Tsai-Wu failure criterion

From Wikipedia, the free encyclopedia

The Tsai-Wu failure criterion[1] is a phenomenological failure theory which is widely used for anisotropic composite materials which have different strengths in tension and compression. This failure criterion is a specialization of the general quadratic failure criterion proposed by Gol'denblat and Kopnov[2] and can be expressed in the form

F_i~\sigma_i + F_{ij}~\sigma_i~\sigma_j \le 1

where i,j=1\dots 6 and repeated indices indicate summation, and Fi,Fij are experimentally determined material strength parameters. The stresses σi are expressed in Voigt notation. If the failure surface is to be closed and convex, the interaction terms Fij must satisfy

F_{ii}F_{jj} - F_{ij}^2 \ge 0

which implies that all the Fii terms must be positive.

Contents

[edit] 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} F_1\sigma_1 + & F_2\sigma_2 + F_3\sigma_3 + F_4\sigma_4 + F_5\sigma_5 + F_6\sigma_6\\ & + F_{11}\sigma_1^2 + F_{22}\sigma_2^2 + F_{33}\sigma_3^2 + F_{44}\sigma_4^2 + F_{55}\sigma_{5}^2 + F_{66}\sigma_6^2 \\ & \qquad + 2F_{12}\sigma_1\sigma_2 + 2F_{13}\sigma_1\sigma_3 + 2F_{23}\sigma_2\sigma_3 \le 1 \end{align}

Let the failure strength in uniaxial tension and compression in the three directions of anisotropy be σ1t,σ1c,σ2t,σ2c,σ3t,σ3c. Also, let us assume that the shear strengths in the three planes of symmetry are τ23,τ12,τ31 (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} F_1 = & \cfrac{1}{\sigma_{1t}}-\cfrac{1}{\sigma_{1c}} ~;~~ F_2 = \cfrac{1}{\sigma_{2t}}-\cfrac{1}{\sigma_{2c}} ~;~~ F_3 = \cfrac{1}{\sigma_{3t}}-\cfrac{1}{\sigma_{3c}} ~;~~ F_4 = F_5 = F_6 = 0 \\ F_{11} = & \cfrac{1}{\sigma_{1c}\sigma_{1t}} ~;~~ F_{22} = \cfrac{1}{\sigma_{2c}\sigma_{2t}} ~;~~ F_{33} = \cfrac{1}{\sigma_{3c}\sigma_{3t}} ~;~~ F_{44} = \cfrac{1}{\tau_{23}^2} ~;~~ F_{55} = \cfrac{1}{\tau_{31}^2} ~;~~ F_{66} = \cfrac{1}{\tau_{12}^2} \\ \end{align}

The coefficients F12,F13,F23 can be determined using equibiaxial tests. If the failure strengths in equibiaxial tension are σ1 = σ2 = σb12,σ1 = σ3 = σb13,σ2 = σ3 = σb23 then

\begin{align} F_{12} &= \cfrac{1}{2\sigma_{b12}^2}\left[1-\sigma_{b12}(F_1+F_2)-\sigma_{b12}^2(F_{11}+F_{22})\right]\\ F_{13} &= \cfrac{1}{2\sigma_{b13}^2}\left[1-\sigma_{b13}(F_1+F_3)-\sigma_{b13}^2(F_{11}+F_{33})\right] \\ F_{23} &= \cfrac{1}{2\sigma_{b23}^2}\left[1-\sigma_{b23}(F_2+F_3)-\sigma_{b23}^2(F_{22}+F_{33})\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 [3].

[edit] Tsai-Wu failure criterion for transversely isotropic materials

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

F_1=F_2 ~;~~ F_4=F_5=F_6=0 ~;~~ F_{11}=F_{22} ~;~~ F_{44}=F_{55} ~;~~ F_{13}=F_{23} ~.

Then the Tsai-Wu failure criterion reduces to

\begin{align} F_2(\sigma_1 + \sigma_2) & + F_3\sigma_3 + F_{22}(\sigma_1^2 + \sigma_2^2) + F_{33}\sigma_3^2 + F_{44}(\sigma_4^2 + \sigma_{5}^2) + F_{66}\sigma_6^2 \\ & \qquad + 2F_{12}\sigma_1\sigma_2 + 2F_{23}(\sigma_1+\sigma_2)\sigma_3 \le 1 \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

F_{22}~F_{33} - F_{23}^2 \ge 0 ~;~~ F_{11}^2-F_{12}^2 \ge 0 ~.

[edit] Tsai-Wu failure criterion in plane stress

For the case of plane stress with σ1 = σ5 = σ6 = 0, the Tsai-Wu failure failure criterion reduces to

F_2\sigma_2 + F_3\sigma_3 + F_{22}\sigma_2^2 + F_{33}\sigma_3^2 + F_{44}\sigma_4^2 + 2F_{23}\sigma_2\sigma_3 \le 1

The strengths in the expressions for Fi,Fij may be interpreted, in the case of a lamina, as σ1c = transverse compressive strength, σ1t = transverse tensile strength, σ3c = longitudinal compressive strength, σ3t = longitudinal strength, τ23 = longitudinal shear strength, τ12 = transverse shear strength.

[edit] Tsai-Wu criterion for foams

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

F_2\sigma_2 + F_3\sigma_3 + F_{22}\sigma_2^2 + F_{33}\sigma_3^2 + 2F_{23}\sigma_2\sigma_3 = 1 - k^2

where

F_{23} = - \cfrac{1}{2}\sqrt{F_{22} F_{33}} ~;~~ k = \cfrac{\sigma_4}{\tau_{23}} ~.

For Divinyl H250 PVC foam (density 250 kg/cu.m.), the values of the strengths are σ2c = 4.6MPa, σ2t = 7.3MPa, σ3c = 6.3MPa, σ3t = 10MPa [4].

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 [5]

3~\tilde{J}_2 + (\eta^2 - 1)~\tilde{I}_1^2 = \eta^2

where

\tilde{J}_2 := \tfrac{1}{3}\left(\cfrac{\sigma_1^2}{\sigma_{1c}^2} - \cfrac{\sigma_1\sigma_2}{\sigma_{1c}\sigma_{2c}} + \cfrac{\sigma_2^2}{\sigma_{2c}^2}\right) ~;~~ \tilde{I}_1 := \cfrac{\sigma_1}{\sigma_{1c}} + \cfrac{\sigma_2}{\sigma_{2c}}

[edit] Tsai-Wu criterion for bone

The Tsai-Wu failure criterion has also been applied to trabecular bone[6] /cancellous bone with varying degrees of success. The quantity F12 has been shown to have a nonlinear dependence on the density of the bone.

[edit] 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. ^ Abrate, S. (2008). Criteria for yielding or failure of cellular materials Journal of Sandwich Structures and Materials, vol. 10, pp. 5-51.
  4. ^ 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.
  5. ^ 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.
  6. ^ 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.

[edit] See also