Tsai–Wu criterion for bone

The Tsai–Wu failure criterion has also been applied to trabecular bone/cancellous bone with varying degrees of success. The quantity

F12

has been shown to have a nonlinear dependence on the density of the bone.

See also

References

[1]

[2]

[3]

[4]

[5]

[6]

Notes and References

  1. Tsai, S. W. and Wu, E. M. (1971). A general theory of strength for anisotropic materials. Journal of Composite Materials. vol. 5, pp. 58–80.
  2. Gol'denblat, I. and Kopnov, V. A. (1966). Strength of glass reinforced plastic in the complex stress state. Polymer Mechanics, vol. 1, pp. 54–60. (Russian: Mechanika Polimerov, vol. 1, pp. 70–78. 1965)
  3. Keaveny, T. M., Wachtel, E. F., Zadesky, S. P., Arramon, Y. P. (1999). Application of the Tsai–Wu quadratic multiaxial failure criterion to bovine trabecular bone. ASME Journal of Biomechanical Engineering, vol. 121, no. 1, pp. 99–107.
  4. Duyoyo, M. and Wierzbicki, T. (2003). Experimental studies on the yield behavior of ductile and brittle aluminum foams. International Journal of Plasticity, vol. 19, no. 8, pp. 1195–1214.
  5. Gdoutos, E. E., Daniel, I. M. and Wang, K-A. (2001). Multiaxial characterization and modeling of a PVC cellular foam. Journal of Thermoplastic Composite Materials, vol.14, pp. 365–373.
  6. Abrate, S. (2008). Criteria for yielding or failure of cellular materials Journal of Sandwich Structures and Materials, vol. 10, pp. 5–51.